# Unwinding Fermionic SPT Phases:
Supersymmetry Extension

Abhishodh Prakasha and Juven Wangb,c
aInternational Centre for Theoretical Sciences, Tata Institute of Fundamental
Research,
Shivakote, Hesaraghatta, Bangalore 560089, India
bCenter of Mathematical Sciences and Applications, Harvard University,
Cambridge, MA 02138, USA
cSchool of Natural Sciences, Institute for Advanced Study, Einstein Drive,
Princeton, NJ 08540, USA
abhishodh.prakash<EMAIL_ADDRESS>
###### Abstract

We show how 1+1-dimensional fermionic symmetry-protected topological states
(SPTs, i.e. nontrivial short-range entangled gapped phases of quantum matter
whose boundary exhibits ’t Hooft anomaly and whose bulk cannot be deformed
into a trivial tensor product state under finite-depth local unitary
transformations only in the presence of global symmetries), indeed can be
_unwound_ to a trivial state by enlarging the Hilbert space via adding extra
degrees of freedom and suitably extending the global symmetries. The extended
projective global symmetry on the boundary can become supersymmetric in a
specific sense, i.e., it contains group elements that do _not_ commute with
the fermion number parity $(-1)^{F}$, while the anti-unitary time-reversal
symmetry becomes fractionalized. This also means we can uplift and remove
certain exotic fermionic anomalies (e.g., “parity” anomaly in time-reversal or
reflection symmetry) via appropriate _supersymmetry extensions_ in terms of
_group extensions_. We work out explicit examples for multi-layers of 1+1d
Majorana fermion chains, then comment on models with Sachdev-Ye-Kitaev (SYK)
interactions, intrinsic fermionic _gapless_ SPTs protected by supersymmetry,
and generalizations to higher spacetime dimensions via a cobordism theory.

###### Contents

  1. 1 Overview and summary of results
    1. 1.1 Classification of gapped phases of matter
    2. 1.2 Summary of results and outline
  2. 2 Warm up: a bosonic example — the Haldane phase
    1. 2.1 The Hamiltonian, symmetry fractionalization, and classification
    2. 2.2 Unwinding by symmetry extension
    3. 2.3 Boundary Hamiltonians and protected degeneracy for the Haldane chain: Sachdev-Ye interaction
    4. 2.4 Formal explanation
  3. 3 Unwinding SPT phases of fermions
    1. 3.1 Toy model Hamiltonian of Majorana fermionic chains
    2. 3.2 Class BDI: $\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F}$ symmetry or Pin-
      1. 3.2.1 Gapping the boundary: Extension to a non-Abelian $\mathbb{D}_{8}^{F,T}$
      2. 3.2.2 Unwinding the bulk: Extension to a non-Abelian $\mathbb{D}_{8}^{F,T}$
    3. 3.3 Class DIII: $\mathbb{Z}_{4}^{TF}$ symmetry or Pin+
      1. 3.3.1 Gapping the boundary: Extension to a non-Abelian $\mathbb{M}^{F,T}_{16}$
      2. 3.3.2 Unwinding the bulk: Extension to a non-Abelian $\mathbb{M}^{F,T}_{16}$
    4. 3.4 Class AIII: ${\rm U}(1)^{F}\times\mathbb{Z}_{2}^{T}$ or $\frac{{\rm U}(1)^{F}\times\mathbb{Z}^{{T}}_{4}}{\mathbb{Z}_{2}^{F}}$ symmetry or Pinc
      1. 3.4.1 Gapping the boundary: Extension to a non-Abelian $\frac{{\rm U}(1)\times\mathbb{D}_{8}^{F,T}}{\mathbb{Z}_{2}}$
      2. 3.4.2 Unwinding the bulk: Extension to a non-Abelian $\frac{{\rm U}(1)\times\mathbb{D}_{8}^{F,T}}{\mathbb{Z}_{2}}$
    5. 3.5 Class CII: ${\rm SU}(2)^{F}\times\mathbb{Z}_{2}^{T}$ symmetry or Pin${}^{-}\times_{\mathbb{Z}_{2}^{F}}{\rm SU}(2)^{F}$
    6. 3.6 Formal explanation on the classifications of invertible phases vs SPT phases
  4. 4 Interacting models for even $\nu\in\mathbb{Z}_{8}$ classes with $\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F}$ symmetry or Pin-
  5. 5 Supersymmetric quantum mechanics
    1. 5.1 General setting for SUSY quantum mechanics
    2. 5.2 ${\cal N}=2$ SUSY on the boundary of 1+1d fermionic SPT phase with $\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F}$ symmetry: Generalized Sachdev-Ye-Kitaev interactions
    3. 5.3 ${\cal N}=2$ SUSY on the boundary of 1+1d fermionic SPT phase with $\mathbb{Z}_{4}^{TF}$ symmetry
    4. 5.4 Systematic framework in 1+1d
  6. 6 Generalizations: from the lattice to continuum
    1. 6.1 1+1d: Lift $\mathrm{Pin}^{-}(d)$ to ${\rm E}\mathrm{Pin}(d)$ for $\nu_{\rm BDI}=4\in\mathbb{Z}_{8}$ class
    2. 6.2 1+1d: Lift ${\rm O}(d)$ to ${\rm E}(d)$ for $\nu_{\rm CII}=1\in\mathbb{Z}_{2}$ or even $\nu_{\rm AIII}\in\mathbb{Z}_{4}$ class
    3. 6.3 1+1d supersymmetry extension: Lift $\mathrm{Pin}^{-}$ to $\mathbb{D}_{8}^{F,T}\mathrm{Pin}$ for even $\nu_{\rm BDI}\in\mathbb{Z}_{8}$ class
    4. 6.4 1+1d supersymmetry extension: Lift $\mathrm{Pin}^{+}$ to $\mathbb{M}_{16}^{F,T}\mathrm{Pin}$ for odd $\nu_{\rm DIII}\in\mathbb{Z}_{2}$ class
  7. 7 Conclusion and Higher Dimensional Generalization
    1. 7.1 1+1d
    2. 7.2 2+1d
    3. 7.3 3+1d
    4. 7.4 4+1d
    5. 7.5 5+1d
    6. 7.6 6+1d
    7. 7.7 Smooth (DIFF) manifolds to topological (TOP) manifolds
    8. 7.8 Intrinsic fermionic gapless topological phase protected by supersymmetry
    9. 7.9 Final comments and future directions
  8. 8 Acknowledgments
  9. A Tantivasadakarn-Vishwanath model and the basis change
  10. B Appendices for Section 6
    1. B.1 Group extension: Formal setup
    2. B.2 Bosonic, Fermionic, Time Reversal, Spacetime Symmetries and Group Extensions

##  1 Overview and summary of results

###  1.1 Classification of gapped phases of matter

A central goal of condensed matter physics is to enumerate and understand the
properties of various phases of matter. The systems of interest are generally
many-body quantum systems in the infinite system size limit described by local
interactions and typically invariant under some group of global symmetries.
Long-range ordering is one common way a system can transit to a non-trivial
phase. The paramagnet-ferromagnet transition of magnets and the superfluid
transition via Bose-Einstein condensation are prominent examples of such an
ordering and are well described by the Ginzburg-Landau framework based on
_spontaneous symmetry breaking_ (SSB) and characterized by local order
parameters. The discovery of quantum Hall phases, which were beyond this
framework heralded a new era in condensed matter and was followed by intense
theoretical and experimental research attempting to get a sense of the the
various possible phases of matter (classification) and their distinguishing
properties (characterization) which continues to this day. This program has
been particularly successful in the case of _gapped phases_ of matter i.e.
when the systems of interest have a non-zero spectral gap separating the
lowest energy states from the excited states.

Gapped phases can be divided into two categories- _non-invertible_ and
_invertible_ phases. Loosely speaking, a physical system $\mathscr{P}$ belongs
to an invertible phase if there exists an inverse system,
$\mathscr{P}^{\text{inv}}$ such that the composite system produced by
_stacking_ $\mathscr{P}$ and $\mathscr{P}^{\text{inv}}$, belongs to the
_trivial_ phase i.e. can be connected, without encountering a phase
transition, to a trivial system $\mathscr{P}_{0}$ which has always has a
unique trivial ground state i.e. product state for bosons or slater
determinant state for fermions. The collection of invertible phases forms an
_Abelian group_ , with group multiplication arising from stacking and the
identity element being the trivial phase. Examples of invertible phases
include paramagnets, integer quantum Hall states, insulators and
superconductors. On the other hand, any phase that is not invertible is called
a non-invertible phase. Examples of non-invertible phases include all gapped
phases with long range order arising through SSB like ferromagnets as well as
fractional quantum Hall phases and spin liquids. In this work, we will be
concerned with a _specific_ class of invertible phases in the presence of
global symmetries.

Given a physical system with some global symmetries $G$ belonging to a non-
trivial invertible phase, if we explicitly break global symmetries, one of two
things can happen- the phase can become trivial (eg: topological insulators),
or the phase can continue to remain non-trivial (eg: integer quantum Hall
phases). The latter phases form a _subgroup_ of the group of invertible phases
and are called Symmetry-Protected-Topological states or Symmetry-Protected-
Trivial states or simply SPT states (SPTs) [1, 2] and will be the focus of
this work.

Let us state the above in the language of groups. Let $\mathscr{I}_{G}$ be the
Abelian group of invertible phases. Explicitly breaking symmetries maps each
element of $\mathscr{I}_{G}$ to $\mathscr{I}_{0}$, the group of invertible
phases with _no symmetries_. It is easy to see that breaking symmetry induces
a _surjective group homomorphism_ , $\pi$

$\displaystyle\mathscr{I}_{G}\xrightarrow{\pi}\mathscr{I}_{0}.$ (1.1)

SPT phases are those elements of $\mathscr{I}_{G}$ that land on the trivial
element of $\mathscr{I}_{0}$ under the symmetry breaking map i.e. $\pi$. In
other words, SPT phases (which we will call $\mathcal{S}_{G}$) correspond to
the _kernel_ of $\pi$.

$\mathcal{S}_{G}=\ker\pi$ (1.2)

These simple arguments reveals a rich structure in the space of gapped
physical systems- for any dimension, for either bosons or fermions with global
symmetry $G$, $\mathcal{S}_{G}$, $\mathscr{I}_{G}$, and $\mathscr{I}_{0}$ can
be arranged in the following _short exact sequence_ 111An exact sequence is a
sequence of homomorphisms $\\{\phi_{n}\\}$ between groups $\\{A_{n}\\}$ of the
form  $\ldots
A_{n}\xrightarrow{\phi_{n}}A_{n+1}\xrightarrow{\phi_{n+1}}\ldots$ such that
$\text{im}\phi_{n}=\ker\phi_{n+1}$. A short exact sequence is an exact
sequence of the form  $1\rightarrow
C\xrightarrow{i}B\xrightarrow{\pi}A\rightarrow 1$ where $i$ is injective and
$\pi$ is surjective as a consequence of exactness.

$1\rightarrow\mathcal{S}_{G}\xrightarrow{i}\mathscr{I}_{G}\xrightarrow{\pi}\mathscr{I}_{0}\rightarrow
1$ (1.3)

Let us briefly review some important properties of SPT phases:

  1. 1.

If symmetries are disregarded, all SPT phases become trivial. This means that
we can adiabatically connect the ground state to a tensor product state using
a finite time-evolution operator generated by a local Hamiltonian which can be
represented as a finite depth unitary circuit (FDUC). Following the
terminology of our previous work [3] based on the method of symmetry extension
[4], we call this _unwinding_.222A more mathematical way to interpret our
approach on unwinding of SPTs [4, 3] can be that pulling back trivialization
of fiber bundle via fibrations [5]. These have physical consequences of (1)
removing ’t Hooft anomalies or (2) constructing symmetry-extended topological
boundaries, via symmetry extensions [4, 3]. The ’t Hooft anomalies are the
obstruction of gauging global symmetries; here we concern the global
symmetries including time-reversal or reflection symmetry, in additional to
spacetime or internal symmetries. These anomalies are generalized “parity”
anomalies – more accurately involving time-reversal $\mathbb{Z}_{2}^{T}$ or
reflection symmetry $\mathbb{Z}_{2}$ instead of the spatial parity. In fact,
Ref. [5, 6] provide further mathematical insights of the problems dealt in [4,
3].  An FDUC is a locality preserving unitary operator that be written as a
product of a finite number of layers of _ultra-local unitaries_ schematically
shown in Fig 1.

  2. 2.

Boundaries of SPT phases are always non-trivial i.e. are gapless or have some
form of long-range order if gapped. This _persistent ordering_ [7] is due to
the presence of an _’t Hooft anomaly_ for the global symmetry $G$ that forbids
a trivial phase.

Figure 1: Finite depth unitary circuit (FDUC). The operator can be expressed
as a produce of finite number of layers (vertically separated in the figure,
indicating the time direction). Each layer is a product of unitary operators
with strictly local support (square blobs in the figure) i.e. acts on the
Hilbert space of a finite number of spins.

Although not immediately obvious, the above two properties are intimately
connected — the different gapped boundary terminations of SPT phases can tell
us how to construct FDUC to unwind an SPT phase. Let us focus on symmetry
breaking boundary terminations of SPT phases. If a $G$-SPT phase can be
terminated by spontaneously breaking $G$ down to one of its subgroups,
$G_{\text{sub}}$ on the boundary, this means that the same SPT phase can be
unwound by a FDUC that is invariant under $G_{\text{sub}}$. Note that breaking
all symmetries is one specific case of the above paradigm corresponding to
$G_{\text{sub}}$ being the trivial group.

A second boundary termination, particularly insightful for 3+1d bulk
systems,333Our convention on the dimensions is that the one, two, three
spatial dimensions are written as 1d, 2d, 3d, etc. We denote their spacetime
dimensions as 1+1d, 2+1d, 3+1d, etc.  is when no global symmetry is broken but
the boundary is topologically ordered. The anomalous nature of the boundary
symmetry is seen in the way it acts on the anyons of the theory which is
impossible in a purely 2+1d system [8, 9] (see the review [2] and references
within for an overview). For a specific class of bosonic SPT phases (those
classified by group cohomology), the authors of [4] were able to place the
understanding of symmetric gapped boundary terminations in a systematic
framework. They prove that given a system belonging to a non-trivial
$d+1$-dimensional bosonic $G$-SPT phase with criteria:

  1. 1.

bosonic SPTs classified by group cohomology.

  2. 2.

$G$ is a finite group, including both unitary and/or anti-unitary symmetries.

  3. 3.

$d\geq 1$ for $d$ is the spatial dimension. (Namely the spacetime dimension
$d+1\geq 2$.)

there always exists a larger group, $\tilde{G}$ such that, starting with a
$(d-1)+1$ dimensional $\tilde{G}$-symmetric invariant theory and gauging a
particular subgroup, $K$ satisfying $\tilde{G}/K\cong G$, we are left with a
$K$ gauge theory with $G$ symmetry realized anomalously and can form a
boundary termination for the G-SPT. This in turn also teaches us something
else very interesting- _the G-SPT phase can be unwound, not by breaking
symmetry but by extending to $\tilde{G}$!_ In other words, the ’t Hooft
anomaly of $G$-symmetry of a boundary theory (in a $d-1$ dimensional space and
1-dimensional time) can be trivialized by extending the system to
$\tilde{G}$-symmetry. In Ref. [3], this was demonstrated explicitly in the
case of 1+1d for various groups $G$ by constructing FDUCs invariant under an
appropriately identified $\tilde{G}$.

This naturally brings about the following questions — _can there be a
systematic framework to understand symmetric gapped boundary terminations for
SPT phases not classified by group cohomology?_ and _can these be obtained by
symmetry extension?_ In particular, this includes all _intrinsically
fermionic_ SPT phases i.e. which cannot be constructed by stacking a non-
trivial bosonic SPT phase with a trivial fermionic phase. In this work, for
1+1d fermionic SPT phases, we provide evidence that the answer to both the
above italicized questions is in the affirmative.

###  1.2 Summary of results and outline

We focus only on on-site, unitary and anti-unitary symmetries (including
various time-reversal symmetries) in this article. We find that just like
bosonic SPT phases, 1+1d fermionic SPT phases can be unwound by an FDUC
invariant under an extended symmetry. For intrinsically fermionic SPT phases,
we find that the extended symmetry has an interesting property — the fermion
parity is not at the center of the symmetry group, i.e. there exists elements
of the symmetry group that _do not commute with fermion parity_ locally. We
term this as supersymmetry extension. This is our main result which we
demonstrate using several examples. Our results are demonstrated using exactly
solvable lattice models and constructing explicit FDUCs.

This article is organized as follows: we warm up by considering a bosonic
example in Sec. 2 which sets the language and the general ideas that are
detailed in Ref. [3]. In Sec. 3, we consider non-trivial free-fermion models
of SPT phases in 1+1d corresponding to classes BDI, AIII and DIII and
demonstrate how they can be unwound to a trivial phase by symmetry extension.
In Sec. 4, we study an interacting example of a fermionic SPT phase with time-
reversal symmetry and demonstrate its unwinding. In Sec. 5, we justify our
usage of the term _supersymmetry extension_ and show that the extended
symmetry has an intimate correspondence to the supersymmetric quantum
mechanics. In Sec. 6, we make comments about generalization in the continuum,
suitable for quantum field theory (QFT) formulation. In Sec. 7, we explore
generalizations to higher dimensions and relations to a cobordism theory (from
Sec. 7.1 to Sec. 7.7) and the implications to intrinsically fermionic
_gapless_ SPTs (Sec. 7.8), and finally conclude with possible future
directions in (Sec. 7.9). Additional details are relegated to the Appendices.

##  2 Warm up: a bosonic example — the Haldane phase

In this section, we warm-up by considering a bosonic example to set the
language as well as list the results that we generalize to fermionic systems
in this work. Additional examples as well as general results for bosonic SPTs
can be found in [3].

Figure 2: Model Hamiltonian belonging to the Haldane phase with open boundary
conditions. The SU(2) doublet written as the SU(2) representation
$J=\frac{1}{2}$ qubits at both ends are indicated.

###  2.1 The Hamiltonian, symmetry fractionalization, and classification

We consider a 1+1d bosonic spin chain belonging to the Haldane phase protected
by an on-site SO(3) symmetry. The Hilbert space on each site is a pair of
qubits each of which transforms as the $J=\frac{1}{2}$ representation of the
SU(2) group. Together, each site forms a $\frac{1}{2}\otimes\frac{1}{2}\cong
1\oplus 0$ reducible representation that is faithful to SO(3). The Hamiltonian
with $\ell$ sites and SO(3) symmetry operators are

$\displaystyle H$
$\displaystyle=-\sum_{j=1}^{\ell}\vec{\sigma}_{B,j}.\vec{\sigma}_{A,j+1},$
(2.1) $\displaystyle U(\hat{n},\theta)$
$\displaystyle=\prod_{j=1}^{\ell}\exp{\left(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\hat{n}.\vec{\sigma}\frac{\theta}{2}\right)_{A,j}}~{}\exp{\left(\hskip
1.0pt\mathrm{i}\hskip
1.0pt\hat{n}.\vec{\sigma}\frac{\theta}{2}\right)_{B,j}}.$ (2.2)

For eq (2.2), two SU(2) symmetry operators (two of $J=\frac{1}{2}$) act on the
same site $j$ makes the SO(3) symmetry operator (for $J=0\oplus 1$). The model
by Affleck, Kennedy, Lieb and Tasaki (AKLT) is obtained by projecting out the
$J=0$ sector on each site and only retaining $J=1$ [10]. We will choose not to
perform this projection and work with the exactly solvable model of eq. 2.2.
With periodic boundary conditions, eq. 2.2 has the following unique ground
state which has SO(3) singlets on each bond connecting the sites.

$\displaystyle|{\rm
GS}\rangle=\prod_{j=1}^{\ell}\frac{|\uparrow_{B,j}\downarrow_{{A,(j|\ell)+1}}\rangle-|\downarrow_{B,j}\uparrow_{A,(j|\ell)+1}\rangle}{\sqrt{2}},$
(2.3)

where, $(j|\ell)\equiv j\mod\ell$. With any symmetry preserving open boundary
conditions, the ground state degeneracy is four (GSD = 4), i.e. any ground
state can be written as a vector in a four-dimensional Hilbert subspace of
ground states:

$\displaystyle|{\rm GS}\rangle$
$\displaystyle=\left(\prod_{j=1}^{\ell}\frac{|\uparrow_{B,j}\downarrow_{A,j+1}\rangle-|\downarrow_{B,j}\uparrow_{A,j+1}\rangle}{\sqrt{2}}\right)\otimes|\psi\rangle,$
(2.4) $\displaystyle|\psi\rangle$
$\displaystyle=\sum_{\alpha=\uparrow,\downarrow}\sum_{\beta=\uparrow,\downarrow}\psi_{\alpha,\beta}|\alpha_{A,1}\rangle|\beta_{B,\ell}\rangle,~{}~{}~{}\sum_{\alpha=\uparrow,\downarrow}\sum_{\beta=\uparrow,\downarrow}|\psi_{\alpha,\beta}|^{2}=1.$
(2.5)

We can define a Hermitian projection operator $\mathbf{P}_{\rm
GS}=\mathbf{P}_{\rm GS}^{\dagger}$ onto the ground space that has support on
the ends of the chain

$\displaystyle\mathbf{P}_{\rm GS}$ $\displaystyle=\mathbf{P}_{{\rm
left}}\otimes|s\rangle\langle s|\otimes\mathbf{P}_{{\rm right}}$ (2.6)
$\displaystyle\mathbf{P}_{{\rm left}}$
$\displaystyle=\sum_{\alpha=\uparrow,\downarrow}|\alpha_{A,1}\rangle\langle\alpha_{A,1}|,~{}~{}~{}\mathbf{P}_{{\rm
right}}=\sum_{\alpha=\uparrow,\downarrow}|\alpha_{B,\ell}\rangle\langle\alpha_{B,\ell}|,$
(2.7) $\displaystyle|s\rangle$
$\displaystyle=\left(\prod_{j=1}^{\ell}\frac{|\uparrow_{B,j}\downarrow_{A,j+1}\rangle-|\downarrow_{B,j}\uparrow_{A,j+1}\rangle}{\sqrt{2}}\right).$
(2.8)

$\mathbf{P}_{\rm GS}$ has support on the boundaries of the chain. This allows
us to study the _fractionalization_ of the SO(3) symmetry on the boundaries at
low energies.

