# Exact solution of the Bose-Hubbard model with unidirectional hopping

Mingchen Zheng  Beijing National Laboratory for Condensed Matter Physics,
Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
School of Physical Sciences, University of Chinese Academy of Sciences,
Beijing 100049, China  Yi Qiao  Institute of Modern Physics, Northwest
University, Xi’an 710127, China  Yupeng Wang  Beijing National Laboratory for
Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences,
Beijing 100190, China  Junpeng Cao  Beijing National Laboratory for Condensed
Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing
100190, China  School of Physical Sciences, University of Chinese Academy of
Sciences, Beijing 100049, China  Songshan lake Materials Laboratory, Dongguan,
Guangdong 523808, China  Peng Huanwu Center for Fundamental Theory, Xian
710127, China  Shu Chen<EMAIL_ADDRESS>Beijing National Laboratory for
Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences,
Beijing 100190, China  School of Physical Sciences, University of Chinese
Academy of Sciences, Beijing 100049, China

###### Abstract

A one-dimensional Bose-Hubbard model with unidirectional hopping is shown to
be exactly solvable. Applying the algebraic Bethe ansatz method, we prove the
integrability of the model and derive the Bethe ansatz equations. The exact
eigenvalue spectrum can be obtained by solving these equations. The
distribution of Bethe roots reveals the presence of a superfluid-Mott
insulator transition at the ground state, and the critical point is
determined. By adjusting the boundary parameter, we demonstrate the existence
of a non-Hermitian skin effect even in the presence of interaction, but it is
completely suppressed for the Mott insulator state in the thermodynamical
limit. Our result represents a new class of exactly solvable non-Hermitian
many-body systems, which has no Hermitian correspondence and can be used as a
benchmark for various numerical techniques developed for non-Hermitian many-
body systems.

Introduction.— Exact solutions of integrable quantum many-body models, e.g.,
the Lieb-Liniger model Lieb2 ; Lieb21 , Yang-Gaudin model Yang ; Gaudin  and
Fermi Hubbard model Lieb-Wu , provide crucial insights into the understanding
of correlation effect and have wide application in cold atomic and condensed
matter physics GuanXW . However, constructing a new quantum integrable model
with simple form and clear physical meanings is a difficult and fascinating
mission. Recently a great deal of progress has been made in the theory of non-
Hermitian physics Bender98 ; Bender07 ; Bergholtz ; Ashida . Some novel
phenomena such as the enriched non-Hermitian topological classification Gong ;
Sato ; Zhou ; LiuCH1 ; LiuCH2 ; LiuCH3 , non-Hermitian skin effect (NHSE) 13 ;
15 ; 16 ; 17 ; 18 ; LeeCH , and scale-free localization Murakami ; LiLH2021 ;
GuoCX2023 ; WangZ2023 ; Bergholtz2023  have attracted intensive studies. At
present, most novel phenomena and concepts about the non-Hermitian effects are
built based on noninteracting systems. Then it is necessary to demonstrate
whether these concepts are applicable and what new effect arises when the
interaction is considered. Recently, there is attracting growing interest in
exploring non-Hermitian phenomena in many-body systems, e.g., the interplay of
non-Hermitian skin effect and interaction ZhangSB ; ZhangDW ; Alsallom , the
fate of the correlated phase in the presence of non-Hermiticity XuZ ; PanL ;
ZhangDW ; Nori2020 ; Fukui ; HeL , and non-Hermitian topological phases in
correlated systems Hatsugai2022 ; Hatsugai2020 ; Ryu .

Most of the existing studies on the non-Hermitian many-body systems are
carried out by numerical diagonalization, which suffers from the many-body
exponential wall problem and the problem of numerical errors and calculation
precision in the diagonalization of non-Hermitian systems Colbrook ; GuoCX2021
. The current understanding of NHSE and scale-free localization in correlated
systems is quite limited. Exact solutions of non-Hermitian integrable many-
body systems can provide benchmarks for understanding novel phenomena due to
the interplay of non-Hermiticity and interaction. In this work, we propose an
integrable non-Hermitian Bose-Hubbard model describing the interacting bosons
in a chain with unidirectional hopping, which is one of conserved quantities
constructed from the expansion coefficients of transfer matrix, whose
integrability is guaranteed by the Yang-Baxter equation. It is well known that
the Bose-Hubbard model is not integrable and cannot be analytically solved
Haldane ; Fisher . Compared to the previously reported exactly solvable non-
Hermitian many-body systems Ueda2021 ; Fukui ; PanL ; Andrei ; WangZ2301 ;
PanL2022 , whose integrability is inherited from their Hermitian corresponding
models by either introducing an imaginary gauge field or complex continuation
of parameters, our model is a new integrable system without a Hermitian
counterpart. By using an algebraic Bethe ansatz method Korepin ; WYCS , we
exactly solve the model and obtain the exact energy spectrum. Although the
spectrum of a non-Hermitian many-body system is generally complex, we find
that the ground state (defined by the minimum of real parts of eigenvalues) of
our model is real. We unveil the occurrence of a superfluid-Mott insulator
transition for integer filling cases by adjusting the ratio of interaction
strength and hopping amplitude. Additionally, based on the obtained exact
eigenstates, we also demonstrate the existence of non-Hermitian skin effect
even in the presence of interaction, which is, however, found to be completely
suppressed for the Mott state in the thermodynamical limit.

Model and its exact solution.— The model Hamiltonian reads

$H=-t\left[\sum_{j=1}^{N-1}b_{j}^{{\dagger}}b_{j+1}+\epsilon
b_{N}^{{\dagger}}b_{1}\right]+\frac{U}{2}\sum_{j=1}^{N}n_{j}(n_{j}-1),$ (1)

where ${b}_{j}^{\dagger}$ and ${b}_{j}$ are bosonic creation and annihilation
operators at $j$th site respectively, ${n}_{j}={b}_{j}^{{\dagger}}{b}_{j}$ is
the particle number operator on the $j$th site, $N$ is the number of sites,
$t$ quantifies the hopping between two sites with the nearest neighbor,
$\epsilon$ is the boundary parameter, and $U$ characterizes the strength of
on-site interaction. For simplicity, we consider $t>0$ and the on-site
interaction is repulsive, i.e., $U>0$. Since the Hamiltonian includes only a
unidirectional hopping term, it is a non-Hermitian model.

First we prove that the model (1) is integrable in the framework of the
quantum inverse scattering method Korepin . A series of conserved quantities
of the Hamiltonian (1) can be generated by a transfer matrix

$t(u)=\text{tr}_{0}[L_{0,N}(u)L_{0,N-1}(u)\cdots L_{0,1}(u)K_{0}],$ (2)

where $u$ is the spectral parameter, $\text{tr}_{0}$ means the trace in the
two-dimensional auxiliary space $V_{0}$, the subscripts $\\{1,...,N\\}$ denote
the physical spaces, $L_{0,j}(u)$ is the Lax operator defined in the tensor
space $V_{0}\otimes V_{j}$, and $K_{0}$ is the $2\times 2$ diagonal matrix
given by $K_{0}=\text{diag}(1,\epsilon)$. In the auxiliary space, the Lax
operator can be expressed by the matrix

$L_{0,j}(u)=\begin{pmatrix}u-n_{j}&gb_{j}\\\
gb_{j}^{{\dagger}}&-g^{2}\end{pmatrix},$ (3)

where the elements are the bosonic operators defined in the $j$th site and $g$
is a constant. The Lax operator (3) satisfies the Yang-Baxter relation

$R_{0,\bar{0}}(u-v){{L}}_{0,j}(u){{L}}_{\bar{0},j}(v)={{L}}_{\bar{0},j}(v){{L}}_{0,j}(u)R_{0,\bar{0}}(u-v),$
(4)

where $R_{0,\bar{0}}(u)$ is the $R$ matrix defined in the auxiliary spaces 0
and $\bar{0}$ with the form of

$R_{0,\bar{0}}(u)=\begin{pmatrix}u-1&0&0&0\\\ 0&u&-1&0\\\ 0&-1&u&0\\\
0&0&0&u-1\end{pmatrix}.$ (5)

Using the Yang-Baxter relation (4), it can be proven that the transfer
matrices with different spectral parameters commute with each other, i.e.,
$[t(u),t(v)]=0$. Expanding the transfer matrix $t(u)$ with respect to $u$,

$t(u)=u^{N}+C_{1}u^{N-1}+C_{2}u^{N-2}+\cdots,$ (6)

then all the expansion coefficients $C_{n}$ $(n=1,...,N)$ are commutative and
can be taken as conserved quantities. Choosing one of them or a certain
combination of them as a Hamiltonian $H$, then $[H,C_{n}]=0$ and the model is
integrable. Direct calculation gives the operators $C_{1}$ and $C_{2}$ as

$\displaystyle C_{1}=-\sum_{j=1}^{N}n_{j},$ (7) $\displaystyle
C_{2}=\frac{1}{2}\sum_{i\neq
j}^{N}n_{i}n_{j}+g^{2}\left(\sum_{j=1}^{N-1}b_{j}^{{\dagger}}b_{j+1}+\epsilon
b_{N}^{{\dagger}}b_{1}\right).$ (8)

The integrable Hamiltonian (1) is constructed as

$H=\frac{t}{2g^{2}}\left(C_{1}^{2}+C_{1}-2C_{2}\right),$ (9)

where the on-site interaction is parameterized as $U=t/g^{2}$.

Using the algebraic Bethe ansatz and considering the number of bosons to be
$M$, we obtain the eigenvalues of the transfer matrix $t(u)$ as SM

$\Lambda(u)=u^{N}\prod_{j=1}^{M}\frac{u-\beta_{j}/U-1}{u-\beta_{j}/U}+\epsilon(-t/U)^{N}\prod_{j=1}^{M}\frac{u-\beta_{j}/U+1}{u-\beta_{j}/U},$
(10)

where the $M$ Bethe roots $\\{\beta_{j}\\}$ should satisfy the Bethe ansatz
equations (BAEs)

$\Big{(}\frac{\beta_{j}}{-t}\Big{)}^{N}=\epsilon\prod_{l\neq
j}^{M}\frac{\beta_{j}-\beta_{l}+U}{\beta_{j}-\beta_{l}-U},\quad j=1,\cdots,M.$
(11)

Taking the logarithm of BAEs (11), we have

$N\ln\Big{(}\frac{\beta_{j}}{-t\sqrt[N]{\epsilon}}\Big{)}=\sum_{l\neq
j}^{M}\Theta\Big{(}\frac{U}{\beta_{j}-\beta_{l}}\Big{)}+2i\pi I_{j},$ (12)

where $\Theta(z)\equiv 2\operatorname{arctanh}(z)$ and $I_{j}$ is the quantum
number. Each set of integers $\\{I_{j}|j=1,2,...,M\\}$ characterizes one
eigenstate. The eigenvalue of Hamiltonian is

$E=\sum_{j=1}^{M}\beta_{j}.$ (13)

The solutions of BAEs (11) or (12) determine the energy spectrum of the
Hamiltonian (1) completely.

