# Long-Range Order and Quantum Criticality in Antiferromagnetic Chains with
Long-Range Staggered Interactions

Jie Ren  Department of Physics, Changshu Institute of Technology, Changshu
215500, China  Zhao Wang  Department of Physics, Changshu Institute of
Technology, Changshu 215500, China  Weixia Chen  Department of Physics,
Changshu Institute of Technology, Changshu 215500, China  Wen-Long You
<EMAIL_ADDRESS>College of Science, Nanjing University of Aeronautics and
Astronautics, Nanjing, 211106, China  MIIT Key Laboratory of Aerospace
Information Materials and Physics, Nanjing University of Aeronautics and
Astronautics, Nanjing 211106, China

###### Abstract

We study quantum phase transitions in Heisenberg antiferromagnetic chains with
a staggered power-law decaying long-range interactions. Employing the density-
matrix renormalization group (DMRG) algorithm and the fidelity susceptibility
as the criticality measure, we establish more accurate values of quantum
critical points than the results obtained from the spin-wave approximation,
quantum Monte Carlo and DMRG in literatures. The deviation is especially
evident for strong long-range interactions. We extend isotropic long-range
interactions to the anisotropic cases and find that kaleidoscope of quantum
phases emerge from the interplay of anisotropy of the long-range exchange
interaction and symmetry breaking. We demonstrate nonfrustrating long-range
interactions induce the true long-range order in Heisenberg antiferromagnetic
chains with a continuous symmetry breaking, lifting the restrictions imposed
by the Mermin-Wagner theorem.

##  I Introduction

As a prototypical model of magnetism, antiferromagnetic (AFM) Heisenberg model
$H$= $\sum_{i,j}J_{i,j}\vec{S}_{i}\cdot\vec{S}_{j}$ has been persistently
investigated for decades Auerbach . Despite being a simplified theoretical
model, the Heisenberg model finds applications in a variety of contexts,
ranging from quantum phase transitions (QPTs) Chen07 ; yi2019 ; Ren20 ; You20a
; You22 , superconductivity Mattis , localization in disordered systems
Znidaric , spin liquid Mei17 , quantum chaos Kharkov20  to quantum information
Nielsen00 . The ground state of the nearest-neighbor AFM Heisenberg model on a
bipartite lattice in $d$ ($\geq$ 2) dimensions is generally expected to host
Néel long-range order (LRO) for any spin magnitude $S$, although a rigorous
proof of the existence of LRO in a two-dimensional quantum-spin-1/2 Heisenberg
magnet is still lacking Tang89 ; Lin92 ; You09 . It was recognized that
imposed by the Mermin-Wagner theorem, the true LRO is prohibited in short-
range interacting Heisenberg model in one spatial dimension. Pioneering work
by Haldane demonstrated that Heisenberg AFM chains of integer spins are
endowed with a symmetry-protected topological gapped ground state Haldane ;
Haldane1 , in stark contrast to the well-known spin-1/2 analog, which supports
a quasi-long-range ordered critical phase, known as the Tomonaga-Luttinger
liquid (TLL). In this regard, higher dimensional magnets provide a testbed for
spin-wave theory, while the spin-wave approximation usually fails in one
dimension. The remarkable difference between one-dimensional (1D) AFM systems
of integer and half-integer spins opens a highly successful avenue in
understanding the low-dimensional strongly correlated electronic materials.
The isotropic Heisenberg AFM model has been unexpectedly coined in a number of
nearly ideal quasi-one-dimensional materials such as Cu(C4H4N2)(NO3)2
PhysRevB.59.1008 , Sr2Cu(PO4)2 PhysRevB.74.174435 , KCuF3
PhysRevLett.111.137205 , CuSO${}_{4}\cdot 5$D2O Mourigal13 , and spin-1 chain
materials like SrNi2V2O8 Bera13 ; Bera15 , Ni(C2H8N2)2NO2(ClO4) Delica91 ;
Avenel92  and NiI2 (C7H9N)4 Williams20 . There have also been attempts to
realize spontaneous symmetry breaking and develop true AFM order in spin-1/2
Heisenberg chains. One scheme under the consideration is the inclusion of the
long-range interactions defenu2021longrange , which effectively increases the
dimensionality and lifts the rigorous restrictions imposed by the Mermin-
Wagner theorem.

