# Topological phases of extended Su-Schrieffer-Heeger-Hubbard model

Pei-Jie Chang  State Key Laboratory of Low-Dimensional Quantum Physics and
Department of Physics, Tsinghua University, Beijing 100084, China  Jinghui Pi
<EMAIL_ADDRESS>State Key Laboratory of Low-Dimensional Quantum Physics and
Department of Physics, Tsinghua University, Beijing 100084, China  Muxi Zheng
State Key Laboratory of Low-Dimensional Quantum Physics and Department of
Physics, Tsinghua University, Beijing 100084, China  Yu-Ting Lei  State Key
Laboratory of Low-Dimensional Quantum Physics and Department of Physics,
Tsinghua University, Beijing 100084, China  Xingbo Pan  State Key Laboratory
of Low-Dimensional Quantum Physics and Department of Physics, Tsinghua
University, Beijing 100084, China  Dong Ruan  State Key Laboratory of Low-
Dimensional Quantum Physics and Department of Physics, Tsinghua University,
Beijing 100084, China  Frontier Science Center for Quantum Information,
Tsinghua University, Beijing 100084, China  Gui-Lu Long
<EMAIL_ADDRESS>State Key Laboratory of Low-Dimensional Quantum Physics
and Department of Physics, Tsinghua University, Beijing 100084, China  Beijing
Academy of Quantum Information Sciences, Beijing 100193, China  Frontier
Science Center for Quantum Information, Tsinghua University, Beijing 100084,
China  Beijing National Research Center for Information Science and
Technology, Beijing 100084, China

###### Abstract

Despite extensive studies on the one-dimensional Su-Schrieffer-Heeger-Hubbard
(SSHH) model, the variant incorporating next-nearest neighbour hopping remains
largely unexplored. Here, we investigate the ground-state properties of this
extended SSHH model using the constrained-path auxiliary-field quantum Monte
Carlo (CP-AFQMC) method. We show that this model exhibits rich topological
phases, characterized by robust edge states against interaction. We quantify
the properties of these edge states by analyzing spin correlation and second-
order Rényi entanglement entropy. The system exhibits long-range spin
correlation and near-zero Rényi entropy at half-filling. Besides, there is a
long-range anti-ferromagnetic order at quarter-filling. Interestingly, an
external magnetic field disrupts this long-range anti-ferromagnetic order,
restoring long-range spin correlation and near-zero Rényi entropy.
Furthermore, our work provides a paradigm studying topological properties in
large interacting systems via the CP-AFQMC algorithm.

††preprint: APS/123-QED

##  I Introduction

Topological phases of matter are a central topic in modern condensed matter
physics due to their rich phenomenology and possible diverse applicability [1,
2, 3, 4].This includes applications in topological spintronics [5], quantum
metrology [6], and quantum computation [7, 8]. Within the independent electron
approximation, topological matter can be characterized by the topological band
theory [9, 10, 11, 12]. Specifically, the degeneracy of zero-energy edge modes
is directly associated with the topology of bulk band through the bulk-edge
correspondence [13, 14]. There are various experimental platforms to explore
the topological states of matter, such as ultracold atoms [15, 16],
artificially engineered solid systems including graphene [17, 18], arrays of
carbon monoxide molecules [19, 20], and exciton-polariton systems [21, 22].

However, in many experiments setups, the interaction between electrons
dominates over the hopping amplitude [23, 24, 25], rendering the independent-
electron approximation invalid. Topological descriptions of strongly
correlated systems are an emerging area of research with significant potential
[26, 27, 28]. In interacting systems, the bulk-edge correspondence is modified
as the relation between bulk topological property and the many-body, ground-
state degeneracy on the edge [29]. Researchers have also developed a range of
metrics to evaluate characterize topological edge states, including the Green
function method [30, 31, 32], entanglement entropy between topological edges
and bulk states [33, 34], and topological phase transition induced by magnetic
flux changes [35].

Su-Schrieffer-Heeger-Hubbard (SSHH) model with the nearest neighbour hopping
has been extensively studied. However, the one with next-nearest neighbour
hopping remains largely unexplored. Here, we discuss the ground-state
properties of a one-dimensional SSHH model with next-nearest neighbour
hopping. In the absence of Hubbard interaction, this model can be well
described by the single-particle theory and exhibits four distinct topological
phases ($\mathcal{W}=-1,0,1,2$) without breaking chiral symmetry [36, 37].
According to the bulk-edge correspondence, these topological phases are
characterized by the degeneracy of zero-energy edge modes. However, in the
presence of onsite Hubbard interaction, we should consider the relation
between bulk topological property and the many-body ground-state. We evaluate
long-range spin correlation and the second-order Rényi entropy of the ground
state, and determine the topological state of the system by measuring the
correlation between the boundary and bulk. The presence of Coulomb interaction
leads to a Mott gap at half-filling and dimerization at quarter-filling [38].
To mitigate the influence of finite site effects on our results, we employ the
constrained-path auxiliary-field quantum Monte Carlo (CP-AFQMC) algorithm.
Furthermore, the applicability of constrained-path method extends beyond the
half-filling case [34] where there is no sign problem. Our analysis reveals
that the system exhibits long-range spin correlation and near-zero Rényi
entropy at half-filling, indicative of a topological order state. Furthermore,
a long-range antiferromagnetic order emerges at quarter-filling. The
application of an external magnetic field disrupts this antiferromagnetic
order, leading to a restoration of long-range spin correlation and near-zero
Rényi entropy.

The rest of the paper is organized as follows: In Sec. II, we simulate the SSH
model incorporating next-nearest neighbour hopping and briefly review the
exact diagonalization and CP-AFQMC methods. In Sec. III, we discuss the exist
of long-range spin correlation and Rényi entanglement entropy at different
filling. A summary is given in Sec. IV. In the Appendix, we give result based
on exact diagonalization as benchmark and result of CP-AFQMC with different
parameters.

##  II MODEL AND METHOD

Figure 1: Su-Schrieffer-Heeger model with next-nearest neighbour hopping. (a)
Schematic diagram of hopping term. (b) Schematic diagram of wave function
rotation in pseudo-spin space. The solid point is the topological singular
point and $\mathcal{W}$ is corresponding winding number. Topological phase
diagram by calculating winding number with (c) $J_{11}=0,J_{1}=1$, (d)
$J_{11}=1,J_{1}=0$ under periodic boundary condition. (e), (f) Under open
boundary condition and along x(y) axis in (c), particle distribution of a
typical bulk state (diamond) and edge state (five-pointed star and inverted
triangle) for $\Delta_{J_{3}(J_{33})-J_{1}}=0.75$. We define that
$\Delta_{J_{3}-J_{1}}=(J_{3}-J_{1})/2$,
$\Delta_{J_{33}-J_{1}}=(J_{33}-J_{1})/2$ and
$\Delta_{J_{33}-J_{11}}=(J_{33}-J_{11})/2$. Inset: Eigen-energies calculated
for a chain with $N=20$ sites. The gapless band represents the topological
edge state.

###  II.1 Next-nearest neighbour hopping SSHH model

Let’s consider a SSHH model with next-nearest neighbour hopping. The
Hamiltonian is given by

$H=H_{0}+H_{U},$ (1)

with hopping term

$\displaystyle H_{0}=$
$\displaystyle-\sum_{i,\sigma}(J_{11}\hat{c}_{i,\sigma}^{A\dagger}\hat{c}_{i,\sigma}^{B}+J_{1}\hat{c}_{i,\sigma}^{B\dagger}\hat{c}_{i+1,\sigma}^{A}+J_{33}\hat{c}_{i,\sigma}^{A\dagger}\hat{c}_{i+1,\sigma}^{B}$
(2)
$\displaystyle+J_{3}\hat{c}_{i\sigma}^{B\dagger}\hat{c}_{i+2,\sigma}^{A}+h.c)\newline
,$

and onsite interaction term

$H_{U}=\sum_{i,s}U\hat{n}^{s}_{i\uparrow}\hat{n}^{s}_{i\downarrow},$ (3)

where $\hat{c}_{i\sigma}^{s\dagger}$ is the fermion creation operator of the
$i$-th site with sub-lattice label $s$ and spin label $\sigma$. The parameter
$J_{1}$ represents inter-cell hopping and $J_{11}$ represents intra-cell
hopping which are similar to SSH model, $J_{3}$ and $J_{33}$ are the next-
nearest neighbour hopping between A and B as shown in Fig. 1(a). $U$ is the
on-site interaction strength with particle number operator
$n^{s}_{i\sigma}=\hat{c}_{i\sigma}^{s\dagger}\hat{c}_{i\sigma}^{s}$.

