# Frustrated mixed-spin ladders:
An intermediate phase between rung-singlet and Haldane phases

N. Ahmadi  Department of Physics, Institute for Advanced Studies in Basic
Sciences (IASBS), Zanjan 45137-66731, Iran  J. Abouie<EMAIL_ADDRESS>Department of Physics, Institute for Advanced Studies in Basic Sciences
(IASBS), Zanjan 45137-66731, Iran  R. Haghshenas  Division of Chemistry and
Chemical Engineering, California Institute of Technology, Pasadena, California
91125, USA

###### Abstract

In frustrated spin ladders composed of antiferromagnetically coupled chains,
homogeneous or inhomogeneous, the interplay of frustration and correlations
causes the emergence of two phases, Haldane (H) phase and rung singlet (RS)
phase, in which the transition between these two phases had been under debate.
In this paper we investigate the ground state phase diagram of frustrated
mixed-spin-(1, 1/2) ladders, using the notions from quantum information theory
such as entanglement entropy, Schmidt gap and entanglement levels
degeneracies, and also defining various local and nonlocal order parameters.
Employing two numerical techniques, the infinite time-evolving block
decimation (iTEBD) and density matrix renormalization group (DMRG) algorithms,
we obtain the ground state phase diagram of the ladder, and demonstrate that
there is an intermediate phase between RS and H phases, where the ground state
is disordered and the entanglement spectrum follow no particular pattern.

##  I Introduction

Low dimensional frustrated spin systems are one of the most active research
fields in physics which have attracted a lot of attentions in the fields of
topological quantum matters and cold atoms systems, aside from their
pioneering importance on the understanding of quantum magnetism and emergent
phenomena Frustration causes different exotic orders such as topological
orders, spin liquids, spin supersolids, helical and chiral orders and so on,
to appear in the ground state phase diagram of the spin systems. Diep (2005);
Zhou  _et al._ (2017); Nisoli  _et al._ (2013); Ortiz-Ambriz  _et al._ (2019);
Zapf  _et al._ (2014a, b) Frustrated spin ladders are one of the low-
dimensional systems that have received special attentions for several reasons;
i) due to their low dimensionality, the interplay of frustration and quantum
correlations causes many topological and nontopological phases such as the H
phase, dimer orders, RS phase, spin liquids and Luttinger liquid to appear in
their ground state phase diagram, Fáth  _et al._ (2001); Kim  _et al._ (2000);
Hijii  _et al._ (2005); Karrasch  _et al._ (2015); Ramos and Xavier (2014);
Aristov  _et al._ (2010); Batchelor  _et al._ (2007); Hikihara and Starykh
(2010); Mishra  _et al._ (2015); Bhaseen and Tsvelik (2003); Sugimoto  _et
al._ (2018); Sasaki  _et al._ (2020); Hung  _et al._ (2006); Barbarino  _et
al._ (2018); Vekua and Honecker (2006); Liu  _et al._ (2008) ii) they are
quasi-one dimension, and it is expected that they show characteristics of both
one- and two-dimensional systemsDagotto and Rice (1996), iii) and finally
because of their low dimensionality there are various known numerical and
analytical methods which can be employed to examine their ground state as well
as their excitations behaviors.Allen  _et al._ (2000); Fouet  _et al._ (2006)
Mixed spin ladders are a group of materials that, despite their great
importance, have received less attentions. The heterogeneity in these systems
causes the emergence of behaviors that are not seen in homogeneous
systems.Abouie and Langari (2004); Abouie  _et al._ (2006); Heydarinasab and
Abouie (2017); Rezai  _et al._ (2010); Heydarinasab and Abouie (2020); Langari
_et al._ (2011) The coexistence of different kinds of spins do not necessarily
yield to magnetic ground states. Depending on the size of spins there could be
three different possibilities. For a mixed-spin system with a unit cell
consists of ($\tau,\sigma$) spins, if $\tau>2\sigma$ then the system mainly
has a ferromagnetic character, if $\tau<2\sigma$ , the system has a
predominantly antiferromagnetic character, and in the case which
$\tau=2\sigma$, the system shows a mixture of both ferromagnetic and
antiferromagnetic characters. Therefore, mixed-spin-(1, 1/2) systems are
expected to have interesting features.