In fact, long-range interactions occur naturally in numerous quantum materials
Guo ; Bramwell ; Castelnovo ; Annica  and versatile quantum simulators Bloch1
; Kim ; Bloch2 ; Senko2015 . Especially it has been suggested that the
existing cavity-mediated cold atom system Esslinger13  or Rydberg dressed
atoms Saffman ; Huse ; Sde ; Zoller2015  could be more ideal experimental
platforms for long-range interactions than solid-state ones. For instance, the
interacting radius of the effective interaction between dressed atoms and the
potential shape can be finely tuned by dressing to different fine-structure
split states Bouchoule ; Zoller ; Pohl ; Li2012 . The typical models have
considered interactions decaying with distance $r$ as a power law $\propto$
$1/r^{\alpha}$ or a staggered power law $\propto$ $(-1)^{r}/r^{\alpha}$,
ranging from dipolar spin chain Noack , Haldane-Shastry chain Haldane88  to
spin-1 chain gong2016 ; gong20161 . The effective exchange interactions
mediated by either photons or Rydberg dressing are generally U(1) or
$\mathbb{Z}_{2}$-symmetric, and a high degree of symmetry, ideally SU(2), can
be achieved by adjusting the laser detunings or increasing bosonic modes. To
be specific, it is found that the long-range interactions of the longitudinal
component results in a Wigner crystal phase Ren ; Li2019 , whereas the
transversal one may break a continuous symmetry, resulting in a continuous
symmetry-breaking phase Ren ; Maghrebi .

Inspired by the rapid development of quantum information science, various
information measurements have been exploited to study of quantum critical
phenomena in spin chains. The well-known and widely studied measures are
entanglement entropy (EE) Amico ; Laflorencie  and fidelity susceptibility
(FS), which diverges at the critical points in the thermodynamic limit You ;
Gu2010 . The ground-state EE and FS were deemed to be capable of qualifying
QPTs in many-body systems with short-range interactions Ren2015 ; Nielsen ;
Ren2018 ; Bwang ; Sun2015 ; Luo ; Lv2022 , even for long-range interacting
system Ren2020 ; Sun2017 ; Sun2018 . In the paper, we will detect the phase
transitions in AFM Heisenberg chain with long-range anisotropic interactions
by the FS and the EE.

The remainder of this paper is organized as follows. We introduce the $S=1/2$
Heisenberg model with long-range anisotropic interactions in Sec. II. The
details of numerical methods and measurements are also introduced. In Sec.
III, effects of long-range interactions on correlation functions, the FS and
the EE are investigated. The discussion and summary are presented in the last
Section.

##  II Hamiltonian and Measurements

In what follows, we are interested in a 1D spin-$1/2$ nearest-neighbor
isotropic AFM chain under the effect of anisotropic long-range Heisenberg
anisotropic interactions, given by

$\displaystyle
H\\!=\\!J\\!\sum_{i}\left\\{\\!\vec{S}_{i}\cdot\vec{S}_{i+1}\\!-\\!\sum_{r\geq
2}\\!\lambda_{i,i+r}[\Delta^{xy}\\!(S^{x}_{i}S^{x}_{i+r}\\!+\\!S^{y}_{i}S^{y}_{i+r})\\!+\\!S^{z}_{i}S^{z}_{i+r}]\right\\},$

where $S_{i}^{\beta}$ ($\beta$ =$x$, $y$, $z$) are spin-1/2 operators at
$i$-th site among total $L$ sites. The AFM coupling $J$ = 1 between the
nearest-neighbor spins is set up as an energy unit for simplicity unless
otherwise stated. The connectivity between two spins at sites $i$ and $i+r$
separated by a distance of $r(\geq 2)$ is given by

$\displaystyle\lambda_{i,i+r}\equiv\lambda(-1)^{r}r^{-\alpha},$ (2)

for the nonfrustrating long-range interactions. Here we always choose the
nearest-neighbor interactions to be isotropic, favoring a TLL ground state.
The deformation of beyond-nearest-neighbour couplings breaking from SU(2)
symmetry down to a U(1) symmetry is characterized by the anisotropy parameter
$\Delta^{xy}$ within the $x-y$ plane, which recovers isotropic interactions
for $\Delta^{xy}=1$ and reduces to $\mathbb{Z}_{2}$-symmetric Ising
interactions for $\Delta^{xy}=0$. In this vein, the interplay of nearest-
neighbor isotropic and longer-range anisotropic interactions admits certain
magnetic symmetry breaking and stabilizes kaleidoscope of quantum phases.
While a solid-state implementation of Hamiltonian Eq.(II) remains challenging,
engineering such graph of interactions can be possibly realized in state-of-
the-art cavity QED Chiocchetta21 . To be explicit, in an array of atomic
ensembles within an optical cavity, the strength of spin-spin interaction
patterns, including the flip-flop and diagonal interactions as well as the
decay exponent, can be delicately resolved in a multimode cavity QED with an
additional drive field Vaidya18 , and the changing sign is determined by the
phase of the corresponding sinusoidal modulation Periwal21 . The unprecedented
controllability of the cavity QED highlight the graph of interactions in
Hamiltonian Eq.(II) becomes programmable.