$\displaystyle\mathbf{P}_{\rm GS}U(\hat{n},\theta)\mathbf{P}_{\rm GS}$
$\displaystyle=\exp{\left(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\hat{n}.\vec{\sigma}\frac{\theta}{2}\right)_{A,1}}\otimes~{}\exp{\left(\hskip
1.0pt\mathrm{i}\hskip
1.0pt\hat{n}.\vec{\sigma}\frac{\theta}{2}\right)_{B,\ell}}$
$\displaystyle=V(\hat{n},\theta)_{{\rm left}}\otimes V(\hat{n},\theta)_{{\rm
right}}.$ (2.9)

The matrix symmetry representation of each of the boundary modes corresponds
to $J=\frac{1}{2}$ which is a fraction of the quantum numbers allowed by the
bulk symmetry group SO(3). Furthermore, the representation is faithful to the
group SU(2) which is a non-trivial extension of SO(3) by $\mathbb{Z}_{2}$.
This is an example demonstration of the emergence of extended symmetries on
the boundaries of SPT phases [4].

Figure 3: Unwinding two-layers of eq. 2.10 via the two-layer FDUC
$W=W_{2}W_{1}$ in eq (2.14).

The non-trivial nature of the Haldane phase comes from the fact that there
does not exist an SO(3) invariant FDUC that can map the ground state to a
product state. If we consider two layers of the Hamiltonian eq. 2.2 however,

$\displaystyle H^{\rm II}$
$\displaystyle=-\sum_{j=1}^{\ell}\vec{\sigma}_{B,j}.\vec{\sigma}_{A,j+1}-\sum_{j=1}^{\ell}\vec{\tau}_{B,j}.\vec{\tau}_{A,j+1},$
(2.10) $\displaystyle U(\hat{n},\theta)$
$\displaystyle=\prod_{j=1}^{\ell}\exp{\left(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\hat{n}.\vec{\sigma}\frac{\theta}{2}\right)_{j,A}}\exp{\left(\hskip
1.0pt\mathrm{i}\hskip
1.0pt\hat{n}.\vec{\sigma}\frac{\theta}{2}\right)_{j,B}}\exp{\left(\hskip
1.0pt\mathrm{i}\hskip
1.0pt\hat{n}.\vec{\tau}\frac{\theta}{2}\right)_{j,A}}\exp{\left(\hskip
1.0pt\mathrm{i}\hskip 1.0pt\hat{n}.\vec{\tau}\frac{\theta}{2}\right)_{j,B}}.$
(2.11)

it is easy to see that it can be _unwound_ to one whose ground state is a
product state as shown in Fig. 3 using a two-layer FDUC $W=W_{2}W_{1}$:

$\displaystyle WH^{\rm II}W^{\dagger}$
$\displaystyle=-\sum_{j=1}^{\ell}\vec{\sigma}_{A,j}.\vec{\sigma}_{B,j}-\sum_{j=1}^{\ell}\vec{\tau}_{A,j}.\vec{\tau}_{B,j},$
(2.12) $\displaystyle W$ $\displaystyle=W_{2}W_{1},$ (2.13) $\displaystyle
W_{1}$
$\displaystyle=\prod_{j=1}^{\ell}\frac{1}{2}\left(\mathds{1}+\vec{\sigma}_{B,j}.\vec{\tau}_{A,j+1}\right),~{}~{}~{}W_{2}=\prod_{j=1}^{\ell}\frac{1}{2}\left(\mathds{1}+\vec{\sigma}_{A,j}.\vec{\tau}_{B,j}\right).$
(2.14)

The bosonic SWAP operator
$\frac{1}{2}\left(\mathds{1}+\vec{\sigma}.\vec{\tau}\right)$ swaps the basis
states of the $\sigma$ and $\tau$ qubits. What we have reproduced is the well-
known $\mathbb{Z}_{2}$ classification of the Haldane phase [1].

###  2.2 Unwinding by symmetry extension

Figure 4: Unwinding the extended Hamiltonian eq. 2.15 state by extending the
bulk on-site symmetry to SU(2), via $W=W_{2}W_{1}$ in two steps: $W_{1}$ and
$W_{2}$ in eq (2.18).

We now demonstrate that by elevating the extended boundary symmetry (SU(2)) to
the bulk, we can unwind the single layer Hamiltonian eq. 2.2. To do this, we
first add an extra $J=\frac{1}{2}$ qubit on each site (which we label “C”). We
also add a trivial Hamiltonian that dimerizes the $C$ qubits in the ground
state 444 An aside: When the on-site representation is half-odd-integer as in
eq. 2.15, the Hamiltonian has to explicitly break the lattice translation
symmetry if we want to have a unique ground state. This is a consequence of
the Lieb-Schultz-Mattis theorem [11, 12].

$H=-\sum_{j=1}^{\ell}\vec{\sigma}_{B,j}.\vec{\sigma}_{A,j+1}-\sum_{\text{odd
j}}\vec{\sigma}_{C,j}.\vec{\sigma}_{C,j+1}.$ (2.15)

This Hamiltonian can be unwound as shown in fig. 4 using a two-layer FDUC

$\displaystyle WHW^{\dagger}$
$\displaystyle=-\sum_{j=1}^{\ell}\vec{\sigma}_{A,j}.\vec{\sigma}_{B,j}-\sum_{\text{even
j}}\vec{\sigma}_{C,j}.\vec{\sigma}_{C,j+1},$ (2.16) $\displaystyle W$
$\displaystyle=W_{2}W_{1}$ (2.17) $\displaystyle W_{1}$
$\displaystyle=\prod_{\text{odd
j}}\frac{1}{2}\left(\mathds{1}+\vec{\sigma}_{C,j}.\vec{\sigma}_{A,j+1}\right),~{}~{}W_{2}=\prod_{\text{odd
j}}\frac{1}{2}\left(\mathds{1}+\vec{\sigma}_{A,j}.\vec{\sigma}_{C,j}\right).$
(2.18)

###  2.3 Boundary Hamiltonians and protected degeneracy for the Haldane chain:
Sachdev-Ye interaction

Figure 5: Multiple copies of the Hamiltonian eq. 2.2

The classification of the Haldane phase tells us that even copies of
Hamiltonian eq. 2.2 belong to the trivial phase and odd number of copies
belong to the Haldane phase. This result can also be obtained by attempting to
lift the boundary degeneracy by introducing interactions. Let us consider $N$
copies of the Hamiltonian eq. 2.2. The boundary Hilbert space consists of $N$
of bosonic $J=\frac{1}{2}$ qubits which we shall label $m=1,\ldots,N$. The
most general symmetry preserving Hamiltonian we can write down is as follows

$\displaystyle H_{{\rm
bdry}}=\sum_{a,b=1}^{N}J_{ab}~{}\vec{\sigma}_{a}.\vec{\sigma}_{b}+\sum_{a,b,c,d=1}^{N}K_{abcd}~{}\left(\vec{\sigma}_{a}.\vec{\sigma}_{b}\right)\left(\vec{\sigma}_{c}.\vec{\sigma}_{d}\right)+\ldots$
(2.19)

The leading term in the above Hamiltonian with $J_{ab}$ drawn from a random
distribution is the model by Sachdev and Ye [13]. We are concerned with the
question of when there is a possibility that eq. 2.19 can have a unique ground
state. This can be answered from some basic representation theoretic facts of
SU(2). The total Hilbert space of $N$ spins forms a reducible representation
of SU(2) corresponding to the Clebsch-Gordan (CG) decomposition

$\displaystyle\frac{1}{2}\otimes\frac{1}{2}\otimes\frac{1}{2}\ldots\otimes\frac{1}{2}~{}~{}(\text{N
terms}).$ (2.20)

The only way to obtain a symmetric unique ground state is if the Hilbert space
contains a $J=0$ irreducible representation (irrep) in the above CG
decomposition. If $N$ is even, this is possible. For example, consider $N=2$:

$\displaystyle\frac{1}{2}\otimes\frac{1}{2}\cong 1\oplus 0.$ (2.21)

If $N$ is odd however, we only obtain half-odd-integer values of $J$ in the CG
decomposition. For example, for $N=3$,

$\displaystyle\frac{1}{2}\otimes\frac{1}{2}\otimes\frac{1}{2}\cong\frac{3}{2}\oplus\frac{1}{2}\oplus\frac{1}{2}.$
(2.22)

This recovers the $\mathbb{Z}_{2}$ classification.

###  2.4 Formal explanation

We now provide a formal way of understanding the above results following for
the mathematically inclined reader [4]. In Sec. 2.1, the boundary spin
$J=\frac{1}{2}$ is a projective representation of SO(3), which is in fact the
faithful representation of SU(2). This can also be interpreted as the group
extension as a short exact sequence:

$\displaystyle 1\longrightarrow\mathbb{Z}_{2}\overset{}{\longrightarrow}{{\rm
SU}(2)}|_{\text{boundary}}\overset{r}{\longrightarrow}{{\rm
SO}(3)}|_{\text{bulk}}\longrightarrow 1,$ (2.23)

as the boundary-bulk symmetry extension, with the following implications:

  1. 1.

_Topological invariant_ : The 1+1d Haldane chain is characterized by a 1+1d
topological invariant given by a nontrivial group element of the cohomology
group $\operatorname{H}^{2}(\mathrm{B}{\rm SO}(3),{\rm U}(1))=\mathbb{Z}_{2}$
[1] also the cobordism group $\Omega_{2}^{{\rm SO}}(\mathrm{B}{\rm
SO}(3))=\mathbb{Z}_{2}$ (see Sec. 5.5.2 of [14]. ) Precisely the 1+1d
topological invariant contains the second Stiefel-Whitney (SW) class $w_{2}$
of the associated vector bundle $V$ of SO(3), as

$\displaystyle\exp(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{M^{2}}w_{2}(V_{{\rm SO}(3)}))$ (2.24)

(which is known as the group cocycle in a group cohomology, or the cobordism
invariant in a cobordism theory). The eq (2.24) is a partition function of the
low energy physics of Haldane chain on a 1+1d spacetime manifold $M^{2}$.

For example, we can construct a nontrivial manifold generator: take the base
manifold $M^{2}=S^{2}$ as a 2-sphere, the map $S^{2}\to\mathrm{B}{\rm SO}(3)$
determines an SO(3) bundle over $S^{2}$. Recall the homotopy group
$\pi_{2}(\mathrm{B}{\rm SO}(3))=\pi_{1}({\rm SO}(3))=\mathbb{Z}_{2}$. The
homotopy class of the map $S^{2}\to\mathrm{B}{\rm
SO}(3)\to\mathrm{B}^{2}\mathbb{Z}_{2}$ is given by a nontrivial generator
exactly $w_{2}(V_{{\rm SO}(3)})$ that can be pull back to the base manifold
$S^{2}$.

  2. 2.

_Trivialization of topological invariant_ : This group extension eq (2.25)
actually solves a problem about the trivialization of topological invariant
via the symmetry extension — once we pull back the SO(3) bundle to the SU(2)
bundle via the pull back $r^{*}$, then the 1+1d topological invariant of
Haldane chain $r^{*}\exp(\hskip 1.0pt\mathrm{i}\hskip 1.0pt\pi\int
w_{2}(V_{{\rm SO}(3)}))\sim 1$ becomes trivialized because $r^{*}w_{2}(V_{{\rm
SO}(3)})=w_{2}(V_{{\rm SU}(2)})=0$ (the trivialization means that it is a
group coboundary as a trivial element in the cohomology group (in
$\operatorname{H}^{2}(\mathrm{B}{\rm SU}(2),{\rm U}(1))=0$), also a trivial
identity invariant in the cobordism theory (in $\Omega_{2}^{{\rm
SO}}(\mathrm{B}{\rm SU}(2))=0$)).

This implies that we can in fact add an extra SU(2) doublet $J=\frac{1}{2}$ on
each of the boundary of Haldane chain, then we can add SU(2)-invariant
interaction terms to the boundary Hamiltonian to gap the degenerate
$\mathrm{GSD}=4$ states, left with only a single unique ground state
($\mathrm{GSD}=1$) preserving the extended SU(2) symmetry (but not preserving
the SO(3) symmetry), see [3].

  3. 3.

_Trivialization of ’t Hooft anomaly_ : The 1+1d bulk topological invariant
from $\operatorname{H}^{2}(\mathrm{B}{\rm SO}(3),{\rm U}(1))$ $=$
$\Omega_{2}^{{\rm SO}}(\mathrm{B}{\rm SO}(3))$ $=\mathbb{Z}_{2}$ classifies
its 0+1d boundary ’t Hooft anomaly of the SO(3) global symmetry. The
$w_{2}(V_{{\rm SO}(3)})$ regarded as the SO(3) background field, once turned
on, can detect the ’t Hooft anomaly.

It is shown that the eq (2.25), with the suitable condition mentioned above,
says that the 0+1d ’t Hooft anomaly of the SO(3) global symmetry becomes
anomaly-free once we pull back the theory to the SU(2) symmetry in 0+1d [4].

  4. 4.

_Bulk trivialization_ : If we consider the trivialization of topological
invariant not only at the boundary but also at the full bulk via the unwinding
method in Sec. 2.2. Then we also write the group extension as the bulk-bulk
symmetry extension:

$\displaystyle 1\longrightarrow\mathbb{Z}_{2}\overset{}{\longrightarrow}{{\rm
SU}(2)}|_{\text{bulk}}\overset{r}{\longrightarrow}{{\rm
SO}(3)}|_{\text{bulk}}\longrightarrow 1.$ (2.25)

The above bosonic SPTs serve as a guiding principle, for the readers to
follow. In the following sections, we will explore the generalization to
fermionic SPTs counterpart. We provide many examples in Sec. 3 and Sec. 4,
then we provide a formal explanation of the fermionic examples in Sec. 6.

##  3 Unwinding SPT phases of fermions

In this section, we present the key results of the paper. Fermionic invertible
phases as well as SPT phases were originally discovered in _non-interacting_
systems i.e. Hamiltonians are quadratic in fermion creation and annihilation
operators. Prominent examples are the integer quantum Hall states, topological
insulators and superconductors with various symmetries. In a monumental series
of works [15, 16], all free-fermion invertible phases have been classified in
all dimensions. The program of extending the classification to interacting
fermions in any dimension is under progress, although so far the best known
mathematical framework is known to be the generalization of group
supercohomology [17, 18, 19, 20, 21, 22], and the cobordism group
classification [23, 24, 25, 26] and [14]

In the coming sections, we will demonstrate unwinding 1+1 d fermionic SPT
phases by extended symmetries similar to section 2.2 and ref.[3] with examples
of models from fermion SPT phases with a _free-limit_ i.e. have
representatives which are quadratic Hamiltonians and whose symmetries can be
placed in one of the symmetry classes (see ref. [27] for an accessible review
on symmetry classes). This is only for convenience. We focus on the following
five symmetry classes — D, BDI, DIII, AII, and CII and pick a particular
representative symmetry for each class. 555The A and C symmetry classes have
no SPT phases in 1+1d. The AI and CI have also SPT phases but they are only
bosonic phases (in fact, Haldane spin-1 chain) up to stacking gapped fermionic
product states. For bosonic SPTs, we can unwind by the approach of [4, 3]. See
the later Table 3 for the systematic web of SPTs related by a symmetry group
embedding. In Table 1, we list the group of invertible phases and the subgroup
of SPT phases. Following Sec. 1.1, we use the convention:

$\begin{array}[]{l}\text{Symmetry}\\\ \text{Class}\end{array}$ | $\mathscr{I}_{G}$ | $\mathcal{S}_{G}$
---|---|---
D | $\mathbb{Z}_{2}$ | $0$
BDI | $\mathbb{Z}$ | $2\mathbb{Z}$
DIII | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}$
AIII | $\mathbb{Z}$ | $\mathbb{Z}$
CII | $\mathbb{Z}$ | $\mathbb{Z}$

$\begin{array}[]{l}\text{Internal}\\\ \text{Symmetry}\end{array}$ | $\begin{array}[]{l}\text{Spacetime-Internal}\\\ \text{Symmetry $G$}\end{array}$ | $\mathrm{TP}_{2}(G)$ | $\frac{\mathrm{TP}_{2}(G)}{\mathrm{TP}_{2}({\rm Spin})}$
---|---|---|---
$\mathbb{Z}_{2}^{F}$ | Spin | $\mathbb{Z}_{2}$ | $0$
$\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F}$ | Pin- | $\mathbb{Z}_{8}$ | $\mathbb{Z}_{4}$
$\mathbb{Z}_{4}^{TF}$ | Pin+ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}$
${{\rm U}(1)^{F}\times\mathbb{Z}^{{T}}_{2}}$ | Pinc | $\mathbb{Z}_{4}$ | $\mathbb{Z}_{4}$
${\rm SU}(2)^{F}\times\mathbb{Z}_{2}^{T}$ | Pin${}^{-}\times_{\mathbb{Z}_{2}^{F}}{\rm SU}(2)$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}$

Table 1:  For 1+1d fermionic systems, on the left-hand side (l.h.s) table, we
show the free fermion classification of invertible phases, $\mathscr{I}_{G}$
and the subgroup of SPT phases, $\mathcal{S}_{G}$ of a given symmetry class.
On the right-hand side (r.h.s) table, we show the corresponding symmetry group
$G$ suitable for the interacting systems (interacting analogous of symmetry
class), their interacting fermion classification of invertible topological
phases (TP) denoted as a version of cobordism group ${\mathrm{TP}_{2}(G)}$ in
Freed-Hopkins’s [24], and interacting fermion classification of SPT phases
denoted as $\frac{\mathrm{TP}_{2}(G)}{\mathrm{TP}_{2}({\rm Spin})}$. The
$\frac{\mathrm{TP}_{2}(G)}{\mathrm{TP}_{2}({\rm Spin})}$ means that the
invertible topological phases (TP) of symmetry group $G$ mod out the
invertible fermionic topological order ${\mathrm{TP}_{2}({\rm Spin})}$ that is
long-range entangled (LRE). Thus the
$\frac{\mathrm{TP}_{2}(G)}{\mathrm{TP}_{2}({\rm Spin})}$ is the short-range
entangled (SRE) fermionic SPT phases. Note that ${{\rm
U}(1)^{F}\times\mathbb{Z}^{{T}}_{2}}$ is equivalent to $\frac{{\rm
U}(1)^{F}\times\mathbb{Z}^{{T}}_{4}}{\mathbb{Z}_{2}^{F}}$ via a redefinition
of the time reversal symmetry (see Sec. 3.4). See also Table 3 for a
systematic web of these 1+1d fermionic SPTs. The cobordism group data is
explained in more details in Sec. 3.6.

  1. (1).

For the free fermion symmetry classes and their group classification of
phases, we denote the group of invertible phases as $\mathscr{I}_{G}$ and the
group of SPT phases, which forms a subgroup of $\mathscr{I}_{G}$, as
$\mathcal{S}_{G}$.

  2. (2).

For the interacting fermion systems and their group classification of phases,
we can obtain the group classification from a cobordism group
$\mathrm{TP}_{d}(G)$ defined in [24] for invertible topological phases with a
$G$-global symmetry. Here $G$ is the full spacetime-internal global symmetry
in the continuum limit, see examples of $G$ listed in Table 1. The low energy
physics of invertible topological phases are described by the invertible
topological field theory (iTQFT) [24] which is also known as SPT invariants
[28]. Based on Wen’s definition [29], part of the invertible topological
phases are long-ranged entangled invertible topological orders, while another
part are short-ranged entangled invertible SPTs. For example, any layer of
Kitaev chains is an invertible topological phase: a single Kitaev chain is a
_long-ranged entangled_ invertible topological order, while two or more layers
of Kitaev chains are _short-ranged entangled_ invertible SPTs.

We can generate a representative of each 1+1d free-fermion invertible phase
for every class by considering an appropriate number of layers of Kitaev’s
Majorana chains [30] and identifying an appropriate symmetry belonging to the
symmetry class in consideration. SPT phases on the other hand turn out to be
those invertible phases which can be represented by an even number of Majorana
chains.666By Wen’s definition [4], a single layer of Kitaev chain is protected
by no symmetry except by the fermion parity, this means that in fact  an odd
number of Kitaev chains is a 1+1d long-range entangled state as a 1+1d
fermionic invertible topological order. Based on the Wen’s definition using
the local unitary quantum circuit, Kitaev chain is long-range entangled (LRE)
but not short-range entangled (SRE) because it cannot be deformed to a trivial
product state by breaking all symmetries (but maintaining the fermion parity
$\mathbb{Z}_{2}^{F}$ as a fermionic system). In contrast,  an even number of
Kitaev chains is a 1+1d short-range entangled state as a 1+1d fermionic
invertible SPT state. because it can be deformed to a trivial product state by
breaking all symmetries (but maintaining the fermion parity
$\mathbb{Z}_{2}^{F}$ as a fermionic system).  We relegate a mathematical and
formal explanation on the classifications of invertible phases vs SPT phases
via the cobordism group data in the end of section in Sec. 3.6.