Figure 1: (a)-(c) Distribution of Bethe roots of the ground state for $U=1,\
N=200$, and $M=\ 200\ $ (a),$\ 199\ $(b),$\ 201\ $(c) with $t$ varying from
0.3 to 0.05 (the step size is 0.01). The inset in (c) displays a part of the
Bethe roots with a finer step size of 0.001. (d) Numerical results of
$\mu_{+}$ and $\mu_{-}$ for the system of $N=M=200$ and $U=1$ with $t$ varying
from 0.4 to 0.05 (step size 0.0001). (e) The ground state energy $E_{g}$ (red
point) and the second derivative of $E_{g}/N$ with respect to $t/U$ (blue
point) for $N=200$ and $M=200$ with $t$ varying from 0.4 to 0.05 (step size
0.0001). (f) The extreme values of ($d^{2}E_{g}/Ndt^{2}$) plotted against
$1/\sqrt{N}$ from $N=20$ to $N=350$ ($\rho=1$). The dashed fitted line is
$1/(d^{2}E_{g}/Ndt^{2})=C_{1}/\sqrt{N}+C_{2}$. The fitting parameters are
$C_{1}=-0.44558$ and $C_{2}=-4.9649\times 10^{-4}$. (g) The value of $t_{m}$
(blue points) and $t_{o}$ (red points) versus $1/N$ with $N=20-350$
($\rho=1$). The dashed lines are fitted to determine the values of $t_{m}$ and
$t_{o}$ as the system size tends to infinity. Fitting results with
$N\rightarrow\infty$ give $t_{m}=0.17155$ and $t_{o}=0.17156$.

The Superfluid-Mott transition.– First, we consider the case of $\epsilon=1$.
At the limit of $U=0$, the Bethe roots take $\beta_{j}=t\exp[i\pi(N+2l)/N]$
with $l=1,...,N$. When $U\neq 0$, the Bethe roots can be obtained by solving
BAEs (11). To verify the accuracy of our analytical results, we calculate the
solutions of BAEs (11) and, thus, obtain all the eigenvalues for a small
system with $N=4$ and $M=4$. We compare the results of BAEs with exact
diagonalization and confirm that they are consistent (see Supplemental
Material SM ).

For the Bose-Hubbard model with integer filling, the system undergoes a
superfluid-Mott insulator transition as the ratio of $U/t$ is varied Greiner .
The critical value of $U/t$ has been determined using various methods such as
mean field theory, quantum Monte Carlo simulation Scalettar91 ; Batrouni92 ,
and density matrix renormalization group method Kuhner98 .

Now we explore the superfluid-Mott phase transition for the Bose-Hubbard model
with unidirectional hopping. We shall focus on the case with unity filling
factor $\rho=M/N=1$ and study its ground state properties. At $U=0$, the
ground state of the system is characterized by $N$ Bethe roots located at
$\beta_{j}=-t$, indicating the condensation of $N$ bosons. As $U/t$ increases
from zero, the Bethe roots gradually move away from this point, forming a
curve on the complex plane. When $U/t$ exceeds a critical value, the curve
closes up and the Bethe roots fill a single ring. To visualize this behavior,
we plot the distribution of the Bethe roots of the ground state for various
$t$ in Fig.1 (a) while fixing $U=1$. It can be seen that as $t$ decreases, the
arc of the curve gradually expands until it closes. When $U/t$ is sufficiently
large, the Bethe roots distribute uniformly on a circle with radius $t$, which
is consistent with the hard-core limit of $U/t\rightarrow\infty$.

When a boson is removed from the system, we have $M=N-1$, and the
distributions of the Bethe roots are similar to those in the case of $M=N$, as
shown in Figure 1 (b). However, adding a boson to the system, i.e., $M=N+1$,
leads to a markedly different behavior in the distribution of Bethe roots when
$U/t$ exceeds a critical value. As shown in Figure 1 (c), the two roots with
the largest real part, denoted as $\beta_{N}$ and $\beta_{N+1}$, approach the
real axis as $t$ decreases. At $t=t_{o}$, they coincide. For $t<t_{o}$, they
repel each other and move oppositely along the real axis. If $U/t$ is large
enough, we have $\beta_{N+1}\approx\beta_{N}+U-2t$, and the other $N$ Bethe
roots are distributed on a circle similar to the $M=N$ case.

Next we reveal that the change in the patterns of Bethe root across $t_{o}$ is
a key indicator of the superfluid-Mott transition. Define $\mu_{+}=E_{N}(\rho
N+1)-E_{N}(\rho N)$ and $\mu_{-}=E_{N}(\rho N)-E_{N}(\rho N-1)$, where
$E_{N}(M)$ is the ground state energy for the system with $M$ bosons and $N$
sites. If the system is in the Mott phase, there exists a nonzero gap
$\Delta\mu=\mu_{+}-\mu_{-}>0$, whereas the excitation in the superfluid phase
is gapless. As the parameter $U/t$ changes, the position where the energy gap
opens or closes marks the superfluid-Mott transition point. In the
thermodynamic limit $N$ $(M)\to\infty$, the Bethe roots of the ground state
distribute continuously. Adding or removing a boson corresponds to adding or
removing a Bethe root at the position with the largest real part on the Bethe
root distribution curve. Therefore, we have

$\mu_{+}=\operatorname{Re}(\beta_{N+1}),~{}~{}~{}~{}\mu_{-}=\operatorname{Re}(\beta_{N}).$
(14)

When $t>t_{o}$, we have
$\operatorname{Re}(\beta_{N+1})=\operatorname{Re}(\beta_{N})$ and, thus,
$\Delta\mu=0$. When $t<t_{o}$,
$\Delta\mu=\operatorname{Re}(\beta_{N+1})-\operatorname{Re}(\beta_{N})>0$.
Since $t_{o}$ ($U/t_{o}$) defines the position where the gap opens [see Fig.1
(d)], the Mott-Superfluid phase transition point can be determined from
$U/t_{o}$ in the thermodynamic limit.

In addition to the Bethe root patterns, the quantum phase transition is also
manifested in the ground state energy $E_{g}$ as a function of $U/t$ (or
equally $t/U$). We demonstrate the change of $E_{g}$ and the second-order
derivative of $E_{g}/N$ versus the scaled interaction $t/U$ in Fig.1 (e),
where $E_{g}$ shows a steep descent when $t$ exceeds a specific value $t_{m}$,
accompanied by a peak in $-(d^{2}E_{g}/Ndt^{2})$ at $t_{m}$. With the finite-
size scaling of the maximum of ($d^{2}E_{g}/Ndt^{2}$) shown in Fig.1 (f), we
get $\left.(d^{2}E_{g}/Ndt^{2})\right|_{t=t_{m}}\propto\sqrt{N},$ indicating
that the second-order derivative of $E_{g}/N$ at $t=t_{m}$ is divergent in the
thermodynamic limit.

The finite-size scaling behavior of $t_{o}$ and $t_{m}$ is depicted in
Fig.1(g), and it is observed that the critical points determined by the
analysis of BAEs and the divergence point of the ground state energy are
nearly identical (with $1/N\to 0,\ |t_{m}-t_{o}|<10^{-5}$). From this, we can
extrapolate that the value of the superfluid-Mott phase transition point is
$t_{c}=0.17155\pm 10^{-5}$ ($\alpha_{c}=U/t_{c}\approx 5.83$). In addition,
numerical calculations of the ground state correlation functions provide
further support for this result (see Supplemental Material SM ). It is
noteworthy that this value is distinct from that in the Bose-Hubbard model
with filling $\rho=1$ where the superfluid-insulator phase transition point is
$U_{c}=3.28\pm 0.04$ obtained by the quantum Monte Carlo simulation 54 . For
the case with filling $\rho=2$, similar analysis on the distribution of Bethe
roots of the ground state and the ground state energy can be carried out SM .
Our result unveils that the value of superfluid-Mott phase transition point is
$t_{c}=0.10100\pm 10^{-5}$.

Figure 2: Exact diagonalization results of the expected values of particle
number at each site $\bar{n}_{j}$ of all eigenstates and the corresponding
eigenvalues with $U=0$ (a), 0.2 (b), 1 (c), 6 (d) respectively. Each point on
the complex plane represents an eigenvalue, while each line corresponds to the
particle number distribution of a single state. The ground state particle
number distribution and the corresponding energies are marked with crosses.
Common parameters: $N=6$, $M=6$, $t=1$, and $\epsilon=0.5$.

Non-Hermitian skin effects.— Now, we investigate the effect of boundary
parameter $\epsilon$ on the eigenstates of the system (1) by considering
$\epsilon\neq 1$, which breaks the translation invariance and leads to scale-
free NHSE. To comprehend the boundary-induced NHSE in the lattice with
unidirectional hopping, it is instructive to see the noninteracting limit of
$U=0$ Guo-CPB2023 , in which all eigenstates accumulated asymmetrically near
the boundaries and exhibit NHSE [see Fig.2 (a)]. From the BAEs (11), for the
Bethe roots pattern or the energy spectrum, the parameter value
$\\{t,\epsilon\\}$ is equivalent to $\\{t/\sqrt[N]{\epsilon},1\\}$. For a
finite nonzero $\epsilon$, when $N\to\infty$, $\sqrt[N]{\epsilon}\to 1$ and
all the eigenvalues are equal to those of the periodic boundary case. It means
that $\epsilon$ does not change the critical point $t_{c}$ in the
thermodynamic limit.

Next we demonstrate that the NHSE exists in the interacting system based on
the exact solution. Using the coordinate Bethe ansatz method, the eigenstates
of the system can be written out explicitly

$\displaystyle\ket{\Psi}_{m}=\sum_{x_{1},...,x_{M}=1}^{N}\psi(x_{1},...,x_{M})b_{x_{1}}^{\dagger}...b_{x_{M}}^{\dagger}\ket{0},$
(15)

where the subscript $m$ takes the values from 1 to the dimensional of Hilbert
space $(N+M-1)!/$$[(N-1)!M!]$, $\psi(x_{1},...,x_{M})$ is the wave function

$\displaystyle\psi=\sum_{p,q}A_{p}(q)\prod_{j=1}^{M}\Big{(}\frac{\beta_{p_{j}}}{-t}\Big{)}^{x_{q_{j}}}\theta(x_{q_{1}}\leq...\leq
x_{q_{M}})$ (16)

with $p=\\{p_{1},...,p_{M}\\}$ and $q=\\{q_{1},...,q_{M}\\}$ being the
permutations of $\\{1,...,M\\}$, $\theta(x_{q_{1}}\leq...\leq x_{q_{M}})$ is
the generalized step function, which is one in the noted regime and zero in
the others, and $\ket{0}$ is the vacuum state. The amplitude $A_{p}$ satisfies

$\frac{A_{p_{1}...p_{j+1}p_{j}...p_{M}}}{A_{p_{1}...p_{j}p_{j+1}...p_{M}}}=\frac{\beta_{p_{j+1}}-\beta_{p_{j}}+U}{\beta_{p_{j+1}}-\beta_{p_{j}}-U}.$
(17)

Each set of Bethe roots $\\{\beta_{j}\\}$ of BAEs (11) completely determines
one eigenstate.

The expected values of particle number operator at the $j$th site
$\bar{n}_{j}={}_{m}\bra{\Psi}n_{j}\ket{\Psi}_{m}$ for all the eigenstates are
shown in Fig.2 (b)-(d), at different values of $U$ while $t$ is fixed to 1. It
is shown that the distributions for most eigenstates are localized
asymmetrically around the boundary, exhibiting the characters of NHSE. This
can be understood directly by analyzing the properties of wave functions.
Multiplying the $M$ BAEs (11), we have

$\prod_{j=1}^{M}|\frac{\beta_{j}}{t}|=\epsilon^{M/N},$ (18)

which gives that

$\displaystyle|\psi(x_{1}+1,...,x_{M}+1)|=\epsilon^{M/N}|\psi(x_{1},...,x_{M})|,$
(19)

where $x_{j}<N$. For the periodic system with $\epsilon=1$, the wave function
has translational symmetry so that the particle number distribution is
uniform. With $\epsilon\neq 1$, the translational symmetry is broken. Because
of the factor $\epsilon^{M/N}$, the wave function will decay if $\epsilon<1$
or increase if $\epsilon>1$ along the sites. Localization lengths of the skin
states at $U=0$ are proportional to the system size and are actually the so-
called scale-free localized states LiLH2021 . The property (18) is independent
on the value of $t/U$. As shown in Fig.2 (b) and (c), the NHSE still survives
when interaction is considered. When $U$ is large enough, the Mott phase
emerges and the skin effect of the ground state is suppressed as shown in
Fig.2 (d). Particularly, the density distribution of a Mott state is uniformly
distributed in the thermodynamical limit, and, thus, the NHSE is completely
suppressed. This can be manifested from our numerical results (see
Supplemental Material SM ) and also the analytical expression of the ground
state wave function in the limit of $U\rightarrow\infty$ given by
$\ket{\Psi}_{G}=b_{1}^{\dagger}b_{2}^{\dagger}...b_{N}^{\dagger}\ket{0}.$ In
SM , we numerically calculate the average deviation of the uniform
distribution defined as $\delta=\frac{1}{N}\sum_{j=1}^{N}|\bar{n}_{j}-\rho|$.
Finite-size analysis of our numerical results indicates that $\delta$
approaches zero in the limit of $N\rightarrow\infty$ for $t/U\leq 0.17$,
whereas it approaches a nonzero value for $t/U\geq 0.18$. These numerical
results suggest that the skin effect of a Mott state is completely suppressed
in the thermodynamical limit.