The simultaneous appearance of long-range interactions and symmetry breaking
leads to quantum critical phenomena that is different from short-range
interactions. In the limit of $\alpha\rightarrow\infty$ or $\lambda\rightarrow
0$, the Hamiltonian in Eq. (II) is reduced to a spin-1/2 chain solely with the
nearest-neighbor interactions, which can be analytically solved by Bethe
ansatz Bethe31 . For generic parameters $\\{\alpha,\lambda\\}$, the system
becomes nonintegrable. It is anticipated that the ground state is still in a
quasi-long-range ordered phase for a sufficiently large $\alpha$, while the
system favors long-range order for a small value of $\alpha$. For the
Heisenberg chain with staggered power-law decaying interactions, the
transition between LRO phase and quasi-long-range order (QLRO) was
successively investigated in literature Parreira ; Affleck ; Sandvik ;
Yang2020 . At first Parreira et al. pointed out that the LRO is absent for
$\alpha>3$ with any $\lambda$ based on spin-wave theory Parreira , indicating
that the critical line $\alpha_{c}<3$ between the AFM Néel order and the QLRO
phase. Lately Laflorencie et al. utilized the staggered structure factor as
order parameter to detect the Néel instability in terms of quantum Monte Carlo
(QMC) simulations. For $\lambda=1$ and $\Delta^{xy}=1$, they obtained
$\alpha_{c}^{\rm QMC}=2.225$, which improved the numerical results
$\alpha_{c}^{\rm SW}=2.46$ given by the lowest order spin-wave approximation
Affleck . Recently Yang et al. studied the QPTs from the perspective of the
fractionalized excitations for chains of length $L=60$ using 400 density-
matrix renormalization group (DMRG) states Yang2020 . The development of the
LRO is associated with the formation of coherent magnons that emerge from
deconfined spinons in the gapless Luttinger liquid, giving rise to
$\alpha_{c}^{\rm DMRG}\approx 2.2$. Thus, it would be interesting to identify
the accurate value of the critical point across this unconventional phase
transition by other observables with enhanced sensitivity.

As a quantum information metric, the FS has proved to be particularly useful
for detecting the critical points of symmetry-knowledge unknown systems Wang18
; PhysRevA.97.013845 ; Liu21 . For a general many-body Hamiltonian $H(g)$, the
ground-state FS per site can be calculated by Gu2010 ; You

$\chi(g)=\lim_{\delta g\rightarrow 0}\frac{-2\textrm{ln}F(g,\delta
g)}{L(\delta g)^{2}},$ (3)

where the fidelity $F$ measures the similarity between the two closest ground
states $|\psi_{0}(g)\rangle$ and $|\psi_{0}(g+\delta g)\rangle$, which is
defined as

$F(g,\delta g)=|\langle\psi_{0}(g)|\psi_{0}(g+\delta g)\rangle|.$ (4)

Here $g$ is the variational parameter of $H(g)$ and $\delta g$ denotes an
infinitesimal deviation. Note that Hamiltonian (II) can not be expressed as a
simple form as $H(\alpha)=H_{0}+\alpha H_{I}$. Subsequently, we obtain the
derivatives of Eq.(2) as $\delta\lambda_{i,i+r}=-\lambda(-1)^{r}r^{-\alpha}\ln
r\delta\alpha$. Due to nonfrustrated characteristics, the average derivatives
of interactions per site is practically considered as an effective tuning
parameter $\delta\bar{\alpha}=\sum_{i<j}\delta\lambda_{i,j}/L$. Therefore, the
FS per site can be calculated numerically by

$\chi(\alpha)=\lim_{\delta\alpha\rightarrow
0}\frac{-2\textrm{ln}F(\alpha,\delta\alpha)}{L(\delta\bar{\alpha})^{2}}.$ (5)

The peak of FS per site is thus used to identify the phase boundary
$\alpha_{c}$ for continuously varying parameters $\\{\lambda,\Delta^{xy}\\}$,
which provides a vital opportunity to testify theoretical predictions with
experimentally accessible results. Another familiar probe to monitor critical
point is the bipartite von Neumann EE, which is defined by