In this paper we study the ground state phase diagram of a mixed-spin-(1, 1/2)
ladder (illustrated in Fig. 1) in the presence of rung, leg and diagonal
interactions. In a wide range of diagonal interactions, the system experiences
two genetically disordered phases; a H phase and a RS phase, but there is no
definite result in the type of phase transition between this two phases. Based
on very few studies on this system, the H and the RS phases are interconnected
by a first order phase transition.Chandra  _et al._ (2010) But whether or not
there is an intermediate phase between these two phases is controversial. It
has been shown that frustrated mixed-spin-(1,1/2) ladders can be mapped onto a
frustrated spin-$1/2$ ladderChandra  _et al._ (2010) where many efforts have
been done in understanding the real nature of the phase transition between the
rung-singlet and the H phase. It is not clear whether it is a first order
phase transitionKim  _et al._ (2008); Hung  _et al._ (2006); Weihong  _et al._
(1998) or there is an intermediate phase between the RS and the H phases.Liu
_et al._ (2008); Vekua and Honecker (2006); Starykh and Balents (2004) In
order to reach a definite conclusion we must carry on more sophisticated
investigations.

In this paper, we examine the ground state phase diagram of the frustrated
mixed-spin-(1,1/2) ladders, using basic notions from quantum information
theory such as entanglement. Recent developments in constructing new
algorithms to efficiently simulate quantum many-body wave functions has
revealed the remarkable role of entanglement and provides astonishing tools to
understand many body systems and quantum phase transitions.White (1992); Vidal
(2003); Schollwöck (2005); Vidal (2003, 2004); Jordan  _et al._ (2008);
Haghshenas and Sheng (2018); Haghshenas  _et al._ (2019a, b) The most well-
known measure of entanglement, is the Von Neumann entanglement entropy (EE)
which is widely used to detect quantum phase transitions Osterloh  _et al._
(2002) as well as topological properties of states.Levin and Wen (2006);
Kitaev and Preskill (2006) Moreover, the entanglement spectrum (ES), the
eigenvalues of reduced density matrices, is a remarkable tool in the
characterization of topological properties of quantum many-body phases.Li and
Haldane (2008); Pollmann  _et al._ (2010); Haghshenas  _et al._ (2014)
Actually, the study of the low-lying part of the ES allows us to detect
topological properties of a state or gives direct access to the excitation
spectrum of edges. Using two numerical techniques, iTEBD and DMRG, we obtain
the EE and ES levels degeneracies, and demonstrate that the H and the RS
phases are separated by an intermediate disordered phase characterized by a
different ES levels degeneracies.

While examining the presence or absence of an intermediate phase in this
system, we also compare both the results of iTEBD and DMRG. We demonstrate
that in the vicinity of the RS-H transition point, i.e., in the intermediate
phase, in contrary to DMRG, the iTEBD is very sensitive to initial conditions,
and the ground state order predicted by iTEBD can not be trusted.

This paper is organized as follows: In sec. II, we will introduce the
Hamiltonian of the mixed-spin ladder and discus its ground state properties by
a perturbation theory. In Sec. III, the numerical iTEBD technique and its
generalization to the mixed-spin-(1, 1/2) ladders will be explained. In Sec.
IV, we will present a comprehensive study of the ground state phase diagram of
the ladder in the absence of diagonal interactions, by introducing different
local order parameters and investigating the behavior of the EE, ES levels
degeneracies, and Schmidt gap. In Sec V, the effects of diagonal interactions
will be discussed. Finally the summary and conclusion are presented in Sec.
VI.

##  II MODEL HAMILTONIAN AND SYMMETRIES

Let us consider a frustrated two-leg spin ladder composed of two different
kinds of spin, $\sigma$ and $\tau$, describing by the following Hamiltonian:

$H=H_{legs}+H_{rung}+H_{diag.},$ (1)

with

$\displaystyle H_{legs}$ $\displaystyle=$ $\displaystyle
J_{l}\sum_{n=1,2}\sum_{i}{(\sigma_{i}^{(n)}\cdot\tau_{i}^{(n)}+\tau_{i}^{(n)}\cdot\sigma_{i+1}^{(n)})},$
$\displaystyle H_{rung}$ $\displaystyle=$ $\displaystyle
J_{r}\sum_{i}{(\sigma_{i}^{(1)}\cdot\sigma_{i}^{(2)}+\tau_{i}^{(1)}\cdot\tau_{i}^{(2)})},$
$\displaystyle H_{diag.}$ $\displaystyle=$ $\displaystyle
J_{d}\sum_{i}(\sigma_{i}^{(1)}\cdot\tau_{i}^{(2)}+\sigma_{i}^{(2)}\cdot\tau_{i}^{(1)}$
$\displaystyle~{}~{}~{}~{}+\tau_{i}^{(1)}\cdot\sigma_{i+1}^{(2)}+\tau_{i}^{(2)}\cdot\sigma_{i+1}^{(1)}).$

where $n$ is the label of legs, and the summations run over unit cells (see
Fig. 1). Here $J_{l}>0$ is the intra-leg exchange interaction between spins
$\sigma$ and $\tau$, and $J_{r}$ and $J_{d}>0$ refer to the inter-legs
interactions, the former is the rung coupling and the latter is the diagonal
interaction between spins $\sigma$ and $\tau$.

