# Removal paths avoiding vertices111The author’s work is supported by NNSF of
China (No.12071260).

Yuzhen Qi,Jin Yan222Corresponding author. E-mail address<EMAIL_ADDRESS>
School of Mathematics, Shandong University, Jinan 250100, China

Abstract: In this paper, we show that for any positive integer $m$ and
$k\in[2]$, let $G$ be a $(2m+2k+2)$-connected graph and let
$a_{1},\ldots,a_{m},s,t$ be any distinct vertices of $G$, there are $k$
internally disjoint $s$-$t$ paths $P_{1},\ldots,P_{k}$ in $G$ such that
$\\{a_{1},\ldots,a_{m}\\}\cap\bigcup^{k}_{i=1}V(P_{i})=\emptyset$ and
$G-\bigcup^{k}_{i=1}V(P_{i})$ is 2-connected, which generalizes the result by
Chen, Gould and Yu [Combinatorica 23 (2003) 185–203], and Kriesell [J. Graph
Theory 36 (2001) 52–58]. The case $k=1$ implies that for any
$(2m+5)$-connected graph $G$, any edge $e\in E(G)$, and any distinct vertices
$a_{1},\ldots,a_{m}$ of $G-V(e)$, there exists a cycle $C$ in
$G-\\{a_{1},\ldots,a_{m}\\}$ such that $e\in E(C)$ and $G-V(C)$ is
2-connected, which improves the bound $10m+11$ of Y. Hong, L. Kang and X. Yu
in [J. Graph Theory 80 (2015) 253–267].

Keywords: connectivity, disjoint paths, rooted graphs

AMS subject classifications. 05C38; 05C40; 05C75

##  1 Introduction

We discuss only finite simple graphs and use standard terminology and notation
from [2] except as indicated. A path $P$ with end vertices $u,v$ is called an
$u$-$v$ path, and for two vertices $x,y\in V(P)$, we denote the subpath of $P$
from $x$ to $y$ by $P[x,y]$, and let $P(x,y)=P[x,y]-\\{x,y\\}$. The following
conjecture is a generalization of an old conjecture made by Lovász [8] in
1975.

###### Conjecture 1.1

[6] There exists a function $f=f(k,l)$ such that the following holds. For
every $f(k,l)$-connected graph $G$ and two distinct vertices $s$ and $t$ in
$G$, there are $k$ internally disjoint $s$-$t$ paths $P_{1},\ldots,P_{k}$ such
that $G-\bigcup^{k}_{i=1}V(P_{i})$ is $l$-connected.

A famous result of Tutte [11] yields $f(1,1)=3$. The case $l=2$ was
independently obtained by [1] and [7], who showed $f(1,2)=5$. It is shown in
[6] that $f(k,1)=2k+1$ and $f(k,2)\leq 3k+2$. Conjecture 1.1 is still (wide)
open for $f(k,l\geq 3)$.

Indeed, Tutte [11] proved a stronger reault. That is, for any 3-connected
graph $G$ and distinct vertices $a$, $s$, $t$ of $G$, there is an $s$-$t$ path
$P$ in $G$ such that $a\notin V(P)$ and $G-V(P)$ is connected. A result of
Jung [5] implies that for any 6-connected graph $G$ and any distinct vertices
$a_{1}$, $a_{2}$, $s$, $t$ of $G$, there is an $s$-$t$ path $P$ in $G$ such
that $a_{1},a_{2}\notin V(P)$ and $G-V(P)$ is connected. These results
motivate us to propose the following question.

###### Question 1.1

Let $m,k,l$ be positive integers. There exists a function $f(m,k,l)$ such that
the following holds. For every $f(m,k,l)$-connected graph $G$ and any distinct
vertices $a_{1},\ldots,a_{m},s,t$ of $G$, there are $k$ internally disjoint
$s$-$t$ paths $P_{1},\ldots,P_{k}$ in $G$ such that
$\\{a_{1},\ldots,a_{m}\\}\cap\bigcup^{k}_{i=1}V(P_{i})=\emptyset$ and
$G-\bigcup^{k}_{i=1}V(P_{i})$ is $l$-connected.

The result in [9] yields $f(m,k,2)\leq 10m+30k+2$. This function may not be
optimal since authors in [9] used the result that $10k$-connected graph is
$k$-linked in [10], and $10k$ is not to optimal for the $k$-linkage problem.
Recently, X. Du, Y. Li , S. Xie and X. Yu [3] verified Question 1.1 in the
$k=l=1$ case by proving the following result.

###### Theorem 1.1

[3] Let $m\geq 1$ be an integer and let $G$ be a $(2m+2)$-connected graph. For
any distinct vertices $a_{1},\ldots,a_{m},s,t$ of $G$, there is an $s$-$t$
path $P$ in $G$ such that $\\{a_{1},\ldots,a_{m}\\}\cap V(P)=\emptyset$ and
$G-V(P)$ is connected.

Our first result gives some progress towards Question 1.1 when $l=2$ and
$k=1,2$.

###### Theorem 1.2

For any positive integer $m$ and $k=1,2$, we have $f(m,k,2)\leq 2m+2k+2$.

Lovász’s conjecture [8] can also be phrased in terms of finding a cycle
containing an arbitrary edge $e$ such that deleting the vertices of the cycle
leaves the graph $l$-connected. Y. Hong, L. Kang and X. Yu [4] thought that in
potential applications one may need the cycle to avoid certain vertices, and
posed the following conjecture.

###### Conjecture 1.2

[4] For any positive integers $m$ and $l$, there exists a smallest positive
integer $g(m,l)$ such that for any $g(m,l)$-connected graph $G$, any edge
$e\in E(G)$, and any distinct vertices $a_{1},\ldots,a_{m}$ of $G-V(e)$, there
exists an induced cycle $C$ in $G-\\{a_{1},\ldots,a_{m}\\}$ such that $e\in
E(C)$ and $G-V(C)$ is $l$-connected.

Theorem 1.1 implies $g(m,1)\leq 2m+3$ (set $e=st$, and then $G-V(e)$ is
$(2m+2)$-connected). Our Theorem 1.3 establishes Conjecture 1.2 for $l=2$,
improves the bound $10m+11$ in [4].

###### Theorem 1.3

For any positive integer $m$, $g(m,2)\leq 2m+5$.

As a natural generalization of Conjecture 1.2, the last result in our paper is
the following:

###### Theorem 1.4

Let $m$ be a positive integer and let $G$ be a $(2m+8)$-connected graph. For
two distinct edges $\\{e_{1},e_{2}\\}\subseteq E(G)$, and any distinct
vertices $a_{1},\ldots,a_{m}$ of $G-V(\\{e_{1},e_{2}\\})$, there exists a
cycle $C$ in $G-\\{a_{1},\ldots,a_{m}\\}$ such that
$\\{e_{1},e_{2}\\}\subseteq E(C)$ and $G-V(C)$ is connected.

The rest of the paper is organized as follows. In Section 2, we introduce
notation, and we give a characterization of “$m$-rooted graphs” in Theorem
2.3, which will be used in the proof of Theorems 1.2 and 1.4. So, the proof of
Theorem 2.3 is given in Section 3. Theorems 1.2 and 1.4 are proved in Section
4.

##  2 Preliminaries and tools

###  2.1 Basic terminology

For an integer $k$, we write $[k]$ for $\\{1,\ldots,k\\}$. A _block_ of a
graph $G$ is a maximal 2-connected subgraph of $G$ or an isolate vertex. For a
graph $G$, we use $V(G)$, $E(G)$, $v(G)$, and $e(G)$ to denote its vertex set,
edge set, the cardinality of its vertex set, and the cardinality of its edge
set, respectively. For any $S\subseteq V(G)$, the subgraph of $G$ induced by
$S$ is denoted by $G[S]$. Let $G-S=G[V\backslash S]$ and
$N_{G}[S]=N_{G}(S)\cup S$.

We follow the notation of [3]. Let $m\geq 0$ be an integer, and we call
$(G,\\{a_{1},\ldots,a_{m}\\},b_{1},b_{2})$ an $m$-_rooted graph_ if $G$ is a
graph and $a_{1},\ldots,a_{m},b_{1},b_{2}$ are distinct vertices of $G$ (note
that $m=0$ means that $\\{a_{1},\ldots,a_{m}\\}=\emptyset$). An $m$-rooted
graph $(G,\\{a_{1},\ldots,a_{m}\\},b_{1},b_{2})$ is said to be _feasible_
(resp. 2-_feasible_) if $G$ contains a $b_{1}$-$b_{2}$ path $P$ and $G-V(P)$
has a component (resp. block) containing $\\{a_{1},\ldots,a_{m}\\}$. Further,
we call an $m$-rooted graph $(G,\\{a_{1},\ldots,a_{m}\\},b_{1},b_{2})$
(2,2)-_feasible_ if $\\{a_{1},\ldots,a_{m}\\}$ and $\\{b_{1},b_{2}\\}$ are
contained in two disjoint blocks, respectively.

Fig. 1: An example of $G/\mathcal{X}$, $\mathcal{G}/\mathcal{X}$ and
$\mathcal{G}|\mathcal{X}$. Dashed lines indicate edges that do not belong to
$E(G)$, but belong to $E(G/\mathcal{X})$ or $E(\mathcal{G}/\mathcal{X})$ or
$E(\mathcal{G}|\mathcal{X})$.

Let $G$ be a graph and $\\{s_{1},\ldots,s_{t}\\}\subseteq V(G)$, we say that
$(G,s_{1},\ldots,s_{t})$ is _planar_ if $G$ can be drawn in a closed disc in
the plane with no edge crossings such that $s_{1},\ldots,s_{t}$ on the
boundary of the disc in clockwise order. For any $S\subseteq V(G)$, an
$\mathcal{S}$-_collection_ in $G$ is a collection $\mathcal{X}\subseteq
V(G)\setminus S$ such that $N_{G}[X_{1}]\cap X_{2}=\emptyset$ for distinct
$X_{1},X_{2}\in\mathcal{X}$. We use $G/\mathcal{X}$ to denote the graph
obtained from $G$ by, for each $X\in\mathcal{X}$, deleting $X$ and adding
edges such that $G[N_{G}(X)]$ is a clique. For an $m$-rooted graph
$\mathcal{G}=(G,\\{a_{1},\ldots,a_{m}\\},b_{1},b_{2})$ and any
$\\{a_{1},\ldots,a_{m},b_{1},b_{2}\\}$-collection $\mathcal{X}$ in $G$, let
$\mathcal{G}/\mathcal{X}$ denote the graph obtained from $G/\mathcal{X}$ by
adding an edge joining every pair of distinct vertices in
$\\{a_{1},\ldots,a_{m},b_{1},b_{2}\\}$ except the pair in $\\{b_{1},b_{2}\\}$,
and let $\mathcal{G}|\mathcal{X}$ denote the graph obtained from
$\mathcal{G}/\mathcal{X}$ by deleting all edges in $\\{uv\in
E(\mathcal{G}/\mathcal{X})\setminus E(G):u\in
V(G)\setminus\\{a_{1},\ldots,a_{m},b_{1},b_{2}\\},v\in\\{a_{1},\ldots,a_{m},b_{1},b_{2}\\}\\}$.
An example of $G/\mathcal{X}$, $\mathcal{G}/\mathcal{X}$ and
$\mathcal{G}|\mathcal{X}$ is shown in Fig. 1. Note that

$e(\mathcal{G}|\mathcal{X})\leq e(\mathcal{G}/\mathcal{X})$ and
$e(G/\mathcal{X})\leq e(\mathcal{G}/\mathcal{X})$.

###  2.2 $m$-rooted graphs

The following result conducted by Seymour [12] gave a characterization of
feasible 2-rooted graphs.

###### Theorem 2.1

[12] Let $(G,\\{a_{1},a_{2}\\},b_{1},b_{2})$ be a $2$-rooted graph. Then
either $(G,\\{a_{1},a_{2}\\},b_{1},b_{2})$ is feasible, or there is some
$\\{a_{1},a_{2},b_{1},b_{2}\\}$-collection $\mathcal{X}$ in $G$ such that
$|N_{G}(X)|\leq 3$ for all $X\in\mathcal{X}$ and
$(G/\mathcal{X},a_{1},b_{1},a_{2},b_{2})$ is planar.

The natural generalization of Theorem 2.1 to $m$-rooted graphs, proved by X.
Du, Y. Li, S. Xie and X. Yu [3] in 2023, is the following:

###### Theorem 2.2

[3] Let $m\geq 0$ be an integer, and let
$(G,\\{a_{1},\ldots,a_{m}\\},b_{1},b_{2})$ be an $m$-rooted graph. Then either
$(G,\\{a_{1},\ldots,a_{m}\\},b_{1},b_{2})$ is feasible, or there is some
$\\{a_{1},\ldots,a_{m},b_{1},b_{2}\\}$-collection $\mathcal{X}$ in $G$ such
that $|N_{G}(X)|\leq m+1$ for all $X\in\mathcal{X}$, and
$e(\mathcal{G/X})\leq(m+1)v(G/\mathcal{X})-m^{2}/2-3m/2-1$.

Now we are able to give a characterization of $m$-rooted graphs, which will be
used in the proof of Theorems 1.2 and 1.4.

###### Theorem 2.3

Let $m\geq 0$ be an integer, and let
$(G,\\{a_{1},\ldots,a_{m}\\},b_{1},b_{2})$ be an $m$-rooted graph of order at
least $m+3$. Suppose that there is no
$\\{a_{1},\ldots,a_{m},b_{1},b_{2}\\}$-collection $\mathcal{X}$ in $G$ such
that $|N_{G}(X)|\leq m+2$ for all $X\in\mathcal{X}$, and
$e(\mathcal{G}|\mathcal{X})\leq(m+2)v(\mathcal{G}|\mathcal{X})-m^{2}/2-5m/2-4$.
Then

(i) either $a_{1},\ldots,a_{m},b_{1},b_{2}$ have a common neighbour in
$G-\\{a_{1},\ldots,a_{m},b_{1},b_{2}\\}$, or $(G,\\{a_{1},\ldots,$
$a_{m}\\},b_{1},b_{2})$ is 2-feasible.