We first discuss the topological phases of the SSH model with next-nearest
neighbour hopping. The Bloch Hamiltonian of this model is

$\mathcal{H}(k)=d_{x}(k)\sigma_{x}+d_{y}(k)\sigma_{y},$ (4)

which is obtained by the Fourier transform of $H_{0}$ under the periodic
boundary condition, where $d_{x}=-J_{11}-(J_{1}+J_{33})\cos k-J_{3}\cos 2k$
and $d_{y}=-J_{11}-(J_{1}-J_{33})\sin k-J_{3}\sin 2k$, $\sigma_{x(y)}$ are the
Pauli matrices.

The model (4) exhibits time-reversal symmetry $\mathcal{T}$, defined as
$\mathcal{T}\mathcal{H}(k)\mathcal{T}^{-1}=\mathcal{H}(-k)$. This symmetry
satisfies $\mathcal{T}^{2}=1$ and takes the form $\mathcal{T}=K$, where $K$ is
the complex conjugation operator. Furthermore, the model preserves chiral
symmetry $\mathcal{C}=\sigma_{z}$, which anticommutes with the Hamiltonian,
$\sigma_{z}\mathcal{H}=-\mathcal{H}\sigma_{z}$. Consequently, each eigenstate
$\ket{\psi}$ with energy $E$ possesses a corresponding partner state
$\sigma_{z}\ket{\psi}$ with energy $-E$. In addition, this model has particle-
hole symmetry $\mathcal{S}$, which is the combination of time-reversal
symmetry and chiral symmetry, $\mathcal{S}=\mathcal{T}\mathcal{C}$. This
symmetry also anticommutes with the Hamiltonian,
$\mathcal{S}\mathcal{H}=-\mathcal{H}\mathcal{S}$. Thus, according to the
tenfold way method, this model belong the BDI topological class as it
possesses the above three symmetry [39, 3].

To characterize the topological property of the interaction free model (4), we
can define the winding number $\mathcal{W}$ as follow

$\mathcal{W}=\frac{1}{2\pi}\int_{\mathrm{FBZ}}\frac{d_{x}\partial_{k}d_{y}-d_{y}\partial_{k}d_{x}}{d_{x}^{2}+d_{y}^{2}}dk,$
(5)

where the integral is a close loop with $k$ varying in the first Brillouin
zone (FBZ), namely $k\in[0,2\pi)$ as we set unit cell length $a=1$. We plot
the typical $\mathcal{W}$ in the parameter space spanned by $d_{x}$ and
$d_{y}$ in Fig. 1(b). This winding number can be associated to the Zak phase
[40, 35]

$\gamma=i\int_{\mathrm{FBZ}}\left\langle\psi|\partial_{k}|\psi\right\rangle
dk$ (6)

via $\gamma=\pi\mathcal{W}$, where
$\left|\psi\right\rangle=\frac{1}{\sqrt{2}}\left(\frac{d_{x}-id_{y}}{\sqrt{d_{x}^{2}+d_{y}^{2}}},-1\right)^{T}$
is the normalized Bloch eigenstate with eigenvalue
$\sqrt{d_{x}^{2}+d_{y}^{2}}$. If the next-nearest neighbour hopping
$J_{3}=J_{33}=0$, the model is the standard SSH model, which have winding
number $\mathcal{W}=0$ or $1$, corresponding to topological trivial phase or
topological nontrivial phase, respectively. When $J_{3}$ and $J_{33}$ is non-
zero, the phase diagram undergoes a modification, allowing $\mathcal{W}$ to
assume the additional values of $-1$ and $2$, as shown in Fig. 1(c) and 1(d).

According to the bulk-edge correspondence principle, the winding number
$\mathcal{W}$ defined by the bulk Hamiltonian (4) is directly related to the
topological edge states under open boundary condition. Specifically, at
$\mathcal{W}=0$, the system lacks topological edge states. Conversely, for
$\mathcal{W}=\pm 1$, two topological edge states with zero energy emerge at
different sites. When $\mathcal{W}=2$, the number of zero energy topological
edge states increases to four. We show the typical topological edge states at
$1_{st},3_{rd},(N-2)_{th},N_{th}$ sites with $\mathcal{W}=2$ in Fig. 1(e), and
topological edge states at $2_{nd},(N-1)_{th}$ sites with $\mathcal{W}=-1$ in
Fig. 1(f).

In the presence of finite on-site Hubbard interaction ($U\neq 0$), the
analytical solution of model (1) via the Bethe ansatz becomes intractable in
general [41]. In the flowing part, we investigate the effect of $U$ using
numerical simulation techniques. For clarity, the subsequent two sections
provide a concise overview of the two numerical methods employed in this work.

###  II.2 Exact diagonalization

Exact diagonalization (ED) serves as a benchmark for solving quantum mechanics
problems numerically. In theory, it provides a path to solve any problem given
sufficient computational resources. However, condensed matter lattice models
often present exponentially difficult complexities. For instance, the fermion
Hubbard model scales as $4^{N}$, where $N$ represents the number of lattice
sites. This rapid growth renders simulations of large-scale models
computationally prohibitive. Despite these limitations, ED’s accuracy and
reliability make it valuable for obtaining reliable results in small-scale
quantum simulations. We consider an open-boundary, one-dimensional chain with
$N=8$ sites. Conserved particle number guarantees a $U(1)$ symmetry in the
system. This symmetry allows for block-diagonalization of the Hamiltonian,
facilitating efficient diagonalization using the Lanczos algorithm. However,
ED scales exponentially, restricting its application to small systems and
inherently limiting studies of finite-size effects. To overcome this
limitation and access the thermodynamic limit, we employ the constrained-path
auxiliary-field quantum Monte Carlo method, detailed in the following section.

###  II.3 Constrained-path auxiliary-field quantum Monte Carlo

In this section, we give a brief introduction to constrained-path auxiliary-
field quantum Monte Carlo (CP-AFQMC), which proves to be a powerful tool for
solving the ground state problem in many-body systems [42, 43, 44, 45, 46].
Analogous to the power method for finding the dominant eigenvalue and
eigenvector of a matrix [47], iterative application of the imaginary time
evolution operator to an initial state $\ket{\psi_{0}}$ recovers the ground
state $|\psi_{g}\rangle$, provided the overlap is nonzero,

$|\psi_{g}\rangle\propto\mathop{lim}\limits_{\beta\to\infty}e^{-\beta
H}|\psi_{0}\rangle,\ if\ \langle\psi_{g}|\psi_{0}\rangle\neq 0.$ (7)

To simplify numerical simulations, we assume the initial state
$\ket{\psi_{0}}$ to be a single Slater determinant, as employed in Ref. [45].
Besides, as the edge states come in degenerate pairs, constructing an
appropriate $|\psi_{0}\rangle$ necessitates a suitable linear superposition of
these states. For instance, if the zeros energy edge state of single particle
states are $\ket{\phi_{1}}=(1,0,\cdots,0)^{T},\
\ket{\phi_{2}}=(0,\cdots,0,1)^{T},$ then the initial state $|\psi_{0}\rangle$
can be chosen to $\ket{\psi_{1}}=(\ket{\phi_{1}}+\ket{\phi_{2}})/\sqrt{2}$ or
$\ket{\psi_{2}}=(\ket{\phi_{1}}-\ket{\phi_{2}})/\sqrt{2}$.