###  3.1 Toy model Hamiltonian of Majorana fermionic chains

Let us start by writing down the Hamiltonian for the Kitaev’s Majorana chain
[30]. The Hilbert space is of one complex fermion per unit site on a 1d
lattice which can be represented using two Majorana operators,
$\gamma,~{}\bar{\gamma}$ satisfying $\gamma_{i}^{\dagger}=\gamma_{i}$,
$\bar{\gamma}_{i}^{\dagger}=\bar{\gamma}_{i}$,
$\\{\gamma_{i},\gamma_{j}\\}=\\{\bar{\gamma}_{i},\bar{\gamma}_{j}\\}=2\delta_{ij}$,
and $\\{\gamma_{i},\bar{\gamma}_{j}\\}=0$. Majorana operators, which are
Hermitian, can be defined in terms of creation and annihilation operators of
the complex fermions, $\psi_{i}$ and $\psi_{i}^{\dagger}$ as 777Throughout the
article, we denote the imaginary number as $\hskip 1.0pt\mathrm{i}\hskip
1.0pt$ with $\hskip 1.0pt\mathrm{i}\hskip 1.0pt^{2}=-1$. The complex fermion
operators obey $\\{\psi_{i},\psi_{j}^{\dagger}\\}=\delta_{ij}$,
$\\{\psi_{i},\psi_{j}\\}=0$, and
$\\{\psi_{i}^{\dagger},\psi_{j}^{\dagger}\\}=0$. We have:
$\psi_{i}=\frac{1}{2}({\gamma}_{i}+\hskip 1.0pt\mathrm{i}\hskip
1.0pt\bar{\gamma}_{i}),\quad\psi_{i}^{\dagger}=\frac{1}{2}({\gamma}_{i}-\hskip
1.0pt\mathrm{i}\hskip 1.0pt\bar{\gamma}_{i}).$

${\gamma}_{i}=(\psi_{i}^{\dagger}+\psi_{i}),~{}~{}~{}\bar{\gamma}_{i}={\hskip
1.0pt\mathrm{i}\hskip 1.0pt}(\psi_{i}^{\dagger}-\psi_{i}).$ (3.1)

Kitaev’s Majorana chain [30] has the Hamiltonian:

$H_{\text{Kitaev}}=-\hskip 1.0pt\mathrm{i}\hskip
1.0pt\sum_{j}\gamma_{j}\bar{\gamma}_{j+1}.$ (3.2)

Figure 6: Graphical representation of Kitaev’s Majorana chain of eq (3.2).

Let us consider two layers of Majorana chains with their flavor indices
$\varsigma=\uparrow,\downarrow$:

$H^{\text{II}}_{\text{Kitaev}}=-\hskip 1.0pt\mathrm{i}\hskip
1.0pt\sum_{\varsigma=\uparrow,\downarrow}\sum_{j}\gamma_{\varsigma,j}\bar{\gamma}_{\varsigma,j+1}.$
(3.3)

Figure 7: Fermionic SWAP operator and two layers of Kitaev’s Majorana chains:
Two layers of Majorana chains before and after change of basis by the
fermionic SWAP operator
$\mathrm{SWAP}_{f}=M=\prod_{j}\frac{1}{\sqrt{2}}\left(1+\gamma_{\downarrow,j}\bar{\gamma}_{\uparrow,j}\right)$
in eq (3.4). The $M$ is the fermionic analog of the SWAP operator of bosonic
spin systems (i.e., the bosonic SWAP operator in Fig. 3 and in Ref. [3]’s Fig.
7). The Hamiltonian before the basis changing is eq (3.3), after the basis
changing is $H_{\text{gen}}=MH^{\text{II}}_{\text{Kitaev}}M^{\dagger}=-\hskip
1.0pt\mathrm{i}\hskip
1.0pt\sum_{j}\left(\gamma_{\uparrow,j}\gamma_{\downarrow,j+1}-\bar{\gamma}_{\uparrow,j}\bar{\gamma}_{\downarrow,j+1}\right)$
in eq (3.5).

For convenience, we now perform a change-of-basis on the Hamiltonian of eq
(3.3) using the unitary operator $M$,

${M=\prod_{j}\exp\left(\frac{\pi}{4}\gamma_{\downarrow,j}\bar{\gamma}_{\uparrow,j}\right)}=\prod_{j}\frac{1}{\sqrt{2}}\left(1+\gamma_{\downarrow,j}\bar{\gamma}_{\uparrow,j}\right).$
(3.4)

The new Hamiltonian $H_{\text{gen}}$ under the change of basis (see Fig. 7)
becomes:888In terms of complex fermion bases eq (3.1), we have
$H^{\text{II}}_{\text{Kitaev}}=+\sum_{\varsigma=\uparrow,\downarrow}\sum_{j}(\psi_{\varsigma,j}^{\dagger}+\psi_{\varsigma,j})(\psi_{\varsigma,j+1}^{\dagger}-\psi_{\varsigma,j+1}),$
while
$H_{\text{gen}}=MH^{\text{II}}_{\text{Kitaev}}M^{\dagger}=-\hskip
1.0pt\mathrm{i}\hskip
1.0pt\sum_{j}\left((\psi_{\uparrow,j}^{\dagger}+\psi_{\uparrow,j})(\psi_{\downarrow,j+1}^{\dagger}+\psi_{\downarrow,j+1})+(\psi_{\uparrow,j}^{\dagger}-\psi_{\uparrow,j})(\psi_{\downarrow,j+1}^{\dagger}-\psi_{\downarrow,j+1})\right).$

$\displaystyle H_{\text{gen}}$ $\displaystyle=$ $\displaystyle
MH^{\text{II}}_{\text{Kitaev}}M^{\dagger}=-\hskip 1.0pt\mathrm{i}\hskip
1.0pt\sum_{j}\left(\gamma_{\uparrow,j}\gamma_{\downarrow,j+1}-\bar{\gamma}_{\uparrow,j}\bar{\gamma}_{\downarrow,j+1}\right).$
(3.5)

The $M$ in eq (3.4) and in Fig. 7 is the fermionic analog of the SWAP operator
of bosonic spin systems (such as the bosonic SWAP operator in Ref. [3]’s Fig.
7).999The bosonic SWAP operator
$\mathrm{SWAP}_{b}\equiv\frac{1}{2}\left({1}+\vec{\sigma}_{A}\cdot\vec{\sigma}_{B}\right)$
exchanges the basis states $|\uparrow\rangle,~{}|\downarrow\rangle$ of two
qubit Hilbert spaces, $A$ and $B$, and satisfies $(\mathrm{SWAP}_{b})^{2}=1$.
On the other hand, the fermionic SWAP operator
$\mathrm{SWAP}_{f}\equiv\frac{1}{\sqrt{2}}\left(1+\gamma_{A}\gamma_{B}\right)=\exp(\frac{\pi}{4}\gamma_{A}\gamma_{B})$
acts on two Majorana operators $(\gamma_{A},\gamma_{B})$ to give
$\mathrm{SWAP}_{f}\cdot(\gamma_{A},\gamma_{B})\cdot\mathrm{SWAP}_{f}^{-1}=(-\gamma_{B},\gamma_{A}).$
Note that $(\mathrm{SWAP}_{f})^{2}=\gamma_{A}\gamma_{B},$
$(\mathrm{SWAP}_{f})^{4}=-1,$ and $(\mathrm{SWAP}_{f})^{8}=1.$ The Hamiltonian
$H_{\text{gen}}$ of eq (3.5) is the proper form of generator of two layers of
Kitaev chains, which will be a primary object of focus of this section and
throughout our work.

###  3.2 Class BDI: $\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F}$ symmetry or
Pin-

We start with class BDI with representative global symmetry
$\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F}$ where $\mathbb{Z}_{2}^{T}$ is
generated by the anti-unitary operator $\mathcal{T}$ satisfying
$\mathcal{T}^{2}=1$ and $\mathbb{Z}_{2}^{F}$ is the fermion parity group
generated by $P_{F}=(-1)^{F}$ where $F$ is the total fermion number. In the
absence of interactions, it is known that there are an infinite number of
invertible phases [16] indexed by an integer $k\in\mathbb{Z}$ representing the
number of ‘dangling’ Majorana modes on each end i.e
$\mathscr{I}_{G}\cong\mathbb{Z}$. This reduces to $\mathbb{Z}_{8}$ in the
presence of interactions [31, 32]. A representative of the generator of these
phases is the Majorana chain of eq (3.2) with symmetry operator

$\displaystyle\mathcal{T}=\mathcal{K}$ (3.6)

which is a complex conjugation which acts as

$\mathcal{K}\gamma\mathcal{K}=\gamma,~{}~{}\quad\quad\mathcal{K}\bar{\gamma}\mathcal{K}=-\bar{\gamma},~{}~{}\quad\quad\mathcal{K}\hskip
1.0pt\mathrm{i}\hskip 1.0pt\mathcal{K}=-\hskip 1.0pt\mathrm{i}\hskip 1.0pt,$
(3.7)

and

$P_{F}=(-1)^{F}=\prod_{j}\hskip 1.0pt\mathrm{i}\hskip
1.0pt\bar{\gamma}_{j}\gamma_{j}.$ (3.8)

The subgroup of SPT phases are those corresponding to even members $k\in
2\mathbb{Z}$ and are non-trivial only in the presence of symmetries
$\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F}$. The SPT phases also correspond
to also the even members $k=0,2,4,6\in\mathbb{Z}_{8}$ in the interacting SPT
classification (See Table 1). A generator of the SPT phases (corresponding to
$k=2$) is two layers of the Majorana chains, eq (3.3) or its equivalent
representation in eq (3.5) with the new time-reversal $\mathbb{Z}_{2}^{T}$
symmetry operators $\tilde{\mathcal{T}}$ under the eq (3.4) basis-change via a
unitary $M$:

$\displaystyle\tilde{\mathcal{T}}$ $\displaystyle\equiv$ $\displaystyle
M\mathcal{T}M^{\dagger}=M\mathcal{K}M^{\dagger}=\left(\prod_{j}(\gamma_{\downarrow,j}\bar{\gamma}_{\uparrow,j})\right)\mathcal{K}.$
(3.9)

The form of fermion parity $P_{F}$ stays unchanged under basis change
$M{P_{F}}={P_{F}}M$.

####  3.2.1 Gapping the boundary: Extension to a non-Abelian
$\mathbb{D}_{8}^{F,T}$

Let us see how we can gap the boundary when we have open boundary conditions
for Hamiltonian (3.5). Fig. 8 shows that the boundary has four fold ground
state degeneracy (GSD = 4) coming from the two dimensional Hilbert space from
two Majorana modes (i.e., one complex fermion) on each boundary site, $L$ and
$R$

Figure 8:  Basis-changed 2-layer Majorana chain with left ($L$) and right
($R$) open boundaries, with a symmetry as in eq (3.9), we have four fold
ground state degeneracy (GSD = 4) with two degenerate Majorana modes on each
end, emphasized in the box frames.  Figure 9: Basis-changed 2-layer Majorana
chain from Fig. 8 with two open boundaries, but now each of the boundary
couples to additional two Majorana modes on both the left $L$ and right $R$
(the $j=1$ site couples to a new site $j=0$, $j={\ell}$ site couples to a new
site $j={\ell}+1$ respectively). We can modify $H_{\text{gen}}$ to
$H_{\text{gen}}+H_{\text{gen,bdry}}$ by adding local Hamiltonian terms to two
boundaries as $H_{\text{gen,bdry}}=-\hskip 1.0pt\mathrm{i}\hskip
1.0pt\left(\gamma_{\uparrow,0}\gamma_{\downarrow,1}-\bar{\gamma}_{\uparrow,0}\bar{\gamma}_{\downarrow,1}\right)-\hskip
1.0pt\mathrm{i}\hskip
1.0pt\left(\gamma_{\uparrow,{\ell}}\gamma_{\downarrow,{\ell}+1}-\bar{\gamma}_{\uparrow,{\ell}}\bar{\gamma}_{\downarrow,{\ell}+1}\right).$
The whole system becomes gapped but with only a unique ground state (GSD = 1).

Fig. 9 shows another boundary termination of the Hamiltonian of eq. 3.5 by
adding two extra Majorana modes on each end ($L$ and $R$) leading to a unique
ground state. From Fig. 9, we can read-off the effective _extended_ boundary
symmetry operators on the left ($L$) and right ($R$) boundaries (on the site
$j=0$ and $j={\ell}+1$) omitting site indices below:

$\displaystyle\mathcal{T}_{L}$ $\displaystyle=$
$\displaystyle\bar{\gamma}_{\uparrow}\mathcal{K},\quad~{}P_{F,L}=\hskip
1.0pt\mathrm{i}\hskip
1.0pt\bar{\gamma}_{\uparrow}\gamma_{\uparrow},\quad~{}\mathcal{T}_{L}^{2}=-1,P_{F,L}^{2}=1,\quad\mathcal{T}_{L}P_{F,L}=-P_{F,L}\mathcal{T}_{L}.$
(3.10) $\displaystyle\mathcal{T}_{R}$ $\displaystyle=$
$\displaystyle\gamma_{\downarrow}\mathcal{K},\quad~{}P_{F,R}=\hskip
1.0pt\mathrm{i}\hskip
1.0pt\bar{\gamma}_{\downarrow}\gamma_{\downarrow},\quad~{}\mathcal{T}_{R}^{2}=P_{F,R}^{2}=1,\quad\quad\quad\mathcal{T}_{L}P_{F,R}=-P_{F,R}\mathcal{T}_{R}.$
(3.11)

We can also show:

$\displaystyle\mathcal{T}_{L}{\gamma}_{\uparrow}\mathcal{T}_{L}^{-1}=-{\gamma}_{\uparrow},\quad\mathcal{T}_{L}\bar{\gamma}_{\uparrow}\mathcal{T}_{L}^{-1}=-\bar{\gamma}_{\uparrow}.$
(3.12)
$\displaystyle\mathcal{T}_{R}{\gamma}_{\downarrow}\mathcal{T}_{R}^{-1}=\gamma_{\downarrow},\quad\;\;\mathcal{T}_{R}\bar{\gamma}_{\downarrow}\mathcal{T}_{R}^{-1}=\bar{\gamma}_{\downarrow}$
(3.13)

It can be easily checked that the boundary symmetry operators generate a
faithful representation of the dihedral group of order 8 known as
$\mathbb{D}_{8}$ which is a group with two generators satisfying the following
relation:

$\displaystyle\mathbb{D}_{8}\equiv\mathbb{Z}_{4}\rtimes\mathbb{Z}_{2}\equiv\langle
a,x|a^{4}=x^{2}=1,xax=a^{-1}\rangle.$ (3.14)

On the left ($L$) end, $a=\mathcal{T}_{L},~{}x=P_{F,L}$, giving us

$\mathbb{D}_{8}^{F,T}|_{\text{boundary}}=\mathbb{Z}_{4}^{T}\rtimes\mathbb{Z}_{2}^{F}\equiv\langle\mathcal{T}_{L},P_{F,L}|\mathcal{T}_{L}^{4}=P_{F,L}^{2}=1,P_{F,L}\mathcal{T}_{L}P_{F,L}=\mathcal{T}_{L}^{-1}\rangle.$

On the right ($R$) end, $a=\mathcal{T}_{R}P_{F,R},~{}x=P_{F,R}$, giving us

$\mathbb{D}_{8}^{F,T}|_{\text{boundary}}=\mathbb{Z}_{4}^{T}\rtimes\mathbb{Z}_{2}^{F}\equiv\langle(\mathcal{T}_{R}P_{F,R}),P_{F,R}|(\mathcal{T}_{R}P_{F,R})^{4}=(P_{F,R})^{2}=1,P_{F,R}(\mathcal{T}_{R}P_{F,R})P_{F,R}=(\mathcal{T}_{R}P_{F,R})^{-1}\rangle.$

Importantly, we observe that the fermion parity ($P_{F,L}$ or $P_{F,R}$) is
_not_ in the center of
$\mathbb{D}_{8}^{F,T}=\mathbb{Z}_{4}^{T}\rtimes\mathbb{Z}_{2}^{F}$ with
$\mathbb{Z}_{4}^{T}$ generated by $a$ and $\mathbb{Z}_{2}^{F}$ generated by
$x$. This is the extended symmetry that will help us to unwind the
bulk.101010Some comments about our conventions and notations:  1. Here the
$\mathbb{D}_{8}^{F,T}$ in our conventions is the left ($L$) or right ($R$)
boundary versions of $\mathbb{D}_{8}$ in eq (3.14). The upper indices
(${F,T}$) emphasize that fermion parity and time reversal are included, but
they do _not_ commute in the extended groups. 2. In general, when we write a
group $G^{F,T}$, the fermion parity ($P_{F}$) and time-reversal ($T$) do not
necessarily commute in $G$. 3. $G^{TF}$ means that $G$ contains time-reversal
($T$) which commutes with and generates fermion parity
$\mathcal{T}^{2}=(-1)^{F}$ . 4. $G^{TB}$ means $G$ contains the time-reversal
($T$) which satisfies $\mathcal{T}^{2}=(-\mathds{1})$ .

In particular, given the bulk onsite symmetry
$\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F}$ (where time reversal
$\mathbb{Z}_{2}^{T}$ and fermion parity $\mathbb{Z}_{2}^{F}$ commutes), we can
extend the symmetry transformation on the boundary to $\mathbb{D}_{8}^{F,T}$
written as a short exact sequence:

$\displaystyle
1\longrightarrow\mathbb{Z}_{2}\overset{}{\longrightarrow}\mathbb{D}_{8}^{F,T}|_{\text{boundary}}\overset{}{\longrightarrow}\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F}|_{\text{bulk}}\longrightarrow
1.$ (3.15)

See footnote 10 for our conventions. The normal subgroup
$N=\mathbb{Z}_{2}=\\{1,-1\\}$ is at the center of $\mathbb{D}_{8}$ on both
ends (left and right), while the quotient group
$\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F}$ contains four commutative
elements: $N\\{1,a,x,ax\\}$ for $\mathbb{D}_{8}^{F,T}$.

Some more comments:

  1. 1.

We can explicitly check that the fully gapped system in Fig. 9 preserves the
global symmetry. Not only the bulk Hamiltonian term $H_{\text{gen}}$ respects
the bulk $\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F}$ symmetry, but the
boundary Hamiltonian
$H_{\text{gen,bdry}}=H_{\text{gen,bdry},L}+H_{\text{gen,bdry},R}=-\hskip
1.0pt\mathrm{i}\hskip
1.0pt\left(\gamma_{\uparrow,0}\gamma_{\downarrow,1}-\bar{\gamma}_{\uparrow,0}\bar{\gamma}_{\downarrow,1}\right)$
$-\hskip 1.0pt\mathrm{i}\hskip
1.0pt\left(\gamma_{\uparrow,\ell}\gamma_{\downarrow,{\ell}+1}-\bar{\gamma}_{\uparrow,{\ell}}\bar{\gamma}_{\downarrow,\ell+1}\right)$
also respects the boundary $\mathbb{D}_{8}^{F,T}$ symmetry. More precisely,
for the time reversal symmetry on the boundary Hamiltonian:

On the left end $L$, we have the symmetry operator from $\mathcal{T}_{L}$ on
the site 0 and $\tilde{\mathcal{T}}$ on the site 1, which together act on the
left end $H_{\text{gen,bdry},L}$.

On the right end $R$, we have the symmetry operator from $\mathcal{T}_{R}$ on
the site $\ell+1$ and $\tilde{\mathcal{T}}$ on the site $\ell$, which together
act on the right end $H_{\text{gen,bdry},R}$.

The system of Fig. 9 is fully gapped with a unique ground state (GSD = 1),
preserving the bulk $\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F}$ symmetry and
the boundary $\mathbb{D}_{8}^{F,T}$ symmetry.

  2. 2.

We should emphasize that there are at least three different ways to look at
$\mathcal{T}$ symmetries on the boundary depending on (1) the boundary
truncation types of quantum systems and (2) the Hilbert space projection
$\mathbf{P}_{\rm GS}$ to ground state subspace (Below we omit the boundary
site $j$ labels, $j=1$ for the left $L$ and $j=\ell$ for the right $R$,
throughout):

    1. (a)

On boundaries of Fig. 8, if we look at $\mathcal{T}_{L}$ and $\mathcal{T}_{R}$
acting on whole boundary sites of $j=1$ and $j=\ell$ respectively without
$\mathbf{P}_{\rm GS}$ projection, then we have:

$\mathcal{T}_{L}=\gamma_{\downarrow}\bar{\gamma}_{\uparrow}\mathcal{K},\quad\mathcal{T}_{R}=\gamma_{\downarrow}\bar{\gamma}_{\uparrow}\mathcal{K}.$

    2. (b)

On boundaries of Fig. 9, if we look at $\mathcal{T}_{L}$ and $\mathcal{T}_{R}$
acting on extra boundary sites of $j=0$ and $j=\ell+1$ respectively (with or
without $\mathbf{P}_{\rm GS}$ projection), then we have:

$\mathcal{T}_{L}=\bar{\gamma}_{\uparrow}\mathcal{K},\quad\mathcal{T}_{R}=\gamma_{\downarrow}\mathcal{K}.$

    3. (c)

On boundaries of Fig. 8, if we look at $\mathcal{T}_{L}$ and $\mathcal{T}_{R}$
acting on whole boundary sites of $j=1$ and $j=\ell$ respectively with
$\mathbf{P}_{\rm GS}$ projection to dangling zero energy modes, then we have:

$\mathcal{T}_{L}=\gamma_{\downarrow}\mathcal{K},\quad\mathcal{T}_{R}=\bar{\gamma}_{\uparrow}\mathcal{K}.$

  3. 3.