(ii) either $a_{1},\ldots,a_{m}$ have a common neighbour in
$G-\\{a_{1},\ldots,a_{m},b_{1},b_{2}\\}$, or $(G,\\{a_{1},\ldots,a_{m}\\},$
$b_{1},b_{2})$ is (2,2)-feasible.

Due to Theorem 2.3, we have the following results.

###### Corollary 2.1

Let $m\geq 0$ be an integer, and let
$(G,\\{a_{1},\ldots,a_{m}\\},b_{1},b_{2})$ be an $m$-rooted graph. Then the
following statements hold.

(i) If $G$ is $(2m+4)$-connected, then
$(G,\\{a_{1},\ldots,a_{m}\\},b_{1},b_{2})$ is 2-feasible.

(ii) If $G$ is $(2m+5)$-connected, then
$(G,\\{a_{1},\ldots,a_{m}\\},b_{1},b_{2})$ is (2,2)-feasible.

######  Proof.

To prove this, we first show that if
$(G,\\{a_{1},\ldots,a_{m}\\},b_{1},b_{2})$ is $(2m+4)$-connected, then $G$
satisfies the condition of Theorem 2.3. That is, there is no
$\\{a_{1},\ldots,a_{m},b_{1},b_{2}\\}$-collection $\mathcal{X}$ in $G$ such
that $|N_{G}(X)|\leq m+2$ for all $X\in\mathcal{X}$, and
$e(\mathcal{G}|\mathcal{X})\leq(m+2)v(\mathcal{G}|\mathcal{X})-m^{2}/2-5m/2-4$.
Suppose not, then $\mathcal{X}=\emptyset$ or
$\mathcal{X}=V(G)\setminus\\{a_{1},\ldots,a_{m},b_{1},b_{2}\\}$ by the
connectivity of $G$. This yields

$e(\mathcal{G}|\mathcal{X})\geq\left\\{\begin{aligned}
&(m+2)v(\mathcal{G}|\mathcal{X}),\ \text{if}\ \mathcal{X}=\emptyset,\\\
&(m+2)(m+1)/2,\ \text{if}\
\mathcal{X}=V(G)\setminus\\{a_{1},\ldots,a_{m},b_{1},b_{2}\\}.\end{aligned}\right.$

In each case,
$e(\mathcal{G}|\mathcal{X})>(m+2)v(\mathcal{G}|\mathcal{X})-m^{2}/2-5m/2-4$, a
contradiction. So if $G$ is $(2m+4)$-connected, then $G$ satisfies the
condition of Theorem 2.3.

(i) If not, then $a_{1},\ldots,a_{m},b_{1},b_{2}$ have a common neighbour in
$G-\\{a_{1},\ldots,a_{m},b_{1},b_{2}\\}$ by Theorem 2.3 (i). So
$G-\\{a_{1},\ldots,a_{m}\\}$ has a $b_{1}$-$b_{2}$ path $P$ of order three,
and then $G-V(P)$ is $(2m+1)$-connected. This implies
$(G,\\{a_{1},\ldots,a_{m}\\},b_{1},b_{2})$ is 2-feasible, a contradiction.
Thus (i) holds.

(ii) If not, then $a_{1},\ldots,a_{m}$ have a common neighbour $v$ in
$G-\\{a_{1},\ldots,a_{m},b_{1},b_{2}\\}$ by Theorem 2.3 (ii). Let
$G^{\prime}=G-\\{v\\}$. Then $G^{\prime}$ is $(2m+4)$-connected. By Theorem
2.3 (ii) again, $a_{1},\ldots,a_{m}$ still have a common neighbour in
$G^{\prime}-\\{a_{1},\ldots,a_{m},b_{1},b_{2}\\}$. This implies that $G$ has a
block $B$ of size $m+2$ containing $\\{a_{1},\ldots,a_{m}\\}$. Since $G$ is
$(2m+5)$-connected, $G-V(B)$ is $(m+3)$-connected, and then
$(G,\\{a_{1},\ldots,a_{m}\\},b_{1},b_{2})$ is (2,2)-feasible, which
contradicts our assumption. Thus (ii) holds.

###### Corollary 2.2

Let $m$ be a positive integer and let $G$ be a $(2m+8)$-connected graph. For
two distinct edges $\\{e_{1},e_{2}\\}\subseteq E(G)$, and any distinct
vertices $a_{1},\ldots,a_{m}$ of $G-V(\\{e_{1},e_{2}\\})$, there exists a
cycle $C$ in $G$ such that $\\{e_{1},e_{2}\\}\subseteq E(C)$ and $G-V(C)$ has
a component containing $\\{a_{1},\ldots,a_{m}\\}$.

######  Proof.

On the contrary, suppose this lemma fails. Write $e_{1}=b_{1}b_{2}$,
$e_{2}=b_{3}b_{4}$, $A=\\{a_{1},\ldots,a_{m}\\}$ and
$B=\\{b_{1},b_{2},b_{3},b_{4}\\}$. Since $G$ is $(2m+8)$-connected, there
exist four distinct vertices, say $u_{1},u_{2},u_{3},u_{4}$, in $G-(A\cup B)$
such that $u_{j}\in N_{G}(b_{j})$ for each $j\in[4]$. For any $i\in\\{1,3\\}$,
let $G^{\prime}$ be the graph obtained from $G$ by deleting all edges between
$b_{i},b_{i+1}$ and $N_{G}(b_{i})\cup
N_{G}(b_{i+1})\cup\\{u_{i},u_{i+1}\\}\setminus(N_{G}(b_{i})\cap
N_{G}(b_{i+1}))$, and contracting $b_{i},b_{i+1}$ to a vertex
$b_{i}^{\prime}$. Then the following results hold.

###### Claim 2.1

$G^{\prime}-\\{b_{1}^{\prime},b_{3}^{\prime}\\}$ is $(2m+4)$-connected, and
for any $i\in\\{1,3\\}$, we have $d_{G^{\prime}-A}(b_{i}^{\prime})\geq 2$.

###### Claim 2.2

For any $x\in\\{b_{1},b_{2}\\}$ and $y\in\\{b_{3},b_{4}\\}$, $x,y$ have no
common neighbour in $G-(A\cup B)$.

######  Proof.

Suppose for a contradiction that there exists $v\in G-(A\cup B)$ such that
$xv,yv\in E(G)$. One may assume $x=b_{1}$ and $y=b_{3}$. Let
$G^{\prime}=G-\\{v,b_{1},b_{3}\\}$. Then $(G^{\prime},A,b_{2},b_{4})$ is
$(2m+5)$-connected. So, this lemma follows from Corollary 2.1 (i) and our
assumption.

###### Claim 2.3

If $G^{\prime}$ (resp. $G$) has a component $H$ containing $A$, then
$G^{\prime}-V(H)$ (resp. $G-V(H)$) has no block containing
$\\{b_{1}^{\prime},b_{3}^{\prime}\\}$ (resp. $B$). That is,
$(G^{\prime},A,b_{1}^{\prime},b_{3}^{\prime})$ is not 2-feasible.

######  Proof.

Suppose not, then by Menger’s Theorem and our construction, $G-V(H)$ has a
cycle $C$ such that $\\{e_{1},e_{2}\\}\subseteq E(C)$, which contradicts our
assumption. Hence this claim holds.

Let $\mathcal{G^{\prime}}=(G^{\prime},A,b_{1}^{\prime},b_{3}^{\prime})$.
Combining Claim 2.3 with Theorem 2.3 (i) and the connectivity of $G$ implies
that there is a $A\cup\\{b_{1}^{\prime},b_{3}^{\prime}\\}$-collection
$\mathcal{X}$ in $G^{\prime}$ such that $|N_{G^{\prime}}(X)|\leq m+2$ for all
$X\in\mathcal{X}$, and
$e(\mathcal{G^{\prime}}|\mathcal{X})\leq(m+2)v(\mathcal{G^{\prime}}|\mathcal{X})-m^{2}/2-5m/2-4$.
By Claim 2.1 ($G^{\prime}-\\{b_{1}^{\prime},b_{3}^{\prime}\\}$ is
$(2m+4)$-connected), $\mathcal{X}=\emptyset$, or
$V(G^{\prime})\setminus\\{A,b_{1}^{\prime},b_{3}^{\prime},u,v\\}\subseteq\mathcal{X}$
with $\\{u,v\\}\subseteq
V(G^{\prime})\setminus\\{A,b_{1}^{\prime},b_{3}^{\prime}\\}$. Further, Claims
2.1 and 2.2 state that
$d_{G^{\prime}-\\{A,b_{1}^{\prime},b_{3}^{\prime},u,v\\}}(b_{1}^{\prime}\cup
b_{3}^{\prime})\geq 2$. So $\mathcal{X}=\emptyset$ or
$\mathcal{X}=V(G^{\prime})\setminus\\{A,b_{1}^{\prime},b_{3}^{\prime}\\}$.
This yields

$e(\mathcal{G^{\prime}}|\mathcal{X})\geq\left\\{\begin{aligned}
&(m+2)(v(\mathcal{G^{\prime}}|\mathcal{X})-2)+4,\ \text{if}\
\mathcal{X}=\emptyset,\\\ &(m+2)(m+1)/2,\ \text{if}\
\mathcal{X}=V(G^{\prime})\setminus\\{A,b_{1}^{\prime},b_{3}^{\prime}\\},\end{aligned}\right.$

a contradiction. So this corollary holds.

##  3 Proof of Theorem 2.3

We will prove Theorem 2.3 by induction on $m$. The main induction step will be
given in this section.

###### Lemma 3.1

Let $(G,\\{a_{1},a_{2}\\},b_{1},b_{2})$ be a 2-rooted graph with $v(G)\geq 5$.
If $a_{1}$ and $a_{2}$ have no common neighbour in $G-\\{b_{1},b_{2}\\}$, then
$G$ has a $\\{a_{1},a_{2},b_{1},b_{2}\\}$-collection $\mathcal{X}$ such that
$|N_{G}(X)|\leq 4$ for all $X\in\mathcal{X}$, and
$e(\mathcal{G}|\mathcal{X})\leq 4v(\mathcal{G}|\mathcal{X})-11$.

######  Proof.

On the contrary, suppose this lemma fails. Let $G^{\prime}$ be a graph
obtained from $G$ by deleting all edges connecting $a_{1},a_{2}$, and
contracting $a_{1},a_{2}$ to a vertex $a^{*}$. Write
$\mathcal{G^{\prime}}=(G^{\prime},\\{a^{*},u\\},b_{1},b_{2})$, where $u\in
V(G)\setminus\\{a_{1},a_{2},b_{1},b_{2}\\}$ ($u$ exists since $v(G)\geq 5$).
According to Theorem 2.1, either $\mathcal{G^{\prime}}$ is feasible, or there
is some $\\{a^{*},u,b_{1},b_{2}\\}$-collection $\mathcal{X}$ in $G^{\prime}$
such that $|N_{G^{\prime}}(X)|\leq 3$ for all $X\in\mathcal{X}$ and
$(G^{\prime}/\mathcal{X},a^{*},b_{1},u,b_{2})$ is planar. By the definition
and construction, $\mathcal{G^{\prime}}$ is not feasible. So $\mathcal{X}$ is
a $\\{a_{1},a_{2},b_{1},b_{2}\\}$-collection in $G$ such that
$|N_{G}(X)|\leq|N_{G^{\prime}}(X)|+1\leq 4$ for all $X\in\mathcal{X}$, and

$\displaystyle E(\mathcal{G}|\mathcal{X})=$ $\displaystyle\\{a_{1}w:w\in
N_{\mathcal{G}|\mathcal{X}}(a_{1})\setminus\\{a_{2},b_{1},b_{2}\\}\\}\cup\\{a_{2}w:w\in
N_{\mathcal{G}|\mathcal{X}}(a_{2})\setminus\\{a_{1},b_{1},b_{2}\\}\\}$
$\displaystyle\cup
E(G^{\prime}/\mathcal{X})\cup\\{a_{1}a_{2},a_{1}b_{1},a_{1}b_{2},a_{2}b_{1},a_{2}b_{2}\\}.$

Because $a_{1}$ and $a_{2}$ have no common neighbour in $G-\\{b_{1},b_{2}\\}$,
it follows that

$|\\{a_{1}w:w\in
N_{\mathcal{G}\mid\mathcal{X}}(a_{1})\setminus\\{a_{2},b_{1},b_{2},u\\}\\}\cup\\{a_{2}w:w\in
N_{\mathcal{G}\mid\mathcal{X}}(a_{2})\setminus\\{a_{1},b_{1},b_{2},u\\}\\}|\leq
v(\mathcal{G}|\mathcal{X})-5,$

and one may assume $a_{1}u\notin E(G)$. Let $G^{\prime\prime}$ be a graph
obtained from $G^{\prime}$ by replacing $a^{*}$ by $a_{1}$, and deleting edges
$a^{*}u$, $a_{1}w$ with $w\in
N_{\mathcal{G}\mid\mathcal{X}}(a_{1})\setminus\\{a_{2},b_{1},b_{2}\\}$.
Further, we can draw edges $a_{1}b_{1}$, $a_{1}b_{2}$, $a_{1}a_{2}$,
$a_{2}b_{1}$, $a_{2}b_{2}$ and $a_{2}u$ outside the disk, without introducing
edge crossings. So
$(G^{\prime\prime}/\mathcal{X})\cup\\{a_{1}a_{2},a_{1}b_{1},a_{1}b_{2},a_{2}b_{1},a_{2}b_{2}\\}$
is planar. By Euler formula,

$e((G^{\prime\prime}/\mathcal{X})\cup\\{a_{1}a_{2},a_{1}b_{1},a_{1}b_{2},a_{2}b_{1},a_{2}b_{2},a_{2}u\\})\leq
3v(G/\mathcal{X})-6.$

This implies $e(\mathcal{G}|\mathcal{X})\leq
3v(G/\mathcal{X})-6+v(G/\mathcal{X})-5=4v(\mathcal{G}|\mathcal{X})-11$, as
required.

