# Weak solutions for stochastic differential equations with additive
fractional noise

Pedro J. Catuogno and Diego S. Ledesma  Departamento de Matemática,
Universidade Estadual de Campinas,
13.081-970 - Campinas - SP, Brazil<EMAIL_ADDRESS>and<EMAIL_ADDRESS>
###### Abstract.

We give a new approach to prove the existence of a weak solution of

$dx_{t}=f(t,x_{t})dt+g(t)dB^{H}_{t}$

where $B^{H}_{t}$ is a fractional Brownian motion with values in a separable
Hilbert space for suitable functions $f$ and $g$. Our idea is to use the
implicit function theorem and the scaling property of the fractional Brownian
motion in order to obtain a weak solution for this equation.

###### Key words and phrases:

Fractional Brownian motion. Stochastic partial differential equations. Weak
solutions.
2010 AMS 2000 Subject classiflcation: 60H15, 60G22, 60G18

Research partially supported by FAPESP 2018/13481-0 and 2020/04426-6 and
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) -
Finance Code 001.

##  1\. Introduction

Recently there has been a great interest in the study of stochastic partial
differential equations (SPDE) driven by coloured noise. The main motivation
comes from finance and physics. In finance, it is well known that the market
evolves according to a geometric fractional $H-$Brownian motion with (see [2]
[10] and [7]). In physics, fractional mechanics and its generalization to
stochastic mechanics have been developed in recent years (see [14] and [15]).

We are interested to deal with the following kind of SPDE

(1) $dx_{t}=f(t,x_{t})~{}dt+g(t)~{}dB_{t}^{H},$

where $B_{t}^{H}$ is a fractional Brownian motion (fBm) with Hurst parameter
$H\in(0,1)$, $f:[0,T]\times\mathbb{H}\rightarrow\mathbb{H}$ and
$g:[0,T]\rightarrow L(\mathbb{H},\mathbb{H})$ are continuous maps and a
Hilbert space $\mathbb{H}$ which will be defined precisely in the section
section 3 below.

From a mathematical point of view, the problem is to guarantee the existence
of solutions for the equation (1). There are, in the literature, different
notions of solution for SPDE of this kind (for example [5], [9] and [20]). In
this article we show the existence of a weak solution for the equation (1).
There are several works in this direction. Here, we highlight the
contributions of D. Nualart and Y. Oukine in [18] that was improved by B.
Boufoussi and Y. Ouknine in [4], J. Snupárková in [21] and, more recently by
Z. Li, W. Zhan and L. Xu in [13]. In the above works the proofs are based on
the Girsanov Theorem and the Picard iteration argument. Here we present an
alternative idea to prove the existence of a weak solution using the scaling
property of the Fractional Brownian motion. We adapt a result about
deterministic dynamical systems of J. Robbin [19] to prove the existence of a
weak solution to of equation (1) under certain hypotheses.

The work is organized as follow: in section 2 we give the basic definitions
about fractional Brownian motion on Hilbert spaces and its integration theory
following the references [3], [6], [10], [16] and [17]. Finally, in section 3
we stablish our main result and give a simple example.

##  2\. Fractional Brownian motion

We start reviewing the basic concepts on fractional Brownian motion (fBm) and
stochastic integral with respect to fBm.

Fix a standard filtered probability space
$(\Omega,\mathcal{F},~{}\\{\mathcal{F}_{t},~{}t\in[0,T]\\},\mathbb{P})$, see
for instance [3], [16] and [17].

A fractional brownian motion with Hurst parameter $H\in(0,1)$ is a continuous
and centered Gaussian process $(B_{t}^{H})_{t\geq 0}$ with covariance function

$\mathbb{E}\left[B_{t}^{H}B_{s}^{H}\right]=\frac{1}{2}(s^{2H}+t^{2H}-|t-s|^{2H}).$

A very important result on fBm theory is the following integral representation

$B^{H}_{t}=\int_{0}^{t}K_{H}(t,s)~{}dB_{s},$

where

  * •

$B_{s}$ is the standard Brownian motion.

  * •

$K_{H}(t,s):=\left\\{\begin{array}[]{lcc}b_{H}\left[\left(\frac{t(t-s)}{s}\right)^{H-1/2}-(H-1/2)s^{1/2-H}\int_{s}^{t}(u-s)^{H-1/2}u^{H-3/2}~{}du.\right]{}&&H<1/2\\\
&&\\\ c_{H}s^{1/2-H}\int_{s}^{t}|u-s|^{H-3/2}u^{H-1/2}~{}du&&H>1/2\\\
\end{array}\right.$

  * •

$b_{H}=\sqrt{\frac{2H}{(1-2H)\beta(1-2H,H+1/2)}},$

  * •

$c_{H}=\sqrt{\frac{H(2H-1)}{\beta(2-2H,H-1/2)}}.$

and

  * •

$\beta(a,b)=\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}$

where $\Gamma(a)=\int_{0}^{\infty}x^{a-1}e^{-x}dx$ is the Gamma function. (see
[3], [16] and [17]).

To construct the fractional Brownian motion we consider the Schwartz space
$\mathcal{S}(\mathbb{R})$ of rapidly decreasing smooth functions on
$\mathbb{R}$ provided with the inner product $<\cdot,\cdot>_{H}$ given by

$<f,g>_{H}:=\left\\{\begin{array}[]{lcc}H(2H-1)\int_{0}^{T}\int_{0}^{T}f(s)g(t)|t-s|^{2H-2}~{}dsdt&\textrm{for}&H>1/2\\\
&&\\\
\int_{0}^{T}K_{H}^{*}f(s)K_{H}^{*}g(s)~{}ds&\textrm{for}&H<1/2.\end{array}\right.$

where

$K_{H}^{*}f(s):=K(T,s)f(s)+\int_{s}^{T}(f(t)-f(s))\partial_{t}K_{H}(t,s)~{}dt.$

Let $\mathcal{S}^{\prime}(\mathbb{R})$ be the dual of
$\mathcal{S}(\mathbb{R})$ the space of tempered distributions on $\mathbb{R}$.
We consider the mapping
$f\in\mathcal{S}(\mathbb{R})\rightarrow\exp(1/2||f||_{H}^{2})\in\mathbb{R}$.
This map is positive definite in $\mathbb{S}(\mathbb{R})$ thus, by the
Bochner-Minlos theorem (see for instance [12, pp. 257] ), we guarantee the
existence of a probability measure $\mathbb{P}^{H}$ on the Borel
$\sigma-$algebra $\mathcal{B}(\mathcal{S}^{\prime}(\mathbb{R}))$ such that

$\int_{\mathcal{S}^{\prime}(\mathbb{R})}e^{i<\omega,f>}~{}d\mathbb{P}^{H}(\omega)=e^{-1/2||f||_{H}^{2}}\quad\quad\textrm{for
all}~{}f\in\mathcal{S}(\mathbb{R}).$

We will denote by $I_{[0,t]}$ the usual characteristic function of the
interval $[0,t]$. We define
$B^{H}:[0,T]\times\mathcal{S}^{\prime}(\mathbb{R})\rightarrow\mathbb{R}$ by

$B^{H}(t,\omega)=B_{t}^{H}(\omega)=<\omega,I_{[0,t]}>$

as an element of $L^{2}(\mathbb{P}^{H})$.

It is a simple consequence of the Kolmogorov continuity theorem that $B^{H}$
is a fractional Brownian motion with Hurst parameter $H$.

In order to construct the Wiener integral with respect to the fBm, we follow
[3]. Consider the functions $R_{H}:[0,T]\times[0,T]\rightarrow[0,\infty)$
given by

$R_{H}(t,s)=\frac{1}{2}(s^{2H}+t^{2H}-|t-s|^{2H}),$

and the closure $\mathcal{H}$ of the vector space spanned by the functions

$\\{R_{H}(t,~{}),~{}t\in[0,T]\\}$

with respect to the scalar product

$<R_{H}(t,\cdot),R_{H}(s,\cdot)>:=R_{H}(t,s)~{}\mbox{ for all }t,s\in[0,T].$

The Wiener integral with respect to the fBm is defined as the linear extension
of the isometric map

$R_{H}(t,~{})\rightarrow B^{H}_{t}\quad\textrm{for all}~{}t\in[0,T],$

from $\mathcal{H}$ in $L^{2}(\mathbb{P}^{H})$.

To define the stochastic integral of a function with respect to the fBm we
follow the usual path [3], [16] and [17]. We start defining it for a simple
function $\phi=\sum_{i}{a_{i}}\chi_{[t_{i},t_{i+1})}$ as

$\int_{0}^{T}\phi(s)~{}dB_{s}^{H}=\sum_{i}{a_{i}}(B_{t_{i+1}}^{H}-B_{t_{i}}^{H}).$

Then, we extend it to $L^{2}(\mathbb{P}^{H})$ and we have the isometry

(2)
$\mathbb{E}\left[\left<\int_{0}^{T}f(s)~{}dB_{s}^{H},\int_{0}^{T}g(s)~{}dB_{s}^{H}\right>\right]=<f,g>_{H},\quad$

see [3] and [11].

Now, we review the construction of a fractional Brownian motion in a Hilbert
space $\mathbb{H}$.

Let $\mathbb{H}$ be a separable Hilbert space. We will denote by
$L(\mathbb{H})$ the space of continuous linear maps from $\mathbb{H}$ to
$\mathbb{H}$. Let $Q\in L(\mathbb{H})$ be a symmetric, non negative trace
class operator.

Let $\\{\lambda_{n}\\}_{n\in\mathbb{N}}$ be a discrete family of eigenvalues
of $Q$, counted with multiplicity, and let $\\{e_{n}\\}_{n\in\mathbb{N}}$ be
the normalized eigenvectors, this is

$Q(e_{n})=\lambda_{n}e_{n}.$

Let $\\{B_{t}^{n}\\}$ be independent 1-dimensional fBm with parameter $H$.
Then the series

$B_{t}^{H}=\sum_{n=1}^{\infty}\sqrt{\lambda_{n}}B_{t}^{n}e_{n},\quad t\geq 0$

converges almost surely and in $L^{p}$ for $p\geq 1$ (see [3], [10], [16] and
[17] ). We say that $B_{t}^{H}$ is a trace-class fractional Brownian motion in
$\mathbb{H}$ with covariance $Q$ and Hurst parameter $H$.

Let $g:[0,T]\rightarrow L(\mathbb{H})$ be a continuous map. We define the
integral of $g$ with respect to the trace-class fBm $B^{H}$ as

$\int_{0}^{t}g(t)~{}dB_{t}^{H}:=\sum_{n,m=1}^{\infty}\left(\int_{0}^{t}\sqrt{\lambda_{m}}<g(t)e_{m},v_{n}>~{}dB_{t}^{m}\right)v_{n},$

where $\\{v_{m}\\}_{m\in\mathbb{N}}$ is any orthonormal basis of $\mathbb{H}$.

Using the isometry (2) we get that

$\displaystyle\mathbb{E}\left|\int_{0}^{t}g(t)~{}dB_{t}^{H}\right|^{2}$
$\displaystyle=$
$\displaystyle\sum_{n,m=1}^{\infty}\mathbb{E}\left|\int_{0}^{t}\sqrt{\lambda_{m}}<g(t)e_{m},v_{n}>~{}dB_{t}^{m}\right|^{2}$
$\displaystyle=$
$\displaystyle\sum_{n,m=1}^{\infty}\lambda_{m}<<g(t)e_{m},v_{n}>,<g(t)e_{m},v_{n}>>_{H}.$

Thus

(3)
$\mathbb{E}\left|\int_{0}^{t}g(t)~{}dB_{t}^{H}\right|^{2}=\left\\{\begin{array}[]{lcc}\sum_{j,k=1}^{\infty}\lambda_{k}H(2H-1)\int_{0}^{T}\int_{0}^{T}<g(t)e_{k},g(s)e_{k}>\times&&\\\
\times|t-s|^{2H-2}~{}dsdt&\textrm{if}&H\geq 1/2,\\\ \\\ {}\\\ {}\\\ {}\\\
\sum_{j,k=1}^{\infty}\lambda_{k}\int_{0}^{T}<K_{H}^{*}g(s)e_{k},K_{H}^{*}g(s)e_{k}>~{}ds&\textrm{if}&H<1/2,\end{array}\right.$

where

$\displaystyle<K_{H}^{*}g(s)e_{k},K_{H}^{*}g(s)e_{k}>$ $\displaystyle=$
$\displaystyle K(T,s)^{2}||g(s)e_{k}||^{2}+$
$\displaystyle+2K(T,s)\int_{s}^{T}<g(s)e_{k},(g(t)-g(s))e_{k}>\partial_{t}K_{H}(t,s)~{}dt+$
$\displaystyle+\int_{s}^{T}\int_{s}^{T}<(g(r)-g(s))e_{k},(g(t)-g(s))e_{k}>\times$
$\displaystyle\times\partial_{t}K_{H}(t,s)\partial_{t}K_{H}(r,s)~{}~{}drdt.$

The next lemma is a consequence of the above set up and it shows estimates
that we will use below.