There is a special _supersymmetric_ feature for a 2 layer of Kitaev chains
which holds when the layer number $N=2\mod 4$ only.

     * •

When the layer number $N=2\mod 4$, we have two Majorana zero modes (with two
operators $\gamma,\bar{\gamma}$) with GSD = 2 on each end of system (with the
left and right ends combined to give GSD = 4). In fact, for this GSD = 2, on
one end, the Hilbert space $\mathcal{H}$ must have one bosonic ground state
$|B\rangle$ and one fermionic ground state $|F\rangle$. We denote the GSD = 2
ground state subspace as
$\mathcal{H}=\\{|B\rangle,|F\rangle\\}=\mathcal{H}_{B}\oplus\mathcal{H}_{F}$,
say $|B\rangle=\begin{pmatrix}1\\\ 0\end{pmatrix}$ and
$|F\rangle=\begin{pmatrix}0\\\ 1\end{pmatrix}$. Consider the $L$ boundary in
Fig. 8, we can write down the operators acting on the two-dimensional Hilbert
space $\mathcal{H}$ explicitly:

$\displaystyle\gamma=\begin{pmatrix}0&1\\\
1&0\end{pmatrix}=\sigma_{x},\quad\bar{\gamma}=\begin{pmatrix}0&-\hskip
1.0pt\mathrm{i}\hskip 1.0pt\\\ \hskip 1.0pt\mathrm{i}\hskip
1.0pt&0\end{pmatrix}=\sigma_{y},\quad P_{F}=-\hskip 1.0pt\mathrm{i}\hskip
1.0pt\gamma\bar{\gamma}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}=\sigma_{z}.$
(3.16) $\displaystyle\psi=\frac{1}{2}({\gamma}+\hskip 1.0pt\mathrm{i}\hskip
1.0pt\bar{\gamma})=\begin{pmatrix}0&1\\\
0&0\end{pmatrix},\quad\psi^{\dagger}=\frac{1}{2}({\gamma}-\hskip
1.0pt\mathrm{i}\hskip 1.0pt\bar{\gamma})=\begin{pmatrix}0&0\\\
1&0\end{pmatrix},\quad\mathcal{T}=\bar{\gamma}\mathcal{K}=\sigma_{y}\mathcal{K}.$
(3.17)

The bosonic $|B\rangle$ and fermionic $|F\rangle$ have the distinct fermion
parity $P_{F}$ (even vs odd).

The GSD = 2 on each end is protected by time-reversal $\mathcal{T}$ symmetry:
because to gap any ground state, the only possible Hamiltonian is

$\displaystyle H^{\prime}\propto-\hskip 1.0pt\mathrm{i}\hskip
1.0pt\gamma\bar{\gamma}=P_{F}$ (3.18)

which is disallowed by a global symmetry $\mathcal{T}$ due to the fact
$H^{\prime}$ does not preserve $\mathcal{T}$. (i.e., $\mathcal{T}$ does not
commute with $P_{F}$, namely $\mathcal{T}P_{F}\mathcal{T}^{-1}=-P_{F}$ or
$P_{F}\mathcal{T}P_{F}=-\mathcal{T}$.)

We will revisit the underlying supersymmetric quantum mechanics in Sec. 5 in
depth.

     * •

When the layer number $N=4$ (or a nonzero $N=0\mod 4$), we have four Majorana
zero modes (with four operators
$\gamma_{1},\bar{\gamma}_{1},\gamma_{2},\bar{\gamma}_{2}$) with GSD = 4 on one
end (with the left and right ends combined to give GSD = 16). However, we can
introduce a four Majorana interaction with a $\mathcal{T}$-symmetric
Hamiltonian

$\displaystyle
H^{\prime}=\alpha\gamma_{1}\bar{\gamma}_{1}\gamma_{2}\bar{\gamma}_{2}=-\alpha(P_{F,1})(P_{F,2})=-\alpha
P_{F}$ (3.19)

where $P_{F,i}=-\hskip 1.0pt\mathrm{i}\hskip 1.0pt\gamma_{i}\bar{\gamma}_{i}$
is the fermion parity for the $i$-th complex fermion $\psi_{i}$ sector. This
$H^{\prime}$ is $\mathcal{T}$-symmetric because
$\mathcal{T}(P_{F,1}P_{F,2})\mathcal{T}^{-1}=(-P_{F,1})(-P_{F,2})=(P_{F,1}P_{F,2})$.
Overall, we can gap two ground states out of GSD = 4 to leave with only GSD =
2 on one end. By turning on
$H^{\prime}=\alpha\gamma_{1}\bar{\gamma}_{1}\gamma_{2}\bar{\gamma}_{2}$,
depend on the sign of $\alpha$, we are left with only either _two bosonic
states_ ($\alpha>0$, so ground states with $P_{F}=+1$) or _two fermionic
states_ ($\alpha<0$, so ground states with $P_{F}=-1$).

In summary, the $N=0\mod 4$ may be left with two ground states but always with
the same fermion parity. In contrast, only $N=2\mod 4$, we have two ground
states with the different fermion parity, hinting the supersymmetric quantum
mechanics to be revealed in Sec. 5.

####  3.2.2 Unwinding the bulk: Extension to a non-Abelian
$\mathbb{D}_{8}^{F,T}$

In order to unwind the bulk, we follow the similar steps as in Sec. 2.2, see
Fig 10:

  1. 1.

The first step, we extend the Hilbert space by adding one complex fermion per
site (which corresponds to Majorana operators $\gamma$ and $\bar{\gamma}$)
with dynamics given by a dimerizing Hamiltonian:

$H_{o}=\hskip 1.0pt\mathrm{i}\hskip
1.0pt\sum_{\text{odd}~{}j}(\gamma_{j}\gamma_{j+1}-\bar{\gamma}_{j}\bar{\gamma}_{j+1}).$
(3.20)

The extended Hamiltonian and symmetry operators are (with $H_{\text{gen}}$ in
eq (3.5))

$\displaystyle H_{\text{ext}}$ $\displaystyle=$ $\displaystyle
H_{\text{gen}}+H_{o}.$ (3.21) $\displaystyle\mathcal{T}_{\text{ext}}$
$\displaystyle=$
$\displaystyle\Big{(}\prod_{\text{odd}~{}j}(\gamma_{\downarrow,j}\bar{\gamma}_{\uparrow,j})\gamma_{j}\Big{)}\Big{(}\prod_{\text{even}~{}j}(\gamma_{\downarrow,j}\bar{\gamma}_{\uparrow,j})\bar{\gamma}_{j}\Big{)}\mathcal{K}.$
(3.22) $\displaystyle P_{F}$ $\displaystyle=$
$\displaystyle\prod_{j}\left(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\bar{\gamma}_{\downarrow,j}\gamma_{\downarrow,j}\right)\left(\hskip
1.0pt\mathrm{i}\hskip
1.0pt\bar{\gamma}_{\uparrow,j}\gamma_{\uparrow,j}\right)\left(\hskip
1.0pt\mathrm{i}\hskip 1.0pt\bar{\gamma}_{j}\gamma_{j}\right).$ (3.23)

In the extended 1+1d bulk, we have the extended $\mathbb{D}_{8}^{F,T}$
symmetry on-site in the bulk:

$\displaystyle\mathbb{D}_{8}^{F,T}|_{\text{bulk}}\equiv\langle\mathcal{T}_{\text{ext}},P_{F}|\mathcal{T}_{\text{ext}}^{4}=P_{F}^{2}=1,P_{F}\mathcal{T}_{\text{ext}}P_{F}=\mathcal{T}_{\text{ext}}^{-1}{=-\mathcal{T}_{\text{ext}}}\rangle.$
(3.24)

Figure 10: Unwinding the Hamiltonian of the basis-changed 2-layer Majorana
chain (eq (3.21) for BDI class) by three steps. We show four sites (as even,
odd, even, and odd sites) from the left to the right. Step 1: we add one extra
complex fermion per site to extend the Hilbert space. Step 2: we perform a
local unitary transformation $W_{1}$ on each of two neighbor sites (a neighbor
pair of odd and even sites). Step 3: we perform a local unitary transformation
$W_{2}$ on each of sites, while the odd and even sites have different
transformations. The final form of Hamiltonian becomes eq (3.25), while the
ground state becomes a tensor product state, which is a trivial tensor product
state respect to effective sites (with each effective site combining two
neighbor [an even and an odd] sites). See Footnote 4: We have the final ground
state breaking the translation symmetry with a double unit cell due to the LSM
constraint.

  2. 2.

The second and third steps, we show the Hamiltonian can be unwound as shown in
Fig 10 using a 2-layer $\mathbb{D}_{8}$-symmetry invariant FDUC $W=W_{2}W_{1}$
as

$WH_{\text{ext}}W^{\dagger}=(W_{2}W_{1})H_{\text{ext}}(W_{2}W_{1})^{\dagger}=-\hskip
1.0pt\mathrm{i}\hskip
1.0pt\sum_{j}\left(\gamma_{\downarrow,j}\gamma_{\uparrow,j}-\bar{\gamma}_{\downarrow,j}\bar{\gamma}_{\uparrow,j}\right)+H_{e},$
(3.25)

such that $\mathcal{T}_{\text{ext}}W_{l}\mathcal{T}_{\text{ext}}^{-1}=W_{l}$
and $P_{F}W_{l}P_{F}^{-1}=W_{l}$ with $l=1,2$, where,

$\displaystyle W_{1}$ $\displaystyle=$
$\displaystyle\prod_{\text{odd}~{}j}\left(\frac{1+\gamma_{j}\gamma_{\downarrow,j+1}}{\sqrt{2}}\right)\left(\frac{1+\bar{\gamma}_{j}\bar{\gamma}_{\downarrow,j+1}}{\sqrt{2}}\right).$
(3.26) $\displaystyle W_{2}$ $\displaystyle=$
$\displaystyle\prod_{\text{odd}~{}j}\left(\frac{1+\gamma_{j}\gamma_{\downarrow,j}}{\sqrt{2}}\right)\left(\frac{1+\bar{\gamma}_{j}\bar{\gamma}_{\downarrow,j}}{\sqrt{2}}\right)\prod_{\text{even}~{}j}\left(\frac{1+\gamma_{\uparrow,j}\gamma_{j}}{\sqrt{2}}\right)\left(\frac{1+\bar{\gamma}_{\uparrow,j}\bar{\gamma}_{j}}{\sqrt{2}}\right).$
(3.27) $\displaystyle H_{e}$ $\displaystyle=$ $\displaystyle\hskip
1.0pt\mathrm{i}\hskip
1.0pt\sum_{\text{even}~{}j}(\gamma_{j}\gamma_{j+1}-\bar{\gamma}_{j}\bar{\gamma}_{j+1}).$
(3.28)

The deformed Hamiltonian eq (3.25) under FDUC is exactly what we aim for,
which shows that under the _enlarged_ Hilbert space and _extended_ global
symmetry, its ground state can be deformed to a trivial tensor product state
(See the outcome state in Fig. 10)! In short, the bulk unwinding follows also
the symmetry extension:

$\displaystyle
1\longrightarrow\mathbb{Z}_{2}\overset{}{\longrightarrow}\mathbb{D}_{8}^{F,T}|_{\text{bulk}}\overset{}{\longrightarrow}\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F}|_{\text{bulk}}\longrightarrow
1.$ (3.29)

###  3.3 Class DIII: $\mathbb{Z}_{4}^{TF}$ symmetry or Pin+

We now consider 1+1d superconductors with a _conventional_
$\mathbb{Z}_{4}^{TF}$ time-reversal invariance, which is known as the Class
DIII for the free fermion systems. We work with the same Hilbert space as the
previous subsection i.e. two species of fermions per unit site. The 2-layer
Majorana chain with a Hamiltonian $H^{\text{II}}_{\text{Kitaev}}$ eq (3.3) is
in fact the generator of $\nu=1\in\mathbb{Z}_{2}$ class of the interacting
SPTs with $\mathbb{Z}_{4}^{TF}$-symmetry (see Table 1). The $\mathcal{T}$
transformation generating $\mathbb{Z}_{4}^{TF}$ obey

$\displaystyle\mathcal{T}$ $\displaystyle=$
$\displaystyle\prod_{j}\left(\frac{1+\gamma_{\uparrow,j}\gamma_{\downarrow,j}}{\sqrt{2}}\right)\left(\frac{1+\bar{\gamma}_{\uparrow,j}\bar{\gamma}_{\downarrow,j}}{\sqrt{2}}\right)\mathcal{K}=\prod_{j}\mathrm{e}^{\frac{\pi}{4}\gamma_{\uparrow,j}\gamma_{\downarrow,j}}\mathrm{e}^{\frac{\pi}{4}\bar{\gamma}_{\uparrow,j}\bar{\gamma}_{\downarrow,j}}\mathcal{K}.$
(3.30) $\displaystyle\mathcal{T}^{2}$ $\displaystyle=$ $\displaystyle
P_{F}=(-1)^{F}{=\prod_{j}\prod_{\varsigma=\uparrow,\downarrow}\hskip
1.0pt\mathrm{i}\hskip
1.0pt\bar{\gamma}_{\varsigma,j}\gamma_{\varsigma,j}}.\quad\quad\quad\mathcal{T}^{4}=1.$
(3.31)

This means $\mathbb{Z}_{2}^{F}\subset\mathbb{Z}_{4}^{TF}$, the fermion parity
generated by $P_{F}$ is a normal subgroup of time reversal
$\mathbb{Z}_{4}^{TF}$, thus these _bulk_ global symmetries form a short exact
sequence:

$\displaystyle
1\to\mathbb{Z}_{2}^{F}\to\mathbb{Z}_{4}^{TF}\to\mathbb{Z}_{2}^{T}\to 1.$
(3.32)

It can easily be checked that the Hamiltonian of eq (3.3) is invariant under
the time-reversal symmetry defined above. Once again, we use the basis change
via eq (3.4)’s $M$ to get the Hamiltonian $H_{\text{gen}}$ of eq (3.5) and the
following basis-changed time-reversal operator:

$\displaystyle{\tilde{\mathcal{T}}}$ $\displaystyle\equiv$ $\displaystyle
M\mathcal{T}M^{\dagger}=\prod_{j}\gamma_{\downarrow,j}\left(\frac{1-\gamma_{\downarrow,j}\bar{\gamma}_{\downarrow,j}}{\sqrt{2}}\right)\bar{\gamma}_{\uparrow,j}\left(\frac{1+\gamma_{\uparrow,j}\bar{\gamma}_{\uparrow,j}}{\sqrt{2}}\right)\mathcal{K}.$
(3.33)

The fermion parity $\tilde{P_{F}}=M{P_{F}}M^{\dagger}=P_{F}$ stays unchanged.

####  3.3.1 Gapping the boundary: Extension to a non-Abelian
$\mathbb{M}^{F,T}_{16}$

To find the symmetry extension, let us now look at the symmetry transformation
on the edges. We have

$\displaystyle\mathcal{T}_{L}$ $\displaystyle=$
$\displaystyle\bar{\gamma}_{\uparrow}\left(\frac{1+\gamma_{\uparrow}\bar{\gamma}_{\uparrow}}{\sqrt{2}}\right)\mathcal{K}{=\bar{\gamma}_{\uparrow}\mathrm{e}^{\frac{\pi}{4}\gamma_{\uparrow}\bar{\gamma}_{\uparrow}}\mathcal{K}},~{}~{}~{}~{}~{}~{}~{}~{}\,P_{F,L}=\hskip
1.0pt\mathrm{i}\hskip 1.0pt\bar{\gamma}_{\uparrow}\gamma_{\uparrow}.$ (3.34)
$\displaystyle\mathcal{T}_{R}$ $\displaystyle=$
$\displaystyle\gamma_{\downarrow}\left(\frac{1-\gamma_{\downarrow}\bar{\gamma}_{\downarrow}}{\sqrt{2}}\right)\mathcal{K}{=\gamma_{\downarrow}\mathrm{e}^{-\frac{\pi}{4}\gamma_{\downarrow}\bar{\gamma}_{\downarrow}}\mathcal{K}},~{}~{}~{}~{}~{}~{}~{}\,P_{F,R}=\hskip
1.0pt\mathrm{i}\hskip 1.0pt\bar{\gamma}_{\downarrow}\gamma_{\downarrow}.$
(3.35)

We can also show:

$\displaystyle\mathcal{T}_{L}{\gamma}_{\uparrow}\mathcal{T}_{L}^{-1}=-\bar{\gamma}_{\uparrow},\quad\mathcal{T}_{L}\bar{\gamma}_{\uparrow}\mathcal{T}_{L}^{-1}={\gamma}_{\uparrow}.$
(3.36)
$\displaystyle\mathcal{T}_{R}{\gamma}_{\downarrow}\mathcal{T}_{R}^{-1}=-\bar{\gamma}_{\downarrow},\quad\;\;\mathcal{T}_{R}\bar{\gamma}_{\downarrow}\mathcal{T}_{R}^{-1}={\gamma}_{\downarrow}.$
(3.37)

The 0+1d edge symmetry operators form a finite non-Abelian group
$\mathbb{M}_{16}$ [33] of order 16:111111Note that $\mathcal{K}^{2}=1$, we can
show $\mathcal{T}_{L}^{2}=\gamma_{\uparrow}\bar{\gamma}_{\uparrow}=-\hskip
1.0pt\mathrm{i}\hskip 1.0ptP_{F,L}$,
$\mathcal{T}_{R}^{2}=\gamma_{\downarrow}\bar{\gamma}_{\downarrow}=-\hskip
1.0pt\mathrm{i}\hskip 1.0ptP_{F,R}$,
$\mathcal{T}_{L}^{4}=\mathcal{T}_{R}^{4}=-1$, and
$\mathcal{T}_{L}^{8}=\mathcal{T}_{R}^{8}=1$, while
$P_{F,L/R}\mathcal{T}_{L/R}P_{F,L/R}=-\mathcal{T}_{L/R}=\mathcal{T}_{L/R}^{5}$

$\mathbb{M}^{F,T}_{16}|_{\text{boundary}}\equiv\langle{\mathcal{T}_{L/R}},{P_{F,L/R}}\mid\mathcal{T}_{L/R}^{4}=-1,\;\mathcal{T}_{L/R}^{8}={P^{2}_{F,L/R}}=1,\;P_{F,L/R}\mathcal{T}_{L/R}P_{F,L/R}=\mathcal{T}_{L/R}^{5}\rangle.$
(3.38)

which again has the fermion parity ${P_{F,L/R}}$ not at its center
$Z(\mathbb{M}^{F,T}_{16})\cong\mathbb{Z}_{4}$ $=\\{1,$
${\mathcal{T}^{2}_{L/R}},$ ${\mathcal{T}^{4}_{L/R}},$
${\mathcal{T}^{6}_{L/R}}\\}$. There is a short exact sequence for
$\mathbb{M}_{16}$ as a central extension

$\displaystyle
1\longrightarrow\Big{(}Z(\mathbb{M}_{16})\cong\mathbb{Z}_{4}\Big{)}\longrightarrow(\mathbb{M}_{16})\longrightarrow(\mathbb{Z}_{2})^{2}\longrightarrow
1,$ (3.39)

where the quotient group can be regarded as
$(\mathbb{Z}_{2})^{2}=Z(\mathbb{M}_{16})\\{1,\;{\mathcal{T}_{L/R}},\;{P_{F,L/R}},\;{\mathcal{T}_{L/R}}{P_{F,L/R}}\\}$.
But the short exact sequence that relates the extended symmetry on the
boundary $\mathbb{M}_{16}^{F,T}|_{\text{boundary}}$ and the original symmetry
group $\mathbb{Z}_{4}^{TF}|_{\text{bulk}}$, is the following

$1\longrightarrow\mathbb{Z}_{4}\overset{}{\longrightarrow}\mathbb{M}_{16}^{F,T}|_{\text{boundary}}\overset{}{\longrightarrow}\mathbb{Z}_{4}^{TF}|_{\text{bulk}}\longrightarrow
1,$ (3.40)

where the normal subgroup $N=\mathbb{Z}_{4}=\\{1,$
${\mathcal{T}^{2}_{L/R}}{P_{F,L/R}},$ ${\mathcal{T}^{4}_{L/R}},$
${\mathcal{T}^{6}_{L/R}}{P_{F,L/R}}\\}$ is not the center
$Z(\mathbb{M}^{F,T}_{16})$. Thus eq (3.40) is the non-central extension. The
quotient group can be regarded as
$\mathbb{Z}_{4}^{TF}=$$N\\{1,\;{\mathcal{T}_{L/R}},\;{\mathcal{T}^{2}_{L/R}},\;{\mathcal{T}^{3}_{L/R}}\\}$.

In fact the boundary fractionalized symmetry with $\mathcal{T}^{4}=-1$ and
$\mathcal{T}^{8}=+1$ has been noticed by a remarkable work by Gu in Ref. [34].
We further point out that the full boundary fractionalized symmetry group
forms a non-Abelian $\mathbb{M}^{F,T}_{16}$.

