# Level spacing statistics of the Haldane chain and its Kennedy-Tasaki dual

Biquan Yang  Department of Physics, Chongqing University, Chongqing, 401331,
China  Zijian Xiong  Corresponding author<EMAIL_ADDRESS>Department
of Physics, Chongqing University, Chongqing, 401331, China  Department of
Applied Physics, University of Tokyo, Tokyo 113-8656, Japan

###### Abstract

We study the level spacing statistics of three spin-1 models in one dimension,
they are the Haldane chain, dual model which is obtained by Haldane chain
under the Kennedy-Tasaki transformation and the combination of the first two
models which contains a solvable self-dual point. Although the first two
models are unitary equivalent, their symmetries are different and this leads
to different level spacing statistics in these two models. Concretely, the
level spacing in the Haldane chain obeys random matrix theory statistics,
while the dual model shows non-random matrix theory statistics. This example
shows that the level spacing statistics in two unitary equivalent systems can
be different. We find that the non-random matrix theory statistics in the dual
model can be ascribed to a non-local hidden symmetry. Under perturbations that
remove the hidden symmetry, the dual model shows random matrix theory
statistics. For the last model, we observe a crossover from chaotic to
integrable around the self-dual point.

##  I Introduction

Level spacing statistics is a standard indicator for distinguishing quantum
chaotic and integrable systems Brody  _et al._ (1981); Edelman  _et al._
(2018); Guhr  _et al._ (1998); Gubin and F. Santos (2012). For quantum
integrable systems, the level spacing distribution typically obeys the Poisson
statistics. While the quantum chaotic systems show random matrix theory
statistics. The characteristic feature of random matrix theory statistics is
the level repulsion Wimberger (2014), i.e. the probability of zero level
spacing vanishes. In contrast, the Poisson statistics have the largest
probability around zero level spacing. These two distinct statistics have been
confirmed to be universal in many different quantum many body systems
Montambaux  _et al._ (1993); Poilblanc  _et al._ (1993); Hsu and Angle‘s
d’Auriac (1993), ranging from nuclei systems Gómez  _et al._ (2011), Hubbard
models Kollath  _et al._ (2010); De Marco  _et al._ (2022); Haderlein  _et
al._ (2023) to spin models Gubin and F. Santos (2012); Santos (2004); Santos
_et al._ (2020); Sanada  _et al._ (2023).

To obtain the level spacing statistics of a given system, an essential step is
to identify the symmetry of the system and separate the eigenvalues into the
symmetry sectors (usually, this symmetry refers to the apparent symmetry, like
translation, inversion and spin rotation). The reason for considering symmetry
is ascribed to the von Neumann-Wigner theorem, which states that the
eigenvalues with the same symmetry quantum number rarely cross each other. In
other words, there is a level repulsion among these eigenvalues in the same
symmetry sector. However, level repulsion vanishes if different symmetry
sectors are mixed even if the system is indeed in a chaotic nature Kudo and
Deguchi (2005); Gubin and F. Santos (2012). Since the level spacing statistics
are sensitive to symmetry, some proposals have been made to probe the
(discrete) symmetry Rosenzweig and Porter (1960); Giraud  _et al._ (2022).

As the level spacing statistics require only eigenvalues, consider two unitary
equivalent Hamiltonians, if they have the same symmetry, their level spacing
data will certainly be the same. When they have different symmetry, naively,
they may show the same level spacing statistics behavior, i.e. if one is
Poisson/ random matrix theory statistics, another will also be Poisson/ random
matrix theory statistics in their own symmetry sectors. But depending on the
unitary transformation $U$, symmetry $G$ in the Hamiltonian $H$ may be
transformed to some highly nonlocal symmetry $\tilde{G}$ in $\tilde{H}$. It
may be impractical or impossible to divide the Hilbert space by such nonlocal
symmetry $\tilde{G}$. Then, we may obtain different level spacing statistics
behaviors in these two unitary equivalent Hamiltonians. In general, it is not
difficult to construct such unitary transformation, but we may restrict
ourselves to the transformation such that a Hamiltonian with short range
interaction is transformed into another Hamiltonian with short range
interaction. In condensed matter physics, a well-studied example- Kennedy-
Tasaki transformation $U_{KT}$ in integer spin systems Kennedy and Tasaki
(1992a, b); Oshikawa (1992); Tasaki (2020) fulfills such requirement: if an
integer spin Hamiltonian $H$ has $Z_{2}\times Z_{2}$ symmetry (which is
generated by $\pi$ rotation about $x,y,z$ axes) as its symmetry subgroup,
then, only this $Z_{2}\times Z_{2}$ symmetry remains as the on-site internal
symmetry in the transformed Hamiltonian $U_{KT}HU_{KT}^{\dagger}$. Previous
studies of Kennedy-Tasaki transformation mainly focus on the low energy
properties, such as symmetry protected topological order- spontaneous symmetry
breaking duality Li  _et al._ (2023a, b), and ground state entanglement
Okunishi (2011).