######  Lemma 1.

Let $g,~{}h:[0,T]\rightarrow L(\mathbb{H},\mathbb{H})$ be continuous functions
such that

(4)
$\sup_{r\in[0,T]}||g(r)e_{j}||^{2}_{H}<\infty\quad\textrm{and}\quad\sup_{r\in[0,T]}||rh(r)e_{j}||^{2}_{H}<\infty.$

Then for any positive real numbers $a$ and $k$,

$\displaystyle\sum_{j}\lambda_{j}\mathbb{E}\left|a^{Hk}\int_{0}^{T}g(a^{k}s)(e_{j})~{}dB_{t}^{H}\right|^{2}$
$\displaystyle<$ $\displaystyle\infty$
$\displaystyle\sum_{j}\lambda_{j}\mathbb{E}\left|a^{Hk}\int_{0}^{T}sh(a^{k}s)~{}dB_{t}^{H}\right|^{2}$
$\displaystyle<$ $\displaystyle\infty.$

###### Proof.

It follows directly from the construction of the fractional Brownian motion
$B^{H}$ and (3) we get

$\displaystyle\sum_{j}\lambda_{j}a^{2Hk}\mathbb{E}\left|\int_{0}^{T}g(a^{k}s)(e_{j})~{}d(B_{t}^{H})^{j}\right|^{2}$
$\displaystyle\leq$ $\displaystyle
C(H,T)a^{2Hk}\sup_{r\in[0,T]}||g(a^{k}r)e_{j}||^{2}_{H}\sum_{j}\lambda_{j}$
$\displaystyle\sum_{j}\lambda_{j}a^{2Hk}\mathbb{E}\left|\int_{0}^{T}sh(a^{K}s)(e_{j})~{}d(B_{t}^{H})^{j}\right|^{2}$
$\displaystyle\leq$ $\displaystyle
C(H,T)a^{2Hk}\sup_{r\in[0,T]}||rh(a^{k}r)e_{j}||^{2}_{H}\sum_{j}\lambda_{j}$

where for $H>1/2$,

$C(H,T)=H(2H-1)\int_{0}^{T}\int_{0}^{T}|t-s|^{2H-2}~{}dsdt$

and for $H<1/2$

$\displaystyle C(H,T)$ $\displaystyle=$
$\displaystyle\int_{0}^{T}K(T,s)^{2}~{}ds+2\int_{0}^{T}\int_{s}^{T}|K(T,s)\partial_{t}K_{H}(t,s)|~{}dt~{}ds$
$\displaystyle+\int_{0}^{T}\int_{s}^{T}\int_{s}^{T}|\partial_{t}K_{H}(t,s)\partial_{t}K_{H}(r,s)|~{}drdtds.$

Now, the hypothesis (4) guarantee the result. ∎

##  3\. Main result

We consider stochastic diferential equations over $\mathbb{H}$ of the
following type

(5) $dx_{t}=f(t,x_{t})~{}dt+g(t)~{}dB_{t}^{H},$

where

  * •

$B_{t}^{H}$ is a trace-class fractional Brownian motion in $\mathbb{H}$ with
covariance $Q$ and Hurst parameter $H$.

  * •

$f:[0,T]\times\mathbb{H}\rightarrow\mathbb{H}$ and $g:[0,T]\rightarrow
L(\mathbb{H},\mathbb{H})$ are continuous maps.

We say that a process $x_{t}$ is a weak solution to equation (5) starting at
$x_{0}\in\mathbb{H}$ if

$x_{t}-x_{0}=\int_{0}^{t}f(s,x_{s})~{}ds+\int_{0}^{t}g(s)~{}d\tilde{B}_{s}^{H},$

for some fractional Brownian motion $\tilde{B}_{s}^{H}$ with Hurst parameter
$H$.

######  Theorem 1.

Let $\mathbb{H}$ and $\tilde{\mathbb{H}}$ be two Hilbert spaces such that
$\mathbb{H}\subset\tilde{\mathbb{H}}$ densely an there is a family of linear
operators $\\{S_{n}\\}_{n\in\mathbb{N}}$ such that

  * •

$S_{n}:\tilde{\mathbb{H}}\rightarrow\tilde{\mathbb{H}}$,

  * •

$S_{n}(x)\in\mathbb{H}$ for all $x\in\tilde{\mathbb{H}}$,

  * •

$||S_{n}-I_{\tilde{\mathbb{H}}}||\rightarrow 0$.

Let $f:[0,T]\times\mathbb{H}\rightarrow\tilde{\mathbb{H}}$ and
$g:[0,T]\rightarrow L(\tilde{\mathbb{H}},\tilde{\mathbb{H}})$ be continuously
differentiable maps such that

  * •

$|f(t,x)|+|\partial_{t}f(t,x)|\leq\phi(t)(1+|x|)$,

  * •

$|\partial_{x}f(t,x)v|^{2}\leq\phi(t)^{2}(1+|x|^{2}+|v|^{2})$,

  * •

$|g(t)(v)|+|\partial_{t}g(t)(v)|\leq\phi(t)|v|$,

for a continuous square integrable function $\phi:[0,T]\rightarrow\mathbb{R}$.

Then there exists a fractional Brownian motion $\tilde{B}^{H}$ in
$\tilde{\mathbb{H}}$ with covariance $Q$ and Hurst parameter $H$, which is a
rescaling of the original $B^{H}$, and
$X:[0,T]\times\Omega\times\mathbb{H}\rightarrow\tilde{\mathbb{H}}$ satisfying
the equation

$X(t)-x_{0}=\int_{0}^{t}f(s,X(s))~{}ds+\int_{0}^{t}g(s)~{}d\tilde{B}^{H}_{s},$

for any $t\in[0,T]$.

Furthermore, the dependence of $X$ in $x_{0}$ is smooth.

###### Proof.

We fix $0<a<1$ and $k\in 2\mathbb{N}$ such that $kH>1.$ For any $X\in
C([0,T],L^{2}(\Omega,\tilde{\mathbb{H}}))$ we define

$F_{n}(X,a,x_{0})(t)=S_{n}X(t)-\int_{0}^{t}a^{k}f(a^{k}s,x_{0}+S_{n}X(s))~{}ds-\int_{0}^{t}a^{Hk}g(a^{k}s)~{}dB_{s}^{H}.$

By Lemma 1 and Fubini’s theorem

$\displaystyle\mathbb{E}[||F_{n}(X,a,x_{0})(t)||^{2}]$ $\displaystyle\leq$
$\displaystyle 8\mathbb{E}[||S_{n}X(t)||^{2}]$
$\displaystyle+8\int_{0}^{t}t(a^{2k}|\mathbb{E}[||f(a^{k}s,x_{0}+S_{n}X(s))||^{2}]~{}dt$
$\displaystyle+8C(H,T)\int_{0}^{t}\sum_{j}\lambda_{j}a^{2Hk}||g(a^{k}s)e_{j}||^{2}_{H}~{}ds.$

Thus,

$F_{n}:C([0,T],L^{2}(\Omega,\tilde{\mathbb{H}}))\times\mathbb{R}\times\mathbb{H}\rightarrow
C([0,T],L^{2}(\Omega,\tilde{\mathbb{H}})),$

is well defined and $F_{n}(0,0,x_{0})=0$.

Now we claim that $F_{n}$ belong to $C^{1}$ and $\partial_{X}F_{n}(0,0,x_{0})$
is invertible. In fact, we have that

$\displaystyle D(F_{n})_{(X,a,x_{0})}(W,b,y)(t)$ $\displaystyle=$
$\displaystyle
S_{n}W(t)-\int_{0}^{t}a^{k}Df_{(a^{k}s,x_{0}+S_{n}X(s))}(S_{n}W(s),b,y)~{}ds$
$\displaystyle-\int_{0}^{t}ka^{Hk+k-1}sb\partial_{t}g(a^{k}s)~{}dB_{s}^{H},$

where $W\in C([0,T],L^{2}(\Omega,\tilde{\mathbb{H}}))$, $b\in\mathbb{R}$ and
$y\in\mathbb{H}$.

We observe that

$Df_{(a^{k}s,x_{0}+S_{n}X)}(S_{n}W,b,y)=\partial_{t}f(a^{k}s,x_{0}+S_{n}X(s))ka^{k-1}b+\partial_{x}f(a^{k}s,x_{0}+S_{n}X(s))(y+S_{n}W),$

is an integrable function for each $n\in\mathbb{N}$ big enough. In fact,

$\displaystyle\int_{0}^{t}|\partial_{t}f(a^{k}s,x_{0}+S_{n}X(s))|^{2}~{}dt$
$\displaystyle\leq$ $\displaystyle\int_{0}^{t}\phi(s)^{2}|S_{n}X(s)|^{2}~{}ds$
$\displaystyle\leq$
$\displaystyle\int_{0}^{t}\phi(s)^{2}|X(s)|^{2}~{}ds+\int_{0}^{t}\phi(s)^{2}|S_{n}X(s)-X(s)|^{2}~{}ds,$

and

$\displaystyle\int_{0}^{t}|\partial_{x}f(a^{k}s,x_{0}+S_{n}X(s))(y+S_{n}W)|^{2}~{}ds$
$\displaystyle\leq$
$\displaystyle\int_{0}^{t}\phi(s)^{2}(1+|x_{0}+S_{n}X(s)|^{2})~{}ds+$
$\displaystyle+\int_{0}^{t}\phi(s)^{2}|y+S_{n}W|^{2}~{}ds.$

Thus $F_{n}$ belong to $C^{1}$.

We have that $F_{n}$ verifies the hypothesis of the the implicit function
theorem (see [1, pp. 121] ) because

$\partial_{X}F_{n}(0,0,x_{0})W(t)=S_{n}W(t).$

and $S_{n}$ is invertible for $n$ big enough. This follows from

$S_{n}=I+(S_{n}-I),$

with $||S_{n}-I||\rightarrow 0$.

By the implicit function theorem, there is a neighborhood $U$ of
$(0,x_{0})\in[0,T]\times\mathbb{H}$ and a differentiable map
$G_{n}:U\rightarrow C([0,T],L^{2}(\Omega,\tilde{\mathbb{H}}))$ such that

$F_{n}(G_{n}(a,x),a,x)=0\quad\textrm{for all}~{}(a,x)\in U.$

We set

$X(t):=S_{n}G_{n}(x_{0},\epsilon)(t/\epsilon^{k}).$

Then, considering the self-similarity property
$\tilde{B}_{a^{k}t}^{H}:=a^{kH}B_{t}^{H}$ (see [16], [17, Prop. 2.2]) we get

$\displaystyle
X(t)-\int_{0}^{t}f(r,x_{0}+X(r))~{}dr-\int_{0}^{t}g(r)~{}d\tilde{B}_{r}^{H}$
$\displaystyle=$ $\displaystyle S_{n}G_{n}(x_{0},\epsilon)(t/\epsilon^{k})$
$\displaystyle-\int_{0}^{t/\epsilon^{k}}\epsilon^{k}f(\epsilon^{k}s,x_{0}+S_{n}G_{n}(x_{0},\epsilon)(s))~{}ds$
$\displaystyle-\int_{0}^{t/\epsilon^{k}}\epsilon^{kH}g(\epsilon^{k}s)~{}dB_{s}^{H}$
$\displaystyle=$ $\displaystyle
F_{n}(G_{n}(x_{0},\epsilon),\epsilon,x_{0})(t/\epsilon^{k})=0.$

Thus

$\tilde{X}=x_{0}+X,$

is a solution of the equation (5) that depends smoothly in $x_{0}$.

∎

######  Remark 1.

To show the result above we just need the estimates

$\int_{0}^{t}t(a^{2k}|\mathbb{E}[f(a^{k}s,x_{0}+X(s))|^{2}]~{}dt<\infty,$

and

$\int_{0}^{t}[\partial_{t}f(a^{k}s,x_{0}+X(s))ka^{k-1}b+\partial_{x}f(a^{k}s,x_{0}+X(s))(y+W)]^{2}~{}ds<\infty.$

######  Example 1.

We consider the quasi-linear equation em $L^{2}(\mathbb{R})$ given by

(6) $\displaystyle du(t,x)$ $\displaystyle=$ $\displaystyle[\alpha\Delta
u(t,x)+\beta u(t,x)+\overrightarrow{\gamma}\cdot\nabla
u(t,x)]~{}dt+dB_{t}^{H}$ $\displaystyle u(0,x)$ $\displaystyle=$
$\displaystyle v(x)$

for $\alpha,\beta$ non-negative constants,
$\overrightarrow{\gamma}\in\mathbb{R}^{n}$ and
$v(x)\in\mathcal{H}^{2}(\mathbb{R})$.

We first observe that $\mathcal{H}^{2}(\mathbb{R})\subset L^{2}(\mathbb{R})$
is densely contained with

$f(t,u)=[\alpha\Delta u+\beta u+\overrightarrow{\gamma}\cdot\nabla u]$

and

$\partial_{u}f(t,u)v=[\alpha\Delta v+\beta
v+\overrightarrow{\gamma}\cdot\nabla v]$

Since the Laplace operator is linear, we get

$\displaystyle||\Delta u||_{L^{2}}$ $\displaystyle\leq$ $\displaystyle
C(1+||u||_{\mathcal{H}^{2}})$ $\displaystyle||u(t,x))||_{L^{2}}$
$\displaystyle\leq$ $\displaystyle C(1+||u||_{\mathcal{H}^{2}})$
$\displaystyle||(|\nabla u(t,x)|)||_{L^{2}}$ $\displaystyle\leq$
$\displaystyle C(1+||u||_{\mathcal{H}^{2}}),$

for a positive constant $C$. It follows that

$|f(t,u)|+|\partial_{t}f(t,u)|\leq C(1+|u|)$

and

$|\partial_{u}f(t,u)|^{2}\leq C(1+|u|^{2}+|v|^{2}).$

By Theorem 1 there is a weak solution for the equation (6).