Figure 1: (Color online) Pictorial representation of a mixed-spin ladder with
different intra- and inter-legs exchange couplings. The blue and red dots are
representing the spin-$\sigma$ and spin-$\tau$. The index $i$ refers to cell
number.

The Hamiltonian in Eq. (1) possesses the SU(2), time-reversal $\mathcal{T}$
and leg-swap $\mathfrak{s}$ symmetries. Under time reversal and leg-swap
symmetries the spin operators transform as:

$\displaystyle\mathcal{T}\sigma_{i}^{(n)}\mathcal{T}^{-1}$ $\displaystyle=$
$\displaystyle-\sigma_{i}^{(n)},~{}~{}~{}\mathcal{T}\tau_{i}^{(n)}\mathcal{T}^{-1}=-\tau_{i}^{(n)},$
$\displaystyle\mathfrak{s}\sigma_{i}^{(1)}\mathfrak{s}^{-1}$ $\displaystyle=$
$\displaystyle\sigma_{i}^{(2)},~{}~{}~{}\mathfrak{s}\tau_{i}^{(1)}\mathfrak{s}^{-1}=\tau_{i}^{(2)}.$

Before examining the ground state phase diagram of this system in detail, it
is intuitive to study the ground state properties of the system in the extreme
limits of strong and weak rung couplings. In the absence of diagonal
interactions the system is unfrustrated. In the limit of vanishing rung
couplings, the ladder is decoupled into two equivalent mixed-spin-(1,1/2)
chains. Mixed-spin chains are a special class of spin models which their
universality class is completely different from the uniform spin models.
According to the Lieb-Mattis theorem, the ground state of a mixed-
spin-($\sigma,\tau$) chain has total spin $S_{tot}=(\tau-\sigma)L$
$(\tau>\sigma)$, with $L$ being the number of unit cells on the chain, and
thus is necessarily long-range ferrimagnetic ordered. Since the elementary
cell consists of two spins, linear spin wave theory yields two types of
magnons: a gapless acoustical branch with $S^{z}=L/2-1$ and a gapped optical
branch with $S^{z}=L/2+1$, respectively with the dispersions $\omega_{k}^{-}$
and $\omega_{k}^{+}$, given by
$\omega_{k}^{\mp}/J_{l}=\mp\frac{1}{2}+(\frac{1}{4}+2\sin^{2}k)^{1/2}$. Though
the antiferromagnetic gap within the lowest-order spin-wave theory is
$\Delta_{opt.}=J_{l}$. Brehmer  _et al._ (1997); Yamamoto  _et al._ (1998,
2000) When rung couplings are switched on, in the case of $J_{r}>0$, the
gapless mode becomes linear reflecting the antiferromagnetic character of the
ladder system, whereas the optical modes moves upward. Trumper and Gazza
(2001) Moreover, as soon as the antiferromagnetic rung couplings become non-
zero the spin gap starts to open, it first increases quadratically till
$J_{r}\sim 0.3$, and then grows linearly by increasing $J_{r}$.Trumper and
Gazza (2001)

In the limit of strong ferromagnetic rung couplings, i.e. for $J_{r}<0$ and
$J_{l}=0$, the spins $\sigma~{}(\tau)$ form rung-triplets (quintets) and the
singlets (triplets) being much higher in energy. In this limit the ground
state is a product of rung-triplet and rung-quintet states, separated by an
energy gap of size $3/4J_{r}$ (per total number of spins) with excited state.
As soon as leg couplings are switched on, i.e. when $|J_{r}|\gg J_{l}$, the
ladder behaves like a ferrimagnetic spin-(1, 2) chain, with a long-range
ferrimagentic ground state of total spin $S_{tot}=L$. In this case, the
antiferromagnetic gap within the lowest-order spin-wave theory is
$\Delta=2J_{l}$. As we mentioned, in the limit of weak ferromagentic rung
couplings there is also an antiferromagntic gap of size $J_{l}$. Therefore, it
appears that weak and strong ferromagnetic couplings are continuously related.