In this paper, we study the level spacing statistics in three spin-1 models in
one dimension. The first one is the standard Haldane chain $H$ with
anisotropy, which is non-integrable. The second one is the dual model
$\tilde{H}=U_{KT}HU_{KT}^{\dagger}$ obtained by Kennedy-Tasaki transformation
$U_{KT}$, which is still a simple model contains only two body interaction. We
show that the level spacing statistics of the dual model take a non-random
matrices distribution although it is unitary equivalent to the Haldane model,
but becomes random matrix theory statistics if we add perturbations. This
finding shows that the level spacing statistics in two unitary equivalent
systems can be different. The third model is a combination of the Haldane
chain and its dual model, which takes the form
$H(\lambda)=(1-\lambda)H+\lambda\tilde{H}$. This model is invariant (self-
dual) under the Kennedy-Tasaki transformation at $\lambda=0.5$. Recently, it
has been found that a self-dual point is exactly solvable where $H$ takes the
spin-1 bilinear-biquadratic model Yang  _et al._ (2023). Here, we show that
there indeed exists a crossover from random matrix theory statistics to
Poisson statistics around the self-dual point of $H(\lambda)$. This crossover
implies the integrability of the self-dual point.

The paper is organized as follows. In Sec.II, we describe the models studied
in this paper and their symmetries. In Sec.III, the level spacing statistics
results of the models are presented. Conclusions are provided in Sec.IV.

##  II Models

In this section, we present the definition of the models and their symmetries.

###  II.1 Haldane model

The spin-1 Haldane chain with anisotropy under the open boundary condition is
given by

$H=\sum_{i=1}^{L-1}(S^{x}_{i}S^{x}_{i+1}+S^{y}_{i}S^{y}_{i+1}+\Delta
S^{z}_{i}S^{z}_{i+1})+D\sum_{i=1}^{L}(S_{i}^{z})^{2}.$ (1)

This model has been well studied, and it exhibits a rich phase diagram
including the Haldane phase and several different symmetry breaking phases
Chen  _et al._ (2003).

The internal symmetry of this model is $U(1)\rtimes Z_{2}$, where the $U(1)$
is spin rotation around the z axis:
$U_{\theta}^{z}=e^{-i\theta\sum_{i=1}^{L}S_{i}^{z}}$. And the $Z_{2}$ is the
spin flip, which is equivalent to the $\pi$ rotation around the x axis
(denoted by $U^{x}_{\pi}$) or y axis. Using the matrix representation of the
spin-1 operators $S^{\alpha},\alpha=x,y,z$ (which is presented in the
appendix), one can find the $\pi$ rotation has a simple form Tasaki (2020):

$U^{\alpha}_{\pi}=1-2(S^{\alpha})^{2},$ (2)

and some useful algebras for spin-1 are provided in the Appendix. It is clear
that the internal symmetry has a subgroup $Z_{2}\times Z_{2}$. This model also
has a spatial symmetry: bond-centered inversion symmetry
$\mathcal{I}:S_{i}^{\alpha}\to S_{L-i+1}^{\alpha}$.

###  II.2 Kennedy-Tasaki transformation and the dual model

Kennedy-Tasaki transformation is defined on the integer spin chain with open
boundary condition, which is given by

$U_{KT}=\prod_{1\leq u<v\leq L}e^{i\pi S^{z}_{u}S^{x}_{v}}.$ (3)

Using the matrix representation of the spin-1 operators, one can find that
every exponential factor in the transformation commutes with each other, and
$[exp(i\pi S^{z}_{u}S^{x}_{v})]^{2}=1$. This implies that $(U_{KT})^{2}=1$ and
$U_{KT}=(U_{KT})^{\dagger}$. Kennedy-Tasaki transformation has also been
generalized to the closed chain recently, but it is not unitary in that case
Li  _et al._ (2023a). Therefore, we only consider the open boundary condition
in this paper.

Under the Kennedy-Tasaki transformation, the spin operators are transformed as

$\displaystyle U_{KT}S_{j}^{x}U_{KT}=$ $\displaystyle
S_{j}^{x}exp(i\pi\sum_{k=j+1}^{L}S^{x}_{k}),$ (4) $\displaystyle
U_{KT}S_{j}^{z}U_{KT}=$ $\displaystyle
exp(i\pi\sum_{k=1}^{j-1}S^{x}_{k})S_{j}^{z},$ $\displaystyle
U_{KT}S_{j}^{y}U_{KT}=$ $\displaystyle
exp(i\pi\sum_{k=1}^{j-1}S^{z}_{k})\,S_{j}^{y}\,exp(i\pi\sum_{m=j+1}^{L}S^{x}_{m}),$

which become complicated and nonlocal. However, all terms in the Haldane model
are transformed into the simple forms Oshikawa (1992)

$\displaystyle U_{KT}S^{z}_{i}S^{z}_{i+1}U_{KT}=$
$\displaystyle-S^{z}_{i}S^{z}_{i+1},$ (5) $\displaystyle
U_{KT}(S^{\alpha}_{i})^{2}U_{KT}=$ $\displaystyle(S^{\alpha}_{i})^{2},$
$\displaystyle U_{KT}S^{x}_{i}S^{x}_{i+1}U_{KT}=$
$\displaystyle-S^{x}_{i}S^{x}_{i+1},$ $\displaystyle
U_{KT}S^{y}_{i}S^{y}_{i+1}U_{KT}=$ $\displaystyle S^{y}_{i}exp(i\pi
S^{z}_{i})exp(i\pi S^{x}_{i+1})S^{y}_{i+1}.$

So the Kennedy-Tasaki dual of the Haldane model eq.(1) is

$\displaystyle\tilde{H}=$ $\displaystyle U_{KT}HU_{KT}$ (6) $\displaystyle=$
$\displaystyle\sum_{i=1}^{L-1}[-S^{x}_{i}S^{x}_{i+1}+S^{y}_{i}exp(i\pi
S^{z}_{i})exp(i\pi S^{x}_{i+1})S^{y}_{i+1}$ $\displaystyle-\Delta
S^{z}_{i}S^{z}_{i+1}]+D\sum_{i=1}^{L}(S_{i}^{z})^{2},$

y component term can be further simplified by using eq.(2). It is obvious that
this model contains only two body interactions.

