# Chiral spin liquid and quantum phase diagram of spin-$1/2$
$J_{1}$-$J_{2}$-$J_{\chi}$ model on the square lattice

Xiao-Tian Zhang1 Yixuan Huang2,3 Han-Qing Wu4<EMAIL_ADDRESS>D. N.
Sheng5<EMAIL_ADDRESS>Shou-Shu Gong6<EMAIL_ADDRESS>1 School
of Physics, Beihang University, Beijing 100191, China
2 Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico
87545, USA
3 Center for Integrated Nanotechnologies, Los Alamos National Laboratory, Los
Alamos, New Mexico 87545, USA
4 Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices,
School of Physics, Sun Yat-sen University, Guangzhou, 510275, China
5 Department of Physics and Astronomy, California State University Northridge,
Northridge, California 91330, USA
6 School of Physical Sciences, Great Bay University, Dongguan 523000, China,
and
Great Bay Institute for Advanced Study, Dongguan 523000, China

###### Abstract

We study the spin-$1/2$ Heisenberg model on the square lattice with the first
and second nearest-neighbor antiferromagnetic couplings $J_{1}$, $J_{2}$, as
well as the three-spin scalar chiral coupling $J_{\chi}$. Using density matrix
renormalization group calculations, we obtain a quantum phase diagram of this
system for $0\leq J_{2}/J_{1}\leq 1.0$ and $0\leq J_{\chi}/J_{1}\leq 1.5$. We
identify the Néel and stripe magnetic order phase at small $J_{\chi}$
coupling. With growing $J_{\chi}$, we identify the emergent chiral spin liquid
(CSL) phase characterized by the quantized spin Chern number $C=1/2$ and
entanglement spectrum with the quasidegenerate group of levels agreeing with
chiral SU(2)1 conformal field theory, which is an analog of the $\nu=1/2$
Laughlin state in spin system. In the vicinity of the Néel and CSL phase
boundary, our numerical results do not find evidence to support the phase
coexistence of Néel order and topological order that was conjectured by mean-
field calculations. In the larger $J_{2}$ and $J_{\chi}$ coupling regime, the
entanglement spectrum of the ground state also exhibits the chiral
quasidegeneracy consistent with a CSL, but the adiabatic flux insertion
simulations fail to obtain the quantized Chern number. By analyzing the
finite-size scaling of magnetic order parameter, we find the vanished magnetic
order suggesting a magnetic disorder phase, whose nature needs further
studies. Different from the spin-$1$ $J_{1}$-$J_{2}$-$J_{\chi}$ model, we do
not find the coexistent stripe magnetic order and topological order. We also
investigate the $J_{\chi}$ dominant regime and find a strong tendency of the
system to develop a dimer order rather than the chiral spin magnetic order
observed in the spin-$1$ model. Our results unveil interesting quantum phases
in this spin-$1/2$ model and also demonstrate the drastic differences between
the spin-$1/2$ and spin-$1$ cases.

##  I Introduction

Quantum spin liquids (QSLs) are exotic states of matter in frustrated quantum
magnets Savary and Balents (2016); Zhou  _et al._ (2017); Broholm  _et al._
(2020); Lancaster (2023); Kivelson and Sondhi (2023), which can escape from
forming the conventional orders even at zero temperature. Remarkably, QSLs
have long-range quantum entanglement as well as fractionalized excitations Wen
(1990, 1991); Senthil and Fisher (2000, 2001); Chen  _et al._ (2010). Among
the various types of QSLs, chiral spin liquid (CSL) is a specific class of
QSLs that is an analog of the fractional quantum Hall state Kalmeyer and
Laughlin (1987); Wen  _et al._ (1989); Baskaran (1989); Yang  _et al._ (1993);
Haldane and Arovas (1995). CSLs break time-reversal and parity symmetries but
preserve lattice translational and spin rotational symmetries. Kalmeyer and
Laughlin first proposed that in a time-reversal invariant spin model with
geometric frustration, one may realize a CSL as the analog of the $\nu=1/2$
Laughlin state through spontaneous time-reversal symmetry breaking Kalmeyer
and Laughlin (1987).

More interestingly, doping a CSL may lead to the exotic anyon
superconductivity Wen  _et al._ (1989); Wilczek (1990). After extensive search
for decades, this Kalmeyer-Laughlin CSL has been identified not only in
various spin-$1/2$ systems with either geometric frustration or competing
interactions He  _et al._ (2014); Gong  _et al._ (2014a); Wietek  _et al._
(2015); Hu  _et al._ (2015); Gong  _et al._ (2015); Hickey  _et al._ (2017);
Gong  _et al._ (2017); Wietek and Läuchli (2017); Nielsen  _et al._ (2013);
Chen  _et al._ (2021), but also in the Hubbard model near the Mott transition
Szasz  _et al._ (2020); Boos  _et al._ (2020); Szasz and Motruk (2021);
Cookmeyer  _et al._ (2021); Chen  _et al._ (2022). Very recently, unbiased
density matrix renormalization group (DMRG) studies have also established a
$d+id$-wave topological superconducting phase near the doped CSL regime Jiang
and Jiang (2020); Huang and Sheng (2022); Huang  _et al._ (2023).

On the other hand, CSL may also play an important role in cuprate
superconductors. Recently, the thermal Hall measurements for cuprate
superconductors have found a giant thermal Hall conductivity at small doping
level even including the half-filled case Grissonnanche  _et al._ (2019,
2020); Boulanger  _et al._ (2020, 2022, 2023), which triggered the theoretical
interests in the origin of the thermal Hall conductivity Samajdar  _et al._
(2019a, b); Han  _et al._ (2019); Li and Lee (2019); Li (2019); Zhang  _et
al._ (2021); Merino and Ralko (2022); Varma (2020); Guo and Sachdev (2021); Ye
_et al._ (2021); Mangeolle  _et al._ (2022a, b); Flebus and MacDonald (2022);
Sun  _et al._ (2022). Based on mean-field calculations, it was proposed that
there is a phase coexistent regime of Néel magnetic order and CSL topological
order Samajdar  _et al._ (2019a); Zhang  _et al._ (2021) in the spin-$1/2$
square-lattice Heisenberg model with the nearest-neighbor (NN) $J_{1}$, next-
nearest-neighbor (NNN) $J_{2}$ antiferromagnetic (AFM) exchange interactions,
as well as the three-spin scalar chiral coupling $J_{\chi}$ describing the
effect of external magnetic fields in the thermal Hall experiments. The
coexistent Néel AFM order and neutral spinon excitations may provide a natural
understanding on the experimental observations Samajdar  _et al._ (2019a);
Zhang  _et al._ (2021).

In previous studies on this microscopic spin model, the Néel AFM order and
Kalmeyer-Laughlin CSL state have been found Merino and Ralko (2022); Nielsen
_et al._ (2013); Liu  _et al._ (2016), but the possible phase coexistence has
not been examined in numerical calculations. Very recently, the similar
spin-$1$ $J_{1}$-$J_{2}$-$J_{\chi}$ model has been comprehensively studied by
DMRG calculations, which find no evidence for coexistent Néel order and CSL
but unveil a phase coexistence of stripe AFM order and a non-Abelian CSL in
the larger $J_{2}$ regime Huang  _et al._ (2022). This phase coexistence is
characterized by a strong magnetic order and the entanglement spectrum that
agrees with the expected CSL state Huang  _et al._ (2022). These reported
results of the spin-$1$ model also stimulate an interest in the similar
phenomenon in the corresponding spin-$1/2$ system.

In this work, we examine the quantum phase diagram of the spin-$1/2$
$J_{1}$-$J_{2}$-$J_{\chi}$ model on the square lattice (see the schematic
figure on the left top corner of Fig. 1) by means of DMRG calculations, with
the focus on exploring the possible coexistence of CSL and magnetic order. In
the studied parameter regime $0\leq J_{2}/J_{1}\leq 1.0$ and $0\leq
J_{\chi}/J_{1}\leq 1.5$, we identify a Néel AFM phase, a stripe AFM phase, a
CSL phase, and a magnetic disorder regime, as shown in Fig. 1. We confirm the
CSL state by showing the quantized topological entanglement entropy
$\gamma\simeq(\ln 2)/2$, chiral entanglement spectrum counting
$\\{1,1,2,3,5,\cdots\\}$, and quantized spin Chern number $C=1/2$.
Furthermore, we carefully investigate the phase transition from the Néel AFM
phase to the CSL with growing chiral coupling. While the transition can be
identified by various quantities, we do not find evidence to support a phase
coexistence of topological order and Néel order near the phase boundary.

On the other hand, at the larger $J_{2}$ side we find either the CSL state or
a magnetic disorder state with increased $J_{\chi}$ coupling. While the
entanglement spectrum of the ground state in the magnetic disorder regime can
exhibit the quasidegenerate group of levels agreeing with chiral SU(2)1
conformal field theory, the adiabatic flux insertion simulations fail to
obtain a half-quantized spin Chern number in the studied system size.
Meanwhile, the finite-size extrapolation of magnetic order parameters strongly
suggest the absent magnetic order in this regime. Therefore, our DMRG results
indicate that the coexistence of topological order and stripe AFM order
observed in the spin-$1$ model Huang  _et al._ (2022) may not exist in the
studied spin-$1/2$ case. In addition, we examine the $J_{\chi}$-dominant
regime where a chiral spin state (CSS) magnetic order Rabson and Trugman
(1995) (see Appendix A) has been found in the spin-$1$
$J_{1}$-$J_{2}$-$J_{\chi}$ model Huang  _et al._ (2022). Nonetheless, we do
not find the CSS magnetic order but a strong tendency of the system to develop
a dimer order spontaneously breaking lattice translational symmetry.

Compared with the spin-$1$ model, our results unveil the big differences
between the spin-$1/2$ and spin-$1$ $J_{1}$-$J_{2}$-$J_{\chi}$ models, showing
the important role of quantum fluctuations in the emergence of the different
quantum states. In particular, the existence and absence of phase coexistence
in the spin-$1$ and $1/2$ systems may shed light on further investigation of
the coexisted topological order and conventional order. We also discuss the
potential nature of the magnetic disorder regime, which we leave for future
study.

The organization of our paper is as follows: In Sec. II, we introduce the
model and the details of DMRG calculations. In Sec. III, we identify the phase
diagram at the smaller $J_{2}$ side. We characterize the CSL state and show
the phase transition from the Néel to the CSL phase. We also explore the
possible phase coexistence between the two phases. In Sec. IV, we discuss the
quantum phases at the larger $J_{2}$ side and carefully examine the properties
of the magnetic disorder regime. In Sec. V, we go beyond the phase diagram
Fig. 1 and discuss the possible dimer order phase in the $J_{\chi}$-dominant
regime. We summarize the results is Sec. VI.

##  II Model Hamiltonian and methods

Figure 1: Model illustration and phase diagram. The inset on the left top
corner is schematic illustration of the spin-$1/2$ $J_{1}$-$J_{2}$-$J_{\chi}$
Heisenberg model on the square lattice. $J_{1}$ and $J_{2}$ are the NN and NNN
AFM interactions, respectively. $J_{\chi}$ is the magnitude of the three-spin
chiral couplings ${\bf S}_{i}\cdot({\bf S}_{j}\times{\bf S}_{k})$ in the
Hamiltonian Eq. (1), which include all the four kinds of triangles in each
plaquette. For each triangle, the sites $i,j,k$ follow the same clockwise
direction. The quantum phase diagram of the spin-$1/2$
$J_{1}$-$J_{2}$-$J_{\chi}$ model is obtained with $0\leq J_{2}/J_{1}\leq 1.0$
and $0\leq J_{\chi}/J_{1}\leq 1.5$. With tuning couplings, we identify the
Néel AFM, stripe AFM, chiral spin liquid, and a magnetic disorder regime. The
blue rhombus symbols indicate the parameter points in which we can obtain a
quantized spin Chern number $C=1/2$ by the adiabatic flux insertion
simulations on the $L_{y}=8$ systems. For the intermediate $J_{2}/J_{1}$
regime with small $J_{\chi}$ couplings, the red asterisks represent the
parameter points where our DMRG simulations do not obtain a Chern number. In
the magnetic disorder regime, the entanglement spectrum of the ground state
shows the chiral counting consistent with a chiral spin liquid, but the flux
insertion simulations fail to obtain a Chern number.

We consider the spin-$1/2$ square-lattice model with the NN ($J_{1}$) and NNN
($J_{2}$) AFM Heisenberg interactions, as well as the three-spin scalar chiral
coupling $J_{\chi}$ for all the four small triangles in each plaquette of the
square lattice. The Hamiltonian is defined as

$H=\sum_{\langle i,j\rangle}J_{1}{\bf S}_{i}\cdot{\bf
S}_{j}+\sum_{\langle\langle i,j\rangle\rangle}J_{2}{\bf S}_{i}\cdot{\bf
S}_{j}+\sum_{\bigtriangleup}J_{\chi}{\bf S}_{i}\cdot({\bf S}_{j}\times{\bf
S}_{k}),$ (1)

where the sums ${\langle i,j\rangle}$ and ${\langle\langle i,j\rangle\rangle}$
run over all the NN and NNN bonds, respectively. The chiral couplings involve
the sum over all the four kinds of triangles in each plaquette, where the
sites $i,j,k$ follow the same clockwise direction as shown in Fig. 1.

Figure 2: Quantum phase transition with vanishing the Néel AFM order by chiral
coupling at $J_{2}=0$. (a) Néel AFM order parameter $m^{2}(\pi,\pi)$ versus
$J_{\chi}/J_{1}$. The data of $m^{2}(\pi,\pi)$ are obtained from the middle
$L_{y}\times L_{y}$ sites of a long cylinder system with $L_{y}=4,6,8$. The
inset is the $J_{\chi}/J_{1}$ dependence of $m^{2}(\pi,\pi)$ and its first-
order derivative for $L_{y}=6$. (b) Finite-size scaling of $m^{2}(\pi,\pi)$
with $L_{y}=4,6,8,10$ for different $J_{\chi}/J_{1}$. The dashed lines denote
the polynomial fittings up to the second order of $1/L_{y}$. (c) Scalar chiral
order $\langle\chi\rangle$ versus $J_{\chi}/J_{1}$ on different system sizes.
The chiral order defined as $\langle\chi\rangle\equiv\langle{\bf
S}_{i}\cdot({\bf S}_{j}\times{\bf S}_{k})\rangle$ is measured in the bulk of
long cylinder and is uniform. The four kinds of triangles in each plaquette
have the same magnitude of chiral order. The inset shows the first-order
derivative of the chiral order with respect to $J_{\chi}/J_{1}$, which is
equivalent to the second-order derivative of ground-state energy. The symbols
in the subfigure (c) and its inset follow the same definitions.