In the limit of strong antiferromagnetic rung couplings ($J_{r}>0$, and
$J_{l}\ll J_{r}$) the ground state is essentially a product of local rung-
singlets, i.e.

$\left|GS^{0}\right\rangle=\prod_{i}\left|S^{\sigma}\right\rangle_{i}\otimes\left|S^{\tau}\right\rangle_{i},$
(2)

where $\left|S^{\sigma}\right\rangle_{i}$ and
$\left|S^{\tau}\right\rangle_{i}$, given by:

$\displaystyle\left|S^{\sigma}\right\rangle_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(|\frac{1}{2}\rangle_{i}^{(1)}|-\frac{1}{2}\rangle_{i}^{(2)}-|-\frac{1}{2}\rangle_{i}^{(1)}|\frac{1}{2}\rangle_{i}^{(2)}\right),$
$\displaystyle\left|S^{\tau}\right\rangle_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}\left(\left|1\right\rangle^{(1)}_{i}\left|-1\right\rangle^{(2)}_{i}-\left|0\right\rangle^{(1)}_{i}\left|0\right\rangle^{(2)}_{i}+\left|-1\right\rangle^{(1)}_{i}\left|1\right\rangle^{(2)}_{i}\right),$

are the rung-singlet states, respectively for the rungs with spins $\sigma$
and $\tau$ in the $i$-th unit cell. Here, $|\pm 1/2\rangle$ are the
eigenstates of $\sigma^{z}$, and $|\pm 1\rangle$ and $|0\rangle$ are the
eigenstates of $\tau^{z}$. The ground state energy is
$E_{g}/N=-\frac{11}{4}J_{r}$, where $N$ refers to the number of unit cells,
and the first excited states are triplets, separated from the ground state by
a finite energy gap of size ($J_{r}/2$). By increasing leg couplings, the
energy gap decreases monotonically. Using the non-degenerate perturbation
theory, to first-order in $J_{l}/J_{r}$, the ground state is readily obtained
as:

$|GS^{1}\rangle=|GS^{0}\rangle+|1\rangle,$ (3)

with

$\displaystyle\left|1\right\rangle$ $\displaystyle=$
$\displaystyle\frac{J_{l}}{J_{r}}\sqrt{\frac{1}{6}}[\sum_{i}\cdots\gamma^{\tau,\sigma}_{i-1,i}\cdots+\sum_{i}\cdots\gamma^{\sigma,\tau}_{i,i+1}\cdots],$

where

$\displaystyle\gamma^{\tau,\sigma}_{i,j}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\left(\left|t^{+}_{\tau}\right\rangle_{i}\left|t^{-}_{\sigma}\right\rangle_{j}-\left|t^{0}_{\tau}\right\rangle_{i}\left|t^{0}_{\sigma}\right\rangle_{j}+\left|t^{-}_{\tau}\right\rangle_{i}\left|t^{+}_{\sigma}\right\rangle_{j}\right),$
$\displaystyle\gamma^{\sigma,\tau}_{i,j}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\left(\left|t^{+}_{\sigma}\right\rangle_{i}\left|t^{-}_{\tau}\right\rangle_{j}-\left|t^{0}_{\sigma}\right\rangle_{i}\left|t^{0}_{\tau}\right\rangle_{j}+\left|t^{-}_{\sigma}\right\rangle_{i}\left|t^{+}_{\tau}\right\rangle_{j}\right).$

Here, $\left|t^{\alpha}_{s}\right\rangle$ with $\alpha=\pm,0$ and
$s=\sigma,\tau$ are the triplet states on the rung with spins $s$ defined as:

$\displaystyle\left|t^{+}_{s}\right\rangle_{i}$ $\displaystyle=$
$\displaystyle-\sqrt{\frac{3}{2}}\sum_{m=-s}^{s-1}\frac{\left({-}\right)^{s-m}\sqrt{s\left(s+1\right)-m\left(m+1\right)}}{\sqrt{s\left(s+1\right)\left(2s+1\right)}}$
$\displaystyle\times$
$\displaystyle\left|m+1\right\rangle_{i}^{(1)}\left|-m\right\rangle_{i}^{(2)},$
$\displaystyle\left|t^{0}_{s}\right\rangle_{i}$ $\displaystyle=$
$\displaystyle\sum_{m=-s}^{s}\frac{\sqrt{3}\left(-1\right)^{s-m}m}{\sqrt{s\left(s+1\right)\left(2s+1\right)}}\left|m\right\rangle_{i}^{(1)}\left|{-m}\right\rangle_{i}^{(2)},$
$\displaystyle{\left|t^{-}_{s}\right\rangle_{i}}$ $\displaystyle=$
$\displaystyle-\sqrt{\frac{3}{2}}\sum_{m=-s}^{s-1}\frac{\left(-1\right)^{s-m}\sqrt{s\left(s+1\right)-m\left(m+1\right)}}{\sqrt{s\left(s+1\right)\left(2s+1\right)}}$
(4) $\displaystyle\times$
$\displaystyle\left|m\right\rangle_{i}^{(1)}\left|-\left(m+1\right)\right\rangle_{i}^{(2)}.$