The on-site internal symmetry of this dual model is $Z_{2}\times Z_{2}$, which
is generated by the $\pi$ rotation around the $x,y,z$ axes. Further, this
symmetry is indeed the same symmetry as the subgroup of $U(1)\rtimes Z_{2}$ in
the Haldane model. This can be seen by
$U_{KT}(S^{\alpha}_{i})^{2}U_{KT}=(S^{\alpha}_{i})^{2}$, which means the $\pi$
rotation is invariant under the transformation:
$U_{KT}U^{\alpha}_{\pi}U_{KT}=U^{\alpha}_{\pi}$.

In fact, the dual model also has a hidden U(1) symmetry, the corresponding
generator is $U_{KT}\sum_{i}S^{z}_{i}U_{KT}$. From eq.(4), we know that this
symmetry is nonlocal and string-like. Thus, it will be impractical to separate
the Hilbert space of the dual model into symmetry sectors according to this
hidden symmetry. Moreover, this hidden symmetry does not commute with
$U^{x}_{\pi}$ and $U^{y}_{\pi}$. So the symmetry sector can not be labeled by
the apparent symmetries and hidden symmetry simultaneously. Nevertheless, we
will still see that this hidden symmetry does play some roles in the level
spacing statistics. In short, the dual model and the Haldane model are unitary
equivalent, but they have different apparent symmetries. We will study their
level spacing statistics in the next section.

Now we consider the spatial symmetry. Under the bond-centered inversion
symmetry,

$\displaystyle\mathcal{I}:S^{y}_{i}exp(i\pi S^{z}_{i})exp(i\pi
S^{x}_{i+1})S^{y}_{i+1}$ (7) $\displaystyle\to$ $\displaystyle
S^{y}_{L-i+1}exp(i\pi S^{z}_{L-i+1})exp(i\pi S^{x}_{L-i})S^{y}_{L-i}$
$\displaystyle=$ $\displaystyle exp(i\pi
S^{x}_{L-i})S^{y}_{L-i}S^{y}_{L-i+1}exp(i\pi S^{z}_{L-i+1}).$

With the matrix representation of the spin-1 operators (which is presented in
Appendix), we can find

$S^{y}_{i}exp(i\pi S^{z}_{i})=exp(i\pi S^{x}_{i})S^{y}_{i},$ (8)

Thus, $\sum_{i}S^{y}_{i}exp(i\pi S^{z}_{i})exp(i\pi S^{x}_{i+1})S^{y}_{i+1}$
is invariant under the bond-centered inversion, and the dual model is
invariant under this symmetry.

The usual eigenstates of $S^{z}$ are denoted by
$|1\rangle,|0\rangle,|-1\rangle$,

$\displaystyle U^{z}_{\pi}|1\rangle$
$\displaystyle=-|1\rangle,U^{z}_{\pi}|0\rangle=|0\rangle,U^{z}_{\pi}|-1\rangle=-|-1\rangle,$
(9) $\displaystyle U^{x}_{\pi}|1\rangle$
$\displaystyle=-|-1\rangle,U^{x}_{\pi}|0\rangle=-|0\rangle,U^{x}_{\pi}|-1\rangle=-|1\rangle.$

To ”diagonalize” the dual model, we can use the ”$Z_{2}\times Z_{2}$” symmetry
basis Kennedy (1994); Yang  _et al._ (2023), which is the simultaneous
eigenstates of $U_{\pi}^{x},U_{\pi}^{y},U_{\pi}^{z}$,

$\displaystyle|\uparrow\rangle=$
$\displaystyle\frac{1}{\sqrt{2}}(|1\rangle-|-1\rangle),$ (10)
$\displaystyle|\downarrow\rangle=$ $\displaystyle|0\rangle,$
$\displaystyle|h\rangle=$
$\displaystyle\frac{1}{\sqrt{2}}(|1\rangle+|-1\rangle).$

The eigenvalues of $U_{\pi}^{x},U_{\pi}^{y},U_{\pi}^{z}$ are listed as

$\displaystyle
U^{x}_{\pi}|\uparrow\rangle=|\uparrow\rangle,\,U^{x}_{\pi}|\downarrow\rangle=-|\downarrow\rangle,\,U^{x}_{\pi}|h\rangle=-|h\rangle,$
(11) $\displaystyle
U^{y}_{\pi}|\uparrow\rangle=-|\uparrow\rangle,\,U^{y}_{\pi}|\downarrow\rangle=-|\downarrow\rangle,\,U^{y}_{\pi}|h\rangle=|h\rangle,$
$\displaystyle
U^{z}_{\pi}|\uparrow\rangle=-|\uparrow\rangle,\,U^{z}_{\pi}|\downarrow\rangle=|\downarrow\rangle,\,U^{z}_{\pi}|h\rangle=-|h\rangle.$

For integer spin, $U^{\alpha}_{\pi}U^{\beta}_{\pi}=U^{\gamma}_{\pi}$ when
$\alpha,\beta,\gamma$ are different. In the following, we will denote the
eigenvalues of $U_{\pi}^{x},U_{\pi}^{y},U_{\pi}^{z}$ as X, Y, Z, respectively.