Using Trotter decomposition, we can separate the imaginary time evolution
process into single-body and multi-body contributions

$e^{-\beta H}=(e^{-\tau H})^{n}=(e^{-\frac{1}{2}\tau H_{0}}e^{-\tau
H_{U}}e^{-\frac{1}{2}\tau H_{0}})^{n}+O(\tau^{2}),$ (8)

where $\beta=\tau n$ is the length of imaginary time and $\tau$ is one step of
imaginary time. In the calculation of AFQMC, we can simply calculate that a
single body operator on the Slater determinant, and the result can still be
expressed by another Slater determinant. As two-body terms are incompatible
with the Slater determinant formalism, then the Hubbard-Stratonovich
transformation is employed. This approach decouples the two-body Hamiltonian
into a superposition of single-body Hamiltonians by introducing an auxiliary
field [48]. Thus far, all operators have been represented as single-body
operators, facilitating efficient computations within Slater determinants. In
AFQMC, the wave function is represented as a linear combination (where is a
single) of Slater determinants (walkers). Eq. (7) is then represented as
random walks in the Slater determinant space by sampling the auxiliary fields.

A significant challenge in simulating quantum systems with the AFQMC method is
the sign problem. This arises from negative weights assigned to certain
configurations during the simulation, leading to a substantial increase in
variance and consequently unreliable results. To address this challenge, CP-
AFQMC utilizes a trial wave function, denoted as $|\psi_{T}\rangle$. A key
point is that walkers have a negative overlap with trial wave function are
excluded, while those with significant overlap are replicated for propagation.
Following extended imaginary time evolution, the converged ground state energy
and its associated wave function can be determined. A limitation of this
method arises from the systematic errors introduced into the simulation
results by the selection of $|\psi_{T}\rangle$. Optimizing $|\psi_{T}\rangle$
is an active research area within the CP-AFQMC framework [49, 50]. Here, we
choose the initial state $\ket{\psi_{0}}$ as the trial wave function. Our
primary focus is the characterization of edge states. Consequently, their
existence or absence plays a crucial role in our simulations. This
necessitates the implementation of a suitable linear superposition of
$\ket{\phi_{1}}$ and $\ket{\phi_{2}}$, as previously established.

Physical quantities can be obtained using the equal-time Green’s function,
formulated as the mixed estimator,

$G_{ij}=\langle\hat{c}_{j}^{\dagger}\hat{c}_{i}\rangle=\frac{\sum_{k}w_{k}\langle\psi_{T}|\hat{c}_{j}^{\dagger}\hat{c}_{i}|\psi_{k}\rangle}{\sum_{k}w_{k}\langle\psi_{T}|\psi_{k}\rangle},$
(9)

where $|\psi_{k}\rangle$ is the $k_{th}$ walker, $\omega_{k}$ is the
corresponding weight of the walker, and $|\psi_{T}\rangle$ is the trail wave
function where same as $\ket{\psi_{0}}$. Specifically, by employing the Wick’s
theorem [51], physical observations like spin correlation and Rényi
entanglement entropy can be calculated via the equal-time Green’s function.

##  III Result

Figure 2: The addition energy spectrum of the next-nearest neighbour hopping
Su-Schrieffer-Heeger-Hubbard model for a $N=8$ sites chain as a function of
Hubbard $U$ with (a) $\Delta_{J_{3}-J_{1}}=-0.75$, (b)
$\Delta_{J_{3}-J_{1}}=0.75$. The circled areas are the gapless states, and the
position circled in b is two lines, it is difficult to distinguish because the
values are very close. Figure 3: Population number of the edge states for
$N=8$ sites chain with $J_{11}=J_{33}=0$, and $n$ is the number of electrons.
(a)-(c) for $U=0$ and (d)-(f) for $U=10$. According to Fig. 1(e), the edge
states appear at the first boundary ($1_{st}$ and $N_{th}$) sites or the
third-boundary ($3_{rd}$ and $(N-2)_{th}$) sites. (a), (d) $n_{edge1}$ is the
sum of electron number on the $1_{st}$ and $N_{th}$ sites; (b), (e)
$n_{edge2}$ is the sum of electron number on the $3_{rd}$ and $(N-2)_{th}$
sites; (c), (f) $n_{edge3}$ is the sum of electron number on above four sites.
In the non-trivial phases without Hubbard $U$, the edge population increases
sharply at half-filling, while with Hubbard $U$ the edge population increases
sharply at quarter-filling and there-quarter-filling. And in the trivial
phase, the edge population increases gradually. There are some wrinkles on the
surface in (d)-(f), which is due to the finite-size effects.  Figure 4: CP-
AFQMC results of next-nearest neighbour hopping SSHH model with OBC at the
half-filling with $J_{11}=J_{33}=0$. (a) Spin correlation $\langle
S_{z,j}S_{z,k}\rangle$ with $\Delta_{J_{3}-J_{1}}=0.75$, topological edge
states at [$1_{st},3_{rd},(N-2)_{th},N_{th}$] sites. (b) Same as (a) with
$\Delta_{J_{3}-J_{1}}=0$ and there is a long-range anti-ferromagnetic order.
(c) Same as (a) with $\Delta_{J_{3}-J_{1}}=-0.75$, and topological edge states
at [$1_{st},N_{th}$] sites. (d) The entanglement Rényi entropy between
subsystem [$3_{rd},(N-2)_{th}$] sites and the remaining sites; (e) Same as (d)
but subsystem [$1_{st},N_{th}$] sites and (f) with subsystem
[$1_{st},3_{rd},(N-2)_{th},N_{th}$] sites. In (d)-(f), the position where the
transition occurs is closer to $\Delta_{J_{3}-J_{1}}=0$ than ED in Appendix C,
which means that the finite-size effects have less impact of the conclusion.
Figure 5: CP-AFQMC results same as Fig 4 but with $J_{1}=J_{3}=0$. (a) Spin
correlation $\langle S_{z,j}S_{z,k}\rangle$ with
$\Delta_{J_{33}-J_{11}}=0.75$, topological edge states at
[$2_{nd},(N-1)_{th}$] sites. (b) Same as (a) with $\Delta_{J_{33}-J_{11}}=0$
and there is a long-range anti-ferromagnetic order. (c) Same as (a) with
$\Delta_{J_{33}-J_{11}}=-0.75$, and there are bulk dimers without topological
edge states as shown in Fig 1(d) $\mathcal{W}=0$. (d) The entanglement Rényi
entropy between subsystem [$2_{nd},(N-1)_{th}$] sites and the remaining sites;
(e) Same as (d) but subsystem [$1_{st},N_{th}$] sites and (f) with subsystem
[$1_{st},2_{nd},(N-1)_{th},N_{th}$] sites. Due to the formation of dimers,
there will be a drop in Rényi entanglement entropy.  Figure 6: Exact
diagonalization results of next-nearest neighbour hopping SSHH model with OBC
at the quarter-filling with $J_{11}=J_{33}=0$ or $J_{1}=J_{3}=0$ for
(b)(d)(f). (a) Spin correlation $\langle S_{z,j}S_{z,k}\rangle$ with
$\Delta_{J_{3}-J_{1}}=-0.75,\ J_{11}=J_{33}=0$, topological edge states are
[$1_{st},N_{th}$] sites, and $\omega$ is degeneracy of ground state. (b) Same
as (a) with $\Delta_{J_{3}-J_{1}}=0.75$ and topological edge states are
[$1_{st},3_{rd},(N-2)_{th},N_{th}$] sites. (c) Rényi entanglement entropy
between the edges [$3_{rd},(N-2)_{th}$] sites and the bulk for various field
strength and $\Delta_{J_{3}-J_{1}}$. And each contour can separate with
different $S_{z}$, where $S_{z}=2$ means the maximally ferromagnetic ground
state. (d) Same as (c) but with topological [$2_{nd},(N-1)_{th}$] sites. (e)
Critical field strength at various Hubbard $U$ and $\Delta_{J_{3}-J_{1}}$. (f)
Same as (e) but changing with $\Delta_{J_{33}-J_{11}}$. If we do a $1D$ slice
for $\Delta_{J_{3}-J_{1}}=0.75$ in (e) or $\Delta_{J_{33}-J_{11}}=0.75$ in
(f), there will be a decrease in the critical field with increasing Hubbard
$U$. (a) and (b) for a $1D$ chain with $N=12$, and (c)-(f) for a 1D chain with
$N=8$.  Figure 7: CP-AFQMC results of next-nearest neighbour hopping SSHH
model with OBC at the quarter-filling with $J_{11}=J_{33}=0$, and $U=10$ for
(a)-(e). (a) Spin correlation $\langle S_{z,j}S_{z,k}\rangle$ with
$\Delta_{J_{3}-J_{1}}=-0.75$, topological edge states at [$1_{st},N_{th}$]
sites. (b) Same as (a) with $\Delta_{J_{3}-J_{1}}=0.75$ and topological edge
states at [$1_{st},3_{rd},(N-2)_{th},N_{th}$] sites. (c) Spin correlation
$\langle S_{z,j}S_{z,k}\rangle$ with $\Delta_{J_{3}-J_{1}}=-0.75$ above the
critical magnetic field. (d) Same as (c) but $\Delta_{J_{3}-J_{1}}=0.75$. (e)
Rényi entanglement entropy with changing external magnetic field and hopping
amplitude difference $\Delta_{J_{3}-J_{1}}$. Each point has error bar in both
x and y direction, and the dotted line is a schematic line connecting
different $\Delta_{J_{3}-J_{1}}$. (f) Critical field strength at various of
the on-site interaction and hopping amplitude difference and it can be seen
that for a certain $\Delta_{J_{3}-J_{1}}$, the value of the critical magnetic
field will decrease as $U$ increases.