## References

  * [1] Abraham, R. ; Marsden, J. E. and Ratiu, T. Manifolds, tensor analysis, and applications. Second edition. Applied Mathematical Sciences, 75. Springer-Verlag, New York, 1988.
  * [2] Bender, C.; Sottinen, T.; Valkeila, E. Fractional processes as models in stochastic finance. Advanced mathematical methods for finance, 75-103, Springer, Heidelberg, 2011.
  * [3] Biagini, F. ; Hu, Y. ; Øksendal, B. ; Zhang, T. Stochastic calculus for fractional Brownian motion and applications. Probability and its Applications (New York). Springer-Verlag London, Ltd., London, 2008.
  * [4] Boufoussi, B. ; Ouknine, Y.  On a SDE driven by a fractional Brownian motion and with monotone drift. Electron. Comm. Probab. 8, 122-134, 2003.
  * [5] Da Prato, G. and Zabczyk, J. Stochastic equations in infinite dimensions, volume 44 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1992.
  * [6] Duncan,T. E. ; Maslowski, B. and Pasik-Duncan, B. Stochastic equations in Hilbert space with a multiplicative fractional Gaussian noise. Stochastic Process. Appl., 115, 8, 1357-1383, 2005.
  * [7] Elliott R.J., van der Hoek J. Fractional Brownian Motion and Financial Modelling. In: Kohlmann M., Tang S. (eds) Mathematical Finance. Trends in Mathematics. Birkhäuser, Basel. (2001)
  * [8] Garrido-Atienza, M. J. ; Lu, K. and Schmalfuß, B. Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion. Discrete Contin. Dyn. Syst. Ser. B, 14, 2, 473-493, 2010\.
  * [9] Gawarecki, L. ; Mandrekar, V. Stochastic differential equations in infinite dimensions with applications to stochastic partial differential equations. Probability and its Applications (New York). Springer, Heidelberg, 2011.
  * [10] Grecksch, W. ; Roth, C. ; Anh, V. V $Q$-fractional Brownian motion in infinite dimensions with application to fractional Black-Scholes market. Stoch. Anal. Appl. 27, 1, 149-175, 2009.
  * [11] Gripenberg, G. and Norros, I. On the prediction of fractional Brownian motion. J. Appl. Probab. 33, 400-410, 1996
  * [12] Holden, H. ; Øksendal, B. ; Uboe, J. ; Zhang, T. Stochastic partial differential equations. A modeling, white noise functional approach. Second edition. Universitext. Springer, New York, 2010.
  * [13] Li, Z. ; Zhan, W. ; Xu, L.  Stochastic differential equations with time-dependent coefficients driven by fractional Brownian motion. Phys. A 530 (2019), 121565, 11 pp.
  * [14] Mandelbrot, B. and Van Ness, J. Fractional Brownian motions, fractional noises and applications. SIAM Rev., 10, 422-437, 1968.
  * [15] Meerschaert, M. ; Sikorskii, A. Stochastic models for fractional calculus. Second edition. De Gruyter Studies in Mathematics, 43. De Gruyter, Berlin, 2019.
  * [16] Mishura, Yuliya S. Stochastic calculus for fractional Brownian motion and related processes. Lecture Notes in Mathematics, 1929. Springer-Verlag, Berlin, 2008.
  * [17] Nourdin, I. Selected aspects of fractional Brownian motion. Bocconi and Springer Series, 4. Springer, Milan; Bocconi University Press, Milan, 2012.
  * [18] Nualart, D. ; Ouknine, Y. Regularization of quasilinear heat equations by a fractional noise. Stoch. Dyn. 4, 2, 201-221, 2004.
  * [19] Robbin, J. On the existence theorem for differential equations. Proc. Amer. Math. Soc. 19, 1005-1006, 1968.
  * [20] Pardoux, E. Stochastic partial differential equations. An introduction. SpringerBriefs in Mathematics. Springer, Cham, (2021).
  * [21] Snupárková, J. Weak solutions to stochastic differential equations driven by fractional Brownian motion. Czechoslovak Math. J. 59, 134, 4, 879-907, 2009.


# Electron correlations and $T$-breaking density wave order in a
$\mathbb{Z}_{2}$ kagome metal

Chandan Setty⊕,† Haoyu Hu⊕ Lei Chen  Qimiao Si  Department of Physics &
Astronomy, Rice Center for Quantum Materials, Rice University, Houston, Texas
77005,USA

###### Abstract

There have been extensive recent developments on kagome metals, such as TmXn
(T= Fe, Co and X= Sn, Ge) and $A$V3Sb5 ($A=$ Cs, K, Rb). An emerging issue is
the nature of correlated phases when topologically non-trivial bands cross the
Fermi level. Here, we consider an extended Hubbard model on the kagome
lattice, involving a Kramers’ pair of bands that have opposite Chern numbers
and are isolated in the band structure. We construct an effective model in a
time-reversal (T) symmetric lattice description. We determine the correlated
phases of this model and identify a density-wave order in the phase diagram.
We show that this order is T-breaking, which originates from the Wannier
orbitals lacking a common Wannier center – a fingerprint of the underlying
$Z_{2}$ topology. Implications of our results for the correlation physics of
the kagome metals are discussed.

Introduction: In standard settings of electron correlations, the effects of
short-range repulsive Coulomb interactions are studied on topologically
trivial Bloch bands. In the perturbative limit, interactions are exactly
marginal and gapless excitations of the Fermi gas are retained in the Fermi
liquid Shankar (1994). In contrast, correlated phases such as Mott insulators,
charge density waves and nematic states typically emerge only when
interactions become comparable to or greater than the overall non-interacting
electron bandwidth ($W_{0}$) Paschen and Si (2021); Imada  _et al._ (1998);
Georges  _et al._ (1996).

Correlated phases in systems such as twisted bilayer graphene Cao  _et al._
(2018); Jiang  _et al._ (2019) on hexagonal boron nitride (TBG/hBN) Sharpe
_et al._ (2019), kagome lattice binary TmXn (T= Fe, Co and X= Sn, Ge) Ye  _et
al._ (2018); Liu  _et al._ (2020); Lin  _et al._ (2018); Liu  _et al._ (2018);
Yao  _et al._ (2018) or ternary $A$V3Sb5 ($A=$ Cs, K, Rb) Jiang  _et al._
(2021); Zhao  _et al._ (2021); Zhou  _et al._ (2021); Liang  _et al._ (2021);
Uykur  _et al._ (2021); Li  _et al._ (2021a); Hu  _et al._ (2021)
intermetallics and beyond Gilmutdinov  _et al._ (2021) demand a fresh
perspective to the aforementioned paradigm. These systems have a collection of
topologically non-trivial bands Ye  _et al._ (2018); Kang  _et al._ (2020a,
b); Li  _et al._ (2018); Ortiz  _et al._ (2020, 2021); Fu  _et al._ (2021)
close to the Fermi level. To isolate the effect of these bands, one may
consider narrow bands (bandwidth $W$) Liu  _et al._ (2020); Lin  _et al._
(2018); Meier  _et al._ (2020); Lin  _et al._ (2020); Hu  _et al._ (2021) that
are separated from the rest of the bands by a gap ($\Delta$). Correlated
phases would already emerge when (see Fig. 1(a))

$\displaystyle U\sim W\ll\Delta<W_{0}\quad,$ (1)

indicating the necessity for low energy effective models obtained by
projecting both the kinetic and interacting parts of the Hamiltonian on to
only the narrow bands.

A key consequence of non-trivial topology for a given band is the well-known
notion of topological obstruction Brouder  _et al._ (2007); Marzari  _et al._
(2012); Soluyanov and Vanderbilt (2012) which is a double-edged sword. While
it can substantially complicate the construction of reliable low-energy
lattice models, it also presents an intriguing playground to isolate and
examine the novel effects of topology Po  _et al._ (2018); Zhang  _et al._
(2019); Zhang and Senthil (2019); Verma  _et al._ (2021); Wang  _et al._
(2021) on well-known phases of matter previously obtained purely from trivial
bands.  These issues have yet to be considered for kagome metals Mielke (1991,
1992); Wen  _et al._ (2010); Feng  _et al._ (2021); Denner  _et al._ (2021);
Lin and Nandkishore (2021), but are expected to be important to their
understanding. They are also general issues that are broadly important to the
physics of electron correlations as it intersects with that of electronic
topology Paschen and Si (2021); Schaffer  _et al._ (2016).

In this Letter, we study the correlated phases of a kagome metal when its
topologically nontrivial bands cross the Fermi level.  Importantly, we advance
a non-perturbative framework that describes correlations in such topological
metals (TMs), especially in the regime of Eq. 1. The applicability of a low
energy effective lattice model in this regime with arbitrary partial fillings
distinguishes this work from earlier attempts (see Ref. Rachel (2018) for a
review) to study interactions in topological models. First, Eq. 1 demands a
non-perturbative treatment of correlations for experimentally relevant
parameter values of $U,W$ and $\Delta$, which we offer here. Second, our
treatment, applied in the context of partially filled Chern bands with
arbitrary fillings, seeks to uncover hitherto unknown topological imprints on
density wave orders. Finally, non-perturbative treatments of partially filled
Chern bands that yield correlated metallic ground states (as we will see
below) are exceedingly rare Rachel (2018); Fogler (2002). These features of
our work stand in stark contrast to earlier attempts at deriving fractional
Chern insulating phases at very specific fillings Tang  _et al._ (2011);
Neupert  _et al._ (2011); Sun  _et al._ (2011); Regnault and Bernevig (2011),
or mean-field orders which are unaffected by the topology of the underlying
bands in the continuum limit (for a review, see Ref. Fogler (2002)).

To this end, we examine an extended Hubbard model on the kagome lattice and
isolate a Kramers’ pair of bands with opposite Chern numbers crossing the
Fermi level. Using Wannier orbitals suitable for topological bands whose
nonzero Chern numbers add up to zero Soluyanov and Vanderbilt (2012), we
project the Coulomb interactions to these bands and construct an effective
lattice model. We identify a T-breaking density wave (DW${}_{\text{t}}$) metal
in the phase diagram, and discuss its implications for the metallic kagome
materials. Correlated metals of this sort have not been identified from
partially filled topological bands in non-perturbative settings Rachel (2018).
We further determine how the unusual nature of this phase is driven by the
topological nature of the underlying bands and why it is absent in analogous
phases derived from trivial bands. Our work highlights how the obstructive
power of topology can be harnessed to drive novel correlated metallic phases.

Extended Hubbard model and topological obstruction: The Hamiltonian is given
by $\mathscr{H}=\mathscr{H}_{0}+\mathscr{H}_{I}$ with

$\displaystyle\mathscr{H}_{0}$ $\displaystyle=$ $\displaystyle-t\sum_{\langle
ij\rangle\alpha\beta}c_{i\alpha}^{\dagger}c_{j\beta}+i\lambda_{1}\sum_{\langle
ij\rangle}\left(\textbf{E}_{ij}\times\textbf{R}_{ij}\right)\cdot
c_{i\alpha}^{\dagger}\bm{\sigma}c_{j\beta}$ $\displaystyle-
t_{2}\sum_{\langle\langle
ij\rangle\rangle\alpha\beta}c_{i\alpha}^{\dagger}c_{j\beta}+i\lambda_{2}\sum_{\langle\langle
ij\rangle\rangle}\left(\textbf{E}_{ij}\times\textbf{R}_{ij}\right)\cdot
c_{i\alpha}^{\dagger}\bm{\sigma}c_{j\beta}$ $\displaystyle\mathscr{H}_{I}$
$\displaystyle=$
$\displaystyle\sum_{ij\alpha\beta}U_{j,\alpha\beta}n_{i\alpha}n_{i+j\beta}\quad.$
(2)

The kinetic part, $\mathscr{H}_{0}$, describes electrons hopping on a kagome
lattice in the presence of spin-orbit couplings (SOC) Guo and Franz (2009);
Tang  _et al._ (2011). Here $t(t_{2})$ and $\lambda_{1}(\lambda_{2})$ are the
nearest neighbor (next-nearest neighbor) hopping and SOC respectively.
$\textbf{R}_{ij}$ is the displacement vector from site $i$ to site $j$ and
$\textbf{E}_{ij}$ is the electric field experienced along $\textbf{R}_{ij}$.
$c_{i\alpha}$ destroys an electron on site $i$ and internal quantum number
$\alpha$, which refers to both sub-lattice and spin indices. The interaction
part, $\mathscr{H}_{I}$, contains local interactions, including the on-site
Hubbard interaction $U$ and the nearest-neighbor (next-nearest neighbor)
density-density interactions $V$ ($V^{\prime}$). They are expressed in terms
of $U_{j,\alpha\beta}$ that couples the number operators $n_{i\alpha}$. We
also denote the three triangular lattice vectors
$\textbf{n}_{1}=2a(\frac{1}{2},\frac{\sqrt{3}}{2})$,
$\textbf{n}_{2}=2a(-\frac{1}{2},\frac{\sqrt{3}}{2})$, $\textbf{n}_{3}=2a(1,0)$
where $a$ is the nearest neighbor inter atomic spacing. For definiteness, we
set $t_{1}=1,t_{2}=-0.3,\lambda_{1}=0.28,\lambda_{2}=0.2$ Tang  _et al._
(2011). These parameters give rise to the non-trivial topology in the band
structure (see Fig. 1(a)) and splits the spectrum into three pairs of
energetically well-separated degenerate bands with overall bandwidth
$W_{0}\sim 6$ and inter-band gaps of order $\Delta\sim 2.33$ (between the
middle and bottom bands). Each pair at the top and bottom has Chern numbers
$\pm 1$ while the middle pair has zero Chern number each. The spinful
$\mathbb{Z}_{2}$ TM is formed by partially filling the lowest pair of
degenerate bands that have a small bandwidth $W\sim 0.04$. To study the
correlation effect in the regime of Eq. 1, a reliable low energy effective
lattice model with exponentially decaying projected interactions is essential.
Topological obstruction complicates the construction of such a model as the
projection hinges on the existence of exponentially localized Wannier
functions. The latter are impossible when a Bloch discontinuity exists in the
BZ that is incurable by a unitary gauge transformation of the band(s).