###  II.3 Self dual and integrable

The model obtained by combing the Haldane model and dual model is invariant
(self-dual) under the Kennedy-Tasaki transformation

$\displaystyle H_{SD}=$ $\displaystyle H+\tilde{H}$ (12) $\displaystyle=$
$\displaystyle\sum_{i=1}^{L-1}(S^{y}_{i}S^{y}_{i+1}+S^{y}_{i}exp(i\pi
S^{z}_{i})exp(i\pi S^{x}_{i+1})S^{y}_{i+1})$
$\displaystyle+2D\sum_{i=1}^{L}(S_{i}^{z})^{2}.$

Obviously, this model has the $Z_{2}\times Z_{2}$ internal symmetry and bond-
centered inversion symmetry. Using the matrix representation of spin-1, one
can also find

$[(S^{\alpha}_{i})^{2},(S^{\beta}_{i})^{2}]=0,$ (13)

hence,

$[H_{SD},(S^{y}_{i})^{2}]=0,$ (14)

$(S^{y}_{i})^{2}$ is conserved on every site.

Following ref.Yang  _et al._ (2023), we define the Pauli operators

$\displaystyle\sigma^{x}=$
$\displaystyle|\uparrow\rangle\langle\downarrow|+|\downarrow\rangle\langle\uparrow|,$
(15) $\displaystyle\sigma^{y}=$
$\displaystyle-i|\uparrow\rangle\langle\downarrow|+i|\downarrow\rangle\langle\uparrow|,$
$\displaystyle\sigma^{z}=$
$\displaystyle|\uparrow\rangle\langle\uparrow|-|\downarrow\rangle\langle\downarrow|,$

and the particle and hole number operators

$\displaystyle n$
$\displaystyle=|\uparrow\rangle\langle\uparrow|+|\downarrow\rangle\langle\downarrow|,$
(16) $\displaystyle h$ $\displaystyle=|h\rangle\langle h|,$

some useful relations can be derived: $(\sigma^{\alpha})^{2}=n$, $n+h=1$,
$\sigma^{\alpha}h=h\sigma^{\alpha}=0$, and
$\sigma^{\alpha}n=n\sigma^{\alpha}=\sigma^{\alpha}$.

Then, the spin-1 operators can be written as

$\displaystyle S^{x}=$
$\displaystyle|h\rangle\langle\downarrow|+|\downarrow\rangle\langle h|,$ (17)
$\displaystyle S^{y}=$ $\displaystyle
i|\downarrow\rangle\langle\uparrow|-i|\uparrow\rangle\langle\downarrow|=\sigma^{y},$
$\displaystyle S^{z}=$
$\displaystyle|h\rangle\langle\uparrow|+|\uparrow\rangle\langle h|,$
$\displaystyle(S^{z})^{2}=$ $\displaystyle|h\rangle\langle
h|+|\uparrow\rangle\langle\uparrow|=h+\frac{1}{2}(n+\sigma^{z}).$

Using the operators listed above, the self-dual model can be reduced to

$H_{SD}=\sum_{i=1}^{L-1}(\sigma^{y}_{i}\sigma^{y}_{i+1}-\sigma^{x}_{i}\sigma^{x}_{i+1})+D\sum_{i=1}^{L}(2h_{i}+n_{i}+\sigma^{z}_{i}),$
(18)

and the hole number is conserved on every site. Therefore, the self-dual model
is equivalent to a spin-1/2 XY chain with immobile holes, which can be solved
in every sector with a fixed hole number on every site. Although the spin-1/2
XY chain is known to be integrable Franchini  _et al._ (2007), it is found
that there are too many degenerate levels and the level spacing statistics
deviate from the Poisson statistics Zangara  _et al._ (2013). Hence we study
the level spacing statistics around the self-dual point, we consider the model

$H(\lambda)=(1-\lambda)H+\lambda\tilde{H},$ (19)

consequently, the self-dual point is located at $\lambda=0.5$. We expect that
there is a crossover from chaotic to integrable around the self-dual point.
The internal symmetry of this model is also $Z_{2}\times Z_{2}$ when
$\lambda\neq 0$ and $\lambda\neq 0.5$. In addition,
$U_{KT}H(\lambda)U_{KT}=H(1-\lambda)$, which tells that $H(\lambda)$ in
$0<\lambda<0.5$ is unitary equivalent to $0.5<\lambda<1$, thus, it is
sufficient to consider the former parameter region. Bond-centered inversion
symmetry also remains in this model.

##  III Level spacing statistics

Denote a set of eigenvalues of the Hamiltonian as $\\{E_{n}\\}_{g}$ , where
$g$ labels a symmetry sector with fixed quantum numbers. And the eigenvalues
are sorted in ascending order, $E_{1}\leq E_{2}\leq\cdots$. Level spacing is
defined as $\delta_{n}=E_{n+1}-E_{n}$. In this paper, the eigenvalues are
obtained by numerical diagonalization Weinberg and Bukov (2017, 2019).

