# Programmable Order by Disorder Effect through Quantum Simulator

Huan-Kuang Wu  Department of Physics, Condensed Matter Theory Center and Joint
Quantum Institute, University of Maryland, College Park, Maryland 20742, USA
Takafumi Suzuki  Graduate School of Engineering, University of Hyogo, Hyogo,
Himeji 670-2280, Japan  Naoki Kawashima  Institute for Solid State Physics,
University of Tokyo, Kashiwa, Chiba 277-8581, Japan  Trans-scale Quantum
Science Institute, The University of Tokyo, Bunkyo, Tokyo 113-0033, Japan
Wei-Lin Tu<EMAIL_ADDRESS>Faculty of Science and Technology, Keio
University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan

###### Abstract

We study the quantum order by disorder effect of the $S=1/2$ system, which is
programmable in quantum simulator composed of Rydberg atoms in the triangular
optical lattice with a controllable diagonal anisotropy. When the total
magnetization is zero, a set of sub-extensive degenerate ground states is
present in the classical limit, composed of continuous strings whose
configuration enjoys a large degree of freedom. Among all possible
configurations, we focus on the stripe (up and down spins align straightly)
and kinked (up and down spins form a zigzag shape) patterns. Adopting the the
real space perturbation theory (RSPT), we estimate the leading order energy
correction when the nearest-neighbor ($nn$) spin-flip coupling $J$ term is
considered, and the overall model becomes an effective XXZ model with spatial
anisotropy. Our calculation demonstrates a lifting of the degeneracy, favoring
the stripe configuration. When $J$ becomes larger, we adopt the infinite
projected entangled-pair state (iPEPS) and numerically check the effect of
degeneracy lifting. The iPEPS results show that even when the spin-flip
coupling term is strong the stripe pattern is still favored. While the above
system is realizable with the deformed optical lattice, at last, we
numerically calculate the hard-core bosonic Hamiltonian with the $nn$ dipole-
dipole interaction for demonstrating the possible underlying phases, using the
cluster mean-field theory. We provide various phase diagrams with different
tilted angles, showing the abundant phases including the supersolid. Our
proposal indicates a realizable scenario through quantum simulator in studying
the quantum effect as well as extraordinary phases.

##  I Introduction

The quantum many-body systems are often represented by Hamiltonian whose
composing parts do not commute with each other. This suggests that although
the properties of low-energy states can be easily identified in a certain
limit, often referred to as the classical limit, considering the whole
Hamiltonian will make the estimation of ground state difficult because one
needs to consider the superposition of every state in the full Hilbert space.
Moreover, the extra quantum or even thermal effect may lead to further
stabilization among some competing low-energy states, causing the energy level
crossing or degeneracy lifting, know as the order by disorder (OBD) effect [1,
2, 3]. Besides, the introduction of frustration, through the Hamiltonian [4,
5, 6, 7] or lattice geometry [8, 9, 10, 11, 12, 13], can also result in the
many-fold degeneracy of ground state in the classical picture and such
degeneracy enriches the quantum or thermal fluctuation.

Among all, the triangular lattice is the simplest lattice structure for
inducing the geometrical frustration. Particularly, the inclusion of
anisotropy could lead to the formation of long strings composed of same spins
[14]. The shape of strings enjoys a great degree of freedom, leading to a
many-fold degeneracy. Classically, such systems and their degeneracies can be
realized, composed of self-assembled colloidal particles in a monolayer [15,
16]. For this soft-matter system, particles move to the opposite walls that
confine them to maximize the free volume, corresponding to the up-and-down
spin scenario and forming a good platform for studying the dynamics of
frustration. In $d$ dimensions, the entropy of degenerate ground states is
proportional to $L^{d-1}$, with the number of sites $N\sim L^{d}$, resulting
in a sub-extensive degeneracy. For the above-mentioned frustrated colloidal
system, its thermal OBD effect has been studied and the authors unveiled the
conclusion that the straight stripe should be favored [17]. On the other hand,
the quantum counterpart for such OBD effect in zero temperature is also of
interest. When an extra non-commutable term, such as the spin-flip coupling,
is introduced to the Ising-like Hamiltonian the quantum fluctuation could
further lower the energy and the amount is dependant on the classical
configurations, resulting in the lifting of degeneracy. While many previous
works have shown that the thermal and quantum fluctuations tend to result in
the same outcome for weakly frustrated magnets, it is not guaranteed for the
general cases [18, 19, 20, 21]. Therefore, it is worthwhile to examine the
quantum version of this OBD effect. However, this effect has not yet been
studied to the best of our knowledge. One possible reason is because such
system can hardly be prepared in real materials.

Figure 1: Configurations for (a) stripe, (b) kinked, and (c) mixed states. Red
(pink) dots indicate that the Rydberg atom occupies the $|R\rangle$
($|G\rangle$) state. And the black (green) bonds represent the bonds for
diagonal (horizontal) interaction, written as $J^{z}_{d}$ ($J^{z}_{h}$) in Eq.
(2). Blue dashed circles represent the region of Rydberg blockade, where in
the deformed triangular lattice only two horizontal nearby sites are included.
The brown dotted boxes indicate the minimum repeated unit cell for the stripe
and kinked patterns, which we will later adopt for the iPEPS calculation.

Thanks to the advance of cold-atom technique [22, 23, 24, 25], many physical
scenarios can now be studied with emphasized quantum effect, so-called the
quantum simulator [26, 27, 28]. One of the commonly studied systems is
composed of Rydberg atoms [29, 30] in the optical lattice. It has been shown
that many models can be realized through the engineering of the above-
mentioned system, including the Ising model [31, 32, 33], XY model [34, 35,
36], XXZ model [37, 38], or even the XYZ model [39]. In such system, each atom
can be regarded as a two-level sub-system, consisting of the ground state
$|G\rangle$ and excited Rydberg state $|R\rangle$. Driven by a coherent laser
field with a Rabi frequency $\Omega$ and a detuning $\delta$, each Rydberg
atom tends to occupy the $|R\rangle$ state. However, in the Rydberg state the
atoms are influenced by a repulsive interaction in the van der Waal form
$V(r)=C/r^{6}$, where $r$ represents the distance between sites. The effective
Hamiltonian can then be written as [40]

$\displaystyle H_{\text{Ryd}}=$
$\displaystyle\sum_{i}\frac{\Omega}{2}(|G\rangle\langle
R|_{i}+|R\rangle\langle G|_{i})-\delta|R\rangle\langle R|_{i}$ (1)
$\displaystyle+\frac{1}{2}\sum_{ij}V(r_{ij})|R\rangle\langle
R|_{i}\otimes|R\rangle\langle R|_{j}.$

Eq. (1) is effectively equal to a quantum Ising model with both longitudinal
and transverse field [40]. Another important feature of the Rydberg physics
comes from the Rydberg blockade [41, 42]. When the nearest-neighbor ($nn$)
interaction is stronger than the Rabi frequency and detuning field
($V(1)\gg|\Omega|,|\delta|$), it will prevent the $nn$ sites from being in the
Rydberg state simultaneously. The blockade radius, $r_{b}$, can be estimated
since the boundary of the blockade is determined following the formula
$V(r_{b})=|\Omega|$; therefore $r_{b}=(C/|\Omega|)^{1/6}$.

When the Rydberg atoms are placed in a deformed optical lattice, with a
properly adjusted $r_{b}$ only two out of the six nearby sites stay within the
region of blockade, as shown in Fig. 1. As a result, atoms in $|R\rangle$ or
$|G\rangle$ state form continuous strings along the diagonal direction. This
system corresponds to the above-mentioned self-assembled colloidal particles
in a monolayer, leading to a sub-extensive degeneracy in the classical limit.
After the introduction of spin-flip coupling, which is realizable through the
microwave engineering [39, 38], the degeneracy is expected to be lifted
through quantum OBD effect. The final effective model then becomes a XXZ model
with external fields.