However, for a subspace of low-energy bands whose sum of Chern numbers
vanishes, there exists a unitary transformation ($U_{T}$) that cures all
discontinuities of the relevant Bloch functions while also respecting their
reciprocal lattice vector periodicity Soluyanov and Vanderbilt (2011, 2012).
Our immediate goal, therefore, is to evaluate $U_{T}$ that “smoothens” out the
lowest pair of degenerate bands lying within an energy window $W$ such that
exponentially localized Wannier orbitals can be obtained while also preserving
the Bravais periodicity.

Figure 1: (a) Plot of the non-interacting kagome bands with SOC. The effective
model is projected on to the lowest pair of bands (blue rectangle) which have
equal and opposite Chern numbers $C=\pm 1$. (b) Exponentially localized
Wannier orbitals (WOs) (blue and red) that no longer share the same Wannier
centers (cyan dots) due to non-trivial topology.

The procedure for evaluating $U_{T}$ using the parallel transport gauge is
briefly summarized in the Supplemental Material (SM) Sec. A.

Wannier orbitals: We denote the final smoothened Bloch states (Figs. 4, 5, 7
of SM) as $|u_{f1}\rangle,|u_{f2}\rangle$, obtained from the initial states
$|u_{1}\rangle$, $|u_{2}\rangle$ with Chern numbers $\pm 1$. The
transformation that achieves this is $U_{T}\equiv U_{H}$, where $U_{H}$ is
defined as the matrix of eigenstates of the Haldane model also properly
smoothened on a cylinder.  In Fig. 3 of SM the geometric effect of applying
$U_{H}$ on the Wannier orbitals of $|u_{1}\rangle,|u_{2}\rangle$ is depicted.
Before the action of $U_{H}$, the two Wannier orbitals share the same Wannier
center, have equal onsite energies, and the corresponding Bloch states satisfy
the Kramers’ relations,
$T|u_{1}\rangle_{\textbf{k}}=|u_{2}\rangle_{-\textbf{k}}$ and
$T|u_{2}\rangle_{\textbf{k}}=-|u_{1}\rangle_{-\textbf{k}}$, where $T$ is the
time reversal operator. After the application of $U_{H}$,
$|u_{f1}\rangle,|u_{f2}\rangle$ no longer satisfy the relations and hence do
not form a Kramers’ pair. The two corresponding Wannier orbitals have
different Wannier centers, as shown in SM Fig. 3 (right panel) or Fig. 1(b),
but are smooth in the momentum space and exponentially localized in the real
space.

Denoting the operators for the Wannier orbitals in the new basis as
$b_{\textbf{k}\mu}$ ($\mu=1,2$), the following transformation properties under
time-reversal hold

$\displaystyle
T\hat{\Psi}_{\textbf{k}}T^{-1}=\hat{\mathscr{U}}(\textbf{k})\hat{\Psi}_{-\textbf{k}}\quad.$
(3)

Here $\hat{\Psi}_{\textbf{k}}$ is a column vector of the operators
$b_{\textbf{k}\mu}$ and $\hat{\mathscr{U}}(\textbf{k})$ is a unitary matrix
formed from sums of products of $U_{H}$ matrix elements (see SM Sec. A). Due
to the aforementioned properties, the set of Bloch states
$|u_{f1}\rangle,|u_{f2}\rangle$ now forms a convenient basis Foo (a) to write
an effective low-energy model with a minimal set of interaction and hopping
parameters that respect translations. We also note that despite
$|u_{f1}\rangle,|u_{f2}\rangle$ having zero Chern numbers, the topological
imprints of the original $\mathbb{Z}_{2}$ TM are now encoded in the lack of a
common Wannier center between the Wannier orbitals and revised definition of
the time-reversal operator Eq. 3. Demanding time-reversal symmetry in the
effective two-orbital Hamiltonian leads to additional non-local terms driven
purely by the topology of the underlying bands and absent for trivial bands.

Projected lattice Hamiltonian: We are now equipped to project the local
interactions $U$, $V$ and $V^{\prime}$ of the original model, Eq. 2, onto the
bands $|u_{f1}\rangle,|u_{f2}\rangle$. We denote $b_{i\mu}$ as the operator
that destroys an electron at site $i$ and Wannier orbital $\mu=1,2$ (WO 1, WO
2 for brevity). The total effective two orbital Hamiltonian ($H$) after
projection is written as a sum of non-interacting band term ($H_{0}$) and
interactions ($H_{I}$) given by

$\displaystyle H$ $\displaystyle=$ $\displaystyle H_{0}+H_{I}$ (4)
$\displaystyle H_{0}$ $\displaystyle=$
$\displaystyle\sum_{ij}\sum_{\mu\nu}b_{i\mu}^{\dagger}t_{ij}^{\mu\nu}b_{j\nu}$
$\displaystyle H_{I}$ $\displaystyle=$
$\displaystyle\sum_{ijkl}\sum_{mu\mu^{\prime}\nu\nu^{\prime}}u_{\mu\mu^{\prime}\nu\nu^{\prime}}(j,k,l)b_{i\mu}^{\dagger}b_{i+j\mu^{\prime}}b_{i+k\nu}^{\dagger}b_{i+l\nu^{\prime}}\quad.$

Here $t_{ij}^{\mu\nu}$ and $u_{\mu\mu^{\prime}\nu\nu^{\prime}}(j,k,l)$ are
respectively the hopping parameters and interaction matrix elements, which are
given in the SM (Tables I-III).

Several points are in order. First, in writing the interaction terms,
transformation properties of the individual WOs under rotations and
reflections must be considered in addition to the Bravais lattice symmetries.
These properties are especially non-trivial since the two WOs no longer share
the same Wannier center. All symmetries of the kagome lattice are respected
once the transformation properties of individual WOs are taken into account.
Second, although the two WOs now no longer form a Kramers’ pair, the
interaction Hamiltonian preserves time-reversal symmetry. The non-trivial
effect of topology is now manifest in the non-local terms that follow from
time-reversal invariance as well as the lack of a common center for the two
WOs. With respect to the hopping parameters, the low-energy band structure can
already be reproduced (see SM Fig. 6) from the effective model by truncating
hopping parameters of the two WOs to the third neighbors.

Figure 2:  Phase diagram of the model Hamiltonian in Eq. 4 expressed in terms
of the interaction parameters of the original model, $U$ and $V^{\prime}$, in
the unit of bandwidth $W$ of the projected bands. The blue and orange  lines
denote first-order and second-order  transitions respectively.  The phase
diagram  involves topological metal (TM), $T$-breaking density wave order
(DW${}_{\text{t}}$), and Mott insulator (MI). In terms of the effective
interactions of Eq. 4 (see Tables II and III, SM), the threshold values are
$\sim 0.35$ ($\sim 0.14$) along the horizontal (vertical) axis.

Phase diagram and the nature of the DW${}_{\text{t}}$ order: We investigate
the correlation effects via a $U(1)$ slave-spin method Yu and Si (2012) (see
SM, Sec. E). In the $U(1)$ slave spin method, we rewrite the electron
operators as $b_{i\mu}=S^{-}_{i\mu}f_{i\mu}$ where $f$ is the fermion operator
and $S^{-}$ is the spin operator. The original Hilbert space can be mapped
onto an extended space of slave spins and $f$ fermions according to the
correspondence: $|1\rangle_{b}=|1\rangle_{f}|\uparrow\rangle_{S}$
$|0\rangle_{b}=|0\rangle_{f}|\downarrow\rangle_{S}$. To project out the
unphysical degree of freedom, we enforce the constraint:
$S^{z}_{i\mu}+\frac{1}{2}=f_{i\mu}^{\dagger}f_{i\mu}$. The slave spins carry
the $U(1)$ charge which allows us to rewrite the four-WO interaction terms in
Eq. 4 as
$b_{i\mu}^{\dagger}b_{i\mu}b_{j\mu^{\prime}}b_{j\mu^{\prime}}=(S_{i\mu}^{z}+\frac{1}{2})(S_{j\mu^{\prime}}^{z}+\frac{1}{2}).$
We then perform a Hubbard-Stratonovich decoupling of the spin terms above and
solve the model at the saddle point level. The full Hamiltonian written in the
slave-spin basis along with the self-consistent parameters is given in Sec. E
of SM . We note that, due to the offset in the Wannier centers (c.f. Eq. 14,
SM), the nearest neighbor interaction $V$ plays a similar role as $U$ in the
effective model and, in particular, its tendency to drive a density wave order
with an ordering wave vector $K$ is reduced. Instead, $V$ only modifies the
phase boundaries, so we have set it to zero for convenience.

The phase diagram at half-filling is shown in Fig. 2. The horizontal and
vertical axes are marked by the onsite and next-nearest neighbor interactions,
$U$ and $V^{\prime}$ respectively. The phase diagram contains three phases: a
TM, Mott insulator (MI), and a density wave phase (DW${}_{\text{t}}$), which
has an ordering wave vector $M$. There are also three phase boundaries, and a
tri-critical phase coexistence occurs around $(U/W,V^{\prime}/W)\sim(6,2.3)$.
In the TM and DW${}_{\text{t}}$ phases, the quasi-particle weights remain
nonzero and the system is metallic. In the MI phase, the quasi-particle
weights vanish due to the interactions, and the electrons are localized. The
Mott phase is also spontaneously T-breaking and characterized by a fully
polarized ferro-orbital order in the WO basis which yields an in-plane spin
order in the original kagome basis of Eq. 2.

We now turn to discussing the nature of the DW${}_{\text{t}}$ phase.  This
phase is a charge order when viewed in the WO basis with a modulation vector
at the $M$ point. This is interesting since, in the absence of topology, a
Kramers pair of WOs share the same Wannier center and a minimization of the
onsite Coulomb repulsion would have instead yielded a (pure) spin order. In
the current situation, due to the lack of a common center from the original
topological bands, the non-local $V^{\prime}$ interaction instead drives a
charge order at $M$ point. Now when viewed in the original kagome basis, the
DWt contains a “collinear charge order” – a phase where the onsite charge
density is modulated along the M direction; i.e., it acquires a gradient along
one of the crystal axis direction but not the other – that is entwined with a
collinear in-plane spin order by the $\mathbb{Z}_{2}$ topology. Due to the
coexistence of the collinear charge and spin orders, the DW${}_{\text{t}}$
phase is naturally T-breaking. Such a property is absent in correlated phases
derived from trivial bands. To see this, we write the charge operator and spin
operator along $x$ direction in the kagome basis as
$\phi_{\textbf{q}}^{c}=\sum_{\alpha}c^{\dagger}_{\textbf{k}\alpha}c_{\textbf{k}+\textbf{q}\alpha}$
and
$\phi_{\textbf{q}}^{x}=\sum_{\alpha\gamma}\sigma^{x}_{\alpha\gamma}c^{\dagger}_{\textbf{k}\alpha}c_{\textbf{k}+\textbf{q}\gamma}$
respectively. Switching to WO basis we obtain (see Sec. D of SM)

$\displaystyle\phi_{q}^{c}$ $\displaystyle\sim$ $\displaystyle
F(\textbf{n}_{1},\textbf{n}_{2})A^{2}(n_{1,q}+n_{2,q})+G(\textbf{n}_{1},\textbf{n}_{2})A^{2}(n_{1,q}-n_{2,q})\quad,$
$\displaystyle\phi_{q}^{x}$ $\displaystyle\sim$ $\displaystyle
G(\textbf{n}_{1},\textbf{n}_{2})A^{2}(n_{1,q}+n_{2,q})+F(\textbf{n}_{1},\textbf{n}_{2})A^{2}(n_{1,q}-n_{2,q})\quad.$

where $A\sim 0.5$ is the overlap constant, $n_{\textbf{q}j}$ is the density
operator for WO $j$ and
$F(\textbf{n}_{1},\textbf{n}_{2})=\frac{4+e^{iq\textbf{n}_{1}}+e^{-iq\textbf{n}_{2}}}{2}$,
$G(\textbf{n}_{1},\textbf{n}_{2})=\frac{2-e^{iq\textbf{n}_{1}}-e^{-iq\textbf{n}_{2}}}{2}$.
Consider the DW${}_{\text{t}}$ appearing in the phase diagram Fig. 2. This
phase is a charge order in the WO basis with
$n_{\textbf{q}1}+n_{\textbf{q}2}\sim\eta\delta_{\textbf{q},M}$ and
$n_{\textbf{q}1}-n_{\textbf{q}2}\sim 0$. Switching to the original kagome
basis, it contributes to both charge and spin channels. A real space
illustration of the intertwined nature of charge and spin degrees of freedom
(defined with respect to the original kagome operators) is shown in Fig. 8 of
SM. The $\textbf{n}_{1}$ and $-\textbf{n}_{2}$ vectors in the phase factors of
Eq. Electron correlations and $T$-breaking density wave order in a
$\mathbb{Z}_{2}$ kagome metal reflect the difference in the Wannier centers of
the two WOs (see SM, Sec. C). Had they vanished, as for the WOs of
topologically trivial bands, the mixing of the spin order into the charge
order would have been absent, despite a non-zero SOC Foo (b).  Hence this
property is a direct consequence of the non-trivial $\mathbb{Z}_{2}$ topology
of the kagome Hamiltonian.