There are several quantities for studying the level spacing statistics. The
first one is based on the ”normalized” level spacing $\tilde{\delta}$ rather
than the raw level spacing Gubin and F. Santos (2012); Edelman  _et al._
(2018). The ”normalization” procedure is called unfolding, which sets the mean
level density to 1. There are different unfolding approaches Gubin and F.
Santos (2012); Kudo and Deguchi (2003). For integrable system, the
distribution of ”normalized” level spacing follows Poisson statistics
$P_{P}(\tilde{\delta})=e^{-\tilde{\delta}}$. The chaotic system with time
reversal symmetry or the real symmetric Hamiltonian (the models in this paper
all belong to this class) can be described by random matrices from the
Gaussian orthogonal ensemble (GOE). And the distribution of ”normalized” level
spacing has the Wigner-Dyson form
$P_{GOE}(\tilde{\delta})=\frac{\pi\tilde{\delta}}{2}exp(-\pi\tilde{\delta}^{2}/4)$,
which reflects the level repulsion by $P_{GOE}(\tilde{\delta}\to 0)\to 0$.

An alternative quantity based on the gap ratio for consecutive levels is
constructed in ref.Oganesyan and Huse (2007),

$r_{n}=\frac{\rm{min}(\delta_{n},\delta_{n+1})}{\rm{max}(\delta_{n},\delta_{n+1})}.$
(20)

In this quantity, Poisson statistics have distribution $P_{P}(r)=2/(1+r)^{2}$,
chaotic system with time reversal symmetry (or the Hamiltonian is real
symmetric) has the distribution Atas  _et al._ (2013)
$P_{GOE}(r)=\frac{27}{4}\frac{r+r^{2}}{(1+r+r^{2})^{5/2}}$. The mean gap ratio
$\langle r\rangle$ is also usually used to characterize the level spacing
statistics, $\langle r\rangle_{P}=0.386$ for Poisson statistics, $\langle
r\rangle_{GOE}=0.536$ for GOE. In the following, we use both the gap ratio
distribution and mean gap ratio as chaotic indicators.

We denote the eigenvalues of $\sum_{i=1}^{L}S^{z}_{i}$, $U^{x}_{\pi}$,
$U^{z}_{\pi}$ and $\mathcal{I}$ as $M^{z}$, X, Z, $I$, respectively.

###  III.1 Haldane model

Though the random matrix theory statistics have been found in some related
spin-1 models, such as Affleck-Kennedy-Lieb-Tasaki model and its variants
Moudgalya  _et al._ (2018); Sanada  _et al._ (2023). To the best of our
knowledge, the level spacing statistics in the Haldane model have not been
investigated, it is believed to be chaotic.

In Fig.1, we show the gap ratio distribution of the Haldane chain with three
different D under the open boundary condition. The system size is $L=12$ and
we fix $\Delta=0.5$. The chosen symmetry sector has quantum numbers
$\\{M^{z}=0,X=-1,I=1\\}$ and size 18300. According to the previous study of
phase diagram Chen  _et al._ (2003), these three D with fixed $\Delta=0.5$
correspond to three different phases, respectively. But there are no
qualitative differences among the gap ratio distribution for these three D,
they all follow random matrix theory statistics nicely. We have also checked
other symmetry sectors, and they all show GOE statistics.

Figure 1: Distribution of the ratios of consecutive level spacings of Haldane
model for $L=12$ with $\Delta=0.5$, (a) $D=-2.5$, (b) $D=0$, (c) $D=3.5$. The
symmetry sector used here has quantum numbers: $\\{M^{z}=0,X=-1,I=1\\}$.

The dependence of level spacing statistics on anisotropies D and $\Delta$ are
studied by mean gap ratio, the results are shown in Fig.2. We find a good
agreement between the mean gap ratio and the prediction from the random matrix
theory statistics for $-2<D<2$ and the mean gap ratio shows a plateau form.
But the mean gap ratio deviates from GOE value for the larger parameter (here,
$|D|$), which is also found in other models Kollath  _et al._ (2010); De Marco
_et al._ (2022); Haderlein  _et al._ (2023). This deviation shows an obvious
system size dependence. For larger system sizes, the deviation decreases and
the mean gap ratio moves toward the GOE value. While the deviation from GOE
value is significant for large $|D|$ in mean gap ratio, the ratio distribution
is still close to GOE distribution, see Fig.2 (b) and Fig.1 (c). We also find
that the width of the plateau increases for larger $\Delta$ (see Fig.2), but
the width increasing is not monotonic.

Figure 2: Mean gap ratio $\langle r\rangle$ of consecutive level spacings of
Haldane model with different $\Delta$: (a)$\Delta=0.2$, (b) $\Delta=0.5$, (c)
$\Delta=0.8$, (d) $\Delta=1.4$. The symmetry sector used here has the quantum
numbers: $\\{M^{z}=0,X=-1,I=1\\}$.

As discussed in the previous section, the internal symmetry group of the
Haldane model is $U(1)\rtimes Z_{2}$, and it has a $Z_{2}\times Z_{2}$
subgroup. In contrast to spin-1/2, $\pi$ rotation
$U^{x}_{\pi},U^{y}_{\pi},U^{z}_{\pi}$ commutes with each other for spin-1, so
we can also label the symmetry sectors according to $Z_{2}\times Z_{2}$
symmetry. Since $U^{\alpha}_{\pi}U^{\beta}_{\pi}=U^{\gamma}_{\pi}$, symmetry
sector can be labeled by the eigenvalues of $U^{z}_{\pi},U^{x}_{\pi}$. The
corresponding level spacing statistics in this symmetry separation is shown in
Fig.3, where we show the mean gap ratio and ratio distribution in two
different symmetry sectors $\\{Z=1,X=1,I=1\\}$ (the sizes of this sector are
861, 7503, 66795 for system sizes $L=8$, 10, 12, respectively) and
$\\{Z=-1,X=-1,I=1\\}$ (the sizes are 820, 7381, 66430 for system sizes $L=8$,
10, 12, respectively). In these two symmetry sectors, the level repulsion
vanishes in the ratio distributions (Fig.3 (b),(d)) and the distributions show
the non-random matrix theory statistics.