####  3.3.2 Unwinding the bulk: Extension to a non-Abelian
$\mathbb{M}^{F,T}_{16}$

The DIII Hamiltonian can be unwound using the _same_ Hilbert space extension
and the _same_ 2-layer FDUC,

$W=W_{2}W_{1}$

as for class BDI (eq (3.25)-eq (3.27)). It is easy to check that $W$ is also
invariant under the extended $\mathbb{M}_{16}^{F,T}$ symmetry generator
$\mathcal{T}_{\text{ext}}$ and the new $P_{F}$ defined below

$\displaystyle\mathcal{T}_{\text{ext}}$ $\displaystyle=$
$\displaystyle\prod_{\text{odd}~{}j}\gamma_{\downarrow,j}\left(\frac{1-\gamma_{\downarrow,j}\bar{\gamma}_{\downarrow,j}}{\sqrt{2}}\right)\bar{\gamma}_{\uparrow,j}\left(\frac{1+\gamma_{\uparrow,j}\bar{\gamma}_{\uparrow,j}}{\sqrt{2}}\right)\gamma_{j}\left(\frac{1-\gamma_{j}\bar{\gamma}_{j}}{\sqrt{2}}\right)$
(3.41)
$\displaystyle\cdot\prod_{\text{even}~{}j}\gamma_{\downarrow,j}\left(\frac{1-\gamma_{\downarrow,j}\bar{\gamma}_{\downarrow,j}}{\sqrt{2}}\right)\bar{\gamma}_{\uparrow,j}\left(\frac{1+\gamma_{\uparrow,j}\bar{\gamma}_{\uparrow,j}}{\sqrt{2}}\right)\bar{\gamma}_{j}\left(\frac{1+\gamma_{j}\bar{\gamma}_{j}}{\sqrt{2}}\right)\mathcal{K}.$
$\displaystyle P_{F}$ $\displaystyle=$ $\displaystyle\prod_{j}\left(\hskip
1.0pt\mathrm{i}\hskip
1.0pt\bar{\gamma}_{\downarrow,j}\gamma_{\downarrow,j}\right)\left(\hskip
1.0pt\mathrm{i}\hskip
1.0pt\bar{\gamma}_{\uparrow,j}\gamma_{\uparrow,j}\right)\left(\hskip
1.0pt\mathrm{i}\hskip 1.0pt\bar{\gamma}_{j}\gamma_{j}\right).$ (3.42)

The extended bulk symmetry operators form again a non-Abelian
$\mathbb{M}_{16}$ [33] of order 16:121212Note that $\mathcal{K}^{2}=1$, we can
show
$\mathcal{T}_{\text{ext}}^{2}=\prod_{j}\left(\gamma_{\downarrow,j}\bar{\gamma}_{\downarrow,j}\right)\left(\gamma_{\uparrow,j}\bar{\gamma}_{\uparrow,j}\right)(\gamma_{j}\bar{\gamma}_{j})$,
$\mathcal{T}_{\text{ext}}^{4}=-1$, and $\mathcal{T}_{\text{ext}}^{8}=1$, also
$P_{F}\mathcal{T}_{\text{ext}}P_{F}=-\mathcal{T}_{\text{ext}}=\mathcal{T}_{\text{ext}}^{5}$.

$\mathbb{M}^{F,T}_{16}|_{\text{bulk}}\equiv\langle{\mathcal{T}_{\text{ext}}},{P_{F}}\mid{\mathcal{T}_{\text{ext}}^{4}=-1,}\;\mathcal{T}_{\text{ext}}^{8}={P^{2}_{F}}=1,\;P_{F}\mathcal{T}_{\text{ext}}P_{F}=\mathcal{T}_{\text{ext}}^{5}\rangle.$
(3.43)

The short exact sequence that relates the extended and original symmetry
groups in the bulk again is the following

$1\longrightarrow\mathbb{Z}_{4}\overset{}{\longrightarrow}\mathbb{M}_{16}^{F,T}|_{\text{bulk}}\overset{}{\longrightarrow}\mathbb{Z}_{4}^{TF}|_{\text{bulk}}\longrightarrow
1.$ (3.44)

where the normal subgroup $\mathbb{Z}_{4}=\\{1,$
${\mathcal{T}^{2}_{\text{ext}}}{P_{F}},$ ${\mathcal{T}^{4}_{\text{ext}}},$
${\mathcal{T}^{6}_{\text{ext}}}{P_{F}}\\}$ is not the center
$Z(\mathbb{M}^{F,T}_{16})$. The quotient group $\mathbb{Z}_{4}^{TF}$ contains
$\\{1,\;{\mathcal{T}_{\text{ext}}},\;{\mathcal{T}^{2}_{\text{ext}}},\;{\mathcal{T}^{3}_{\text{ext}}}\\}$.

###  3.4 Class AIII: ${\rm U}(1)^{F}\times\mathbb{Z}_{2}^{T}$ or $\frac{{\rm
U}(1)^{F}\times\mathbb{Z}^{{T}}_{4}}{\mathbb{Z}_{2}^{F}}$ symmetry or Pinc

Let us look at another example for free fermion Hamiltonian AIII class, which
can be chosen to be ${\rm U}(1)^{F}\times\mathbb{Z}_{2}^{T}$ or $\frac{{\rm
U}(1)^{F}\times\mathbb{Z}^{{T}}_{4}}{\mathbb{Z}_{2}^{F}}$ symmetry, see Table
1.131313The time reversal $\mathcal{T}$ with $\mathcal{T}^{2}=+1$ in ${\rm
U}(1)^{F}\times\mathbb{Z}_{2}^{T}$ can be redefined as a new time reversal
$\mathcal{T}^{\prime}=\exp(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi/2)\mathcal{T}=\hskip 1.0pt\mathrm{i}\hskip 1.0pt\mathcal{T}$ with
${\mathcal{T}^{\prime}}^{2}=-1$ and ${\mathcal{T}^{\prime}}^{4}=+1$ in
$\frac{{\rm U}(1)^{F}\times\mathbb{Z}^{{T}}_{4}}{\mathbb{Z}_{2}^{F}}$. Here
the $\exp(\hskip 1.0pt\mathrm{i}\hskip 1.0pt\pi/2)=\hskip
1.0pt\mathrm{i}\hskip 1.0pt$ is a $\pi/2$ rotation of the U(1) symmetry
transformation which is also related to the $\pi$ rotation of fermion since
the $2\pi$ rotation of fermion gives a spin-statistics $(-1)$ from
${\mathbb{Z}_{2}^{F}}$. For ${\rm U}(1)^{F}\times\mathbb{Z}_{2}^{T}$ symmetry,
a representative of class AIII systems is a superconductor with time reversal
$\mathcal{T}=\mathcal{K}$ and a spin-rotation ${\rm U}(1)$ symmetry

$V(\theta)=\prod_{j}\exp\Big{(}\frac{\theta}{2}\left(\gamma_{\downarrow,j}\gamma_{\uparrow,j}+\bar{\gamma}_{\downarrow,j}\bar{\gamma}_{\uparrow,j}\right)\Big{)}.$
(3.45)

It can be easily checked that the Hamiltonian $H^{\text{II}}_{\text{Kitaev}}$
of eq (3.3) is invariant under the symmetries defined above. Once again, we
use the basis change to get the Hamiltonian $H_{\text{gen}}$ of eq (3.5) and
the following basis-changed symmetry operators:

$\displaystyle{\tilde{\mathcal{T}}}$ $\displaystyle\equiv$ $\displaystyle
M\mathcal{T}M^{\dagger}=M\mathcal{K}M^{\dagger}=\left(\prod_{j}(\gamma_{\downarrow,j}\bar{\gamma}_{\uparrow,j})\right)\mathcal{K}.$
(3.46) $\displaystyle{\tilde{V}{(\theta)}}$ $\displaystyle\equiv$
$\displaystyle
MV(\theta)M^{\dagger}=\prod_{j}\exp\Big{(}\frac{\theta}{2}\left(\bar{\gamma}_{\downarrow,j}\gamma_{\downarrow,j}-\bar{\gamma}_{\uparrow,j}\gamma_{\uparrow,j}\right)\Big{)}{=\prod_{j}\exp\Big{(}\hskip
1.0pt\mathrm{i}\hskip
1.0pt{\theta}\big{(}(N_{F,\downarrow})_{j}-(N_{F,\uparrow})_{j}\big{)}\Big{)}.}\quad\quad\quad$
(3.47)

Here we use the fact for the flavor $\uparrow$ or $\downarrow$ (abbreviated as
$\uparrow/\downarrow$),141414We remark that the left-handed side expression
fermion number $(N_{F,\uparrow/\downarrow})_{j}$ has eigenvalues in
$\\{0,1\\}$ for states with a complex fermion empty or filled. In contrast,
$\hskip 1.0pt\mathrm{i}\hskip
1.0pt\bar{\gamma}_{\uparrow/\downarrow,j}\gamma_{\uparrow/\downarrow,j}=(P_{F,\uparrow/\downarrow})_{j}$
is the fermion parity on the site $j$ for the flavor $\uparrow$ or
$\downarrow$, which $(P_{F,\uparrow/\downarrow})_{j}$ has eigenvalues in
$\\{1,-1\\}$. So the right-handed side
$\frac{1-(P_{F,\uparrow/\downarrow})_{j}}{2}$ also indeed has eigenvalues in
$\\{0,1\\}$ exactly matching $(N_{F,\uparrow/\downarrow})_{j}$.

$(N_{F,\uparrow/\downarrow})_{j}\equiv\psi^{\dagger}_{\uparrow/\downarrow,j}\psi_{\uparrow/\downarrow,j}=\frac{1-\hskip
1.0pt\mathrm{i}\hskip
1.0pt\bar{\gamma}_{\uparrow/\downarrow,j}\gamma_{\uparrow/\downarrow,j}}{2}=\frac{1-(P_{F,\uparrow/\downarrow})_{j}}{2}.$

Note that eq (3.47) is in the form of $S_{z}$ spin-rotational U(1) symmetry
generator $\prod_{j}\exp(-\hskip 1.0pt\mathrm{i}\hskip
1.0pt{\theta}(S_{z})_{j})$ of the spin-1/2 system (a fundamental
representation 2 of SU(2)), with a periodicity $\theta\sim\theta+2\pi$, so
$\theta\in[0,2\pi)$.

####  3.4.1 Gapping the boundary: Extension to a non-Abelian $\frac{{\rm
U}(1)\times\mathbb{D}_{8}^{F,T}}{\mathbb{Z}_{2}}$

Let us now look at the symmetry transformation on the edges, on the left ($L$)
and right ($R$) boundaries:

$\displaystyle V_{L}(\theta)$ $\displaystyle=$
$\displaystyle\exp\Big{(}\frac{\theta}{2}\bar{\gamma}_{\downarrow}\gamma_{\downarrow}\Big{)},~{}~{}~{}~{}~{}~{}~{}~{}~{}V_{R}(\theta)=\exp\Big{(}-\frac{\theta}{2}\bar{\gamma}_{\uparrow}\gamma_{\uparrow}\Big{)},$
(3.48) $\displaystyle\mathcal{T}_{L}$ $\displaystyle=$
$\displaystyle\gamma_{\downarrow}\mathcal{K},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mathcal{T}_{R}=\bar{\gamma}_{\uparrow}\mathcal{K},$
(3.49) $\displaystyle P_{F,L}$ $\displaystyle=$ $\displaystyle\hskip
1.0pt\mathrm{i}\hskip
1.0pt\bar{\gamma}_{\downarrow}\gamma_{\downarrow},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}P_{F,R}=\hskip
1.0pt\mathrm{i}\hskip 1.0pt\bar{\gamma}_{\uparrow}\gamma_{\uparrow}.$ (3.50)

Some comments are in order:

  * •

The bulk ${\rm U}(1)^{F}$ symmetry eq (3.47) acting on the two complex
fermions per site has a $2\pi$ periodicity: $\theta\in[0,2\pi)$. Note that eq
(3.47) at $\theta=\pi$, we have effectively ${\tilde{V}{(\theta=\pi)}}=P_{F}$;
this generates a
$\mathbb{Z}_{2}^{F}=\\{\tilde{V}{(\theta=0)},\tilde{V}{(\theta=\pi)}\\}=\\{1,P_{F}\\}$,
which is a normal subgroup of ${\rm U}(1)^{F}$.

  * •

The boundary U(1) symmetry eq (3.48) acting on the one complex fermion per
site has a $4\pi$ periodicity: $\theta\in[0,4\pi)$. Note that this boundary
U(1) does _not_ contain $\mathbb{Z}_{2}^{F}=\\{1,P_{F}\\}$, since eq (3.48)’s
generator
$\exp(\frac{\theta}{2}\bar{\gamma}{\gamma})=\cos(\frac{\theta}{2})+\sin(\frac{\theta}{2})\bar{\gamma}{\gamma}$
can _not_ contain eq (3.50)’s $P_{F}=\hskip 1.0pt\mathrm{i}\hskip
1.0pt\bar{\gamma}\gamma$. Instead eq (3.48) of the $L$/$R$ boundary contains a
$\mathbb{Z}_{2}=\\{1,V_{L/R}(\theta=2\pi)\\}=\\{1,-1\\}$.

  * •

On the boundary, still there is a $\mathbb{D}_{8}^{F,T}$ finite group symmetry
generated by $\mathcal{T}_{L/R}$ and $P_{F,L/R}$, similar to Sec. 3.2.1. There
is a normal subgroup $\mathbb{Z}_{2}=\\{1,-1\\}$ shared by both the boundary
symmetry ${\rm U}(1)$ and
$\mathbb{D}_{8}^{F,T}\equiv\mathbb{Z}_{2}^{F}\ltimes\mathbb{Z}_{4}^{T}$ with
the new fractionalized $\mathbb{Z}_{4}^{T}$. We can explicitly check the
boundary U(1) commutes with other generators of $\mathbb{D}_{8}^{F,T}$.

In summary of the above, so the edge symmetry operators form the boundary
extended symmetry group $\tilde{G}|_{\text{boundary}}$, which can be written
into the short exact sequence:151515To check some of the above mentioned
equalities, we use the fact $\exp(\hskip 1.0pt\mathrm{i}\hskip 1.0pt\theta
X)=\cos(\theta X)+\hskip 1.0pt\mathrm{i}\hskip 1.0pt\sin(\theta X)$ for a
matrix operator $X$. For example, we have
$\exp(\frac{\theta}{2}\bar{\gamma}{\gamma})=\cos(\frac{\theta}{2})+\sin(\frac{\theta}{2})\bar{\gamma}{\gamma}$.
$\bullet$ Note that for eq (3.47) of the bulk,
$\tilde{V}{(\theta=\pi)}=\prod_{j}\exp(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi((N_{F,\uparrow})_{j}+(N_{F,\downarrow})_{j}\big{)})=(-1)^{((N_{F,\uparrow})_{j}+(N_{F,\downarrow})_{j})}=P_{F}=\prod_{j}\left(\hskip
1.0pt\mathrm{i}\hskip
1.0pt\bar{\gamma}_{\uparrow,j}\gamma_{\uparrow,j}\right)\left(\hskip
1.0pt\mathrm{i}\hskip
1.0pt\bar{\gamma}_{\downarrow,j}\gamma_{\downarrow,j}\right)$ generates the
fermion parity $\mathbb{Z}_{2}^{F}$ of the bulk system.
$\bullet$ Note that for eq (3.48) of the $L$ or $R$ boundary,
$V_{L/R}(\theta=2\pi)=\exp\Big{(}{\pi}\bar{\gamma}_{\downarrow/\uparrow}\gamma_{\downarrow/\uparrow}\Big{)}=(-1)^{\hskip
1.0pt\mathrm{i}\hskip
1.0pt(\bar{\gamma}_{\downarrow/\uparrow}\gamma_{\downarrow/\uparrow})}=(-1)^{P_{F,L/R}}=-1$,
generates a $\mathbb{Z}_{2}$ but _not_ the fermion parity $\mathbb{Z}_{2}^{F}$
of the effective boundary.

$\displaystyle
1\to\mathbb{Z}_{2}\to\underset{\quad\supseteq\mathbb{D}_{8}^{F,T}=\mathbb{Z}_{2}^{F}\ltimes\mathbb{Z}_{4}^{T}}{\frac{{\rm
U}(1)\times\mathbb{D}_{8}^{F,T}}{\mathbb{Z}_{2}}}|_{\text{boundary}}\to\underset{\quad\supseteq\mathbb{Z}_{2}^{F}\times\mathbb{Z}_{2}^{T}}{{\rm
U}(1)^{F}\times\mathbb{Z}_{2}^{T}}|_{\text{bulk}}\to 1.$ (3.51)

In the underset of groups $\tilde{G}|_{\text{boundary}}$ and
$G|_{\text{bulk}}$, we indicate that they contain the subgroups
($\supseteq\dots$) established in Sec. 3.2.1.

####  3.4.2 Unwinding the bulk: Extension to a non-Abelian $\frac{{\rm
U}(1)\times\mathbb{D}_{8}^{F,T}}{\mathbb{Z}_{2}}$

The AIII Hamiltonian can be unwound using the same Hilbert space extension and
2-layer FDUC, $W=W_{2}W_{1}$ as for class BDI (eq (3.25)-eq (3.28)). It is
easy to check that $W$ is also invariant under the extended $\frac{{\rm
U}(1)\times\mathbb{D}_{8}^{F,T}}{\mathbb{Z}_{2}}$ symmetry generators defined
below

$\displaystyle V_{\text{ext}}(\theta)$ $\displaystyle=$
$\displaystyle\prod_{\text{odd}~{}j}\exp\Big{(}\frac{\theta}{2}\left(\bar{\gamma}_{\downarrow,j}\gamma_{\downarrow,j}-\bar{\gamma}_{\uparrow,j}\gamma_{\uparrow,j}+\bar{\gamma}_{j}\gamma_{j}\right)\Big{)}\prod_{\text{even}~{}j}\exp\Big{(}\frac{\theta}{2}\left(\bar{\gamma}_{\downarrow,j}\gamma_{\downarrow,j}-\bar{\gamma}_{\uparrow,j}\gamma_{\uparrow,j}-\bar{\gamma}_{j}\gamma_{j}\right)\Big{)},\quad\quad\;$
(3.52) $\displaystyle\mathcal{T}_{\text{ext}}$ $\displaystyle=$
$\displaystyle\prod_{\text{odd}~{}j}\gamma_{\downarrow,j}\bar{\gamma}_{\uparrow,j}\gamma_{j}\prod_{\text{even}~{}j}\gamma_{\downarrow,j}\bar{\gamma}_{\uparrow,j}\bar{\gamma}_{j}\mathcal{K},$
(3.53) $\displaystyle P_{F}$ $\displaystyle=$
$\displaystyle\prod_{j}\left(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\bar{\gamma}_{\downarrow,j}\gamma_{\downarrow,j}\right)\left(\hskip
1.0pt\mathrm{i}\hskip
1.0pt\bar{\gamma}_{\uparrow,j}\gamma_{\uparrow,j}\right)\left(\hskip
1.0pt\mathrm{i}\hskip 1.0pt\bar{\gamma}_{j}\gamma_{j}\right).$ (3.54)

In short, the bulk unwinding follows also the symmetry extension:

$\displaystyle
1\longrightarrow\mathbb{Z}_{2}\overset{}{\longrightarrow}\frac{{\rm
U}(1)\times\mathbb{D}_{8}^{F,T}}{\mathbb{Z}_{2}}|_{\text{bulk}}\overset{}{\longrightarrow}{\rm
U}(1)^{F}\times\mathbb{Z}_{2}^{T}|_{\text{bulk}}\longrightarrow 1.$ (3.55)

###  3.5 Class CII: ${\rm SU}(2)^{F}\times\mathbb{Z}_{2}^{T}$ symmetry or
Pin${}^{-}\times_{\mathbb{Z}_{2}^{F}}{\rm SU}(2)^{F}$

The 1+1d CII fermion phase has a nontrivial class
$\nu_{\text{CII}}\in\mathbb{Z}_{2}$ which actually corresponds to a bosonic
SPT phase with a $\mathbb{Z}_{2}^{T}$ time reversal symmetry tensor product
with a trivial fermionic gapped phase, and can be unwound by an extended
symmetry that has a fermion parity at the center, see the detailed discussions
in [3]. In terms of the CII class symmetry realization, other than ${\rm
SU}(2)^{F}\times\mathbb{Z}_{2}^{T}$, it can also be realized as $\frac{({{\rm
U}}(1)\rtimes{\mathbb{Z}}_{4}^{C})}{{\mathbb{Z}}_{2}^{F}}$ symmetry [25, 3].
Let us interpret how to gap the boundary and unwind the bulk via the following
symmetry extensions.