Discussions: Several remarks are in order. First, the DW${}_{\text{t}}$ order
is collinear and has an ordering wave vector $M$. At this level of energetics,
all three $M$ point orders in the BZ are feasible and hence degenerate.
Further breaking of this degeneracy can occur once perturbations from coupling
to the lattice via structural distortions is accounted for. In particular,
such subleading energetics can lock the system into a 3$Q$ order, which has
the $2\times 2$ real space pattern as observed in AV3Sb5 Jiang  _et al._
(2021); Zhao  _et al._ (2021). We can additionally expect such an order to
form a composite field that serves as an accompanying $C_{3}$-breaking nematic
order parameter, as has been found in recent experiments in KV3Sb5 Li  _et
al._ (2021b) and studied in the context of the moire systems Chen  _et al._
(2020). We reserve a systematic study of the nematicity for a later work.

Second, we have considered a regime of parameters which facilitates a
controlled theoretical analysis, with a particular form for SOC. Because the
DW${}_{\text{t}}$ order appears as a phase, it is expected to be robust over
an extended range of parameters and SOC types. Thus, even though in
$3d$-electron systems such as TmXn (T= Fe, Co and X= Sn, Ge) and $A$V3Sb5
($A=$ Cs, K, Rb), the SOC is typically several percents of the overall
bandwidth, we expect DW${}_{\text{t}}$ to be relevant. Finally, due to the
involvement of Haldane eigenstates in the construction of the Hamiltonian Eq.
4, for ground states that feature non-trivial boundary physics one must treat
the boundary physics by reverting to the original kagome basis.

Implications for Experiments: Several correlated phases Jiang  _et al._
(2021); Zhao  _et al._ (2021); Zhou  _et al._ (2021); Liang  _et al._ (2021);
Uykur  _et al._ (2021); Li  _et al._ (2021a); Hu  _et al._ (2021) have been
observed in $A$V3Sb5. These systems have topological bands near their Fermi
level, a key ingredient for the relevance of the DW${}_{\text{t}}$ phase.
However,  the width of these topological bands  is intermediate and the
strength of the correlations appears to be relatively weak, which suggests
that the mixing of the spin order in the underlying charge density wave order
will be relatively weak. These features are qualitatively consistent with the
experimental observations. The charge order is indeed centered at the $M$
point of the BZ Jiang  _et al._ (2021); Zhao  _et al._ (2021); Zhou  _et al._
(2021); Li  _et al._ (2021a). Both scanning tunneling microscopy (STM) Jiang
_et al._ (2021) and anomalous Hall transport Yang  _et al._ (2020)
measurements find evidence of chiral topological charge order that breaks
time-reversal symmetry. In addition, $\mu$SR has so far found no evidence of
static moments Kenney  _et al._ (2021). Thus, any spin order must be below the
$\mu$SR resolution limit  (see, however, “Note added” below). In the kagome
lattice binary TmXn (T= Fe, Co and X= Sn, Ge) compounds Ye  _et al._ (2018);
Liu  _et al._ (2020); Lin  _et al._ (2018); Liu  _et al._ (2018), flat bands
are located about $\lesssim 0.3$ eV from the Fermi level Kang  _et al._
(2020a, b); Liu  _et al._ (2020), and the topological nature of the Fermi-
level-crossing bands remains to be established. Because the correlation
strength here is expected to be considerably larger than the $0.3$ eV flat-
band energy scale, we expect the latter to significantly influence the
mechanism and nature of the correlated phases. Our work motivates the search
for T-breaking charge-density-wave phases in the TmXn and related systems.

Conclusions: We studied correlation effects of a kagome metal when a pair of
$Z_{2}$ topological bands with nonzero but opposite Chern numbers cross the
Fermi level. Using a time-reversal symmetric lattice description, we
constructed an effective lattice model and determined its correlated phase
diagram. We identified a density-wave order that breaks time-reversal symmetry
(DW${}_{\text{t}}$) in the phase diagram. The phase originates from a Coulomb
avoidance in a way that respects the topology of the underlying bands. We
expect this mechanism to operate in broader contexts of correlated topological
systems. The T-breaking density-wave order has important implications for
understanding rapidly emerging experiments in the ternary kagome metals. Such
phases may also be explored by future experiments in the binary kagome systems
as well as in other correlated metals whose crystalline lattice nurtures
topological bands.

###### Acknowledgements.

Acknowledgements: We thank P. C. Dai,  Z. Guguchia and M. Yi for useful
discussions. This work has in part been supported by the U.S. Department of
Energy, Office of Science, Basic Energy Sciences, under Award No. DE-SC0018197
and the Robert A. Welch Foundation Grant No. C-1411 (Q.S.). Q.S. acknowledges
the hospitality of the Aspen Center for Physics, which is supported by NSF
grant No. PHY-1607611.

Note added: After this manuscript became available on the arXiv, two new
$\mu$SR au2  _et al._ (2021); Yu  _et al._ (2021) and polar Kerr Wu  _et al._
(2021) measurements found evidence of broken time-reversal symmetry in the
density wave phase of ternary Kagome systems. The $\mu$SR data detect both in-
plane and out-of-plane components of the broken time reversal symmetry
supporting our findings.

$\oplus$ These authors contributed equally to this work.
$\dagger$<EMAIL_ADDRESS>
## References