The non-random matrix theory statistics behavior can be understood from the
fact that every sector in $Z_{2}\times Z_{2}$ symmetry mixes levels from
different sectors in $U(1)\rtimes Z_{2}$ symmetry. For example,
$\\{Z=1,X=1\\}$ sector in $Z_{2}\times Z_{2}$ symmetry mixes levels from
$M^{z}$ and $-M^{z}$ sectors with $M^{z}=even$.

The mean ratio tends to be flat when increasing system size (see Fig.3
(a),(c)), this is also observed in the half-filled Hubbard model when mixing
particle-hole symmetry sectors De Marco  _et al._ (2022). The mean ratio in
$\\{Z=1,X=1,I=1\\}$ sector ($\langle r\rangle\approx 0.406$) is smaller than
the ratio in $\\{Z=-1,X=-1,I=1\\}$ sector ($\langle r\rangle\approx 0.4167$).
This is consistent with the fact that the statistics will approach Poisson
($\langle r\rangle_{P}=0.386$) when mixing more and more levels with different
quantum numbers, and the size of $\\{Z=1,X=1,I=1\\}$ sector is indeed slightly
larger than $\\{Z=-1,X=-1,I=1\\}$ sector.

Figure 3: (a) (c) Mean gap ratio $\langle r\rangle$ of consecutive level
spacings of Haldane model with $\Delta=0.5$. (b) (d) Distribution of ratio of
consecutive level spacing of Haldane model for $L=12$, D=0, $\Delta=0.5$. The
symmetry sectors used here: (a) (b) $\\{Z=1,X=1,I=1\\}$, (c) (d)
$\\{Z=-1,X=-1,I=1\\}$.

Here, we conclude this section that the statistics of the gap ratio of
consecutive level spacing of the Haldane model show the expected GOE behavior
and indicate the non-integrable nature. To see the correct statistics, we need
the finer symmetry separation $U(1)\rtimes Z_{2}$ rather than its subgroup
$Z_{2}\times Z_{2}$.

###  III.2 Dual model

From Sec.II, we know that the dual model is unitary equivalent to the non-
integrable Haldane model, so the dual model is also non-integrable. The
apparent symmetries of the dual model are $Z_{2}\times Z_{2}$ and bond-
centered inversion.

The mean ratios and ratio distributions in the sectors $\\{Z=1,X=1,I=1\\}$ and
$\\{Z=-1,X=-1,I=1\\}$ are shown in Fig.4, they all show non-random matrix
theory statistics. And there is no level repulsion in the gap ratio
distribution. These results are against the expectations, since the dual model
is non-integrable.

Before discussing the reason of the non-random matrix theory statistics, it is
worthy to mention that the bond-centered inversion symmetry is not invariant
under the unitary Kennedy-Tasaki transformation
$U_{KT}\mathcal{I}U_{KT}\mathcal{I}\neq 1$. Hence, the $I=\pm 1$ sectors in
the Haldane chain can be different from $I=\pm 1$ sectors in the dual model,
while their full $Z_{2}\times Z_{2}$ symmetry sectors are the same. This can
be seen as follows. Compare the mean ratio between Haldane model (Fig.3 (a))
and dual model (Fig.4 (a)) in the symmetry sector: $\\{Z=1,X=1,I=1\\}$, we
find the mean ratio values are exactly the same in these two models for every
D. But they are different in the symmetry sector $\\{Z=-1,X=-1,I=1\\}$, see
Fig.3 (c) and Fig.4 (c). Regardless of these detailed values, their statistics
behavior are consistent, namely, they show neither GOE behavior nor Poisson
behavior.

Figure 4: (a) (c) Mean gap ratio $\langle r\rangle$ of consecutive level
spacings of dual model with $\Delta=0.5$. (b) (d) Distribution of the ratio of
consecutive level spacing of dual model for $L=12$, D=0, $\Delta=0.5$. The
symmetry sectors used here: (a) (b) $\\{Z=1,X=1,I=1\\}$, (c) (d)
$\\{Z=-1,X=-1,I=1\\}$.

As mentioned in Sec.II, there is a hidden U(1) symmetry in the dual model.
Since the apparent symmetries of the dual model are only $Z_{2}\times Z_{2}$
internal symmetry and bond-centered inversion, we now consider the level
spacing statistics under the perturbation which preserves these two apparent
symmetries but breaks the hidden U(1) symmetry. A simple choice can be
$D_{x}\sum_{i=1}^{L}(S^{x}_{i})^{2}$ which is invariant under the Kennedy-
Tasaki transformation and uniform. Once the hidden U(1) symmetry is removed by
small perturbation $D_{x}$ term (see Fig.5 (a)), the mean ratio moves toward
the GOE value. With increasing system size, the mean ratio reaches GOE value
with smaller $D_{x}$, one would expect that only infinitesimal perturbation is
needed in the thermal dynamical limit. With perturbation, the mean ratio
(Fig.5 (b)) shows less fluctuation than in Haldane model (Fig.2) when the
system size $L\geq 10$. These results imply that the non-random matrix theory
statistics in the dual model can be ascribed to the hidden symmetry. One may
regard the effect from the hidden symmetry in the dual model as a kind of
accidental degeneracy. And this degeneracy is not related to the (apparent)
symmetries, since it is easily lifted by (apparent) symmetries preserving
perturbations.