If we extend the $\mathbb{Z}_{2}^{T}$ via a new species of fermions with a
fermion parity $\mathbb{Z}_{2}^{F_{+}}$ to $\mathbb{Z}_{4}^{T{F_{+}}}$ with
$\mathcal{T}^{2}=(-1)^{F_{+}}$, we have the symmetry extensions:

$\displaystyle\begin{array}[]{ccccccccc}1&\longrightarrow&\mathbb{Z}_{2}^{F_{+}}&\longrightarrow&{\rm
SU}(2)^{F}\times\mathbb{Z}_{4}^{T{F_{+}}}&\longrightarrow&{\rm
SU}(2)^{F}\times\mathbb{Z}_{2}^{T}&\longrightarrow&1.\\\
1&\longrightarrow&\mathbb{Z}_{2}^{F_{+}}&\longrightarrow&\frac{({{\rm
U}}(1)\rtimes{\mathbb{Z}}_{4}^{C})}{{\mathbb{Z}}_{2}^{F}}\times\mathbb{Z}_{4}^{T{F_{+}}}&\longrightarrow&\frac{({{\rm
U}}(1)\rtimes{\mathbb{Z}}_{4}^{C})}{{\mathbb{Z}}_{2}^{F}}\times\mathbb{Z}_{2}^{T}&\longrightarrow
1.\\\
1&\longrightarrow&\mathbb{Z}_{2}^{F_{+}}&\longrightarrow&\mathbb{Z}_{4}^{T{F_{+}}}&\longrightarrow&\mathbb{Z}_{2}^{T}&\longrightarrow&1.\end{array}$
(3.59)

If we extend the $\mathbb{Z}_{2}^{T}$ via a bosonic $\mathbb{Z}_{2}^{B}$ to
$\mathbb{Z}_{4}^{T{B}}$ symmetry with $\mathcal{T}^{2}=-1$ as a bosonic
$(-1)$, we have the symmetry extensions:

$\displaystyle\begin{array}[]{ccccccccc}1&\longrightarrow&\mathbb{Z}_{2}^{B}&\longrightarrow&{\rm
SU}(2)^{F}\times\mathbb{Z}_{4}^{TB}&\longrightarrow&{\rm
SU}(2)^{F}\times\mathbb{Z}_{2}^{T}&\longrightarrow&1.\\\
1&\longrightarrow&\mathbb{Z}_{2}^{B}&\longrightarrow&\frac{({{\rm
U}}(1)\rtimes{\mathbb{Z}}_{4}^{C})}{{\mathbb{Z}}_{2}^{F}}\times\mathbb{Z}_{4}^{TB}&\longrightarrow&\frac{({{\rm
U}}(1)\rtimes{\mathbb{Z}}_{4}^{C})}{{\mathbb{Z}}_{2}^{F}}\times\mathbb{Z}_{2}^{T}&\longrightarrow
1.\\\
1&\longrightarrow&\mathbb{Z}_{2}^{B}&\longrightarrow&\mathbb{Z}_{4}^{TB}&\longrightarrow&\mathbb{Z}_{2}^{T}&\longrightarrow&1.\end{array}$
(3.63)

The quantum mechanical lattice constructions of the above are done in [3].
Formally, we can also understand it from trivializing the 2d topological term
in the bordism group
$\Omega_{2}^{\mathrm{Pin}^{-}\times_{\mathbb{Z}_{2}^{F}}{\rm SU}(2)^{F}}$,
which will be explained in Sec. 6.

###  3.6 Formal explanation on the classifications of invertible phases vs SPT
phases

  * •

_Long-ranged entangled (LRE) invertible fermionic topological orders_
(protected by no other symmetry, except the fermion parity
$\mathbb{Z}_{2}^{F}$ essential for fermionic systems): To obtain its
classification, we can read from $\mathrm{TP}_{d}({\rm Spin})$ with the group
$G={\rm Spin}(d)$ symmetry. Note that ${\rm Spin}(d)/{\rm
SO}(d)=\mathbb{Z}_{2}^{F}$ with the ${\rm SO}(d)$ is the rotational symmetry
for the continuum limit of the tangent manifold of spacetime manifold.

  * •

_Short-ranged entangled (SRE) invertible SPTs_ : To obtain its classification,
we can consider the quotient group
$\frac{\mathrm{TP}_{d}(G)}{\mathrm{TP}_{d}({\rm Spin})}$. Namely, the
invertible fermionic phases classified by $\mathrm{TP}_{d}(G)$ mod out the LRE
invertible fermionic topological orders in $\mathrm{TP}_{d}({\rm Spin})$,
indeed are left with only SRE invertible SPTs [26]. In fact, for interacting
systems, for the finite group subclasses of the SRE SPT classification (known
as the torsion [tors] classes in mathematics), we can rewrite the SRE
fermionic SPT finite-subgroup-part classification following [26] as the
torsion part of the quotient group
$\frac{\mathrm{TP}_{d}(G)}{\mathrm{TP}_{d}({\rm Spin})}$:

$\displaystyle\big{[}\frac{\mathrm{TP}_{d}(G)}{\mathrm{TP}_{d}({\rm
Spin})}\big{]}_{{\rm tors}}=\frac{[\mathrm{TP}_{d}(G)]_{\rm
tors}}{[\mathrm{TP}_{d}({\rm Spin})]_{\rm tors}}=\frac{[\Omega_{d}^{G}]_{\rm
tors}}{[\Omega_{d}^{\rm Spin}(pt)]_{\rm tors}}.$ (3.64)

Here we use the fact that the torsion (finite subgroup) part of cobordism and
bordism groups are the same ${[\mathrm{TP}_{d}(G)]_{\rm
tors}}={[\Omega_{d}^{G}]_{\rm tors}}$; this eq (3.64) relates to the $d$-th
bordism group $\Omega_{d}$. This expression in eq (3.64) left out other
infinite group $\mathbb{Z}$ classes:
$\bullet$ Some of $\mathbb{Z}$ classes are known as the free classes in
mathematics correspond to LRE invertible topological orders by Wen’s
definition [29], captured by

$\displaystyle\big{[}{\mathrm{TP}_{d}({\rm Spin})}\big{]}_{{\rm free}}.$
(3.65)

$\bullet$ Other $\mathbb{Z}$ classes are captured by the free part of quotient
group $\frac{\mathrm{TP}_{d}(G)}{\mathrm{TP}_{d}({\rm Spin})}$ as

$\displaystyle\big{[}\frac{\mathrm{TP}_{d}(G)}{\mathrm{TP}_{d}({\rm
Spin})}\big{]}_{{\rm free}},$ (3.66)

which happen especially in an odd dimensional spacetime and correspond to the
$\mathbb{Z}$ classes of SRE invertible SPTs whose boundary exhibits the
_perturbative local anomalies_. For example, in 2+1d, there is a $\mathbb{Z}$
class of SRE invertible SPTs protected by U(1) symmetry, for both bosonic or
fermionic systems [14].
$\bullet$ In contrast, the other $\mathbb{Z}_{n}$ classes (the torsion) from
$\big{[}{\mathrm{TP}_{d}(G)}\big{]}_{{\rm tors}}$ correspond to invertible
topological phases whose boundary exhibits the _non-perturbative global
anomalies_. This $\big{[}{\mathrm{TP}_{d}(G)}\big{]}_{{\rm tors}}$ includes
both the $\big{[}{\mathrm{TP}_{d}({\rm Spin})}\big{]}_{{\rm tors}}$ as classes
of LRE invertible topological orders and the
$\big{[}\frac{\mathrm{TP}_{d}(G)}{\mathrm{TP}_{d}({\rm Spin})}\big{]}_{{\rm
tors}}$ as SRE invertible SPTs. The $pt$ is a point as a certain classifying
space.

##  4 Interacting models for even $\nu\in\mathbb{Z}_{8}$ classes with
$\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F}$ symmetry or Pin-

We now consider an interacting model of time-reversal invariant fermions with
$\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F}$ symmetry belonging to the same
$k=2$ phase as the model in section 3.2 of the $\mathbb{Z}_{8}$
classification. This is a basis-changed version of the model introduced by
Tantivasadakarn and Vishwanath [35] (see Appendix A for details of the basis-
change operations). The Hilbert space for the model consists of two species of
fermions ($\varsigma=\uparrow,\downarrow$: each complex fermions beget two
Majorana fermions, ${\gamma}$ and $\bar{\gamma}$) on the sites and qubits
($\sigma$) on the links of a one-dimensional lattice. Using the Majorana
operators as defined in the previous sections, the Hamiltonian is written as

$H=-\hskip 1.0pt\mathrm{i}\hskip
1.0pt\sum_{j}\left[\sigma^{z}_{j,j+1}\left(\bar{\gamma}_{\uparrow,j}\gamma_{\uparrow,j}+\bar{\gamma}_{\downarrow,j+1}\gamma_{\downarrow,j+1}\right)+{\sigma^{x}_{j,j+1}\bar{\gamma}_{\uparrow,j}\gamma_{\downarrow,j+1}}\right].$
(4.1)

The above Hamiltonian is also represented graphically in Fig 11.

Figure 11: The interacting $\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F}$
invariant SPT Hamiltonian of eq. 4.1. The squares consist of two complex
fermions, labeled $\uparrow$ and $\downarrow$ and reside on the sites of the
lattice. The diamonds represent the qubits $\sigma$ and reside on the links of
the lattice.

The time-reversal and fermion parity operators are

$\mathcal{T}=\left(\prod_{j}\left(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\gamma_{\downarrow,j}\bar{\gamma}_{\uparrow,j}\right)\sigma^{x}_{j,j+1}\right)~{}\mathcal{K},~{}~{}~{}~{}P_{F}=\prod_{j}\prod_{\varsigma=\uparrow,\downarrow}\hskip
1.0pt\mathrm{i}\hskip 1.0pt\bar{\gamma}_{\varsigma,j}\gamma_{\varsigma,j},$
(4.2)

where $\mathcal{K}$ is the complex conjugation operation as before. The
boundary symmetry operators are as follows:

$\displaystyle\mathcal{T}_{L}$ $\displaystyle=$
$\displaystyle\gamma_{\downarrow}\mathcal{K},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mathcal{T}_{R}=\hskip
1.0pt\mathrm{i}\hskip 1.0pt\bar{\gamma}_{\uparrow}\mathcal{K}$ (4.3)
$\displaystyle P_{F,L}$ $\displaystyle=$ $\displaystyle\hskip
1.0pt\mathrm{i}\hskip
1.0pt\bar{\gamma}_{\downarrow}\gamma_{\downarrow},~{}~{}~{}~{}~{}~{}P_{F,R}=\hskip
1.0pt\mathrm{i}\hskip 1.0pt\bar{\gamma}_{\uparrow}\gamma_{\uparrow}$ (4.4)

On both ends, the symmetry operators generate the group $\mathbb{D}_{8}^{F,T}$
which does not have fermion parity at the center

${\mathbb{D}_{8}^{F,T}\equiv\langle(\mathcal{T}_{L/R}P_{F,L/R}),P_{F,L/R}\;|\;\mathcal{T}_{L/R}^{2}=P_{F,L/R}^{2}=1,P_{F,L/R}\mathcal{T}_{L/R}P_{F,L/R}=-\mathcal{T}_{L/R}\rangle.}$
(4.5)

We now proceed to extend the Hilbert space and unwind the Hamiltonian using a
$\mathbb{D}_{8}^{F,T}$ invariant quantum circuit. The extended Hilbert space
we choose consists of an additional qubit per link ($\tau_{j,j+1}$), an
additional qubit per site ($\sigma_{j}$) and an additional fermion per site
($\gamma_{j},\bar{\gamma_{j}}$). The extended Hamiltonian we consider is

$H_{\text{ext}}=H-\sum_{j}\sigma^{x}_{j}-\hskip 1.0pt\mathrm{i}\hskip
1.0pt\sum_{\text{odd}~{}j}\left[\tau^{z}_{j,j+1}\left(\bar{\gamma}_{j}\gamma_{j}+\bar{\gamma}_{j+1}\gamma_{j+1}\right)+{\tau^{x}_{j,j+1}\gamma_{j}\bar{\gamma}_{j+1}}\right]-\sum_{\text{even}~{}j}\tau^{x}_{j,j+1}.$
(4.6)

This extended Hamiltonian is schematically shown in Fig 12.

Figure 12: Extended $\mathbb{D}_{8}\equiv\mathbb{D}_{8}^{F,T}$ invariant SPT
Hamiltonian of eq. 4.6. The squares consist of three complex fermions and one
qubit and reside on the sites of the lattice. Two other qubits (diamonds not
enclosed in squares) reside on the links of the lattice.

The Hamiltonian $H_{\text{ext}}$ is unwound to $H_{\text{triv}}$ as shown in
Fig 13 using the FDUC, $W$ as follows

$\displaystyle WH_{\text{ext}}W^{\dagger}$ $\displaystyle=$ $\displaystyle
H_{\text{triv}}.$ (4.7) $\displaystyle H_{\text{triv}}$ $\displaystyle=$
$\displaystyle-\sum_{j}\left[\sigma^{z}_{j}\Big{(}\sum_{\varsigma{=\uparrow,\downarrow}}\hskip
1.0pt\mathrm{i}\hskip
1.0pt\bar{\gamma}_{\varsigma,j}\gamma_{\varsigma,j}\Big{)}+\hskip
1.0pt\mathrm{i}\hskip
1.0pt\sigma^{x}_{j}\gamma_{\downarrow,j}\bar{\gamma}_{\uparrow,j}+\sigma^{x}_{j,j+1}\right]$
(4.8) $\displaystyle-\hskip 1.0pt\mathrm{i}\hskip
1.0pt\sum_{\text{odd}~{}j}\left[\tau^{z}_{j,j+1}\left(\bar{\gamma}_{j}\gamma_{j}+\bar{\gamma}_{j+1}\gamma_{j+1}\right)+\tau^{x}_{j,j+1}\bar{\gamma}_{j}\gamma_{j+1}\right]-\sum_{\text{even}~{}j}\tau^{x}_{j,j+1}.$
$\displaystyle W$ $\displaystyle=$ $\displaystyle W_{2}W_{1}.$ (4.9)
$\displaystyle W_{1}$ $\displaystyle=$
$\displaystyle\prod_{\text{odd}~{}j}\mathrm{SWAP}\left(\sigma_{j},\sigma_{j,j+1}\right)\mathrm{SWAP}\left(\tau_{j,j+1},\sigma_{j+1}\right)\left(\frac{1+\gamma_{j}\gamma_{\downarrow,j+1}}{\sqrt{2}}\right)\left(\frac{1+\bar{\gamma}_{j}\bar{\gamma}_{\downarrow,j+1}}{\sqrt{2}}\right).$
$\displaystyle W_{2}$ $\displaystyle=$
$\displaystyle\prod_{\text{even}~{}j}\left[\mathrm{SWAP}\left(\sigma_{j,j+1},\tau_{j,j+1}\right)\left(\frac{1+\gamma_{j}\gamma_{\uparrow,j}}{\sqrt{2}}\right)\left(\frac{1+\bar{\gamma}_{j}\bar{\gamma}_{\uparrow,j}}{\sqrt{2}}\right)\right]$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}\prod_{\text{odd}~{}j}\left[\left(\frac{1+\gamma_{j}\gamma_{\downarrow,j}}{\sqrt{2}}\right)\left(\frac{1+\bar{\gamma}_{j}\bar{\gamma}_{\downarrow,j}}{\sqrt{2}}\right)\right].$

Figure 13: Unwinding the interacting
$\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F}$ invariant SPT Hamiltonian eq.
4.6. The unwinding is done in two steps using a series of bosonic and
fermionic swap operators which commute with the extended symmetry.

It can be verified that the extended Hamiltonian as well as the unwinding FDUC
are invariant under the extended symmetry operator transformation

$\displaystyle\mathcal{T}_{\text{ext}}$ $\displaystyle=$
$\displaystyle\prod_{\text{odd}~{}j}\left(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\gamma_{\downarrow,j}\bar{\gamma}_{\uparrow,j}\gamma_{j}~{}\sigma^{x}_{j}\sigma^{x}_{j,j+1}\tau^{x}_{j,j+1}\right)\prod_{\text{even}~{}j}\left(\hskip
1.0pt\mathrm{i}\hskip
1.0pt\gamma_{\downarrow,j}\bar{\gamma}_{\uparrow,j}\hskip
1.0pt\mathrm{i}\hskip
1.0pt\bar{\gamma}_{j}~{}\sigma^{x}_{j}\sigma^{x}_{j,j+1}\tau^{x}_{j,j+1}\right)\mathcal{K}$
(4.10) $\displaystyle=$ $\displaystyle{\prod_{\text{all}~{}j}\left(\hskip
1.0pt\mathrm{i}\hskip
1.0pt\gamma_{\downarrow,j}\bar{\gamma}_{\uparrow,j}~{}\sigma^{x}_{j}\sigma^{x}_{j,j+1}\tau^{x}_{j,j+1}\right)(\prod_{\text{odd}~{}j}\gamma_{j})(\prod_{\text{even}~{}j}\hskip
1.0pt\mathrm{i}\hskip 1.0pt\bar{\gamma}_{j})\mathcal{K}.}$ (4.11)

##  5 Supersymmetric quantum mechanics

In this section, we detail the relationship between the extended symmetries
and supersymmetric quantum mechanics. More details can be found in ref [36].
Supersymmetric systems are characterized by the presence of _supercharges_ \-
symmetry operators which are fermionic i.e. anti-commute with fermion parity
and square to the generators of space-time translations [37]. The simplest of
supersymmetric systems are supersymmetric quantum-mechanical (SUSY QM) systems
where supercharges only square to the generator of time-translations i.e. the
Hamiltonian. More specifically, a Hamiltonian $H$ is a SUSY QM system if there
exists $\alpha=1\ldots{\cal N}$ Hermitian supercharges $Q_{\alpha}$ satisfying
the following algebra [38, 39, 40, 41]

$\displaystyle\\{Q_{\alpha},Q_{\beta}\\}=2H\delta_{\alpha,\beta},~{}[Q_{\alpha},H]=\\{Q_{\alpha},P_{F}\\}=0.$
(5.1)

If we consider the _fractionalized_ time-reversal symmetry on the boundaries,

  * •

from $\mathbb{Z}_{2}^{T}$ ($\mathcal{T}^{2}=+1$ for BDI) to a fractionalized
$\mathbb{Z}_{4}^{T}$ ($\mathcal{T}^{4}=+1$ in Sec. 3.2), and

  * •

from $\mathbb{Z}_{4}^{TF}$ ($\mathcal{T}^{4}=+1$ for DIII) to a fractionalized
$\mathbb{Z}_{8}^{T}$ ($\mathcal{T}^{8}=+1$ in Sec. 3.3),

their anti-commutation with the fermion parity $P_{F}$ operator,

$\\{\mathcal{T},P_{F}\\}\equiv\mathcal{T}P_{F}+P_{F}\mathcal{T}=0,$

will result in any compatible Hamiltonian being a SUSY QM system. As a result,
we refer to the presence of any symmetry operators which do not commute with
fermion parity as supersymmetric.

###  5.1 General setting for SUSY quantum mechanics

Let us see the emergence of SUSY QM from global symmetries. Consider a
fermionic system described by some Hamiltonian operator $H$. This means that
the Hamiltonian commutes with a fermion parity $\mathbb{Z}_{2}^{F}$ symmetry,
generated by $P_{F}\equiv(-1)^{F}$, so $P_{F}HP_{F}^{-1}=H$. Let $\\{V_{g}\\}$
be a collection of symmetry operators, including $P_{F}$ that commute with
$H$. Recall that $P_{F}$ also grades states and operators as bosonic and
fermionic as follows

$\displaystyle P_{f}|b\rangle$
$\displaystyle=+|b\rangle,~{}~{}P_{F}O_{b}P_{F}=+O_{b}$ (5.2) $\displaystyle
P_{f}|f\rangle$ $\displaystyle=+|f\rangle,~{}~{}P_{F}O_{f}P_{F}=-O_{f}$ (5.3)

We show that if there exist symmetry operators which are fermionic i.e. anti-
commutes with $P_{F}$, $H$ is a SUSY QM system with at least ${\cal N}=2$
supercharges. Let $\mathcal{T}$ be a such a symmetry of $H$ i.e.
$\mathcal{T}H\mathcal{T}^{-1}=H$, and $P_{F}\mathcal{T}P_{F}=-\mathcal{T}$.
This implies that any eigenstate with non-zero eigenvalue (ensured by a
constant shift of $H$) is degenerate and contains an even number of bosonic
and fermionic states. The diagonalized Hamiltonian can be schematically
written, with $\mu$ the eigenvalue labels, as

$H=\sum_{\mu}\sum_{a_{\mu}}E_{\mu}(|\mu,a_{\mu},+\rangle\langle\mu,a_{\mu},+|+|\mu,a_{\mu},-\rangle\langle\mu,a_{\mu},-|).$
(5.4)

The $+$ or $-$ specifies the $(-1)^{F}$ eigenvalue which correspond to bosonic
or fermionic sector and $a_{\mu}$ keeps track of additional degeneracies. Let
us consider the minimal setting when there are no other degeneracies in the
system. For concreteness, let us also take $\mathcal{T}^{2}=1$ so we have a
$\mathbb{Z}_{2}^{\mathcal{T}}$ symmetry (in fact,
$\mathbb{Z}_{2}^{\mathcal{T}}$ can be more general, _either unitary or anti-
unitary_ in this section). We assume that the Hamiltonian is shifted so as to
ensure that all energies are positive-definite. The eigenstates are organized
as

$H|\mu,\pm\rangle=E_{\mu}|\mu,\pm\rangle,~{}~{}~{}P_{F}|\mu,\pm\rangle=\pm|\mu,\pm\rangle.$
(5.5)

It is clear that $\mathcal{T}$ switches the parity

$\mathcal{T}|\mu,\pm\rangle=|\mu,\mp\rangle.$ (5.6)

Following ref [42], we can now define the following operators

$Q_{\mu}\equiv\sqrt{E_{\mu}}|\mu,+\rangle\langle\mu,-|,\quad
Q^{\dagger}_{\mu}\equiv\sqrt{E_{\mu}}|\mu,-\rangle\langle\mu,+|,$ (5.7)

using which we can define the following two Hermitian _supercharges_

$\displaystyle Q_{+}$
$\displaystyle\equiv\sum_{\mu}Q_{\mu}+Q^{\dagger}_{\mu}.$ (5.8) $\displaystyle
Q_{-}$ $\displaystyle\equiv\sum_{\mu}{-}\hskip 1.0pt\mathrm{i}\hskip
1.0pt\left(Q_{\mu}-Q^{\dagger}_{\mu}\right).$ (5.9)

We also have the fermion parity $P_{F}$ and identity $\mathds{1}$ operators
as:

$\displaystyle
P_{F}\equiv\sum_{\mu}|\mu,+\rangle\langle\mu,+|-|\mu,-\rangle\langle\mu,-|,\quad\mathds{1}\equiv\sum_{\mu}|\mu,+\rangle\langle\mu,+|+|\mu,-\rangle\langle\mu,-|.$
(5.10)

It can be checked that the supercharges satisfy the algebra of eq. 5.1 (below
$\alpha,\beta\in\\{\pm\\}$) [41, 38] i.e.