  * Shankar (1994) R. Shankar, “Renormalization-group approach to interacting fermions,” Reviews of Modern Physics 66, 129 (1994).
  * Paschen and Si (2021) Silke Paschen and Qimiao Si, “Quantum phases driven by strong correlations,” Nature Reviews Physics 3, 9 (2021).
  * Imada  _et al._ (1998) Masatoshi Imada, Atsushi Fujimori, and Yoshinori Tokura, “Metal-insulator transitions,” Rev. Mod. Phys. 70, 1039–1263 (1998).
  * Georges  _et al._ (1996) Antoine Georges, Gabriel Kotliar, Werner Krauth, and Marcelo J Rozenberg, “Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions,” Reviews of Modern Physics 68, 13 (1996).
  * Cao  _et al._ (2018) Yuan Cao, Valla Fatemi, Ahmet Demir, Shiang Fang, Spencer L Tomarken, Jason Y Luo, Javier D Sanchez-Yamagishi, Kenji Watanabe, Takashi Taniguchi, Efthimios Kaxiras, _et al._ , “Correlated insulator behaviour at half-filling in magic-angle graphene superlattices,” Nature 556, 80–84 (2018).
  * Jiang  _et al._ (2019) Yuhang Jiang, Xinyuan Lai, Kenji Watanabe, Takashi Taniguchi, Kristjan Haule, Jinhai Mao, and Eva Y Andrei, “Charge order and broken rotational symmetry in magic-angle twisted bilayer graphene,” Nature 573, 91–95 (2019).
  * Sharpe  _et al._ (2019) Aaron L Sharpe, Eli J Fox, Arthur W Barnard, Joe Finney, Kenji Watanabe, Takashi Taniguchi, MA Kastner, and David Goldhaber-Gordon, “Emergent ferromagnetism near three-quarters filling in twisted bilayer graphene,” Science 365, 605–608 (2019).
  * Ye  _et al._ (2018) Linda Ye, Mingu Kang, Junwei Liu, Felix Von Cube, Christina R Wicker, Takehito Suzuki, Chris Jozwiak, Aaron Bostwick, Eli Rotenberg, David C Bell, _et al._ , “Massive dirac fermions in a ferromagnetic kagome metal,” Nature 555, 638–642 (2018).
  * Liu  _et al._ (2020) Zhonghao Liu, Man Li, Qi Wang, Guangwei Wang, Chenhaoping Wen, Kun Jiang, Xiangle Lu, Shichao Yan, Yaobo Huang, Dawei Shen, _et al._ , “Orbital-selective dirac fermions and extremely flat bands in frustrated kagome-lattice metal cosn,” Nature communications 11, 1–7 (2020).
  * Lin  _et al._ (2018) Zhiyong Lin, Jin-Ho Choi, Qiang Zhang, Wei Qin, Seho Yi, Pengdong Wang, Lin Li, Yifan Wang, Hui Zhang, Zhe Sun, _et al._ , “Flatbands and emergent ferromagnetic ordering in fe 3 sn 2 kagome lattices,” Physical review letters 121, 096401 (2018).
  * Liu  _et al._ (2018) Enke Liu, Yan Sun, Nitesh Kumar, Lukas Muechler, Aili Sun, Lin Jiao, Shuo-Ying Yang, Defa Liu, Aiji Liang, Qiunan Xu, _et al._ , “Giant anomalous hall effect in a ferromagnetic kagome-lattice semimetal,” Nature physics 14, 1125–1131 (2018).
  * Yao  _et al._ (2018) M Yao, H Lee, N Xu, Y Wang, J Ma, OV Yazyev, Y Xiong, M Shi, G Aeppli, and Y Soh, “Switchable Weyl nodes in topological Kagome ferromagnet Fe3Sn2,” arXiv:1810.01514 (2018).
  * Jiang  _et al._ (2021) Yu-Xiao Jiang, Jia-Xin Yin, M Michael Denner, Nana Shumiya, Brenden R Ortiz, Junyi He, Xiaoxiong Liu, Songtian S Zhang, Guoqing Chang, Ilya Belopolski, _et al._ , “Unconventional chiral charge order in kagome superconductor kv3sb5,” Nature Materials 20, 1353 (2021).
  * Zhao  _et al._ (2021) He Zhao, Hong Li, Brenden R Ortiz, Samuel ML Teicher, Taka Park, Mengxing Ye, Ziqiang Wang, Leon Balents, Stephen D Wilson, and Ilija Zeljkovic, “Cascade of correlated electron states in a kagome superconductor csv3sb5,” Nature (2021), https://doi.org/10.1038/s41586-021-03946-w.
  * Zhou  _et al._ (2021) Xiaoxiang Zhou, Yongkai Li, Xinwei Fan, Jiahao Hao, Yaomin Dai, Zhiwei Wang, Yugui Yao, and Hai-Hu Wen, “Origin of the Charge Density Wave in the Kagome Metal CsV3Sb5 as Revealed by Optical Spectroscopy,” arXiv e-prints , arXiv:2104.01015 (2021), arXiv:2104.01015 [cond-mat.supr-con] .
  * Liang  _et al._ (2021) Zuowei Liang, Xingyuan Hou, Wanru Ma, Fan Zhang, Ping Wu, Zongyuan Zhang, Fanghang Yu, J-J Ying, Kun Jiang, Lei Shan, _et al._ , “Three-dimensional charge density wave and robust zero-bias conductance peak inside the superconducting vortex core of a kagome superconductor csv $\\_3$ sb $\\_5$,” arXiv preprint arXiv:2103.04760 (2021).
  * Uykur  _et al._ (2021) E Uykur, BR Ortiz, SD Wilson, M Dressel, and AA Tsirlin, “Optical detection of charge-density-wave instability in the non-magnetic kagome metal kv $\\_3$ sb $\\_5$,” arXiv preprint arXiv:2103.07912 (2021).
  * Li  _et al._ (2021a) HX Li, TT Zhang, Y-Y Pai, C Marvinney, A Said, T Yilmaz, Q Yin, C Gong, Z Tu, E Vescovo, _et al._ , “Observation of unconventional charge density wave without acoustic phonon anomaly in kagome superconductors av3sb5 (a= rb, cs),” arXiv preprint arXiv:2103.09769 (2021a).
  * Hu  _et al._ (2021) Yong Hu, Samuel ML Teicher, Brenden R Ortiz, Yang Luo, Shuting Peng, Linwei Huai, JZ Ma, NC Plumb, Stephen D Wilson, J-F He, _et al._ , “Charge-order-assisted topological surface states and flat bands in the kagome superconductor csv $\\_3$ sb $\\_5$,” arXiv preprint arXiv:2104.12725 (2021).
  * Gilmutdinov  _et al._ (2021) I. F. Gilmutdinov, R. Schónemann, D. Vignolles, C. Proust, I. R. Mukhamedshin, L. Balicas, and H. Alloul, “ Evidence for the coexistence between linearly dispersing bands and strong electronic correlations in the underlying kagome lattice of Na2/3CoO2,” arXiv:2101.05252 (2021).
  * Kang  _et al._ (2020a) Mingu Kang, Linda Ye, Shiang Fang, Jhih-Shih You, Abe Levitan, Minyong Han, Jorge I Facio, Chris Jozwiak, Aaron Bostwick, Eli Rotenberg, _et al._ , “Dirac fermions and flat bands in the ideal kagome metal fesn,” Nature materials 19, 163–169 (2020a).
  * Kang  _et al._ (2020b) Mingu Kang, Shiang Fang, Linda Ye, Hoi Chun Po, Jonathan Denlinger, Chris Jozwiak, Aaron Bostwick, Eli Rotenberg, Efthimios Kaxiras, Joseph G Checkelsky, _et al._ , “Topological flat bands in frustrated kagome lattice cosn,” Nature communications 11, 1–9 (2020b).
  * Li  _et al._ (2018) Zhi Li, Jincheng Zhuang, Li Wang, Haifeng Feng, Qian Gao, Xun Xu, Weichang Hao, Xiaolin Wang, Chao Zhang, Kehui Wu, _et al._ , “Realization of flat band with possible nontrivial topology in electronic kagome lattice,” Science advances 4, eaau4511 (2018).
  * Ortiz  _et al._ (2020) Brenden R Ortiz, Samuel ML Teicher, Yong Hu, Julia L Zuo, Paul M Sarte, Emily C Schueller, AM Milinda Abeykoon, Matthew J Krogstad, Stephan Rosenkranz, Raymond Osborn, _et al._ , “Cs v 3 sb 5: A z 2 topological kagome metal with a superconducting ground state,” Physical Review Letters 125, 247002 (2020).
  * Ortiz  _et al._ (2021) Brenden R Ortiz, Paul M Sarte, Eric M Kenney, Michael J Graf, Samuel ML Teicher, Ram Seshadri, and Stephen D Wilson, “Superconductivity in the z 2 kagome metal kv 3 sb 5,” Physical Review Materials 5, 034801 (2021).
  * Fu  _et al._ (2021) Yang Fu, Ningning Zhao, Zheng Chen, Qiangwei Yin, Zhijun Tu, Chunsheng Gong, Chuanying Xi, Xiangde Zhu, Yuping Sun, Kai Liu, _et al._ , “Quantum transport evidence of topological band structures of kagome superconductor csv3sb5,” arXiv preprint arXiv:2104.08193 (2021).
  * Meier  _et al._ (2020) William R Meier, Mao-Hua Du, Satoshi Okamoto, Narayan Mohanta, Andrew F May, Michael A McGuire, Craig A Bridges, German D Samolyuk, and Brian C Sales, “Flat bands in the cosn-type compounds,” Physical Review B 102, 075148 (2020).
  * Lin  _et al._ (2020) Zhiyong Lin, Chongze Wang, Pengdong Wang, Seho Yi, Lin Li, Qiang Zhang, Yifan Wang, Zhongyi Wang, Hao Huang, Yan Sun, _et al._ , “Dirac fermions in antiferromagnetic fesn kagome lattices with combined space inversion and time-reversal symmetry,” Physical Review B 102, 155103 (2020).
  * Brouder  _et al._ (2007) Christian Brouder, Gianluca Panati, Matteo Calandra, Christophe Mourougane, and Nicola Marzari, “Exponential localization of wannier functions in insulators,” Physical review letters 98, 046402 (2007).
  * Marzari  _et al._ (2012) Nicola Marzari, Arash A Mostofi, Jonathan R Yates, Ivo Souza, and David Vanderbilt, “Maximally localized wannier functions: Theory and applications,” Reviews of Modern Physics 84, 1419 (2012).
  * Soluyanov and Vanderbilt (2012) Alexey A Soluyanov and David Vanderbilt, “Smooth gauge for topological insulators,” Physical Review B 85, 115415 (2012).
  * Po  _et al._ (2018) Hoi Chun Po, Liujun Zou, Ashvin Vishwanath, and T Senthil, “Origin of mott insulating behavior and superconductivity in twisted bilayer graphene,” Physical Review X 8, 031089 (2018).
  * Zhang  _et al._ (2019) Ya-Hui Zhang, Dan Mao, Yuan Cao, Pablo Jarillo-Herrero, and T Senthil, “Nearly flat chern bands in moiré superlattices,” Physical Review B 99, 075127 (2019).
  * Zhang and Senthil (2019) Ya-Hui Zhang and T Senthil, “Bridging hubbard model physics and quantum hall physics in trilayer graphene/h- bn moiré superlattice,” Physical Review B 99, 205150 (2019).
  * Verma  _et al._ (2021) Nishchhal Verma, Tamaghna Hazra, and Mohit Randeria, “Optical spectral weight, phase stiffness and tc bounds for trivial and topological flat band superconductors,” arXiv preprint arXiv:2103.08540 (2021).
  * Wang  _et al._ (2021) Jie Wang, Jennifer Cano, Andrew J Millis, Zhao Liu, and Bo Yang, “Exact landau level description of geometry and interaction in a flatband,” arXiv preprint arXiv:2105.07491 (2021).
  * Mielke (1991) Andreas Mielke, “Ferromagnetic ground states for the hubbard model on line graphs,” Journal of Physics A: Mathematical and General 24, L73 (1991).
  * Mielke (1992) A Mielke, “Exact ground states for the hubbard model on the kagome lattice,” Journal of Physics A: Mathematical and General 25, 4335 (1992).
  * Wen  _et al._ (2010) Jun Wen, Andreas Rüegg, C.-C. Joseph Wang, and Gregory A. Fiete, “Interaction-driven topological insulators on the kagome and the decorated honeycomb lattices,” Phys. Rev. B 82, 075125 (2010).
  * Feng  _et al._ (2021) Xilin Feng, Kun Jiang, Ziqiang Wang, and Jiangping Hu, “Chiral flux phase in the kagome superconductor av3sb5,” Science Bulletin (2021).
  * Denner  _et al._ (2021) M Michael Denner, Ronny Thomale, and Titus Neupert, “Analysis of charge order in the kagome metal $a$ v $\\_3$ sb $\\_5$($a=$ k, rb, cs),” arXiv preprint arXiv:2103.14045 (2021).
  * Lin and Nandkishore (2021) Yu-Ping Lin and Rahul M Nandkishore, “Complex charge density waves at van hove singularity on hexagonal lattices: Haldane-model phase diagram and potential realization in kagome metals $a$ v $\\_3$ sb $\\_5$,” arXiv preprint arXiv:2104.02725 (2021).
  * Schaffer  _et al._ (2016) Robert Schaffer, Eric Kin-Ho Lee, Bohm-Jung Yang, and Yong Baek Kim, “Recent progress on correlated electron systems with strong spin-orbit coupling,” Rep. Prog. Phys. 79, 094504 (2016).
  * Rachel (2018) Stephan Rachel, “Interacting topological insulators: a review,” Reports on Progress in Physics 81, 116501 (2018).
  * Fogler (2002) Michael M Fogler, “Stripe and bubble phases in quantum hall systems,” in  _High Magnetic Fields_ (Springer, 2002) pp. 98–138.
  * Tang  _et al._ (2011) Evelyn Tang, Jia-Wei Mei, and Xiao-Gang Wen, “High-temperature fractional quantum hall states,” Physical review letters 106, 236802 (2011).
  * Neupert  _et al._ (2011) Titus Neupert, Luiz Santos, Claudio Chamon, and Christopher Mudry, “Fractional quantum hall states at zero magnetic field,” Physical review letters 106, 236804 (2011).
  * Sun  _et al._ (2011) Kai Sun, Zhengcheng Gu, Hosho Katsura, and S Das Sarma, “Nearly flatbands with nontrivial topology,” Physical review letters 106, 236803 (2011).
  * Regnault and Bernevig (2011) Nicolas Regnault and B Andrei Bernevig, “Fractional chern insulator,” Physical Review X 1, 021014 (2011).
  * Guo and Franz (2009) H.-M. Guo and M. Franz, “Topological insulator on the kagome lattice,” Phys. Rev. B 80, 113102 (2009).
  * Soluyanov and Vanderbilt (2011) Alexey A Soluyanov and David Vanderbilt, “Wannier representation of z 2 topological insulators,” Physical Review B 83, 035108 (2011).
  * Foo (a) It is worth emphasizing that, since $U_{H}$ is a unitary gauge transformation that is smooth only on a cylinder due to the non-trivial topology of the Haldane model, observables of two Hamiltonians formed by the two sets of Wannier orbitals are only invariant in the absence of a boundary. Hence the predictions of the two models – provided one is able to construct a low energy Hamiltonian from $|u_{1}\rangle,|u_{2}\rangle$ – are the same as long as only bulk properties are probed (a).
  * Yu and Si (2012) Rong Yu and Qimiao Si, “$u(1)$ slave-spin theory and its application to mott transition in a multiorbital model for iron pnictides,” Phys. Rev. B 86, 085104 (2012).
  * Foo (b) Since the order parameter for our charge order is a scalar, it does not couple to spin degrees through SOC. This is unlike other vectorial density-wave order parameters studied in the context of loop current orders. (b).
  * Li  _et al._ (2021b) Hong Li, He Zhao, Brenden R Ortiz, Takamori Park, Mengxing Ye, Leon Balents, Ziqiang Wang, Stephen D Wilson, and Ilija Zeljkovic, “Rotation symmetry breaking in the normal state of a kagome superconductor kv3sb5,” arXiv preprint arXiv:2104.08209 (2021b).
  * Chen  _et al._ (2020) Lei Chen, Haoyu Hu, and Qimiao Si, “Fragile insulator and electronic nematicity in a graphene moiré system,” arXiv preprint arXiv:2007.06086 (2020).
  * Yang  _et al._ (2020) Shuo-Ying Yang, Yaojia Wang, Brenden R Ortiz, Defa Liu, Jacob Gayles, Elena Derunova, Rafael Gonzalez-Hernandez, Libor Šmejkal, Yulin Chen, Stuart SP Parkin, _et al._ , “Giant, unconventional anomalous hall effect in the metallic frustrated magnet candidate, kv3sb5,” Science advances 6, eabb6003 (2020).
  * Kenney  _et al._ (2021) Eric M Kenney, Brenden R Ortiz, Chennan Wang, Stephen D Wilson, and Michael Graf, “Absence of local moments in the kagome metal kv3sb5 as determined by muon spin spectroscopy,” Journal of Physics: Condensed Matter (2021).
  * au2  _et al._ (2021) C. Mielke III au2, D. Das, J. X. Yin, H. Liu, R. Gupta, C. N. Wang, Y. X. Jiang, M. Medarde, X. Wu, H. C. Lei, J. J. Chang, P. Dai, Q. Si, H. Miao, R. Thomale, T. Neupert, Y. Shi, R. Khasanov, M. Z. Hasan, H. Luetkens, and Z. Guguchia, “Time-reversal symmetry-breaking charge order in a correlated kagome superconductor,” (2021), arXiv:2106.13443 [cond-mat.mtrl-sci] .
  * Yu  _et al._ (2021) Li Yu, Chennan Wang, Yuhang Zhang, Mathias Sander, Shunli Ni, Zouyouwei Lu, Sheng Ma, Zhengguo Wang, Zhen Zhao, Hui Chen, Kun Jiang, Yan Zhang, Haitao Yang, Fang Zhou, Xiaoli Dong, Steven L. Johnson, Michael J. Graf, Jiangping Hu, Hong-Jun Gao, and Zhongxian Zhao, “Evidence of a hidden flux phase in the topological kagome metal csv3sb5,” (2021), arXiv:2107.10714 [cond-mat.supr-con] .
  * Wu  _et al._ (2021) Qiong Wu, Z. X. Wang, Q. M. Liu, R. S. Li, S. X. Xu, Q. W. Yin, C. S. Gong, Z. J. Tu, H. C. Lei, T. Dong, and N. L. Wang, “The large static and pump-probe kerr effect with two-fold rotation symmetry in kagome metal csv3sb5,” (2021), arXiv:2110.11306 [cond-mat.supr-con] .

##  I Supplemental Material

###  I.1 A. Parallel transport

Topological obstruction is the lack of a smooth Bloch representation for a
given band throughout the Brillouin zone (BZ) which also respects symmetries
of the Bravais lattice. Such obstruction, depending on the type, can
substantially complicate the construction of reliable low-energy lattice
models that are necessary to treat the correlated system’s need to avoid the
Coulomb repulsion. In situations which the obstruction can be resolved, like
the specific case we discuss in this work, a key task is to determine the
unitary band transformation matrix $U_{T}$ for each k point in the Brillouin
zone grid that untangles the topology. Below we outline the steps for
evaluating the form of $U_{T}$ and applying it to the Bloch states in our
problem.

Determining the form of $U_{T}$ for specific systems can be a non-trivial task
that depends on the lattice structure and the degree of Wannier localization
desired. The simplest procedure Soluyanov and Vanderbilt (2011) involves
projecting the $n$ Bloch states onto an ansatz of $n$ trial states and using
the determinant of the resulting $n\times n$ matrix to renormalize the
original Bloch functions. This procedure was used to derive reliable tight-
binding description in moire superlattices Po  _et al._ (2018); Zhang  _et
al._ (2019); Zhang and Senthil (2019) and recently implemented in Lieb
lattices Verma  _et al._ (2021) to derive bounds on superfluid stiffness. An
obvious difficulty with the projection method, however, is the requirement of
a suitable ansatz with the correct symmetries. The initial guess becomes
harder with more quantum numbers, increased size of Hilbert space and complex
lattice geometry. In the case of the kagome lattice, even if the correct
symmetry of the wave functions is used as a guess, non-singular wave function
normalization throughout the BZ is not assured. To circumvent this difficulty,
we instead use the more intuitive parallel transport gauge Soluyanov and
Vanderbilt (2012) from which $U_{T}$ can be obtained more readily.