To benchmark the CP-AFQMC calculations, we perform exact diagonalization on
the chain with $N=8$ sites. Subsequently, CP-AFQMC simulations are carried out
for a larger chain of $N=30$ sites. The system exhibits no sign problem at
half-filling, allowing us to employ parameters of $\tau=0.05,\
N_{walker}=100,$ and $\beta=n\tau=15.$ For systems away from half-filling,
where a sign problem arises, we set $\tau=0.01,\ N_{walker}=2000,$ and
$\beta=n\tau=64.$

###  III.1 Edge population

In the absence of Hubbard interaction ($U=0$), as discussed in Sec. II.1, the
chiral symmetry ensures that the eigenvalues appear in $(E,-E)$ pairs, leading
to the emergence of topological edge states at the $E=0$. According to Fig.
1(e) and 1(f), while electrons progressively fill the bulk bands, they do not
populate the edge states until the half-filling regime is reached. The
occupation of the edge state commences, as illustrated in Fig. 3(a). Under the
parameters in Fig. 3 (where $J_{1}>J_{11}$), the two outermost sites $1_{st}$
and $N_{th}$ always support edge states. Consequently, modifying the
$\Delta_{J_{3}-J_{1}}$ will not affect the filling of this specific pair of
edge states.

The difference between nearest neighbour hopping and next-nearest hopping
$\Delta_{J_{3}-J_{1}}$ affects the topological properties of $3_{rd}$ and
$(N-2)_{th}$ sites according to Fig. 1(c). When $\Delta_{J_{3}-J_{1}}$ is
negative , the $3_{rd}$ and $(N-2)_{th}$ sites do not support edge states.
Consequently, the electron population at these sites continuously varies
across the entire filling range, mirroring the behavior of bulk states.
Conversely, for positive $\Delta_{J_{3}-J_{1}}$, a pair of edge states appears
at the $3_{rd}$ and $(N-2)_{th}$ sites. This scenario, as illustrated in Fig.
3(b), leads to a sharp change in the electron population at half-filling. By
focusing on the combined behavior of these four potential edge state sites and
considering the electron filling, we observe that only all four sites support
topological edge state does a sharp population change occur at half-filling,
as shown in Fig. 3(c).

In the strong interaction limit (large Hubbard $U$), the Coulomb repulsion
between spin-up and spin-down electrons leads to the emergence of a Mott gap
in the energy spectrum. This is reflected in the addition energy spectrum,
$E_{ad}(n)=E(n)-E(n-1)$ where $n$ represents the number of electrons. As
expected for the Hubbard model, $E_{ad}(n)$ separates the lower and upper
bands, as illustrated in Fig. 2(a) and 2(b). Notably, when Hubbard $U$ is
sufficiently large, the edge states begin to be filled with electrons at
quarter-filling. When considering different electron filling at this time,
there will be a difference from no interaction, that is, electrons begin to
occupy the edge state at quarter-filling as shown in Fig. 3(d)-3(f). This
discrepancy arises due to the inherent gapless nature of symmetry-protected
topological edge states. When the system is quarter-filling with large Hubbard
$U$, introducing an additional electron at any bulk site incurs a significant
energy penalty due to the Hubbard $U$ interaction. This additional electron
can occupy the edge sites without costing energy.

There are some wrinkles on the surface in Fig. 3(d)-3(f). This is due to the
fact that $3_{rd}$ and $(N-2)_{th}$ sites to support new topological edge
states following a change in the topological number $\mathcal{W}$, which
allows for an additional electron to occupy these two sites without costing
additional energy. And filling the sites of $1_{st}$ and $N_{th}$ extends
backward by one electron. As established in Fig. 1(b), the system undergoes a
topological phase transition at $J_{3}=J_{1}$, corresponding to
$\Delta_{J_{3}-J_{1}}=0$. However, the wrinkles on the surface in Fig.
3(d)-3(f) deviate from zero. This is actually the result of the finite-size
effects in the system.

###  III.2 Bulk-edge correspondence

The bulk-edge correspondence means that the topological properties of a system
can be extracted by analyzing the characteristics of its edge states. The
topological winding number, determined under periodic boundary conditions,
captures the nature of the bulk states. Conversely, well-defined boundaries,
enforced through open boundary conditions in simulations, are necessary for
the emergence of edge states. This correspondence dictates that a change in
the winding number induces the appearance of localized, gapless edge states
that typically come in pairs. As illustrated in Fig. 2 and 3, regardless of
the presence or absence of strong interaction, the edge state of the system is
always occupied by electrons when it is half-filling.

We choose $U=10,\ J_{11}=J_{33}=0$ for the investigation of the impact of
$\Delta_{J_{3}-J_{1}}$ on the system , as illustrated in Fig. 4. In this
configuration, $1_{st}$ and $N_{th}$ sites consistently support topological
edge states. Additionally, for $\Delta_{J_{3}-J_{1}}>0$, $3_{rd}$ and
$(N-2)_{th}$ sites also support topological non-trivial edge states. These
edge states are decoupled from the bulk, and each pair of edge states existing
in either a singlet state
$\ket{S}=(\ket{\uparrow\downarrow}-\ket{\downarrow\uparrow})/\sqrt{2}$ or one
of the three triplet states
$\ket{T_{0}}=(\ket{\uparrow\downarrow}+\ket{\downarrow\uparrow})/\sqrt{2}$,
$\ket{T_{1}}=\ket{\uparrow\uparrow}$ and
$\ket{T_{2}}=\ket{\downarrow\downarrow}$. All there states are zero-energy and
gapless. When $\Delta_{J_{3}-J_{1}}<0$, the ground state of half-filling
exhibits fourfold degeneracy, and for $\Delta_{J_{3}-J_{1}}>0$, the degeneracy
of the ground state at half-filling increases to $16$, as confirmed by the
exact diagonalization results presented in Appendix C. The existence of
Hubbard $U$ reduces the ground state degeneracy. This arises due to the
exclusion principle enforced by U, forbidding the co-occupation of a single
site by electrons with opposing spins.