We determine the quantum phases of the model by using DMRG White (1992)
calculations. In DMRG simulation, we study the system on the cylinder geometry
with the periodic boundary conditions along the circumference direction ($y$
direction) and the open boundary conditions along the axis direction ($x$
direction). We denote $L_{y}$ and $L_{x}$ as the site numbers in the two
directions, respectively. We implement the spin SU(2) symmetry McCulloch and
Gulácsi (2002) in the finite DMRG simulations, which allows us to study the
systems with circumference up to $L_{y}=12$. We keep the bond dimensions up to
$6000$ SU(2) multiplets, which are equivalent to about $24000$ U(1) states.
For the $L_{y}=12$ systems, the DMRG truncation error is smaller than $5\times
10^{-5}$, while for smaller $L_{y}$ systems, we can obtain very accurate
results with the truncation error smaller than $1\times 10^{-6}$. In addition,
we use the TeNPy Hauschild and Pollmann (2018) package to perform the infinite
DMRG simulations on the systems with the circumference up to $L_{y}=8$, in
which we keep the bond dimensions up to $3000$ U(1) states, with the
truncation error near $10^{-5}$.

We obtain the quantum phase diagram of the model at $0\leq J_{2}/J_{1}\leq
1.0$ and $0\leq J_{\chi}/J_{1}\leq 1.5$, as shown in Fig. 1. We set
$J_{1}=1.0$ as the energy unit. In the presence of a moderate chiral coupling,
we identify two magnetic order phases, a CSL phase, and a magnetic disorder
regime in the studied parameter region. For the intermediate non-magnetic
regime $J_{2}/J_{1}\sim 0.55$, we find that the CSL state can be easily driven
by a small chiral coupling.

##  III Néel order, chiral spin liquid, and the phase transition

In this section, we will demonstrate our numerical results to identify the
Néel AFM phase, CSL phase, and the phase transition. We will also discuss our
investigation of the possible phase coexistence.

###  III.1 Néel phase and the phase transition

For $J_{2}=J_{\chi}=0$, the system possesses a Néel AFM order, which would be
suppressed by the increased $J_{\chi}$ coupling. With growing chiral coupling,
a transition from the Néel order to the CSL has been proposed at
$J_{\chi}/J_{1}\simeq 0.189$ by exact diagonalization calculation Nielsen  _et
al._ (2013), showing that the AFM order will be easily melted by chiral
coupling. On the other hand, a recent exact diagonalization calculations of
quantum fidelity on a $4\times 4$ cluster suggests the transition at
$J_{\chi}/J_{1}\simeq 0.7$ Merino and Ralko (2022), and the spinon mean-field
study finds the transition at $J_{\chi}/J_{1}\simeq 1.35$ Liu  _et al._
(2016).

To accurately determine the phase transition with vanishing the Néel order, we
first compute the Néel order parameter $m^{2}(\pi,\pi)$ defined as

$m^{2}(\vec{k})=\frac{1}{N^{2}_{m}}\sum_{i,j}\langle{\bf S}_{i}\cdot{\bf
S}_{j}\rangle e^{i\vec{k}\cdot(\vec{r}_{i}-\vec{r}_{j})}$ (2)

at the momentum $\vec{k}=(\pi,\pi)$. In the summation of the above equation,
we only consider the middle $N_{m}=L_{y}\times L_{y}$ sites on a long cylinder
to avoid the edge effects. In Fig. 2(a), we show $m^{2}(\pi,\pi)$ with growing
$J_{\chi}/J_{1}$ for $L_{y}=4,6,8$ ($J_{2}=0$), which all have a behavior
change at $J_{\chi}/J_{1}\simeq 0.85$ that can be clearly identified in the
derivative as shown in the inset of Fig. 2(a). In Fig. 2(b), we further
analyze the finite-size scaling of $m^{2}(\pi,\pi)$ with $L_{y}=4,6,8,10$,
which are extrapolated using the polynomial fitting up to the second order of
$1/L_{y}$. The extrapolated results support a transition with vanished Néel
AFM order at $J_{\chi}/J_{1}\simeq 0.85$, which is consistent with the finding
in Fig. 2(a).

Since the phase transition may be characterized by the ground-state energy, we
also study the the scalar chiral order parameter $\langle\chi\rangle$ in the
bulk of the cylinder versus $J_{\chi}/J_{1}$. The chiral order is defined as
$\langle\chi\rangle\equiv\langle{\bf S}_{i}\cdot({\bf S}_{j}\times{\bf
S}_{k})\rangle$, where the sites $i,j,k$ follow the definition of the small
triangle shown in the inset of Fig. 1. Thus, this chiral order is equivalent
to the first-order derivative of the ground-state energy with respect to the
chiral coupling. The obtained chiral orders on different system sizes [Fig.
2(c)] and their derivatives shown in the inset also characterize a transition
at $J_{\chi}/J_{1}\simeq 0.85$. Therefore, our DMRG results indicate that a
chiral coupling as large as the Heisenberg interaction $J_{1}$ is required to
suppress the Néel order. Following the same way, we can determine the phase
boundary with vanishing the Néel order in Fig. 1. More data can be found in
Fig. 12 (see Appendix B).

Figure 3: Identification of the chiral spin liquid state at $J_{2}=0$. (a)
Finite-size scaling of spin-triplet gap, which is obtained for the middle
$L_{y}\times L_{y}$ sites on the $L_{y}=4,6,8$ systems. We show the results in
both Néel AFM phase ($J_{\chi}/J_{1}=0.8$) and chiral spin liquid phase
($J_{\chi}/J_{1}=1.0,1.2$), which are clearly extrapolated to vanished and
finite, respectively. (b) Entanglement entropy $S$ versus cylinder
circumference $L_{y}$ at $J_{\chi}/J_{1}=1.0,1.2$. We fit the data by the
formula $S=aL_{y}-\gamma$. The red asterisk shows the theoretical prediction
$\gamma=(\ln 2)/2$ for the $\nu=1/2$ Laughlin state. The inset shows the
entanglement entropy obtained for different subsystem length $d$, which follow
the area law of entropy. (c) Entanglement spectrum labeled by the quantum
numbers of total spin $S^{z}=0$ and relative momentum along the $y$ direction
$\triangle{k_{y}}$. $\lambda_{i}$ is the eigenvalue of reduced density matrix.
We show the results for $J_{\chi}/J_{1}=1.0$ at the $L_{y}=8$ system. The
numbers $\\{1,1,2,3,5\\}$ denote the near-degenerate low-lying levels
separated by an entanglement gap from higher levels, which agrees with the
prediction of the SU(2)1 conformal-field theory.

###  III.2 Identification of the chiral spin liquid

With suppressed Néel order by increasing the chiral coupling, we can identify
an emergent gapped Kalmeyer-Laughlin CSL by DMRG calculations, which is
equivalent to the $\nu=1/2$ bosonic fractional quantum Hall state Kalmeyer and
Laughlin (1987). First of all, we calculate the spin-triplet gap. We obtain
the gap in the bulk by calculating the lowest-energy states in the total spin
$S=0$ and $S=1$ sectors for the middle $L_{y}\times L_{y}$ sites by DMRG. We
first calculate the ground state in the $S=0$ sector on a long cylinder and
then sweep the $S=1$ sector for the middle $L_{y}\times L_{y}$ sites, which
can avoid the edge excitations Yan  _et al._ (2011). In Fig. 3(a), we show the
system size dependence of the triplet gap obtained on the $L_{y}=4,6,8$
systems at different $J_{\chi}$ and $J_{2}=0$. One can see that the gaps drop
fast versus $1/L_{y}$ and go to zero in the thermodynamic limit at
$J_{\chi}/J_{1}=0.8$, in consistent with the Néel AFM order with gapless
Goldstone modes. In contrast, for $J_{\chi}/J_{1}=1.0$ and $1.2$, the gaps are
extrapolated to finite values, showing the gapped spin-triplet excitations.

Next, we identify the quantized topological entanglement entropy of the
Kalmeyer-Laughlin CSL by analyzing the circumference dependence of
entanglement entropy Jiang  _et al._ (2012). For gapped topological states,
the entanglement entropy follows the area law and thus mainly depends on the
system circumference $L_{y}$ Eisert  _et al._ (2010). In particular, the
entropy $S(L_{y})$ of the minimum entangled state with a smooth boundary of
length $L_{y}$ will take the form

$S(L_{y})\simeq aL_{y}-\gamma,$ (3)

where $\gamma$ is the topological entanglement entropy, which is a universal
additive constant characterizing the long-range entanglement in the ground
state and is related to the total quantum dimension $D$ of a topological order
as $\gamma=\ln D$ Kitaev and Preskill (2006); Levin and Wen (2006). The
coefficient $a$ is non-universal due to the short-distance physics near the
boundary. In DMRG calculation, it is straightforward to obtain the
entanglement entropy by bipartiting the whole system into two subsystems. The
entropy can be obtained as $S=-\sum_{i}\lambda_{i}\ln\lambda_{i}$, where
$\lambda_{i}$ are the eigenvalues of the reduced density matrix of the
subsystems. In the inset of Fig. 3(b), the entropy versus subsystem length
shows that the entropy is almost independent of the subsystem length, which
agrees with the area law of entanglement entropy for such a quasi-one-
dimensional gapped system Eisert  _et al._ (2010). Since DMRG simulation on
topological order states would naturally find the minimum entangled state
Jiang  _et al._ (2012), in Fig. 3(b), we can study the scaling of the obtained
entropy $S(L_{y})$ versus $L_{y}$ ($L_{y}=6,8,10$) for $J_{\chi}/J_{1}=1.0$
and $1.2$. Notice that the entropy data for each $L_{y}$ is obtained on long
cylinders, which is almost independent of cylinder length and can be taken as
the result in the $L_{x}\rightarrow\infty$ limit. Following Eq. (3), we fit
our entropy data linearly with the red star showing the theoretical prediction
$\gamma=(\ln 2)/2$ for the Kalmeyer-Laughlin CSL Levin and Wen (2006). One can
see that although the finite-size effects may exist, the extrapolated $\gamma$
at $J_{\chi}/J_{1}=1.0,1.2$ are very close to $(\ln 2)/2$, which highly agrees
with the Kalmeyer-Laughlin CSL state.

Furthermore, we calculate the entanglement spectrum to characterize the CSL
state because the entanglement spectrum has a one-to-one correspondence with
the physical edge spectrum Li and Haldane (2008); Qi  _et al._ (2012). We use
the quantum number total $S^{z}$ of the half system and the relative momentum
quantum number along the $y$ direction $\triangle{k_{y}}$ to label the
obtained eigenvalues of the reduced density matrix Cincio and Vidal (2013);
Zaletel  _et al._ (2013). In Fig. 3(c), we present the entanglement spectrum
of the ground state at $J_{\chi}/J_{1}=1.0$ on the $L_{y}=8$ cylinder, which
is obtained for the total $S^{z}=0$ sector. The spectrum clearly shows the
quasidegenerate pattern $\\{1,1,2,3,5,\cdots\\}$ with increasing
$\triangle{k_{y}}$, and the quasidegenerate levels are well separated from the
higher levels by an entanglement gap. Such a chiral structure of entanglement
spectrum agrees with the SU(2)1 conformal field theory prediction and supports
a Kalmeyer-Laughlin CSL Di Francesco  _et al._ (1997).

Figure 4: Adiabatic flux insertion simulation and quantized spin Chern number
for $J_{\chi}/J_{1}=1.0$, $J_{2}=0$ on the $L_{y}=8$ cylinder. (a)
Entanglement spectrum flow with adiabatically inserted flux $\theta$, which is
obtained by iDMRG smulation with $1000$ bond dimensions. The eigenvalues in
the $S_{z}=0,\pm 1$ sectors are labelled by different symbols. (b) Spin
pumping with adiabatically inserted flux $\theta$, which is obtained by iDMRG
smulation with $3000$ bond dimensions. In a period of inserted flux
$\theta=0\rightarrow 2\pi$, a quantized magnetization moments $0.5$ is
accumulated at one open boundary of the cylinder, indicating a fractionally
quantized spin Chern number $C=1/2$.

At last, we compute the spin Chern number through adiabatic flux insertion and
the change of entanglement spectrum with increased flux Haldane and Arovas
(1995); Gong  _et al._ (2014a). Since inserting a flux along the cylinder is
equivalent to imposing the twist boundary conditions in the circumference
($y$) direction, we consider the flux by replacing the spin flipping term
$S^{+}_{i}S^{-}_{j}+h.c.$ to $S^{+}_{i}S^{-}_{j}e^{i\theta}+h.c.$ for these
Heisenberg interacting bonds crossing the $y$ boundary. For the chiral
interactions crossing the $y$ boundary, we introduce the similar changes in
the Hamiltonian terms. We take one component
$S^{+}_{i}S^{-}_{j}S^{z}_{k}+h.c.$ as an example, which is replaced as
$S^{+}_{i}S^{-}_{j}S^{z}_{k}e^{i\theta}+h.c.$ in the presence of flux. We
adiabatically increase the flux by taking the obtained ground state at a given
flux $\theta$ as the initial state for the next-step simulation of the added
flux $\theta+\Delta\theta$. The entanglement spectrum versus the adiabatically
added flux is shown in Fig. 4(a) for $J_{2}=0,J_{\chi}/J_{1}=1.0$, where the
eigenvalues $-\ln\lambda_{i}$ in the $S^{z}=-1,0,1$ sectors are demonstrated.
By analyzing the distributions of the eigenvalues at each given flux, one can
find that while the spectrum is symmetric about $S^{z}=0$ at zero flux
$\theta=0$, it becomes symmetric about $S^{z}=1/2$ at $\theta=2\pi$ flux,
which agrees with a spin-$1/2$ spinon at each end of the cylinder for the
system with $\theta=2\pi$. Correspondingly, the system evolves from the vacuum
sector ($\theta=0$) to the spinon sector ($\theta=2\pi$). With further
increase of the flux from $2\pi$ to $4\pi$, the entanglement spectrum becomes
symmetric about $S^{z}=1$, indicating that the system evolves back to the
vacuum sector at $\theta=4\pi$.