Summary and outlook.– In the framework of the quantum inverse scattering
method, we proved the integrability of the Bose-Hubbard model with
unidirectional hopping and derived the Bethe ansatz equations for the system.
Based on the exact solution, we investigated the energy spectrum and
distribution of Bethe roots of the ground state, which allows us to identify
the existence of a superfluid-Mott insulating transition for integer fillings.
Furthermore, we studied the effect of boundary parameter $\epsilon$ on the
system and demonstrated the presence of NHSE when $\epsilon$ departs from $1$.
We also elucidated the effect of interaction on NHSE and showed that the NHSE
is completely suppressed for the Mott insulator state in the thermodynamical
limit. Our work introduces a new class of integrable models without Hermitian
correspondence and sheds light on the interplay between interaction and NHSE.
Additionally, our study provides a benchmark for testing numerical techniques
developed for non-Hermitian many-body systems.

Theoretically, it has been unveiled that the combination of loss and
multichannel interference with tunable phases can generate nonreciprocal
hopping and even unidirectional transmission LiuYC . Dissipative chain model
with nonreciprocal hopping has been implemented in cold atomic optical
lattices ExpColdLiang , where nonreciprocal hopping is effectively generated
by introducing an auxiliary lattice with on-site loss ExpColdLapp ;
ExpColdLiang ; LiuYC . Therefore, the Bose-Hubbard model with unidirectional
hopping is principally realizable in current cold atomic experiments.

###### Acknowledgements.

We thank Yanxia Liu for helpful discussions on the numerical study of non-
Hermitian models and Kang Wang for his support in density matrix
renormalization group numerical calculations. The financial supports from
National Key R$\&$D Program of China (Grant No. 2021YFA1402104), the National
Natural Science Foundation of China (Grants No. 12074410, No. 12174436, No.
11934015, and No. T2121001), and Strategic Priority Research Program of the
Chinese Academy of Sciences (Grant No. XDB33000000) are gratefully
acknowledged.

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Supplemental Material for “Exact solution of Bose-Hubbard model with
unidirectional hopping”

##  I Derivation of Bethe Ansatz equations by the algebraic Bethe ansatz
method

In this section of the supplementary material, we present the detailed
procedures of deriving the Bethe ansatz equations (BAEs) by the algebraic
Bethe ansatz method Korepin ; WYCS .

We first consider the $R$ matrix $R_{0,\bar{0}}(u)\in\text{End}(V_{0}\otimes
V_{\bar{0}})$ given by

$R_{0,\bar{0}}(u)=\begin{pmatrix}u-\eta&0&0&0\\\ 0&u&\eta&0\\\ 0&\eta&u&0\\\
0&0&0&u-\eta\end{pmatrix},$ (S1)

where $u$ is the spectral parameter, $\eta=-1$, $V_{0}$ and $V_{\bar{0}}$
represent $2\times 2$ auxiliary spaces. Then we introduce the Lax operator
Kuznetsov89 . In the auxiliary space, the Lax operator is written as

$L_{0,j}(u)=\begin{pmatrix}u-n_{j}&gb_{j}\\\
gb_{j}^{{\dagger}}&-g^{2}\end{pmatrix}.$ (S2)

It can be verified that $R$ matrix and Lax operator satisfy the Yang-Baxter
equation

$R_{0,\bar{0}}(u-v)L_{0,j}(u)L_{\bar{0},j}(v)=L_{\bar{0},j}(v)L_{0,j}(u)R_{0,\bar{0}}(u-v).$
(S3)

The monodromy matrix $T(u)$ is constructed as the product of the $L$ operators
at $N$ different sites of one chain and a $c$-number matrix $K_{0}$:

$\begin{split}T_{0}(u)=L_{0,N}(u)L_{0,N-1}(u)\cdots L_{0,1}(u)K_{0}.\\\
\end{split}$ (S4)

In the auxiliary space, $K_{0}$ is a diagonal matrix given by
$K_{0}=diag(1,\epsilon)$.

From the Yang-Baxter equation (S3), we can further derive the matrix $T(u)$
satisfies the Yang-Baxter relation (the $RTT$ relation)

$R_{0,\bar{0}}(u-v)T_{0}(u)T_{\bar{0}}(v)=T_{\bar{0}}(v)T_{0}(u)R_{0,\bar{0}}(u-v).$
(S5)

Using the $RTT$ relation (S5), it can be proven that the transfer matrices
with different spectral parameters commute with each other, i.e.,
$[t(u),t(v)]=0$. Taking the trace over the auxiliary space of monodromy
matrix, we obtain the transfer matrix

$\begin{split}t(u)&=\text{tr}_{0}[L_{0,N}(u)L_{0,N-1}(u)\cdots
L_{0,1}(u)K_{0}]\\\ &=u^{N}+C_{1}u^{N-1}+C_{2}u^{N-2}+\cdots.\end{split}$ (S6)

The transfer matrix $t(u)$ is an Nth-degree polynomial with respect to $u$
which gives $N$ different conserved quantities $C_{n}$ ($n=1,...,N$). Direct
calculation gives the operators $C_{1}$ and $C_{2}$ as

$\displaystyle C_{1}=-\sum_{j=1}^{N}n_{j},$ (S7) $\displaystyle
C_{2}=\frac{1}{2}\sum_{i\neq
j}^{N}n_{i}n_{j}+g^{2}\left(\sum_{j=1}^{N-1}b_{j}^{{\dagger}}b_{j+1}+\epsilon
b_{N}^{{\dagger}}b_{1}\right).$ (S8)

The integrable Hamiltonian is constructed as

$H=\frac{t}{2g^{2}}\left(C_{1}^{2}+C_{1}-2C_{2}\right),$ (S9)

where the on-site interaction is parameterized as $U=t/g^{2}$. It is easy to
see $[H,C_{n}]=0$, so that $H$ is integrable.

Next we need to diagonalize the transfer matrix to get the eigenstates and
eigenvalues with the help of the $RTT$ relation (S5). In the auxiliary tensor
space, the monodromy matrix can be expressed as

$T(u)=\begin{pmatrix}A(u)&B(u)\\\ C(u)&D(u)\end{pmatrix},$ (S10)

where $A(u)$, $B(u)$, $C(u)$, $D(u)$ are operators depending on the spectral
parameter $u$. Then the transfer matrix is

$t(u)=A(u)+D(u).$ (S11)

From the $RTT$ relation (S5), we can get the commutation relations between
these operators:

$\begin{split}[A(u),A(v)]&=[B(u),B(v)]=[C(u),C(v)]=[D(u),D(v)]=0,\\\
A(u)C(v)=&\frac{u-v-1}{u-v}C(v)A(u)+\frac{1}{u-v}C(u)A(v),\\\
D(u)C(v)=&\frac{u-v+1}{u-v}C(v)D(u)-\frac{1}{u-v}C(u)D(v).\end{split}$ (S12)

The reference state we choose is the vacuum state

$\ket{0}=\ket{0}_{1}\otimes\ket{0}_{2}\otimes\dots\otimes\ket{0}_{N}.$ (S13)

Applying the monodromy matrix $T(u)$ to the reference state, we have

$\begin{split}&A(u)\ket{0}=u^{N}\ket{0},\\\
&D(u)\ket{0}=\epsilon(-\frac{t}{U})^{N}\ket{0},\\\ &B(u)\ket{0}=0.\end{split}$
(S14)

The operator $C(u)$ contains bosonic creation operators and can be used to
construct the Bethe states

$\ket{\mu_{1},...,\mu_{M}}=\prod_{j=1}^{M}C(\mu_{j})\ket{0},$ (S15)

where $M$ is the particle number and $\\{\mu_{j}\\}$ is a set of parameters.
Based on the relation (S12), the following useful formulae can be derived:

$\begin{split}A(u)C(\mu_{1})...C(\mu_{M})=&\prod_{j=1}^{M}\frac{u-\mu_{j}-1}{u-\mu_{j}}C(\mu_{1})...C(\mu_{M})A(u)\\\
&+\sum_{j=1}^{M}\frac{1}{u-\mu_{j}}\prod_{l\neq
j}^{M}\frac{\mu_{j}-\mu_{l}-1}{\mu_{j}-\mu_{l}}C(\mu_{1})...C(\mu_{j-1})C(u)C(\mu_{j+1})...C(\mu_{M})A(\mu_{j}),\\\
D(u)C(\mu_{1})...C(\mu_{M})=&\prod_{j=1}^{M}\frac{u-\mu_{j}+1}{u-\mu_{j}}C(\mu_{1})...C(\mu_{M})D(u)\\\
&-\sum_{j=1}^{M}\frac{1}{u-\mu_{j}}\prod_{l\neq
j}^{M}\frac{\mu_{j}-\mu_{l}+1}{\mu_{j}-\mu_{l}}C(\mu_{1})...C(\mu_{j-1})C(u)C(\mu_{j+1})...C(\mu_{M})D(\mu_{j}).\end{split}$
(S16)

Applying $t(u)$ to the Bethe state and using relations (S12) (S16), we have

$\begin{split}t(u)\ket{\mu_{1},...,\mu_{M}}&=\left(A\left(u\right)+D\left(u\right)\right)\prod_{j=1}^{M}C(\mu_{j})\ket{0}\\\
&=\Lambda(u)\prod_{j=1}^{M}C(\mu_{j})\ket{0}+\sum_{j=1}^{M}\Lambda_{j}(u)C(\mu_{1})...C(\mu_{j-1})C(u)C(\mu_{j+1})...C(\mu_{M})\ket{0},\end{split}$
(S17)

where

$\begin{split}&\Lambda(u)=u^{N}\prod_{j=1}^{M}\frac{u-\mu_{j}-1}{u-\mu_{j}}+\epsilon(-\frac{t}{U})^{N}\prod_{j=1}^{M}\frac{u-\mu_{j}+1}{u-\mu_{j}},\\\
&\Lambda_{j}(u)=\frac{1}{u-\mu_{j}}\left(\mu_{j}^{N}\prod_{l\neq
j}^{M}\frac{\mu_{j}-\mu_{l}-1}{\mu_{j}-\mu_{l}}-\epsilon(-\frac{t}{U})^{N}\prod_{l\neq
j}^{M}\frac{\mu_{j}-\mu_{l}+1}{\mu_{j}-\mu_{l}}\right).\end{split}$ (S18)

To ensure the Bethe state to be an eigenstate of the transfer matrix, the
unwanted terms must vanish, i.e., $\Lambda_{j}(u)=0$. This induces the BAEs:

$\frac{U^{N}}{(-t)^{N}}\mu_{j}^{N}=\epsilon\prod_{l\neq
j}^{M}\frac{\mu_{j}-\mu_{l}+1}{\mu_{j}-\mu_{l}-1},\quad j=1,\cdots,M.$ (S19)

From (S16) and (S17), we can see that the solutions {$\mu_{j}|j=1,...,M$}
should satisfy $\mu_{j}\neq\mu_{l}$ for $j\neq l$.