  1. 1.

Supersymmetry anti-commutes with the $\mathbb{Z}_{2}^{F}$ fermion parity
symmetry, generated by $P_{F}\equiv(-1)^{F}$:

$\\{Q_{\alpha},P_{F}\\}\equiv Q_{\alpha}P_{F}+P_{F}Q_{\alpha}=0.$ (5.11)

  2. 2.

Supercharges commute with Hamiltonian thus supersymmetry is a global symmetry:

$[Q_{\alpha},H]\equiv Q_{\alpha}H-HQ_{\alpha}=0.$ (5.12)

  3. 3.

The anti-commutator for supersymmetric quantum mechanics obeys:

$\\{Q_{\alpha},Q_{\beta}\\}\equiv
Q_{\alpha}Q_{\beta}+Q_{\alpha}Q_{\beta}=2H\delta_{\alpha\beta}.$ (5.13)

This means that we have an ${\cal N}=2$ SUSY QM system at hand. The value of
${\cal N}$ could be larger depending on the additional symmetries but some
SUSY is guaranteed. The symmetry $\mathcal{T}$ acts on the supercharges as
follows.161616In general, for each energy $E_{\mu}$ sector, we have a
2-dimensional Hilbert space
$\mathcal{H}_{\mu}=\mathcal{H}_{B,{\mu}}\oplus\mathcal{H}_{F,{\mu}}$ where the
bosonic sector $\mathcal{H}_{B,{\mu}}$ is spanned by the vector
$\\{|\mu,+\rangle\\}$, the fermionic sector $\mathcal{H}_{F,{\mu}}$ is spanned
by the vector $\\{|\mu,-\rangle\\}$. Overall, the total Hilbert space is even
dimensional:
$\displaystyle\mathcal{H}=\mathcal{H}_{B}\oplus\mathcal{H}_{F}=\bigoplus_{\mu}\mathcal{H}_{\mu}=\bigoplus_{\mu}\big{(}\mathcal{H}_{B,{\mu}}\oplus\mathcal{H}_{F,{\mu}}\big{)}.$
(5.14) We can write all the aforementioned operators $H,P_{F},\mathcal{T}$,
and $Q_{\pm}$ respect to the 2-dimensional Hilbert space $\mathcal{H}_{\mu}$
associated to each energy $E_{\mu}$. We can denote the projection of the
$H,P_{F},\mathcal{T}$, and $Q_{\pm}$ to a 2-dimensional Hilbert space
$\mathcal{H}_{\mu}\equiv\\{|\mu,+\rangle,|\mu,-\rangle\\}$ as
$(H)_{{\mu}},(P_{F})_{\mu}$, $(Q_{\pm})_{\mu}$, and $(\mathcal{T})_{\mu}$.
Then we have:  $\displaystyle(H)_{{\mu}}=E_{\mu}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}_{\mu}=E_{\mu}(\sigma_{0})_{\mu},\;\;(P_{F})_{\mu}=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}_{\mu}\equiv(\sigma_{z})_{\mu},\;\;$ (5.15)
$\displaystyle(Q_{+})_{\mu}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix}_{\mu}\equiv(\sigma_{x})_{\mu},\;\;(Q_{-})_{\mu}=\begin{pmatrix}0&-\hskip
1.0pt\mathrm{i}\hskip 1.0pt\\\ \hskip 1.0pt\mathrm{i}\hskip
1.0pt&0\end{pmatrix}_{\mu}\equiv(\sigma_{y})_{\mu},\;\;Q_{\mu}=\begin{pmatrix}0&1\\\
0&0\end{pmatrix}_{\mu},\;\;Q_{\mu}^{\dagger}=\begin{pmatrix}0&0\\\
1&0\end{pmatrix}_{\mu}.$ (5.16)
$\displaystyle(\mathcal{T})_{\mu}=\begin{cases}(n_{x}(Q_{+})_{\mu}+n_{y}(Q_{-})_{\mu})=(n_{x}(\sigma_{x})_{\mu}+n_{y}(\sigma_{y})_{\mu}),&\text{
if $\mathcal{T}$ is unitary}.\\\
(n_{x}(Q_{+})_{\mu}+n_{y}(Q_{-})_{\mu})K=(n_{x}(\sigma_{x})_{\mu}+n_{y}(\sigma_{y})_{\mu})K,&\text{
if $\mathcal{T}$ is anti-unitary}.\end{cases}$ (5.17) Here $(n_{x},n_{y})\in
S^{1}\subset\mathbb{R}^{2}$ is a unit vector so $n_{x}^{2}+n_{y}^{2}=1$, and
$K$ is a complex conjugation operator. It is amusing to notice such a
$(\mathcal{T})_{\mu}$ is in fact a linear combination
$(n_{x}(Q_{+})_{\mu}+n_{y}(Q_{-})_{\mu})$. In the case if $\mathcal{T}$ is an
anti-unitary time-reversal symmetry, interestingly we have
$(\mathcal{T})_{\mu}$ proportional to SUSY charge linear combination
$(Q_{+})_{\mu}$ and $(Q_{-})_{\mu}$ up to a complex conjugation $K$. In the
main text, we choose $(n_{x},n_{y})=(1,0)$ to derive eq (5.18) and eq (5.19).
If $\mathcal{T}$ is a unitary $\mathbb{Z}_{2}$ symmetry, we can choose

$\mathcal{T}:Q_{\pm}\mapsto\mathcal{T}Q_{\pm}\mathcal{T}^{-1}=\pm Q_{\pm}.$
(5.18)

If the $\mathcal{T}$ is an anti-unitary $\mathbb{Z}_{2}$ symmetry, we can
choose $\mathcal{T}$ acts as an identity operator on the supercharges:

$\mathcal{T}:Q_{\pm}\mapsto\mathcal{T}Q_{\pm}\mathcal{T}^{-1}=+Q_{\pm}.$
(5.19)

Generally, the original symmetries acts by permuting the supercharges.

###  5.2 ${\cal N}=2$ SUSY on the boundary of 1+1d fermionic SPT phase with
$\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F}$ symmetry: Generalized Sachdev-Ye-
Kitaev interactions

Recall that one-dimensional SPT phases have symmetries that are realized
_projectively_. For _intrinsically_ fermionic SPT phases i.e phases that
cannot be thought of as a stack of a bosonic SPT phase, tensor product with a
trivial gapped fermionic system, some of the projective symmetry generators do
not commute with fermion parity [43]. Due to the arguments of the previous
subsection, this leads to the boundary Hamiltonian being supersymmetric. Let
us consider a specific phase for concreteness. We again consider the
$\nu=2\mod 8$ phase of time-reversal invariant superconductors where
$\mathcal{T}$ which acts anti-unitarily and has the property
$\mathcal{T}^{2}=1$. This belongs to class BDI and (in the presence of
interactions) has a $\mathbb{Z}_{8}$ classification. We again consider the
Hamiltonian in eq (3.5):

$H=-\hskip 1.0pt\mathrm{i}\hskip
1.0pt\sum_{j=1}^{\ell}\left(\gamma_{\uparrow,j}\gamma_{\downarrow,j+1}-\bar{\gamma}_{\uparrow,j}\bar{\gamma}_{\downarrow,j+1}\right).$
(5.20)

Recall the symmetries are

$\displaystyle\mathcal{T}=\left(\prod_{j}(\gamma_{\downarrow,j}\bar{\gamma}_{\uparrow,j})\right)\mathcal{K}.\quad\quad
P_{F}=\left(\prod_{j}\prod_{\varsigma=\uparrow,\downarrow}\hskip
1.0pt\mathrm{i}\hskip
1.0pt\bar{\gamma}_{\varsigma,j}\gamma_{\varsigma,j}\right).$ (5.21)

$\mathcal{K}$ is the complex conjugation which acts on the Majorana operators
as

$\mathcal{K}\gamma\mathcal{K}=\gamma,~{}~{}\mathcal{K}\bar{\gamma}\mathcal{K}=-\bar{\gamma},~{}~{}\mathcal{K}\hskip
1.0pt\mathrm{i}\hskip 1.0pt\mathcal{K}=-\hskip 1.0pt\mathrm{i}\hskip 1.0pt.$
(5.22)

Let us now consider a boundary termination as shown in Fig. 8. There are 2
dangling Majorana zero modes on each (left or right) end of Fig. 8, each end
has a 2-dimensional Hilbert space. For Fig. 8, the symmetries acting on the
boundary sites are the same as the bulk sites. However, if we only consider
how the _effective symmetries_ acting on the ground state subspace which
associated with the boundary zero modes, we can do the ground state (GS)
projection $\mathbf{P}_{\rm GS}=\mathbf{P}_{\rm GS}^{\dagger}$ on
$\mathcal{T}$ which factorize eq (5.21) to left and right boundary symmetry
operators:

$\mathbf{P}_{\rm GS}\mathcal{T}\mathbf{P}_{\rm
GS}\simeq\mathcal{T}_{L}\otimes\mathcal{T}_{R}.$

Here _effective_ $\mathcal{T}_{L}$ and $\mathcal{T}_{R}$ act only on the left
and right Majorana zero modes of Fig. 8 as:

$\displaystyle\mathcal{T}_{L}$ $\displaystyle=$
$\displaystyle\gamma_{\downarrow}\mathcal{K},~{}P_{F,L}=i\bar{\gamma}_{\downarrow}\gamma_{\downarrow},~{}\mathcal{T}_{L}^{2}=P_{F,L}^{2}=1,\mathcal{T}_{L}P_{F,L}=-P_{F,L}\mathcal{T}_{L}.$
(5.23) $\displaystyle\mathcal{T}_{R}$ $\displaystyle=$
$\displaystyle\bar{\gamma}_{\uparrow}\mathcal{K},~{}P_{F,R}=i\bar{\gamma}_{\uparrow}\gamma_{\uparrow},~{}\mathcal{T}_{R}^{2}=-1,P_{F,R}^{2}=1,\mathcal{T}_{R}P_{F,R}=-P_{F,R}\mathcal{T}_{R}.$
(5.24)

It can be easily checked that the boundary symmetry operators generate a
faithful representation of dihedral group of order 8,
$\mathbb{D}_{8}\equiv\mathbb{D}_{8}^{F,T}$, which is a group with two
generators satisfying the following relation: $\langle
a,x|a^{4}=x^{2}=1,xax=a^{-1}\rangle$. On the left end,
$a=\mathcal{T}_{L}P_{F,L},~{}x=P_{F,L}$ and on the right,
$a=\mathcal{T}_{R},~{}x=P_{F,R}$.171717Readers may be puzzled by why that the
boundary $\mathcal{T}_{L}$ and $\mathcal{T}_{R}$ symmetries act oppositely in
comparison between those of eq (3.10) and eq (3.11) versus those of eq (5.23)
and eq (5.24). Recall Sec. 3.2.1’s remark 2 that there are three different
ways to look at $\mathcal{T}$ symmetries on the boundary depending on (1) the
boundary truncation types of quantum systems and (2) the Hilbert space
projection $\mathbf{P}_{\rm GS}$ to ground state subspace. Here we do the
remark 2’s (c) case, with the $\mathbf{P}_{\rm GS}$ projection to dangling
zero energy modes.  Observe that fermion parity is not in the center of
$\mathbb{D}_{8}^{F,T}$. This is going to be crucial. From now on, we focus
only on the left boundary and drop the $\downarrow$ index. The effective
symmetry can be simply written as

$\mathcal{T}=\gamma\mathcal{K},~{}~{}~{}P_{F}=\hskip 1.0pt\mathrm{i}\hskip
1.0pt\bar{\gamma}\gamma.$ (5.25)

The symmetry acts on the Majorana operators trivially:

$\mathcal{T}:\begin{pmatrix}\gamma\\\
\bar{\gamma}\end{pmatrix}\mapsto\begin{pmatrix}\gamma\\\
\bar{\gamma}\end{pmatrix}.$ (5.26)

However, it is anti-linear $\mathcal{T}:\hskip 1.0pt\mathrm{i}\hskip
1.0pt\mapsto-\hskip 1.0pt\mathrm{i}\hskip 1.0pt$. As a result, the only
possible boundary Hamiltonian term in this Hilbert space,

$H{\propto-\hskip 1.0pt\mathrm{i}\hskip 1.0pt\gamma\bar{\gamma}=P_{F}},$
(5.27)

is disallowed by $\mathcal{T}$ symmetry. Therefore, the only allowed boundary
Hamiltonian is proportional to the identity operator,

$H=c\mathds{1}.$ (5.28)

As shown in [36], with $c$ being positive (with no loss of generality), this
has an ${\cal N}=2$ SUSY with supercharges

$Q_{+}=\sqrt{c}\gamma,~{}Q_{-}=\sqrt{c}\bar{\gamma}.$ (5.29)

Figure 14: Representative of the $\nu=2$ phase with
$\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F}$ symmetry consisting of multiple
layers — here $(4N+1)$ layers for the $\nu=2$ phase. We can introduce SYK
model Hamiltonian on the boundary, similar to the setup of [44].

We can increase the possible boundary dynamics by _layering_ trivial phases on
the bulk and therefore increasing the boundary by mod 8 layers of Kitaev
chains. Since the Hamiltonian eq. 5.20 is the $\nu=2\mod 8$ member, this means
that we can add multiples of four copies of the Hamiltonian eq. 5.20 to the
bulk and preserve the same phase. The bulk Hamiltonian and symmetry operators
correspond to this layered system shown in Fig. 14 is

$\displaystyle H$ $\displaystyle=-\hskip 1.0pt\mathrm{i}\hskip
1.0pt\sum_{j=1}^{\ell}\sum_{a=1}^{4N+1}\left(\gamma_{\uparrow,a,j}\gamma_{\downarrow,a,j+1}-\bar{\gamma}_{\uparrow,a,j}\bar{\gamma}_{\downarrow,a,j+1}\right).$
(5.30) $\displaystyle\mathcal{T}$
$\displaystyle=\left(\prod_{a=1}^{4N+1}\prod_{j=1}^{\ell}(\gamma_{\downarrow,a,j}\bar{\gamma}_{\uparrow,a,j})\right)\mathcal{K}.$
(5.31) $\displaystyle P_{F}$
$\displaystyle=\left(\prod_{a=1}^{4N+1}\prod_{j=1}^{\ell}\prod_{\varsigma=\uparrow,\downarrow}\hskip
1.0pt\mathrm{i}\hskip
1.0pt\bar{\gamma}_{\varsigma,a,j}\gamma_{\varsigma,j}\right).$ (5.32)

We now focus on the boundary. The Hilbert space can be acted by a total $8N+2$
Majorana operators: $\\{\gamma_{a},\bar{\gamma}_{a}\\}$ for $a=1,\ldots,4N+1$.
The symmetry operators are

$\mathcal{T}=\prod_{a=1}^{4N+1}\gamma_{a}\mathcal{K},~{}~{}~{}P_{F}=\prod_{a=1}^{4N+1}\hskip
1.0pt\mathrm{i}\hskip 1.0pt\bar{\gamma}_{a}\gamma_{a}.$ (5.33)

It can be checked that the operators again form a $\mathbb{D}_{8}^{F,T}$ group
where the fermion parity is not at the center subgroup because $\mathcal{T}$
anti-commutes with $P_{F}$.

$\mathcal{T}^{2}=P_{F}^{2}=1,~{}~{}~{}~{}\mathcal{T}P_{F}=-P_{F}\mathcal{T}.$
(5.34)

However, we can have a boundary Hamiltonian that is not quadratic, but can
consist of Majorana operators coupled in multiples of 4:

$H=\sum_{a,b,c,d}J^{1}_{a,b,c,d}\gamma_{a}\gamma_{b}\gamma_{c}\gamma_{d}+J^{2}_{a,b,c,d}\bar{\gamma}_{a}\gamma_{b}\gamma_{c}\gamma_{d}+J^{3}_{a,b,c,d}\bar{\gamma}_{a}\bar{\gamma}_{b}\gamma_{c}\gamma_{d}+J^{4}_{a,b,c,d}\bar{\gamma}_{a}\bar{\gamma}_{b}\bar{\gamma}_{c}\gamma_{d}+J^{5}_{a,b,c,d}\bar{\gamma}_{a}\bar{\gamma}_{b}\bar{\gamma}_{c}\bar{\gamma}_{d}\\\
{+K^{1}_{a,b,c,d,e,f,g,h}\gamma_{a}\gamma_{b}\gamma_{c}\gamma_{d}\gamma_{e}\gamma_{f}\gamma_{g}\gamma_{h}+K^{2}_{a,b,c,d,e,f,g,h}\bar{\gamma}_{a}\gamma_{b}\gamma_{c}\gamma_{d}\gamma_{e}\gamma_{f}\gamma_{g}\gamma_{h}+\dots}.$
(5.35)

This Hamiltonian $H$ is time reversal invariant:
$\mathcal{T}H\mathcal{T}^{-1}=H$. A shown in [44], if the couplings are
restricted to the quartic operators and $J^{i}_{abcd}$ are drawn from a random
distribution, this is the famous Sachdev-Ye-Kitaev (SYK) model ([13, 45] also
the later expositions in [46, 47]). The SUSY nature of the SYK model was shown
by the authors of [42]. This SUSY nature still holds if we add any number of
$4n$-Majorana operators (such as quartic, or octatonic as shown in eq. 5.35)
to the Hamiltonian all of which preserve the $\mathcal{T}$ and $P_{F}$
symmetry. The logic is precisely what was sketched out in the previous
subsection and detailed in [42]: once we have the anti-commutative
$\mathcal{T}P_{F}=-P_{F}\mathcal{T}$, this implies that we must have two-fold
degenerate spectra from the equal number of bosonic sectors
$\\{|\mu,+\rangle\\}$ and fermionic sectors $\\{|\mu,-\rangle\\}$ at every
energy level $E_{\mu}$, each with a distinct fermion parity $P_{F}=+1$ and
$P_{F}=-1$. We denote the number of states

$\displaystyle n_{B,\mu}=n_{F,\mu}=1$ (5.36)

for each $E_{\mu}$. By a constant shift, we can ensure that the Hamiltonian
has positive definite energies and can be written as

$H=\sum_{\mu}E_{\mu}(|\mu,+\rangle\langle\mu,+|+|\mu,-\rangle\langle\mu,-|),$
(5.37)

where

$H|\mu,\pm\rangle=E_{\mu}|\mu,\pm\rangle,~{}~{}~{}P_{F}|\mu,\pm\rangle=\pm|\mu,\pm\rangle.$
(5.38)

and $\mathcal{T}$ switches the parity

$\mathcal{T}|\mu,\pm\rangle=|\mu,\mp\rangle.$ (5.39)

We can now define the two _supercharges_

$\displaystyle Q_{+}$
$\displaystyle=\sum_{\mu}Q_{\mu}+Q^{\dagger}_{\mu},~{}~{}~{}~{}~{}Q_{-}=\sum_{\mu}{-}\hskip
1.0pt\mathrm{i}\hskip 1.0pt\left(Q_{\mu}-Q^{\dagger}_{\mu}\right),$ (5.40)

where $Q_{\mu}\equiv\sqrt{E_{\mu}}|\mu,+\rangle\langle\mu,-|$ and
$Q^{\dagger}_{\mu}\equiv\sqrt{E_{\mu}}|\mu,-\rangle\langle\mu,+|$, and check
that the supercharges satisfy the algebra of eq. 5.1

$\\{Q_{\alpha},Q_{\beta}\\}=2H\delta_{\alpha\beta},~{}[Q_{\alpha},H]=\\{Q_{\alpha},P_{F}\\}=0$
(5.41)

This means that we have an ${\cal N}=2$ SUSY QM system on the boundary of
these fermionic SPT phases. The action of $\mathcal{T}$ on the supercharges is
as shown in eq. 5.19 i.e.

$\mathcal{T}:Q_{\pm}\mapsto\mathcal{T}Q_{\pm}\mathcal{T}^{-1}=+Q_{\pm}.$
(5.42)

###  5.3 ${\cal N}=2$ SUSY on the boundary of 1+1d fermionic SPT phase with
$\mathbb{Z}_{4}^{TF}$ symmetry

In Sec. 5.2, we had shown that the Sec. 3.2’s model with a bulk
$\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F}$-symmetric $8N+2$ layers of 1+1d
Kitaev chains, the boundary exhibits the ${\cal N}=2$ SUSY quantum mechanics.
In this section, we can also take the Sec. 3.3’s model with a bulk
$\mathbb{Z}_{4}^{TF}$-symmetric $8N+2$ layers of Kitaev chains, to analyze its
boundary energy spectrum property. Follow the Remark 3 in Sec. 3.2.1, when the
number of Kitaev chains is $2\mod 4$ and when
$\mathcal{T}P_{F}=-P_{F}\mathcal{T}$, we have at least two fold generate
ground states (with GSD = 2), one bosonic $|B\rangle$ and one fermionic
$|F\rangle$. In general, we have the equal number of bosonic eigenstate
$\\{|\mu,+\rangle\\}$ and fermionic eigenstate $\\{|\mu,-\rangle\\}$ for each
eigenvalue $E_{\mu}$. We again have $n_{B,\mu}=n_{F,\mu}=1$ as in eq (5.36)
for each $E_{\mu}$. Thus we can repeat the analysis Sec. 5.2 again reaching
the same conclusion that we have ${\cal N}=2$ SUSY quantum mechanics, but with
a different bulk symmetry $\mathbb{Z}_{4}^{TF}$ and the boundary has
$\mathcal{T}^{4}=-1$ and $\mathcal{T}^{8}=+1$. This would also result in a
different action on the supercharges.