In the following content, we will study the quantum OBD effect using the real
space perturbation theory (RSPT) [43, 44, 19, 45, 46, 47] and infinite
projected entangled-pair state (iPEPS) [48, 49, 50]. Since the two-level
Rydberg system can be mapped to a $S=1/2$ scenario, we treat the $|R\rangle$
state as the up spin and $|G\rangle$ state as the down spin. When the number
of up spins ($n_{\uparrow}$) is equal to that of down spins
($n_{\downarrow}$), a sub-extensive degeneracy is present in the classical
limit. Introducing the spin-flip coupling, energy correction can be considered
with RSPT and we focus on the leading-order correction between the stripe and
kinked configurations. Our calculation shows that the energy correction favors
the stripe pattern, in agreement with previous results of thermal OBD [17]. As
a further confirmation we apply the iPEPS calculation for a simplified toy
model, showing that the stripe state is indeed more stable. Also, we provide
numerical simulations using the cluster mean-field theory (CMFT) for dipolar
hard-core bosons in the triangular lattice with tilted polar angle. We unveil
fruitful phase diagrams with various underlying phases, including the
supersolid. We anticipate the future realization for the quantum OBD effect
and each potential phase with the real quantum simulators.

##  II Quantum OBD effect

###  II.1 Real space perturbation theory

We first introduce the RSPT and derive the formula in this section. In this
work, our effective Hamiltonian can be written as

$\displaystyle H$ $\displaystyle=J\sum_{\langle
ij\rangle}(\hat{S}^{+}_{i}\hat{S}^{-}_{j}+\hat{S}^{-}_{i}\hat{S}^{+}_{j})+\sum_{\langle
ij\rangle}J^{z}_{ij}\hat{S}^{z}_{i}\hat{S}^{z}_{j}-\sum_{i}\textbf{B}\cdot\hat{\textbf{S}}_{i},$
(2)

where $J^{z}_{ij}\gg|J|,|\textbf{B}|$ and B represents the external Zeeman
field. To effectively reflect the anisotropy caused by the deformed lattice,
we assign that $J^{z}_{ij}$ is equal to $J^{z}_{d}$ when the bond is in the
diagonal direction (black bonds in Fig. 1) and $J^{z}_{h}$ when the bond is in
the horizontal direction (green bonds in Fig. 1). When $J^{z}_{d}<J^{z}_{h}$
and $n_{\uparrow}=n_{\downarrow}$, the ground state configuration enjoys a
many-fold degeneracy because same spins tend to align along the diagonal
direction. However, due to the fact that there are two diagonal bonds, we can
have many different degenerate configurations and the total number is equal to
$2^{L_{y}-1}$, where $L_{y}$ is the number of rows along $\hat{y}$ direction.
As a result, the ground state entropy $S_{GS}\sim L_{y}-1$, leading to a sub-
extensive degeneracy. Among these degenerate configurations there are two
special patterns, the stripe and kinked states. The stripe state is composed
of parallel straight lines of up and down spins (Fig. 1(a)), while for the
kinked state there are three sub-lattices occupied by up or down spins within
each vertical diamond of the triangular lattice (Fig. 1(b)). The rest
degenerate states, called the mixed states, are simply the combinations of
stripe and kinked states in a specific proportion (one example shown in Fig.
1(c)).

For applying RSPT, one needs to rotate the local coordinate on each site so
that every spin points along the $\hat{z}$-direction. Thus, we re-write the
corresponding spin-1/2 Hamiltonian in the following form

$\displaystyle H=\sum_{\langle
ij\rangle}\textbf{S}_{i}\cdot\textbf{J}\cdot\textbf{S}_{j}-\sum_{i}\textbf{B}\cdot\textbf{S}_{i},$
(3)

with

$\textbf{J}=\left\\{\begin{aligned}
&\text{diag}(2J,2J,J^{z}_{h})\;\;\;\text{if}\>\langle
ij\rangle\parallel\text{horizontal direction}\\\
&\text{diag}(2J,2J,J^{z}_{d})\;\;\;\text{if}\>\langle
ij\rangle\parallel\text{diagonal direction}.\end{aligned}\right.$ (4)

We can extend this to a more generalized Hamiltonian with longer-range hopping
or even off-diagonal terms if needed, simply by modifying the form of coupling
matrices. In Eq. (3),
$\textbf{S}_{i}=S^{+}_{i}\hat{e}^{-}_{i}+S^{-}_{i}\hat{e}^{+}_{i}+S^{z}_{i}\hat{e}^{z}_{i}$
with $\hat{e}^{\pm}_{i}=\frac{1}{2}(\hat{e}^{x}_{i}\pm i\hat{e}^{y}_{i})$ and
the classical configuration on each site is
$\textbf{S}^{(0)}_{i}=S\hat{e}^{z}_{i}$. Note that now the $S^{\pm}$ and
$S^{z}$ are no longer operators. They simply represent the corresponding
operation for the classical spin. Our choice of the ”unit” vectors guarantees
the transformation to be unitary, although its length is not necessarily the
unity. This also explains why we have a pre-factor of 2 for $J$ in Eq. (4).
Next, we expand the equation and the result becomes

$\displaystyle H=$ $\displaystyle\sum_{\langle
ij\rangle}(\hat{e}^{+}_{i}\cdot\textbf{J}\cdot\hat{e}^{+}_{j}S^{-}_{i}S^{-}_{j}+H.C.)$
(5)
$\displaystyle\;\;+(\hat{e}^{+}_{i}\cdot\textbf{J}\cdot\hat{e}^{-}_{j}S^{-}_{i}S^{+}_{j}+H.C.)$
$\displaystyle\;\;+(\hat{e}^{+}_{i}\cdot\textbf{J}\cdot\hat{e}^{z}_{j}S^{-}_{i}S^{z}_{j}+\hat{e}^{z}_{i}\cdot\textbf{J}\cdot\hat{e}^{+}_{j}S^{z}_{i}S^{-}_{j}+H.C.)$
$\displaystyle\;\;+\hat{e}^{z}_{i}\cdot\textbf{J}\cdot\hat{e}^{z}_{j}S^{z}_{i}S^{z}_{j}$
$\displaystyle-\sum_{i}\textbf{B}\cdot\hat{e}^{-}_{i}S^{+}_{i}+\textbf{B}\cdot\hat{e}^{+}_{i}S^{-}_{i}+\textbf{B}\cdot\hat{e}^{z}_{i}S^{z}_{i}.$

After introducing the quantum fluctuation, the spin configuration does no
longer align along the local $\hat{e}^{z}_{i}$ direction and thus we can
replace $S^{z}_{i}$ by $S-\delta S_{i}$. Then, Eq. (5) becomes

$\displaystyle H=H_{0}+H_{\text{unperturbed}}+H_{\text{p}},$ (6)

where

$\displaystyle H_{0}=S^{2}\sum_{\langle
ij\rangle}\hat{e}^{z}_{i}\cdot\textbf{J}\cdot\hat{e}^{z}_{j}-S\sum_{i}\textbf{B}\cdot\hat{e}^{z}_{i},$
(7)

representing the classical energy. And $H_{\text{unperturbed}}$ is

$\displaystyle H_{\text{unperturbed}}=$ $\displaystyle-S\sum_{\langle
ij\rangle}\hat{e}^{z}_{i}\cdot\textbf{J}\cdot\hat{e}^{z}_{j}(\delta
S_{i}+\delta S_{j})$ (8) $\displaystyle+\sum_{\langle
ij\rangle}\hat{e}^{z}_{i}\cdot\textbf{J}\cdot\hat{e}^{z}_{j}\delta S_{i}\delta
S_{j}+\sum_{i}\textbf{B}\cdot\hat{e}^{z}_{i}\delta S_{i}.$

Since Eq. (8) does not change the spin configuration, it will not contribute
to the perturbation. At last, the perturbative Hamiltonian $H_{p}$ is written
as

$\displaystyle H_{\text{p}}=$ $\displaystyle\sum_{\langle
ij\rangle}(J^{(1)}_{ij}+J^{(2)}_{ij}+J^{(3)}_{ij})-\sum_{i}B_{i},$ (9)

where

$\displaystyle
J^{(1)}_{ij}=\hat{e}^{+}_{i}\cdot\textbf{J}\cdot\hat{e}^{+}_{j}S^{-}_{i}S^{-}_{j}+H.C.$
(10) $\displaystyle
J^{(2)}_{ij}=\hat{e}^{+}_{i}\cdot\textbf{J}\cdot\hat{e}^{-}_{j}S^{-}_{i}S^{+}_{j}+H.C.$
$\displaystyle
J^{(3)}_{ij}=\hat{e}^{+}_{i}\cdot\textbf{J}\cdot\hat{e}^{z}_{j}S^{-}_{i}S^{z}_{j}+\hat{e}^{z}_{i}\cdot\textbf{J}\cdot\hat{e}^{+}_{j}S^{z}_{i}S^{-}_{j}+H.C.$
$\displaystyle
B_{i}=\textbf{B}\cdot\hat{e}^{-}_{i}S^{+}_{i}+\textbf{B}\cdot\hat{e}^{+}_{i}S^{-}_{i}.$