###### Lemma 3.2

Theorem 2.3 holds for $m=0,1,2$.

######  Proof.

Let $\mathcal{G}=(G,\\{a_{1},\ldots,a_{m}\\},b_{1},b_{2})$ be an $m$-rooted
graph. We first prove that Theorem 2.3 (ii) holds for $m=0,1,2$. If not, then
$m=0,1$ by Lemma 3.1 and the condition of Theorem 2.3, and $G$ has no two
internally $b_{1}$-$b_{2}$ paths in $G-\\{a_{m}\\}$. So, there exists a vertex
$u$ in $G-\\{b_{1},b_{2},a_{m}\\}$ such that $G-\\{u,a_{m}\\}$ has no
$b_{1}$-$b_{2}$ path ($u$ exists since $v(G)\geq m+3$). Let $D$ be the
component of $G-\\{u,a_{m}\\}$ containing $b_{1}$; so $b_{2}\notin V(D)$.
Write $\mathcal{X}=\\{V(D)-\\{b_{1}\\},V(G-D)-\\{u,b_{2},a_{m}\\}\\}$. Then
$|N_{G}(X)|\leq m+2$ for all $X\in\mathcal{X}$ and

$e(\mathcal{G}|\mathcal{X})\leq\left\\{\begin{aligned} &2=2\cdot
v(G|\mathcal{X})-4,\ \text{if}\ m=0,\\\ &5=3\cdot v(G|\mathcal{X})-7,\
\text{if}\ m=1.\end{aligned}\right.$

This contradicts the condition of Theorem 2.3. So Theorem 2.3 (ii) holds for
$m=0,1,2$. It remains to show that Theorem 2.3 (i) also holds for $m=0,1,2$.

On the contrary, suppose that $\mathcal{G}$ does not satisfy Theorem 2.3 (i).
For $m=0,1$, $G$ has no $b_{1}$-$b_{2}$ path in $G-\\{a_{m}\\}$. Let $D$ be
the component of $G-\\{a_{m}\\}$ containing $b_{1}$; so $b_{2}\notin V(D)$.
Let $\mathcal{X}=\\{V(D)-\\{b_{1}\\},V(G-D)-\\{a_{m},b_{2}\\}\\}$. Then
$|N_{G}(X)|\leq m+2$ for all $X\in\mathcal{X}$ and

$e(\mathcal{G}|\mathcal{X})\leq\left\\{\begin{aligned} &0=2\cdot
v(G\mid\mathcal{X})-4,\ \text{if}\ m=0,\\\ &2=3\cdot v(G\mid\mathcal{X})-7,\
\text{if}\ m=1.\end{aligned}\right.$

This contradicts the condition of Theorem 2.3. So Theorem 2.3 (i) holds for
$m=0,1$.

For $m=2$, Lemma 3.1 and the condition of Theorem 2.3 state that $a_{1}$ and
$a_{2}$ have a common neighbour $v$ in $G-\\{b_{1},b_{2}\\}$. By our
assumption, one may assume $vb_{1}\notin E(G)$. Let $G^{\prime}=G-\\{v\\}$ and
$\mathcal{G^{\prime}}=(G^{\prime},\\{a_{1},a_{2}\\},b_{1},b_{2})$. According
to Theorem 2.1, either $\mathcal{G^{\prime}}$ is feasible, or there is some
$\\{a_{1},a_{2},b_{1},b_{2}\\}$-collection $\mathcal{X}$ in $G^{\prime}$ such
that $|N_{G^{\prime}}(X)|\leq 3$ for all $X\in\mathcal{X}$ and
$(G^{\prime}/\mathcal{X},a_{1},b_{1},a_{2},b_{2})$ is planar. If the former
holds, then $G$ is 2-feasible. If the latter holds, then
$|N_{G}(X)|\leq|N_{G^{\prime}}(X)|+1\leq 4$ for all $X\in\mathcal{X}$, and we
can draw edges $a_{1}b_{1}$, $a_{1}b_{2}$, $a_{1}a_{2}$, $a_{2}b_{1}$,
$a_{2}b_{2}$, $a_{2}v$, $a_{1}v$ and $b_{2}v$ such that
$(\mathcal{G}^{\prime}/\mathcal{X})\cup\\{a_{2}v,a_{1}v,b_{2}v\\}$ is planar.
It follows from Euler formula that
$e((\mathcal{G}^{\prime}/\mathcal{X})\cup\\{a_{2}v,a_{1}v,b_{2}v\\})\leq
3v(G/\mathcal{X})-6$. Then

$\displaystyle e(\mathcal{G}|\mathcal{X})\leq e(\mathcal{G}/\mathcal{X})\leq
e(\mathcal{G^{\prime}/X})+d_{\mathcal{G}/\mathcal{X}}(v)\leq
3v(G/\mathcal{X})-6+v(G/\mathcal{X})-5=4v(G|\mathcal{X})-11.$

This contradicts the condition of Theorem 2.3 again. This completes the proof
of this lemma.

We now proceed by induction on $m$. Owing to Lemma 3.2, Theorem 2.3 holds for
$m=0,1,2$. Suppose $m\geq 3$, Theorem 2.3 hold for all $(m-1)$-rooted graphs.
Let $G^{\prime}$ be the graph obtained from $G$ by deleting all edges between
$a_{1},a_{2}$ and $N_{G}(a_{1})\cup N_{G}(a_{2})\setminus(N_{G}(a_{1})\cap
N_{G}(a_{2}))$, and contracting $a_{1},a_{2}$ to a vertex $a^{*}$. Let
$\mathcal{G^{\prime}}=(G^{\prime},\\{a^{*},a_{3},\ldots,a_{m}\\},b_{1},b_{2})$.
We see that if $\mathcal{G^{\prime}}$ satisfies the condition of Theorem 2.3,
then by induction, the following two statements hold.

(i) Either $a^{*},a_{3},\ldots,a_{m},b_{1},b_{2}$ have a common neighbour in
$G-\\{a^{*},a_{3},\ldots,a_{m},b_{1},b_{2}\\}$, or $\mathcal{G^{\prime}}$ is
$2$-feasible.

(ii) Either $a^{*},a_{3},\ldots,a_{m}$ have a common neighbour in
$G-\\{a^{*},a_{3},\ldots,a_{m},b_{1},b_{2}\\}$, or $\mathcal{G^{\prime}}$ is
$(2,2)$-feasible.

By our construction, Theorem 2.3 holds. Note that $v(G^{\prime})\geq m+2$, so
$G^{\prime}$ has a $\\{a^{*},a_{3},\ldots,a_{m},b_{1},b_{2}\\}$-collection
$\mathcal{X}$ such that $|N_{G^{\prime}}(X)|\leq m+1$ for all
$X\in\mathcal{X}$, and

$\displaystyle e(\mathcal{G}^{\prime}|\mathcal{X})$
$\displaystyle\leq(m+1)v(\mathcal{G}^{\prime}|\mathcal{X})-(m-1)^{2}/2-5(m-1)/2-4$
$\displaystyle=(m+1)v(\mathcal{G}|\mathcal{X})-m^{2}/2-5m/2-3.$

It is easy to see that $\mathcal{X}$ is a
$\\{a_{1},a_{2},a_{3},\ldots,a_{m},b_{1},b_{2}\\}$-collection in $G$ such that
$|N_{G}(X)|\leq|N_{G^{\prime}}(X)|+1\leq m+2$ for all $X\in\mathcal{X}$. By
the definition of $\mathcal{G}|\mathcal{X}$, we obtain

$E(\mathcal{G}|\mathcal{X})=E(\mathcal{G^{\prime}}|\mathcal{X})\cup\\{a_{i}w:w\in
N_{\mathcal{G}|\mathcal{X}}(a_{i})\cup\\{a_{3},\ldots,a_{m},b_{1},b_{2}\\}\text{
with }i\in[2]\\}.$

Note that

$\displaystyle|\\{a_{i}w:w\in
N_{\mathcal{G}|\mathcal{X}}(a_{i})\cup\\{a_{3},\ldots,a_{m},b_{1},b_{2}\\}\text{
with }i\in[2]\\}|$ $\displaystyle\leq$
$\displaystyle(v(\mathcal{G}|\mathcal{X})-1)+|\\{a_{j}w:w\in
N_{\mathcal{G}|\mathcal{X}}(a_{1})\cap
N_{\mathcal{G}|\mathcal{X}}(a_{2})\text{ for some }j\in[2]\\}|$

and

$\\{a_{j}w:w\in N_{\mathcal{G}|\mathcal{X}}(a_{1})\cap
N_{\mathcal{G}|\mathcal{X}}(a_{2})\text{ for some }j\in[2]\\}\subseteq
E(\mathcal{G^{\prime}|X}).$

This implies $e(\mathcal{G}|\mathcal{X})\leq
e(\mathcal{G^{\prime}}|\mathcal{X})+v(\mathcal{G}|\mathcal{X})-1\leq(m+2)v(\mathcal{G}|\mathcal{X})-m^{2}/2-5m/2-4$,
a contradiction. So Theorem 2.3 holds.

##  4 Proof of Theorems 1.2 and 1.4

In this section, we will show Theorems 1.2 and 1.4. We first give a
definition. For two sequences $(a_{1},\cdots,a_{l})$ and
$(b_{1},\cdots,b_{l^{\prime}})$ with $l<l^{\prime}$ and $a_{i}$ = $b_{i}$ for
any $i\in[l]$, we regard that $(b_{1},\cdots,b_{l^{\prime}})$ is larger than
$(a_{1},\cdots,a_{l})$ _in lexicographic order_.

In what follows, let $G$ be a graph and $\\{a_{1},\ldots,a_{m}\\}\subseteq
V(G)$, where $m$ is a positive integer. Write $A=\\{a_{1},\ldots,a_{m}\\}$.
Now we show the following result holds.

###### Lemma 4.1

Let $G$ be a $(2m+6)$-connected graph and let $e_{1},e_{2}$ be two distinct
edges of $G-A$. If there exists a cycle $C$ in $G$ such that
$\\{e_{1},e_{2}\\}\subseteq E(C)$ and $G-V(C)$ has a block (resp. component)
$B$ containing $A$, then there exists a cycle $C^{\prime}$ in $G-A$ such that
$\\{e_{1},e_{2}\\}\subseteq E(C^{\prime})$ and $G-V(C^{\prime})$ is
2-connected (resp. connected).

######  Proof.

On the contrary, suppose this lemma fails. Let $B_{1},\ldots,B_{t}$ denote the
components of $G-V(C\cup B)$ such that $v(B_{1})\geq\cdots\geq v(B_{t})$. We
choose $C$ and $B$ such that

$(v(B),v(B_{1}),\cdots,v(B_{t}))\text{ is as large as possible in
lexicographic order.}$

By deleting $e_{1},e_{2}$, we have two disjoint paths, which denoted by
$P_{1}$ and $P_{2}$. Further, for each $i\in[2]$, assume that $P_{i}$ is a
path from $b_{i}$ to $b_{i}^{\prime}$. By our choice, the following result
holds.

###### Claim 4.1

Suppose that $u_{1},v_{1},u_{2},v_{2}$ are four distinct vertices and
$u_{i},v_{i}$ occur along the orientation of $P_{i}$ for each $i\in[2]$. If
$\\{u_{1}v_{2},u_{2}v_{2}\\}\subseteq E(G)$, then $u_{i}v_{i}\in E(P_{i})$ for
each $i\in[2]$. Furthermore, $P_{1},P_{2}$ are induced.

For each $i\in[2]$, let $c_{i},d_{i}\in N_{P_{i}}(B_{t})$ with
$P_{i}[c_{i},d_{i}]$ maximal. Next we will perform the following operation,
and we shall update the vertices $c_{i}$ and $d_{i}$ for some $i\in[2]$ at
each step.
Operation 1. For each $i\in[2]$, we define

$Q_{i}=\left\\{\begin{aligned} &P_{i}[b_{i},c_{i})\cup
P_{i}(d_{i},b_{i}^{\prime}],\ \text{if}\ c_{i}\text{ and }d_{i}\text{
exist},\\\ &P_{i},\ \text{otherwise}.\end{aligned}\right.$

Suppose that there exists an edge $uv$ connecting $Q_{i}$ and
$P_{j}(c_{j},d_{j})$ for $\\{i,j\\}=\\{1,2\\}$ with $u\in V(Q_{i})$ and $v\in
V(P_{j}(c_{j},d_{j}))$. If $u\in V(P_{i}[b_{i},c_{i}))$, then we regard $u$ as
the new $c_{i}$, and if $u\in V(P_{i}(d_{i},b_{i}^{\prime}])$, then we regard
$u$ as the new $d_{i}$. If $c_{i}$ and $d_{i}$ do not exist, then let
$c_{i}=d_{i}=u$ as the new $c_{i}$ and $d_{i}$.

We perform Operation 1 as many times as possible. Note that it must stop,
since $V(Q_{1}\cup Q_{2})$ becomes smaller in each step. For the last
$c_{i}$’s and $d_{i}$’s, let $C=\\{c_{i}:i\in[2]\text{ and }c_{i}\text{
exists}\\}$ and $D=\\{d_{i}:i\in[2]\text{ and }d_{i}\text{ exists}\\}$. By the
definition of $C$ and $D$, we obtain the following claim.