Expanding $\Lambda(u)$ and comparing with the polynomial of $t(u)$, we can get

$\begin{split}&C_{1}\ket{\mu_{1},...,\mu_{M}}=-M\ket{\mu_{1},...,\mu_{M}},\\\
&C_{2}\ket{\mu_{1},...,\mu_{M}}=\left(-\sum_{j=1}^{M}\mu_{j}+\frac{M(M-1)}{2}\right)\ket{\mu_{1},...,\mu_{M}}.\end{split}$
(S20)

Thus, the eigenvalue of Hamiltonian is

$E=U\sum_{j=1}^{M}\mu_{j}.$ (S21)

For convenience, we put $\beta_{j}=U\mu_{j}$. The BAEs can be rewritten as

$\left(\frac{\beta_{j}}{-t}\right)^{N}=\epsilon\prod_{l\neq
j}^{M}\frac{\beta_{j}-\beta_{l}+U}{\beta_{j}-\beta_{l}-U},\quad j=1,\cdots,M,$
(S22)

where $\beta_{j}\neq\beta_{l}$ for $l\neq j$. The eigenvalue of the
Hamiltonian becomes

$E=\sum_{j}^{M}\beta_{j}.$ (S23)

##  II Solutions of Bethe roots

Here we provide more detailed analysis of BAEs (S22). We consider the case of
$\epsilon=1$ and mainly concern the Bethe root patterns in the two limits
$t/U\to\infty$ and $t/U\to 0$.

When $U$ is small ($t/U$ is large), the right side of BAEs (S22) tends to 1,
so that the solution $\\{\beta_{j}\\}$ is

$\beta_{j}=te^{\frac{i\pi(N+2l)}{N}}+\delta_{j},l=1,\cdots,N,j=1,\cdots,M,$
(S24)

where $\delta_{j}$ is a perturbation induced by $U$. When $U=0$,
$\delta_{j}=0$ and $\beta_{j}$ is the solution for the free boson with
unidirectional hopping. The corresponding spectrum, which is the sum of each
set of Bethe roots, resembles regular polygons.

As $U$ becomes large ($t/U$ approaches zero), the interaction term begins to
dominate and causes a change in the Bethe roots pattern. If there exists a
Bethe roots $\beta_{j}\in\\{\beta_{j}\\}$ satisfying $|\beta_{j}|>t$, the left
side of BAEs (S22) tends to infinity in the thermodynamic limit, i.e.,
$|\beta_{j}/t|^{N}\to\infty$, which requires that the denominator of right
hand side should tend to zero. Therefore, there must exist another Bethe root
$\beta_{k}\in\\{\beta_{j}\\}$ with $\beta_{k}=\beta_{j}-U$. If
$|\beta_{k}|<t$, both sides of BAEs (S22) tend to zero with $N\to\infty$, and
the BAEs can be satisfied. Conversely, if $|\beta_{k}|>t$, there must exist
another Bethe root $\beta_{l}\in\\{\beta_{j}\\}$ with $\beta_{l}=\beta_{k}-U$.
Based on singularity analysis, we know that for a given $M$, there must at
least exist one $\beta_{j,1}$, which satisfies $|\beta_{j,1}|<t$ in each set
of solutions. Then we conclude that if $U$ is large enough, the Bethe roots
form the strings along the real axis with a length of $k$,

$\beta_{j,k}=\beta_{j,1}+(k-1)U,\quad k=1,\cdots,M,$ (S25)

here $|\beta_{j,k}/U|\approx k-1$. Since the quantum number $I_{j}$
corresponding to each Bethe root has only $N$ different values, $\beta_{j,1}$
has $N$ different values, i.e. there are $N$ different strings.

The string structure reflects the unique properties of bosonic systems. When
the interaction strength $U$ is large, the hard-core boson exclusion principle
leads to the restriction of $\beta_{j,1}$ to $N$ distinct values. The Bethe
roots form strings with gaps $U$, which describe the potential energy required
to add a boson to one site already occupied by $k$ bosons.

In this way, the pattern of Bethe roots can be determined. For a given $M$,
suppose that there are $m_{l}$ Bethe roots in the $l$-th string

$\sum_{l=1}^{N}m_{l}=M,\ l=1,...,N.$ (S26)

Each set of $\\{m_{l}\\}$ corresponds exactly to one solution of BAEs (S22)

$\\{\beta_{j}\\}=\\{\beta_{1,1},\beta_{1,2},...,\beta_{1,m_{1}},\beta_{2,1},...,\beta_{2,m_{2}},...,\beta_{N,1},...,\beta_{N,m_{N}}\\}.$
(S27)

The corresponding energy is $E\approx\sum_{l=1}^{N}m_{l}(m_{l}-1)U/2$.

Figure S1: The distributions of Bethe roots and the energy spectrum with
$U=0.1$ in (A, B), $U=5$ in (C, D). Each set of Bethe roots is color-coded in
(A, C), with corresponding eigenvalues marked in (B, D) using the same color.
The hollow circles denote the Bethe roots for the ground state, while blue
asterisks represent exact diagonalization results. Common parameters: $N=4$,
$M=4$, $t=1$, $\epsilon=1$.

The numerical verification of above results are given in Fig.S1. By solving
the BAEs (S22) under different values of $U$, the corresponding Bethe roots
are obtained (Figs.S1(A) and (C)). When $t/U$ is large, all $\beta_{j}$s are
distributed around $t\exp[i\pi(N+2l)/N]$ ($l=1,\cdots,N$). When $t/U$ is
small, all $\beta_{j}$s are localized around discrete regimes satisfying
$\operatorname{Re}(\beta_{j})/U\approx k-1$ $(k=1,...M)$. Based on the above
Bethe roots, we obtain the spectrum of the system and show them in Fig.S1(B)
and (D). The spectrum can also be calculated by using the numerical exact
diagonalization method (shown in Fig.S1(B) and (D) as the asterisks). We can
see that the results obtained by the two methods are in good agreement with
each other.

We define the ground state energy as the one with minimum real part. It is
worth noting that the $M$ Bethe roots at the ground state are always
symmetrically distributed around the real axis, ensuring that the ground state
energy is always real. When $t/U\to\infty$, the excitation of the system is
gapless, which is independent of the filling factor $\rho=M/N$. When $t/U\to
0$, the solutions of $\beta_{j}$s are described by the string solution. The
solution corresponding to the ground state is the one among all possible
solutions with the smallest sum of the real parts of the roots. Thus, the
solution is $\\{\beta_{j,k}\\}$ with $k$ as small as possible. Considering
that each $k$ layer in the string structure has at most $N$ possible
positions, the pattern of the solution of the ground state is related to
$\rho$. Especially, when $\rho=1$, the solution $\\{\beta_{j}\\}$
correspondings to the ground state is composed of $M$ different $\beta_{j,1}$.
In this case, the right side of BAEs (S22) tends to $(-1)^{M-1}$, so that

$\beta_{j}=te^{\frac{i\pi(N+M+2l-1)}{N}},l=1,\cdots,N,j=1,\cdots,M.$ (S28)

Now, adding one boson to this system results in a change of the ground state
energy by $\mu_{+}=\operatorname{Re}{(\beta_{N+1})}\approx U$, while removing
one boson results in a change of the energy by
$\mu_{-}=\operatorname{Re}{(\beta_{N})}=t$. We have $\mu_{+}>\mu_{-}$. It is
evident that this phenomenon of energy gap arising from particle addition or
removal emerges only in systems with integer fillings.

##  III More numerical analysis on the Superfluid-Mott Phase Transition

In this section, we present numerical analysis on the superfluid-Mott phase
transition based on the calculation of density-density correlation functions.
Using the non-Hermitian density matrix renormalization group (DMRG) algorithm,
we calculate the correlation function $C(r)$ of the ground state, which is
defined as

$C(r)=\braket{n_{r}n_{0}}-\braket{n_{r}}\braket{n_{0}}.$ (S29)

In references Haldane81 ; Giamarchi81 ; Kuhner00 ; Ejima11 , it has been
unveiled that the density-density correlation function exhibits a power-law
decay with distance the superfluid state, whereas it follows an exponential
decay in the Mott phase.

Figure S2: Density-density correlation function $C(r)$ calculated by the non-
Hermitian DMRG algorithm with periodic boundary condition $\epsilon=1$.
$E_{D}$ is the ground state energy calculated by the non-Hermitian DMRG
algorithm and $E$ corresponds to the ground state energy obtained from the
exact solution. It can be observed that the ground state energy values
provided by these two methods are nearly identical. Here, we fix $N=M=128$,
$U=1$, $\epsilon=1$.

In Fig.S2, we display the correlation function $C(r)$ versus $r$ calculated by
the non-Hermitian DMRG algorithm under the periodic boundary condition for
various values of $t$ around the transition point. We compare the ground state
energy obtained by the non-Hermitian DMRG algorithm with the exact result
obtained by Bethe ansatz and find they agree very well. From Fig.S2, we can
observe that at $t/U=0.18$, 0.19, and 0.20, the behavior of the $C(r)$ curves
in terms of decay is entirely distinct from that observed at $t/U=0.15$ and
0.16. The upper three curves in the graph either approach or decay more slowly
than a power-law, while the lower two curves significantly deviate from the
linear behavior observed in the double-logarithmic plot, indicating a faster
decay rate than a power-law. At $t/U=0.17$, the decay pattern closely aligns
with and slightly exceeds a power-law decay rate. It is noticeable that the
critical point for the superfluid-to-Mott phase transition in the system is
slightly greater than $t/U=0.17$. This observation is consistent with our
previous conclusion.

Figure S3: (A, B, C) Distribution of Bethe roots of the ground state for
$N=100,\ M=\ 200\ \text{(A)},\ 199\ \text{(B)},\ 201\ \text{(C)}$ with $t$
varying from 0.4 to 0.05 (the step size is 0.01). (D) Numerical results of
$\mu_{+}$, $\mu_{-}$ of the system with $N=100$,$M=200$, $U=1$ with $t$
varying from 0.4 to 0.05 (The step size is 0.0001). (E) The ground state
energy (blue point) and the second derivative of $E_{g}/N$ with respect to
$t/U$ (red point) for $N=100$, $M=200$ with $t$ varying from 0.4 to 0.05 (step
size 0.0001). (F) The extreme values of
$\frac{\mathrm{d^{2}}E_{g}}{N\mathrm{d}t^{2}}$ plotted against $1/\sqrt{N}$
from $N=20$ to $N=150$ ($\rho=2$). The dashed fitted lines is
$1/\frac{\mathrm{d^{2}}E_{g}}{N\mathrm{d}t^{2}}=C_{1}/\sqrt{N}+C_{2}$. The
fitting parameters are $C_{1}=-0.15374$, $C_{2}=-6.6690\times 10^{-6}$. (G)
The value of $t_{m}$ (blue points) and $t_{o}$ (red points) versus $1/N$ with
$N=20$ to $150$ ($\rho=2$). The dashed lines are fitted to determine the
values of $t_{m}$ and $t_{o}$ as the system size tends to infinity. The
fitting results are $t_{m}=0.10100$ and $t_{o}=0.10101$. Here we fix $U=1$,
$\epsilon=1$.

##  IV The case of $\bf{\rho=2}$

From the analysis of BAEs, we can see that there exists quantum phase
transition in this model as long as $\rho=$ integer. Here we discuss the case
of $\rho=2$.