Figure 5: Mean gap ratio $\langle r\rangle$ of consecutive level spacings of
dual model with perturbation $D_{x}\sum_{i=1}^{L}(S^{x}_{i})^{2}$ and fixed
$\Delta=0.5$, (a) $D=0.5$, (b) $D_{x}=0.05$. The symmetry sector used here has
the quantum numbers: $\\{Z=1,X=1,I=1\\}$.

We further consider two other different perturbations which are usually used
for studying quantum chaos and integrability. The first one is considering the
model in the presence of a single defect on a site Santos  _et al._ (2020);
Santos (2004). The second one is adding a random field Kudo and Deguchi (2004,
2018). These two perturbations both break bond-centered inversion symmetry
explicitly.

In Fig.6 (a) (b), we consider a single defect $D_{x}(S^{x}_{1})^{2}$ located
at the edge of the chain. In Fig.6 (c) (d), the defect $D_{x}(S^{x}_{i})^{2}$
is put in the bulk and close to the center of the chain ($i=L/2$). These two
kinds of defect both modify the level spacing statistics significantly and
lead to the GOE statistics even if the strength is weak ($D_{x}=0.05$). The
dual model with the bulk defect has better GOE behavior than the boundary
defect, where the width of the plateau is wider and the deviation from the GOE
value is smaller when $L=10$.

Figure 6:  Mean gap ratio $\langle r\rangle$ of consecutive level spacings of
the dual model (a) with a defect $D_{x}(S^{x}_{1})^{2}$ at the boundary, (c)
with a defect $D_{x}(S^{x}_{i})^{2}$ ($i=L/2$) in the bulk. (b) (d)
Distribution of ratios of consecutive level spacings of the dual model for
$L=10$, D=1 with the single-site defect (b) on the boundary, (d) in the bulk.
Here, $\Delta=0.5,D_{x}=0.05$. And the quantum numbers are $\\{Z=1,X=1\\}$.

In Fig.7, we consider the random field $\sum_{i=1}^{L}h_{i}(S_{i}^{x})^{2}$,
where $h_{i}$’s are uncorrelated random numbers with a Gaussian
distribution:$\langle h_{i}\rangle=0$ and $\langle
h_{i}h_{j}\rangle=h^{2}\delta_{ij}$. We choose $h=0.1$ as a perturbation. The
number of samples used here is 200, and we have checked the results are stable
with more samples up to 400. Compared to the uniform perturbation (Fig.5 (b))
and defect (Fig.6), the random field perturbation has better GOE behavior and
less fluctuation since it is obtained by averaging over samples.

Figure 7: (a) (c) Mean gap ratio $\langle r\rangle$ of consecutive level
spacings of the dual model with random magnetic field. (b) (d) Distribution of
ratio of consecutive level spacing of dual model with random magnetic field
for L=10, D=-2. The symmetry sectors used are (a) (b) $\\{Z=1,X=1\\}$, (c) (d)
$\\{Z=-1,X=-1\\}$. Here, h=0.1, $\Delta=0.5$.

Here, this dual model provides an example that a simple non-integrable model
with only two-body interaction shows non-random matrix theory statistics. The
reason for this statistics can be traced back to a hidden U(1) symmetry, but
this is still different from the known examples Kudo and Deguchi (2005); Gubin
and F. Santos (2012) where the non-random matrix theory statistics appear in
non-integrable systems due to mix some apparent symmetry sectors. Firstly, it
is impractical to separate the levels into symmetry sectors by this nonlocal
symmetry in the dual model. And it is very nontrivial to identify such hidden
symmetry if one begins with the dual model but not the Haldane chain and
Kennedy-Tasaki transformation. Secondly, this hidden U(1) symmetry does not
commute with the apparent internal symmetries, thus it is impossible to label
the symmetry sectors with both hidden symmetry and apparent symmetries. In
this sense, no apparent symmetry sectors are mixed.

###  III.3 Crossover from chaotic to integrable

In this section, we study the level spacing statistics of the model
$H(\lambda)$ (eq.(19)) in $0\leq\lambda\leq 0.5$. As discussed in the previous
section, the internal symmetry of this model is $Z_{2}\times Z_{2}$ when
$\lambda\neq 0$ and 0.5. We expect this model will show chaotic statistics for
$0<\lambda<0.5$. The model reduces to the Haldane model when $\lambda=0$, and
the internal symmetry recovers to $U(1)\rtimes Z_{2}$. For $\lambda=0.5$, the
model is equivalent to a spin-1/2 XY model with immobile holes, which is
solvable in every sector with fixed hole number on every site.

We use the mean ratio as an indicator for studying the crossover behavior. In
Fig.8 (a), a very good agreement with the GOE value is found for
$0.01<\lambda<0.45$ with system size $L>8$. Apart from this regime, the mean
ratio deviates from GOE value for $\lambda\geq 0.45$ and decreases around the
self-dual point $\lambda=0.5$ and the Haldane point $\lambda=0$. Compared to
the self-dual point, the mean ratio decreases more rapidly around the Haldane
point.