By means of this adiabatic flux insertion, we can also compute the spin Chern
number. In fractional quantum Hall states, a quantized net charge transfer
would appear as $\Delta N=C$ from one edge of the sample to the other edge
after inserting a period of flux $\theta=0\rightarrow 2\pi$, corresponding to
a fractionally quantized topological invariant Chern number $C$ Hatsugai
(1993). Following this method, we can measure the local spin magnetization
moment $\langle S^{z}_{i,j}\rangle$ for each site $(i,j)$ at each flux Gong
_et al._ (2014a). In the process of inserting flux, one can find that the
magnetizations in the bulk remain unchanged but only change near the open
boundaries, which is equivalent to spin transfer from one edge to the other
one. In a period of flux insertion, the transferred total spin gives the spin
Chern number. As shown in Fig. 4(b) for $J_{2}=0$ and $J_{\chi}/J_{1}=1.0$,
the fractionally quantized spin transfer $0.5$ characterizes the spin Chern
number $C=1/2$, which agrees with a Kalmeyer-Laughlin CSL state. An additional
example at $J_{2}>0$ is demonstrated in Fig. 13 (see Appendix C).

Figure 5: Exploring the possible phase coexistence. (a) Entanglement spectrum
labeled by the quantum numbers of total spin $S^{z}=0$ and relative momentum
along the $y$ direction $\triangle{k_{y}}$. $\lambda_{i}$ is the eigenvalue of
reduced density matrix. (b) Spin pumping with adiabatically inserted flux
$\theta$. We consider the parameter at $J_{2}/J_{1}=0$, $J_{\chi}/J_{1}=0.8$
on the $L_{y}=8$ cylinder.

###  III.3 Investigation on the possible phase coexistence

Between the Néel AFM and CSL phase, a possible phase coexistence regime has
been proposed based on the mean-field theory Samajdar  _et al._ (2019a), which
is expected to exhibit both the Néel AFM order and topological order. In a
recent DMRG study of the spin-$1$ square-lattice $J_{1}$-$J_{2}$-$J_{\chi}$
model, a phase coexistence of stripe AFM order and non-Abelian topological
order is suggested, which is characterized by the finite magnetic order and
chiral entanglement spectrum Huang  _et al._ (2022). Such a combination of
numerical results could be taken as the primary evidence of this kind of
coexistence of magnetic order and topological order. In our study, we follow
the same strategy to investigate the possible phase coexistence.

We focus on the small parameter regime below the Néel order boundary in Fig.
1, in which the system still possesses the Néel AFM order. Here, we
investigate the possible topological order by studying the entanglement
spectrum and Chern number. By DMRG simulations, we find that even we choose
the parameter point very close to the boundary such as $J_{2}=0$,
$J_{\chi}/J_{1}=0.8$, either the quasidegenerate chiral pattern of
entanglement spectrum or the quantized Chern number is not obtained, as shown
in Fig. 5. The entanglement spectrum we obtain should be interpreted as
evolving towards that of the CSL with increased chiral coupling. On the other
hand, above the boundary when the Néel AFM order is vanished, both the
entanglement spectrum and Chern number simulation results are consistent with
an emergent CSL state. Near this phase boundary at finite $J_{2}$, our DMRG
results lead to the same conclusion. Therefore, our results do not support the
coexistence of Néel order and CSL that was conjectured by mean-field
calculations Samajdar  _et al._ (2019a).

##  IV Stripe phase and the magnetic disorder regime

For the intermediate $J_{2}/J_{1}\sim 0.55$ regime in the $J_{1}$-$J_{2}$
model, we find that a small $J_{\chi}$ coupling can lead to the emergent CSL
state with the characteristic Chern number $C=1/2$, as shown in Fig. 14 of
Appendix C. For the stripe AFM phase at larger $J_{2}/J_{1}$, our DMRG results
identify either a CSL state or a magnetic disorder state while the stripe
order is suppressed by the increased $J_{\chi}$ coupling. In this magnetic
disorder state, the entanglement spectrum of the ground state shows the
quasidegenerate pattern $\\{1,1,2,3,\cdots\\}$, but the flux insertion
simulations fail to obtain a spin Chern number. Next, we will demonstrate the
details of the DMRG results.

###  IV.1 Stripe phase and the phase transition

For $J_{\chi}=0$, the $J_{1}$-$J_{2}$ model exhibits a stripe AFM order for
$J_{2}/J_{1}\gtrsim 0.61$ Liu  _et al._ (2022). With increased chiral
coupling, the stripe order will be suppressed. To identify the phase
transition with vanishing the stripe order, we investigate the spin structure
factor and chiral order parameter.

Figure 6: Quantum phase transition with vanishing the stripe AFM order by
tuning chiral coupling. The spin structure factor $S(0,\pi)$ which
characterizes the stripe order and the scalar chiral order parameter
$\langle\chi\rangle$ are shown versus $J_{\chi}/J_{1}$ on the $L_{y}=8$
systems for (a) $J_{2}=0.7$ and (b) $J_{2}=0.8$. The shades denote the phase
transitions.

For the stripe AFM order on the square lattice, the spin structure factor on a
finite symmetric lattice should have the peaks at both $\vec{k}=(0,\pi)$ and
$(\pi,0)$. In our DMRG calculations on long cylinder geometry, the spin
correlations form the stripe pattern along a certain direction due to the
geometry reason, and the spin structure factor selects the peak at
$\vec{k}=(0,\pi)$. Therefore, we study $S(0,\pi)$ versus the tuning chiral
coupling. In Fig. 6, we show the structure factor $S(0,\pi)$ and chiral order
parameter $\langle\chi\rangle$ for $J_{2}/J_{1}=0.7$ and $0.8$ on the
$L_{y}=8$ cylinder. As marked by the shades, both quantities consistently
characterize a phase transition with the vanished stripe AFM order. One can
find that while the $J_{\chi}$ dependence of chiral order exhibits a clear
change characterizing a continuous-like phase transition (the red curve), the
stripe AFM structure factor $S(0,\pi)$ drops fast to quite small values (the
black curve), which is consistent with the transition described by the change
of chiral order $\langle\chi\rangle$.

In the spin-$1$ $J_{1}$-$J_{2}$-$J_{\chi}$ model, the DMRG calculations find
evidence for the coexistence of stripe order and non-Abelian topological order
Huang  _et al._ (2022). Therefore, in the stripe AFM phase of the studied
spin-$1/2$ model, we also examine the entanglement spectrum and Chern number,
but our results do not support a coexistent topological order in the stripe
phase (not shown here).

###  IV.2 The magnetic disorder regime

Figure 7: Entanglement spectrum and adiabatic flux insertion simulations in
the magnetic disorder regime. (a) and (b) show the entanglement spectrum
labeled by the quantum numbers of total spin $S^{z}=0$ and relative momentum
along the $y$ direction $\triangle{k_{y}}$ for the ground states of
$J_{2}/J_{1}=J_{\chi}/J_{1}=0.8$ and $J_{2}/J_{1}=J_{\chi}/J_{1}=1.0$,
respectively, on the $L_{y}=8$ cylinder systems. $\lambda_{i}$ are the
eigenvalues of reduced density matrix. The results are obtained by keeping
$2500$ U(1) states. The insets show the spin pumping with adiabatically
inserted flux $\theta$. At a flux smaller than $2\pi$, the spin accumulation
collapses. As a result, the obtained ground state at $\theta=2\pi$ is the same
as that at zero flux.

Next, we focus on the discussion on the magnetic disorder regime based on the
DMRG results. In this magnetic disorder regime, the entanglement spectrum of
the obtained ground state also exhibits the quasidegenerate chiral counting
$\\{1,1,2,3,5,\cdots\\}$ and a large entanglement gap that separates the
quasidegenerate levels from the higher levels, as shown in Fig. 7 of the
$L_{y}=8$ systems. The entanglement spectrum results may indicate the obtained
ground state as the vacuum sector of a CSL state. Nonetheless, the spin
accumulation in the adiabatic flux insertion simulation is found to collapse
at a flux $\theta$ smaller than $2\pi$, as demonstrated in the insets of Fig.
7. Consequently, the ground state at $\theta=2\pi$ is the same as that at
$\theta=0$, namely the flux insertion simulations fail to find the spinon
topological sector as expected for a CSL.

Furthermore, we explore possible magnetic order in this regime and provide
DMRG data to support the absent magnetic order. First of all, we carefully
examine the spin structure factor in the region of $0.6\leq J_{2}/J_{1}\leq
1.0$ and $0\leq J_{\chi}/J_{1}\leq 1.5$ on the $L_{y}=8$ cylinder. We find
that when the stripe AFM order is suppressed as we identify in Fig. 6, the
spin structure factor shows small peaks at $\vec{k}=(0,\pi)$, $(\pi,0)$, and
$(\pi/2,\pi/2)$ in the magnetic disorder regime (see Fig. 15 in Appendix D),
which implies a possible weak CSS magnetic order Rabson and Trugman (1995). To
detect such a possible magnetic order, we calculate the corresponding magnetic
order parameters $m^{2}(0,\pi)$, $m^{2}(\pi,0)$, $m^{2}(\pi/2,\pi/2)$ and make
the finite-size extrapolations to estimate the results in the thermodynamic
limit. Notice that we only consider the results on the $L_{y}=4,8,12$ systems,
which are compatible with the structure factor peak at
$\vec{k}=(\pi/2,\pi/2)$.

Figure 8: Analyses of magnetic order parameters at
$J_{2}/J_{1}=J_{\chi}/J_{1}=1.0$. (a) Bond dimension dependence of the
magnetic order parameters $m^{2}(0,\pi)$, $m^{2}(\pi,0)$, and
$m^{2}(\pi/2,\pi/2)$ with bond dimensions $M=2000$, $4000$, and $6000$ on the
$L_{y}=12$ cylinder. We estimate the error bar of the extrapolation by using
the $M=6000$ result and the linearly extrapolated result by using $M=4000$ and
$6000$ data (the dashed line). (b) Finite-size scaling of the magnetic order
parameters $m^{2}(0,\pi)$, $m^{2}(\pi,0)$, and $m^{2}(\pi/2,\pi/2)$ with
$L_{y}=4,8,12$. The error bars of the $L_{y}=12$ data determined from (a) are
also shown, which are very small.

We first examine the coupling dependence of the spin structure factor at
$\vec{k}=(0,\pi)$, $(\pi,0)$, and $(\pi/2,\pi/2)$ on the $L_{y}=8$ system (see
Fig. 15 in Appendix D), which shows us the parameter points with the strongest
structure factors on this finite-size system. Furthermore, we make the finite-
size scaling of these magnetic order parameters at the parameter points with
strong spin structure factor. Due to the computation cost of the complex
wavefunction, we keep the bond dimensions up to $6000$ SU(2) multiplets, which
can give the well converged results for $L_{y}=4$ and $8$, but less converged
results for $L_{y}=12$. Therefore, we analyze the bond dimension dependence of
the magnetic order parameters for $L_{y}=12$ to estimate the results in the
infinite-bond-dimension limit, as shown in Fig. 8(a) for
$J_{2}/J_{1}=J_{\chi}/J_{1}=1.0$. We keep $M=2000$, $4000$, and $6000$ SU(2)
multiplets to obtain the magnetic order parameters, and plot the results
versus $1/M$. While $m^{2}(0,\pi)$ and $m^{2}(\pi/2,\pi/2)$ seem to approach
convergence with bond dimension, the extrapolated $m^{2}(\pi,0)$ may have the
relatively larger error bar. We estimate the error bar of the extrapolation by
using the $M=6000$ result and the linearly extrapolated result by considering
$M=4000$ and $6000$, which are taken as the lower and upper boundaries of the
$M\rightarrow\infty$ result. In Fig. 8(b), we make the finite-size scaling for
the three order parameters with $L_{y}=4$, $8$, $12$, which all decay fast
with system width and are smoothly extrapolated to vanished. Notice that since
the error bars of the $L_{y}=12$ results are quite small, the uncertainties of
the results do not change our conclusion. We have also checked the size-
scaling results at $J_{2}/J_{1}=0.8$, $J_{\chi}/J_{1}=0.7$ and
$J_{2}/J_{1}=0.8$, $J_{\chi}/J_{1}=1.0$ (see Fig. 16 in Appendix D), which all
agree with no magnetic order. Therefore, our finite-size scaling analyses of
the magnetic order parameters suggest no magnetic ordering in this regime,
which could be consistent with the ground state that appears like a CSL state
in the vacuum topological sector, as characterized by the entanglement
spectrum in Fig. 7.

Figure 9: Spin structure factor for $J_{1}=0$, $J_{\chi}=1.0$. The structure
factor is obtained from the Fourier transformation of spin correlations of the
middle $L_{y}\times L_{y}$ sites on the $L_{y}=8,L_{x}=32$ cylinder, which
shows round peaks at ${\bf k}=(\pi,0)$, $(0,\pi)$, and
$(\pm\frac{\pi}{2},\pm\frac{\pi}{2})$. These peak momenta agree with that in
the CSS magnetic order state, but the very broad peaks indicate the vanished
magnetic order.

##  V Strong tendency to form a dimer order at large chiral coupling

In this section, we extend our discussion to the large chiral coupling regime,
which is beyond the parameter region of Fig. 1. Classically, the dominant
chiral coupling $J_{\chi}$ in this system will lead to the magnetically
ordered CSS Rabson and Trugman (1995). In a recent DMRG study of the spin-$1$
$J_{1}$-$J_{2}$-$J_{\chi}$ model, the CSS has also been identified in the
large-$J_{\chi}$ regime Huang  _et al._ (2022).

Here, we examine the spin-$1/2$ case. We first measure the spin structure
factor to detect the possible CSS magnetic order. Since the spin pattern of
the CSS order has the translation period of $4$ along both the $x$ and $y$
directions (see Fig. 11 in Appendix A), we choose the system circumference
$L_{y}=4,8$ to accommodate this spin pattern. Indeed, the spin structure
factor $S(\bf{k})$ exhibits peaks at ${\bf k}=(\pi,0),(0,\pi)$ and
$(\pm\frac{\pi}{2},\pm\frac{\pi}{2})$, as shown in Fig. 9 of the pure
$J_{\chi}$ model on the $L_{y}=8$ system. The peak momenta fully agree with
the feature of the CSS state Huang  _et al._ (2022). Nonetheless, one can find
in Fig. 9 that all the peaks of $S(\bf{k})$ are very broad, which indicates
that the CSS magnetic order is likely to be melted by the stronger quantum
fluctuations in the spin-$1/2$ case while this order can persist in the
spin-$1$ case Huang  _et al._ (2022). One possibility of this non-magnetic
state could be the CSL, which however is not supported by the DMRG results.

Figure 10: Logarithmic-linear plot of the horizontal bond dimer order $D_{x}$
versus the distance $d$ away from the open boundary. The system is the pure
$J_{\chi}$ model with $J_{\chi}=1.0$, $J_{1}=J_{2}=0$ on the
$L_{y}=4,L_{x}=24$, $L_{y}=8,L_{x}=32$, and $L_{y}=12,L_{x}=48$ cylinders. The
exponential fitting of $D_{x}\sim e^{-d/\xi_{x}}$ gives the decay length
$\xi_{x}\approx 2.02$ and $5.09$ for the $L_{y}=4$ and $8$ cylinders,
respectively. The $L_{y}=12$ results are obtained with the bond dimensions
$M=5000$.