With these transverse perturbative terms, we can evaluate the energy
correction with $H_{p}$ in $n$-th order with

$\displaystyle\delta E^{(n)}=\sum_{\\{\psi_{i}\\}}\frac{\langle
0|H_{p}|\psi_{1}\rangle\langle\psi_{1}|H_{p}|\psi_{2}\rangle\cdot\cdot\cdot\langle\psi_{n-1}|H_{p}|0\rangle}{(E_{0}-E_{\psi_{1}})\cdot\cdot\cdot(E_{0}-E_{\psi_{n-1}})},$
(11)

where $E_{0}$ is the energy for the classical configuration, $|0\rangle$, and
$E_{\psi_{i}}$ is the energy for intermediate state denoted by
$|\psi_{i}\rangle$. The summation will run over every possible intermediate
processes for a given order. For any order of perturbation, its energy
correction is obtained under several rules:

(1) Each perturbation is evaluated within a linked cluster to ensure the
extensiveness of energy correction. On each bond of such cluster $H_{p}$ can
be acted for more than one time and the total number of execution of $H_{p}$
determines the order of perturbation.

(2) Any series must start and end with $J^{(1)}_{ij}$ and $J^{(3)}_{ij}$,
since we have rotated the local coordinate so that for every site,
$\textbf{S}^{(0)}_{i}=S\hat{e}^{z}_{i}$.

(3) $J^{(3)}_{ij}$ and $B_{i}$ only consider one spin flip and thus there must
be an even number for such process in a series. Because of this constraint,
one can always find a lower order series without these two terms and therefore
we can ignore them in search of leading order correction.

Figure 2: (a) The four-site diamond cluster that distinguishes kinked and
stripe states. Red and pink dots indicate sublattice $a$ and $b$ separately.
For kinked state it has two configurations with three sublattice $a$ or $b$
while for stripe state the numbers of $a$ and $b$ sublattice are equal. (b) An
example for the fourth-order tunneling process within the diamond cluster. The
perturbative process corresponds to $J^{(1)}_{12}\rightarrow
J^{(2)}_{14}\rightarrow J^{(2)}_{34}\rightarrow J^{(1)}_{23}$. Black dots
indicate the places where spins have been flipped.

Because of the above rules, we know that an effective leading-order
perturbative series must start and end with $J^{(1)}_{ij}$ and is composed of
$J^{(1)}_{ij}$ and $J^{(2)}_{ij}$. In Eq. (11), the denominator considers the
energy difference between classical and intermediate configurations. It is
given by

$\displaystyle E_{0}-E_{\psi}=\langle
0|H|0\rangle-\langle\psi|H|\psi\rangle=-\langle\psi|H_{\text{unperturbed}}|\psi\rangle.$
(12)

To adopt RSPT, first we rotate the local coordinates so that
$\textbf{S}^{(0)}_{i}=S\hat{e}^{z}_{i}$. Our classical configurations are
composed of two different sub-lattices representing the up and down spins

$\displaystyle\hat{e}^{z}_{a}=(\sin{\phi_{a}},0,\cos{\phi_{a}}),\;\hat{e}^{z}_{b}=(\sin{\phi_{b}},0,\cos{\phi_{b}}),$
(13)

and then we choose

$\displaystyle\hat{e}^{\pm}_{a}=\frac{1}{2}(\cos{\phi_{a}},\pm
i,-\sin{\phi_{a}}),$ (14)
$\displaystyle\hat{e}^{\pm}_{b}=\frac{1}{2}(\cos{\phi_{b}},\pm
i,-\sin{\phi_{b}}),$

where $a$ and $b$ denote two different sub-lattices. Without the loss of
generality, we choose the planar spin orienting along $\hat{x}$ axis. In fact,
in the classical limit we can simply assign $\phi_{a}=0$ for up spin and
$\phi_{b}=\pi$ for down spin. Since we hope that our derivation can also be
useful in the more general scenarios breaking the $U(1)$ symmetry, we take the
off-diagonal order into our consideration. It is also clear to see that by
adopting the same coordinate of sub-lattice $a$ or $b$ for every site in Eq.
(5), we can resume the original Hamiltonian, Eq. (2), where the translation
invariance is accompanied.

To estimate the leading-order correction, we need to first look for the
minimal closed cluster which differentiates the stripe and kinked patterns,
and it is the vertical diamonds whose longer diagonal line is parallel to
$\hat{y}$ in Fig. 1. In Fig. 2(a) we demonstrate the minimal different cluster
for kinked and stripe states. Then, we need to flip the spins through the
intermediate process on the diamond bonds to evaluate the energy correction.
Since there are four bonds on each diamond, the leading-order correction which
we are looking for is of the fourth order. One of the possible intermediate
processes is shown in Fig. 2(b).

The detailed equations and derivation of RSPT are shown in Appendix A and here
we only demonstrate the final results. While the degeneracy stays intact when
$J=0$, turning on the spin-flip coupling will immediately introduce an energy
correction to different patterns. For the stripe and kinked patterns, we have
$\phi_{a}=0$ and $\phi_{b}=\pi$ for the classical configurations. Under this
assumption our equations can be largely simplified. More importantly,
$J^{d(2)}_{3}$ becomes zero and thus only a few terms remain in Eqs. (28) and
(29). The energy correction terms then become

$\displaystyle\delta
E^{(4)}_{\text{kinked}}=\frac{2J^{4}}{(\Gamma_{3})^{2}\Gamma_{4}},$ (15)
$\displaystyle\delta
E^{(4)}_{\text{stripe}}=\frac{2J^{4}}{(\Gamma_{3})^{2}\Gamma_{4}}+\frac{2J^{4}}{(\Gamma_{3})^{2}\Gamma_{5}}.$

Since now $\Gamma_{5}=-8S^{2}J^{z}_{h}<0$ (see Appendix A) because
$J^{z}_{h}>0$, the stripe configuration is favored and would be selected. This
conclusion coincides with the previous one for the frustrated colloidal soft
matter system, whose thermal OBD has been studied through the entropy
estimation [17]. There, the authors have shown that the straight stripe
configuration possesses lower free energy, despite the fact that during the
cooling process the energy barrier among different configurations will
eventually lead to a mixed pattern. The effective Hamiltonian of such system
can be written down as an antiferromagnetic Ising model with deformation in
the triangular lattice, similar to our system considered here. At last, we
emphasize that since the leading-order correction is of the fourth order, the
sign of $J$ does not affect the final conclusion. In Section III, we will
consider an effective model with the ferromagnetic $J$.

###  II.2 iPEPS calculation

|  |  $D=2$
---
|  |  |  $D=3$ |  |  |
$J/J_{z}$ | $\langle H_{s}\rangle$ | $m$ | $\tilde{S}^{z}$ | $\tilde{S}^{+}$ | $\langle H_{s}\rangle$ | $m$ | $\tilde{S}^{z}$ | $\tilde{S}^{+}$
$0.1$ | -0.26327 | 0.47966 | 0.47966 | 0.00002 | -0.26336 | 0.47880 | 0.47880 | 0.00004
$0.2$ | -0.29746 | 0.43363 | 0.43363 | 0.00004 | -0.29843 | 0.42404 | 0.42404 | 0.00007
$0.3$ | -0.34539 | 0.38197 | 0.38197 | 0.00004 | -0.34853 | 0.35545 | 0.35545 | 0.00032
$0.4$ | -0.40150 | 0.33782 | 0.33782 | 0.00007 | -0.40753 | 0.29673 | 0.29673 | 0.00108
$0.5$ | -0.46177 | 0.30808 | 0.30808 | 0.00001 | -0.47140 | 0.25309 | 0.25308 | 0.00174
$0.6$ | -0.52501 | 0.28392 | 0.28392 | 0.00003 | -0.53779 | 0.22126 | 0.22122 | 0.00372
$0.7$ | -0.58987 | 0.26586 | 0.26586 | 0.00005 | -0.60623 | 0.18589 | 0.18586 | 0.00359
Table 1: We provide the values of energies and orders for several $J/J_{z}$,
with bond dimension $D=2$ or 3.