Figure 8:  (a) (b) Mean gap ratio $\langle r\rangle$ of consecutive level
spacings of model $H(\lambda)$ (eq(19)). The symmetry sector used here has the
quantum numbers: $\\{Z=1,X=1,I=1\\}$.

In Fig.8 (b), we further investigate the mean ratio with $\lambda$ which are
very close to the self-dual point. The mean ratio approaches the Poisson value
as the system size increases (see the inset in Fig.8 (b)).

The change of the mean gap ratio in Fig.8 actually indicates a crossover from
chaotic to integrable around the self-dual point. To better show the
crossover, we use the gap ratio distribution, the results are shown in Fig.9.
Level repulsion is suppressed when $\lambda$ gets close to the self-dual
point. And there is a large distribution around $r=0$ for $\lambda\geq 0.499$
which deviates from standard Poisson statistics, this is very similar to the
XY model Zangara  _et al._ (2013).

Figure 9: Distribution of ratios of consecutive level spacings of model
$H(\lambda)$ (eq(19)) for L=12 and (a) $\lambda=0.45$, (b) $\lambda=0.49$, (c)
$\lambda=0.499$, (d) $\lambda=0.4999$. The symmetry sector used here has the
quantum numbers: $\\{Z=1,X=1,I=1\\}$.

##  IV Conclusions

In this paper, we studied the level spacing statistics of three spin-1 models
in one dimension which are related to Kennedy-Tasaki transformation. The first
model is the standard spin-1 Haldane chain with single ion anisotropy, which
is believed to be chaotic. We find clear random matrix theory statistics in
this model in the symmetry sector corresponding to $U(1)\rtimes Z_{2}$
symmetry. Non-random matrix theory statistics is found in this model when we
use its symmetry subgroup $Z_{2}\times Z_{2}$ to label the sector. This non-
random matrix behavior can be understood by the fact that $Z_{2}\times Z_{2}$
symmetry sector mixes levels from different symmetry sectors in $U(1)\rtimes
Z_{2}$ symmetry.

The second model is obtained by the first model after applying Kennedy-Tasaki
transformation, which is also a simple model with only two-body interaction.
The apparent internal symmetry of this model is $Z_{2}\times Z_{2}$ which is
the subgroup of the first model. Although this model is unitary equivalent to
the Haldane model, the level spacing statistics results show non-random
matrices behavior since these two models have different symmetries. This is
our main result. The reason for this statistics can be traced back to a hidden
non-local U(1) symmetry, which is hard to identify if one begins with this
model. And it is also impractical to use this symmetry to label the sectors.
We further study the level spacing statistics of this model under three
different perturbations which preserve the $Z_{2}\times Z_{2}$ symmetry but
remove the hidden symmetry. Under these perturbations, the level spacing
statistics show the GOE behavior.

The third model is a combination of the first and second models. It contains a
special parameter point (self-dual point) where the model is invariant under
Kennedy-Tasaki transformation. At this point, the model is equivalent to a
spin-1/2 XY chain doped by immobile holes and thus it is integrable. We study
the level spacing statistics around this point and find a crossover from
random matrix theory statistics to Poisson like statistics where the level
repulsion vanishes.

Here we mainly focus on the level spacing statistics under the Kennedy-Tasaki
transformation, where only the eigenvalues are considered. It is known that
there are other indicators that involve the eigenstates or both eigenvalues
and eigenstates, such as entanglement entropy, and survival probability. In
future study, it would be interesting to consider the behaviors of these
indicators in unitary equivalent systems as in this work.

###### Acknowledgements.

We thank Linhao Li, Han-Qing Wu, Haruki Watanabe, Zongping Gong and Hosho
Katsura for helpful discussions. This work is supported by the International
Postdoctoral Exchange Fellowship Program 2022 by the Office of China
Postdoctoral Council: No.PC2022072 and the National Natural Science Foundation
of China: No.12147172.

##  Appendix A Some useful algebras for spin-1 system

In this appendix, we provide some useful algebras for spin-1 system which are
used in this paper.

The well known matrix representations of spin-1 operators are given by

$\displaystyle S^{z}$ $\displaystyle=\left(\begin{array}[]{ccc}1&0&0\\\
0&0&0\\\
0&0&-1\end{array}\right),\,S^{x}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccc}0&1&0\\\
1&0&1\\\ 0&1&0\end{array}\right),$ (21) $\displaystyle S^{y}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccc}0&-i&0\\\ i&0&-i\\\
0&i&0\end{array}\right),$

and one can find that

$(S^{\alpha})^{3}=S^{\alpha}.$ (22)

$\alpha=x,y,z$. With this property, the rotation operator can be simplified to

$U^{\alpha}_{\theta}=1-i\sin(\theta)S^{\alpha}+(\cos(\theta)-1)(S^{\alpha})^{2},$
(23)

and the $\pi$ rotation is $U^{\alpha}_{\pi}=1-2(S^{\alpha})^{2}$.

Some identities can be found

$\displaystyle S^{\alpha}S^{\beta}S^{\alpha}=0,$ (24) $\displaystyle
i\epsilon_{\alpha\beta\gamma}S^{\alpha}S^{\beta}+S^{\gamma}(S^{\beta})^{2}=0,$

where $\alpha,\beta,\gamma$ are different. The second identity and its
hermitian conjugate imply that $S^{y}(S^{z})^{2}=(S^{x})^{2}S^{y}$, which
leads to eq.(8).

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