Therefore, we turn to explore a possible translational symmetry breaking for
large-$J_{\chi}$ coupling. To accommodate the short-range spin configurations
as shown in Fig. 9, we study the $L_{y}=4,8,12$ cylinders. However, at large
$J_{\chi}$, the DMRG calculations are harder to converge, which makes the good
convergence for $L_{y}=12$ more difficult. Thus, we will mainly rely on the
well converged $L_{y}=4,8$ results and also refer to the less converged
results for $L_{y}=12$.

We study the decay length of the boundary-induced dimer order and its
circumference dependence, which has been found efficient to identify a bond
dimer state even if the dimer order is very weak Sandvik (2012); Gong  _et
al._ (2014b). Here, we briefly introduce the strategy of this method. Since
our DMRG calculations are performed on the cylinder geometry with open
boundary conditions, the translational symmetry along the $x$ direction has
already been broken, which leads to a dimer order of the bond energy
$\langle{\bf S}_{i}\cdot{\bf S}_{j}\rangle$. We define the dimer order
parameter $D_{x}$ as the bond energy difference between the two neighboring
horizontal bonds

$D_{x}(d)=\langle{\bf S}_{(d,y)}\cdot{\bf S}_{(d+1,y)}\rangle-\langle{\bf
S}_{(d+1,y)}\cdot{\bf S}_{(d+2,y)}\rangle,$ (4)

where ${\bf S}_{(d,y)}$ denotes the spin on the position $(d,y)$. Therefore,
on finite-size systems $D_{x}$ would decay exponentially from the edge to the
bulk as $D_{x}\sim e^{-d/\xi_{x}}$. If the ground state is a dimer order state
in two dimensions, the decay length $\xi_{x}$ would grow rapidly with
increased system circumference and eventually diverge in a certain width when
spontaneous symmetry broken can happen. Otherwise, in a state without a dimer
order, the decay length $\xi_{x}$ would be a finite number in the
thermodynamic limit.

Following this strategy, we compute the dimer order $D_{x}$ on different
systems. In Fig. 10, we show the log-linear plot of $D_{x}$ for
$J_{\chi}=1.0$, $J_{1}=0$ on different sizes. On the smaller $L_{y}=4,8$
systems, $D_{x}$ decays exponentially and $\xi_{x}$ grows rapidly from
$\xi_{x}\simeq 2.02$ to $5.09$. On the wider $L_{y}=12$ system, we push the
calculations up to $5000$ SU(2) multiplets, and the results show a strong
dimer order. Although the $L_{y}=12$ results are less converged, the strong
dimer order and the quickly growing $\xi_{x}$ from $L_{y}=4$ to $L_{y}=8$
strongly suggest a dimer order in two-dimensional limit. This tendency to
develop a dimer order at large chiral coupling is also suggested by the
growing $\xi_{x}$ with increased $J_{\chi}$ coupling (see Fig. 17 in Appendix
E). While the CSS is robust in the spin-$1$ system, the magnetic order seems
to give way to a magnetically disordered dimer state in the spin-$1/2$ system
due to the stronger quantum fluctuations.

##  VI Summary and discussion

We have studied the quantum phase diagram of the spin-$1/2$
$J_{1}$-$J_{2}$-$J_{\chi}$ model on the square lattice using DMRG
calculations. This model was expected to give some insight for the giant
thermal Hall conductivity in cuprate superconductors Samajdar  _et al._
(2019a, b); Zhang  _et al._ (2021); Merino and Ralko (2022); Han  _et al._
(2019); Li and Lee (2019); Li (2019). By mean-field calculation, a phase
coexistence of Néel AFM order and topological CSL was proposed in this model,
which could be used to explain the enhanced thermal Hall response in some
cuprate materials Samajdar  _et al._ (2019a); Zhang  _et al._ (2021). On the
other hand, a recent DMRG study of the spin-$1$ $J_{1}$-$J_{2}$-$J_{\chi}$
model identifies the phase coexistence of stripe AFM order and non-Abelian
CSL. The reported coexistence of magnetic order and topological order
motivates us to examine the quantum phases and possible phase coexistence in
this spin-$1/2$ model.

By tuning the interactions $0\leq J_{2}/J_{1}\leq 1.0$ and $0\leq
J_{\chi}/J_{1}\leq 1.5$, we identify the Néel AFM phase, stripe AFM phase, CSL
phase, and a magnetic disorder regime, as shown in Fig. 1. For the
intermediate non-magnetic region $J_{2}/J_{1}\simeq 0.5$ of the
$J_{1}$-$J_{2}$ model, we find that a small chiral coupling can drive the
emergence of CSL state. However, the related phase transition requires
numerical studies of much larger length scales, which is beyond the scope of
this work.

With growing chiral coupling, we identify the phase transition with vanishing
the Néel AFM order by examining the Néel order parameter $m^{2}(\pi,\pi)$ and
scalar chiral order parameter $\langle\chi\rangle$. Above this phase boundary,
we can verify the Kalmeyer-Laughlin CSL phase by the characteristic features
including the quantized topological entanglement entropy $\gamma=(\ln 2)/2$,
chiral entanglement spectrum agreeing with the SU(2)1 conformal field theory,
and spin Chern number $C=1/2$. Below the phase boundary when the Néel order
exists, we explore the possible phase coexistence. Nonetheless, either the
entanglement spectrum or Chern number result does not support a coexistent
topological order. Even near the phase boundary with a very weak Néel order,
the entanglement spectrum does not form the chiral structure of the CSL but
exhibits an evolution towards the spectrum of a CSL, which suggests no phase
coexistence between the Néel and CSL phase.

For the stripe phase at larger $J_{2}/J_{1}$, we find two regimes when the
stripe AFM order is suppressed by increased chiral coupling. One regime is
identified as the same CSL phase, and the other one is a magnetic disorder
regime. In the stripe and CSL phase, our DMRG results do not support
coexistence of magnetic and topological order. In the magnetic disorder
regime, the spin structure factor shows broad peaks at ${\bf k}=(0,\pi)$,
$(\pi,0)$ and $(\pm\pi/2,\pm\pi/2$), which agrees with the spin configuration
of the CSS magnetic order. However, finite-size scaling of magnetic order
parameters suggests no magnetic ordering in the thermodynamic limit. On the
other hand, the entanglement spectrum of the ground state exhibits the
quasidegenerate levels that is consistent with the CSL. Therefore, the
obtained ground state appears like the vacuum-sector ground state of a CSL.
The absent spinon sector in the flux insertion simulation implies that the
state corresponding to the ground state in the spinon sector may intersect
with higher-energy states. The exact nature of this magnetic disorder regime
needs further studies on larger system size or by investigating other
properties.

Beyond the phase diagram Fig. 1, we also study the $J_{\chi}$-dominant regime.
While the spin-$1$ $J_{1}$-$J_{2}$-$J_{\chi}$ model still possesses the CSS
state in this regime, this magnetic order seems to give way to a disordered
dimer state in the studied spin-$1/2$ system due to the stronger quantum
fluctuations. Our results show the striking differences between the spin-1/2
and spin-1 systems, and provide numerical insights for further understanding
on the coexistence of conventional order and topological order.

Note added. At the stage of finalizing our work, we became aware of an article
by Yang et al. Yang  _et al._ (2024), who studied the same model by using
numerical simulations. Both of the two works consistently find the Néel AFM,
stripe AFM, chiral spin liquid, and magnetic disorder (named as nematic spin
liquid in their paper) phases. While Ref. Yang  _et al._ (2024) proposes a
magnetically ordered chiral spin state based on the enhanced spin structure
factor, our finite-size extrapolation results suggest that the chiral spin
state is unlikely in the studied parameter regime.

###### Acknowledgements.

We acknowledge the stimulating discussions with Jian-Wei Yang, Wei Zhu, and
Ling Wang. X.T.Z. and S.S.G. were supported by the National Natural Science
Foundation of China Grants No. 11874078 and No. 11834014. Y.H. was supported
by the U.S. DOE NNSA under Contract No. 89233218CNA000001 and by the Center
for Integrated Nanotechnologies, a DOE BES user facility, in partnership with
the LANL Institutional Computing Program for computational resources. H.Q.W.
was supported by Guangdong Provincial Key Laboratory of Magnetoelectric
Physics and Devices (Grant No. 2022B1212010008) and Guangzhou Basic and
Applied Basic Research Foundation (Grant No.202201011569). D.N.S. was
supported by National Science Foundation through the Partnership in Research
and Education in Materials Grant DMR-1828019. S.S.G. and H.Q.W. also
acknowledge the support from the Dongguan Key Laboratory of Artificial
Intelligence Design for Advanced Materials.

##  Appendix A Magnetically ordered chiral spin state

To demonstrate the magnetic chiral spin state (CSS), we plot the spin
configuration of the classical CSS in Fig. 11, where the different arrows
denote the spins pointing to the different directions. One can find that the
spin configuration has the translation period of $4$ along both the $x$ and
$y$ directions. Classically, the square-lattice system with only the three-
spin scalar chiral interaction $J_{\chi}{\bf S}_{i}\cdot({\bf S}_{j}\times{\bf
S}_{k})$ can lead to this CSS.

In the quantum case, the spin-$1$ square-lattice $J_{1}$-$J_{2}$-$J_{\chi}$
model has been studied by DMRG calculation Huang  _et al._ (2022), which
focuses on the intermediate $0.4\leq J_{2}/J_{1}\leq 0.6$. In this region, a
relatively small chiral coupling $J_{\chi}/J_{1}\gtrsim 0.4$ can induce a
robust CSS magnetic order Huang  _et al._ (2022).

Figure 11: Spin configuration of the classical magnetic chiral spin state. The
arrows denote the spins pointing to the different directions.

##  Appendix B More data about phase transition

In Fig. 2 of the main text, we have shown the DMRG results for determining the
transition with vanishing the Néel AFM order at $J_{2}=0$. Here, we
demonstrate more DMRG data to support the similar transition with increasing
$J_{\chi}$ at finite $J_{2}$ in Fig. 12, where the derivative of the bulk
chiral order $\langle\chi\rangle$ with respect to $J_{\chi}/J_{1}$ clearly
characterizes the phase transition.

Figure 12: The bulk chiral order $\langle\chi\rangle$ versus $J_{\chi}/J_{1}$
and its derivative $\partial\langle\chi\rangle/\partial J_{\chi}$ for
$J_{2}/J_{1}=0.1$ and $0.3$ on the $L_{y}=6$ systems.

##  Appendix C More data of entanglement spectrum and adiabatic flux insertion
results in the chiral spin liquid phase

In Fig. 3 and Fig. 4 of the main text, we have shown the DMRG results to
identify the CSL state at $J_{\chi}/J_{1}=1.0$, $J_{2}=0$. Here, we
demonstrate additional DMRG data to support the CSL phase at $J_{2}\neq 0$ in
Fig. 13, for $J_{2}/J_{1}=0.3$, $J_{\chi}/J_{1}=0.7$ on the $L_{y}=8$
cylinder. The entanglement spectrum versus flux and the quantized spin Chern
number $C=1/2$ identify the CSL state with double-degenerate ground states.

In the intermediate $J_{2}/J_{1}\simeq 0.5$ regime, the $J_{1}$-$J_{2}$ model
is a non-magnetic state Liu  _et al._ (2022). With growing chiral coupling, we
find that the system will be easily driven to the CSL phase by a small chiral
coupling. In Fig. 14, we show the entanglement spectrum and adiabatic flux
insertion results for $J_{2}/J_{1}=0.5$, $J_{\chi}/J_{1}=0.1$ and $0.2$. While
the DMRG results at $J_{\chi}/J_{1}=0.1$ do not support a CSL, the CSL
evidence is robust at $J_{\chi}/J_{1}=0.2$. By using the similar simulations,
we identify the CSL state in the intermediate $J_{2}/J_{1}$ regime with a
small chiral coupling, as shown in Fig. 1.

Figure 13: Adiabatic flux insertion simulation and quantized spin Chern number
for $J_{2}/J_{1}=0.3$, $J_{\chi}/J_{1}=0.7$ on the $L_{y}=8$ cylinder. (a)
Entanglement spectrum flow with adiabatically inserted flux $\theta$, which is
obtained by iDMRG smulation with $1000$ bond dimensions. The eigenvalues in
the $S_{z}=0,\pm 1$ sectors are labelled by different symbols. (b) Spin
pumping with adiabatically inserted flux $\theta$, which is obtained by iDMRG
smulation with $3000$ bond dimensions. In a period of inserted flux
$\theta=0\rightarrow 2\pi$, a quantized magnetization moments $0.5$ is
accumulated at one open boundary of the cylinder, indicating a fractionally
quantized spin Chern number $C=1/2$.

Figure 14: Entanglement spectrum and adiabatic flux insertion simulations in
the intermediate $J_{2}/J_{1}$ regime. (a) and (b) show the entanglement
spectrum labeled by the quantum numbers of total spin $S^{z}=0$ and relative
momentum along the $y$ direction $\triangle{k_{y}}$ for the ground states of
$J_{2}/J_{1}=0.5$, $J_{\chi}/J_{1}=0.1$ and $0.2$, respectively, on the
$L_{y}=8$ cylinder systems. $\lambda_{i}$ are the eigenvalues of reduced
density matrix. The results are obtained by keeping $2000$ U(1) states. The
insets show the spin pumping with adiabatically inserted flux $\theta$. The
results at $J_{\chi}/J_{1}=0.2$ characterize the ground state as the CSL
state.

##  Appendix D Analyses of magnetic order parameters

In order to explore a possible weak CSS magnetic order in the magnetic
disorder regime shown in Fig. 1, we have studied spin structure factor
$S(0,\pi)$, $S(\pi,0)$ and $S(\pi/2,\pi/2)$ for $0.6\leq J_{2}/J_{1}\leq 1.0$
and $0\leq J_{\chi}/J_{1}\leq 1.5$ on the $L_{y}=8$ cylinder, as shown in Fig.
15. In Fig. 15(a), $S(0,\pi)$ generally becomes weaker with increased chiral
coupling. While $S(0,\pi)$ is strongly suppressed with vanishing stripe
magnetic order, $S(\pi,0)$ shows the relatively stronger intensity for
$0.8\lesssim J_{2}/J_{1}\lesssim 1.0$ and $0.7\lesssim J_{\chi}/J_{1}\lesssim
1.3$ as shown in Fig. 15(b). Notice that in this parameter region, $S(0,\pi)$
and $S(\pi,0)$ have the similar magnitudes. On the contrary, as demonstrated
in Fig. 15(c), $S(\pi/2,\pi/2)$ is slightly enhanced with increased chiral
coupling, in particular when the stripe order is suppressed.