###### Claim 4.2

There exists no edge connecting $Q_{1}\cup Q_{2}$ and $P_{1}(c_{1},d_{1})\cup
P_{2}(c_{2},d_{2})\cup B_{t}$.

The following claim is crucial for our proof.

###### Claim 4.3

The following statements hold.

(i) For any $j\in[t-1]$, there exists no edge connecting $B_{j}$ and
$P_{1}(c_{1},d_{1})\cup P_{2}(c_{2},d_{2})$.

(ii) For any $i\in[2]$, there exists a vertex $z_{i}$ in $B$ such that there
exist no edges connecting $B-\\{z_{i}\\}$ and $P_{i}(c_{i},d_{i})$.

######  Proof.

For each $i\in[2]$, let $u_{i,1},u_{i,2}\in N_{P_{i}}(B_{t})$ with
$P_{i}[u_{i,1},u_{i,2}]$ maximal. On the contrary, suppose that (i) and (ii)
fail. Now we prove that there exists some $j\in[2]$ such that $u_{j,1}\in C$.
If not, then Operation 1 yields that there exist two vertices
$c_{i}^{\prime}\in P_{i}[b_{i},u_{i,1})$ and $w_{j}\in
P_{j}(u_{j,1},b_{j}^{\prime})$ such that $c_{i}^{\prime}w_{j}\in E(G)$ with
$\\{i,j\\}=\\{1,2\\}$. One may assume $i=1$ and $j=2$. Let $c_{2}^{\prime}$ be
the first vertex we get by updating $u_{2,1}$. Then $c_{2}^{\prime}\in
P_{2}[b_{2},u_{2,1})$ and there exists $w_{1}\in
P_{1}(c_{1}^{\prime},b_{1}^{\prime})$ such that $c_{2}^{\prime}w_{1}\in E(G)$,
which contradicts Claim 4.1. Therefore, there exists some $j\in[2]$ such that
$u_{j,1}\in C$. Similarly, there also exists some $j^{\prime}\in[2]$ such that
$u_{j^{\prime},2}\in D$. Then the proof will be divided into the following two
cases.

Case 1: $j\neq j^{\prime}$.

One may assume $j=2$ and $j^{\prime}=1$. By our assumption and symmetry,
assume $u_{1,1}\notin C$. It follows from Operation 1 that $c_{1}$ has an
neighbour $w_{2}$ in $P_{2}(u_{2,1},d_{2})$. So, $G[V(B_{t}\cup
C)-P_{1}(c_{1},u_{1,2})]$ has a cycle $C^{\prime}$ such that
$\\{e_{1},e_{2}\\}\subseteq E(C^{\prime})$. By our choice, this claim holds.

Case 2: $j=j^{\prime}$.

One may assume $j=j^{\prime}=1$. Then this claim holds for $i=1$. It follows
from Case 1 that $u_{2,1}\notin C$ and $u_{2,2}\notin D$. By Operation 1, both
$c_{2}$ and $d_{2}$ have an neighbour in $P_{1}(u_{1,1},u_{1,2})$. Suppose
that $\\{w_{1},w_{2}\\}\subseteq V(P_{1}(u_{1,1},u_{1,2}))$ and
$\\{c_{2}w_{1},d_{2}w_{2}\\}\subseteq E(G)$. Owing to Claim 4.1, $w_{1},w_{2}$
occur along the orientation of $P_{1}$. So, $G[V(B_{t}\cup
C)-P_{2}(c_{2},d_{2})]$ has a cycle $C^{\prime}$ such that
$\\{e_{1},e_{2}\\}\subseteq E(C^{\prime})$. By our choice, this claim holds
for $i=2$.

For each $i\in[2]$, we take the vertex $z_{i}$ as in Claim 4.3 (ii), and let
$z$ be a cut vertex of $G$ which separates $B_{t}$ and $B$. By Claims 4.2 and
4.3, $C\cup D\cup\\{z_{1},z_{2},z\\}$ separates $P_{1}(c_{1},d_{1})\cup
P_{2}(c_{2},d_{2})\cup B_{t}$ from the other part (Possibly, $z_{1}=z_{2}$ and
$z\in\\{z_{1},z_{2}\\}$). Combining this with the connectivity of $G$ implies

$G-(C\cup D\cup\\{z_{1},z_{2},z\\})-(P_{1}(c_{1},d_{1})\cup
P_{2}(c_{2},d_{2})\cup B_{t})=\emptyset.$

So $\\{b_{1},b_{1}^{\prime}\\}\subseteq C$ and
$\\{b_{2},b_{2}^{\prime}\\}\subseteq D$. By the connectivity of $G$, there
exists an edge connecting $B_{j^{\prime}}$ and
$P_{1}(b_{1},b_{1}^{\prime})\cup P_{2}(b_{2},b_{2}^{\prime})$ for some
$j^{\prime}\in[t-1]$ or there are at least two edges connecting $B$ and
$P_{j}(b_{j},b_{j}^{\prime})$ for some $j\in[2]$, which contradicts Claim 4.3.
This proves this lemma.

######  Proof.

of Theorem 1.4. Theorem 1.4 follows easily from Corollary 2.2 and Lemma 4.1.

######  Proof.

of Theorem 1.2. Let us first show $f(m,2,2)\leq 2m+6$. Let $G$ be a
$(2m+6)$-connected graph. Owing to Corollary 2.1 (ii), $(G,A,b_{1},b_{2})$ is
(2,2)-feasible. Therefore, there exists a cycle $C$ containing $b_{1},b_{2}$,
and $G-V(C)$ has a block containing $A$. By the connectivity of $G$ and Lemma
4.1, this result holds.

Now we will prove $f(m,1,2)\leq 2m+4$. Let $G$ be a $(2m+4)$-connected graph
and suppose that $G$ does not satisfy Theorem 1.2. Owing to Corollary 2.1 (i),
$(G,A,\\{b_{1},b_{2}\\})$ is $2$-feasible. This implies that there is a
$b_{1}$-$b_{2}$ path $P$ and $A$ is contained in a block $B$ of $G-V(P)$. Let
$B_{1},\ldots,B_{t}$ denote the components of $G-V(P\cup B)$ such that
$v(B_{1})\geq\cdots\geq v(B_{t})$. We choose $P$ and $B$ such that
($v(B),v(B_{1}),\ldots,v(B_{t})$) is as large as possible in lexicographic
order.

It suffices to show that $t=0$. On the contrary, suppose $t\geq 1$. Since $G$
is $(2m+4)$-connected, there exist $2m+4$ disjoint paths from $B_{t}$ to
$B,B_{1},\ldots,B_{t-1}$. Let $u_{1},u_{2}\in N_{P}(B_{t})$ with
$P[u_{1},u_{2}]$ maximal, and either $u_{1}^{\prime},u_{2}^{\prime}\in
N_{P(u_{1},u_{2})}(B)$ or $u_{1}^{\prime}\in N_{P(u_{1},u_{2})}(B_{j})$ for
some $j\in[t-1]$. Then we get a contradiction against our choice. This
completes the proof.

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# Phase transitions in the Haldane-Hubbard model

Wan-Xiu He  Beijing Computational Science Research Center, Beijing 100084,
China  Rubem Mondaini<EMAIL_ADDRESS>Beijing Computational Science
Research Center, Beijing 100084, China  Hong-Gang Luo  School of Physical
Science and Technology, Lanzhou University, Lanzhou 730000, China  Lanzhou
Center for Theoretical Physics $\&$ Key Laboratory of Theoretical Physics of
Gansu Province, Lanzhou University, Lanzhou 730000, China  Beijing
Computational Science Research Center, Beijing 100084, China  Xiaoqun Wang
School of Physics, Zhejiang University, Hangzhou, 310058, China  Shijie Hu
<EMAIL_ADDRESS>Beijing Computational Science Research Center, Beijing
100084, China  Department of Physics, Beijing Normal University, Beijing,
100875, China

###### Abstract

The Haldane-Hubbard model is a prime example of the combined effects of band
topology and electronic interaction. We revisit its spinful phase diagram at
half-filling as a consensus on the presence of SU($2$) symmetry is currently
lacking. To start, we utilize the Hartree-Fock mean-field method, which offers
a direct understanding of symmetry breaking through the effective mass term
that can acquire spin dependence. Our results, in agreement with previous
studies, provide an instructive insight into the regime where the Chern number
$C=1$, with only one spin species remaining topological. Besides that, we
numerically study the phase diagram of the Haldane-Hubbard model via a large-
scale infinite-density matrix renormalization group (iDMRG) method. The phase
boundaries are determined by the Chern number and the correlation lengths
obtained from the transfer-matrix spectrum. Unlike previous studies, the iDMRG
method investigates the Haldane-Hubbard model on a thin and infinitely long
cylinder and examines scenarios consistent with the two-dimensional
thermodynamic limit. Here, the phase diagram we obtained qualitatively goes
beyond the Hartree-Fock scope, particularly in the $C=1$ region, and serves as
a quantitative benchmark for further theoretical and experimental
investigations.

##  I Introduction

The study of topologically ordered states has garnered significant attention
in condensed matter physics, especially since the discovery of the quantum
Hall effect (QHE) [1, 2, 3]. This field aims to understand the intricate
interplay between topology and electronic properties in materials [4, 5],
leading to the emergence of novel phenomena and potential applications in
quantum computing [6] and spintronics [7, 8]. Haldane’s original proposal
suggested that a honeycomb lattice model with electrons hopping on it could
exhibit QHE without requiring Landau levels, indicating the potential for non-
trivial topology in basic band insulators [9]. This discovery culminated in
its experimental realization using ultracold atoms [10] and sparked a surge of
interest in studying systems with both topologically non-trivial band
structures and electron-electron interactions. In recent years, there has been
a particular focus on the Haldane model [11, 12, 13, 14, 15, 16, 17, 18, 19,
20, 21, 22, 23, 24], as well as other models such as the Kane-Mele [25, 26,
27, 28] and Bernevig-Hughes-Zhang models [29, 30, 31].

Concerning the spinful Haldane-Hubbard model and its variations, the
exploration of the topological phases and phase transitions has been performed
by various numerical methods, including the static mean-field (MF) theory [13,
14, 17, 20, 19], exact diagonalization (ED) [17, 20, 24], dynamical mean-field
theory (DMFT) [17] as well as the cluster variations [16, 15], and bold
diagrammatic Monte Carlo (BDMC) technique [32]. When the ground state exhibits
non-trivial topology, a Chern number $C\neq 0$ exists, mapping the number of
protected chiral edge modes in the presence of open boundary conditions (OBC).
When the interacting (Hubbard) term is turned off, the topological insulator
of the spinful Hamiltonian leads to a total Chern number $C=2$ (CI2), wherein
each spin species contributes with a Chern number $C=1$. Remarkably, an exotic
insulator with a Chern number $C=1$ (CI1) can also emerge, which has been
considered as a result of spontaneous spin-rotation symmetry breaking.
Although the observation of this state has reached a certain consensus [17,
32, 24], there are qualitative and quantitative inconsistencies in the phase
diagrams obtained by different methods. Moreover, a debate persists regarding
the presence of CI1 in the extended Haldane-Hubbard model [20, 19]. As such,
the phase diagram of the Haldane-Hubbard model and the comprehensive
characterization of the associated phase transitions remain largely
unresolved.

Here, we mainly focus on the model at half-filling on a thin and infinitely
long cylinder to address these questions. The infinite density matrix
renormalization group (iDMRG) method is employed to better understand phase
transitions in the model. In addition, we revisit the Hartree-Fock method and
present results that help set the stage for the unbiased ones from iDMRG and
can be used to readily understand its mapping onto the familiar non-
interacting phase diagram of the model [9]. Furthermore, we investigate the
symmetry-broken topological phase CI1 with a Chern number $C=1$.

Our paper is structured as follows. In Sec. II, we introduce the Haldane-
Hubbard model on a honeycomb lattice and provide a detailed explanation of the
two distinct lattice structures that were examined using the iDMRG
simulations. In Sec. III, we present the Hartree-Fock approximation and
results, which are used to benchmark and introduce all phases and phase
transitions. Section IV showcases our main results, specifically the phase
diagram at half-filling with the iDMRG method. Lastly, we conclude our paper
in Sec. V. The appendices contain additional numerical results for other
lattice structures obtained through the iDMRG method, as well as other
parameter sets in mean-field results.