$\displaystyle S_{A}=-\textrm{Tr}(\rho_{A}\ln\rho_{A}).$ (6)

Here $\rho_{A}$ is the reduced density matrix of subsystem $A$ with respect to
the whole system. The EE can also be extracted from the ground-state
wavefunction $|\psi_{0}\rangle$ and hence properly characterize the QPTs. The
ground states of short ranged Hamiltonians usually satisfy an area law
according to which the EE $S_{A}$ of a subregion $A$ of the system is
proportional to the size of its boundary area. This area-law conjecture is be
derived from the power-law decay of the bipartite correlations Hastings06  and
numerically verified in various quantum many-body systems, and is expected to
be true in all noncritical phases Amico , even for long-range interacting
systems Kuwahara20 . However, a logarithmic violation of the area law is
usually known to occur in critical ground states, as is coined by conformal
field theory (CFT), where the system size $L$ is related to the correlation
length $\xi$ near the critical point such as $L\sim\xi$ and the gap decays as
$1/L$. In this case, a coefficient proportional to the central charge of the
underlying CFT, the half-chain EE of 1D critical systems of finite size $L$
with open boundary condition satisfy

$\displaystyle S_{\rm h}(L)=\frac{c}{6}\ln L+S_{0},$ (7)

where $c$ is the central charge, and $S_{0}$ is a nonuniversal constant.
However, the area law for long-range interacting systems is still elusive. The
conformal symmetry will break down under the long-range interactions when
$\alpha$ is small Vodola2014 ; Vodola2016 ; Maghrebi , as the long-range
interactions results in correlation patterns similar to those in critical
phases. To this end, we calculate the effective central charge $c_{\rm eff}$
as a function of $\alpha$, which is obtained by calculating the half-chain EE
for two chains with different $L_{1}$ and $L_{2}$. By using finite-size DMRG
algorithm, the effective central charge can be obtained by

$c_{\rm eff}=\frac{6[S_{\rm h}(L_{2})-S_{\rm
h}(L_{1})]}{\ln(L_{2})-\ln(L_{1})}.$ (8)

We emphasize that $c_{\rm eff}$ may not have the meaning of the central charge
for the short-range interacting cases with conformal symmetries, although we
find the half-chain EE always obeys the scaling form in (7).

A precise numerical determination of $\alpha_{c}$ poses significant technical
challenges in terms of various criticality measures. Theoretically, the
treatment of quantum many-body systems is notoriously complicated so that many
investigations are still accessible by numerical techniques like the DMRG
method white ; KWHP ; U01 , the present studies of Hamiltonian (II) can be
simulated with very high accuracy. Based on matrix product states, we adopt
both infinite-size DMRG (iDMRG) McCulloch  and finite-size DMRG U02  where up
to $m=2000$ in the truncation of bases are kept, and this allows the
truncation error to be smaller than $10^{-9}$. The long-range interactions can
be approximated by a summation of finite exponential terms Vidal ; Pirvu ,
which inevitably introduces additional systematic error and corrupts the
numerical results of FS.In our calculations of finite-size DMRG algorithm, we
handle with the long-range interactions using a summation over matrix product
operators (MPOs). Our codes are mainly based on iTensor C++ library tensor .
Since the $z$-component of the total spins for the present system commutes
with the Hamiltonian (II), the ground-state energy is obtained by comparing
the lowest energies for each subspace of $S^{z}_{\rm
tot}=\sum_{i=1}^{L}\langle S^{z}_{i}\rangle$. The ground state resides in the
sector $S^{z}_{\rm tot}=0$ as a consequence of the continuous U(1) symmetry
therein.

##  III Results

With the DMRG algorithm at hand, we analyze the kaleidoscope of quantum phases
that emerge in this system for different types of long-range exchange
interactions. In the following, we will consider isotropic ($\Delta^{xy}=1$),
Ising-type ($\Delta^{xy}=0$) and XY-type ($\Delta^{xy}=1.5$) anisotropic
cases, respectively. Using the powerful tools, the phases of long-range
interacting systems are numerically diagnosed and the corresponding phase
diagrams are determined.