In this section, we briefly outline the various steps involved in obtaining
the parallel transport gauge used for the kagome problem and refer the reader
to Ref. Soluyanov and Vanderbilt (2012) for further details. First, through a
linear transformation, the two low-lying Bloch functions,
$|u_{1}\rangle,|u_{2}\rangle$, with Chern numbers $\pm 1$ and bandwidth $W$
are redefined on a rectangular BZ where $\tilde{k}_{x}\in[0,2\pi]$,
$\tilde{k}_{y}\in[-\pi,\pi]$. Next, the states are parallel transported along
$\tilde{k}_{y}=0$ and the continuity between $\tilde{k}_{x}=0$ and
$\tilde{k}_{x}=2\pi$ is restored by redistributing the total phase along the
$\tilde{k}_{y}=0$ line. In the third step, for every $\tilde{k}_{x}$, two
directionally opposite parallel transports are carried out – one from
$(\tilde{k}_{x},0)$ to $(\tilde{k}_{x},\pi)$, and another from
$(\tilde{k}_{x},0)$ to $(\tilde{k}_{x},-\pi)$. In the next step, periodicity
along $\tilde{k}_{y}$ is restored at $\tilde{k}_{x}=0$ and
$\tilde{k}_{x}=2\pi$ while an off-diagonal unitary matrix, $V(\tilde{k}_{x})$,
connects the two states for every remaining $\tilde{k}_{x}$. In the
penultimate step, the two Bloch states are untangled by diagonalizing the
matrix $V(\tilde{k}_{x})$ without disturbing their smoothness in rest of the
BZ. At this stage, one is left with a Kramers’ pair of bands with Chern
numbers $\pm 1$ that is periodic with respect to reciprocal vectors along
$\tilde{k}_{x}$ but not along $\tilde{k}_{y}$ for each $\tilde{k}_{x}$ except
$\tilde{k}_{x}=0,2\pi$, i.e., they are smooth on a cylinder. In the final
step, the Bloch states are rotated into a gauge that is smooth on a torus by
using the eigenstates of the Haldane model ($U_{H}$) that have Chern numbers
$\pm 1$. The Haldane matrix $U_{H}$, properly smoothened on a cylinder, plays
the role of $U_{T}$ and rotates the original Bloch states into a trivial basis
$|u_{f1}\rangle,|u_{f2}\rangle$ with zero Chern numbers each. The topological
obstruction is thus resolved. Certain real and imaginary components of
$|u_{f1}\rangle$ and $|u_{f2}\rangle$ are plotted in Fig. 4 and the
exponential decay of the Wannier functions is shown in Fig. 5.

Figure 3: Wannier orbitals before (left) and after (right) Haldane rotation.
On the left panel, the Wannier orbitals share a common Wannier center and form
a Kramers’ pair. On the right panel, the Wannier orbitals lack a common
Wannier center and break Kramers’ relations, but their Bloch states instead
satisfy the more general Eq. 3.

Figure 4: Example plots of Bloch functions in the rescaled and rotated
rectangular Brillouin zone after the Wannierization process. (Top row) Real
(left) and imaginary (right) parts of fourth component of $|u_{f1}\rangle$.
(Bottom row) Same as top row but for $|u_{f2}\rangle$.

Figure 5: Plots of probability density along the six projections of the first
(left) and second (right) Bloch states as a function of distance from Wannier
center.

The linear transformation matrix $U_{T}$ is defined as the matrix ($U_{H}$) of
eigenvectors of the Haldane model properly smoothened on a cylinder according
to the prescription detailed above. We denote these matrix elements as
$g_{ij}(\textbf{k})$. Using this matrix for $U_{T}$

$\displaystyle U_{T}\equiv
U_{H}=\begin{bmatrix}g_{11}(\textbf{k})&g_{12}(\textbf{k})\\\
g_{21}(\textbf{k})&g_{22}(\textbf{k})\end{bmatrix},$ (5)

the unitary matrix $\mathscr{\hat{U}}(\textbf{k})$ which defines the action of
time reversal operator $T$ on the operators $d_{\textbf{k}\mu}$ in the main
text is given by

$\displaystyle\mathscr{\hat{U}}(\textbf{k})=\begin{bmatrix}\left(g_{11}^{*}(\textbf{k})g_{21}(-\textbf{k})+g_{12}^{*}(\textbf{k})g_{22}(-\textbf{k})\right)&\left(-g_{11}^{*}(\textbf{k})g_{11}(-\textbf{k})-g_{12}^{*}(\textbf{k})g_{12}(-\textbf{k})\right)\\\
\left(g_{21}^{*}(\textbf{k})g_{21}(-\textbf{k})+g_{22}^{*}(\textbf{k})g_{22}(-\textbf{k})\right)&\left(-g_{21}^{*}(\textbf{k})g_{11}(-\textbf{k})-g_{22}^{*}(\textbf{k})g_{12}(-\textbf{k})\right)\end{bmatrix}.$
(6)

###  I.2 B. Projected Hamiltonian in the Wannier basis

In this section, we derive the effective model in the WO basis. We start from
the extended Hubbard model with Coulomb interactions on the kagome lattice.

$\displaystyle\mathscr{H}$ $\displaystyle=$
$\displaystyle\mathscr{H}_{0}+\mathscr{H}_{I}$ $\displaystyle\mathscr{H}_{0}$
$\displaystyle=$
$\displaystyle\sum_{ij\alpha\beta}c_{i\alpha}^{\dagger}T_{ij,\alpha\beta}c_{j\beta}$
(7) $\displaystyle\mathscr{H}_{I}$ $\displaystyle=$
$\displaystyle\sum_{ij\alpha\beta}U_{j,\alpha\beta}n_{i\alpha}n_{i+j\beta}\quad.$
(8)

The hopping matrix can be diagonalized by $V_{k}^{\dagger}$ with
$V_{k}T_{k}V_{k}^{\dagger}=\text{diag}(\epsilon_{k,1},\epsilon_{k,2},...)$. We
then introduce the band basis $d_{\bm{k}}=V_{\textbf{k}}c_{\textbf{k}}$ and
projection operator
$P=\prod_{\textbf{k},\mu>2}(1-d^{\dagger}_{\textbf{k}\mu}d^{\dagger}_{\textbf{k}\mu})$.
It’s not difficult to show $Pd_{\textbf{k}\mu}P=d_{\textbf{k}\mu}$ for
$\mu=1,2$ and $Pd_{\textbf{k}\mu}P=0$ for $\mu>2$. The projection operator
would only keep the lowest two bands and project out the high energy degrees
of freedom. The effective Hamiltonian is defined as

$H=P\mathscr{H}P$ (9)

Since the lowest two bands have non-trivial topology, the Hamiltonian in the
band basis contains long-range interactions. In order to derive a local
Hamiltonian with exponentially decayed interactions, we introduce the WOs as
described in previous section $b_{\textbf{k}}=U_{H,\textbf{k}}d_{\textbf{k}}$.
In the WO basis, it’s not difficult to show:

$\displaystyle Pc_{\textbf{k}\mu}P$ $\displaystyle=$
$\displaystyle\sum_{\mu}(V_{\textbf{k}}^{\dagger}U_{H.\textbf{k}}^{\dagger})_{\mu\nu}b_{\textbf{k}\nu}$
(10)

Then we can replace all the terms in the Hamiltonian with their projected
formula and then the effective Hamiltonian reads

$\displaystyle H$ $\displaystyle=$ $\displaystyle H_{0}+H_{I}$ (11)
$\displaystyle H_{0}$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}ij\\\
\mu\nu\end{subarray}}b_{i\mu}^{\dagger}t_{ij}^{\mu\nu}b_{j\nu}$ $\displaystyle
H_{I}$ $\displaystyle=$ $\displaystyle\sum_{\begin{subarray}{c}ijkl\\\
\mu\mu^{\prime}\nu\nu^{\prime}\end{subarray}}u_{\mu\mu^{\prime}\nu\nu^{\prime}}(j,k,l)b_{i\mu}^{\dagger}b_{i+j\mu^{\prime}}b_{i+k\nu}^{\dagger}b_{i+l\nu^{\prime}}$

where

$\displaystyle t_{ij}^{\mu\nu}$ $\displaystyle=$
$\displaystyle\sum_{\mu^{\prime}\nu^{\prime}i^{\prime}j^{\prime}}T_{i^{\prime}j^{\prime}}^{\mu^{\prime}\nu^{\prime}}w_{\mu^{\prime}\mu,i^{\prime}-i}^{*}w_{\nu^{\prime}\nu,j^{\prime}-j}$
$\displaystyle u_{\mu\nu\mu^{\prime}\nu^{\prime}}(j,k,l)$ $\displaystyle=$
$\displaystyle\sum_{{e,i,\alpha\gamma}}U_{e,\alpha\gamma}w_{\alpha\mu,i}^{*}w_{\alpha\nu,i-j}w_{\gamma\mu^{\prime},i-k+e}^{*}w_{\gamma\nu^{\prime},i-l+e}\quad,$
(12)

and
$w_{\alpha\nu,i}=\frac{1}{N}\sum_{k}(V_{\textbf{k}}^{\dagger}U_{H,\textbf{k}}^{\dagger})_{\mu\nu}e^{i\textbf{k}\cdot\textbf{r}_{i}}$
denotes the real space wave functions of WOs.

Figure 6: A comparison of dispersions along high symmetry lines of the lowest
degenerate doublet of kagome bands. Dashed line is the original kagome band
and solid line is from the effective model with Wannier orbitals.

The hopping parameters of the original kagome lattice are:
$t_{1}=1,t_{2}=-0.3,\lambda_{1}=0.28,\lambda_{2}=0.2$. The effective hopping
takes form of $t_{ij}^{\mu\nu}=\delta_{\mu\nu}t_{j-i}$. We truncate to the
third-neighbor hopping and list the hopping strength in Table 1. In Fig. 6, we
compare the original band structures with the dispersion generated by Table 1
and a good consistency is seen.

$n,m$ | (0,1) | (1,0) | (0,-1) | (-1,0) | (-1,-1) | (1,1)
---|---|---|---|---|---|---
$t_{n\textbf{n}_{1}+m\textbf{n}_{2}}$ | -0.000698 | -0.000698 | -0.000698 | -0.000698 | -0.000698 | -0.000698
$n,m$ | (-2,0) | (2,0) | (0,2) | (0,-2) | (-2,-2) | (2,2)
$t_{n\textbf{n}_{1}+m\textbf{n}_{2}}$ | -0.001945 | -0.001945 | -0.001945 | -0.001945 | -0.001945 | -0.001945
$n,m$ | (-2,-1) | (-1,-2) | (-1,1) | (1,-1) | (1,2) | (2,1)
$t_{n\textbf{n}_{1}+m\textbf{n}_{2}}$ | -0.003026 | -0.003026 | -0.003026 | -0.003026 | -0.003026 | -0.003026
Table 1: Hopping parameters in the WO basis

For the interaction term, we consider the onsite-Hubbard interactions $U$, the
nearest-neighbor Coulomb interactions $V$ and next-nearest-neighbor Coulomb
interactions $V^{\prime}$ in the original kagome models. These would generate
various interaction terms in the effective model. The numerical values of each
term are shown in Table 2 and Table 3. For the terms generated by $U$, we omit
ones that are less than $10\%$ of the leading-order contributions. For the
terms generated by $V,V^{\prime}$, we only keep the leading order
contributions, The sub-leading part is about $30\%$ of the leading-order
contributions. Even though various types of interactions could be induced, all
the terms in the Hamiltonian are exponentially decayed.