To make finite-size scaling for magnetic order parameters, we choose three
parameter points in the region where the spin structure factors $S(0,\pi)$,
$S(\pi,0)$ and $S(\pi/2,\pi/2)$ are all relatively strong on the 8-leg system.
In Fig. 8, we have shown the finite-size scaling of magnetic order parameters
at $J_{2}/J_{1}=J_{\chi}/J_{1}=1.0$. Here, we show the results for
$J_{2}/J_{1}=0.8$, $J_{\chi}/J_{1}=0.7,1.0$ in Fig. 16. Similar to the results
in Fig. 8, the finite-size scaling of the magnetic order parameters in Fig. 16
all decays fast with system width and is smoothly extrapolated to vanished,
indicating the absent magnetic order.

Figure 15: Coupling dependence of spin structure factors on the 8-leg systems.
(a)-(c) show the results $S(0,\pi)$, $S(\pi,0)$, and $S(\pi/2,\pi/2)$,
respectively.

Figure 16: Finite-size scaling of the magnetic order parameters
$m^{2}(0,\pi)$, $m^{2}(\pi,0)$ and $m^{2}(\pi/2,\pi/2)$ on the $L_{y}=4,8,12$
systems for (a) $J_{2}/J_{1}=0.8$, $J_{\chi}/J_{1}=0.7$, and (b)
$J_{2}/J_{1}=0.8$, $J_{\chi}/J_{1}=1.0$. The results for $L_{y}=12$ are
obtained by keeping $4000$ SU(2) multiplets.

##  Appendix E Coupling dependence of the dimer order decay length

In Fig. 10, we show the log-linear plot of the horizontal dimer order
parameter $D_{x}$ for the pure $J_{\chi}$ model on different system widths,
which demonstrates a strong tendency of the system to develop a bond dimer
order.

Here, we supplement the results at $J_{2}/J_{1}=0.5$ with different $J_{\chi}$
couplings on the $L_{y}=12$ cylinder. As shown in Fig. 17, the decay length
$\xi_{x}$ of the dimer order parameter $D_{x}$ keeps growing with increased
chiral coupling, which consistently suggests the stronger tendency to develop
a dimer order at larger chiral coupling.

Figure 17: Logarithmic-linear plot of the horizontal dimer order parameter
$D_{x}$ versus the distance $d$ for $J_{2}/J_{1}=0.5$ and different
$J_{\chi}/J_{1}$ on the $L_{y}=12$ cylinder. $d$ is the distance of the bond
from the open edge. The exponential fitting of $D_{x}\sim e^{-d/\xi_{x}}$
gives the decay length $\xi_{x}$. The results are obtained by keeping $4000$
SU(2) multiplets.

## References

  * Savary and Balents (2016) Lucile Savary and Leon Balents, “Quantum spin liquids: a review,” Reports on Progress in Physics 80, 016502 (2016).
  * Zhou  _et al._ (2017) Yi Zhou, Kazushi Kanoda, and Tai-Kai Ng, “Quantum spin liquid states,” Rev. Mod. Phys. 89, 025003 (2017).
  * Broholm  _et al._ (2020) C. Broholm, R. J. Cava, S. A. Kivelson, D. G. Nocera, M. R. Norman, and T. Senthil, “Quantum spin liquids,” Science 367 (2020), 10.1126/science.aay0668.
  * Lancaster (2023) T. Lancaster, “Quantum spin liquids,” (2023), arXiv:2310.19577 [cond-mat.str-el] .
  * Kivelson and Sondhi (2023) Steven Kivelson and Shivaji Sondhi, “50 years of quantum spin liquids,” Nature Reviews Physics 5, 368–369 (2023).
  * Wen (1990) X.-G. Wen, “Topological orders in rigid states,” Int. J. Mod. Phys. B 4, 239 (1990).
  * Wen (1991) X. G. Wen, “Mean-field theory of spin-liquid states with finite energy gap and topological orders,” Phys. Rev. B 44, 2664 (1991).
  * Senthil and Fisher (2000) T. Senthil and M. P. A. Fisher, “${Z}_{2}$ gauge theory of electron fractionalization in strongly correlated systems,” Phys. Rev. B 62, 7850 (2000).
  * Senthil and Fisher (2001) T. Senthil and M. P. A. Fisher, “Fractionalization in the cuprates: Detecting the topological order,” Phys. Rev. Lett. 86, 292 (2001).
  * Chen  _et al._ (2010) Xie Chen, Zheng-Cheng Gu, and Xiao-Gang Wen, “Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order,” Phys. Rev. B 82, 155138 (2010).
  * Kalmeyer and Laughlin (1987) V. Kalmeyer and R. B. Laughlin, “Equivalence of the resonating-valence-bond and fractional quantum hall states,” Phys. Rev. Lett. 59, 2095–2098 (1987).
  * Wen  _et al._ (1989) X. G. Wen, Frank Wilczek, and A. Zee, “Chiral spin states and superconductivity,” Phys. Rev. B 39, 11413–11423 (1989).
  * Baskaran (1989) G. Baskaran, “Novel local symmetries and chiral-symmetry-broken phases in s=1/2 triangular-lattice heisenberg model,” Phys. Rev. Lett. 63, 2524–2527 (1989).
  * Yang  _et al._ (1993) Kun Yang, L. K. Warman, and S. M. Girvin, “Possible spin-liquid states on the triangular and kagomé lattices,” Phys. Rev. Lett. 70, 2641–2644 (1993).
  * Haldane and Arovas (1995) F. D. M. Haldane and Daniel P. Arovas, “Quantized spin currents in two-dimensional chiral magnets,” Phys. Rev. B 52, 4223–4225 (1995).
  * Wilczek (1990) Frank Wilczek,  _Fractional statistics and anyon superconductivity_ , Vol. 5 (World scientific, 1990).
  * He  _et al._ (2014) Yin-Chen He, D. N. Sheng, and Yan Chen, “Chiral spin liquid in a frustrated anisotropic kagome heisenberg model,” Phys. Rev. Lett. 112, 137202 (2014).
  * Gong  _et al._ (2014a) Shou-Shu Gong, Wei Zhu, and D. N. Sheng, “Emergent chiral spin liquid: Fractional quantum hall effect in a kagome heisenberg model,” Scientific Reports 4 (2014a), 10.1038/srep06317.
  * Wietek  _et al._ (2015) Alexander Wietek, Antoine Sterdyniak, and Andreas M. Läuchli, “Nature of chiral spin liquids on the kagome lattice,” Phys. Rev. B 92, 125122 (2015).
  * Hu  _et al._ (2015) Wen-Jun Hu, Wei Zhu, Yi Zhang, Shoushu Gong, Federico Becca, and D. N. Sheng, “Variational monte carlo study of a chiral spin liquid in the extended heisenberg model on the kagome lattice,” Phys. Rev. B 91, 041124 (2015).
  * Gong  _et al._ (2015) Shou-Shu Gong, Wei Zhu, Leon Balents, and D. N. Sheng, “Global phase diagram of competing ordered and quantum spin-liquid phases on the kagome lattice,” Phys. Rev. B 91, 075112 (2015).
  * Hickey  _et al._ (2017) Ciarán Hickey, Lukasz Cincio, Zlatko Papić, and Arun Paramekanti, “Emergence of chiral spin liquids via quantum melting of noncoplanar magnetic orders,” Phys. Rev. B 96, 115115 (2017).
  * Gong  _et al._ (2017) Shou-Shu Gong, W. Zhu, J.-X. Zhu, D. N. Sheng, and Kun Yang, “Global phase diagram and quantum spin liquids in a spin-$\frac{1}{2}$ triangular antiferromagnet,” Phys. Rev. B 96, 075116 (2017).
  * Wietek and Läuchli (2017) Alexander Wietek and Andreas M. Läuchli, “Chiral spin liquid and quantum criticality in extended $s=\frac{1}{2}$ heisenberg models on the triangular lattice,” Phys. Rev. B 95, 035141 (2017).
  * Nielsen  _et al._ (2013) Anne E. B. Nielsen, Germán Sierra, and J. Ignacio Cirac, “Local models of fractional quantum hall states in lattices and physical implementation,” Nature Communications 4, 2864 (2013).
  * Chen  _et al._ (2021) Ji-Yao Chen, Jheng-Wei Li, Pierre Nataf, Sylvain Capponi, Matthieu Mambrini, Keisuke Totsuka, Hong-Hao Tu, Andreas Weichselbaum, Jan von Delft, and Didier Poilblanc, “Abelian ${{\mathrm{SU(N)}_{1}}}$ chiral spin liquids on the square lattice,” Phys. Rev. B 104, 235104 (2021).
  * Szasz  _et al._ (2020) Aaron Szasz, Johannes Motruk, Michael P. Zaletel, and Joel E. Moore, “Chiral spin liquid phase of the triangular lattice hubbard model: A density matrix renormalization group study,” Phys. Rev. X 10, 021042 (2020).
  * Boos  _et al._ (2020) C. Boos, C. J. Ganahl, M. Lajkó, P. Nataf, A. M. Läuchli, K. Penc, K. P. Schmidt, and F. Mila, “Time-reversal symmetry breaking abelian chiral spin liquid in mott phases of three-component fermions on the triangular lattice,” Phys. Rev. Res. 2, 023098 (2020).
  * Szasz and Motruk (2021) Aaron Szasz and Johannes Motruk, “Phase diagram of the anisotropic triangular lattice hubbard model,” Phys. Rev. B 103, 235132 (2021).
  * Cookmeyer  _et al._ (2021) Tessa Cookmeyer, Johannes Motruk, and Joel E. Moore, “Four-spin terms and the origin of the chiral spin liquid in mott insulators on the triangular lattice,” Phys. Rev. Lett. 127, 087201 (2021).
  * Chen  _et al._ (2022) Bin-Bin Chen, Ziyu Chen, Shou-Shu Gong, D. N. Sheng, Wei Li, and Andreas Weichselbaum, “Quantum spin liquid with emergent chiral order in the triangular-lattice hubbard model,” Phys. Rev. B 106, 094420 (2022).
  * Jiang and Jiang (2020) Yi-Fan Jiang and Hong-Chen Jiang, “Topological superconductivity in the doped chiral spin liquid on the triangular lattice,” Phys. Rev. Lett. 125, 157002 (2020).
  * Huang and Sheng (2022) Yixuan Huang and D. N. Sheng, “Topological chiral and nematic superconductivity by doping mott insulators on triangular lattice,” Phys. Rev. X 12, 031009 (2022).
  * Huang  _et al._ (2023) Yixuan Huang, Shou-Shu Gong, and D. N. Sheng, “Quantum phase diagram and spontaneously emergent topological chiral superconductivity in doped triangular-lattice mott insulators,” Phys. Rev. Lett. 130, 136003 (2023).
  * Grissonnanche  _et al._ (2019) G. Grissonnanche, A. Legros, S. Badoux, E. Lefrançois, V. Zatko, M. Lizaire, F. Laliberté, A. Gourgout, J.-S. Zhou, S. Pyon, T. Takayama, H. Takagi, S. Ono, N. Doiron-Leyraud, and L. Taillefer, “Giant thermal hall conductivity in the pseudogap phase of cuprate superconductors,” Nature 571, 376–380 (2019).
  * Grissonnanche  _et al._ (2020) G Grissonnanche, S Thériault, A Gourgout, M-E Boulanger, E Lefrançois, A Ataei, F Laliberté, M Dion, J-S Zhou, S Pyon, _et al._ , “Chiral phonons in the pseudogap phase of cuprates,” Nature Physics 16, 1108–1111 (2020).
  * Boulanger  _et al._ (2020) Marie-Eve Boulanger, Gaël Grissonnanche, Sven Badoux, Andréanne Allaire, Étienne Lefrançois, Anaëlle Legros, Adrien Gourgout, Maxime Dion, CH Wang, XH Chen, _et al._ , “Thermal hall conductivity in the cuprate mott insulators $\mathrm{Nd_{2}CuO_{4}}$ and $\mathrm{Sr_{2}CuO_{2}Cl_{2}}$,” Nature communications 11, 5325 (2020).
  * Boulanger  _et al._ (2022) Marie-Eve Boulanger, Gaël Grissonnanche, Étienne Lefrançois, Adrien Gourgout, Ke-Jun Xu, Zhi-Xun Shen, Richard L. Greene, and Louis Taillefer, “Thermal hall conductivity of electron-doped cuprates,” Phys. Rev. B 105, 115101 (2022).
  * Boulanger  _et al._ (2023) Marie-Eve Boulanger, Lu Chen, Vincent Oliviero, David Vignolles, Gaël Grissonnanche, Kejun Xu, Zhi-Xun Shen, Cyril Proust, Jordan Baglo, and Louis Taillefer, “Thermal hall conductivity of electron-doped cuprates: Electrons and phonons,” (2023), arXiv:2310.15892 [cond-mat.str-el] .
  * Samajdar  _et al._ (2019a) Rhine Samajdar, Mathias S Scheurer, Shubhayu Chatterjee, Haoyu Guo, Cenke Xu, and Subir Sachdev, “Enhanced thermal hall effect in the square-lattice néel state,” Nature Physics 15, 1290–1294 (2019a).
  * Samajdar  _et al._ (2019b) Rhine Samajdar, Shubhayu Chatterjee, Subir Sachdev, and Mathias S. Scheurer, “Thermal hall effect in square-lattice spin liquids: A schwinger boson mean-field study,” Phys. Rev. B 99, 165126 (2019b).
  * Han  _et al._ (2019) Jung Hoon Han, Jin-Hong Park, and Patrick A. Lee, “Consideration of thermal hall effect in undoped cuprates,” Phys. Rev. B 99, 205157 (2019).
  * Li and Lee (2019) Zi-Xiang Li and Dung-Hai Lee, “The thermal hall conductance of two doped symmetry-breaking topological insulators,” (2019), arXiv:1905.04248 [cond-mat.str-el] .
  * Li (2019) Tao Li, “Theory of the giant thermal hall effect in the high temperature superconductors,” (2019), arXiv:1906.01712 [cond-mat.str-el] .
  * Zhang  _et al._ (2021) Yunchao Zhang, Yanting Teng, Rhine Samajdar, Subir Sachdev, and Mathias S. Scheurer, “Phonon hall viscosity from phonon-spinon interactions,” Phys. Rev. B 104, 035103 (2021).
  * Merino and Ralko (2022) Jaime Merino and Arnaud Ralko, “Majorana chiral spin liquid in a model for mott insulating cuprates,” Phys. Rev. Res. 4, 023122 (2022).
  * Varma (2020) Chandra M. Varma, “Thermal hall effect in the pseudogap phase of cuprates,” Phys. Rev. B 102, 075113 (2020).
  * Guo and Sachdev (2021) Haoyu Guo and Subir Sachdev, “Extrinsic phonon thermal hall transport from hall viscosity,” Phys. Rev. B 103, 205115 (2021).
  * Ye  _et al._ (2021) Mengxing Ye, Lucile Savary, and Leon Balents, “Phonon hall viscosity in magnetic insulators,” (2021), arXiv:2103.04223 [cond-mat.str-el] .
  * Mangeolle  _et al._ (2022a) Léo Mangeolle, Leon Balents, and Lucile Savary, “Phonon thermal hall conductivity from scattering with collective fluctuations,” Phys. Rev. X 12, 041031 (2022a).
  * Mangeolle  _et al._ (2022b) Léo Mangeolle, Leon Balents, and Lucile Savary, “Thermal conductivity and theory of inelastic scattering of phonons by collective fluctuations,” Phys. Rev. B 106, 245139 (2022b).
  * Flebus and MacDonald (2022) B. Flebus and A. H. MacDonald, “Charged defects and phonon hall effects in ionic crystals,” Phys. Rev. B 105, L220301 (2022).
  * Sun  _et al._ (2022) Xiao-Qi Sun, Jing-Yuan Chen, and Steven A. Kivelson, “Large extrinsic phonon thermal hall effect from resonant scattering,” Phys. Rev. B 106, 144111 (2022).
  * Liu  _et al._ (2016) Xiong-Jun Liu, Zheng-Xin Liu, K T Law, W Vincent Liu, and T K Ng, “Chiral topological orders in an optical raman lattice,” New Journal of Physics 18, 035004 (2016).
  * Huang  _et al._ (2022) Yixuan Huang, W. Zhu, Shou-Shu Gong, Hong-Chen Jiang, and D. N. Sheng, “Coexistence of non-abelian chiral spin liquid and magnetic order in a spin-1 antiferromagnet,” Phys. Rev. B 105, 155104 (2022).
  * Rabson and Trugman (1995) David A Rabson and Stuart A Trugman, “A spin model for investigating chirality,” Journal of Physics: Condensed Matter 7, 9005 (1995).
  * White (1992) Steven R. White, “Density matrix formulation for quantum renormalization groups,” Phys. Rev. Lett. 69, 2863–2866 (1992).
  * McCulloch and Gulácsi (2002) I. P McCulloch and M Gulácsi, “The non-abelian density matrix renormalization group algorithm,” Europhysics Letters (EPL) 57, 852–858 (2002).
  * Hauschild and Pollmann (2018) Johannes Hauschild and Frank Pollmann, “Efficient numerical simulations with Tensor Networks: Tensor Network Python (TeNPy),” SciPost Phys. Lect. Notes , 5 (2018), code available from https://github.com/tenpy/tenpy, arXiv:1805.00055 .
  * Yan  _et al._ (2011) Simeng Yan, David A Huse, and Steven R White, “Spin-liquid ground state of the s= 1/2 kagome heisenberg antiferromagnet,” Science 332, 1173–1176 (2011).
  * Jiang  _et al._ (2012) Hong-Chen Jiang, Zhenghan Wang, and Leon Balents, “Identifying topological order by entanglement entropy,” Nature Physics 8, 902–905 (2012).
  * Eisert  _et al._ (2010) J. Eisert, M. Cramer, and M. B. Plenio, “Colloquium: Area laws for the entanglement entropy,” Reviews of Modern Physics 82, 277–306 (2010).
  * Kitaev and Preskill (2006) Alexei Kitaev and John Preskill, “Topological entanglement entropy,” Phys. Rev. Lett. 96, 110404 (2006).
  * Levin and Wen (2006) Michael Levin and Xiao-Gang Wen, “Detecting topological order in a ground state wave function,” Phys. Rev. Lett. 96, 110405 (2006).
  * Li and Haldane (2008) Hui Li and F. D. M. Haldane, “Entanglement spectrum as a generalization of entanglement entropy: Identification of topological order in non-abelian fractional quantum hall effect states,” Phys. Rev. Lett. 101, 010504 (2008).
  * Qi  _et al._ (2012) Xiao-Liang Qi, Hosho Katsura, and Andreas W. W. Ludwig, “General relationship between the entanglement spectrum and the edge state spectrum of topological quantum states,” Phys. Rev. Lett. 108, 196402 (2012).
  * Cincio and Vidal (2013) L. Cincio and G. Vidal, “Characterizing topological order by studying the ground states on an infinite cylinder,” Phys. Rev. Lett. 110, 067208 (2013).
  * Zaletel  _et al._ (2013) Michael P. Zaletel, Roger S. K. Mong, and Frank Pollmann, “Topological characterization of fractional quantum hall ground states from microscopic hamiltonians,” Phys. Rev. Lett. 110, 236801 (2013).
  * Di Francesco  _et al._ (1997) P. Di Francesco, P. Mathieu, and D. Senechal,  _Conformal Field Theory_, Graduate Texts in Contemporary Physics (Springer-Verlag, New York, 1997).
  * Hatsugai (1993) Yasuhiro Hatsugai, “Chern number and edge states in the integer quantum hall effect,” Phys. Rev. Lett. 71, 3697–3700 (1993).
  * Liu  _et al._ (2022) Wen-Yuan Liu, Shou-Shu Gong, Yu-Bin Li, Didier Poilblanc, Wei-Qiang Chen, and Zheng-Cheng Gu, “Gapless quantum spin liquid and global phase diagram of the spin-1/2 $J_{1}-J_{2}$ square antiferromagnetic heisenberg model,” Science Bulletin 67, 1034–1041 (2022).
  * Sandvik (2012) Anders W. Sandvik, “Finite-size scaling and boundary effects in two-dimensional valence-bond solids,” Phys. Rev. B 85, 134407 (2012).
  * Gong  _et al._ (2014b) Shou-Shu Gong, Wei Zhu, D. N. Sheng, Olexei I. Motrunich, and Matthew P. A. Fisher, “Plaquette ordered phase and quantum phase diagram in the spin-$\frac{1}{2}$ ${J}_{1}\text{$-$}{J}_{2}$ square heisenberg model,” Phys. Rev. Lett. 113, 027201 (2014b).
  * Yang  _et al._ (2024) Jianwei Yang, Zhao Liu, and Ling Wang, “Ground state phase diagram and the exotic phases in the spin-$1/2$ square lattice $J_{1}$-$J_{2}$-$J_{\chi}$ model,” arXiv e-prints , arXiv:2401.03434 (2024), arXiv:2401.03434 [cond-mat.str-el] .