As it is seen, to first order in $J_{l}/J_{r}$, the ground state of the ladder
is given in terms of the rung-singlet and the rung-triplet states, and there
is no contribution from the rung-quintets. This means that in the limit of
weak leg coupling the quintet states can be projected out and the mixed-
spin-(1,1/2) ladder is mapped onto a uniform spin-1/2 ladder with
antiferromagnetic rung and leg couplings. However, when leg couplings become
stronger we should take the rung-quintet states into account to write the true
ground state of the ladder. To second order in $J_{l}/J_{r}$, the ground state
of the ladder is given in terms of rung-singlets, rung-triplets and rung-
quintets (see the Appendix A). According to the form of the second order
ground state (see Eq. (17)) we can see that the inhomogeneity of the system
appears at stronger values of leg couplings. To see the ground state magnetic
orders, we study the behavior of the expectation values of $\sigma_{i}^{z}$
and $\tau_{i}^{z}$ on the first and second order ground states. In these
states all the local order parameters are vanishing and the ground state has
no magnetic order.

As rung couplings between the mixed-spin chains are turned on, in the case of
ferromagnetic rung couplings, the ground state order of the chains does not
change essentially, and we still have a ferrimagnetic long-range order along
the chains. However, when rung couplings are antiferromagentic, even very
small, according to the Marshall’s theorem the ferrimagnetic order of the
chains is destroyed immediately. In next sections, we will carry out iTEBD and
DMRG computations and investigate the ground state properties of our mixed-
spin ladder in the vicinity of this point. We will show that in the limit of
weak antiferromagnetic rung couplings these two techniques lead to different
ground state orders.

Like the uniform spin ladders, our mixed-spin ladder is invariant under the
exchange of the leg and diagonal couplings, Weihong  _et al._ (1998) i.e.
$H\left(J_{l},J_{r},J_{d}\right)=H\left(J_{d},J_{r},J_{l}\right)$. Thus all
the above discussions can be entirely applied to the case of $J_{l}=0$ and
$J_{d}\neq 0$, by replacing $J_{l}$ with $J_{d}$.

##  III Methods

Matrix product states (MPSs)Östlund and Rommer (1995) provide an efficient
representation for the ground state of one-dimensional systems which obey the
area law in which the entanglement entropy grows with the boundary of a
specific area rather than its volume.Eisert  _et al._ (2010) To use MPSs
representation for our mixed-spin ladder, we map the ladder onto a chain and
consider each rung as a supersite with a larger Hilbert space. In Vidal’s
representationVidal (2003) a generic translational invariant state of a one-
dimensional system in thermodynamic limit can be described by the use of two
sets of matrices in infinite system. However, In our case, we have considered
a larger unit cell consists of four pairs of matrices
$(\Gamma_{A},\Gamma_{B},\Gamma_{C},\Gamma_{D})$ and
$(\lambda_{A},\lambda_{B},\lambda_{C},\lambda_{D})$, to make sure that the
unit cell is sufficiently large to detect the dimer phases or phases with
broken translational symmetry. In this representation an arbitrary many-body
quantum state of our ladder can be written as

$\displaystyle\left|\psi\right\rangle$ $\displaystyle=$
$\displaystyle\sum_{i_{1},\cdots,i_{N}}[\cdots\Gamma_{A}^{i_{m}}\lambda_{A}\Gamma_{B}^{i_{m+1}}\lambda_{B}$
(5)
$\displaystyle\Gamma_{C}^{i_{m+2}}\lambda_{C}\Gamma_{D}^{i_{m+3}}\lambda_{D}\cdots]\left|i_{1},i_{2},...,i_{N}\right\rangle,$

where $|i_{1},i_{2},...,i_{N}\rangle$ is a product state with $i_{n}$ being
the $i$-th state of the rung $n$, 4 states for spin-$\frac{1}{2}$ rungs while
9 for spin-1. This state is diagrammatically shown in Fig. 2,