As the edge states forming a decoupled single state or a triplet state, the
topological properties of the bulk can be observed by simulating spin
correlation between different sites. We define spin correlation
$C_{j,k}=\langle S_{z,j}S_{z,k}\rangle$, where
$S_{z,j}=\frac{1}{2}(\hat{n}_{j,\uparrow}-\hat{n}_{j,\downarrow})$ represents
the z-component spin operator at site $j$. According to the equation (9) and
Wick’s theorem [51], the calculation of the spin correlation function can be
achieved through the two-point correlation Green function, as detailed in
Appendix 11.

As the $1_{st}$ and $N_{th}$ sites can always support topological edge states,
the $1_{st}$ site only has spin correlation with $N_{th}$ site as shown in
Fig. 4(a)-4(c). When $\Delta_{J_{3}-J_{1}}>0$ [Fig. 4(a)], a similar long-
range spin correlation emerges between $3_{rd}$ site and $(N-2)_{th}$ site
same as $1_{st}$ site and $N_{th}$ site. This is because the $3_{rd}$ and
$(N-2)_{th}$ can also support topological non-trivial edge states. Conversely,
when $3_{rd}$ and $(N-2)_{th}$ sites only support topological trivial states,
long-range spin correlation are no longer observed, and a long range anti-
ferromagnetic order is shown in Fig. 4(b). This observation is consistent with
Hubbard model’s conclusion in the large $U$ limit, which can be attributed to
second-order perturbation theory.

Long-range correlation effect can be observed not only through spin
correlation, but also by quantifying entanglement entropy. The paired
appearance of topological edge states means the existence of near-zero
entanglement. Since the positions of edge states are located at the two
boundaries of the system, the formation of topological edge states can be
determined by calculating the long-range entanglement entropy between the edge
sites and bulk sites.

Define the reduced density matrix of subsystem A as
$\rho_{A}=Tr_{B}\ket{\Psi}\bra{\Psi}$, where A is $1_{st}$ and $N_{th}$ sites
or $3_{rd}$ and $(N-2)_{th}$ sites. Rényi entropy can be calculated by the
reduced density matrix of subsystem A as
$S_{n}=-\frac{1}{n-1}\log_{2}[Tr(\rho_{A}^{n})]$, where $n$ is a positive
integer representing the order of the Renyi entropy. And the entanglement
between subsystems A and B can be measured by Rényi entropy. In CP-AFQMC, the
second-order Rényi entropy can be efficiently computed through the Green’s
function under different auxiliary fields [52, 53]:

$\displaystyle S_{2}=$
$\displaystyle-\log\biggl{[}\sum_{\\{s\\},\\{s^{\prime}\\}}P_{s}P_{s^{\prime}}\\{\mathrm{Det}(G_{s,A}G_{s^{\prime},A}$
(10) $\displaystyle+(1-G_{s,A})(1-G_{s^{\prime},A}))\biggr{]}$

where $G_{s}$ is equal-time Green matrix, and
$G_{s}^{ij}=\langle\pmb{\hat{c}}^{\dagger}_{j}\pmb{\hat{c}}_{i}\rangle$ with
$\pmb{\hat{c}}_{i}=\left(\begin{matrix}\hat{c}_{i\uparrow},\hat{c}_{i\downarrow}\end{matrix}\right)$.

The presence of topological edge states can be judged using the second-order
Rényi entanglement entropy. When topological edge states exist, the
entanglement between the sites where the topological edge states appear and
the remaining system will be significantly reduced, as illustrated in Fig.
4(d)- 4(f). Since $1_{st},N_{th}$ sites can always support topological edge
states, leading to near-zero entanglement with the bulk, as depicted in Fig.
4(e). Conversely, when $\Delta_{J_{3}-J_{1}}$ is changed until
$3_{rd},(N-2)_{th}$ can support a pair of topological edge states, the
entanglement entropy also decrease close to zero. In all other configurations,
the entanglement remains non-zero. Notably, the observed decrease in
entanglement entropy does not occur precisely at $\Delta_{J_{3}-J_{1}}=0$ due
to finite-size effects within the system. This effect will decrease as the
system size increases. This observation aligns with the exact diagonalization
results presented in the Appendix C. As larger system sizes calculated by CP-
AFQMC, the position where the entanglement entropy decreases is closer to
$\Delta_{J_{3}-J_{1}}=0$.

Our calculations show that $J_{3}=J_{1}=0$ in Fig. 5, and the conclusion
obtained is consistent with Fig. 4. When $\Delta_{J_{33}-J_{11}}<0$, $1_{st},\
2_{nd},\ (N-1)_{th}$ and $\ N_{th}$ sites cannot support topological edge
states. This is because $J_{11}>J_{1}$ indicates a stronger intra-cell hopping
compared to inter-cell hopping. Consequently, electrons favor forming dimer-
like states. Furthermore, as $J_{11}>J_{33}$, next neighbour hopping does not
disrupt the dimer state. Consequently, this explains the sequential emergence
of dimers observed in Fig. 5(c). Additionally, the Rényi entanglement entropy
between the two outermost dimers and the remaining dimers remains near zero,
as depicted in the left panel of Fig. 5(f). When $\Delta_{J_{33}-J_{11}}>0$,
the $2_{nd},\ (N-1)_{th}$ sites become topological edge states. This scenario,
illustrated in Fig. 5(a), exhibits long-range spin correlation, and the Rényi
entanglement entropy between $2_{nd},\ (N-1)_{th}$ sites and other sites
approaches zero as shown in Fig. 5(d).

###  III.3 Magnetic-field-induced transition at quarter-filling

In contrast to the isolated edge states observed at half-filling, the system
at quarter-filling exhibits long-range antiferromagnetic (AFM) order across
all topological phases. For the system parameters corresponding to Fig. 6(a)
and 6(b), the sites where the edge states appear are $(1_{st},N_{th})$ sites
and $(1_{st},3_{rd},(N-2)_{th},N_{th})$ sites, the long-range spin correlation
characteristic of half-filling are absent at quarter-filling. Instead, a
short-range AFM order, characterized by adjacent or sub-adjacent spin
correlation, is observed. This short-range order, arising due to the existence
of AFM order within the bulk, leads to a strong spin correlation between the
edge states and the bulk. Consequently, the Rényi entanglement entropy does
not exhibit a significant drop to zero as the $\Delta_{J}$ is changed, as
illustrated in Fig. 6(c) and 6(d) along the x-axis where $E_{B}=0$.

The Hamiltonian describing the system under a magnetic field is given by
$H_{1}=H-(E_{B}/2)\Sigma_{i}(\hat{n}_{i\uparrow}^{s}-\hat{n}_{i\downarrow}^{s})$
with $E_{B}=g\mu_{B}B$, where $\mu_{B}$ is the Bohr magneton and $g$ is the
g-factor of the electron in the material. According to the above discussion,
the system exhibits no long-range spin correlation at quarter-filling.
However, a sufficiently strong magnetic field can induce a transition to a
maximally ferromagnetic ground state, as evidenced by the emergence of long-
range spin correlation in Fig. 6(c) and 6(d). This can be attributed to the
magnetic field effectively transforming the quarter-filling state into a quasi
half-filling state at high field strengths, which is similar to the true half-
filling under zero magnetic field. Additionally, the Rényi entropy drops to
zero for $\Delta_{J}>0$ due to the formation of new topological edge states.

The contours depicted in Fig. 6(c) and 6(d) separate regions with distinct
total spin $S_{z}$. For a one-dimensional chain with $N=8$, a maximally
ferromagnetic ground state with four spin-up electrons at quarter-filling
corresponds to $S_{z}=2$. Notably, the contour for $S_{z}=0$ disappears in
Fig. 6(c). This is a consequence of the degeneracy between $S_{z}=1$ and
$S_{z}=0$ under zero field ($\omega=8$). When a magnetic field, regardless of
its strength, is applied, $S_{z}=1$ becomes the ground state of the system,
leading to the disappearance of the $S_{z}=0$ contour.