###  III.1 $\Delta^{xy}=1$

Figure 1: Phase diagram of Hamiltonian Eq. (II) as functions of $\alpha$ and
$\lambda$ with $\Delta^{xy}=1$. The boundary (-$\bullet$-) between QLRO and
LRO is computed by the large scale QMC simulation Affleck , and the results
(-$\diamond$-) is obtained by the FS. It is noted that the LRO phase is
equivalent to Néel phase in 1D spin systems. The symbols ($\ast$,$\times$,$+$)
mark the positions of parameters used in Fig.2. Figure 2:  Loglog-plot
correlations $\langle S^{+}_{i}S^{-}_{i+r}\rangle$, $\langle
S^{z}_{i}S^{z}_{i+r}\rangle$ versus the distance $r$ with $\Delta^{xy}=1$ for
(a) $\lambda=1$, $\alpha=2.10$; (b) $\lambda=0.5$, $\alpha=2$ and (c)
$\lambda=0.01$, $\alpha=1$.  Figure 3: (a) Half-chain EE versus $\ln L$ for
different $\alpha$ with $\Delta^{xy}=1$, $\lambda=\infty$. Symbols show
numerical results obtained by DMRG calculations, and solid lines are linear
fits of the data. (b) Finite-size scaling of the energy gap $\Delta$ with
various $\alpha$. Symbols show numerical results obtained by DMRG
calculations, and solid lines are fits of the data by quadratic polynomials in
1/$L$.

For the long-range isotropic Heisenberg interactions, i.e., $\Delta^{xy}=1$,
by using a combination of QMC and analytic methods, Laflorencie et al. have
studied the phase diagram in the $\lambda$-$\alpha$ plane Affleck , as is
shown in Fig. 1. It is shown that the critical point between the Néel phase
and the QLRO phase increases sharply from $\alpha^{c}(\lambda=0^{+})=2$ to
$\alpha^{c}(\lambda=8)\approx 2.7$. To further understand two phases, we
investigate the correlation functions of the system using iDMRG, which can
avoid the boundary effects. The correlation functions $\langle
S^{z}_{i}S^{z}_{i+r}\rangle$ and $\langle S^{+}_{i}S^{-}_{i+r}\rangle$ with
respect to the distance $r$ for $\alpha=2.1$, $\lambda=1$ are shown in Fig.
2(a). As we know, for 1D spin-1/2 short-range AFM Heisenberg system, the spin-
spin correlation function

$\displaystyle\langle\vec{S}_{i}\cdot\vec{S}_{i+r}\rangle\propto\frac{(-1)^{r}\sqrt{\ln
r}}{r},$ (9)

is expected to characterize the QLRO phase, and

$\displaystyle\lim_{r\rightarrow+\infty}\langle\vec{S}_{i}\cdot\vec{S}_{i+r}\rangle=\pm
m^{2}_{c}$ (10)

is capable of identifying the Néel phase Affleck1989 . The power-law decay of
$\langle S^{z}_{i}S^{z}_{i+r}\rangle$, $\langle S^{+}_{i}S^{-}_{i+r}\rangle$
in Fig. 2(a), implies
$\lim_{r\rightarrow\infty}\langle\vec{S}_{i}\cdot\vec{S}_{i+r}\rangle$ would
approach zero and the ground state for $\\{\alpha=2.1$, $\lambda=1\\}$ under
consideration is within the QLRO phase and thus the critical point of Néel-to-
QLRO transition should be below $2.1$ for $\lambda=1$. The spatial correlation
functions for $\\{\alpha=2$, $\lambda=0.5\\}$ are also calculated, and the QMC
results showcase the system should be in Néel phase. One finds $\langle
S^{z}_{i}S^{z}_{i+r}\rangle$ and $\langle S^{+}_{i}S^{-}_{i+r}\rangle$ also
exhibit a power-law decay, as is observed in Fig. 2(b), which means the ground
state remains the QLRO phase. These above mentioned discoveries indicate the
critical points retrieved by the QMC are not accurate. A more creditable
measure should be adopted to determine the phase boundaries.

Figure 4: (a) Fidelity susceptibility per site is plotted as a function of the
parameter $\alpha$ on different system sizes $L$ with $\Delta^{xy}=1$,
$\lambda$= $\infty$. (b) Scaling of the peak positions of $\chi$. Figure 5:
(a) Fidelity susceptibility per site is plotted as a function of the parameter
$\alpha$ on different system sizes $L$ with $\Delta^{xy}=1$, $\lambda=1$. (b)
The corresponding scaling of the peak positions of $\chi$.