For the system with $M=2N$, for small values of $U$, all the Bethe roots tend
to concentrate around $(-t,0)$ in the complex plane. As $U$ increases, $N$ of
these Bethe roots gradually move towards the origin $(0,0)$, while the
remaining $N$ Bethe roots move towards $(U,0)$. If a boson is removed from the
system, $\beta_{2N}$ disappears, and the pattern of Bethe roots resembles that
of $M=2N$. Differently, when a boson is added to the system, the $2N$ Bethe
roots in the first two layers ($k=1,2$) are already occupied, so
$\beta_{2N+1}$ will be squeezed out as $U$ increases, eventually approaching
$(2U,0)$.

The solutions of the BAEs (S22) with $M=2N$ and $M=2N\pm 1$ are given in
Figs.S3 (A, B, C). Similar to the $\rho=1$ case, there exists an odd point
$t_{o}$ where $\beta_{2N}$ and $\beta_{2N+1}$ coincide. The diagram of
$\mu_{+}$ and $\mu_{-}$ (Fig.S3 (D)) shows that the energy gap of the system
undergoes a noticeable change near $t/U=t_{o}$. Moreover, the ground state
energy curve versus $t/U$ of the $\rho=2$ system has a derivative
discontinuity point $t_{m}$ (Fig. S3(E)). By analyzing the finite size scaling
of the maximum of $\frac{d^{2}E_{g}}{Ndt^{2}}$ (Fig. S3 (F)), we observe that
the second order derivative of $E_{g}/N$ at $t=t_{m}$ diverges as
$N\to\infty$, with the scaling behavior given by
$\left.\frac{d^{2}E_{g}}{Ndt^{2}}\right|_{t=t_{m}}\propto\sqrt{N}$.

The finite size scaling behaviors of $t_{o}$ and $t_{m}$ are shown in
Fig.S3(G). The value of $t_{o}$ and $t_{m}$ converge to nearly the same value
in the thermodynamic limit. We can confirm that the superfluid-Mott phase
transition critical point value is $t_{c}=0.10100\pm 10^{-5}$
($\alpha_{c}=U/t_{c}\approx 9.90$) for the case of $\rho=2$.

##  V Numerical analysis on the suppression of the skin effect

Through the analysis of the change of spectrum and the correlation, we have
demonstrated that the ground state of the system undergoes a superfluid-Mott
transition. To see clearly how the density distribution changes with the
change of $t/U$, we consider the case $\rho=1$ and calculate the density
distribution of the ground for various $t$ by fixing $U=1$. Using the non-
Hermitian DMRG algorithm, we demonstrate the density distributions of the
ground state for the system with $N=128$ and various $t$ in Fig.S4. Here
$\bar{n}_{j}$ is the expected values of particle number operator at the $j$th
site $\bar{n}_{j}={}_{G}\bra{\Psi}n_{j}\ket{\Psi}_{G}$. For fixed $N$ and
$\epsilon$, the boson distribution gradually approaches a uniform distribution
with decrease of the value of $t/U$.

Figure S4: The density distribution of the ground state $\bar{n}_{j}$ with
various $t$. The numerical results are obtained by using the non-Hermitian
density matrix renormalization group algorithm. Here $j$ refers to the $j$-th
site, and we have taken $N=128$, $\epsilon=0.5$ and $U=1$.

To see how the density distributions change with the lattice size $N$ in
different parameter regions, we display the density distributions of systems
with different sizes in Fig.S5. While Fig.S5 (A) with $t/U=0.20$ corresponds
to the superfluid phase, Fig.S5 (B) with $t/U=0.15$ corresponds to the Mott
phase. For $t/U=0.15$, it is shown that the density distribution gradually
approaches a uniform distribution with the increase of the size. However, for
$t/U=0.20$, the density distribution does not approach a uniform distribution
even in the large size limit.

Figure S5: The density distribution of the ground state $\bar{n}_{j}$ with
different sizes for (A) $t=0.20,U=1$ and (B) $t=0.15,U=1$. Here $j$ refers to
the $j$-th site and $\epsilon=0.5$.  Figure S6: The value of
$\delta=\frac{1}{N}\sum_{j=1}^{N}|\bar{n}_{j}-\rho|$ versus $1/N$ with $N=32$
to $128$ at intervals of $8$ under different values of $t/U$. The gray lines
are fitted with a first-degree polynomial using data of $N=96,\ 104,\ 112,\
120,\ 128$ to determine the values of $\delta$ as the system size tends to
infinity. Here $\epsilon=0.5$

In order to characterize quantitatively the change of density distribution
around the transition point, we define

$\delta=\frac{1}{N}\sum_{j=1}^{N}|\bar{n}_{j}-\rho|$ (S30)

to quantify an average deviation of the uniform distribution. The finite size
scaling results of $\delta$ with different $t/U$ are presented in Fig.S6. For
$t/U\leq 0.17$, it is observed that the fitting results give $\delta=0$ in the
thermodynamic limit. However, for $t/U\geq 0.18$, the fitting value of
$\delta>0$ is evident in the thermodynamic limit. Since $\delta=0$ implies the
complete suppression of the deviation from the uniform distribution, the skin
effect of the ground state is completely suppressed in the thermodynamic limit
when the system is in the Mott phase for $t/U\leq 0.17$. This is in sharp
contrast with the nonzero value of $\delta$ for $t/U\geq 0.18$. The different
behaviors of $\delta$ in the thermodynamic limit suggest that the critical
value lies between 0.17 and 0.18, consistent with value of the superfluid-Mott
phase transition given by $t_{c}/U=0.17155$.

## References

  * (1) V.E. Korepin, A.G. Izergin, and N.M. Bogoliubov, Quantum Inverse Scattering Method, Correlation Functions and Algebraic Bethe Ansatz (Cambridge University Press, Cambridge, England, 1993).
  * (2) Y. Wang, W.-L. Yang, J. Cao, and K. Shi, Off-Diagonal Bethe Ansatz for Exactly Solvable Models (Springer, New York, 2015).
  * (3) V B Kuznetsov and A V Tsiganov, A special case of Neumann’s system and the Kowalewski-Chaplygin-Goryachev top,  J. Phys. A: Math. Gen. 22 L73, (1989).
  * (4) F.D.M. Haldane, Effective Harmonic-fluid approach to low-energy properties of one-dimensional quantum fluids, Phys. Rev. Lett. 47, 1840 (1981).
  * (5) T. Giamarchi and H. J. Schulz, Correlation functions of one-dimensional quantum systems, Phys. Rev. B 39, 4620 (1981).
  * (6) Kühner T D, White S R, Monien H. One-dimensional Bose-Hubbard model with nearest-neighbor interaction, Phys. Rev. B 61, 12474 (2000).
  * (7) S. Ejima, H. Fehske, and F. Gebhard, Dynamic properties of the one-dimensional Bose-Hubbard model, Europhys. Lett. 93, 30002 (2011).


# An Alexander method for infinite-type surfaces

Roberta Shapiro

###### Abstract

The Alexander method is a combinatorial tool used to determine when two
elements of the mapping class group are equal. We extend the Alexander method
to include the case of infinite-type surfaces. Versions of the Alexander
method were proven by Hernández–Morales–Valdez and Hernández–Hidber. As sample
applications, we verify a particular relation in the mapping class group, show
that the centralizers of many twist subgroups of the mapping class group are
trivial, and provide a simple basis for the topology of the mapping class
group.

##  1 Introduction

The Alexander method is a tool used to determine whether two homeomorphisms of
a surface $S$ are isotopic. It states that two homeomorphisms of $S$ are
isotopic if and only if they have identical actions on the isotopy classes of
the curves in an Alexander system, which is a collection of curves with
certain properties described below.

The mapping class group of a surface $S$, denoted $\operatorname{MCG}(S)$, is
the group of isotopy classes of homeomorphisms of $S$. (When $S$ is
orientable, this group is usually called the extended mapping class group;
this distinction will be of no consequence in this paper.) There is an analogy
between mapping class groups and general linear groups. Via the Alexander
method, Alexander systems for surfaces play the role of basis vectors in a
vector space in the sense that isotopy classes of homeomorphisms are
determined by their actions on such a set of curves.

Most earlier results surrounding the Alexander method work with surfaces of
finite-type–that is, surfaces whose fundamental groups are finitely generated.
The most general statement of the Alexander method for finite-type surfaces
was formulated by Farb–Margalit [7, Proposition 2.8], but the method can be
traced back to the works of Dehn [4, 5] and Thurston [1, 8].

In this paper, we extend the notion of an Alexander system to include curve
systems on infinite-type surfaces (surfaces whose fundamental groups are
infinitely generated), and then prove our main result: a generalization of the
Alexander method of Farb–Margalit to include the case of infinite-type
surfaces (including non-orientable surfaces) [7].

A version of the Alexander method was proven for orientable surfaces by
Hernández–Morales–Valdez [11] and for non-orientable surfaces by
Hernández–Hidber [10]. In both of these papers, the authors construct a family
of Alexander systems such that any homeomorphism that permutes the isotopy
classes of curves and arcs in an Alexander system by the identity is isotopic
to the identity homeomorphism. In this paper, we prove the Alexander method in
the case of arbitrary permutations and arbitrary Alexander systems. As such,
our main theorem is a more direct analogue of the Alexander method for finite-
type surfaces given by Farb–Margalit [7].

Alexander systems. Let $S$ be a (possibly infinite-type and possibly non-
orientable) surface. Although $S$ may have punctures, it will be convenient to
treat these punctures as marked points. For the remainder of this paper, we
will refer to marked points and punctures (isolated planar ends)
interchangeably. We define a curve to be the image of an embedding
$\gamma:S^{1}\hookrightarrow S$. In this paper, an arc is the image of a
proper embedding $\gamma:(0,1)\hookrightarrow S$ with the additional
characteristic of being finite-type—that is, every arc has a regular
neighborhood that is a compact surface (treating punctures as marked points).

A curve in $S$ is essential if it is not nullhomotopic, not homotopic to a
puncture, and not homotopic to the boundary of a Möbius band; simple if it
does not self-intersect; and non-peripheral if it is not isotopic to a
boundary component. Our definition of curves implies that all curves are
simple. An arc is essential and non-peripheral if it and (a subset of) a
boundary component do not jointly bound an unpunctured disk in $S$.

A subsurface exhaustion of surface $S$ is a sequence of finite-type
subsurfaces $\\{S_{n}\\}$ with $S_{i}\subset S_{i+1}$ such that $\cup
S_{n}=S$.

We say that a collection $\Gamma=\\{\gamma_{i}\\}_{i\in I}$ of essential, non-
peripheral, simple closed curves and finite-type arcs on $S$ is an Alexander
system if it satisfies the following properties:

  1. 1.

(minimal position) $\gamma_{i}$ and $\gamma_{j}$ are in minimal position for
all $i,j\in I,$

  2. 2.

(distinct isotopy classes) no two elements of $\Gamma$ are isotopic,

  3. 3.

(no triple intersections) for distinct $i,j,k\in I,$ at least one of
$\gamma_{i}\cap\gamma_{j},$ $\gamma_{i}\cap\gamma_{k},$ or
$\gamma_{j}\cap\gamma_{k}$ is empty, and

  4. 4.

(local finiteness) for a fixed subsurface exhaustion $\\{S_{n}\\}$ of $S$,

$|\\{\gamma_{i}\in\Gamma\ :\ \gamma_{i}\cap S_{n}\neq\emptyset\\}|<\infty$

for all $n$.

Properties (1), (2), and (3) of Alexander systems are inherited from the
classical finite-type Alexander method. Property (4) is only needed for
infinite-type surfaces and is automatically satisfied for finite-type
surfaces. Property (4) is also equivalent to the following: every finite-type
subsurface of $S$ intersects finitely many elements of $\Gamma.$ In fact,
property (4) is a necessary condition; a counterexample to the Alexander
method with property (4) not satisfied is given in Section 4.