Figure 2: (Color online) MPSs representation for mixed-spin-(1,1/2) ladder.
The rungs with spins $\sigma=1/2$ are described by $A$ and $C$ index and the
rungs with spins $\tau=1$ by $B$ and $D$. Spin-$1/2$ (1) rungs have a local
Hilbert space equals to $d^{A}=d^{C}=4$ $(d^{B}=d^{D}=9)$.

where $\lambda$’s are positive, real and square diagonal matrices. The matrix
elements of $\lambda$ are nothing but the Schmidt coefficients, and
$\lambda_{i}^{2}$ are the eigenvalues of the reduced density matrix
($\rho_{red}=Tr_{B(A)}(\left|\psi\right\rangle\left\langle\psi\right|)$, where
A and B are two halves of the ladder). The full set of these eigenvalues are
the ES, and the EE is directly connected to this eigenvalues thorough

$S=-\sum_{i}\lambda_{i}^{2}\log\lambda_{i}^{2}.$ (6)

The tensors $\Gamma$ are matrices of dimension $\chi$ which correspond to
changes of basis between the different Schmidt basis. $\chi$ is a key
parameter in tensor network states, called bond dimension, and the accuracy of
the state (5) is controlled by this quantity. For partially entangled states
by choosing a fairly small bond dimension one can reach to valid results.

One of the efficient MPSs-based algorithm for simulating one-dimensional
quantum many-body systems is iTEBD technique.Vidal (2003, 2004); White and
Feiguin (2004); Daley  _et al._ (2004); Orús and Vidal (2008) In iTEBD, by
considering the imaginary time evolution of a quantum state we can find the
ground state of the Hamiltonian through the relation

$\left|GS\right\rangle=\lim_{T\rightarrow\infty}\exp(-TH)\left|\psi_{0}\right\rangle,$
(7)

where $\left|\psi_{0}\right\rangle$ is an arbitrary state which we call it
initial guess state. In the vicinity of the phase transition points, the final
results obtained by iTEBD are strictly dependent to this initial guess, and it
is of crucially important to choose suitable guess states to reach the true
ground state of the system. By dividing the time ($T$) into $N$ infinitesimal
periods ($\delta t$), rewriting the Hamiltonian as $H=H_{odd}+H_{even}$, where

$\displaystyle H_{odd}$ $\displaystyle=$ $\displaystyle
J_{r}(\sigma_{i}^{1}\cdot\sigma_{i}^{2}+\tau_{i}^{1}\cdot\tau_{i}^{2})+J_{l}(\sigma_{i}^{1}\cdot\tau_{i}^{1}+\sigma_{i}^{2}\cdot\tau_{i}^{2}),$
$\displaystyle H_{even}$ $\displaystyle=$ $\displaystyle
J_{r}(\tau_{i}^{1}\cdot\tau_{i}^{2}+\sigma_{i}^{1}\cdot\sigma_{i}^{2})+J_{l}(\tau_{i}^{1}\cdot\sigma_{i}^{1}+\tau_{i}^{2}\cdot\sigma_{i}^{2}),$

and performing Suzuki-Trotter decomposition, the time evolution operator can
be approximated as:

$U(\delta t)\approx\prod_{i_{odd}}e^{-i\delta
th[i,i+1]}\prod_{i_{even}}e^{-i\delta th[i,i+1]},$ (8)

where $h[i,i+1]$s are the two-rung operators which their summation will
construct $H_{odd}(H_{even})$. This operator is applied severally to the state
$|\psi_{0}\rangle$ in Eq. (5) in order to update the infinite MPSs
representation, until the ground state energy or entropy converge. A key
feature of the iTEBD algorithm is that it directly treats the infinite system
by exploiting the translational invariance, so it is free of finite-size
effects.

In our ladder system, we used second order Suzuki-Trotter decomposition, and
the simulations are started with a time step $\delta t=0.5$, then it is
gradually decreased up to $\delta t=10^{-5}$. As we mentioned, close to the
RS-H phase transition point, the iTEBD can lead to the results, not compatible
with fundamental theories in condensed matter physics such as Mermin-Wagner
and Lieb-Mattis theorems. Therefore, we use DMRG technique to obtain the
correct ground state of the system. Unlike iTEBD, DMRG is a variational method
but these two methods have many steps in common. We iteratively optimize the
MPSs of two neighboring sites to minimize the ground state energy. Then
project the Hamiltonian onto the variational space and use an iterative
algorithm such as Lanczos to lower the energy. Repeating this two-site update
for each pair of neighboring sites, the wave function converges to the ground
state. In our computations, we used a finite DMRG technique, and considered
systems with 120, 180 and 240 rungs. Also, the bond dimension has been
increased up to 1000.