To alleviate the controversy by the discrepancy between the QMC results (cf.
Fig.1) and correlations (cf. Fig.2), we consider a limiting case, i.e.,
$\lambda$$\to$$\infty$, which can be equivalently implemented by switching off
the nearest-neighbor isotropic interactions in Hamiltonian (II) with finite
$\lambda$. The absence of the LRO has been rigorously proven for $\alpha>3$
with $\lambda=1$ Parreira ; Roscilde , and was lately extended to arbitrary
$\lambda$ Affleck . The critical point $\alpha_{c}^{\rm
SW}$($\lambda$$\to$$\infty$)=2.9032 was inferred by the spin-wave
approximation Affleck . In this case, we use the EE to speculate the critical
point. We find the EE decreases monotonically with increasing $\alpha$. In
particular, the EE always shows a logarithmic growth with the system size as
$S_{\rm h}\propto\ln L$ [Fig. 3(a)], which can be treated as reminiscent of
gapless ground state in both the QLRO phase and the Néel phase Yang2020 , as
is indicated in Fig. 3(b). Consequently the signal of the QPT is hardly
discerned from the EE.

The impetus to identify the precise position of the quantum critical
point(QCP) was given by the FS, which has been proven to be capable of
detecting the phase transition successfully between two gapless phases Ren .
To this end, we will adopt the FS to identify the QCP between the Néel phase
with LRO and the QLRO phase for $\lambda=\infty$ as a glimpse. The numerical
results are shown in Fig. 4(a). One can observe the peak of the FS increases
with the system size $L$ nearby $\alpha=3.1$. In order to locate the critical
points $\alpha_{c}$ in the thermodynamic limit, we have used the finite-size
scaling theory Fisher , which can be used in finite systems with long-range
interactions Loscar . The position of the maximal points of the FS can be
fitted by the following formula:

$|\alpha_{c}(L)-\alpha_{c}|\sim L^{-b},$ (11)

where $b$ is a constant and $\alpha_{c}$ is the QCP in the thermodynamic
limit. For properly chosen values of $\alpha_{c}=3.00$, $b=0.85$, we can see
from Fig. 4(b) that a linear relation following Eq.(11) for different $L$ is
verified. Our results indicate the critical points $\alpha_{c}$ would approach
3.0 as $\lambda\rightarrow\infty$. Recall that Parreira et al. pointed out the
nonexistence of the Néel phase at zero temperature for $\alpha>3$ for
$\lambda=1$ Parreira  and a straightforward extension for all $\lambda$
Affleck . In this sense, the surprising consistence between our result with
the previous results confirm that the FS shows high accuracy and reliability
in detecting the critical point of the Néel-to-QLRO transitions.

Next we investigate the case of $\lambda=1$. The FS results for various system
sizes are shown in Fig. 5(a). The corresponding finite-size scaling according
to Eq. (11) is illustrated in Fig. 5(b), giving rise to $\alpha_{c}=1.955$,
$b=1.0$. In contrast to the QMC result $\alpha_{c}^{\rm QMC}=2.225\pm 0.025$
and the spin-wave result $\alpha_{c}^{\rm SW}=2.46$, the obtained value of
$\alpha_{c}$ indicates that the ground state for $\alpha=2.1$ is within the
QLRO phase. This is consistent with the correlations in Fig. 2(a). To this
end, the FS is calculated for different $\lambda$ and the positions of
critical points can be precisely retrieved from the FS results for
$\lambda\geq 0.02$. One finds $\alpha_{c}\simeq 1.12$ for $\lambda=0.02$,
whereas the positions of the critical points become elusive through the peak
of the FS for $\lambda<0.02$. As is observed in Fig. 2(c), it is found that
the correlations $\langle S^{z}_{i}S^{z}_{i+r}\rangle$, $\langle
S^{+}_{i}S^{-}_{i+r}\rangle$ tend to a constant for $\\{$$\alpha=1$,
$\lambda=0.01$$\\}$, implying that the critical point $\alpha_{c}\geq 1$ when
$\lambda\rightarrow 0^{+}$, which is consistent with spin-wave result. Based
on the above analysis of correlation functions and the FS, we obtain the
critical points and establish the ground-state phase diagram in Fig. 1. It is
clear that the critical values $a_{c}(\lambda)$ get lower than those obtained
by the large scale QMC simulation. The deviation is extremely prominent for
small $\lambda$ but negligible for large $\lambda$.