Since the RSPT applies well only for small $J$, we would like to see if the
conclusion still holds when the quantum fluctuation is strong. For this
purpose, we adopt the two dimensional tensor network ansatz, the infinite
projected entangled-pair state (iPEPS), to numerically calculate the
simplified Eq. (2)

$\displaystyle H_{s}$ $\displaystyle=J\sum_{\langle
ij\rangle}(\hat{S}^{+}_{i}\hat{S}^{-}_{j}+\hat{S}^{-}_{i}\hat{S}^{+}_{j})+J^{z}\sum_{\langle
ij\rangle\in h}\hat{S}^{z}_{i}\hat{S}^{z}_{j},$ (16)

where we ignore the magnetic field and consider only the constant $J^{z}$
coupling along the horizontal direction, since the diagonal sites are located
outside the blockade in a deformed triangular lattice (Fig. 1). By properly
choosing the pre-designated unit-cell size (in this work $2\times 2$ or
$4\times 2$ as shown in Fig. 1) and optimizing the $d\times D\times D\times
D\times D$ tensors, where $d$ represents the dimension of local Hilbert space
and $D$ is the virtual bond dimension, this tensor network ansatz serves as a
good variational wavefunction for the quantum many-body systems. We provide
some details of iPEPS in Appendix B.

To numerically estimate the ground state ansatz of Eq. (16), we need to adopt
numerous trials starting from different initial setups. It is because we
already recognize that there are many competing states (stripe, kinked, or
mixed states) so that we need to avoid the simulation being trapped in some
undesirable local minima. As a result, besides the random initial tensors we
also start our calculation by constructing the distinct product states,
corresponding to the stripe and kinked configurations separately. Since the
product state is of $D=1$, we then enlarge our tensors to the assigned $D$,
with extra tensor elements being small random numbers. Through this measure,
we guide the optimization toward obtaining the stripe or kinked state. After
the convergence is reached, the averaged magnetization is defined as

$\displaystyle m=\frac{1}{N_{C}}\sum_{i\in
C}\sqrt{\langle\hat{S}^{x}_{i}\rangle^{2}+\langle\hat{S}^{y}_{i}\rangle^{2}+\langle\hat{S}^{z}_{i}\rangle^{2}},$
(17)

where $C$ means the repeating unit cell and $N_{C}$ is the number of lattice
points within the cell. For the stripe pattern since there are two possible
choices for the iPEPS unit cell and they correspond to the wave numbers
$(0,\pi)$ and $(\pi,\pi)$ after Fourier transformation, we thus define the two
related orders for the stripe pattern. The first one is the diagonal magnetic
moment

$\displaystyle\tilde{S}^{z}=\frac{1}{N_{C}}\sum_{i\in
C}\sum_{\textbf{k}\in\textbf{k}_{1},\textbf{k}_{2}}\langle\hat{S}^{z}_{i}\rangle
e^{i\textbf{k}\cdot\textbf{r}_{i}},$ (18)

where $\textbf{k}_{1}=(\pi,\pi)$ and $\textbf{k}_{2}=(0,\pi)$.
$\textbf{r}_{i}$ represents the coordinate $(r_{i}^{x},r_{i}^{y})$ for each
site within the unit cell. For the off-diagonal order, we define

$\displaystyle\tilde{S}^{+}=\frac{1}{N_{C}}\sum_{i\in
C}|\langle\hat{S}^{+}_{i}\rangle|.$ (19)

In Table 1 we provide the lowest energies as well as the orders for several
different $J$. We can clearly see that for each $J/J_{z}$ we have
$m\approx\tilde{S}^{z}$, meaning that despite the different initial trials the
states of lowest energy always belong to the stripe state. Although the off-
diagonal orders are almost zero, $\langle\hat{S}^{z}\rangle$ is highly
supressed due to the quantum effect, suggesting that the initial setup for
RSPT might not be well applicable. However, our simulation indicates that the
conclusion does not change and the stripe state keeps being the ground state,
selected by the quantum OBD effect. It is also worth mentioning that for $D=3$
and $J>0.3$ no matter what initial states we adopt the ground state ansatz
always evolves toward the stripe state. We then conclude that our system,
where the Rydberg atoms are placed in a deformed triangular optical lattice,
favors the existence of stripe state.

##  III Dipolar hard-core bosons

Figure 3: Schematic demonstration of dipolar interaction in the triangular
lattice. Dots denote the lattice site and arrows represent the dipole
polarization. Such polarization can be parametrized with polar ($\theta$) and
azimuthal ($\phi$) angles. The lower right panel shows the projection of
lattice from above for a better demonstration of the azimuthal angle. Figure
4: CMFT phase diagrams with $nn$ dipolar interaction with polar angle equal to
(a) 0.4, (b) 0.5, and (c) 0.6, which approximate to 0.127$\pi$, 0.159$\pi$,
and 0.191$\pi$. The inset in (b) enlarges the area for two supersolid phases.
Solid (dashed) phase boundaries indicate the first- (second-) order phase
transition.

In the previous section we have revealed the quantum OBD effect for the
targeted Rydberg system and identified that the stripe state is more stable.
Next, we aim to unveil the possible underlying phases for the artificial
Hamiltonian. Instead of the original Rydberg Hamiltonian, however, we turn our
attention to the bosonic system with dipole-dipole interaction, which has been
shown to host many different phases in earlier works [51, 52, 53, 54, 55, 56,
57, 58, 59, 60, 61, 62]. In our previous work, we have studied the effect by
tilting the polar angle of dipole and realized a competing scenario between
different lattice structures [57]. Within the original square lattice, we
discovered that a three sub-lattice supersolid phase ought to show up by
tilting the polar angle. Related studies have demonstrated the potential of
such techniques in fabricating artificial physical scenarios of interest.
Therefore, in this section we study the hard-core bosons with dipole-dipole
interaction in the triangular lattice with different tilted dipolar angles,
mimicking the anisotropy. The general form of the dipole-dipole interaction is
written as

$\displaystyle V_{ij}=\frac{V}{r_{ij}^{3}}(1-3\text{cos}^{2}\alpha_{ij}),$
(20)

where $V$ is the interactive strength and $r_{ij}=|\vec{r}_{i}-\vec{r}_{j}|$.
$\alpha_{ij}$ is the included angle by the dipole moment and $\vec{r}_{ij}$.
For simplicity, we consider the scenario with only the $nn$ coupling and write
down the following Hamiltonian

$\displaystyle H=-t\sum_{\langle
ij\rangle}(\hat{b}^{\dagger}_{i}\hat{b}_{j}+H.C.)+\sum_{\langle
ij\rangle}V_{nn}\hat{n}_{i}\hat{n}_{j}-\mu\sum_{i}\hat{n}_{i},$ (21)

where $\hat{b}^{\dagger}_{i}$ and $\hat{b}_{i}$ represent the creation and
annihilation operators of the hard-core boson, with the number operator being
$\hat{n}_{i}=\hat{b}^{\dagger}_{i}\hat{b}_{i}$. $V_{nn}$ denotes the $nn$
dipole-dipole interaction among bosons. The interaction in Eq. (20) can be
easily tilted with an external magnetic field. In this work we study the case
when the dipole moments are tilted as $\phi=0$ (see Fig. 3) and thus for the
$nn$ interaction, the two interactive terms along diagonal direction are equal
and become attractive when polar angle $\theta$ is large enough. Therefore,
there are two different terms for $V_{nn}$

$\displaystyle V_{nn}^{h}=V$ (22) $\displaystyle
V_{nn}^{d}=V(1-\frac{9}{4}\text{sin}^{2}\theta),$

where indices $h$ and $d$ indicate the interaction along the horizontal and
diagonal directions, as shown in Fig. 1 labeled by green and black bonds,
separately. As a result, Eq. (21) is akin to Eq. (2) with negative $J$ since
there is a one-to-one mapping by simply replacing
$\hat{b}^{\dagger}_{i}\rightarrow\hat{S}^{+}_{i}$,
$\hat{b}_{i}\rightarrow\hat{S}^{-}_{i}$, and
$\hat{n}_{i}\rightarrow\hat{S}^{z}_{i}+\frac{1}{2}$ [7]. Therefore, by
numerically studying Eq. (21) with different tilting angles the resulting
phase diagrams cover a wide range of XXZ-like model with anisotropy in the
triangular lattice.