$m,n$ | $\mu,\nu$ | $u_{\mu\mu\nu\nu}(0,m\textbf{n}_{1}+n\textbf{n}_{2},m\textbf{n}_{1}+n\textbf{n}_{2})$
---|---|---
0,0 | 0,1 | $0.078U+0.320V+0.146V^{\prime}$
-1,0 | 0,1 | $0.078U+0.320V+0.146V^{\prime}$
0,1 | 0,1 | $0.078U+0.320V+0.146V^{\prime}$
-1,-1 | 0,1 | $0.033V+0.142V^{\prime}$
1,1 | 0,1 | $0.033V+0.142V^{\prime}$
-1,1 | 0,1 | $0.033V+0.142V^{\prime}$
1,0 | 0,0 | $0.066V+0.142V^{\prime}$
0,1 | 0,0 | $0.066V+0.142V^{\prime}$
1,1 | 0,0 | $0.066V+0.142V^{\prime}$
1,0 | 1,1 | $0.066V+0.142V^{\prime}$
0,1 | 1,1 | $0.066V+0.142V^{\prime}$
1,1 | 1,1 | $0.066V+0.142V^{\prime}$
Table 2: Interaction terms in the WO basis (part I). $m,n,k,l$ | $\mu,\nu$ | $u_{\mu\mu\nu\nu}(0,0,m\textbf{a}_{1}+n\textbf{n}_{2},k\textbf{a}_{1}+l\textbf{n}_{2})$ | $m,n,k,l$ | $\mu,\nu$ | $u_{\mu\mu\nu\nu}(0,0,m\textbf{a}_{1}+n\textbf{n}_{2},k\textbf{a}_{1}+l\textbf{n}_{2})$
---|---|---|---|---|---
0,1,0,0 | 0,1 | $0.02U$ | 1,0,0,0 | 0,1 | $0.016U$
0,0,0,-1 | 1,0 | $0.02U$ | 0,0,-1,0 | 1,0 | $0.016U$
0,-1,0,0 | 1,0 | $0.02U$ | -1,0,0,0 | 1,0 | $0.016U$
0,0,0,1 | 1,0 | $0.02U$ | 0,0,1,0 | 1,0 | $0.016U$
0,0,0,1 | 0,1 | $0.02U$ | 0,0,1.0 | 0,1 | $0.016U$
0,1,0,0 | 0,1 | $0.02U$ | 1.0,0,0 | 0,1 | $0.016U$
0,0,0,1 | 0,1 | $0.02U$ | 0,0,1,0 | 0,1 | $0.016U$
0,-1,0,0 | 1,0 | $0.02U$ | -1,0,0,0 | 1,0 | $0.016U$
1,0,0,1 | 1,0 | $0.018U$ | 0,1,1,0 | 0,1 | $0.018U$
0,-1,1,0 | 1,0 | $0.018U$ | -1,0,0,1 | 0,1 | $0.018U$
1,0,0,-1 | 1,0 | $0.018U$ | 0,1,-1,0 | 0,1 | $0.018U$
0,-1,1,0 | 1,0 | $0.018U$ | -1,0,0,1 | 0,1 | $0.018U$
Table 3: Interaction terms in the WO basis (part II).

###  I.3 C. Real space Wannier functions and approximate Hamiltonian

Figure 7: Probability distributions of two maximally localized WOs.

We now turn to the real-space pattern of WOs and the approximate Hamiltonian.
After constructing the WOs from the smooth gauge, we also use the standard
maximal-localization procedure to make the Wannier orbitals even more
localized. The final amplitude of two WOs are shown in Fig. 7

Approximately, the leading contributions of each WO come from three sites
(marked as blue points in Fig. 7). It allows us to construct the approximate
formula of WOs

$\displaystyle b_{r,1}^{\dagger}$ $\displaystyle\sim
A(c_{r,a+}^{\dagger}+c_{r,b+}^{\dagger}+c_{r,c+}^{\dagger})$ (13)
$\displaystyle b_{r,2}^{\dagger}$ $\displaystyle\sim
A(c_{r+\textbf{n}_{1},a-}^{\dagger}+c_{r-\textbf{n}_{2},b-}^{\dagger}+c_{r,c-}^{\dagger})$

Here we are using the eigenstates of $S^{x}$:
$|\pm\rangle=\frac{1}{\sqrt{2}}|\uparrow\pm\downarrow\rangle$. $a,b,c$ denotes
three sub-lattices located at
$(-1/4,\sqrt{3}/4),(1/4,\sqrt{3}/4),(0,\sqrt{3}/2)$. Numerically, we find
$A\sim 0.5$ and the sub-leading contributions from each site is less than
$10\%$.

We want to emphasize that the two WOs don’t have a common center in the
topological non-trivial case (even after the maximal-localization procedure).
The sum of two Wannier centers $\bar{r}$ is a gauge-invariant quantity and
won’t change during the localization procedure. It lies at the center of the
hexagon (setting one center of the hexagon as the origin) in the topological
non-trivial case. If we maximally localized one of the Wannier orbitals, its
Wannier center lies at the center of the triangular. Consequently, the center
of another Wannier orbitals has to deviate from the first one in order to make
$\bar{r}$ stay at the center of the hexagon. However, in a topological trivial
case, we can always make two WOs be Kramers doublets; in turn, the two WOs
have a common center due to the time-reversal symmetry.

According to the approximate formula in Eq. 13, we can also write the
approximate interactions as:

$\displaystyle H_{I}$ $\displaystyle=$
$\displaystyle\sum_{ij}(U+4V+4V^{\prime})A^{4}b_{1i}^{\dagger}b_{1i}(\delta_{j,i}+\delta_{j,i-\bm{n}_{1}}+\delta_{j,i+\bm{n}_{2}})b_{2j}^{\dagger}b_{2j}$
(14) $\displaystyle+$ $\displaystyle\sum_{ij,\mu}(V+2V^{\prime})A^{4}b_{\mu
i}^{\dagger}b_{\mu
i}(\delta_{j,i+\bm{n}_{1}}+\delta_{j,i+\bm{n}_{2}}+\delta_{j,i+\bm{n}_{1}+\bm{n}_{2}})b_{\mu
j}^{\dagger}b_{\mu j}$ $\displaystyle+$
$\displaystyle\sum_{ij,\mu}2V^{\prime}A^{4}b_{1i}^{\dagger}b_{1i}(\delta_{j,i+\bm{n}_{1}+\bm{n}_{2}}+\delta_{j,i-\bm{n}_{1}-\bm{n}_{2}}+\delta_{j,i-\bm{n}_{1}+\bm{n}_{2}})b_{2j}^{\dagger}b_{2j}$

The above expressions are also consistent with the leading contributions from
our numerical results shown in Table 2 and Table 3.

We now discuss the effect of sub-leading terms that are not included in Eq.
14. We divide the sub-leading interactions into two parts: density-density and
normalized hopping. The density-density part takes the form of
$v_{ij,\mu\nu}n_{i\mu}n_{j\nu}$. We found such terms to favor the same order
as the leading interactions, This part would only modify the phase boundaries.
The normalized hopping term can be written as
$V_{ijk,\mu\nu\alpha}n_{i\mu}d_{j\nu}^{\dagger}d_{k\alpha}$. At the mean-field
level, we can decouple this term as $V_{ijk,\mu\nu\alpha}(\langle
n_{i\mu}\rangle d_{j\nu}^{\dagger}d_{k\alpha}+n_{i\mu}\langle
d_{j\nu}^{\dagger}d_{k\alpha}\rangle)$. It would normalize the hopping
strength and favors some bond orders. However, the average strength of the
normalized hopping term is about $80\%$ smaller than the dominant terms. In
summary, the approximate interactions in Eq. 14 already faithfully describe
the original kagome system.

###  I.4 D. Entwined order parameters

In this section, we discuss the connections between order parameters in WO
basis and kagome basis. The charge and spin order parameters along $x$
direction in the kagome basis can be written as

$\displaystyle\phi_{q}^{c}$ $\displaystyle=$ $\displaystyle\langle
Pc^{\dagger}_{k}c_{k+q}P\rangle\sim\langle\sum_{\alpha}w_{\alpha\nu
k}w^{*}_{\alpha\mu k+q}b_{k\nu}^{\dagger}b_{k+q\mu}\rangle$
$\displaystyle\phi_{q}^{x}$ $\displaystyle=$ $\displaystyle\langle
Pc^{\dagger}_{k}\sigma^{x}c_{k+q}P\rangle\sim\langle\sum_{\alpha}\sigma^{x}_{\alpha\gamma}w_{\alpha\nu
k}w^{*}_{\gamma\mu k+q}b_{k\nu}^{\dagger}b_{k+q\mu}\rangle$ (15)

Combining the above expressions with Eq. 13, we find the following approximate
formula as shown in the main text

$\displaystyle\phi_{q}^{c}$ $\displaystyle\sim$
$\displaystyle\frac{4+e^{iq\textbf{n}_{1}}+e^{-iq\textbf{n}_{2}}}{2}A^{2}(n_{1,q}+n_{2,q})+\frac{2-e^{iq\textbf{n}_{1}}-e^{-iq\textbf{n}_{2}}}{2}A^{2}(n_{1,q}-n_{2,q})$
$\displaystyle\phi_{q}^{x}$ $\displaystyle\sim$
$\displaystyle\frac{2-e^{iq\textbf{n}_{1}}-e^{-iq\textbf{n}_{2}}}{2}A^{2}(n_{1,q}+n_{2,q})+\frac{4+e^{iq\textbf{n}_{1}}+e^{-iq\textbf{n}_{2}}}{2}A^{2}(n_{1,q}-n_{2,q})$
(16)

This suggests a single charge order (or spin order) in the WO basis produces
entwined charge and spin orders in the kagome basis. In Fig. 8, we illustrate
this by plotting the real-space pattern of the DW${}_{\text{t}}$ phase (charge
order in the WO basis).

Figure 8: Real space pattern of the DW${}_{\text{t}}$. The color represents
the filling of each site and the arrow denotes the magnitude and direction of
in-plane local moment. We can observe a collinear order in the charge channel
that is decorated by a collinear order in the spin channel.

###  I.5 E. U(1) Slave-spin method

In the $U(1)$ slave spin method, we rewrite the electron operators as
$b_{i\mu}=S^{-}_{i\mu}f_{i\mu}$ where $f$ is the fermion operator and $S^{-}$
is the spin operator. The original Hilbert space can be mapped onto an
extended space of slave spins and $f$ fermions according to the following
correspondence:

$\displaystyle|1\rangle_{b}=|1\rangle_{f}|\uparrow\rangle_{S}$
$\displaystyle|0\rangle_{b}=|0\rangle_{f}|\downarrow\rangle_{S}$ (17)

To project out the unphysical degree of freedom, we enforce the constraint:
$S^{z}_{i\mu}+\frac{1}{2}=f_{i\mu}^{\dagger}f_{i\mu}$

The slave spins carry the $U(1)$ charge which allows us to rewrite the
interactions as

$\displaystyle
b_{i\mu}^{\dagger}b_{i\mu}b_{j\mu^{\prime}}b_{j\mu^{\prime}}=(S_{i\mu}^{z}+\frac{1}{2})(S_{j\mu^{\prime}}^{z}+\frac{1}{2})$
(18)

For the non-local interactions with $i\neq j$, we further decouple it via

$(S_{i\mu}^{z}+\frac{1}{2})(S_{j\mu^{\prime}}^{z}+\frac{1}{2})\sim-\langle
S_{i\mu}^{z}+\frac{1}{2}\rangle\langle
S_{j\mu^{\prime}}^{z}+\frac{1}{2}\rangle+\langle
S_{i\mu}^{z}+\frac{1}{2}\rangle(S_{j\mu^{\prime}}^{z}+\frac{1}{2})+(S_{i\mu}^{z}+\frac{1}{2})\langle
S_{j\mu^{\prime}}^{z}+\frac{1}{2}\rangle$ (19)

Then the effective Hamiltonian is

$\displaystyle H_{f}$
$\displaystyle=\sum_{ij}t_{ij}^{\mu\mu^{\prime}}\eta_{ij}f_{i\mu}^{\dagger}f_{j\mu^{\prime}}+\sum_{i}\lambda_{i\mu}f_{i\mu}^{\dagger}f_{i\mu}$
$\displaystyle H_{S}$
$\displaystyle=\sum_{ij,\mu\mu^{\prime}}t_{ij}^{\mu\mu^{\prime}}(\chi_{ij}^{\mu\mu^{\prime}}\langle
z^{\dagger}_{i\mu}\rangle
z_{j\mu^{\prime}}+\text{h.c.})-\sum_{i,\mu}\lambda_{i\mu}S_{i\mu}^{z}+\sum_{i}u_{1122}(0,0,0)(S_{i,1}^{z}+\frac{1}{2})(S_{i,2}^{z}+\frac{1}{2})+\sum_{i\mu}m_{i\mu}(S_{i\mu}^{z}+\frac{1}{2})$

where $\lambda_{\mu}$ is the Lagrangian multiplier and operators
$z_{i\mu}=\frac{1}{\sqrt{1/2+S_{i\mu}^{z}}}S^{-}_{i\mu}\frac{1}{\sqrt{1/2-S_{i\mu}^{z}}}$
are introduced to obtain the correct non-interacting limit. Other parameters
are determined self-consistently by

$\displaystyle\eta_{ij}^{\mu\mu^{\prime}}=\langle
z_{i\mu}^{\dagger}\rangle_{H_{S}}\langle z_{j\mu^{\prime}}\rangle_{H_{S}}$
$\displaystyle\chi_{ij}^{\mu\mu^{\prime}}=\langle
f_{i\mu}^{\dagger}f_{j\mu^{\prime}}\rangle_{H_{f}}$ $\displaystyle
m_{i\mu}=\sum_{j,\nu}u_{\mu\mu\nu\nu}(0,j,j)\langle
S^{z}_{i+j\nu}+\frac{1}{2}\rangle_{H_{S}}$ (21)

Effectively, our slave-spin method introduces a wave-function Ansatz:
$P_{G}|HF\rangle$ and variationally find the state with the lowest energy.
$P_{G}$ is a Gutwiller projector that disfavors double occupancy and
$|HF\rangle$ denotes the ground state of a free electrons model with Hartree-
Fock contributions.

At each $(U,V^{\prime})$, we randomly initialized 15 sets of variables
$(\eta_{ij}^{\mu\mu^{\prime}},\chi_{ij}^{\mu\mu^{\prime}},m_{i\mu},\lambda_{i\mu})$
and find self-consistent solutions of each initialization. Any type of three-
sublattice and two-sublattice orders are allowed in the calculations, and
other types of orders are not favored by interactions. The final ground state
corresponds to the solution with the lowest free energy. In addition, we set
the temperature to be $\beta W=1000$.