##  II MODEL AND METHOD

On a honeycomb lattice composed of two nested triangular sublattices, A and B,
the Hamiltonian of the Haldane-Hubbard model (HHM) reads

$\displaystyle\hat{H}$
$\displaystyle=-t_{1}\sum_{\braket{l,l^{\prime}},\sigma}\hat{c}_{l\sigma}^{{\dagger}}\hat{c}_{l^{\prime}\sigma}^{\phantom{{\dagger}}}-t_{2}\sum_{\langle\\!\langle
l,l^{\prime}\rangle\\!\rangle,\sigma}e^{{\rm
i}\phi_{ll^{\prime}}}\hat{c}_{l\sigma}^{{\dagger}}{\hat{c}}_{l^{\prime}\sigma}^{\phantom{{\dagger}}}$
(1)
$\displaystyle+U\sum_{l}\left({\hat{n}}^{\phantom{{\dagger}}}_{l\uparrow}-\frac{1}{2}\right)\left({\hat{n}}^{\phantom{{\dagger}}}_{l\downarrow}-\frac{1}{2}\right)+\delta\sum_{l,\sigma}s^{\phantom{{\dagger}}}_{l}\hat{n}^{\phantom{{\dagger}}}_{l\sigma},$

where $\hat{c}_{l\sigma}^{{\dagger}}$
($\hat{c}_{l\sigma}^{\phantom{{\dagger}}}$) is the creation (annihilation)
operator for an electron with spin $\sigma=\uparrow$, $\downarrow$ at
site-$l$,
$\hat{n}^{\phantom{{\dagger}}}_{l\sigma}=\hat{c}^{{\dagger}}_{l\sigma}{\hat{c}}^{\phantom{{\dagger}}}_{l\sigma}$
the particle number operator, $t_{1}$ and $t_{2}$ the hopping amplitudes on
the nearest-neighbor (NN) and next-nearest-neighbor (NNN) links respectively,
$U>0$ the strength of the onsite repulsion, and $\delta>0$ the staggered
chemical potential with the sign $s^{\phantom{{\dagger}}}_{l}=+1$ ($-1$) for
the sublattice A (B). Lastly, $l$ sums over $N$ unit cells or equivalently
$2N$ sites; $\braket{l,l^{\prime}}$ and $\langle\\!\langle
l,l^{\prime}\rangle\\!\rangle$ run over all NN and NNN links – see Fig. 1(a)

Figure 1: (a1) Schematic cartoon marking the relevant terms in the
Hamiltonian, where the NNN links get a phase $e^{+{\rm i}\phi}$ ($e^{-{\rm
i}\phi}$) for clockwise (counterclockwise) directions. This lattice shape is
used in the mean-field calculations, here shown for a linear size
$L_{x}=L_{y}=L=3$. (a2) FBZ for the diamond-shaped cluster and corresponding
valid momenta in the case of $L=6$. The lattice structures for two distinct
cylinders (b1) tZT$L_{y}$-$2$ and (c1) ZT$L_{y}$-$1$ used in the iDMRG
simulations. Both structures have a rolling $n$-site shift, as marked by the
red lattice sites. (b2) According to the condition (7), valid momentum cutting
lines (grey), that thread FBZ, are plotted for the cylinder tZT$4$-$2$.
Similarly, (c2) for ZT$3$-$1$. The high-symmetry points, $\bf\Gamma$, ${\bf
M}_{1}$, ${\bf M}_{2}$, ${\bf M}_{3}$, and ${\bf K}^{\pm}$, in (a2-c2) are
annotated.

In this model, the time-reversal symmetry is broken due to the presence of
complex phase factors $e^{{\rm i}\phi_{ll^{\prime}}}$. Specifically, the sign
of the phase angle $\phi_{ll^{\prime}}$ depends on the hopping direction on
the triangular lattice consisting of NNN links, with a positive (negative)
value for the clockwise (counterclockwise) hopping. Hereafter, $t_{1}=t$ sets
the energy unit, and we choose $t_{2}=0.2t$, and
$|\phi_{ll^{\prime}}|=\phi=\pi/2$. Additionally, we focus our calculations on
the half-filling, meaning there is one electron per site on average. This is
represented by $N_{\uparrow}=N_{\downarrow}=N$, where
$N_{\sigma}=\sum_{l}\braket{\hat{n}^{\phantom{{\dagger}}}_{l\sigma}}$ denotes
the number of electrons for species-$\sigma$.

To start, we describe a few symmetries of the model. First, the Hamiltonian
(1) is invariant under an arbitrary choice of phase $\varphi_{\sigma}$, i.e.,

$\displaystyle\hat{c}^{\phantom{{\dagger}}}_{l\sigma}\rightarrow\hat{c}^{\phantom{{\dagger}}}_{l\sigma}e^{{\rm
i}\varphi_{\sigma}}\quad\text{and}\quad\hat{c}^{{\dagger}}_{l\sigma}\rightarrow{\hat{c}}^{{\dagger}}_{l\sigma}e^{-{\rm
i}\varphi_{\sigma}}\ ,$ (2)

denoting a conservation of U(1) symmetry, i.e., the number of electrons
$N_{\sigma}$ for each spin species. Besides that, under a rotation

$\mathcal{R}(\varphi_{r})=\begin{pmatrix}\phantom{-}\cos{\varphi_{r}}&\sin{\varphi_{r}}\\\
-\sin{\varphi_{r}}&\cos{\varphi_{r}}\end{pmatrix}$ (3)

with an arbitrary angle $\varphi_{r}$, we get a new pair of annihilation
operators
$(\hat{c}_{l\uparrow}^{\prime},\hat{c}_{l\downarrow}^{\prime})^{\text{T}}=\mathcal{R}(\varphi_{r})(\hat{c}^{\phantom{{\dagger}}}_{l\uparrow},\hat{c}^{\phantom{{\dagger}}}_{l\downarrow})^{\text{T}}$,
while the Hamiltonian remains unchanged. Combining the above-mentioned double
U($1$) symmetries with the rotation symmetry yields a total symmetry of the
model (1), characterized by the U($1$)$\times$SU($2$) group for the charge and
spin degrees of freedom, separately. In particular, when $\varphi_{r}=\pi/2$,
the rotation $\mathcal{R}(\pi/2)$ gives

$\displaystyle\hat{c}^{\prime}_{l\uparrow}=\hat{c}^{\phantom{{\dagger}}}_{l\downarrow}\quad\text{and}\quad\hat{c}^{\prime}_{l\downarrow}=-\hat{c}^{\phantom{{\dagger}}}_{l\uparrow}\
.$ (4)

This corresponds to the “time-reversal” operation defined in the space of spin
degrees of freedom, which is anti-unitary, i.e.,
$\mathcal{R}^{2}(\pi/2)=-\mathbb{1}$, so we call it the spin time-reversal
symmetry. Lastly, particle-hole symmetry is also preserved at half-filling.
Regardless of whether the staggered potential and the phase $\phi$ change
sign, the overall physics remains unchanged [17].

On the honeycomb lattice, each sublattice site A has three neighboring
sublattice sites B, connected by vectors ${\bf c}_{1}=(0$, $\sqrt{3}/3)$,
${\bf c}_{2}=(1/2$, $-\sqrt{3}/6)$, ${\bf c}_{3}=(-1/2$, $-\sqrt{3}/6)$ in
two-dimensional coordinates, where $|{\bf c}_{1}|=|{\bf c}_{2}|=|{\bf
c}_{3}|=\sqrt{3}/3$ – see Fig. 1(a1). We can roll the honeycomb lattice onto a
sphere if three edges along ${\bf c}_{1}$, ${\bf c}_{2}$ and ${\bf c}_{3}$
directions share an equal side length. After choosing primitive vectors

$\displaystyle\bm{a}_{1}=\left(\frac{1}{2},\
\frac{\sqrt{3}}{2}\right)\quad\text{and}\quad\bm{a}_{2}=\left(1,\ 0\right)\ ,$
(5)

the first Brillouin zone (FBZ), expanded in reciprocal vectors

$\displaystyle\bm{b}_{1}=\left(0,\
\frac{4\pi}{\sqrt{3}}\right)\quad\text{and}\quad\bm{b}_{2}=\left(2\pi,\
-\frac{2\pi}{\sqrt{3}}\right)\ ,$ (6)

has high-symmetry points relevant to the point group symmetry $D_{6h}$ of the
lattice: two inequivalent $K$-points ${\bf K}^{\pm}$, one $\Gamma$-point
${\bf\Gamma}=(0,0)$, three distinct $M$-points (i.e., ${\bf M}_{1}$, ${\bf
M}_{2}$ and ${\bf M}_{3}$) - see Fig. 1(a2). As both $\delta$ and $\phi$ are
finite in the Hamiltonian (1), the $D_{6h}$ symmetry reduces to a lower
$C_{3}$ one, specifically $120^{\circ}$ rotational symmetry. However, on a
cylinder that we study by the iDMRG method [33, 34, 35, 36, 37], the $C_{3}$
symmetry is also absent.

To carry out the iDMRG simulations, one obtains the cylindrical geometry by
rolling the sheared two-dimensional lattice onto a thin and infinitely long
cylinder. The allowed cutting lines in the FBZ encompass the momentum points
where the gap closure of the low-energy excitations is most relevant for
capturing phase transitions. This procedure is specifically performed when
high-symmetry points are available [12, 20, 24]. Figures 1(b1, c1) illustrate
this process for two different cylinders: zigzag-top (ZT) and tilted-zigzag-
top (tZT), respectively, distinguished by the orientation along the short
direction and the way they are rolled up. For example, we plot the cylinder
tZT$L_{y}$-$2$ in Fig. 1(b1), which has zigzag short-edges and a shift of two
sites. For the cylinder tZT$L_{y}$-$n$, a translation by $L_{y}$ sites along
the short circumference ($y$-axis) is equivalent to a translation by $n$ sites
along the $x$-axis, which selects the momentum ${\bf k}=(k_{x}$, $k_{y})$ as
follows:

$\displaystyle L_{y}\left(\frac{k_{x}}{2}+\frac{\sqrt{3}k_{y}}{2}\right)=2\pi
p+nk_{x}\ ,$ (7)

with an arbitrary integer $p$. Each selected value of $p$ gives a cutting line
of momentum in the FBZ. Previous studies [20, 38] have shown that various
cylinders with distinct $L_{y}$ and $n$ yield momentum cutting lines in FBZ.
This work mainly considers two cylinders, that is, ZT$3$-$1$ and tZT$4$-$2$.
As illustrated in Fig. 1(b2, c2), these strike a balance of encompassing all
the necessary high-symmetry points relevant to the phase transitions, i.e.,
two $K$-points, one $\Gamma$-point, and three $M$-points, while having
sufficiently small variational unit cell (encoded in the values $n=1$ and $2$)
that result in amenable computing times in iDMRG. In Appendix A, we also study
one extra geometry, the cylinder tZT$2$-$1$, which we show to be significantly
impacted by finite-size effects precisely because some of the high-symmetry
points are unavailable. In Sec. IV.4, we further analyze the momentum
resolution of the correlation-length spectra for cylinders tZT$L_{y}$-$1$.

##  III Mean-field analysis

Figure 2: Mean-field results for staggered potential $\delta/t=6$. (a) The
positive-$\phi$ phase diagram in the non-interacting limit ($U=0$) for each
spin species. (b) The effective staggered potential $\delta^{\sigma}_{\rm MF}$
for $\sigma=\uparrow$, $\downarrow$ defined in (21), as a function of the
interaction strength $U$. The shaded region marks the topological regime. (c)
The staggered magnetization order parameter $\mathcal{M}$ and the sublattice
density imbalance $\cal I$ as a function of $U/t$. (d) The charge gap
$\Delta_{c}({\bf k})$ defined in (25) as a function of $U/t$. Here,
contrasting two different system sizes, $L=30$ and $180$ – finite-size effects
are marginal. Inset: FBZ with two $K$-points $\mathbf{K}^{\pm}$ annotated [and
similarly shown in (a) and (b)], and where a gap closing occurs at values of
the interactions $U_{c1}/t=11.28$ and $U_{c2}/t=14.3$, respectively, marking
the two topological transitions. The right vertical axis gives the location
$k_{x}$ of the smallest gap $\Delta_{c}$.

Let us begin by describing some general features of the original Hamiltonian
(1). Setting $U=0$, the model reduces to the original Haldane model. In the
case of $\delta=0$, the phase $\phi=\pi/2$ guarantees a Chern insulator CI2
with a Chern number $C=2$ in the ground state ($C_{\sigma}=1$ for each spin
species). Such Chern insulator survives until the closure of the gap at
$|\delta/t_{2}|=3\sqrt{3}$, evolving into a band-insulator with a trivial
charge density wave (CDW) state, manifesting a finite density imbalance
between sublattices A and B [9], given by

$\displaystyle{\cal I}=\left\lvert
n_{\text{A}\uparrow}+n_{\text{A}\downarrow}-n_{\text{B}\uparrow}-n_{\text{B}\downarrow}\right\rvert\
,$ (8)

where

$\displaystyle\left\\{\begin{array}[]{cl}n_{\text{A}\uparrow}=\braket{\hat{n}_{l\uparrow}}\
\text{and}\
n_{A\downarrow}=\braket{\hat{n}_{l\downarrow}}&\quad\text{for}\quad
l\in\text{A}\\\ \\\ n_{\text{B}\uparrow}=\braket{\hat{n}_{l\uparrow}}\
\text{and}\
n_{B\downarrow}=\braket{\hat{n}_{l\downarrow}}&\quad\text{for}\quad
l\in\text{B}\end{array}\right.\ ,$ (12)

denote the number of electrons for distinct sublattice sites and species. We
stress that ${\cal I}\neq 0$ once $\delta>0$ and thus the transition at
$\lvert\delta/t_{2}\rvert=3\sqrt{3}$ occurs without any relevant symmetry
breaking.

In the other limit of $\delta=0$ and $U\gg t_{1}$, $t_{2}$, the presence of
charge excitations is suppressed: The second-order hopping processes result in
an effective Heisenberg model, which includes NN and NNN antiferromagnetic
Heisenberg couplings, denoted as $J_{1}$ and $J_{2}$, respectively. When the
ratio of $J_{2}/J_{1}=t^{2}_{2}/t^{2}_{1}\lesssim 0.27$ [39, 40, 41], a Néel
phase can be achieved in the honeycomb lattice. In this phase, the ground
state exhibits SU($2$) symmetry breaking, and the low-energy excitation
spectrum contains gapless Goldstone modes [42, 43]. Nonetheless, when hopping
energy scales are compatible with the repulsive interactions, the emergence of
CI1 with spontaneous spin-rotation symmetry breaking has also been observed in
this model using various methods [17, 32, 24].