###  5.4 Systematic framework in 1+1d

The original fractionalized global symmetry such as $\mathcal{T}$ acts on the
supercharges $Q_{\alpha}$, thus the $\mathcal{T}$ is within a subgroup of the
R symmetry that permutes the supercharges $Q_{\alpha}$. Note that the usual R
symmetry is a unitary symmetry, here we need to generalize R symmetry to
include both unitary and anti-unitary symmetry (to include time-reversal).

Cartan class | $k=1$ | $k=2$ | $k=3$ | $k=4$ | $k=5$ | $k=6$ | $k=7$
---|---|---|---|---|---|---|---
BDI ($\nu\in\mathbb{Z}_{8}$) | - | $\begin{matrix}\mathcal{N}=2\cr{(\nu=2)}\end{matrix}$ | - | $\begin{matrix}\mathcal{N}=0\cr{(\nu=4)}\end{matrix}$ | - | $\begin{matrix}\mathcal{N}=2\cr{(\nu=6)}\end{matrix}$ | -
DIII ($\nu\in\mathbb{Z}_{2}$) | - | $\begin{matrix}\mathcal{N}=2\cr{(\nu=1)}\end{matrix}$ | - |  | - |  | -
AI, CI, CII ($\nu\in\mathbb{Z}_{2}$) | - |  | - | $\begin{matrix}\mathcal{N}=0\cr{(\nu=1)}\end{matrix}$ | - |  | -
AIII ($\nu\in\mathbb{Z}_{4}$) | - | $\begin{matrix}\mathcal{N}=2\cr{(\nu=1)}\end{matrix}$ | - | $\begin{matrix}\mathcal{N}=0\cr{(\nu=2)}\end{matrix}$ | - | $\begin{matrix}\mathcal{N}=2\cr{(\nu=3)}\end{matrix}$ | -
Table 2: We show the relations between the SUSY number ${\cal N}$, the global
symmetry (Cartan symmetry class or Wigner-Dyson-Altland-Zirnbauer symmetry
class) and their SPT classes $\nu$, and the number $k$ of Majorana zero modes.
For $k\in\mathbb{Z}_{{\text{odd}}}^{+}$, the - implies that those models are
not studied in our framework. The blanks mean those classes are either not the
generators of the SPT classification, or they are already determined by the
smaller $k$ of SPT class (other filled data).

In general, we only have the ${\cal N}=2$ SUSY quantum mechanics when the
layer number $k=2\mod 4$. We can summarize our result in Table 2.

We remark again that the fact that the SYK model can be thought of as a
termination of fermionic SPT phases has been observed by You, Ludwig and Xu
[44]. Also the SUSY QM property of the SYK model was studied by Behrends and
Béri [42]. However, our interpretation of SUSY is tightly connected to the
generalization of the symmetry extension.

C: ${{\rm SU}}(2)^{F}$, No class $\Omega^{2}_{{\mathrm{Spin}^{h}}}$ A: ${\rm
U}(1)^{F}$, No class $\Omega^{2}_{{\mathrm{Spin}^{c}}}$ D:
$\mathbb{Z}_{2}^{F}$, $(\nu_{\rm D})$ $\in$ $\mathbb{Z}_{2}$-class Arf 1
Kitaev chain $\Omega^{2}_{{\mathrm{Spin}}}$ CI: $\frac{{{\rm
SU}}(2)\times\mathbb{Z}_{4}^{TF}}{\mathbb{Z}_{2}^{F}}$, $(\alpha)$ $\in$
$\mathbb{Z}_{2}$-class $w_{1}^{2}\sim$ 4ABK 4 Kitaev chains
$\Omega^{2}_{\mathrm{Pin}^{+}\times_{\mathbb{Z}_{2}^{F}}{\rm SU}(2)^{F}}$
AI: ${\rm U}(1)^{F}\rtimes\mathbb{Z}^{{T}}_{2}$, $(\alpha)\in$
$\mathbb{Z}_{2}$-class $w_{1}^{2}\sim$ 4ABK 4 Kitaev chains
$\Omega^{2}_{\mathrm{Pin}^{\tilde{c}-}}\equiv$
$\Omega^{2}_{\mathrm{Pin}^{-}\ltimes_{\mathbb{Z}_{2}^{F}}{\rm U}(1)^{F}}$ BDI:
$\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F}$, $(\nu_{\rm BDI})$ $\in$
$\mathbb{Z}_{8}$-class ABK 1 Kitaev chain
$\Omega^{2}_{\mathrm{Pin}^{-}}$ AIII: $\frac{{\rm
U}(1)^{F}\times\mathbb{Z}^{{TF}}_{4}}{\mathbb{Z}_{2}^{F}}=$ ${{\rm
U}(1)^{F}\times\mathbb{Z}^{{T}}_{2}}$, $(\nu_{\text{AIII}})$
$\in\mathbb{Z}_{4}$-class 2ABK 2 Kitaev chains
$\Omega^{2}_{\mathrm{Pin}^{c}}$ DIII: $\mathbb{Z}_{4}^{TF}$, $(\nu_{\rm
DIII})\in$ $\mathbb{Z}_{2}$-class $\tilde{\eta}\mathrm{PD}(w_{1})$ 2 Kitaev
chains
$\Omega^{2}_{\mathrm{Pin}^{+}}$ CII: $\frac{({{\rm
U}}(1)\rtimes\mathbb{Z}_{4}^{C})}{\mathbb{Z}_{2}^{F}}\times\mathbb{Z}_{2}^{T}$,
$(\nu_{\text{CII}})$ $\in$ $\mathbb{Z}_{2}$-class $w_{1}^{2}\sim$ 4ABK 4
Kitaev chains
$\Omega^{2}_{\mathrm{Pin}^{-}\times_{\mathbb{Z}_{2}^{F}}{\rm SU}(2)^{F}}$
AII: $\frac{{\rm U}(1)^{F}\rtimes\mathbb{Z}^{{TF}}_{4}}{\mathbb{Z}_{2}^{F}}$,
No class $\Omega^{2}_{\mathrm{Pin}^{\tilde{c}+}}\equiv$
$\Omega^{2}_{\mathrm{Pin}^{+}\ltimes_{\mathbb{Z}_{2}^{F}}{\rm U}(1)^{F}}$
Table 3:  The symmetry embedding web of 1+1d fermionic invertible topological
phases relevant for 10 Cartan symmetry classes (we follow the notations of
Table 2 in [3] and Table 4 in [25] for 3+1D cases). The web suggests the maps
between the nontrivial classes of their classifications of SPTs (namely,
cobordism invariants). The web can also suggest a possible symmetry group
extension to unwind the SPT states. We include topological invariants from the
2d Pin- bordism invariant Arf-Brown-Kervaire (ABK), and 1d spin bordism
invariant $\tilde{\eta}$. The PD means the Poincaré dual. The “$n$ Kitaev
chains” mean the generator of invertible phases of given symmetries.

##  6 Generalizations: from the lattice to continuum

In Sec. 6, We will make some refinement about the group extension (for a quick
overview see Appendix B.1) and properties of bosonic/fermionic/time-
reversal/spacetime symmetries and group extensions (for a quick overview see
Appendix B.2) that we used.181818We relegate the two subsections B.1 and B.2
to Appendices due to their heavy mathematical formality. However, the readers
are strongly encouraged to read Appendix B.1 and B.2 first before proceed to
Sec. 6. In Table 3, we show a symmetry embedding web of 1+1d fermionic
invertible topological phases with their symmetry groups (relevant for 10
Cartan symmetry classes), and the number of layers of Kitaev chains to
construct the minimal phase. In Table 4, we summarize the result of fermionic
symmetry group extension presented in Sec. 3 and 4, also those in [3], for
fermionic SPT phases in 1+1d.

Cartan class |  Symmetry $G$ | $N\to\tilde{G}\to G$ | $\begin{array}[]{c}\text{Reduced classification}\\\ \text{of 1+1d $G$-iTQFT or}\\\ \text{0+1d $G$-anomaly in $\tilde{G}$}\end{array}$
---|---|---|---
BDI | $\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F}$ | $\mathbb{Z}_{2}\to\mathbb{Z}_{4}^{T}\times\mathbb{Z}_{2}^{F}\to G$ | $\mathbb{Z}_{8}\Rightarrow\mathbb{Z}_{4}$
$\mathbb{Z}_{2}\to\mathbb{D}_{8}^{F,T}\to G$ | $\mathbb{Z}_{8}\Rightarrow\mathbb{Z}_{2}$
DIII | $\mathbb{Z}_{4}^{TF}$ | $\mathbb{Z}_{4}\to\mathbb{M}_{16}^{F,T}\to G$ | $\mathbb{Z}_{2}\Rightarrow 0$
AIII | ${{\rm U}}(1)^{F}\times\mathbb{Z}_{2}^{T}$ | $\mathbb{Z}_{2}\to{{\rm U}}(1)^{{F}}\times\mathbb{Z}_{4}^{T}\to G$ | $\mathbb{Z}_{4}\Rightarrow\mathbb{Z}_{2}$
$\mathbb{Z}_{2}\to\frac{{\rm U}(1)\times\mathbb{D}_{8}^{F,T}}{\mathbb{Z}_{2}}\to G$ | $\mathbb{Z}_{4}\Rightarrow 0$
CII | $\frac{({{\rm U}}(1)^{F}\rtimes{\mathbb{Z}}_{4}^{C})}{{\mathbb{Z}}_{2}^{F}}\times{\mathbb{Z}}_{2}^{T}$ | $\mathbb{Z}_{2}\to\frac{({{\rm U}}(1)^{F}\rtimes{\mathbb{Z}}_{4}^{C})}{{\mathbb{Z}}_{2}^{F}}\times\mathbb{Z}_{4}^{T}\to G$ | $\mathbb{Z}_{2}\Rightarrow 0$
${\rm SU}(2)^{F}\times\mathbb{Z}_{2}^{T}$ | $\mathbb{Z}_{2}\to{\rm SU}(2)^{F}\times\mathbb{Z}_{4}^{T}\to G$ | $\mathbb{Z}_{2}\Rightarrow 0$
Table 4: Summary of unwinding 1+1d fermionic $G$-SPT of $G$-invertible topological field theory (iTQFT) phases via the symmetry extension $1\to N\to\tilde{G}\to G\to 1$ and the change of classification by symmetry extension. It turns out that the group extension of $G$ by $\mathbb{Z}_{2}$ has the $\mathbb{Z}_{2}=\mathbb{Z}_{2}^{B}$ as a bosonic symmetry acting on any state by multiplying a sign $\\{1,-1\\}$. So the $\mathbb{Z}_{4}^{T}$ in the $\tilde{G}$ is in fact $\mathbb{Z}_{4}^{T}=\mathbb{Z}_{4}^{TB}$ obtained from $1\to\mathbb{Z}_{2}^{B}\to\mathbb{Z}_{4}^{TB}\to\mathbb{Z}_{2}^{T}\to 1$, which by including (Lorentz to Euclidean) spacetime symmetry becomes $1\to\mathbb{Z}_{2}^{B}\to{\rm E}(d)\to{\rm O}(d)\to 1$, with the ${\rm E}(d)=\mathbb{Z}_{2}^{B}\rtimes{\rm O}(d)={\rm SO}(d)\rtimes\mathbb{Z}_{4}^{TB}$, see details in Sec. B.2. Cartan class | $\begin{array}[]{c}\text{Spacetime-Internal}\\\ \text{Symmetry $G_{\text{Tot}}$}\end{array}$ | $\begin{array}[]{c}\text{Symmetry Extension}\\\ \text{to $\tilde{G}_{\text{Tot}}$}\end{array}$ | $\begin{array}[]{c}\text{Reduced classification}\\\ \text{of 1+1d $G$-iTQFT or}\\\ \text{0+1d $G$-anomaly in $\tilde{G}$}\end{array}$
---|---|---|---
BDI | $\mathrm{Pin}^{-}$ | $\mathrm{EPin}$ | $\mathbb{Z}_{8}\Rightarrow\mathbb{Z}_{4}$
$\mathbb{D}_{8}^{F,T}\mathrm{Pin}$ | $\mathbb{Z}_{8}\Rightarrow\mathbb{Z}_{2}$
DIII | $\mathrm{Pin}^{+}$ | $\mathbb{M}_{16}^{F,T}\mathrm{Pin}$ | $\mathbb{Z}_{2}\Rightarrow 0$
AIII |  $\begin{array}[]{r}\mathrm{Pin}^{c}=\mathrm{Pin}^{-}\times_{\mathbb{Z}_{2}^{F}}{\rm U}(1)^{F}\\\ =\mathrm{Pin}^{+}\times_{\mathbb{Z}_{2}^{F}}{\rm U}(1)^{F}\end{array}$ | ${\rm E}\mathrm{Pin}\times_{\mathbb{Z}_{2}^{F}}{\rm U}(1)^{F}$ | $\mathbb{Z}_{4}\Rightarrow\mathbb{Z}_{2}$
${\mathbb{D}_{8}^{F,T}\mathrm{Pin}}\times_{\mathbb{Z}_{2}^{F}}{\rm U}(1)^{F}$ | $\mathbb{Z}_{4}\Rightarrow 0$
CII | $\mathrm{Pin}^{-}\times_{\mathbb{Z}_{2}^{F}}\frac{({{\rm U}}(1)^{F}\rtimes{\mathbb{Z}}_{4}^{C})}{{\mathbb{Z}}_{2}^{F}}$ | ${\rm E}\mathrm{Pin}\times_{\mathbb{Z}_{2}^{F}}\frac{({{\rm U}}(1)^{F}\rtimes{\mathbb{Z}}_{4}^{C})}{{\mathbb{Z}}_{2}^{F}}$ | $\mathbb{Z}_{2}\Rightarrow 0$
$\mathrm{Pin}^{-}\times_{\mathbb{Z}_{2}^{F}}{\rm SU}(2)^{F}$ | ${\rm E}\mathrm{Pin}\times_{\mathbb{Z}_{2}^{F}}{\rm SU}(2)^{F}$ | $\mathbb{Z}_{2}\Rightarrow 0$
Table 5: The _continuum_ version description of Table 4. The extended
$\tilde{G}_{\text{Tot}}$ includes the EPin group (firstly introduced in Ref.
[48] and reviewed in Sec. 6.1), the $\mathbb{D}_{8}^{F,T}\mathrm{Pin}$ and
$\mathbb{M}_{16}^{F,T}\mathrm{Pin}$ (that we will introduce in Sec. 6.3 and
Sec. 6.4).

We also summarize the bundle constraints for various symmetry groups and
corresponding spacetime structures in Table 6 following Ref. [48].

Group | $w_{1}$ | $w_{1}^{2}$ | $w_{2}$
---|---|---|---
${\rm SO}(d)$ | 0 | 0 | free
${\rm Spin}(d)$ | 0 | 0 | 0
${\rm O}(d)$ | free | free | free
${\rm E}(d)$ | free | 0 | free
$\mathrm{Pin}^{+}(d)$ | free | free | 0
$\mathrm{Pin}^{-}(d)$ | free | $w_{1}^{2}+w_{2}=0$
$\mathrm{EPin}(d)$ | free | 0 | 0
Table 6: Follow Ref. [48], we list down the (spacetime) bundle constraints for
various spacetime structures. Here we denote $w_{j}\equiv w_{j}(TM)$ are the
$j$-th Stiefel-Whitney (SW) classes of the tangent bundle $TM$ of the
spacetime manifold $M$. The product of cohomology classes as $w_{1}^{2}\equiv
w_{1}(TM)\cup w_{1}(TM)$ is the cup product. The free here means no
restriction which can be a nontrivial cohomology class.

In the following subsections, we show by examples, based on the knowledge of
Appendix B.1 and B.2, how to convert the discrete group version of symmetry
extensions (Table 4) to the continuous group version of symmetry extensions
(Table 5), suitable for the continuum quantum field theory [QFT]).

###  6.1 1+1d: Lift $\mathrm{Pin}^{-}(d)$ to ${\rm E}\mathrm{Pin}(d)$ for
$\nu_{\rm BDI}=4\in\mathbb{Z}_{8}$ class

The symmetry extension
$\mathbb{Z}_{2}\to\tilde{G}=\mathbb{Z}_{4}^{T}\times\mathbb{Z}_{2}^{F}\to{G}=\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F}$
in Table 4, reduces BDI class from the interacting $\mathbb{Z}_{8}$
classification to $\mathbb{Z}_{4}$ classification. The
$\tilde{G}=\mathbb{Z}_{4}^{T}\times\mathbb{Z}_{2}^{F}$ in fact can have at
least three physical interpretations of the symmetry realization, which in
turn give also three physical interpretations of $\mathbb{Z}_{2}\to{\rm
E}\mathrm{Pin}(d)\to\mathrm{Pin}^{-}(d)$. The ${\rm E}\mathrm{Pin}(d)$ is
firstly introduced by Ref. [48].191919Below we encounter and define three
versions of EPin called ${\rm E}\mathrm{Pin}^{BF_{-}}(d)$, ${\rm
E}\mathrm{Pin}^{BF_{+}}(d)$, and ${\rm E}\mathrm{Pin}^{F_{+}F_{-}}(d)$. In
fact the ${\rm E}\mathrm{Pin}^{F_{+}F_{-}}(d)$ is denoted as ${\rm
E}\mathrm{Pin}(d)$ in Ref. [48]. It is shown in Ref. [48] and in this section
that the three versions of EPin are the same group. The three interpretations
are:

  1. (1).

One interpretation of Table 4’s involves a $\mathcal{T}^{2}=-1$ boson $B$ and
a $\mathcal{T}^{2}=1$ fermion $F_{-}$:

$\mathbb{Z}_{2}^{B}\to\mathbb{Z}_{4}^{TB}\times\mathbb{Z}_{2}^{F_{-}}\to\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F_{-}},$

by including SO$(d)$ which becomes Table 5’s

$\displaystyle\mathbb{Z}_{2}^{B}\to{\rm
E}\mathrm{Pin}^{BF_{-}}(d)\equiv\Big{(}(\mathbb{Z}_{2}^{F_{-}}\rtimes{\rm
SO}(d))\times\mathbb{Z}_{2}^{B}\Big{)}\rtimes\mathbb{Z}_{2}^{T_{\rm
E}}\to\mathrm{Pin}^{-}(d).$ (6.1)

Note that ${\rm
E}\mathrm{Pin}^{BF_{-}}(d)\equiv\Big{(}(\mathbb{Z}_{2}^{F_{-}}\rtimes{\rm
SO}(d))\times\mathbb{Z}_{2}^{B}\Big{)}\rtimes\mathbb{Z}_{2}^{T_{\rm
E}}=\Big{(}{\rm
Spin}(d)\times\mathbb{Z}_{2}^{B}\Big{)}\rtimes\mathbb{Z}_{2}^{T_{\rm E}}$ has
the properties _specifically_ here:

     * •

$(\mathbb{Z}_{2}^{F_{-}}\rtimes{\rm SO}(d))\rtimes\mathbb{Z}_{2}^{T_{\rm
E}}={\rm Spin}(d)\rtimes\mathbb{Z}_{2}^{T_{\rm E}}=\mathrm{Pin}^{-}(d)$.

     * •

In Euclidean signature: $\mathbb{Z}_{2}^{B}\rtimes\mathbb{Z}_{2}^{T_{\rm
E}}=\mathbb{Z}_{4}^{{T_{\rm E}}B}$ and
$\mathbb{Z}_{2}^{F_{-}}\rtimes\mathbb{Z}_{2}^{T_{\rm
E}}=\mathbb{Z}_{4}^{{T_{\rm E}}F_{-}}$.

     * •

In Lorentzian signature:
$\mathbb{Z}_{2}^{B}\rtimes\mathbb{Z}_{2}^{T}=\mathbb{Z}_{4}^{TB}$ and
$\mathbb{Z}_{2}^{F_{-}}\rtimes\mathbb{Z}_{2}^{T}=\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F_{-}}$.

     * •

$\big{(}{\rm
SO}(d)\times\mathbb{Z}_{2}^{B}\big{)}\rtimes\mathbb{Z}_{2}^{T_{\rm
E}}=\mathbb{Z}_{2}^{B}\rtimes{\rm O}(d)={\rm
SO}(d)\rtimes\mathbb{Z}_{4}^{T_{\rm E}B}={\rm E}(d).$

  2. (2).

One interpretation of Table 4’s involves two types of fermions, $F_{+}$ with
$\mathcal{T}^{2}=-1$ and $F_{-}$ with $\mathcal{T}^{2}=1$:

$\mathbb{Z}_{2}^{F_{+}}\to\mathbb{Z}_{4}^{TF_{+}}\times\mathbb{Z}_{2}^{F_{-}}\to\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F_{-}},$

by including SO$(d)$ which becomes Table 5’s

$\displaystyle\mathbb{Z}_{2}^{F_{+}}\to{\rm
E}\mathrm{Pin}^{F}(d)\equiv\Big{(}\mathbb{Z}_{2}^{F_{+}}\times\mathbb{Z}_{2}^{F_{-}}\Big{)}\rtimes\Big{(}{\rm
SO}(d)\rtimes\mathbb{Z}_{2}^{T}\Big{)}\to\mathrm{Pin}^{-}(d).$ (6.2)

Note that ${\rm
E}\mathrm{Pin}^{F}(d)\equiv\Big{(}\mathbb{Z}_{2}^{F_{+}}\times\mathbb{Z}_{2}^{F_{-}}\Big{)}\rtimes\Big{(}{\rm
SO}(d)\rtimes\mathbb{Z}_{2}^{T}\Big{)}$ has the properties _specifically_
here:

     * •

$\mathbb{Z}_{2}^{F_{+}}\rtimes{\rm
SO}(d)\cong\mathbb{Z}_{2}^{F_{-}}\rtimes{\rm SO}(d)\cong{\rm Spin}(d)$.

     * •