To achieve a maximally ferromagnetic state, the last spin-down electron needs
to be pumped to the next unoccupied spin-up single-particle state. The
existence of Hubbard $U$ causes that at quarter-filling, the energy of only
one spin-down electron is higher than $U=0$ case. Consequently, a smaller
critical magnetic field is required to overcome the energy barrier for
flipping this spin-down electron to a spin-up state. Since it is very
difficult to realize a strong magnetic field in experiment, the existence of
Hubbard $U$ allows us to observe the above transformation under a much smaller
magnetic field, as demonstrated in Fig. 6(e) and 6(f).

Fig. 6(e) suggests a potential demarcation line for $\Delta_{J_{3}-J_{1}}<0$.
We infer that this is likely the result of a competition between the energy-
lowering hopping term and the second-order process under the large U limit.
However, a complete understanding of the underlying mechanism remains elusive.
Further clarification regarding the demarcation line can be found in Appendix
A.

Although there will be a sign problem at quarter-filling, we can calculate the
above results under different electron numbers using CP-AFQMC for a larger
$1D$ chain with $N=28$, as shown in Fig. 7. The presence of a magnetic field
introduces topological edge states component to the ground state at quarter-
filling. However, the trial wave function at quarter-filling obtained from a
single-particle model does not contain edge component. Therefore, we can no
longer simply use the single-particle wave function as the trial wave
function. Here, we propose a new trial wave function by linearly superimposing
the single-particle wave function at quarter-filling with the topological edge
states. This new trial wave function at quarter-filling can be used to reduce
the system’s bias.

In Fig. 7(a)-7(d), we employ the formula (11) to calculate the spin
correlation. The results consistent with Fig. 6, that is, the long-range AFM
order manifested under the critical magnetic field and the long-range spin
correlation above the critical magnetic field. Subsequently, the Rényi
entanglement entropy of the system is calculated using formulas (10), as
depicted in Fig. 7(e). We can observe that for $\Delta_{J_{3}-J_{1}}<0$, the
entanglement entropy remains nonzero regardless of the applied magnetic field
. Conversely, when $\Delta_{J_{3}-J_{1}}>0$, the formation of long-range spin
correlation leads to a drop in entanglement entropy towards zero as the
magnetic field strength increases. This behavior is consistent with Fig. 6(c)
and 6(d), the entanglement entropy within each contour is a constant value.
The calculated entanglement entropy for different spin configurations is
connected by a polyline fit. The critical magnetic field, corresponding to the
formation of the maximally ferromagnetic state, is defined as the magnetic
field strength at which the entanglement entropy reaches its minimum for
different values of $\Delta_{J_{3}-J_{1}}$. As illustrated in Fig. 7(f), the
relationship between the critical magnetic field and the Hubbard $U$ is
determined using CP-AFQMC. Similarly, a substantial decrease in the critical
field with increasing Hubbard $U$ is observed. The case $J_{1}=J_{3}=0$ is
detailed in the Appendix D.

##  IV SUMMARY AND DISCUSSION

Topological edge states are a fundamental concept in topological physics, and
their characterization is crucial for understanding the properties of
topological materials. Spin correlation and entanglement entropy serve as
essential measures for evaluating these edge states, even in strongly
correlated systems.

In this work, we employ two computational techniques, exact diagonalization
and CP-AFQMC, to simulate the SSHH model incorporating next-nearest neighbour
hopping. Through exact diagonalization calculations, we observed that in small
systems with Hubbard $U$, spin correlation and Rényi entanglement entropy
exhibit variations in response to the relative magnitude of system hopping. To
extrapolate to the thermodynamic limit, we utilize the CP-AFQMC algorithm and
employ the method outlined in [52] to calculate the system’s Rényi
entanglement entropy. The results obtained align with those from exact
diagonalization but exhibit reduced finite-size effects.

When the system is at quarter-filling, long-range AFM order emerges, leading
to the disappearance of long-range spin correlation and a significant increase
of the entanglement entropy. Interestingly, if an external magnetic field is
applied at this case, the system reverts to a quasi-half-filling state,
reintroducing long-range spin correlation and reducing entanglement entropy to
near zero.

During the simulation of quarter-filling, the presence of a sign problem
necessitates the employment of the CP-AFQMC method to mitigate the issue. In
this process, we utilize modified trial wave functions to achieve enhanced
convergence. By calculating spin correlation and entanglement entropy using
CP-AFQMC, we can extrapolate the results to other interacting topological
systems, obtaining findings that closely approximate the thermodynamic limit.

###### Acknowledgements.

The authors would like to thanks Yuhang Lu and Pengyu Wen for help on
computing resources. The authors also would like to thanks Wenjia Rao, Shuai
Chen, and Zhen Guo for their helpful discussion. This work is supported by the
National Natural Science Foundation of China under Grants No. 11974205, and
No. 61727801, the Key Research and Development Program of Guangdong province
(2018B030325002), and National Natural Science Foundation of China under Grant
62131002.

##  Appendix A Comparison of ED and CP-AFQMC results with 1D $N=8$ chain and a
puzzle

To demonstrate the accuracy of CP-AFQMC results especially when there is a
sign problem, we calculate the magnitude of the critical field for a quarter-
filling 1D chain with $N=8$ sites as a function of Hubbard $U$, selecting
$\Delta_{J_{3}-J_{1}}=-0.75$ and $\Delta_{J_{3}-J_{1}}=0.75$, which correspond
to the leftmost and rightmost parameters in Fig. 6(e). Due to the reduced
system size, the imaginary time evolution length for CP-AFQMC calculations was
appropriately shortened. This optimization improve computational efficiency
without compromising accuracy for small systems, like the chain with $N=8$
sites and electron number $n=4$. The parameters we choose here are:
$\tau=0.01,\ N_{walker}=200,\ \beta=40\times 16\times\tau=6.4.$ Under the
above parameters, we can calculate a relatively accurate value of the critical
magnetic field, as shown in Fig. 8. Compared with the exact diagonalization
results, CP-AFQMC gives consistent calculation results.

Figure 8: The magnitude of the critical field for a quarter-filling 1D chain
with $N=8$ sites and electron numbers $n=4$ as a function of Hubbard $U$.
Figure 9: ED results same as Fig 4. (a) Spin correlation $\langle
S_{z,j}S_{z,k}\rangle$ with $\Delta_{J_{3}-J_{1}}=0.7$, topological edge
states appear at [$1_{st},3_{rd},6_{th},8_{th}$] sites. (b) Same as (a) with
$\Delta_{J_{3}-J_{1}}=0$ and there is a long-range anti-ferromagnetic order.
(c) Same as (a) with $\Delta_{J_{3}-J_{1}}=-0.7$, and topological edge states
appear at [$1_{st},8_{th}$] sites. And $\omega$ is the degeneracy of ground
state. (d) The entanglement Rényi entropy between subsystem [$3_{rd},6_{th}$]
sites and the remaining sites. (e) Same as (d) but subsystem [$1_{st},8_{th}$]
sites and (f) with subsystem [$1_{st},3_{rd},6_{th},8_{th}$] sites. In
(d)-(f), the position where the transition occurs is far from
$\Delta_{J_{3}-J_{1}}=0$, which is result of finite-size effects.  Figure 10:
ED results same as Fig 5. (a) Spin correlation $\langle S_{z,j}S_{z,k}\rangle$
with $\Delta_{J_{33}-J_{11}}=0.7$, topological edge states appear at
[$2_{nd},7_{th}$] sites. (b) Same as (a) for $\Delta_{J_{33}-J_{11}}=0$ and
there is a long-range anti-ferromagnetic order. (c) Same as (a) with
$\Delta_{J_{33}-J_{11}}=-0.7$, and there are bulk dimers without topological
edge states. (d) The entanglement Rényi entropy between subsystem
[$2_{nd},7_{th}$] sites and the remaining sites. (e) Same as (d) but subsystem
[$1_{st},8_{th}$] sites and (f) with subsystem [$1_{st},2_{nd},7_{th},8_{th}$]
sites. And due to the formation of dimers, there will be a drop in Rényi
entanglement entropy.  Figure 11: Same as Fig 7 for $J_{1}=J_{3}=0$, and
$U=10$ for (a)-(e). (a) Spin correlation $\langle S_{z,j}S_{z,k}\rangle$ with
$\Delta_{J_{33}-J_{11}}=-0.75$. (b) Same as (a) with
$\Delta_{J_{33}-J_{11}}=0.75$ and topological edge states appear at
[$2_{nd},(N-1)_{th}$] sites. (c) Spin correlation $\langle
S_{z,j}S_{z,k}\rangle$ with $\Delta_{J_{33}-J_{11}}=-0.75$ above the critical
magnetic field. (d) Same as (c) but $\Delta_{J_{33}-J_{11}}=0.75$. (e) Rényi
entanglement entropy with changing external magnetic field and hopping
amplitude difference $\Delta_{J_{33}-J_{11}}$. (f) Critical field strength at
various of the on-site interaction and hopping amplitude difference and it can
be seen that for a certain $\Delta_{J_{33}-J_{11}}$, the value of the critical
magnetic field will decrease as U increases.