We say that a set of curves and arcs in minimal position in $S$ fills $S$ if
$S\setminus\Gamma$ is a union of disks, once punctured disks, and Möbius bands
(each possibly with noncompact boundary). Equivalently, the components of
$S\setminus\Gamma$ have a trivial mapping class group.

Statement of the main theorem. To state the main theorem, we require several
more definitions. Define a surface graph to be an abstract graph with an
embedding into some surface. Given an Alexander system on a surface $S$, let
$G(S,\Gamma)=\cup\gamma_{i}$ be the surface graph (an abstract graph with an
embedding into a surface) in $S$ whose vertex set, denoted $V(G),$ is
comprised of the points of intersection of curves and arcs in $\Gamma$ and the
endpoints of arcs (using marked points in lieu of punctures), and whose edges
are the connected components of $\cup\gamma_{i}\setminus V(G)$. Let
$G^{\prime}(S,\Gamma)$ be the barycentric subdivision of $G(S,\Gamma)$, which
is also a surface graph.

An automorphism of a surface graph is an automorphism of the associated
abstract graph that arises from a homeomorphism of the surface.

######  Main theorem.

Let $S$ be any surface, $\Gamma=\\{\gamma_{i}\\}_{i\in I}$ an Alexander system
in $S$, and $\phi:S\to S$ a homeomorphism. Suppose $\sigma\in\Sigma_{I}$ is a
permutation such that $\phi(\gamma_{i})$ is isotopic to $\gamma_{\sigma(i)}$
for all $i.$ Then,

  1. 1.

there exists a homeomorphism $\psi:S\to S$ isotopic to $\phi$ rel $\partial S$
such that $\psi(\gamma_{i})=\gamma_{\sigma(i)}$ for all $\gamma_{i}\in\Gamma,$

  2. 2.

$\phi$ induces a unique automorphism $\phi_{*}$ of $G^{\prime}(S,\Gamma),$ and

  3. 3.

if $\Gamma$ is filling, then $\phi_{*}$ is the identity if and only if $\phi$
is isotopic to the identity.

The statement that $\phi_{*}$ acts by the identity on the barycentric
subdivision of $G(S,\Gamma)$ is equivalent to saying that $\phi$ induces the
identity automorphism of $G(S,\Gamma)$ and preserves the orientation of each
loop edge.

We further note that $\phi$ need not be a self-homeomorphism of $S$, as our
proof will not use this at all. That is, our proof applies to a homeomorphism
$\phi:S\to S^{\prime}$ and Alexander systems $\Gamma$ (indexed by $I$) and
$\Gamma^{\prime}$ (indexed by $I^{\prime}$) on $S$ and $S^{\prime}$,
respectively, with a bijection $\sigma:S\to S^{\prime}.$ In the case of
distinct surfaces, $\phi_{*}$ is a graph isomorphism and the last statement of
the main theorem does not apply.

When we work with the domain of $\phi,$ we will say we are working with the
source and when we work with the range of $\phi$, we will name it the target.

A useful version: the case of the identity mapping class. A stable Alexander
system is an Alexander system such that any mapping class that acts by the
identity on the set of isotopy classes of curves in this system is the
identity mapping class.

The existence of a stable Alexander system is proved for orientable infinite-
type surfaces by Hernández–Morales–Valdez [11] and for non-orientable surfaces
by Hernández–Hidber [10], both without noncompact boundary components. Both
papers provide explicit constructions for stable Alexander systems. Given the
constructions, proof of stability directly follows from the Alexander method
for finite-type surfaces. The existence of a stable Alexander system is
further proved for surfaces with noncompact boundary by Dickmann [6].

In this paper, we provide another criterion for determining whether a
homeomorphism is isotopic to the identity based on how it acts on an Alexander
system, but this time without a significant restriction on the Alexander
system; this is item (3) in the Alexander method.

Applications of the Alexander method. The Alexander method and the existence
of stable Alexander systems are key to proving multiple fundamental results
about mapping class groups of finite-type surfaces, including the computation
of the center of the mapping class group, the Dehn–Nielsen–Baer theorem, the
solvability of the word problem for the mapping class group, and the existence
of certain relations in the mapping class group [7].

Some of the above applications were extended to mapping class groups of
infinite-type surfaces. For instance, Lanier and Loving [12] compute the
center of the mapping class group of a surface of infinite type using the
results of Hernández–Morales–Valdez; their approach is analogous to the
finite-type case.

In Section 2 of this paper, we include several sample applications of our
Alexander method: verifying relations in the mapping class group, computing
centralizers of subgroups of the mapping class group, and describing the
topology of the mapping class group using a simpler basis.

Paper outline. In Section 2, we prove the applications of our main theorem
from the previous paragraph. In Section 3, where we prove the main theorem. We
conclude with Section 4, where we provide counterexamples to the Alexander
method with each of the hypotheses altered.

Acknowledgments. The author is infinitely grateful to Dan Margalit for his
support and for many helpful conversations. The author would further like to
thank Jesús Hernández Hernández for extensive disucssions on a draft of this
paper. The author would also like to thank Ferrán Valdez for correspondences
regarding the subject matter and Daniel Minahan, Jorge Aurelio Víquez,
Katherine Booth, Noah Caplinger, Yvon Verberne, Wade Bloomquist, and Brian
Pinsky for many discussions. Thank you to Justin Lanier, John Etnyre, and
Jennifer Hom for comments on a draft of this paper. Finally, thank you to an
anonymous referee for pointing out a hole in an earlier draft of this paper.
The author is supported by NSF grant DMS 1745583.

##  2 Sample applications

In this section, we discuss three applications of the Alexander method:
proving relations in the mapping class group of a surface, finding the
centralizer of subgroups of the mapping class group, and describing the
topology (and, more precisely, the non-discreteness and local non-compactness)
of the mapping class group using a simpler basis. These applications represent
the many types of questions the Alexander method can answer.

###  2.1 Relations in the mapping class group

Let $S$ be the surface with three ends accumulated by genus and no boundary
components nor punctures (the tripod surface). We label the ends $a,\ b,$ and
$c,$ as in Figure 1 and let $\Gamma$ be the Alexander system pictured in
Figure 1.

$b$$c$$a$$h_{ca}=h_{ac}^{-1}$$h_{bc}$$h_{ab}$ Figure 1: Surface $S$ with ends
$a,\ b,$ and $c$, along with an Alexander system $\Gamma$ on $S$. The handle
shifts used in this example are pictured as well.
$a$$c$$b$$id$$h_{ab}$$h_{bc}$$h_{ca}$ Figure 2: Computations for the action of
$h_{ca}h_{bc}h_{ab}$ on $S$ and $\Gamma$.

Let $h_{ab}$ be a handle shift from end $a$ to end $b$, $h_{bc}$ a handle
shift from end $b$ to end $c,$ and $h_{ca}$ a handle shift from end $c$ to end
$a.$ (Handle shifts are not determined by a pair of ends; we take the arrows
in Figures 1 and 2 as the definitions of the handle shifts.) We verify a the
following relation, which is a special case of a relation initially proven by
Afton–Freedman–Lanier–Yin [2] using the Alexander method for finite-type
surfaces.

######  Proposition 2.1 (Afton–Freedman–Lanier–Yin).

Let $S$ be the surface and $h_{ab},\ h_{bc}$ and $h_{ca}$ the handle shifts in
Figure 2. Then,

$h_{ca}h_{bc}h_{ab}=id,$

where we compose from right to left.

The computations are shown in Figure 2. We only need to keep track of the
curves within one genus of the center of $S$, as all other curves are not
distorted by the handle shifts.

The orientation of loop edges in $G(S,\Gamma)$ does not change upon
application of $h_{ca}h_{bc}h_{ab}$. We conclude that $h_{ca}h_{bc}h_{ab}=id$
is a valid relation.

Afton–Freedman–Lanier–Yin also observe that in the tripod surface, $h_{ab},\
h_{bc},$ and $h_{ca}^{-1}$ are conjugate in $\operatorname{MCG}(S)$, implying
that in the abelianization of the $\operatorname{MCG}(S),$ handle shifts are
equal to the identity [2].

Similar relations are true in more general settings, including non-orientable
handle shifts. Such relations can be verified using the version of the
Alexander method in Hernández–Morales–Valdez [11]; our Alexander method
provides more flexibility with the choice of the Alexander system. We note
that the action on the above Alexander system alone (without considering the
orientations of curves) does not show that the above homeomorphism is isotopic
to the identity, as there is a second, orientation-reversing homeomorphism of
$S$ that fixes the isotopy classes of all curves in $\Gamma.$

###  2.2 Centralizer of subgroups of mapping class groups

In this section, we will show a result relating to the centralizers of twist
subgroups of mapping class groups.

Let $S$ be a surface with filling Alexander system
$\Gamma=\\{\gamma_{i}\\}_{i\in I}$ comprised of two-sided curves. We define
$T=\langle\\{T_{\gamma_{i}}^{k_{i}}\\}_{i\in I}\rangle$ to be the subgroup
generated by the $k_{i}$th powers of the Dehn twists about the $\gamma_{i}$.
The propositions in this section concern this subgroup.

Let $G_{r}(S,\Gamma)$ be the ribbon graph associated with $G(S,\Gamma)$. Since
$\Gamma$ is comprised of two-sided curves without triple intersections, we
have that the geometric realization of $G_{r}(S,\Gamma)$ is a union of annuli
$\\{A_{i}\\}$ with core curves $\\{\gamma_{i}\\}$ glued together along disks.
We define an automorphism of a ribbon graph to be an automorphism of the
underlying abstract graph that preserves the $A_{i}$ setwise. We note that
this is not the usual definition of an automorphism of a ribbon graph.

If $f:S\to S$ induces an automorphism of $G_{r}(S,\Gamma)$ that fixes the
curves corresponding the elements of $\Gamma$, then each $A_{i}$ is mapped to
itself and its orientation is either preserved or reversed. In particular, for
the orientation of $A_{i}$ to be preserved, the core curve $\gamma_{i}$ of
$A_{i}$ must be mapped to itself and the orientations of both the core and
cocore of $A_{i}$ must both be preserved or must both be reversed.

Let $G^{\prime}_{a}(S,\Gamma)$ be the abstract graph associated with
$G^{\prime}(S,\Gamma).$ We note that automorphisms of $G^{\prime}(S,\Gamma)$
induce automorphisms of $G_{r}(S,\Gamma)$ by considering $G_{r}(S,\Gamma)$ to
be embedded in $S$ with core curves $\gamma_{i}$. Similarly, automorphisms of
$G_{r}(S,\Gamma)$ induce automorphisms of $G^{\prime}_{a}(S,\Gamma)$ by
forgetting the annular structure. That is, there are natural inclusion (and,
in particular, injective) maps

$\operatorname{Aut}(G^{\prime}(S,\Gamma))\xhookrightarrow{F}\operatorname{Aut}(G_{r}(S,\Gamma))\xhookrightarrow{H}G^{\prime}_{a}(S,\Gamma).$

We note that if we replace $\operatorname{Aut}(G^{\prime}(S,\Gamma))$ by
$\operatorname{Aut}(G(S,\Gamma))$, this map need not be injective.

We call an Alexander system $\Gamma$ on $S$ weakly stable if: 1) it is filling
and 2) any automorphism of $G_{r}(S,\Gamma)$ in the image of $F$ above that
preserves each $A_{i}$ with orientation is the image of the trivial
automorphism of $G^{\prime}(S,\Gamma)$.

######  Proposition 2.2.

Let $S$ be a surface and $\Gamma$ a weakly stable Alexander system comprised
of two-sided curves. Let $\\{k_{i}\\}_{i\in I}$ be a collection of non-zero
integers. Then, the centralizer in $\operatorname{MCG}(S)$ of the subgroup
$T=\langle\\{T_{\gamma_{i}}^{k_{i}}\\}_{i\in I}\rangle$ is trivial.