##  IV Results: unfrustrated mixed-spin ladders

In the absence of the diagonal interactions, i.e., for $J_{d}=0$, our mixed-
spin ladder (1) is unfrustrated and, as seen by perturbation theory, in the
two extreme limits of strong FM and vanishing rung couplings, the ground state
is ferrimagnetically long-range ordered, while in the limit of strong AF rung
couplings it is RS disordered. In order to obtain the ground sate phase
diagram we introduce a dimensionless parameter as $R=J_{r}/(J_{l}+|J_{r}|)$,
where $R=-1~{}(+1)$ indicates the strong FM (AF) rung couplings regime, and
$R=0$ corresponds to two decoupled mixed-spin chains.

Before finding the ground state phase diagram, it is insightful to study the
behavior of the ground state energy. We have plotted in Fig. 3, the ground
state energy per unit cell versus the parameter $R$, obtained by iTEBD with
two different bond dimensions, $\chi=100$ and 120. As it is obviously seen,
the energy obtained with $\chi=100$ and $\chi=120$ are almost the same,
meaning that $\chi=120$ is a suitable bond dimension.

Figure 3: (Color online) The ground state energy of the unfrustrated mixed-
spin-(1,1/2) ladder per unit cell as a function of $R$, obtained by iTEBD with
$\chi=100$ and $120$. At $R=-1~{}(J_{l}=0,~{}J_{r}=-1)$ the ground state
energy is equal to $-5/4$ which is exactly the total energy of a spin-1/2
rung-triplet and a spin-1 rung-quintet, at $R=0~{}(J_{l}=1,~{}J_{r}=0)$ the
energy coincides with the energy of a mixed-spin-(1,1/2) chain, and at
$R=1~{}(J_{l}=0,~{}J_{r}=1)$ the ground state energy is $-\frac{11}{4}$, the
total energy of spin-1/2 and spin-1 rung-singlets.

The discontinuity at $R=0$ implies that a phase transition of first order
occurs in the system. In order to capture the ground state orders in both
sides of this point we introduce the following order parameters:

$\displaystyle\mathcal{O}_{FiM-F}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left|\left\langle(\tau_{i}^{(1)z}-\sigma_{i}^{(1)z})+(\tau_{i}^{(2)z}-\sigma_{i}^{(2)z})\right\rangle\right|,$
$\displaystyle\mathcal{O}_{FiM-N}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left|\left\langle(\tau_{i}^{(1)z}-\sigma_{i}^{(1)z})-(\tau_{i}^{(2)z}-\sigma_{i}^{(2)z})\right\rangle\right|.$
(9)

where $\langle\dots\rangle$ is the expectation value of ”$\dots$” on the
ground state of the system. In the phase with non-vanishing FiM-F (FiM-N)
order parameter, the system orders ferrimagnetically along the legs (as in
ferrimagnetic chains) and ferromagnetically (antiferromagnetically) along the
rungs (see the schematic pictures in the inset of Fig. 4). In both these
phases the time reversal symmetry is broken, but the leg-swap breaks only in
the FiM-N phase. We have plotted in Fig. 4 the ground state order parameters
versus $R$, obtained by iTEBD. In the entire region of $-1\leq R<0$, the FiM-F
order parameter is non-zero and the ladder is ferrimagnetically ordered. In
this region the ground state is in the sector $[1,2,1,2,\dots,1,2]$, where the
numbers are the values of $\sigma$ and $\tau$ in the sector
$[\sigma_{1},\tau_{1},\sigma_{2},\tau_{2},\dots,\sigma_{N-1},\tau_{N-1},\sigma_{N},\tau_{N}]$
of the Hilbert space. At $R=0$, the ladder is decoupled to two equivalent
mixed-spin-(1,1/2) chains and the system is ferrimagnetically ordered, and the
ground state is in the sector $[1/2,1,\dots,1/2,1]$. For large values of
$R>0$, all local order parameters are exactly zero, and the system is in the
RS disordered phase. By decreasing $R$, although the leg couplings change the
ground state of the system (as seen by our perturbation theory), but according
to the Mermin-Wagner theorem, we expect the ground state order does not change
and the ladder shows no long-range order in the entire region of $R>0$. But
amazingly we see that in a narrow interval near the $R=0$ point, i.e., in the
region of $0<R\leq 0.08$, our iTEBD predicts a ground state with FiM-N long-
range order.