The trend of the critical field change exhibits a clear segmentation effect in
the figure below Fig. 8. This is consistent with the demarcation line observed
in Fig. 6(e) as described in the main text. We speculate that this is caused
by the competition between two processes, but we do not yet fully understand
the underlying physical mechanism. Additionally, if we perform a vertical
cross-section of Fig. 7(f), we can also observe this phenomenon for
$\Delta_{J_{3}-J_{1}}<0$. This indicates that the phenomenon is not caused by
the finite-size effects of the system and requires further investigation.

##  Appendix B Calculation of spin correlation in CP-AFQMC

As before, we define the spin correlation as $C_{j,k}=\langle
S_{z,j}S_{z,k}\rangle$ with
$S_{z,j}=\frac{1}{2}(\hat{n}_{j,\uparrow}-\hat{n}_{j,\downarrow})$. And in CP-
AFQMC we can calculate the spin correlation through the Green function

$\displaystyle C_{j,k}$
$\displaystyle=\frac{1}{4}\langle(\hat{n}_{j,\uparrow}-\hat{n}_{j,\downarrow})(\hat{n}_{k,\uparrow}-\hat{n}_{k,\downarrow})\rangle$
(11)
$\displaystyle=\frac{1}{4}(G_{up}^{jj}G_{up}^{kk}-G_{up}^{kj}G_{up}^{jk}+G_{up}^{jk}\delta_{j,k}+G_{dn}^{jj}G_{dn}^{kk}$
$\displaystyle\quad-
G_{dn}^{kj}G_{dn}^{jk}+G_{dn}^{jk}\delta_{j,k}-G_{dn}^{jj}G_{up}^{kk}-G_{up}^{jj}G_{dn}^{kk}).$

##  Appendix C Spin correlation and entropy base on Exact Diagonalization

In this above study, we employ the CP-AFQMC method to compute the spin
correlation and second-order Rényi entropy at half-filling. To validate the
accuracy of the CP-AFQMC results, we perform additional calculations using
exact diagonalization on a small system to obtain the long-range spin
correlation and second-order Rényi entanglement entropy, as shown in Fig. 9
and Fig. 10. The second-order Rényi entanglement entropy values obtained via
ED exhibits a larger deviation from zero compared to the CP-AFQMC results.
This discrepancy suggests a more pronounced finite-size effects in the
calculations using ED, likely attributable to the inherent limitations of
exact diagonalization when applied to small systems.

##  Appendix D quarter-filling with changing $\Delta_{J_{33}-J_{11}}$

To further substantiate our conclusions, we employ the CP-AFQMC method to
investigate the behavior of the system under varying values of
$\Delta_{J_{33}-J_{11}}$ as depicted in Fig. 11. The results obtained
corroborate the findings presented in Fig. 7, demonstrating the restoration of
long-range spin correlation and the disappearance of Rényi entanglement
entropy above the critical magnetic field. Additionally, the critical field
strength exhibits a gradual decrease with increasing values of the Hubbard
$U$.