In this work, we adopt the CMFT [53, 54, 57] for the construction of phase
diagram. Some Details of applying CMFT are indicated in Appendix C. When the
strength of hopping term is small, the ground state is an ordered state with
$U(1)$ symmetry preserved. As we increase $t$, the solid order will dissolve
and phase transits to the superfluid state through a first-order phase
transition. However, in some small windows between solid and superfluid phases
two orders, the structural and superfluid orders, can co-exist and a
supersolid phase manifests. To identify distinct phases we first introduce the
order parameters adopted here. For solid and supersolid phases, they both
possess structural order which is defined as

$\displaystyle\tilde{n}(\textbf{k})=\frac{1}{N_{C}}\sum_{i\in
C}\langle\hat{n}_{i}\rangle e^{i\textbf{k}\cdot\textbf{r}_{i}},$ (23)

where $C$ again represents the repeating unit cell whose size is equal to
$3\times 3$ or $4\times 4$ in our CMFT calculation, and thus $N_{C}=9$ or 16.
As for the superfluid order, we calculate its condensate density defined as

$\displaystyle\rho_{0}=\frac{1}{N_{C}}\sum_{i\in
C}|\langle\hat{b}_{i}\rangle|^{2}.$ (24)

The above two order parameters help identify different phases in our numerical
phase diagrams.

We then conduct the CMFT calculation and plot the phase diagrams for several
polar angles in Fig. 4. When tilting is small (Fig. 4(a)), the diagonal solid
(blue) and supersolid (purple) dominate the phase diagram before entering the
superfluid phase. Its structural factor has the modulating momentum
$(2\pi/3,2\pi/3)$, and is named after the diagonal stripe in our earlier work
[57]. In the supersolid phase the $\tilde{n}(2\pi/3,2\pi/3)$ and $\rho_{0}$
orders co-exist. This phase appears after a continuous transition from the
diagonal solid. Further enlarging $t$, solid order disappears and the phase
transits into superfluid (red) phase discontinuously.

On the other hand, the central orange lobe indicates the classically highly
degenerate solid configurations (stripe, kinked, and mixed). Thanks to the
diagnosis of RSPT and iPEPS in the previous section, we have learned that the
true ground state must be in the stripe configuration, although the numerical
energies of stripe and kinked configurations by CMFT are very close to each
other due to the small $t$. The green phase in Fig. 4(b) and (c) represents a
supersolid phase coming from the stripe solid and thus we call it the stripe
supersolid. Note that our results indicate that the transition from stripe
solid to stripe supersolid, as well as the one from stripe supersolid to
superfluid are both continuous.

One can see from Fig. 4 that as we further tilt the polar angle, the central
lobe grows and its corresponding supersolid phase also starts to appear (Fig.
4(b)). Finally, as the polar angle is large enough, dominant ordered phases in
the phase diagram will be replaced by the stripe solid and supersolid (Fig.
4(c)). The scenario described here is just opposite to the one in our previous
work, where a square lattice effectively shifts to a triangular one by tilting
the polar angle [57]. Here, we starts from the triangular lattice where the
corresponding phase possesses three sub-lattices (see, e.g., Fig. 3 in Ref.
[57] for explanation), and eventually ends up with a square one with only two
sub-lattices, according to the phase diagram. At last, we emphasize that
including the long-range tail of dipole-dipole interaction will automatically
lift the degeneracy even in the classical limit. In such scenario we expect a
even more fruitful phase diagram with many different ground-state
configurations. We will leave the further studies for this issue in the future
work.

##  IV Conclusion

In the first part of this work we study the quantum OBD effect of $S=1/2$
system in the triangular lattice, which can be realized through quantum
simulator and we have illustrated the possible setup in the Introduction. When
a diagonal anisotropy is considered, there is a sub-extensive degeneracy of
lowest-energy configuration in the classical limit. Once the spin-flip term is
turned on, the overall Hamiltonian is akin to the XXZ model with anisotropic
interactive potential. With RSPT and iPEPS, we predict that the stripe
configuration is the true ground state, in agreement with the thermal
counterpart of this OBD effect according to the previous work [17]. For a
better demonstration of the possible underlying phases in our artificial
model, we then provide phase diagrams for the $nn$ dipolar hard-core bosonic
Hamiltonian with different tilting angles in the triangular lattice, using
CMFT. We exploit the competing scenario proposed in our early work [57] and
demonstrate that the effective lattice structure shifts from triangular
lattice to the square one, as well as various different phases including the
supersolid.

Our results provide a playground for the modern optical lattice with the cold-
atom technique. Placing the dipolar or Rydberg atoms in the optical lattice,
many artificial Hamiltonian can be realized and studied with experiments. We
anticipate that future experiment could help confirm our conclusion and
realize the potential phases through the programmable quantum simulator.
Controlling the size of blockade of Rydberg atom in an anisotropic triangular
lattice, moreover, we might reach a novel regime where the order disappears
and a state with larger entanglement entropy manifests [40]. It benefits a
further investigation using quantum Monte Carlo or tensor networks, and we
will leave it for the future consideration.

##  V Acknowledgement

W.-L.T. acknowledge the useful discussion with Mike Zhitomirsky during the
International Conference on Strongly Correlated Electron Systems 2023 (SCES
2023), Hyun-Yong Lee, Tsuyoshi Okubo, S. R. Ghazanfari, and Xinliang Lyu.
Authors also acknowledge the Advanced Study Group (ASG) Tensor Network
Approaches to Many-Body Systems organized by the Center for Theoretical
Physics of Complex Systems (PCS) of the Institute for Basic Science (IBS) in
Daejeon, Korea. H.-K.W. is supported by JQI-NSF-PFC (supported by NSF grant
PHY-1607611). N.K. is supported by JSPS KAKENHI Grants No. JP19H01809 and No.
JP23H01092. T.S. is supported by JSPS KAKENHI Grant. No. 21K03390. W.-L.T. is
supported by the Center of Innovations for Sustainable Quantum AI (JST Grant
Number JPMJPF2221).

Figure 5: Possible configurations during the perturbation process for
(a)kinked and (b)stripe states. For the ground state there are two
configurations in kinked state and thus their evolving must be considered
separately. Different configurations that give equal denominator contribution
are labeled with the same $\Gamma$.