###  III.2 $\Delta^{xy}=1.5$

Figure 6:  Phase diagram of Hamiltonian Eq. (II) as functions of $\alpha$ and
$\lambda$ with $\Delta^{xy}=1.5$. Inset: Correlations $\langle
S^{+}_{i}S^{-}_{i+r}\rangle$, $\langle S^{z}_{i}S^{z}_{i+r}\rangle$ versus the
distance $r$ for $\lambda=1$, $\alpha=2$ with $\Delta^{xy}=1.5$. Figure 7: (a)
Fidelity susceptibility per site is plotted as a function of the parameter
$\alpha$ on different system sizes $L$ with $\lambda=1$, $\Delta^{xy}=1.5$.
(b) The corresponding scaling of the peak positions of $\chi$.  Figure 8: (a)
Half-chain EE versus $\ln L$ for different $\alpha$ with $\Delta^{xy}=1.5$,
$\lambda=1$. (b) Finite-size scaling of the energy gap $\Delta$ with various
$\alpha$. Symbols show numerical results obtained by DMRG calculations, and
solid lines are linear fits of the data in 1/$L$.

Next, we begin to study the effect of anisotropy of long-range exchange
interactions. First, the XY-type ($\Delta^{xy}>1$) exchange interactions are
considered. The phase diagram of Hamiltonian Eq. (II) with $\Delta^{xy}=1.5$
as functions of $\alpha$ and $\lambda$ is shown in Fig. 6. For sufficiently
large $\alpha$, the system would be in the QLRO phase. As the decay exponent
$\alpha$ gets smaller, the long-range interactions will become dominated. The
correlation functions for $\\{\alpha=2$, $\lambda=1\\}$ are shown in the inset
of Fig. 6, where $\langle S^{z}_{i}S^{z}_{i+r}\rangle$ tends to vanish as
$r\to\infty$, while $\langle S^{+}_{i}S^{-}_{i+r}\rangle$ will alternate
between $-0.1$ and $0.1$, which means that the $xy$-Néel phase is stabilized
with breaking of the continuous U(1) symmetry in the $x-y$ plane. In a sense,
the correlation function $\langle S^{+}_{i}S^{-}_{i+r}\rangle$ can act as an
order parameter for the QPT between the QLRO and U(1)-symmetric broken phase.
The ground-state FS per site $\chi$ for $\lambda=1$ is exhibited in Fig. 7(a).
Following the similar strategy as SU(2) symmetric model, the critical point
$\alpha_{c}=2.42$ between the $xy$-Néel and QLRO phase is identified from Fig.
7(b). Similarly, the EE scales logarithmically with the system size in the
$xy$-Néel phase, as is disclosed in Fig. 8(a), suggesting that the $xy$-Néel
phase remains gapless. In order to validate the gapless nature, the finite-
size energy gap $\Delta(L)$ is calculated for $\alpha<\alpha_{c}$ in Fig.
8(b). The linear fitting with respect to $1/L$ designates that
$\Delta(\infty)$ will vanish in the thermodynamic limit and the dynamical
exponent $z=1$.

###  III.3 $\Delta^{xy}=0$

Figure 9:  (a) Phase diagram of Hamiltonian Eq. (II) as functions of $\alpha$
and $\lambda$ with $\Delta^{xy}=0$. Inset: Correlations $\langle
S^{+}_{i}S^{-}_{i+r}\rangle$, $\langle S^{z}_{i}S^{z}_{i+r}\rangle$ versus the
distance $r$ for $\lambda=1$, $\alpha=3$ with $\Delta^{xy}=0$. Figure 10: (a)
Fidelity susceptibility per site is plotted as a function of the parameter
$\alpha$ on different system sizes $L$ with $\Delta^{xy}=0$, $\lambda=1$. (b)
The corresponding scaling of the peak positions of $\chi$. Figure 11: (a)
Half-chain EE is plotted as a function of the decay exponent $\alpha$ on
different system sizes $L$ with $\Delta^{xy}=0$, $\lambda=1$. (b) Half-chain
EE versus $\ln L$ for different $\alpha$. (c) The effective central charge in
Eq. (8), versus $\alpha$ with $L_{1}=400$ and $L_{2}=500$. (d) The
corresponding scaling of peak positions of $S_{\rm h}$.