## References

  * Hasan and Kane [2010] M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys. 82, 3045 (2010).
  * Qi and Zhang [2011] X.-L. Qi and S.-C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83, 1057 (2011).
  * Chiu  _et al._ [2016] C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, Classification of topological quantum matter with symmetries, Rev. Mod. Phys. 88, 035005 (2016).
  * Wen [2017] X.-G. Wen, Colloquium: Zoo of quantum-topological phases of matter, Rev. Mod. Phys. 89, 041004 (2017).
  * Nagaosa and Tokura [2013] N. Nagaosa and Y. Tokura, Topological properties and dynamics of magnetic skyrmions, Nature nanotechnology 8, 899 (2013).
  * Pekola  _et al._ [2013] J. P. Pekola, O.-P. Saira, V. F. Maisi, A. Kemppinen, M. Möttönen, Y. A. Pashkin, and D. V. Averin, Single-electron current sources: Toward a refined definition of the ampere, Rev. Mod. Phys. 85, 1421 (2013).
  * Beenakker [2013] C. Beenakker, Search for majorana fermions in superconductors, Annu. Rev. Condens. Matter Phys. 4, 113 (2013).
  * Li  _et al._ [2014] L. Li, Z. Xu, and S. Chen, Topological phases of generalized su-schrieffer-heeger models, Phys. Rev. B 89, 085111 (2014).
  * Fu  _et al._ [2007] L. Fu, C. L. Kane, and E. J. Mele, Topological insulators in three dimensions, Phys. Rev. Lett. 98, 106803 (2007).
  * Hsieh  _et al._ [2008] D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava, and M. Z. Hasan, A topological dirac insulator in a quantum spin hall phase, Nature 452, 970 (2008).
  * Zhang  _et al._ [2009] H. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang, and S.-C. Zhang, Topological insulators in bi2se3, bi2te3 and sb2te3 with a single dirac cone on the surface, Nature physics 5, 438 (2009).
  * Bansil  _et al._ [2016] A. Bansil, H. Lin, and T. Das, Colloquium: Topological band theory, Rev. Mod. Phys. 88, 021004 (2016).
  * Ryu and Hatsugai [2002] S. Ryu and Y. Hatsugai, Topological origin of zero-energy edge states in particle-hole symmetric systems, Phys. Rev. Lett. 89, 077002 (2002).
  * Qi  _et al._ [2006] X.-L. Qi, Y.-S. Wu, and S.-C. Zhang, General theorem relating the bulk topological number to edge states in two-dimensional insulators, Phys. Rev. B 74, 045125 (2006).
  * Atala  _et al._ [2013] M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, Direct measurement of the zak phase in topological bloch bands, Nature Physics 9, 795 (2013).
  * Cooper  _et al._ [2019] N. Cooper, J. Dalibard, and I. Spielman, Topological bands for ultracold atoms, Rev. Mod. Phys. 91, 015005 (2019).
  * Rizzo  _et al._ [2018] D. J. Rizzo, G. Veber, T. Cao, C. Bronner, T. Chen, F. Zhao, H. Rodriguez, S. G. Louie, M. F. Crommie, and F. R. Fischer, Topological band engineering of graphene nanoribbons, Nature 560, 204 (2018).
  * Pesin and MacDonald [2012] D. Pesin and A. H. MacDonald, Spintronics and pseudospintronics in graphene and topological insulators, Nature materials 11, 409 (2012).
  * Gomes  _et al._ [2012] K. K. Gomes, W. Mar, W. Ko, F. Guinea, and H. C. Manoharan, Designer dirac fermions and topological phases in molecular graphene, Nature 483, 306 (2012).
  * Kempkes  _et al._ [2019] S. Kempkes, M. Slot, J. van Den Broeke, P. Capiod, W. Benalcazar, D. Vanmaekelbergh, D. Bercioux, I. Swart, and C. Morais Smith, Robust zero-energy modes in an electronic higher-order topological insulator, Nature Materials 18, 1292 (2019).
  * Klembt  _et al._ [2018] S. Klembt, T. Harder, O. Egorov, K. Winkler, R. Ge, M. Bandres, M. Emmerling, L. Worschech, T. Liew, M. Segev, _et al._ , Exciton-polariton topological insulator, Nature 562, 552 (2018).
  * Su  _et al._ [2021] R. Su, S. Ghosh, T. C. Liew, and Q. Xiong, Optical switching of topological phase in a perovskite polariton lattice, Science Advances 7, eabf8049 (2021).
  * Le  _et al._ [2017] N. H. Le, A. J. Fisher, and E. Ginossar, Extended hubbard model for mesoscopic transport in donor arrays in silicon, Phys. Rev. B 96, 245406 (2017).
  * Dusko  _et al._ [2018] A. Dusko, A. Delgado, A. Saraiva, and B. Koiller, Adequacy of si: P chains as fermi–hubbard simulators, npj Quantum Information 4, 1 (2018).
  * Hu  _et al._ [2023] H. Hu, J. Wang, R. Lalor, and X.-J. Liu, Two-dimensional coherent spectroscopy of trion-polaritons and exciton-polaritons in atomically thin transition metal dichalcogenides, AAPPS Bulletin 33, 12 (2023).
  * Rachel [2018] S. Rachel, Interacting topological insulators: a review, Reports on Progress in Physics 81, 116501 (2018).
  * Jünemann  _et al._ [2017] J. Jünemann, A. Piga, S.-J. Ran, M. Lewenstein, M. Rizzi, and A. Bermúdez, Exploring interacting topological insulators with ultracold atoms: The synthetic creutz-hubbard model, Phys. Rev. X 7, 031057 (2017).
  * Nawa  _et al._ [2019] K. Nawa, K. Tanaka, N. Kurita, T. J. Sato, H. Sugiyama, H. Uekusa, S. Ohira-Kawamura, K. Nakajima, and H. Tanaka, Triplon band splitting and topologically protected edge states in the dimerized antiferromagnet, Nature communications 10, 2096 (2019).
  * Tang and Wen [2012] E. Tang and X.-G. Wen, Interacting one-dimensional fermionic symmetry-protected topological phases, Phys. Rev. Lett. 109, 096403 (2012).
  * Manmana  _et al._ [2012] S. R. Manmana, A. M. Essin, R. M. Noack, and V. Gurarie, Topological invariants and interacting one-dimensional fermionic systems, Phys. Rev. B 86, 205119 (2012).
  * Wagner  _et al._ [2023a] N. Wagner, L. Crippa, A. Amaricci, P. Hansmann, M. Klett, E. König, T. Schäfer, D. D. Sante, J. Cano, A. Millis, _et al._ , Mott insulators with boundary zeros, Nature Communications 14, 7531 (2023a).
  * Wagner  _et al._ [2023b] N. Wagner, D. Guerci, A. J. Millis, and G. Sangiovanni, Edge zeros and boundary spinons in topological mott insulators (2023b), arXiv:2312.13226 [cond-mat.str-el] .
  * Jiang  _et al._ [2012] H.-C. Jiang, Z. Wang, and L. Balents, Identifying topological order by entanglement entropy, Nature Physics 8, 902 (2012).
  * Wang  _et al._ [2015] D. Wang, S. Xu, Y. Wang, and C. Wu, Detecting edge degeneracy in interacting topological insulators through entanglement entropy, Phys. Rev. B 91, 115118 (2015).
  * Xiao  _et al._ [2010] D. Xiao, M.-C. Chang, and Q. Niu, Berry phase effects on electronic properties, Rev. Mod. Phys. 82, 1959 (2010).
  * Pérez-González  _et al._ [2019] B. Pérez-González, M. Bello, A. Gómez-León, and G. Platero, Interplay between long-range hopping and disorder in topological systems, Phys. Rev. B 99, 035146 (2019).
  * Cinnirella  _et al._ [2024] E. G. Cinnirella, A. Nava, G. Campagnano, and D. Giuliano, Fate of high winding number topological phases in the disordered extended su-schrieffer-heeger model, Phys. Rev. B 109, 035114 (2024).
  * Le  _et al._ [2020] N. H. Le, A. J. Fisher, N. J. Curson, and E. Ginossar, Topological phases of a dimerized fermi–hubbard model for semiconductor nano-lattices, npj Quantum Information 6, 24 (2020).
  * Ryu  _et al._ [2010] S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. W. Ludwig, Topological insulators and superconductors: tenfold way and dimensional hierarchy, New Journal of Physics 12, 065010 (2010).
  * Zak [1989] J. Zak, Berry’s phase for energy bands in solids, Phys. Rev. Lett. 62, 2747 (1989).
  * Bethe [1997] H. Bethe, On the theory of metals, i. eigenvalues and eigenfunctions of a linear chain of atoms, in  _Selected Works Of Hans A Bethe: (With Commentary)_ (World Scientific, 1997) pp. 155–183.
  * Zhang  _et al._ [1995] S. Zhang, J. Carlson, and J. E. Gubernatis, Constrained path quantum monte carlo method for fermion ground states, Phys. Rev. Lett. 74, 3652 (1995).
  * Zhang  _et al._ [1997] S. Zhang, J. Carlson, and J. E. Gubernatis, Constrained path monte carlo method for fermion ground states, Phys. Rev. B 55, 7464 (1997).
  * Zhang [2013] S. Zhang, 15 auxiliary-field quantum monte carlo for correlated electron systems, Emergent Phenomena in Correlated Matter (2013).
  * Nguyen  _et al._ [2014] H. Nguyen, H. Shi, J. Xu, and S. Zhang, Cpmc-lab: A matlab package for constrained path monte carlo calculations, Computer Physics Communications 185, 3344 (2014).
  * He  _et al._ [2019] Y.-Y. He, M. Qin, H. Shi, Z.-Y. Lu, and S. Zhang, Finite-temperature auxiliary-field quantum monte carlo: Self-consistent constraint and systematic approach to low temperatures, Phys. Rev. B 99, 045108 (2019).
  * Panza [2018] M. J. Panza, Application of power method and dominant eigenvector/eigenvalue concept for approximate eigenspace solutions to mechanical engineering algebraic systems, American Journal of Mechanical Engineering 6, 98 (2018).
  * Hirsch [1983] J. E. Hirsch, Discrete hubbard-stratonovich transformation for fermion lattice models, Phys. Rev. B 28, 4059 (1983).
  * Qin [2023] M. Qin, Self-consistent optimization of the trial wave function within the constrained path auxiliary field quantum monte carlo method using mixed estimators, Phys. Rev. B 107, 235124 (2023).
  * Vitali  _et al._ [2019] E. Vitali, P. Rosenberg, and S. Zhang, Calculating ground-state properties of correlated fermionic systems with bcs trial wave functions in slater determinant path-integral approaches, Phys. Rev. A 100, 023621 (2019).
  * Wick [1950] G. C. Wick, The evaluation of the collision matrix, Phys. Rev. 80, 268 (1950).
  * Grover [2013] T. Grover, Entanglement of interacting fermions in quantum monte carlo calculations, Phys. Rev. Lett. 111, 130402 (2013).
  * Assaad  _et al._ [2014] F. F. Assaad, T. C. Lang, and F. Parisen Toldin, Entanglement spectra of interacting fermions in quantum monte carlo simulations, Phys. Rev. B 89, 125121 (2014).