#  Understanding Partial $\mathcal{PT}$ symmetry as Weighted Composition
Conjugation in Reproducing Kernel Hilbert Space :An application to Non-
hermitian Bose-Hubbard Type Hamiltonian in Fock space

Arindam Chakraborty
Department of Physics, Heritage Institute of Technology, Kolkata-700107, India
<EMAIL_ADDRESS>

###### Abstract

A new kind of symmetry behaviour introduced as partial
${\mathcal{PT}}$-symmetry(henceforth $\partial_{\mathcal{PT}}$) is
investigated in a typical Fock space setting understood as a Reproducing
Kernel Hilbert Space (RKHS). The same kind of symmetry is understood for a
non-hermitian Bose-Hubbard type Hamiltonian involving two boson operators as
well as its eigenstates. The phenomenon of symmetry breaking has also been
considered.

Keywords: Non-hermitian operator, Bose-Hubbard model, Partial
${\mathcal{PT}}$-symmetry, Fock space, Reproducing Kernel Hilbert Space.

##  1\. Introduction

Non-hermitian Hamiltonian with real eihenvalues in the context of
${\mathcal{PT}}$ symmetry has become an interesting area of investigation for
last couple of decades [1, 2, 3, 4, 5, 6, 7, 8]. The present article stems
from a recent study of partial $\mathcal{PT}$ symmetry by Beygi et. al. [9]
where a variable specific action of symmetry operator is understood
considering a model of an N-coupled harmonic oscillator Hamiltonian with
purely imaginary coupling terms. It has also been observed that the reality
and partial reality of the spectrum have direct correspondences with the
classical trajectories.The present formulation attempts to explore the
possibility of partial $\mathcal{PT}$ symmetry in a Bose-Hubberd hamiltonian
operator[10] as well as in its eigenstates in a typical Fock space
environment. The relevant Fock space [11] has been viewed as a Reproducing
Kernel Hilbert Space [12, 13, 14, 15, 16, 17]and the symetry operators are
understood as Weighted Composition Conjugation [13, 14, 15] acting on it.

We begin with the following definition of Fock space involving functions of
$n$ complex variable.

  * Definition 1.

A Fock (or Segal-Bargmann) space $(\mathcal{F}^{2}({C}^{n}))$ is a separable
complex Hilbert space of entire functions (of the complex variables
$\\{\zeta_{j}:j=1\dots n\\}$ ) equipped with an inner-product

$\displaystyle\langle\psi,\phi\rangle=\prod_{j=1}^{n}\int_{W(\zeta_{j})}\psi(\zeta_{j}:j=1\dots
n)\overline{\phi(\zeta_{j}:j=1\dots n)}$
$\displaystyle\;\;{\rm{with}}\prod_{j=1}^{n}\int_{W(\zeta_{j})}\equiv\int\int
d{W(\zeta_{1})}\dots d{W(\zeta_{n})}.$ (1)

Here, ${d}{W(u)}=\frac{1}{\pi}e^{-|u|^{2}}{d}({\rm{Re}}(u))d({\rm{Im}}(u))$
represents the relevant Gaussian measure relative to the complex variable $u$.

In a Fock space of one complex variable ${\mathcal{PT}}$ symmetry is often
understood as a consequence of the more general notion of weighted composition
conjugation [13] defined as follows.

  * Definition 2.

Let, $\zeta$ is a complex variable and $\\{\vartheta,\eta,\upsilon\\}$ are
complex numbers satisfying the set of necessary and sufficient conditions :
$|\vartheta|=1,\bar{\vartheta}\eta+\bar{\eta}=0$ and
$|\upsilon|^{2}e^{|\eta|^{2}}=1$.The weighted composition conjugation is
defined as

$\displaystyle\mathcal{C}_{(\vartheta,\eta,\upsilon)}\psi(\zeta)=\upsilon
e^{\eta\zeta}\overline{\psi(\overline{\vartheta\zeta+\eta})}.$ (2)

The anti-linear operator $\mathcal{C}_{(\vartheta,\eta,\upsilon)}$ is a
conjugation since it is involutive and isometric. The action of the operator
$\mathcal{PT}$ is equivalent to the choice : $\vartheta=-1=-\upsilon,\eta=0$
which results to the following equation

$\displaystyle\mathcal{C}_{(\vartheta,0,1)}|_{\vartheta=-1}\psi(\zeta)=\overline{\psi(\overline{-\zeta})}.$
(3)

Similarly, the action of $\mathcal{T}$ is indicative of the choice :
$\vartheta=1,\eta=0,\upsilon=1$ giving

$\displaystyle\mathcal{C}_{(\vartheta,0,1)}|_{\vartheta=1}\psi(\zeta)=\overline{\psi(\overline{\zeta})}.$
(4)

If $\psi$ is a function of several complex variables $\\{\zeta_{j}:j=1\dots
n\\}$ one can define an operator
$\mathcal{C}_{(\vartheta_{j},\eta_{j},\upsilon_{j}:j=1\dots n)}$ with the
action

$\displaystyle\mathcal{C}_{(\vartheta_{j},\eta_{j}=0,\upsilon_{j}=1:j=1\dots
n)}\psi(\zeta_{1},\dots,\zeta_{j},\dots,\zeta_{n})=\overline{\psi(\overline{\vartheta_{1}\zeta_{1}},\dots,\overline{\vartheta_{j}\zeta_{j}},\dots,\overline{\vartheta_{n}\zeta_{n}})}.$
(5)

Let us introduce an operator
$\mathcal{C}^{(j)}_{n}=\mathcal{C}_{(\vartheta_{j},\eta_{j}=0,\upsilon_{j}=1;j=1\dots
n)}|_{\vartheta_{1}=1,\dots,\vartheta_{j}=-1,\dots,\vartheta_{n}=1}$ as $j$-th
partial $\mathcal{PT}$ symmetry ($\partial_{\mathcal{PT}}$) operator through
the following action

$\displaystyle\mathcal{C}^{(j)}_{n}\psi(\zeta_{1},\dots,\zeta_{j},\dots,\zeta_{n})=\overline{\psi(\bar{\zeta_{1}},\dots,\overline{-\zeta_{j}},\dots,\bar{\zeta_{n}})}.$
(6)

and a global $\mathcal{PT}$ symmetry operator $\mathcal{C}_{n}$ through the
action

$\displaystyle\mathcal{C}_{n}\psi(\zeta_{1},\dots,\zeta_{j},\dots,\zeta_{n})=\overline{\psi(\overline{-\zeta_{1}},\dots,\overline{-\zeta_{j}},\dots,\overline{-\zeta_{n}})}.$
(7)

For our present purpose we shall only consider the operators $\mathcal{C}_{2}$
and $\\{\mathcal{C}_{2}^{(j)}:j=1,2\\}$. Now, global and partial
$\mathcal{PT}$ symmetries of any function $\psi(\zeta_{1},\zeta_{2})$ are
understood through the following equations

$\displaystyle\mathcal{C}_{2}\psi(\zeta_{1},\zeta_{2})=\psi(\zeta_{1},\zeta_{2})\>\>{\rm
and}\>\>\mathcal{C}_{2}^{(j)}\psi(\zeta_{1},\zeta_{2})=\psi(\zeta_{1},\zeta_{2})\>\>\forall\>\>j=1,2$
(8)

respectively.