In the simplest mean-field treatment, the on-site interaction is approximated
by

$\displaystyle{\hat{n}}_{l\uparrow}{\hat{n}}_{l\downarrow}\approx{\hat{n}}_{l\uparrow}\braket{\hat{n}_{l\downarrow}}+\braket{\hat{n}_{l\uparrow}}{\hat{n}}_{l\downarrow}-\braket{\hat{n}_{l\uparrow}}\braket{\hat{n}_{l\downarrow}}\
.$ (13)

This approximation assumes that one species of electrons move in the
background of the site-dependent chemical potential provided by the other
species. The superconducting term
$\braket{\hat{c}_{l\uparrow}\hat{c}_{l\downarrow}}$, which breaks U($1$)
symmetry, is also neglected here because $U>0$, following the BCS paradigm
[44, 45]. We notice that spin-flipping terms
$\braket{\hat{c}^{\dagger}_{l\uparrow}\hat{c}^{\phantom{\dagger}}_{l\downarrow}}$,
describing gapless transverse spin excitations in the Néel phase, would
artificially assign a finite mass to the relevant Goldstone modes, indirectly
breaking SU($2$) symmetry. Therefore, the Néel phase is characterized by the
spin-spin correlation function along the $z$-axis, given by

$\displaystyle
S_{\text{SDW}}=\frac{1}{N^{2}}\sum_{l,l^{\prime}}\braket{(\hat{n}_{l\uparrow}-\hat{n}_{l\downarrow})(\hat{n}_{l^{\prime}\uparrow}-\hat{n}_{l^{\prime}\downarrow})}\
.$ (14)

This correlation function describes the long-range spin staggering in space,
corresponding to a spin density wave (SDW) state. However, we observe that
including spin-flipping terms in the mean-field approximation does not alter
our conclusions below, and we remove them for simplicity.

In principle, a total of $2N$ variational parameters, specifically
$\langle\hat{n}_{l\uparrow}\rangle$ and $\langle\hat{n}_{l\downarrow}\rangle$
for all sites, are necessary. In practice, considering the translation
symmetry, only four variational parameters, that is, $n_{\text{A}\uparrow}$,
$n_{\text{A}\downarrow}$, $n_{\text{B}\uparrow}$, and
$n_{\text{B}\downarrow}$, are sufficient for representing the density
imbalance of charge and spin between the two inequivalent sublattices in four
distinct phases of the quantum phase diagram [17].

In reciprocal space, the creation operators are defined as follows

$\displaystyle\left\\{\begin{array}[]{cc}\hat{c}^{\dagger}_{l\sigma}=\frac{1}{\sqrt{N}}\sum_{{\bf
k}\in\text{FBZ}}\hat{d}^{\dagger}_{{\bf k}\sigma}e^{-{\rm i}{\bf
k}\cdot\mathbf{r}_{l}}&\quad\text{for}\quad l\in\text{A}\\\ \\\
\hat{c}^{\dagger}_{l\sigma}=\frac{1}{\sqrt{N}}\sum_{{\bf
k}\in\text{FBZ}}\hat{g}^{\dagger}_{{\bf k}\sigma}e^{-{\rm i}{\bf
k}\cdot\mathbf{r}_{l}}&\quad\text{for}\quad l\in\text{B}\end{array}\right.$
(18)

with $\hat{d}^{\dagger}_{\mathbf{k}\sigma}$ and
$\hat{g}^{\dagger}_{\mathbf{k}\sigma}$ denoting the creation operators in
reciprocal space. The resulting mean-field Hamiltonian is given by

$\displaystyle\hat{H}^{\text{MF}}=\sum_{{\bf k}\in\text{FBZ}}\hat{\psi}_{{\bf
k}}^{{\dagger}}H^{\text{MF}}_{\bf k}\hat{\psi}_{\bf k}\ ,$ (19)

where $\hat{\psi}_{\bf k}^{\dagger}=(\hat{d}^{\dagger}_{{\bf k}\uparrow}$,
$\hat{g}^{\dagger}_{{\bf k}\uparrow}$, $\hat{d}^{\dagger}_{{\bf
k}\downarrow}$, $\hat{g}^{{\dagger}}_{{\bf k}\downarrow})$ is the spinor
notation used as a basis for each lattice momentum ${\bf k}$. The $4\times 4$
matrix $H^{\text{MF}}_{\bf k}$ is given by

$\left(\begin{array}[c]{cccc}m^{+}_{{\bf
k}}+\delta_{\text{A}\downarrow}&f_{{\bf k}}&&\\\ f^{\ast}_{{\bf
k}}&m^{-}_{{\bf k}}+\delta_{\text{B}\downarrow}&&\\\ &&m^{+}_{{\bf
k}}+\delta_{\text{A}\uparrow}&f_{{\bf k}}\\\ &&f^{\ast}_{{\bf k}}&m^{-}_{{\bf
k}}+\delta_{\text{B}\uparrow}\end{array}\right)$ (20)

with the NN structure factor $f_{{\bf k}}=t_{1}(1+e^{-i{\bf
k}\cdot\mathbf{a}_{1}}+e^{-i{\bf k}\cdot\mathbf{a}_{2}})$, the NNN ones
$m^{\pm}_{{\bf k}}=\pm 2t_{2}(\sin({\bf k}\cdot\mathbf{a}_{1})-\sin({\bf
k}\cdot\mathbf{a}_{2})-\sin({\bf k}\cdot(\mathbf{a}_{1}-\mathbf{a}_{2})))$,
and finally, $\delta_{\text{A}\sigma}=\delta+Un_{\text{A}\bar{\sigma}}$ and
$\delta_{\text{B}\sigma}=-\delta+Un_{\text{B}\bar{\sigma}}$. This leads to an
effective spin-dependent staggered potential

$\displaystyle\delta^{\sigma}_{\text{MF}}\equiv\frac{1}{2}\left(\delta_{\text{A}\sigma}-\delta_{\text{B}\sigma}\right)=\delta+\frac{U}{2}\left(n_{\text{A}\bar{\sigma}}-n_{\text{B}\bar{\sigma}}\right)\
,$ (21)

where $\bar{\sigma}$ denotes the reverse of $\sigma$.

In the nonlinear variational process, to avoid the problem of metastable
states, we minimize the energy by starting from various initial guesses of the
ground state. More concretely, each of the four parameters in an initial guess
can be randomly chosen from the range of $[0,\ 1]$. However, given that the
exchanging symmetry between species potentially breaks, we restrict the
initial states to have $n_{\text{A}\uparrow}>n_{\text{A}\downarrow}$ and
$n_{\text{B}\downarrow}>n_{\text{B}\uparrow}$ 111The rationale behind this
choice is that if a constraint was not imposed, the behavior of the two spin-
species are often reversed in what concern their topological characteristics
and a smooth variation of $\delta_{\text{MF}}^{\sigma}$ is not seen, but
rather a rapid oscillation when increasing $U/t$ between the current values
shown in Fig. 2(b). Apart from the normal CDW state, the staggered
magnetization is represented by
$\mathcal{M}=\sqrt{\mathcal{M}^{2}_{x}+\mathcal{M}^{2}_{y}+\mathcal{M}^{2}_{z}}$,
where

$\displaystyle\mathcal{M}_{x/y/z}=\frac{1}{2}\sqrt{\left\langle\left(\hat{S}^{x/y/z}_{\text{A}}-\hat{S}^{x/y/z}_{\text{B}}\right)^{2}\right\rangle}$
(22)

represent the components of staggered magnetization in the x, y, and z-axes,

$\displaystyle\hat{\mathbf{S}}_{\text{A}/\text{B}}=\sum_{\sigma\sigma^{\prime}}\hat{c}_{l\sigma}^{\dagger}\mathbb{S}^{\phantom{{\dagger}}}_{\sigma\sigma^{\prime}}\hat{c}^{\phantom{{\dagger}}}_{l\sigma^{\prime}}$
(23)

gives the vector operators of the spin at site-$l$ belonging to the
sublattices A and B, and the vector $\mathbb{S}=(S^{x}$, $S^{y}$, $S^{z})$
includes the matrix representations $S^{x}$, $S^{y}$ and $S^{z}$ for the
spin-$1/2$ operators.

To characterize the different phases and associated transitions, we now
consider the case of fixed $\delta/t=6$ in Fig. 2 as an example. At small
interactions $U$, due to the strong alternating chemical potential, electrons
mostly occupy sublattice B, resulting in ${\cal I}>0$ [Fig. 2(c)], namely
$n_{\text{B}\sigma}\gg n_{\text{A}\sigma}$. Besides that, no staggered spin
polarization is observed, resulting in $\mathcal{M}=0$; such a regime
corresponds to a normal CDW state with a band gap. In contrast, the ground
state favors a Mott insulator with SDW in the large-$U$ region, where double
occupancies are disfavored and spin-up/down electrons separately reside on
sublattices A/B. This results in a reduced imbalance ${\cal I}\to 0$ while
$\mathcal{M}$ turns finite.

Interestingly, a topologically nontrivial Chern insulator with finite
staggered spin polarization emerges in an intermediate region, the spin-
rotation symmetry-broken CI1 state. This is easy to see by mapping to the
effective staggered potential for each spin species
$\delta_{\text{MF}}^{\sigma}$. Compared to the phase diagram of the original
non-interacting Haldane model [9], because of the effective chemical potential
[Fig. 2(b)]

$\displaystyle\left\lvert\delta_{\text{MF}}^{\uparrow}/t_{2}\right\rvert<3\sqrt{3}\quad\text{and}\quad\delta_{\text{MF}}^{\downarrow}/t_{2}>3\sqrt{3}\
,$ (24)

only spin-up species contributes with a finite Chern number $C=1$ [see Fig.
2(a)].

Two consecutive topological transitions, CDW-CI1 and CI1-SDW, occur as the
interaction strength $U$ increases. These transitions can be characterized by
the closure of the charge gap

$\displaystyle\Delta_{c}({\bf k})=\min_{{\bf k}_{1},{\bf k}_{2}}[E_{2}({\bf
k}_{2})-E_{1}({\bf k}_{1})]\ ,$ (25)

which is defined as the minimum energy difference between the first unoccupied
band $E_{2}$ at momentum ${\bf k}_{2}$ and the last occupied band $E_{1}$ at
momentum ${\bf k}_{1}$. They occur approximately at transition points
$U_{c1}/t\simeq 11.28$ and $U_{c2}/t\simeq 14.3$, with gap closing at two
distinct Dirac points ${\bf k}={\bf k}_{2}-{\bf k}_{1}={\bf K}^{\pm}$,
respectively. The magnetic order parameter $M$ behavior suggests a first-order
transition for CDW-CI1, while the CI1-SDW transition appears to be continuous.
We consider a fully periodic diamond-shaped lattice with a linear dimension
$L$ to study these transitions. When $L$ is a multiple of 3, all the high-
symmetry momentum points are captured, as illustrated in Fig. 1(a2) for $L=6$.

In summary, we obtain the phase diagram within the mean-field approximation in
the parameter space $(U,\ \delta)$ quantitatively agreeing with existing
results [17]. Here, at large $\delta$, there are three distinct phases with
increasing the interaction strength $U$: band-insulating with trivial CDW in
the weak-$U$ region, Chern insulator CI1 with SDW in the intermediate region,
trivial SDW in the strong-$U$ region. If the original staggered potential is
small, one also recovers a Chern insulator CI2 phase when the interaction
strength $U$ is sufficiently small (see Appendix B). In the next section, we
depart from this approximation, considering all quantum fluctuations, and
advance that the microscopic physical scenario of the spin part qualitatively
changes in the CI1 and SDW regions.

##  IV iDMRG calculations

Having established the mean-field results, particularly on the emergence of
the symmetry-broken phase with a Chern number $C=1$, in this section, we use
the iDMRG method to supplement them. In what follows, Secs. IV.1 and IV.2 are
used to briefly introduce the algorithm for evaluating characteristic
correlation lengths and the Chern number, respectively, which are then
employed to identify the different phases of the Hamiltonian (1) in Sec. IV.3.

###  IV.1 Correlation lengths

In the iDMRG method for an infinitely long cylinder, a “snake-like” matrix
product state (MPS) is translation invariant along the longitudinal direction
[47, 48]. Within this representation, we can immediately obtain the transfer
matrix for a large variational unit cell along the circumference, whose
eigenvalues indicate the contribution of different excitations to the
correlation functions [49, 50]. By normalizing the MPS, we obtain the dominant
eigenvalue $\gamma_{\text{max}}=1$ and sort the eigenvalues in descending
order of their amplitudes.

Using U($1$) symmetry for each spin species individually, we divide the
transfer matrix into multiple subspaces. For a subspace $\cal S$, we get
conserved numbers of electrons for each spin species $N^{\cal S}_{\uparrow}$
and $N^{\cal S}_{\downarrow}$. By taking the ground state at half-filling as a
reference, we can define the deviation in the number of electrons and the spin
polarization along the z-axis as follows

$\displaystyle\delta N=N^{\cal S}_{\uparrow}+N^{\cal S}_{\downarrow}-2N\
\text{and}\ S^{z}=\frac{1}{2}\left(N^{\cal S}_{\uparrow}-N^{\cal
S}_{\downarrow}\right)\ .\quad$ (26)

Within the subspace $\cal S$ of the transfer matrix, which is labeled by
$(\delta N,S^{z})$, the $j$-th eigenvalue can be expressed as

$\displaystyle\gamma^{\cal S}_{j}=\exp\left({\rm i}k^{\cal
S}_{j}-\left\lvert\mathbf{a}_{x}\right\rvert/\xi^{\cal S}_{j}\right)\ ,$ (27)

where real numbers $k^{\cal S}_{j}$ and $\xi^{\cal S}_{j}$ represent the
momentum and correlation length, respectively, and $\mathbf{a}_{x}$ is the
translation vector between two NN variational unit cells. For cylinders types
tZT$L_{y}$-$n$ and ZT$L_{y}$-$n$ described in Sec. II,
$\left|\mathbf{a}_{x}\right|=1$ and $\sqrt{3}$, respectively. Thus, the
correlation lengths in the subspace $\cal S$ can be defined as

$\displaystyle\xi^{\cal S}_{j}=-|\mathbf{a}_{x}|/\ln\left|\gamma^{\cal
S}_{j}\right|\ .$ (28)

For the case of $n=1$, the variational unit cell in practice reduces to one
$L_{y}$-th of the above-mentioned size, i.e., $2$ and $4$ for the geometry
tZT$L_{y}$-$1$ and ZT$L_{y}$-$1$, respectively. In such cases, we have

$\displaystyle\xi^{\cal S}_{j}=-|\mathbf{a}_{x}|/(L_{y}\ln\left|\gamma^{\cal
S}_{j}\right|)\ .$ (29)

Here, we focus on three representative correlation lengths in the subspaces
specified by ${\cal S}=(0,0)$, $(1,1/2)$, and $(0,1)$:
$\xi_{c}\equiv\xi^{(1,1/2)}_{0}$, associated with the charge excitation where
a single spin-up electron is added; $\xi_{s}\equiv\xi^{(0,1)}_{0}$, associated
with the spin-flipping excitation where a spin-up electron is added after a
spin-down electron is removed; and $\xi_{n}\equiv\xi^{(0,0)}_{1}$, associated
with a neutral excitation that potentially indicates a combination of the
charge and spin excitations.