##  Appendix A Details for the real space perturbation theory

Kinked |  |
---|---|---
$\Gamma_{k}(\leavevmode\hbox to11.37pt{\vbox to11.37pt{\pgfpicture\makeatletter\hbox{\hskip 5.68657pt\lower-5.68657pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{1}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})$ | $4S^{2}\Gamma_{a}+8S^{2}(J^{z}_{d}+J^{z}_{h})\cos\phi_{a}\cos\phi_{b}+32S^{2}J\sin\phi_{a}\sin\phi_{b}-2E_{\textbf{B}}^{a}$ |
$\Gamma_{k}(\leavevmode\hbox to11.37pt{\vbox to11.37pt{\pgfpicture\makeatletter\hbox{\hskip 5.68657pt\lower-5.68657pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{2}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})$ | $4S^{2}(\Gamma_{a}+\Gamma_{b})+4S^{2}(2J^{z}_{d}+J^{z}_{h})\cos\phi_{a}\cos\phi_{b}+24S^{2}J\sin\phi_{a}\sin\phi_{b}-E_{\textbf{B}}^{a}-E_{\textbf{B}}^{b}$ |
$\Gamma_{k}(\leavevmode\hbox to11.37pt{\vbox to11.37pt{\pgfpicture\makeatletter\hbox{\hskip 5.68657pt\lower-5.68657pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{3}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})$ | $4S^{2}(\Gamma_{a}+\Gamma_{b})+4S^{2}(J^{z}_{d}+2J^{z}_{h})\cos\phi_{a}\cos\phi_{b}+24S^{2}J\sin\phi_{a}\sin\phi_{b}-E_{\textbf{B}}^{a}-E_{\textbf{B}}^{b}$ |
$\Gamma_{k}(\leavevmode\hbox to11.37pt{\vbox to11.37pt{\pgfpicture\makeatletter\hbox{\hskip 5.68657pt\lower-5.68657pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{4}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})$ | $8S^{2}\Gamma_{a}+8S^{2}(J^{z}_{d}+J^{z}_{h})\cos\phi_{a}\cos\phi_{b}+32S^{2}J\sin\phi_{a}\sin\phi_{b}-2E_{\textbf{B}}^{a}$ |
$\Gamma^{\prime}_{k}(\leavevmode\hbox to11.37pt{\vbox to11.37pt{\pgfpicture\makeatletter\hbox{\hskip 5.68657pt\lower-5.68657pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{1}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})$ | $4S^{2}\Gamma_{b}+8S^{2}(J^{z}_{d}+J^{z}_{h})\cos\phi_{a}\cos\phi_{b}+32S^{2}J\sin\phi_{a}\sin\phi_{b}-2E_{\textbf{B}}^{b}$ |
$\Gamma^{\prime}_{k}(\leavevmode\hbox to11.37pt{\vbox to11.37pt{\pgfpicture\makeatletter\hbox{\hskip 5.68657pt\lower-5.68657pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{4}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})$ | $8S^{2}\Gamma_{b}+8S^{2}(J^{z}_{d}+J^{z}_{h})\cos\phi_{a}\cos\phi_{b}+32S^{2}J\sin\phi_{a}\sin\phi_{b}-2E_{\textbf{B}}^{b}$ |
Stripe |  |
$\Gamma_{s}(\leavevmode\hbox to11.37pt{\vbox to11.37pt{\pgfpicture\makeatletter\hbox{\hskip 5.68657pt\lower-5.68657pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{1}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})$ | $4S^{2}\Gamma_{b}+8S^{2}(J^{z}_{d}+J^{z}_{h})\cos\phi_{a}\cos\phi_{b}+32S^{2}J\sin\phi_{a}\sin\phi_{b}-2E_{\textbf{B}}^{b}$ |
$\Gamma_{s}(\leavevmode\hbox to11.37pt{\vbox to11.37pt{\pgfpicture\makeatletter\hbox{\hskip 5.68657pt\lower-5.68657pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{2}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})$ | $4S^{2}\Gamma_{a}+8S^{2}(J^{z}_{d}+J^{z}_{h})\cos\phi_{a}\cos\phi_{b}+32S^{2}J\sin\phi_{a}\sin\phi_{b}-2E_{\textbf{B}}^{a}$ |
$\Gamma_{s}(\leavevmode\hbox to11.37pt{\vbox to11.37pt{\pgfpicture\makeatletter\hbox{\hskip 5.68657pt\lower-5.68657pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{3}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})$ | $4S^{2}(\Gamma_{a}+\Gamma_{b})+4S^{2}(J^{z}_{d}+2J^{z}_{h})\cos\phi_{a}\cos\phi_{b}+24S^{2}J\sin\phi_{a}\sin\phi_{b}-E_{\textbf{B}}^{a}-E_{\textbf{B}}^{b}$ |
$\Gamma_{s}(\leavevmode\hbox to11.37pt{\vbox to11.37pt{\pgfpicture\makeatletter\hbox{\hskip 5.68657pt\lower-5.68657pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{4}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})$ | $4S^{2}(\Gamma_{a}+\Gamma_{b})+4S^{2}(2J^{z}_{d}+J^{z}_{h})\cos\phi_{a}\cos\phi_{b}+24S^{2}J\sin\phi_{a}\sin\phi_{b}-E_{\textbf{B}}^{a}-E_{\textbf{B}}^{b}$ |
$\Gamma_{s}(\leavevmode\hbox to11.37pt{\vbox to11.37pt{\pgfpicture\makeatletter\hbox{\hskip 5.68657pt\lower-5.68657pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{5}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})$ | $4S^{2}(\Gamma_{a}+\Gamma_{b})+8S^{2}(J^{z}_{d}+J^{z}_{h})\cos\phi_{a}\cos\phi_{b}+32S^{2}J\sin\phi_{a}\sin\phi_{b}-E_{\textbf{B}}^{a}-E_{\textbf{B}}^{b}$ |
Table 2: Different $\Gamma$ for the kinked configuration.
$E_{\textbf{B}}^{a}=2S\textbf{B}\cdot\hat{e}^{z}_{a}$ and
$E_{\textbf{B}}^{b}=2S\textbf{B}\cdot\hat{e}^{z}_{b}$.
$\Gamma_{a}=J^{z}_{d}\cos^{2}\phi_{a}+2J\sin^{2}\phi_{a}$ and
$\Gamma_{b}=J^{z}_{d}\cos^{2}\phi_{b}+2J\sin^{2}\phi_{b}$.

In this Appendix, we work on the details for RSPT. With the choice for the
local coordinate shown in the main text (Eqs. (13) and (14)), we can now write
down the corresponding $J^{(1)}_{ij}$ and $J^{(2)}_{ij}$ terms. For diagonal
bonds we have

$\displaystyle
J^{d(1)}_{aa}=\frac{1}{4}\sin^{2}\phi_{a}(J^{z}_{d}-2J)S^{+}_{a}S^{+}_{a}+H.C.$
(25) $\displaystyle
J^{d(1)}_{bb}=\frac{1}{4}\sin^{2}\phi_{b}(J^{z}_{d}-2J)S^{+}_{b}S^{+}_{b}+H.C.$
$\displaystyle
J^{d(1)}_{ab}=\frac{1}{4}[2J(\cos\phi_{a}\cos\phi_{b}-1)+J^{z}_{d}\sin\phi_{a}\sin\phi_{b}]S^{+}_{a}S^{+}_{b}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;+H.C.,$

where we assign $J^{d(1)}_{1}=\frac{1}{4}\sin^{2}\phi_{a}(J^{z}_{d}-2J)$,
$J^{d(1)}_{2}=\frac{1}{4}\sin^{2}\phi_{b}(J^{z}_{d}-2J)$, and
$J^{d(1)}_{3}=\frac{1}{4}[2J(\cos\phi_{a}\cos\phi_{b}-1)+J^{z}_{d}\sin\phi_{a}\sin\phi_{b}]$.
And

$\displaystyle
J^{d(2)}_{aa}=\frac{1}{4}[2J(1+\cos^{2}\phi_{a})+J^{z}_{d}\sin^{2}\phi_{a}]S^{+}_{a}S^{-}_{a}+H.C.$
(26) $\displaystyle
J^{d(2)}_{bb}=\frac{1}{4}[2J(1+\cos^{2}\phi_{b})+J^{z}_{d}\sin^{2}\phi_{b}]S^{+}_{b}S^{-}_{b}+H.C.$
$\displaystyle
J^{d(2)}_{ab}=\frac{1}{4}[2J(1+\cos\phi_{a}\cos\phi_{b})+J^{z}_{d}\sin\phi_{a}\sin\phi_{b}]S^{+}_{a}S^{-}_{b}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;+H.C.,$