It is not straightforward to see how Proposition 2.2—and, by extension,
Corollary 2.3 that follows—could be achieved using the version of the
Alexander method of Hernández–Morales–Valdez [11], as the graphs $G(S,\Gamma)$
and $G_{r}(S,\Gamma)$ play a crucial role in the proofs.

###### Proof of Proposition 2.2.

Suppose $[f]\in\operatorname{MCG}(S)$ is in the centralizer of $T$. Since
$[f]$ commutes with each $T_{\gamma_{i}}^{k_{i}},$ it follows that $f$ fixes
the isotopy classes of all curves in $\Gamma$ as well as the orientation of
the annular neighborhood of each curve. We then have that $f$ induces an
automorphism $\tilde{\alpha}$ of $G^{\prime}(S,\Gamma)$ by statement 2 of the
Alexander method. Furthermore, $\alpha=F(\tilde{\alpha})$ is an automorphism
of $G_{r}(S,\Gamma)$ that preserves the orientations of the $A_{i}$.

Since $\Gamma$ is weakly stable, $\tilde{\alpha}$ is the identity. The third
statement of the Alexander method implies that $[f]$ is the identity mapping
class. ∎

We use the above proposition to prove the following example.

######  Corollary 2.3.

Let $S$ and $\Gamma$ be the surface and Alexander system in Figure 3. Let
$\\{k_{i}\\}_{i\in I}$ be a collection of non-zero integers. Then, $\Gamma$ is
weakly stable and hence the centralizer in $\operatorname{MCG}(S)$ of the
subgroup $T=\langle\\{T_{\gamma_{i}}^{k_{i}}\\}_{i\in I}\rangle$ is trivial.

Figure 3: The Alexander system $\Gamma=\\{\gamma_{i}\\}_{I}$ in Corollary 2.3.
Figure 4: The graph $G_{a}(S,\Gamma)$. Any automorphisms of $G_{a}(S,\Gamma)$
must preserve loop edges. Due to the lack of symmetry in $G_{a}(S,\Gamma),$
every automorphism of $G(S,\Gamma)$ must fix all loop edges, possibly
reversing orientation. As such, all monochromatic loops must be fixed setwise.

###### Proof.

Let $\alpha\in\operatorname{Aut}G_{r}(S,\Gamma)$ be induced by an automorphism
$\tilde{\alpha}$ of $G^{\prime}(S,\Gamma)$ and suppose $\alpha$ maps the
annuli $A_{i}$ to themselves and preserves their orientations. Let
$\alpha_{*}\in\operatorname{Aut}G^{\prime}_{a}(S,\Gamma)$ be the induced
automorphism of the abstract graph. That is,

$\displaystyle\operatorname{Aut}(G^{\prime}(S,\Gamma))$
$\displaystyle\xhookrightarrow{F}\operatorname{Aut}(G_{r}(S,\Gamma))$
$\displaystyle\xhookrightarrow{H}G^{\prime}_{a}(S,\Gamma)$
$\displaystyle\tilde{\alpha}\quad$
$\displaystyle\mapsto\quad\quad\quad\alpha\quad$
$\displaystyle\mapsto\quad\alpha_{*}.\quad$

We note that $\tilde{\alpha}$ induces $\alpha_{*}$ as well by forgetting the
surface structure. We want to show that $\tilde{\alpha}$ is the identity
automorphism.

Let $G_{a}(S,\Gamma)$ be the abstract graph associated to $G(S,\Gamma)$. If we
orient the loop edges of $G_{a}(S,\Gamma)$ and keep track of their
orientations, then the elements of $\operatorname{Aut}(G_{a}(S,\Gamma))$ are
in a natural bijection with elements of
$\operatorname{Aut}(G^{\prime}_{a}(S,\Gamma))$ (automorphisms of
$G_{a}(S,\Gamma)$ need not repsect orientation). Let
$\alpha^{\prime}_{*}\in\operatorname{Aut}(G_{a}(S,\Gamma))$ be the
automorphism corresponding to $\alpha_{*}.$

We first show that $\alpha^{\prime}_{*}$ is a product of automorphisms that
reverse the directions of loop edges or swap pairs of edges in 2-cycles. Any
automorphism of $G_{a}(S,\Gamma)$ must preserve same-colored cycles setwise
such that no two cycles are interchanged. This property of
$\alpha^{\prime}_{*}$ is also inherited from $\alpha.$ Moreover, all 3-cycles
are fixed by any automorphism of $G_{a}(S,\Gamma).$ It follows that
$\alpha_{*}$ may only reverse orientations of loop edges (equivalent to
$a_{*}$ swapping edges in 2-cycles in $G^{\prime}_{a}(S,\Gamma)$) or swap
pairs of edges in 2-cycles.

Since $\alpha$ preserves the $A_{i}$ with orientations, the orientations of
all curves that intersect the curves corresponding to the red 3-cycles are
preserved. As such, the orientations of loop edges cannot be reversed.
Similarly, by working our way sequentially toward the ends, the orientations
of all other curves must be preserved as well. It follows that
$\alpha^{\prime}_{*}$—and similarly $\alpha_{*}$—are the trivial
automorphisms. It follows from injectivity that $\tilde{\alpha}$ is the
trivial automorphism of $G^{\prime}(S,\Gamma)$.

We now have that $\Gamma$ is weakly stable and Proposition 2.2 gives us that
the centralizer of $T$ is trivial. ∎

###  2.3 Topology of mapping class groups

Aramayona–Vlamis discuss the topology on the mapping class group of infinite-
type surfaces and show that the mapping class groups of infinite-type surfaces
are not discrete and not locally compact [3]. As we describe below, the
topology can be described in a simpler manner using the Alexander method. The
applications in this section were suggested by Jesús Hernández Hernández [9].

The topology on the mapping class group via the Alexander method. There is a
natural topology on $\operatorname{MCG}(S)$ arising from the compact-open
topology. The permutation topology is an equivalent and more combinatorial
description of the topology on mapping class groups of infinite-type surfaces
(see [3, Section 4.1] for more details). We give a new perspective on the
permutation topology in light of the Alexander method.

For any finite set $A$ of isotopy classes of curves in $S$, let $U(A)$ be the
set of mapping classes that fix all of the curves in $A.$ Define the
permutation topology on $\operatorname{MCG}(S)$ to be the topology whose basis
elements are all $\operatorname{MCG}(S)-$translates of all $U(A)$.

Let $S$ be an infinite-type surface and $\Gamma$ a stable Alexander system in
$S.$ Such a system exists by the works of Hernández–Morales–Valdez [11],
Hernández–Hidber [10], and Dickmann [6]. We define the Alexander topology
$\mathcal{T}_{\Gamma}$ on $\operatorname{MCG}(S)$ as follows. For any finite
subset $B$ of (the isotopy classes of) curves in $\Gamma,$ let $U(A)$ be the
set of mapping classes that fix all the curves in $B.$ Then, the basis for
$\mathcal{T}_{A}$ is the set of all $\operatorname{MCG}(S)-$translates of all
$U(B).$

$\mathcal{T}_{\Gamma}$ is indeed a topology. The basis elements cover
$\operatorname{MCG}(S)$ since $h\in h\cdot U(B)$ for every $B\subseteq\Gamma$
(abusing notation, as the elements of $B$ are isotopy classes of curves and
$\Gamma$ contains curves). We also have that if $h\in g_{1}\cdot U(B_{1})\cap
g_{2}\cdot U(B_{2}),$ then $h\in h\cdot(B_{1})\cap
h\cdot(B_{2})=h\cdot(B_{1}\cup B_{2}),$ another basis element.

We show that the permutation topology and the Alexander topology are
equivalent. This result is to be expected since, by the Alexander method, a
mapping class is determined by its action on an Alexander system.

######  Proposition 2.4.

Let $S$ be an infinite-type surface and $\Gamma$ a stable Alexander system on
$S$. Then, $\mathcal{T}_{\Gamma}$ is equal to the permutation topology
$\mathcal{T}$.

###### Proof.

We have that $\mathcal{T}$ is a priori finer than $\mathcal{T}_{\Gamma}$ since
$\mathcal{T}_{\Gamma}\subset\mathcal{T},$ so it remains to show the opposite
inclusion. Let $h\in\operatorname{MCG}(S)$ and let $A$ be a set of isotopy
classes curves such that $g\cdot U(A)$ is a basis element that contains $h$.
Then, $h(\alpha)=g(\alpha)$ for $\alpha\in A,$ so $g\cdot U(A)=h\cdot U(A).$
Abusing notation, let $\Gamma_{A}\subset\Gamma$ be a finite set of curves such
that the subsurface of $S$ filled by $\Gamma_{A}$ contains all the curves in
$A.$ We then have that $h\in g\cdot U(A)=h\cdot U(A)\subseteq h\cdot
U(\Gamma_{A}),$ completing the proof. ∎

The mapping class group is not discrete. We now use our characterization of
the topology on $\operatorname{MCG}(S)$ to verify that the mapping class group
is not discrete.

Let $\Gamma$ be a stable Alexander system. Take $\\{c_{n}\\}$ to be a set of
distinct curves in $\Gamma.$ It follows from the local finiteness of Alexander
systems and our characterization of the topology of the mapping class group in
Proposition 2.4 that $\\{T_{c_{n}}\\}_{n\in\mathbb{N}}$ limits to the
identity. This is so because every finite-type subsurface of $S$ is eventually
fixed by the $T_{c_{n}}.$

This is akin to the example provided by Aramayona–Vlamis in their proof of the
fact that mapping class groups of infinite-type surfaces are not discrete.
Further discussion of this result is provided in Aramayona–Vlamis [3].

The mapping class group is not locally compact. Aramayona–Vlamis further
discuss why the mapping class group of an infinite-type surface is not locally
compact; this discussion is rooted in the permutation topology. We show the
same result, but using the Alexander topology.

To begin, fix a curve $c$ and consider the mapping class $T_{c}.$ Then, any
neighborhood $N$ of $T_{c}$ must contain all the powers of $T_{c}$ (if $T_{c}$
fixes a curve, then so does $T^{k}_{c}$ for all $k$). As such,
$\\{T^{k}_{c}\\}$ is contained in $N$.

Suppose $h$ is the limit of $\\{T^{k}_{c}\\}$. Applying Proposition 2.4, we
see that $\\{T^{k}_{c}\\}$ must agree with $h$ on increasing and exhaustive
subsets of $\Gamma$ (and therefore $S$). However, $\\{T^{k}_{c}\\}$ does not
stabilize on a neighborhood of $c$: let $\gamma$ be a curve that intersects
$c.$ Then, $T^{k}(\gamma)\neq T^{j}(\gamma)$, implying that $\\{T^{k}_{c}\\}$
is not eventually in $h\cdot U(\\{\gamma\\}),$ a contradiction. Therefore,
$\operatorname{MCG}(S)$ is not locally compact.

The same sequence is used by Aramayona–Vlamis to show that the mapping class
group is not locally compact [3]. The novelty in our approach is that the
Alexander method provides a simpler way to check convergence.

##  3 Proof of the Alexander method

In this section, we prove the main theorem. Let $S$ be a surface and $\Gamma$
an Alexander system on $S$. We restate the Alexander method for convenience.

Recall that $G(S,\Gamma)$ is the surface graph on $S$ whose vertices
correspond to intersections between curves and arcs in $\Gamma$ as well as
endpoints of arcs and whose edges correspond to the connected components of
$\cup\gamma_{i}\setminus V(G)$. The barycentric subdivision of $G(S,\Gamma)$
is denoted $G^{\prime}(S,\Gamma).$

With that notation, the main theorem states the following. Suppose $\phi:S\to
S$ is a homeomorphism that permutes the isotopy classes of curves in $\Gamma$
according to permutation $\sigma.$ Then

  1. 1.

there exists a homeomorphism $\psi:S\to S$ isotopic to $\phi$ rel $\partial S$
such that $\psi(\gamma_{i})=\gamma_{\sigma(i)}$ for all $\gamma_{i}\in\Gamma,$

  2. 2.