In the ground-state phase diagram of HHM, the lowest-energy charge excitations
in all phases have a finite gap, implying the insulating nature of all phases,
topologically trivial or not. Assuming that they are Lorentz-invariant, the
inverse of the correlation length $1/\xi_{c}$ is proportional to the charge
gap $\Delta_{c}$. At a continuous phase transition point, where the charge gap
closes in the thermodynamic limit, the relevant correlation length $\xi_{c}$
becomes infinite. However, due to the finite truncated bond dimension $m$ and
the small circumference of the cylinder, sharp peaks in $\xi_{c}$ arise
instead. Besides that, the spin and neutral excitations have an “extra
degeneracy” in the CI1 state, specifically a spin-triplet state with
degenerate $S^{z}=0$, $\pm 1$ states, where $\xi_{s}=\xi_{n}$, exemplified
with cylinder tZT$2$-$1$ in Appendix A. That degeneracy implies that the
lowest-energy elementary excitation behaves as a magnon carrying a total spin
of $1$, similar to the Goldstone modes in the SDW phase (or precisely, Néel
phase) [51]. Therefore, when $U$ is fixed, we determine the phase boundaries
according to two features in the correlation lengths as a function of
$\delta$: the sharp peaks of $\xi_{c}$ and the overlaps between $\xi_{s}$ and
$\xi_{n}$.

Figure 3: (a) Phase diagram of the HHM (1) in the $(U,\delta)$ space of
parameters determined by the iDMRG simulations with a cylindrical geometry and
a truncated bond dimension $m=2048$. Here we contrast the phase boundaries
calculated from two different cylinders ZT$3$-$1$, and tZT$4$-$2$ exhibiting
overall minimal discrepancy. (b) Correlation lengths for neutral ($\xi_{n}$)
and spin ($\xi_{s}$) excitations for $U/t=8$, showing the SDW-CI2 transition.
(c) The charge excitation correlation length ($\xi_{c}$) at $U/t=18$ whose
peaks mark the SDW-CI1 and CI1-CDW transitions. In (a), (b), and (c), dashed
(solid) lines are data for the ZT$3$-$1$ (tZT$4$-$2$) cylinder with $m=4096$.
The colored regions in (b) and (c) represent the corresponding phases
classified for the tZT$4$-$2$ cylinder in (a). (d) Accumulative discrepancy
$Q$ as a function of the adiabatically inserted magnetic flux $\theta$ with a
small interval $\delta\theta/\pi=0.1$ for the tZT$4$-$2$ cylinder, using a
truncated bond dimension $m=4096$. The markers and color codes are also
inserted in (a).

###  IV.2 Chern number

To substantiate the phase diagram determined by the correlation lengths, we
further calculate the Chern number in different phases by the iDMRG method
using a charge pumping scheme [52]. Specifically, one can envision that if a
magnetic flux $\theta$ is adiabatically inserted along the $x$-axis, charges
would be pumped from the left to the right side of a cylinder. The
accumulative discrepancy of the charges between two sides of the cylinder can
be obtained with

$\displaystyle
Q(\theta)=\sum_{p}\Lambda^{2}_{p}(\theta)[Q^{L}_{p}(\theta)-Q^{R}_{p}(\theta)],$
(30)

where $\Lambda_{p}$ is the singular value obtained after decomposing the whole
cylinder into two semi-infinite parts, $Q^{L/R}_{p}$ is the charge degree of
freedom on the left/right side marked for the $p$-th renormalized basis, and
$p$ runs over all truncated bond dimension $m$.

Upon inserting the magnetic flux from $\theta$ to $\theta+\delta\theta$, all
eigenstates of the system undergo adiabatic evolution, meaning that the system
stays at its targeted state, indicated by the fidelity
$F(\theta)=\left\lvert\braket{\Psi_{0}(\theta)}{\Psi_{0}(\theta+\delta\theta)}\right\rvert$
being approximately fixed at $1$. As a result, the Chern number can be
calculated using the accumulated change with

$\displaystyle C=\frac{1}{2}\left\lvert Q(2\pi)-Q(0)\right\rvert\ .$ (31)

###  IV.3 Phase transitions

We begin by presenting the phase diagram of HHM in the space of parameters
$(U,\delta)$, as shown in Fig. 3(a). The phase boundaries are determined using
the correlation lengths and the Chern number. In particular, we note that the
phase boundaries for cylinders ZT$3$-$1$ and tZT$4$-$2$ are fairly consistent.
Four phases exist: the CI1 phase with a Chern number $C=1$ and the CI2 phase
with a Chern number $C=2$, when $U/2$ and $\delta$ are comparable; a CDW and
an SDW insulator phase, both with a zero Chern number, when either $\delta$ or
$U/2$ prevails. Other methods have also predicted such phases, e.g., MF, ED,
DMFT [17], and BDMC [32], although quantitative agreement regarding the
locations of the phase transitions was lacking.

Several remarks on Fig. 3(a) are in order: i) The CI2 phase can extend to the
lobe at $U/t\approx 11$ and $\delta/t\approx 4.6$, in closer quantitative
agreement to the DMFT outcomes [17]. In addition, for $10\leq U/t\leq 11$, the
CI2 phase survives in the middle of CI1, exhibiting a reentrant behavior. ii)
The CI1 phase exists even at the small-$U$ region, making the transition
between the CI2 and CDW phase indirect, in qualitative agreement with the ED
results [17]. iii) At $\delta=0$, the transition between the CI2 and SDW phase
occurs around $U/t\sim 6.8$, comparable to those of ED, DMFT, and BDMC [17,
32]. However, when increasing $U$ with a small nonzero $\delta$, both ED and
BDMC predict two consecutive transitions from CI2 to CI1, and then to SDW,
instead of the CI2-SDW transition predicted by us.

Having established those main features unifying the picture for the phase
diagram obtained from different methods, we now illustrate how the phase
boundaries in Fig. 3(a) are determined by the correlation lengths introduced
in Sec. IV.1. In Fig. 3(b), the charge correlation length $\xi_{c}$ exhibits
two pronounced peaks for $U/t=18$ when sweeping the staggered potential
$\delta$, indicative of the critical points of the phase transitions SDW-CI1
and CI1-CDW. In Fig. 3(c) for $U/t=8$, one more feature emerges compared to
those in Fig. 3(c): $\xi_{n}$ and $\xi_{s}$ overlap for $\delta/t\lesssim
1.68$ and they exhibit a sharp discontinuity at this point. This is a
signature of the phase transition SDW-CI2, since in the SDW phase, the spin
and neutral excitation form a spin-triplet excitation.

By computing such correlation lengths for different sets of parameters, we
compile the final phase diagram, Fig. 3(a). Lastly, we note that despite minor
quantitative differences, the results for the smaller cylinder geometry
ZT$3$-$1$ [dashed lines in Figs. 3(a), (b), and (c)] result in similar
behavior to the ones for the tZT$4$-$2$ (solid lines).

After computing the correlation lengths and obtaining the phase diagram, our
analysis progresses towards determining the Chern number in different phase
regions with the charge pumping scheme introduced in Sec. IV.2; results for
the tZT$4$-$2$ cylinder geometry are summarized in Fig. 3(d). Specifically, in
the case of $U/t=8$ and $\delta/t=1.4$, the adiabatic insertion of a
$2\pi$-flux pumps a charge $\Delta Q=Q(2\pi)-Q(0)=0$, confirming that the
Chern number is given by $C=0$, which is indicative of the system being in a
topologically trivial SDW phase. On the other hand, for $U/t=8$ and
$\delta/t=2.9$, a continuous curve with an adiabatic process yields a pumping
charge $\Delta Q=4$, which corresponds to a topologically nontrivial phase
with a Chern number of $C=2$. Lastly, the CI1 phase is exemplified in the set
of parameters $U/t=15$ and $\delta/t=7.1$ – a clear accumulated charge $\Delta
Q=2$ exposes the CI1 ground-state.

Overall, the phase characteristics revealed by the Chern number calculations
for different parameters agree with the expected correlation lengths.
Therefore, the phase diagram of HHM can be accurately identified through the
combined analysis of correlation lengths and Chern numbers with the iDMRG
method.

###  IV.4 Correlation-length spectrum

Further characterization can be performed by describing the low-energy
excitations with momentum resolution, achieved via a particular type of
adiabatic driving. For the charge excitations, for example, we add a uniform
phase factor in front of hopping terms
$\hat{c}^{\dagger}_{l\sigma}\hat{c}^{\phantom{{\dagger}}}_{l^{\prime}\sigma}$
for both species, which equivalently threads a gauge flux $\theta$ through the
thin and infinitely-long cylinder adiabatically. For the cylinder
tZT$L_{y}$-$1$, the corresponding momentum cutting lines, mentioned in Eq. (7)
of Sec. II, follow a set of equations

$\displaystyle k^{\cal S}_{j,1}=k^{\cal S}_{j}+\theta/L_{y}\ \text{and}\
k^{\cal S}_{j,2}=L_{y}k^{\cal S}_{j}\ \text{modulo}\ 2\pi\ ,\quad$ (32)

which sweeps the FBZ as $\theta$ grows from $0$ to $2\pi$. Here, $k^{\cal
S}_{j,1}$ and $k^{\cal S}_{j,2}$ denote the momenta in the basis of two
reciprocal vectors $\bm{b}_{1}$ and $\bm{b}_{2}$. As a result, we can obtain
the momentum-dependence features of the correlation-length spectrum
$1/\xi_{c}$ for the charge excitations in the subspace $(1,1/2)$ of the
associated transfer matrix.

Figure 4: The correlation-length spectrum as a function of the inserted flux
$\theta$, the momenta $k_{1}$ and $k_{2}$ in the reciprocal vectors
$\bm{b}_{1}$ and $\bm{b}_{2}$, respectively. We consider two distinct
excitations for the cylinder tZT$4$-$1$ when $U/t=15$ and $\delta/t=0$ in the
SDW region. (a.1-3) The inverse correlation lengths $1/\xi_{c}$ for the charge
excitations in the subspace $(1,1/2)$ of the transfer matrix. The lowest-
energy charge excitations in the positive (red) and negative (magenta)
branches hit $K$\- points $\mathbf{K}^{\pm}$ as $\theta$ approaches $\pm
2\pi/3$, individually. (b.1-3) The inverse correlation lengths $1/\xi_{s}$ for
the spin-flipping excitations in the subspace $(0,1)$. Both of the lowest-
energy spin excitations hit the $\Gamma$-point as $\theta$ approaches $0$ and
$\pm\pi$. (a.4, b.4) We annotate the momenta of the lowest-energy excitations
for various $\theta$, color marked according to the previous panels. The
momentum-cutting lines (grey) for the charge and spin-flipping excitations are
indicated by Eq. (32) and Eq. (33), respectively, when $\theta=0$. Here, we
set the truncated bond dimension $m=4096$.

In contrast, to produce the spectrum for the spin-flipping excitations in the
subspace $(0,1)$, it is necessary to adiabatically thread a $-\theta$
($+\theta$) flux to the cylinder for the spin-up (spin-down) electrons.
Considering the exchanging symmetry between two spin species, it is sufficient
to increase $\theta$ from $0$ to $\pi$. Similarly, for the cylinder
tZT$L_{y}$-$1$, the corresponding momentum is resolved as

$\displaystyle k^{\cal S}_{j,1}=k^{\cal S}_{j}-2\theta/L_{y}\ \text{and}\
k^{\cal S}_{j,2}=L_{y}k^{\cal S}_{j}\ \text{modulo}\ 2\pi\ .\quad$ (33)

Figure 5: Finite circumference effect of the lowest-lying correlation-length
spectrum level $\xi^{(1,1/2)}$ for the charge excitation (two left columns),
and $\xi^{(0,1)}$ for the spin excitation (two right columns). We choose four
typical parameter sets: (a1-4) $U=15$ and $\delta=0$, (b1-4) $U=18$ and
$\delta=8$, (c1-4) $U=18$ and $\delta=8.5$, and (d1-4) $U=15$ and
$\delta=8.5$. Besides, we systematically select four circumferences: $L_{y}=2$
(brown), $3$ (black), $4$ (red), $5$ (blue) in the cylinder tZT$L_{y}$-$1$.
The truncated bond dimension is given by $m=4096$.