##  2\. The model Hamiltonian and $\partial_{\mathcal{PT}}$ symmetry in Fock
space

In the present discussion, following [10] a Bose-Hubbard type Hamiltonian has
been considered. Such a Hamiltonian has been invoked as a two mode version for
a second quantized many particle system showing Bose-Einstein Condensation
(BEC) in a double well potential at low temperature. The said Hamiltonian
becomes non-hermitian if one of the interaction terms present in it is taken
as purely imaginary.The model Hamiltonian under consideration is given by

$\displaystyle
H=\epsilon_{0}(a_{1}^{\dagger}a_{1}-a_{2}^{\dagger}a_{2})+\epsilon(a_{1}^{\dagger}a_{2}+a_{2}^{\dagger}a_{1})+\alpha(a_{1}^{\dagger}a_{1}-a_{2}^{\dagger}a_{2})^{2}.$
(9)

Here $\epsilon_{0}$ represents the on site energy difference, $\epsilon$ is
the single particle tunneling and $\alpha$ stands for the interaction
strength. $\\{a_{j},a_{j}^{\dagger}:j=1,2\\}$ are boson operators satisfying
the condition
$[a_{j},a^{\dagger}_{k}]-\delta_{jk}=[a_{j},a_{k}]=[a^{\dagger}_{j},a^{\dagger}_{k}]=0$.
The Hamiltonian commutes with the number operator
$N=a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}$ indicating particle
conservation. For the time being we consider $\epsilon_{0}=1,\epsilon=i\gamma$
and $\gamma$ and $\alpha$ to be real.

In order to understand $\partial_{\mathcal{PT}}$ symmetry in Fock space we
rewrite the Hamiltonian using Bargmann-Fock correspondence :
$a_{j}^{\dagger}=\zeta_{j},a_{j}=\partial_{\zeta_{j}}$ as follows

$\displaystyle
H=(\zeta_{1}\partial_{\zeta_{1}}-\zeta_{2}\partial_{\zeta_{2}})+i\gamma(\zeta_{1}\partial_{\zeta_{2}}+\zeta_{2}\partial_{\zeta_{1}})+\alpha(\zeta_{1}\partial_{\zeta_{1}}-\zeta_{2}\partial_{\zeta_{2}})^{2}.$
(10)

Following [13, 14, 15] we shall demonstrate the actions of weighted
composition conjugations $\mathcal{C}_{2}$ and $\mathcal{C}_{2}^{(j)}$ on $H$
via the notion of Reproducing Kernel Hilbert Space (RKHS).

  * Definition 3.

A function of the form $K^{[m_{j}]}_{\\{\zeta_{j}\\}}(u_{j}:j=1\dots
n)=\prod_{j=1}^{n}u_{j}^{m_{j}}e^{u_{j}\overline{\zeta_{j}}}$($m_{j}\in{N}$
and $\zeta_{j},u_{j}\in{C}\>\>\forall\>\>j=1\dots n$) is called a kernel
function (or a reproducing kernel) which satisfies the condition

$\displaystyle\psi^{(m_{1},m_{2},\dots,m_{n})}(\zeta_{1},\zeta_{2},\dots,\zeta_{n})=\langle\psi,K^{[m_{j}]}_{\\{\zeta_{1}\\}}\rangle$
$\displaystyle=\int_{W(u_{1})}\int_{W(u_{2})}\dots\int_{W(u_{n})}\psi(u_{1},u_{2},\dots,u_{n})\overline{K^{[m_{j}]}_{\\{\zeta_{1}\\}}}.$
(11)

Considering the case with two complex variables the following proposition is
immediate.

  * Proposition 1.

$H^{\star}\neq H$ where $H^{\star}$ is defined as $\langle
H\psi(u_{1},u_{2}),K^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}\rangle=\langle\psi(u_{1},u_{2}),H^{\star}K^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}\rangle$.

Proof : We shall first show that $\langle
u_{1}\partial_{u_{1}}\psi(u_{1},u_{2}),K^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}\rangle=\langle\psi(u_{1},u_{2}),u_{1}\partial_{u_{1}}K^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}\rangle$.

$\displaystyle\langle
u_{1}\partial_{u_{1}}\psi(u_{1},u_{2}),K^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}\rangle=\langle
u_{1}\partial_{u_{1}}\psi{(u_{1},u_{2})},u_{1}^{m_{1}}u_{2}^{m_{2}}e^{u_{1}\overline{\zeta_{1}}+u_{2}\overline{\zeta_{2}}}\rangle$
$\displaystyle=\int_{W(u_{1})}\int_{W(u_{2})}u_{1}\partial_{u_{1}}\psi(u_{1},u_{2})\overline{u_{1}}^{m_{1}}\overline{u_{2}}^{m_{2}}e^{\overline{u_{1}}{\zeta_{1}}+\overline{u_{2}}{\zeta_{2}}}$
$\displaystyle=\int_{W(u_{1})}\int_{W(u_{2})}u_{1}[\int_{W(v_{1})}\int_{W(v_{2})}\overline{v_{1}}\psi(v_{1},v_{2})e^{\overline{v_{1}}{u_{1}}+\overline{v_{2}}{u_{2}}}]\overline{u_{1}}^{m_{1}}\overline{u_{2}}^{m_{2}}e^{\overline{u_{1}}{\zeta_{1}}+\overline{u_{2}}{\zeta_{2}}}$
$\displaystyle=\int_{W(v_{1})}\int_{W(v_{2})}\overline{v_{1}}\psi(v_{1},v_{2})[\int_{W(u_{1})}\int_{W(u_{2})}u_{1}\overline{u_{1}}^{m_{1}}\overline{u_{2}}^{m_{2}}e^{\overline{u_{1}}{\zeta_{1}}+\overline{u_{2}}{\zeta_{2}}}e^{\overline{v_{1}}{u_{1}}+\overline{v_{2}}{u_{2}}}]$
$\displaystyle=\int_{W(v_{1})}\int_{W(v_{2})}\overline{v_{1}}\psi(v_{1},v_{2})\langle
u_{1}e^{\overline{v_{1}}{u_{1}}+\overline{v_{2}}{u_{2}}},{u_{1}}^{m_{1}}{u_{2}}^{m_{2}}e^{{u_{1}}\overline{\zeta_{1}}+{u_{2}}\overline{\zeta_{2}}}\rangle$
$\displaystyle=\int_{W(v_{1})}\int_{W(v_{2})}\overline{v_{1}}\psi(v_{1},v_{2})\partial_{\zeta_{1}}^{m_{1}}\partial_{\zeta_{2}}^{m_{2}}(\zeta_{1}e^{\overline{v_{1}}{\zeta_{1}}+\overline{v_{2}}{\zeta_{2}}})$
$\displaystyle=\int_{W(v_{1})}\int_{W(v_{2})}\overline{v_{1}}\psi(v_{1},v_{2})\partial_{\zeta_{1}}^{m_{1}}\partial_{\zeta_{2}}^{m_{2}}\partial_{\overline{v_{1}}}(e^{\overline{v_{1}}{\zeta_{1}}+\overline{v_{2}}{\zeta_{2}}})$
$\displaystyle=\int_{W(v_{1})}\int_{W(v_{2})}\psi(v_{1},v_{2})\overline{v_{1}}\partial_{\overline{v_{1}}}\partial_{\zeta_{1}}^{m_{1}}\partial_{\zeta_{2}}^{m_{2}}(e^{\overline{v_{1}}{\zeta_{1}}+\overline{v_{2}}{\zeta_{2}}})$
$\displaystyle=\langle\psi(v_{1},v_{2}),v_{1}\partial_{v_{1}}K^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}(v_{1},v_{2})\rangle.$
(12)

An identical argument holds for $u_{2}\partial_{u_{2}}$ and similar
calculations justify forms of adjoints for the operators
$iu_{j}\partial_{u_{k}}$ for $j\neq k$. For example, it can be shown that

$\displaystyle\langle
iu_{1}\partial_{u_{2}}\psi(u_{1},u_{2}),u_{1}^{m_{1}}u_{2}^{m_{2}}e^{u_{1}\overline{\zeta_{1}}+u_{2}\overline{\zeta_{2}}}\rangle$
$\displaystyle=\langle\psi(v_{1},v_{2}),-iv_{2}\partial_{v_{1}}v_{1}^{m_{1}}v_{2}^{m_{2}}e^{v_{1}\overline{\zeta_{1}}+v_{2}\overline{\zeta_{2}}}\rangle.$
(13)

Using these results in the expression of Hamiltonian the proposition is
verified.

  * Proposition 2.

$H$ is $\mathcal{C}_{2}$ self-adjoint i. e.;
$\mathcal{C}_{2}H^{\star}\mathcal{C}_{2}=H$

Proof : We shall first show the case with $u_{1}\partial_{u_{1}}$. Considering
the conjugation operator $\mathcal{C}_{2}$ we find

$\displaystyle\mathcal{C}_{2}(u_{1}\partial_{u_{1}})^{\star}\mathcal{C}_{2}K^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}(u_{1},u_{2})$
$\displaystyle=\mathcal{C}_{2}u_{1}\partial_{u_{1}}\mathcal{C}_{2}u_{1}^{m_{1}}u_{2}^{m_{2}}e^{u_{1}\overline{\zeta_{1}}+u_{2}\overline{\zeta_{2}}}$
$\displaystyle=\mathcal{C}_{2}u_{1}\partial_{u_{1}}\overline{\overline{(-u_{1})}^{m_{1}}}\overline{\overline{(-u_{2})}^{m_{2}}}\overline{e^{\overline{-u_{1}}\overline{\zeta_{1}}+\overline{-u_{2}}\overline{\zeta_{2}}}}$
$\displaystyle=\mathcal{C}_{2}u_{1}\partial_{u_{1}}(-1)^{m_{1}+m_{2}}u_{1}^{m_{1}}u_{2}^{m_{2}}e^{-u_{1}{\zeta_{1}}-u_{2}{\zeta_{2}}}$
$\displaystyle=\mathcal{C}_{2}u_{1}(-1)^{m_{1}+m_{2}}[u_{1}^{m_{1}}u_{2}^{m_{2}}(-{\zeta_{1}})e^{-u_{1}{\zeta_{1}}-u_{2}{\zeta_{2}}}+m_{1}u_{1}^{m_{1}-1}u_{2}^{m_{2}}e^{-u_{1}{\zeta_{1}}-u_{2}{\zeta_{2}}}]$
$\displaystyle=-u_{1}(-1)^{m_{1}+m_{2}}[(-1)^{m_{1}+m_{2}}u_{1}^{m_{1}}u_{2}^{m_{2}}(-\overline{\zeta_{1}})$
$\displaystyle+m_{1}(-1)^{m_{1}+m_{2}-1}u_{1}^{m_{1}-1}u_{2}^{m_{2}}]e^{u_{1}\overline{\zeta_{1}}+u_{2}\overline{\zeta_{2}}}$
$\displaystyle=u_{1}\partial_{u_{1}}K^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}(u_{1},u_{2}).$
(14)

Similarly one can prove
$\mathcal{C}_{2}(iu_{1}\partial_{u_{2}})^{\star}\mathcal{C}_{2}K^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}(u_{1},u_{2})=(iu_{2}\partial_{u_{1}})K^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}(u_{1},u_{2})$
and
$\mathcal{C}_{2}(iu_{2}\partial_{u_{1}})^{\star}\mathcal{C}_{2}K^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}(u_{1},u_{2})=(iu_{1}\partial_{u_{2}})K^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}(u_{1},u_{2})$.
Using these result in the expression of the Hamiltonian the above proposition
is verified.

  * Proposition 3.

$H$ has $\partial_{\mathcal{PT}}$ symmetry i. e.;
$\mathcal{C}_{2}^{(j)}H\mathcal{C}_{2}^{(j)}=H$, for $j=1,2$ but it lacks
global $\mathcal{PT}$ symmetry i. e.; $\mathcal{C}_{2}H\mathcal{C}_{2}\neq H$.

Proof : We shall only verify the that
$\mathcal{C}_{2}^{(1)}u_{1}\partial_{u_{1}}\mathcal{C}_{2}^{(j)}K^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}(u_{1},u_{2})=u_{1}\partial_{u_{1}}K^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}(u_{1},u_{2})$
through the following steps

$\displaystyle\mathcal{C}_{2}^{(1)}u_{1}\partial_{u_{1}}\mathcal{C}_{2}^{(1)}K^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}(u_{1},u_{2})$
$\displaystyle=\mathcal{C}_{2}^{(1)}u_{1}\partial_{u_{1}}[(-u_{1})^{m_{1}}(u_{2})^{m_{2}}e^{-u_{1}\zeta_{1}+u_{2}\zeta_{2}}]$
$\displaystyle=\mathcal{C}_{2}^{(1)}u_{1}(-1)^{m_{1}}[m_{1}u_{1}^{m_{1}-1}u_{2}^{m_{2}}+(-\zeta_{1})u_{1}^{m_{1}}u_{2}^{m_{2}}]e^{-u_{1}\zeta_{1}+u_{2}\zeta_{2}}$
$\displaystyle=-u_{1}(-1)^{m_{1}}[m_{1}(-u_{1})^{m_{1}-1}u_{2}^{m_{2}}+\overline{(-\zeta_{1})}(-u_{1})^{m_{1}}u_{2}^{m_{2}}]e^{u_{1}\overline{\zeta_{1}}+u_{2}\overline{\zeta_{2}}}$
$\displaystyle=u_{1}\partial_{u_{1}}K^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}(u_{1},u_{2}).$
(15)

Similarly,

$\displaystyle\mathcal{C}_{2}^{(1)}iu_{1}\partial_{u_{2}}\mathcal{C}_{2}^{(1)}K^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}(u_{1},u_{2})$
$\displaystyle=\mathcal{C}_{2}^{(1)}iu_{1}\partial_{u_{2}}[(-u_{1})^{m_{1}}(u_{2})^{m_{2}}e^{-u_{1}\zeta_{1}+u_{2}\zeta_{2}}]$
$\displaystyle=\mathcal{C}_{2}^{(1)}iu_{1}(-1)^{m_{1}}[(u_{1})^{m_{1}}m_{2}(u_{2})^{m_{2}-1}+(u_{1})^{m_{1}}(u_{2})^{m_{2}}(\zeta_{2})]e^{-u_{1}\zeta_{1}+u_{2}\zeta_{2}}$
$\displaystyle=iu_{1}\partial_{u_{2}}K^{[m_{1},m_{2}]}_{\zeta_{1},\zeta_{2}}(u_{1},u_{2}).$
(16)

Using these and similar results for the expression of the Hamiltonian the
proposition can be verified.

##  3\. $\partial_{\mathcal{PT}}$ symmetry of the eigenstates of the
Hamiltonian $H$

First we shall show that the reality of the eigenvalues is directly related to
the $\partial_{\mathcal{PT}}$ symmetry of the eigen states of the Hamiltonian.
It is readily observed that the present Hamiltonian leaves the homogeneous
polynomial space of two indeterminates $(\zeta_{1},\zeta_{2})$ invariant.
Considering such a space of degree of homogeneity $m$ and polynomial bases
$\\{f_{k}=\zeta_{1}^{m-k}\zeta_{2}^{k}:k=0\dots m\\}$ the operator $H$ has the
following tridiagonal representation

$\displaystyle
H=\left(\begin{array}[]{cccccccc}\beta_{m}&i\gamma&0&0&0&\dots&0&0\\\
im\gamma&\beta_{m-2}&2i\gamma&0&0&\dots&0&0\\\
0&i\gamma(m-1)&\beta_{m-4}&3i\gamma&0&\dots&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\dots&\beta_{-(m-2)}&im\gamma\\\
0&0&0&0&0&\dots&i\gamma&\beta_{-m}\end{array}\right).$ (22)

Here, $\beta_{\mu}=\mu+\alpha{\mu}^{2}$. The the eigenvalues of such a matrix
can be found out with the help of the following theorem [18].