$\phi$ induces a unique automorphism $\phi_{*}$ of $G^{\prime}(S,\Gamma),$ and

  3. 3.

if $\Gamma$ is filling, then $\phi_{*}$ is the identity if and only if $\phi$
is isotopic to the identity.

###### Proof of main theorem..

We prove the three statements of the Alexander method in turn.

Statement 1: finding a homeomorphism $\psi$ isotopic to $\phi$. We first
notice that $\phi(\Gamma)$ is an Alexander system. Since we will be isotoping
$\phi(\Gamma)$ to $\Gamma,$ it is sufficient to consider isotoping an
Alexander system to another Alexander system such that each curve (or arc) in
the first is isotopic to a curve (or arc) in the second.

With that in mind, let $\Gamma$ and $\Gamma^{\prime}$ be Alexander systems in
$S$ such that $\gamma_{i}\in\Gamma$ is isotopic to
$\gamma_{i}^{\prime}\in\Gamma^{\prime}.$ We will construct an isotopy of $S$
such that each $\gamma_{i}$ is taken to $\gamma_{i}^{\prime}.$

Both Alexander systems are indexed by $\mathbb{N}$ or $\\{1,\ldots,n\\}$. We
present the proof in the former case and the latter case follows the same
outline.

We claim that there exists a subsurface exhaustion $\\{S_{i}\\}_{\mathbb{N}}$
of $S$ as follows: $S_{i}$ contains $\gamma_{i}\cup\gamma^{\prime}_{i}.$ That
is, $\gamma_{i}$ and $\gamma^{\prime}_{i}$ are isotopic in $S_{i}$ (rel
$\partial S_{i}$). This is so because each curve and arc exists on a finite-
type subsurface. The work of Richards [13] gives us that there exists an
exhaustion of $S$ by finite-type subsurfaces, so each
$\gamma_{i}\cup\gamma_{i}^{\prime}$ lives on a subsurface in the exhaustion.
We take a subsequence of the exhaustion to obtain $\\{S_{i}\\}$ as required
above.

We will now define a sequence of isotopies that we will perform sequentially
to produce an isotopy $H:S\times I\to S$ such that $H(S,0)=\operatorname{id}$
and $\psi(S):=H(S,1),$ with $\psi(\gamma_{i})=\gamma^{\prime}_{i}.$

We claim that there exists a sequence of isotopies $H_{i}:S\times I\to S$
supported on $S_{i}$ such that

  1. 1.

the $i$th curve or arc is corrected:
$H_{i}(\gamma_{i},1)=\gamma^{\prime}_{i}$,

  2. 2.

previously corrected curves or arcs are not moved: for all $j<i$ and
$p\in\gamma_{j},$ $H_{i}(p,t)=p$ for all $t\in I,$

  3. 3.

the intersections with other curves (and arcs) are preserved:
$H_{i}(\gamma_{i}\cap\gamma_{j},1)=\gamma_{i}^{\prime}\cap\gamma^{\prime}_{j}$
for all $j$ in such a way that each connected component of
$H_{i}(\gamma_{j}\setminus\gamma_{i},1)$ is isotopic to a component of
$\gamma^{\prime}_{j}\setminus\gamma^{\prime}_{i}$ rel $\gamma^{\prime}_{i},$

  4. 4.

non-intersected subsurfaces are fixed: if
$(\gamma_{i}\cup\gamma^{\prime}_{i})\cap S_{j}=\emptyset,$ then $H_{i}$ is
supported on $S_{i}\setminus S_{j}$ (for $j<i$), and

  5. 5.

isotopies are gluable: $H_{i}(x,1)=H_{i+1}(x,0)$ for all $x\in S.$

To prove the claim, we address each of the five points. Point (1) is possible
by directly applying the Alexander method for finite-type surfaces on $S_{i}$
[7, Proposition 2.8]. Point (2) is achievable since we can perform the isotopy
on $S_{i}$ cut along $\gamma_{j}$ for $j<i$ (retaining the boundary arising
from $\gamma_{j}$). Such an isotopy is the identity on the boundary—and, as
such, it extends to an isotopy of $S$ that fixes $\gamma_{j}.$ Point (3) is
possible because we can perform an isotopy that rotates, stretches, and/or
shrinks along $\gamma_{i}$ until the intersections line up.

We now prove that point (4) is achievable. Let $\Delta_{j}$ be the set of
nonperipheral curves in $\partial S_{j}.$ Then,
$\Delta_{j}\cup\\{\gamma_{i}\\}$ and $\Delta_{j}\cup\\{\gamma^{\prime}_{i}\\}$
are Alexander systems in $S_{i}$ that satisfy the hypotheses for the finite-
type Alexander method. Following the proof of Lemma 2.9 in Farb–Margalit [7],
there is an isotopy of $S_{i}$ that takes $\gamma_{i}$ to
$\gamma^{\prime}_{i}$ and fixes $\Delta_{j}.$ As such, we may take $S_{j}$ to
be fixed as well.

Property (5) is achieved by defining the isotopies sequentially. We have
therefore completed the proof the claim.

We define a sequence of functions to reparametrize the intervals.

$\displaystyle\zeta_{i}:\left[1-\frac{1}{2^{i-1}},1-\frac{1}{2^{i}}\right]$
$\displaystyle\to[0,1]$ $\displaystyle x$ $\displaystyle\mapsto
2^{i}\left[x-\left(1-\frac{1}{2^{i-1}}\right)\right]$

Finally, we define our overall isotopy $H:S\times I\to S$.

$H(p,t)=\begin{cases}H_{i}(p,\zeta_{i}(t))&t\in\left[1-\frac{1}{2^{i-1}},1-\frac{1}{2^{i}}\right]\\\
\psi(p)&t=1\end{cases}$

The map $H$ is continuous for all $(x,t)$ if $t<1$ due to the continuity of
the $H_{i}$, point (5) above, and the pasting lemma from point-set topology.
It remains to show that $H$ is continuous at $t=1.$ In other words, we want to
show that, for all $p\in S,$ there exists an open neighborhood $U_{p}$ of $p$
and $t_{0}\in[0,1)$ such that $H(x,t)=x$ for all $x\in U_{p}$ for all
$t>t_{0}$ for some $t_{0}.$ Suppose $p\in\operatorname{int}(S_{i})$ for some
$i$. Let $J_{0}$ be the largest $j$ such that
$(\gamma_{J_{0}}\cup\gamma^{\prime}_{J_{0}})\cap S_{i}\neq\emptyset.$ Such a
largest value exists due to the local finiteness of Alexander systems. Then,
$H_{J}(x,t)|_{S_{i}\times I}=x$ for all $J>J_{0}$ and for all $t$ by point (4)
above. Since $p\in\operatorname{int}(S_{i}),$ we have that there is an open
neighborhood $U_{p}$ of $p$ such that $H(x,t)=\psi(x)$ for all $x\in U_{p}$
for all $t>t_{0}$ for some $t_{0}.$

The homeomorphism $\psi(x):=H(x,1)$ has the property
$\psi(\gamma_{i})=\gamma_{i}^{\prime}$ by points (1), (2), and (3) above, so
$\gamma_{i}$ is isotoped to $\gamma^{\prime}_{i}$ and is not moved afterward.

Statement 2: $\phi$ induces a unique automorphism of $G(S,\Gamma).$ We now
return to our original notational conventions. That is, the $\Gamma$ in the
proof of statement 1 is now $\phi(\Gamma)$ and the $\Gamma^{\prime}$ in the
proof of statement 1 is now $\Gamma.$ It follows from Statement 1 that $\phi$
induces an automorphism of $G(S,\Gamma)$. It remains to show that this
automorphism is unique.

It is sufficient to show that any automorphism of $G(S,\Gamma)$ induced by an
isotopy of $S$ is the identity. The universal cover of $S$ is the hyperbolic
plane $\mathbb{H}^{2}$. The preimage of $G(S,\Gamma)$ in $\mathbb{H}^{2}$ is a
graph whose vertices are the preimages of vertices of $G(S,\Gamma)$ and edges
are preimages of edges in $G(S,\Gamma).$ An isotopy of $S$ lifts to an isotopy
of $\mathbb{H}^{2}$.

As each arc and curve is isotoped on a finite-type surface, the induced graph
automorphism of the preimage of $G(S,\Gamma)$ must preserve the endpoints of
the lifts of the arcs and curves at $\partial\mathbb{H}^{2}$, and therefore
preserve every elevation of every curve and arc in $\Gamma.$ As such, all
intersections between elevations of curves in $\Gamma$ must be preserved by
the isotopy, and it follows that the induced automorphism of the preimage
graph in $\mathbb{H}^{2}$ is the identity. We conclude that $\phi$ induces a
unique automorphism of $G(S,\Gamma).$

Statement 3: inducing the identity automorphism on $G^{\prime}(S,\Gamma)$
implies being isotopic to the identity. Let $\Gamma$ be a filling Alexander
system in $S$. Suppose $\phi_{*}$ is the identity automorphism of
$G^{\prime}(S,\Gamma)$; that is, such that $\phi_{*}$ preserves the
orientation of loop edges in $G(S,\Gamma)$. By our work above, $\phi_{*}$
takes vertices to vertices and edges to edges; thus, if an edge has distinct
vertices, its orientation is preserved by $\phi_{*}.$ Then, up to isotopy,
$\phi$ fixes each (potentially once-punctured or Möbius band) disk in
$S\setminus\Gamma$. Furthermore, $\phi$ fixes, with orientation, each edge
(and thus the boundary) of each disk. It follows that, up to isotopy, $\phi$
is equal to the identity on every disk, once-punctured disk, and Möbius band.
We then apply the pasting lemma from point-set topology to obtain that $\phi$
is isotopic to the identity. ∎

##  4 Non-example of an Alexander system

In this section, we illustrate why the local finiteness condition on Alexander
systems is necessary.

The main idea of this restriction is that we do not want any limit points of
distinct elements of our Alexander system. That is, any sequence $\\{x_{n}\\}$
of points on distinct $\gamma_{i}$ does not converge. Let us consider an
example with seemingly minimal convergence.

Let $S$ be the plane with punctures at the origin and at angles $\frac{1}{n}$
and $0$ on the unit circle. Let $\Gamma$ consist of arcs connecting 0 to the
punctures on the unit circle (except to the puncture at angle 0). It suffices
to show that $\Gamma$ is not well-defined up to homeomorphism.

Let $\Gamma_{1}$ be the collection of arcs that appear as straight rays from
the origin to the punctures on the unit circle (except at angle 0). Let
$\Gamma_{2}$ be the same collection of arcs, except the arc at angle $1$ is
isotoped counterclockwise to hit the positive $x$-axis before $x=1$. As such,
there is a sequence of points on the arcs in $\Gamma_{2}$ that converges to a
point in an arc; this is a property preserved by homeomorphism and it is not
true of $\Gamma_{1}.$ An example of such arrangements is shown in Figure 5.

Figure 5: An example of an Alexander system $\Gamma$ on a surface $S$ where
local finiteness is broken. We note that any disk about the origin (isolated
puncture connected to all other punctures) intersects infinitely many arcs in
$\Gamma.$ The key point is that although all arcs in the left image are
isotopic to their corresponding arcs in the right image, the two
configurations are not homeomorphic.

We note that we require that arcs be of finite type in order for the above
proof technique to work. However, there may be a version of the Alexander
method which allows for all arcs.

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