First, we exemplify the correlation-length spectrum for a typical parameter
$U/t=15$ and $\delta/t=0$ in the SDW phase region. As shown in Fig. 4(a1), the
lowest-lying branch of charge excitations reaches its minimum at $\theta=\pm
2\pi/3$, where one of the momentum cutting lines goes through either of

$\displaystyle\mathbf{K}^{\pm}=\left(k_{1}=\pm\frac{2\pi}{3},\
k_{2}=\pm\frac{4\pi}{3}\right)\ \text{modulo}\ 2\pi\ ,$ (34)

which equivalently corresponds to the momenta $k_{x}=\pm 4\pi/3$ and $k_{y}=0$
in Cartesian coordinates, marked in Fig. 4. For the spin-flipping excitations,
the lowest-lying branch, as shown in Figs. 4(b1-3), gets the minimum at the
high-symmetry point $\mathbf{\Gamma}=(0,0)$ modulo $2\pi$ as $\theta=0$. The
cone structure in the vicinity of the $\Gamma$-point implies the emergence of
Goldstone modes, providing a hallmark of gapless excitations in the
SU($2$)-symmetry broken Néel phase [53, 54]. The small but finite
$L_{y}/\xi_{s}$ at the $\Gamma$-point is caused by the finite truncated bond
dimension $m=4096$ and circumference $L_{y}$.

Second, we check the effects of the finite circumference of the cylinder on
the spectrum as we vary $L_{y}$ from 2 to 5, while choosing a sufficiently
large $m\leq 4096$. In the SDW phase region, at $U/t=15$ and $\delta/t=0$, as
shown in Figs. 5(a1-2), all minima of the lowest-lying branches of charge
excitations are located at the $K$-points $\mathbf{K}^{\pm}$ and $1/\xi_{c}$
approaches a constant as $L_{y}$ grows. In turn, for the minima of spin-
flipping excitations at $\mathbf{\Gamma}$, shown in Figs. 5(a3-4), the inverse
correlation length $1/\xi_{s}$, theoretically associated to the gapless
Goldstone modes, is expected to decrease as the circumference $L_{y}$ grows.
We notice that the approach to the thermodynamic limit is highly influenced by
large finite-size effects when $L_{y}\leq 5$. In contrast, in the CDW phase,
at $U/t=15$ and $\delta/t=8.5$, all the minima of the inverse correlation
lengths $1/\xi_{c}$ and $1/\xi_{s}$ for the charge and spin-flipping
excitations in Figs. 5(d1-4) approach convergence as the circumference $L_{y}$
increases.

Lastly, in the CI1 region, such as $\delta/t=8$ and $8.5$ when $U/t=18$, the
equivalent charge gap $\Delta_{c}\propto 1/\xi_{c}$ keeps finite [Figs.
5(b1-2) and 5(c1-2)]. Up to relatively small $L_{y}=5$, we can not conclude
that the gap for the spin-flipping excitations closes in the thermodynamic
limit of infinite cylinder circumference [Figs. 5(b3-4) and 5(c3-4)]. Yet,
compared to the spin excitation spectrum in the SDW phase for $U/t=15$ and
$\delta/t=0$ [Figs. 5(a3-4)], we find that the low-energy spectrum of the
spin-flipping excitations in the CI1 phase region has a similar qualitative
profile, based on which we believe that the resulting ground state behaves as
an Néel state with SU($2$) symmetry breaking. However, when the iDMRG
simulations are applied to a cylinder with a finite circumference $L_{y}$, we
consistently obtain the lowest-energy state with a total spin of $0$. This is
because the energy spacing between quasi-degenerate joint states in distinct
total spin sectors is approximately proportional to $1/L_{y}$ [53, 54]. The
resulting spin-spin correlation functions preserve the spin rotation symmetry,
as shown in Fig. S4 in Appendix C for the cylinder circumferences studied.

##  V Summary and conclusions

We have investigated the ground-state phase diagram of HHM at half-filling
using the MF and iDMRG methods. Within MF, we examine four physical
quantities: the effective mass term for each spin species, the staggered
magnetization, the sublattice density imbalance, and the excitation gap,
extracting a phase diagram consistent with previous MF calculations [17].
Remarkably, the calculated effective mass term, as a function of the
interaction $U$, clearly demonstrates a scenario where only one of the two
spin species contributes to the Chern number, thereby unequivocally
identifying the CI1 phase.

Within an unbiased scheme using iDMRG, we have employed the Chern number and
three different characteristic correlation lengths to determine the phase
diagram in the $(U,\delta)$ space, with $\pm\delta$ the staggered potential
energies of the two sublattices of the honeycomb lattice. In addition, we
unveil a detailed analysis of the low-energy charge and spin excitations by
resolving the momentum information in the correlation-length spectrum. Our
state-of-the-art phase diagram not only analyzes the discrepancies between the
phase boundaries predicted by different approaches as MF, ED, DMFT [17], and
BDMC [32], but also excludes the emergence of any new phase other than the
SDW, CDW, CI1, and CI2 phases, therefore establishing a reliable benchmark for
future theoretical and experimental investigations of HHM.

###### Acknowledgements.

We thank Wei Pan, Ji-Lu He, and Tian-Cheng Yi for the helpful discussions. We
acknowledge funding from the Ministry of Science and Technology of the
People’s Republic of China (Grant No. 2022YFA1402700) and the National Natural
Science Foundation of China (Grants No. U2230402). R. M. acknowledges support
from the NSFC Grants No. 12111530010, 12222401, and No. 11974039. H.-G. L.
acknowledges funding from the NSFC Grants No. 11834005 and No. 12247101. X. Q.
W. acknowledges funding from the NSFC Grant No. 11974244. S. H. acknowledges
funding from NSFC Grant No. 12174020. The computations were performed on the
Tianhe-2JK at the Beijing Computational Science Research Center (CSRC) and the
Quantum Many-body I cluster at the School of Physics and Astronomy, Shanghai
Jiaotong University.

##  Appendix A tZT2-1 cylinder

In Sec. IV, we have presented a detailed account of the quantitatively
consistent phase diagrams of ZT$3$-$1$ and tZT$4$-$2$ with the iDMRG method.
In this section, we extend our investigation to the cylinder tZT$2$-$1$ and
demonstrate the significance of the proper choice of the lattice structure in
the iDMRG simulation.

Figure S1: Phase diagram of the HHM in the space $(U,\delta)$ of parameters
for cylinder tZT$2$-$1$. Unlike in Fig. 3(a), the CI2 phase does not exhibit a
reentrant behavior in the CI1 phase, a feature to which we attribute being
related to finite-size effects – see text. (b,c) Correlation lengths as a
function of the potential energy $\delta$ by setting onsite interaction (b)
$U/t=6$ and (c) $15$. All results are obtained by the iDMRG simulations with a
truncated bond dimension $m=2048$.

To summarize these results, Fig. S1(a) shows the phase diagram in the
$(U,\delta)$ space for the tZT$2$-$1$ cylinder. The phase diagram is
qualitatively similar to the one before, exhibiting four different phases. The
most remarkable difference between Fig. 3 and Fig. S1(a) is the direct
transition between CI2 and SDW, which leads to the absence of CI2 phase when
$U/t=10\thicksim 11$. In other words, the CI2 phase misses the reentrant
behavior in the CI1 phase. Ref. [24] obtained the precise same result on the
$12$A cluster. Again, we attribute this difference to a finite-size effect,
i.e., $L_{y}=2$ is too small to adequately capture fine features of the phase
diagram in the thermodynamic limit. Overall, an accurate selection of the
lattice structure on the infinite cylinder in our work effectively preserves
the physical properties of the ground state related to the high-symmetry
points.

Figure S1(b) and (c) shows various correlation lengths at $U/t=6$ and $15$,
respectively, which qualitatively agrees with the conclusions in the large-$U$
region for the cylinder ZT$3$-$1$ mentioned in the main text. We also obtained
the SDW-CI1 and CI1-CDW transition points by observing the peaks of the
correlation length for the lowest-energy charge excitation $\xi_{c}$. The
correlation lengths for the spin and neural excitations exhibit an “extra
degeneracy” of $\xi_{s}=\xi_{c}$ in the CI1 state. This implies the presence
of an excitation “magnon” with degenerate $S^{z}=0$, $\pm 1$ states in the
lowest excited energy level. On the other hand, the multiplet vanishes in the
CI2 state. Hence, we distinguish the CI1 and CI2 states by examining the
overlaps between $\xi_{s}$ and $\xi_{n}$.

For the lattice structure under consideration, the allowed momentum values
appear as a discrete set of lines tangent to two edges of FBZ [see Fig.
S2(a)], owing to the finite number of sites on the circumference of the
cylinder. Notably, despite being a lattice encompassing the inequivalent
Brillouin zone corners, it does not capture other essential symmetry points,
as it misses some of the inequivalent $M$ points.

Figure S2: (a) Excitation momentum in the first Brillouin zone – several
excitation momenta are superimposed and color-coded according to (b). (b) The
corresponding excitation momentum $k_{x}$ in the x-axis as a function of the
potential energy $\delta$ by setting onsite interaction $U/t=15$ for cylinder
tZT$2$-$1$. (d) A zoom-in of the excitation momentum $k_{x}$ after it
transitioned to close $\mathbf{K}^{+}$, the x-axis projection of which is
equal to $4\pi/3$ (black dashed line). We use a truncated bond dimension
$m=2048$ in the iDMRG simulations.

The x-axis projection of the excitation momentum $k_{x}$ in FBZ with
increasing $\delta$ is illustrated by the empty blue hexagons positioned near
the momentum points $\mathbf{K}^{-}$ and $\mathbf{K}^{+}$. The momentum
associated with the spin and neutral excitations, represented by red and black
empty hexagons, respectively, are located at the $\mathbf{\Gamma}$-point
instead. Generally, the momentum $k_{x}$ of the low-energy excitations along
the longitudinal direction of the cylinder can be obtained according to Eq.
(27).

To give an intuitive understanding of the low-energy excitation momentum
points, Figs. S2(b) and (c) show the $k_{x}$ dependence of $\delta$. The
momentum point jumps from $\mathbf{K}^{-}$ to $\mathbf{K}^{+}$ within the CI1
phase region in Fig. S2(b). We have also checked the phase transitions SDW-
CI1/CI2 related to the $\mathbf{K}^{+}$ point, while the others are about
$\mathbf{K}^{-}$, by sweeping the parameter space. Figure S2(c) gives a zoom-
in of the excitation momentum for the charge excitation around
$\mathbf{K}^{+}$ for $\delta/t\gtrsim 6.8$, close to where the momentum point
jumping occurs. Unlike the MF results in the inset of Fig. 2(c), the energy
gap closure here does not strictly happen at the Dirac point. We emphasize
that, similarly, for ED studies in small clusters, charge excitation momentum
may not coincide with the $K$-points [24]. As here, such $12$-site clusters
(often labeled as $12$A) do not present all point-group symmetries also
missing many of the inequivalent $M$-points.

Figure S3: Mean-field results for $\delta/t=1$, in a lattice with $L=30$. (a)
The effective staggered potential for each spin species after the interactions
are accounted for on a mean-field level; the shaded region marks the
topological regime. (b) the staggered magnetic order parameter, the sublattice
density imbalance, and the excitation gap (c) vs. $U/t$.
Figure S4: Spin-spin correlation function
$O^{\alpha}_{l}=\braket{\hat{S}^{\alpha}_{0}\hat{S}^{\alpha}_{l}}$
($\alpha=x$, $y$, and $z$) for the cylinder tZT$L_{y}$-$1$ with different
circumferences, i.e., $L_{y}=2$ (brown), $3$ (black), $4$ (red), $5$ (blue).
We choose $U/t=18$ and $\delta/t=8$ in the CI1 phase region, which share the
same parameter set of Fig. 5(b1-4). All results are obtained by the iDMRG
simulations with a truncated bond dimension $m=4096$.

##  Appendix B Other $\delta$ values in the mean-field theory

The main text shows a line cut in the $(U,\delta)$ phase diagram using the MF
method, with $\delta/t=6$ fixed. While illustrative of the CI1 phase and the
topologically trivial ones, either band-insulating CDW or SDW, it does not
show the fourth phase in the phase diagram, corresponding to CI2. For
completeness, we include in Fig. S3 data for $\delta/t=1$, which with
increasing interactions $U$, drive the system across the phases CI2, CI1, and
SDW. As for the $\delta/t=6$ case shown in the main text, the topological
character of the different phases can be readily inferred from the value of
the effective staggered potential $\delta_{\text{MF}}^{\sigma}$ [Fig. S3(a)],
checking whether $|\delta_{\text{MF}}^{\sigma}/t_{2}|<3\sqrt{3}$. For
small-$U$, both $\delta_{\text{MF}}^{\uparrow}$ and
$\delta_{\text{MF}}^{\downarrow}$ satisfy this condition, and the resulting
phase exhibits a Chern number of $2$. The spontaneous symmetry-breaking that
occurs when $\delta_{\text{MF}}^{\uparrow}\neq\delta_{\text{MF}}^{\downarrow}$
is marked as a first-order transition with a discontinuous magnetic order
parameter $M$ [Fig. S3(b)], whereas a continuous transition occurs from the
CI1 to the SDW regime. Nevertheless, in both cases, the gap excitation
$\Delta_{c}({\bf k})\to 0$ identifies the topological transition.

##  Appendix C Spin-rotation symmetry

Here, we present the spin-spin correlation function
$O^{\alpha}_{l}=\braket{\hat{S}^{\alpha}_{0}\hat{S}^{\alpha}_{l}}$
($\alpha=x$, $y$, and $z$) along a “snake-like” path for the cylinder
tZT$L_{y}$-$1$ in the CI1 phase region. We use a parameter set of $U/t=18$ and
$\delta/t=8$. For four different circumferences $L_{y}=2$, $3$, $4$, and $5$,
we demonstrate that the gapless lowest-energy spin excitations, indicated by
the inverse of the correlation lengths in Fig. 5(b3-4), do not arise from
spin-rotation symmetry breaking at these lattice sizes. Also note that the
decay rate, non-monotonic in $L_{y}$, is tied to the non-monotonic spin gap
structure observed in Fig. 5(b3-4).

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