where we again set
$J^{d(2)}_{1}=\frac{1}{4}[2J(1+\cos^{2}\phi_{a})+J^{z}_{d}\sin^{2}\phi_{a}]$,
$J^{d(2)}_{2}=\frac{1}{4}[2J(1+\cos^{2}\phi_{b})+J^{z}_{d}\sin^{2}\phi_{b}]$,
and
$J^{d(2)}_{3}=\frac{1}{4}[2J(1+\cos\phi_{a}\cos\phi_{b})+J^{z}_{d}\sin\phi_{a}\sin\phi_{b}]$.
For the horizontal bonds we simply need to replace $J^{z}_{d}$ in Eq. (25) and
(26) with $J^{z}_{h}$, but in fact it does not affect the leading-order
correction since our cluster of interest is the vertical diamond (Fig. 2(a)).
Moreover, all possible intermediate configurations as well as their
corresponding $-\langle\psi|H_{\text{unperturbed}}|\psi\rangle$, labeled by
different $\Gamma$ and composing the denominator of energy correction, also
need to be considered. For this purpose, we first list all the possible
intermediate configurations during the spin flipping process, shown in Fig. 5,
with each configuration’s corresponding minus unperturbed energy,
$E_{0}-E_{\psi}$, labeled by different $\Gamma$. We list their formula in
Table 2. Note that $\Gamma^{\prime}_{k}(\leavevmode\hbox to11.37pt{\vbox
to11.37pt{\pgfpicture\makeatletter\hbox{\hskip 5.68657pt\lower-5.68657pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{2}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})=\Gamma_{k}(\leavevmode\hbox
to11.37pt{\vbox to11.37pt{\pgfpicture\makeatletter\hbox{\hskip
5.68657pt\lower-5.68657pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{2}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})$,
$\Gamma^{\prime}_{k}(\leavevmode\hbox to11.37pt{\vbox
to11.37pt{\pgfpicture\makeatletter\hbox{\hskip 5.68657pt\lower-5.68657pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{3}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})=\Gamma_{k}(\leavevmode\hbox
to11.37pt{\vbox to11.37pt{\pgfpicture\makeatletter\hbox{\hskip
5.68657pt\lower-5.68657pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{3}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})$, and
$\Gamma^{\prime}_{s}(\leavevmode\hbox to11.37pt{\vbox
to11.37pt{\pgfpicture\makeatletter\hbox{\hskip 5.68657pt\lower-5.68657pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{1}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})=\Gamma_{s}(\leavevmode\hbox
to11.37pt{\vbox to11.37pt{\pgfpicture\makeatletter\hbox{\hskip
5.68657pt\lower-5.68657pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{2}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})$,
$\Gamma^{\prime}_{s}(\leavevmode\hbox to11.37pt{\vbox
to11.37pt{\pgfpicture\makeatletter\hbox{\hskip 5.68657pt\lower-5.68657pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{2}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})=\Gamma_{s}(\leavevmode\hbox
to11.37pt{\vbox to11.37pt{\pgfpicture\makeatletter\hbox{\hskip
5.68657pt\lower-5.68657pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{1}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})$,
$\Gamma^{\prime}_{s}(\leavevmode\hbox to11.37pt{\vbox
to11.37pt{\pgfpicture\makeatletter\hbox{\hskip 5.68657pt\lower-5.68657pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{3}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})=\Gamma_{s}(\leavevmode\hbox
to11.37pt{\vbox to11.37pt{\pgfpicture\makeatletter\hbox{\hskip
5.68657pt\lower-5.68657pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{3}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})$,
$\Gamma^{\prime}_{s}(\leavevmode\hbox to11.37pt{\vbox
to11.37pt{\pgfpicture\makeatletter\hbox{\hskip 5.68657pt\lower-5.68657pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{4}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})=\Gamma_{s}(\leavevmode\hbox
to11.37pt{\vbox to11.37pt{\pgfpicture\makeatletter\hbox{\hskip
5.68657pt\lower-5.68657pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{4}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})$,
$\Gamma^{\prime}_{s}(\leavevmode\hbox to11.37pt{\vbox
to11.37pt{\pgfpicture\makeatletter\hbox{\hskip 5.68657pt\lower-5.68657pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{5}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})=\Gamma_{s}(\leavevmode\hbox
to11.37pt{\vbox to11.37pt{\pgfpicture\makeatletter\hbox{\hskip
5.68657pt\lower-5.68657pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{5}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})$. For simplicity, we
further conclude that

$\displaystyle\Gamma_{1}=\Gamma_{k}(\leavevmode\hbox to11.37pt{\vbox
to11.37pt{\pgfpicture\makeatletter\hbox{\hskip 5.68657pt\lower-5.68657pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{1}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})=\Gamma_{s}(\leavevmode\hbox
to11.37pt{\vbox to11.37pt{\pgfpicture\makeatletter\hbox{\hskip
5.68657pt\lower-5.68657pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{2}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})=\Gamma^{\prime}_{s}(\leavevmode\hbox
to11.37pt{\vbox to11.37pt{\pgfpicture\makeatletter\hbox{\hskip
5.68657pt\lower-5.68657pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{1}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}),$ (27)
$\displaystyle\Gamma_{2}=\Gamma^{\prime}_{k}(\leavevmode\hbox to11.37pt{\vbox
to11.37pt{\pgfpicture\makeatletter\hbox{\hskip 5.68657pt\lower-5.68657pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{1}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})=\Gamma_{s}(\leavevmode\hbox
to11.37pt{\vbox to11.37pt{\pgfpicture\makeatletter\hbox{\hskip
5.68657pt\lower-5.68657pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{1}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})=\Gamma^{\prime}_{s}(\leavevmode\hbox
to11.37pt{\vbox to11.37pt{\pgfpicture\makeatletter\hbox{\hskip
5.68657pt\lower-5.68657pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{2}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}),$
$\displaystyle\Gamma_{3}=\Gamma_{k}(\leavevmode\hbox to11.37pt{\vbox
to11.37pt{\pgfpicture\makeatletter\hbox{\hskip 5.68657pt\lower-5.68657pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{3}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})=\Gamma^{\prime}_{k}(\leavevmode\hbox
to11.37pt{\vbox to11.37pt{\pgfpicture\makeatletter\hbox{\hskip
5.68657pt\lower-5.68657pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{3}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})=\Gamma_{s}(\leavevmode\hbox
to11.37pt{\vbox to11.37pt{\pgfpicture\makeatletter\hbox{\hskip
5.68657pt\lower-5.68657pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{3}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})=\Gamma^{\prime}_{s}(\leavevmode\hbox
to11.37pt{\vbox to11.37pt{\pgfpicture\makeatletter\hbox{\hskip
5.68657pt\lower-5.68657pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{3}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}),$
$\displaystyle\Gamma_{4}=\Gamma_{k}(\leavevmode\hbox to11.37pt{\vbox
to11.37pt{\pgfpicture\makeatletter\hbox{\hskip 5.68657pt\lower-5.68657pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{2}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})=\Gamma^{\prime}_{k}(\leavevmode\hbox
to11.37pt{\vbox to11.37pt{\pgfpicture\makeatletter\hbox{\hskip
5.68657pt\lower-5.68657pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{2}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})=\Gamma_{s}(\leavevmode\hbox
to11.37pt{\vbox to11.37pt{\pgfpicture\makeatletter\hbox{\hskip
5.68657pt\lower-5.68657pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{4}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})=\Gamma^{\prime}_{s}(\leavevmode\hbox
to11.37pt{\vbox to11.37pt{\pgfpicture\makeatletter\hbox{\hskip
5.68657pt\lower-5.68657pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{4}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}),$
$\displaystyle\Gamma_{5}=\Gamma_{s}(\leavevmode\hbox to11.37pt{\vbox
to11.37pt{\pgfpicture\makeatletter\hbox{\hskip 5.68657pt\lower-5.68657pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{5}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})=\Gamma^{\prime}_{s}(\leavevmode\hbox
to11.37pt{\vbox to11.37pt{\pgfpicture\makeatletter\hbox{\hskip
5.68657pt\lower-5.68657pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{5}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}).$

Along with $\Gamma_{k}(\leavevmode\hbox to11.37pt{\vbox
to11.37pt{\pgfpicture\makeatletter\hbox{\hskip 5.68657pt\lower-5.68657pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{4}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})$ and
$\Gamma^{\prime}_{k}(\leavevmode\hbox to11.37pt{\vbox
to11.37pt{\pgfpicture\makeatletter\hbox{\hskip 5.68657pt\lower-5.68657pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{4}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})$, we have obtained
all the ingredients needed for the denominators of the energy corrections.

Finally, through considering all possible intermediate processes akin to the
one in Fig. 2(b), we obtain the energy correction. For the kinked state,