We can consider the Ising-type ($\Delta^{xy}<1$) long-range interactions. Here
we exhibit a special case, i.e., $\Delta^{xy}=0$. The phase diagram of
Hamiltonian Eq. (II) as functions of $\alpha$ and $\lambda$ is shown in Fig.
9. For sufficiently large $\alpha$, the system also would enter the QLRO
phase. As the decay exponent $\alpha$ decreases, the long-range Ising
interactions will become dominated. The correlation functions for
$\\{\alpha=3$, $\lambda=1\\}$ are shown in the inset of Fig. 9, where $\langle
S^{+}_{i}S^{-}_{i+r}\rangle$ exhibits an oscillating decay until vanishes as
$r\to\infty$, while $\langle S^{z}_{i}S^{z}_{i+r}\rangle$ alternates between
$-0.053$ and $0.053$, implying the characteristic of ${\mathbb{Z}}_{2}$
symmetry broken $z$-Néel phase.

Moreover, we find that the phase transitions between $z$-Néel and QLRO can be
sensitively detected by both the FS and the EE. In Fig. 10(a), the FS per site
with respect to $\alpha$ for different system sizes $L$ is presented and the
peak of the ground-state FS becomes pronounced with increasing system sizes,
which signals the occurrence of the QPT. Regarding the finite-size scaling in
Eq.(11), $\alpha_{c}=3.88$ and $b=0.40$ can be extracted from Fig. 10(b).
Further evidence for indicating the $z$-Néel-to-QLRO transition is provided by
the EE, which is shown in Fig. 11(a). Upon increasing the system size $L$, the
EE shows a logarithmic growth for $\alpha>\alpha_{c}$ but saturates quickly
otherwise [see Fig. 11(b)], suggesting the $z$-Néel phase is gapped and the
breaking of conformal symmetry. The effective central charge $c_{\rm eff}$
would be zero [cf. 11(c)]. Similar to that of the FS, the finite-size scaling
of the EE yields $\alpha_{c}=3.88$, $b=0.457$, as is exhibited in Fig. 11(d).
It is worthy noting that the critical point $\alpha_{c}$ with $\Delta^{xy}=0$
becomes vanishing when the parameter $\lambda$ tends to zero, and diverges
when $\lambda$ increases to the infinity.

##  IV Discussion

In this paper, we have studied the quantum phase transitions (QPTs) in the
one-dimensional spin-$1/2$ chains with modulated long-range power-law-decaying
interactions in terms of the density-matrix renormalization group technique.
Together with the correlations and the entanglement entropy (EE), the ground-
state fidelity susceptibility (FS) are employed to determine the phase
boundary. The XY-type long-range interactions lead to the emergence of
U(1)-symmetric broken $xy$-Néel phase with long-range order (LRO) along easy
axes ZehanLi , akin to the SU(2) symmetry broken Néel phase induced by
isotropic long-range interactions, while the Ising-type long-range
interactions prompt the ${\mathbb{Z}}_{2}$ symmetry broken $z$-Néel phase. The
FS can detect the QPT between the gapless quasi-long-range order (QLRO) phase
and three different Néel phase, whether it is gapless or not. The FS proved to
be a reliable tool to determine the ground-state phase diagram. An area-law
scaling is still valid in the gapped phase in the presence of the long-range
interactions, although it was originally derived for the short-range
interacting Hamiltonian. Figures 3(a) and 8(a) demonstrate that the half-chain
EE satisfies a logarithmic scaling with respect to the system size in gapless
phases. In this respect, the half-chain EE can faithfully seize the QPT
between the gapless QLRO phase and the gapped $z$-Néel phase, while it is
insensitive to QPTs between two gapless phases, such as QLRO to $xy$-Néel
phase transition, QLRO to Néel phase transition. The insensitivity of the EE
at quantum critical points between gapless phases may be traced back to the
gapless mode associated with the spontaneous breaking of the continuous
symmetry, sparking the challenge to demand much larger-scale computation for
the effective central charge. In this context, using the maximum of bipartite
EE as an indicator of a QPT from a gapless phase to another gapless phase is
still elusive. The models under consideration could be envisioned in quantum
simulation in ultracold atoms Periwal21 ; Takahashi , opening the prospect for
experimental investigation of the issues confronted here.

###### Acknowledgements.

W.-L. You appreciates the valuable discussion with Gaoyong Sun and Ming Xue.
This work is supported by the National Natural Science Foundation of China
(NSFC) under Grants No. 11104021, No. 12174194. J.R kindly acknowledge support
from Open Project of Key Laboratory of Artificial Structures and Quantum
Control(Ministry of Education), Shanghai Jiao Tong University. W.-L. Y
acknowledges the startup fund of Nanjing University of Aeronautics and
Astronautics under Grant No. 1008-YAH20006, Top-notch Academic Programs
Project of Jiangsu Higher Education Institutions and stable supports for basic
institute research (Grant No. 190101).

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