  * Theorem 1.

Given a tri-diagonal matrix

${\mathcal{M}}=\left(\begin{array}[]{cccccccc}b_{0}&d_{0}&0&0&0&\dots&0&0\\\
c_{0}&b_{1}&d_{1}&0&0&\dots&0&0\\\ 0&c_{1}&b_{2}&d_{2}&0&\dots&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\dots&b_{l-2}&d_{l-2}\\\
0&0&0&0&0&\dots&c_{l-2}&b_{l-1}\end{array}\right)$ (23)

with $d_{i}\neq 0\>\>\forall\>\>i$, let us consider a polynomial
$P_{n}(\lambda)$ that follows the well-known three term recursion relation
[19, 20]

$P_{n+1}(\lambda)=\frac{1}{d_{n}}[(\lambda-
b_{n})P_{n}(x)-c_{n-1}P_{n-1}(\lambda)].$ (24)

If $P_{-1}(\lambda)=0$ and $P_{0}(\lambda)=1$ the eigenvalues are given by the
zeros of the polynomial $P_{l}(\lambda)$ and eigenvector corresponding to
$j$-th eigenvalue $\lambda_{j}$ is given by the vector

$\left(\begin{array}[]{c}P_{0}(\lambda_{j})\\\ P_{1}(\lambda_{j})\\\ \vdots\\\
P_{l-2}(\lambda_{j})\\\ P_{l-1}(\lambda_{j})\end{array}\right).$ (25)

Proof : Let $\left(\begin{array}[]{c}v_{0}\\\ v_{1}\\\ \vdots\\\ v_{l-2}\\\
v_{l-1}\end{array}\right).$be the eigen vector corresponding to the eigenvalue
$\lambda$. Then the eigenvalue equation gives us

$\displaystyle b_{0}v_{0}+d_{0}v_{1}=\lambda v_{0}$ $\displaystyle
c_{0}v_{0}+b_{1}v_{1}+d_{1}v_{2}=\lambda v_{1}$ $\displaystyle\vdots$
$\displaystyle c_{l-2}v_{l-2}+b_{l-1}v_{l-1}=\lambda v_{l-1}$ (26)

Since $P_{0}(\lambda)=1$, one can write $v_{0}=P_{0}(\lambda)v_{0}$. Now in
view of the recurrence relation (equation-24)

$\displaystyle v_{1}=\frac{1}{d_{0}}(\lambda-
b_{0})P_{0}(\lambda)v_{0}=P_{1}(\lambda)$ (27)

Continuing the substitution recursively we get

$v_{n}=P_{n}(\lambda)v_{0}:0\leq n<l$ (28)

Substituting $n=l-1$ in equation-28 and using the three term relation
(equation-24) we get

$P_{l}(\lambda)v_{0}=0$ (29)

giving the characteristic equation $P_{l}(\lambda)=0$.

Now going back to the matrix in equation-22 and comparing the matrices in
equation-22 and equation-23 we get
$\\{b_{0},\dots,b_{l-1}\\}=\\{\beta_{m},\dots,\beta_{-m}\\}$,
$\\{c_{0},\dots,c_{l-2}\\}=\\{im\gamma,\dots,i\gamma\\}$ and
$\\{d_{0},\dots,d_{l-2}\\}=\\{i\gamma,\dots,im\gamma\\}$, one can determine
the eigenvectors and eigenvalues of the matrix using the above algorithm.

###  3.1. Reality of the eigenvalues and $\partial_{\mathcal{PT}}$ symmetry of
the eigenstates

Without calculating the eigenvalues explicitly, an immediate inference
regarding the symmetry behaviour of the eigenfunctions is possible in view of
the reality of the eigenvalues. We shall consider the following two cases.

Case-I : $|m|\in\\{2s:s\in{Z^{+}}\\}$ Let us begin from the value $|m|=2$. The
matrix of $H$ is a $3\times 3$ matrix. Correspondingly the matrix
${\mathcal{M}}$ has the diagonal elements
$b_{0}=2+4\alpha,b_{1}=0,b_{2}=-2+4\alpha$. This implies $l=3$. When $|m|=2s$
for some fixed $s$, $l=2s+1$ and the matrix ${\mathcal{M}}$ becomes a
$(2s+1)\times(2s+1)$ matrix. As a consequence a polynomial equation
$P_{l}(x)=0$ implies $P_{2s+1}(x)=0$ and roots of this polynomial equation
gives us the eigenvalues. The eigenvector corresponding to an eigenvalue
$\lambda_{0}$ would be given by the vector

$\displaystyle(P_{0}(\lambda_{0})\>\>\>P_{1}(\lambda_{0})\dots
P_{2s}(\lambda_{0})).$ (30)

The first term is equal to one for all values of $\lambda_{0}$. Now for real
$\lambda_{0}$ it is easy to verify that starting from the purely imaginary
second term the subsequent terms are alternatively real and purely imaginary
leading to the following equivalence

$\displaystyle(P_{0}(\lambda_{0})\>\>\>P_{1}(\lambda_{0})\dots
P_{2s}(\lambda_{0}))=(A_{0}(\lambda_{0})\>\>\>iA_{1}(\lambda_{0})\dots
A_{2s}(\lambda_{0})).$ (31)

Here, $\\{A_{l}:l=0\dots 2s\\}$ are real functions of $\lambda_{0}$ with
$A_{0}=1$. In the Fock space setting the eigenfunction in
$({\zeta_{1},\zeta_{2}})$ can be given by

$\displaystyle\psi^{(I)}_{2s}(\zeta_{1},\zeta_{2})=A_{0}\zeta_{1}^{2s}+iA_{1}\zeta_{1}^{2s-1}\zeta_{2}+A_{2}\zeta_{1}^{2s-2}\zeta_{2}^{2}+\dots+A_{2s}\zeta_{2}^{2s}.$
(32)

Now the action of partial parity operator $\\{\mathcal{C}_{2}^{(j)}:j=1,2\\}$
on $\psi^{(I)}_{2s}(\zeta_{1},\zeta_{2})$ can be understood through the
following equation

$\displaystyle\mathcal{C}_{2}^{(1)}\psi^{(I)}_{2s}(\zeta_{1},\zeta_{2})$
$\displaystyle=\overline{A_{0}\overline{(-\zeta_{1})}^{2s}+iA_{1}\overline{(-\zeta_{1})}^{2s-1}\overline{\zeta_{2}}+A_{2}\overline{(-\zeta_{1})}^{2s-2}\overline{\zeta_{2}}^{2}+\dots+A_{2s}\overline{(\zeta_{2})}^{2s}}$
$\displaystyle=\psi^{(I)}_{2s}(\zeta_{1},\zeta_{2}).$ (33)

Similarly,
$\mathcal{C}_{2}^{(2)}\psi^{(I)}_{2s}(\zeta_{1},\zeta_{2})=\psi^{(I)}_{2s}(\zeta_{1},\zeta_{2})$.

Case-II : $|m|\in\\{2s-1:s\in{Z^{+}}\\}$ Similar argument as given above can
lead to the following eigenfunction for odd $m$

$\displaystyle\psi^{(II)}_{2s-1}(\zeta_{1},\zeta_{2})=B_{0}\zeta_{1}^{2s-1}+iB_{1}\zeta_{1}^{2s-2}\zeta_{2}+B_{2}\zeta_{1}^{2s-3}\zeta_{2}^{3}+\dots+iB_{2s-1}\zeta_{2}^{2s-1}$
(34)

where, $\\{B_{l}:l=0\dots 2s-1\\}$ are real functions of some real eigenvalue
$\lambda_{1}$ with $B_{0}=1$.

Now actions of $\\{\mathcal{C}_{2}^{(j)}:j=1,2\\}$ on
$\psi^{(II)}_{2s-1}(\zeta_{1},\zeta_{2})$ are given by

$\displaystyle\mathcal{C}_{2}^{(1)}\psi^{(II)}_{2s-1}(\zeta_{1},\zeta_{2})=-\psi^{(II)}_{2s-1}(\zeta_{1},\zeta_{2})\>\>{\rm
and}\>\>\mathcal{C}_{2}^{(2)}\psi^{(II)}_{2s-1}(\zeta_{1},\zeta_{2})=\psi^{(II)}_{2s-1}(\zeta_{1},\zeta_{2}).$
(35)

Remark-I : It is observed that the eigen functions for even $m$ are
$\partial_{\mathcal{PT}}$ symmetric in either of the variables whereas, for
odd $m$ the eigenfunctions are symmetric in one variable $(\zeta_{2})$ and
anti-symmetric in the other $(\zeta_{1})$. It can also be shown that
$\mathcal{C}_{2}\psi^{(I)}_{2s}(\zeta_{1},\zeta_{2})\neq\psi^{(I)}_{2s}(\zeta_{1},\zeta_{2})$
and
$\mathcal{C}_{2}\psi^{(II)}_{2s-1}(\zeta_{1},\zeta_{2})\neq\psi^{(II)}_{2s-1}(\zeta_{1},\zeta_{2})$
implying the fact that neither of them has any global $\mathcal{PT}$ symmetry.

Remark-II : If the eigenvalue has any nonzero complex part the antilinear
action of the operators representing $\partial_{\mathcal{PT}}$ symmetry
replaces the coefficients $\\{A_{l}\\}$ or $\\{B_{l}\\}$ by their respective
complex conjugates thus destroying the symmetry of the eigenfunctions. This
phenomenon may termed as $\partial_{\mathcal{PT}}$ symmetry breaking.

It is obvious that the eigenvalues have a strong parametric dependence on the
values of the parameters $\gamma$ and $\alpha$ an issue that may be discussed
elsewhere. Considering $m=2$ and $\gamma=\alpha=\frac{1}{2}$ the eigenvalues
are $\lambda_{1}=3.87513,\lambda_{2,3}=0.06244\pm 0.71569i$. This means,
according to the above discussion, the states corresponding to $\lambda_{1}$
is a $\partial_{\mathcal{PT}}$ symmetric state but those corresponding to
$\lambda_{2}$ and $\lambda_{3}$ are symmetry breaking states. On the other
hand, for $\alpha=4=8\gamma$ the eigenvalues are $(0.06375,13.96387,17.97237)$
and consequently, all the states are $\partial_{\mathcal{PT}}$ symmetric.

In the above discussion only the cases with non-degenerate eigenvalues have
been considered. The situation with degenerate eigenvalues may be an
interesting area of investigation.

##  4\. Conclusion

The present investigation deals with a new kind of symmetry operator that acts
on operators or functions in a variable specific way. In this article, the
presence of such symmetry known as Partial $\mathcal{PT}$ symmetry is
understood in a typical Fock space setting for a two-boson Bose-Hubbard type
Hamiltonian with one purely imaginary interaction term. The symmetry operators
are represented as Weighted Composition Conjugations acting on a Reproducing
Kernel Hilbert Space. The existence of such symmetry as well as the
possibility of breaking of such symmetry are found to have direct
correspondence with the reality of eigenvalues of the Hamiltonian.

## References

  * [1] C. M. Bender and S. Boettcher : Phys. Rev. Lett. $\bf{80}$, 5243 (1998).
  * [2] C. M. Bender , S. Boettcher and P. N. Meisinger : J. Math. Phys. $\bf{40}$, 2201 (1999).
  * [3] C. M. Bender , D. C. Brody and H. F. Jones : Phys. Rev. Lett. $\bf{89}$, 27040 (2002).
  * [4] D. C. Brody : J. Phys. A: Math. Theor. $\bf{49}$, 10LT03 (2016).
  * [5] A. Mostafazadeh : J. Math. Phys. $\bf{43}$, 205 (2002).
  * [6] D. C. Brody : J. Phys. A: Math. Theor. $\bf{47}$, 035305 (2014).
  * [7] N. Moiseyev : Non-Hermitian Quantum Mechanics (chapter-3, 4) (Chambridge(UK) : Chembridge University Press) (2011).
  * [8] Bagarello F, Gazeau J P, Szafraniec F H, Znojil M (Editors) : Non-Selfadjoint Operators in Quantum Physics (chapter-3 pp 123-124 and pp 277-278 regarding basis construction), (chapter-6 specifically pp 323-324 regarding ${\mathcal{PT}}$ symmetry ) (New Jersry : John Wiley and Sons, Inc.) (2015)
  * [9] A. Beygi ,S. P. Klevansky and C. M. Bender : Phys. Rev. A. $\bf{91}$, 062101 (2015).
  * [10] E. M. Graefe , U. G$\ddot{\rm u}$nther, H. J. Korsch , A. E. Niederle : J. Phys. A: Math. Theor. $\bf{41}$, 255206 (2008).
  * [11] K. Zhu : Analysis on Fock Spaces(Springer(NY) )(2012).
  * [12] V. I. Paulsen and M. Raghupathi : An Introduction To The Theory of Reproducing Kernel Hilbert Space(Chambridge(UK) : Chembridge University Press)(2016).
  * [13] P. V. Hai and M. Putinar : J. Diff. Equations $\bf{265}$, 4213 (2018).
  * [14] P. V. Hai and L. H. Khoi : J. Math. Anal. Appl. $\bf{433}$, 1757 (2016).
  * [15] P. V. Hai and L. H. Khoi : Complex Variables and Elliptic Equations $\bf{63(3)}$, 391 (2018).
  * [16] S. R. Garcia and M. Putinar : Trans. Am. Math. Soc. $\bf{358}$, 1285 (2006).
  * [17] S. R. Garcia and M, Putinar : Trans. Am. Math. Soc. $\bf{359}$,3913 (2007).
  * [18] A. Sandryhaila ,M. F. Moura : (arXiv:1306.0217v1[math.SP]2 June) (2013) .
  * [19] C. F. Dunkl and Xu. Y. : Orthogonal Polynomials of Several Variables, second edition (chapter-1 pp 9 proposition-1.3.7 and corollary-1.3.8) (Cambridge : Cambridge University Press ) (2014).
  * [20] M. E. H. Ismail and W. V. Assche : Classical and Quantum Orthogonal Polynomials in One Variable (chapter-2 pp 22 theorem-2.2.1) (Cambridge : Cambridge University Press) (2005).