$\displaystyle\delta
E^{(4)}_{\text{kinked}}=\frac{(J^{d(1)}_{1})^{2}(J^{d(2)}_{3})^{2}}{(\Gamma_{1})^{2}\Gamma_{4}}+\frac{(J^{d(1)}_{2})^{2}(J^{d(2)}_{3})^{2}}{(\Gamma_{2})^{2}\Gamma_{4}}$
(28)
$\displaystyle+\frac{(J^{d(1)}_{3})^{2}(J^{d(2)}_{1})^{2}}{(\Gamma_{3})^{2}\Gamma_{4}}+\frac{(J^{d(1)}_{3})^{2}(J^{d(2)}_{2})^{2}}{(\Gamma_{3})^{2}\Gamma_{4}}$
$\displaystyle+\frac{4J^{d(1)}_{1}J^{d(1)}_{3}J^{d(2)}_{1}J^{d(2)}_{3}}{\Gamma_{1}\Gamma_{3}\Gamma_{k}(\leavevmode\hbox
to11.37pt{\vbox to11.37pt{\pgfpicture\makeatletter\hbox{\hskip
5.68657pt\lower-5.68657pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{4}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})}+\frac{4J^{d(1)}_{2}J^{d(1)}_{3}J^{d(2)}_{2}J^{d(2)}_{3}}{\Gamma_{2}\Gamma_{3}\Gamma^{\prime}_{k}(\leavevmode\hbox
to11.37pt{\vbox to11.37pt{\pgfpicture\makeatletter\hbox{\hskip
5.68657pt\lower-5.68657pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{4}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})}$
$\displaystyle+\frac{2J^{d(1)}_{1}J^{d(1)}_{3}J^{d(2)}_{1}J^{d(2)}_{3}}{\Gamma_{1}\Gamma_{3}\Gamma_{4}}+\frac{2J^{d(1)}_{2}J^{d(1)}_{3}J^{d(2)}_{2}J^{d(2)}_{3}}{\Gamma_{2}\Gamma_{3}\Gamma_{4}},$

and for stripe state,

$\displaystyle\delta
E^{(4)}_{\text{stripe}}=\frac{2(J^{d(1)}_{3})^{2}J^{d(2)}_{1}J^{d(2)}_{2}}{(\Gamma_{3})^{2}\Gamma_{4}}+\frac{2J^{d(1)}_{1}J^{d(1)}_{2}(J^{d(2)}_{3})^{2}}{\Gamma_{1}\Gamma_{2}\Gamma_{4}}$
(29)
$\displaystyle+\frac{2(J^{d(1)}_{3})^{2}J^{d(2)}_{1}J^{d(2)}_{2}}{(\Gamma_{3})^{2}\Gamma_{5}}+\frac{2J^{d(1)}_{1}J^{d(1)}_{2}(J^{d(2)}_{3})^{2}}{\Gamma_{1}\Gamma_{2}\Gamma_{5}}$
$\displaystyle+\frac{2J^{d(1)}_{1}J^{d(1)}_{3}J^{d(2)}_{2}J^{d(2)}_{3}}{\Gamma_{1}\Gamma_{3}\Gamma_{4}}+\frac{2J^{d(1)}_{2}J^{d(1)}_{3}J^{d(2)}_{1}J^{d(2)}_{3}}{\Gamma_{2}\Gamma_{3}\Gamma_{5}}$
$\displaystyle+\frac{2J^{d(1)}_{1}J^{d(1)}_{3}J^{d(2)}_{2}J^{d(2)}_{3}}{\Gamma_{1}\Gamma_{3}\Gamma_{5}}+\frac{2J^{d(1)}_{2}J^{d(1)}_{3}J^{d(2)}_{1}J^{d(2)}_{3}}{\Gamma_{2}\Gamma_{3}\Gamma_{4}},$

where $\delta E^{(4)}$ is the energy correction per site. Eqs. (28) and (29)
will then be used for studying the degeneracy lifting.

##  Appendix B Infinite projected entangled-pair state

Figure 6: (a) Composition of the rank-5 tensors into the formation of
triangular lattice. Grey solid bonds represent the auxiliary bonds with
dimension $D$, and blue open legs are the local indices with dimension $d=2$
for the $S=1/2$ system. Dashed bonds are not the real legs for the tensors but
we act the
$H_{ij}=\hat{S}^{+}_{i}\hat{S}^{-}_{j}+\hat{S}^{-}_{i}\hat{S}^{+}_{j}$ on
these bonds too for simulating the triangular lattice. (b) The double layered
tensors with bond dimension $D^{2}$ in the bulk, enclosed by the blue dashed
box, and the environment tensors. The dimension of bonds interconnecting $C$
and $T$ tensors is labeled by $\chi$, with $\chi\geq D^{2}$ to guarantee the
accuracy.

To study our effective model when the fluctuation is strong, we have no other
option but to turn to the assistance of numerical tool. Due to the frustration
that hinders the usage of quantum Monte Carlo, we adopt the infinite projected
entangled-pair state (iPEPS) [48, 49, 50] for our purpose. There are two parts
for iPEPS tensor network. One is the tensors within the repeating unit cell,
where each composing unit is a rank-5 tensor with four auxiliary bonds
(dimension $D$) and one physical bond representing the size of local Hilbert
space (dimension $d=2$ for $S=1/2$). Since originally iPEPS was designed for
the square lattice, in the triangular lattice the coordinate number for each
site is equal to six. Instead of increasing the number of auxiliary bonds
which would make the computation cumbersome, however, we act the local
Hamiltonian also along the virtual bonds (dashed bonds in Fig. 6(a)) when
calculating the energy. As a result, the overall lattice structure is
effectively equivalent to the triangular lattice.

Another important part for iPEPS lies on the environment tensors. Since the
thermodynamics limit can be extrapolated through the corner transfer matrix
renormalization group (CTMRG) procedure [63, 64, 65], we need to first trace
out the physical legs of each tensor with its adjoint one, forming the so-
called double layered tensor where each bond is of dimension $D^{2}$. We then
adopt the CTMRG and after the fixed-point tensors have been found, they act as
the effective “bath” surrounding the bulk tensors, where we label the ones on
the corners with $C$ and ones on the edges with $T$. One example with the bulk
size equal to $4\times 2$ is provided in Fig. 6(b).

At last, we need to optimize the tensors so that the overall network can
represent the ground state ansatz of the target Hamiltonian. Here we adopt the
variational optimization with the usage of automatic differentiation, first
introduced by Liao et al. [66]. We adopt the widely applicable package, peps-
torch [67], for our calculation. For readers aiming for some further
information, we refer to our previous works in Refs. [68, 69].

##  Appendix C Cluster mean-field theory

In this Appendix we briefly introduce the CMFT method that we use for
numerically solving Eq. (21). For adopting this method, we first divide our
Hamiltonian into two parts: $H_{C}$ within the chosen cluster, and
$H_{\partial C}$ that contains the terms connecting the bulk to the
environment on the boundary of the cluster. $H_{C}$ possesses the exact form
of Eq. (21), and the mean-field decoupling only takes place in $H_{\partial
C}$, written as

$\displaystyle H_{\partial C}=$
$\displaystyle-t{\sum_{ij}}^{\prime}(b^{\dagger}_{i}\langle
b_{j}\rangle+H.C.)+{\sum_{ij}}^{\prime}V_{ij}n_{i}\langle n_{j}\rangle,$ (30)

where the prime symbol indicates that this summation is between site $i$ on
the boundary of the cluster and site $j$ connected to $i$ outside the cluster.
Our effective Hamiltonian is then written as $H_{\text{eff}}=H_{C}+H_{\partial
C}$. Although we write down the most general form in Eq. (30), in this work we
apply CMFT for the case with only the nearest-neighbor coupling. We then
exactly diagonalize the effective Hamiltonian and obtain the ground state to
calculate the mean-field parameters, $\langle b_{j}\rangle$ and $\langle
n_{j}\rangle$, for the following iteration. After the mean-field parameters
converge to consistent values, our calculation reaches its self-consistent
solution. In CMFT, exact diagonalizations are performed within the chosen
cluster. Therefore, it is not expected to include the long-range entanglement
beyond the cluster size. To study this finite size effect and infer the phases
in the thermodynamic limit, a common practice is to compute the order
parameters at varying cluster sizes and extrapolate it to the infinity.

CMFT is considered to be more accurate than the regular single-site mean-field
theory, for that it can capture the correct correlation within the selected
cluster. Thus it is suitable for states whose correlation length is small,
meaning that deep inside the ordered phase (or away from transition boundary)
we can use CMFT and obtain quite accurate results. Because of the above-
mentioned reason, for the boundary lines shown in Fig. 4, we have extrapolated
the energies and order parameters for both the first- and second-order
transition boundaries to give a better outcome. In addition, due to the
randomness from the initialization process, CMFT could converge to distinct
phases with the same set of parameters. In this case, one should compare the
energies among them to determine the correct ground state. Accordingly, it
becomes a challenge when two stable phases appear with energy differences less
than or comparable to the finite size effect.

In our CMFT calculation for Eq. (21), it provides definitive results on most
of the phases shown in Fig. 4. However, it is nontrivial to resolve the ground
state phase between the stripe and kinked supersolid due to the fact that the
energy difference between these two lies in the forth-order energy correction
from the perturbation theory. For the $4\times 4$ cluster we employed, this
difference is comparable to the finite size effect coming from the boundary,
as can be estimated by investigating the changes in energy upon displacements
of the solid patterns. However, by checking the order parameters we can still
diagnose different phases.

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