Figure 4: (Color online) The iTEBD results for FiM-M and FiM-N order
parameters, defined in Eq. (IV). Here, the bond dimension is $\chi=120$.
Inset: schematic configurations of the FiM-F and FiM-N long-range orders.

The existence of a magnetic order in the small region in the vicinity of
$R=0$, is not consistent with Mermin-Wagner theorem. Hohenberg (1967); Mermin
and Wagner (1966) This might be caused by the dependency of the results in
iTEBD to the initial guess we choose (for more details see the Appendix B). To
overcome this inconsistency, we carried out DMRG algorithm which are less
dependent to initial guess states. We have plotted in Fig.5, our DMRG results
on the ground state EE, for different system sizes. The EE is obtained by
dividing the ladder (with size $N$) into two halves of size $N/2$, and tracing
one of the halves degrees of freedom out. As it is obviously seen, the EE
shows a discontinuity only at $R=0$ where a transition of first order occurs
from the ferrimagnetically ordered phase to the rung-singlet disordered phase.
In the RS phase all the local magnetizations are zero, and the SU(2) symmetry
of the Hamiltonian is preserved.

Figure 5: (Color online ) The EE for the unfrustrated mixed-spin-(1, 1/2)
ladder versus $R$, obtained by DMRG with $L=60$, 90, and 120 rungs, for $\chi$
increasing from 300 to 1000. The discontinuity at $R=0$ shows the occurrence
of a phase transition of first order in the system. The EE is vanishing in the
two extreme limits of $R=\pm 1$.

Although the discontinuous change in EE indicates exactly the phase transition
point from the FiM-F state to the RS, it fails to characterize the ground
state phases, individually. Since changes in the degeneracy of the ES levels
is associated with a quantum phase transition,Pollmann  _et al._ (2010, 2012);
Huang and Lin (2011); Ye  _et al._ (2016); Haghshenas  _et al._ (2014) we use
the ES as a tool to identify these phases.

Figure 6: (Color online) The ES levels degeneracies for the unfrustrated
mixed-spin-(1, 1/2) ladder versus $R$, obtained by DMRG with $\chi=800$ for a
ladder of 120 rungs. The filled circles show the degeneracy of ES levels. The
RS phase is characterized by odd degeneracies of ES levels, whereas in the
FiM-F phase ES levels have no degeneracies.

We have shown in Fig. 6, the degeneracy of the ES levels in both the FiM-F and
the RS phases. As seen, by changing the parameter $R$ from $-1$ to $+1$, the
degeneracy of the ES levels varies such that in the RS phase ($0<R\leq 1$) the
ES levels have odd degeneracies, while in the FiM-F phase ($-1<R\leq 0$), they
have no degeneracy.

##  V Results: frustrated mixed-spin ladder

In this section, we introduce the diagonal interactions ($J_{d}>0$) to the
system, and investigate the impact of frustration on the ground state phase
diagram of the mixed-spin-(1, 1/2) ladder. In the case of ferromagnetic rung
couplings, in the limits of weak leg and weak diagonal couplings, i.e., for
$J_{d},J_{l}\ll|J_{r}|$, the ground state can be obtained using a perturbation
theory. To this end, we write the legs and rungs Hamiltonians as:

$H_{legs}+H_{diag}=\sum_{m=+,-}J^{m}\sum_{i}[\sigma_{i}^{m}\cdot\tau_{i}^{m}+\tau_{i}^{m}\cdot\sigma_{i+1}^{m}],$
(10)

with $\sigma_{i}^{\pm}=\sigma_{i}^{(1)}\pm\sigma_{i}^{(2)}$ and
$\tau_{i}^{\pm}=\tau_{i}^{(1)}\pm\tau_{i}^{(2)}$, and
$J^{\pm}=\frac{1}{2}(J_{l}\pm J_{d})$. Here, the ”+” term preserves the total
spin in each rung while the ”-” term changes the triplets and quintets to
singlets. In the case of antiferromagnetic legs and diagonal interactions the
”+” term gives a nonzero contribution at the first order perturbation and
lifts the ground state degeneracy of the Hamiltonian $H_{rung}$. The effective
Hamiltonian turns out to be the mixed-spin-(1,2) Heisenberg chain

${\cal H}=J\sum_{i}(S_{i}\cdot T_{i}+T_{i}\cdot S_{i+1}),$ (11)

where $S_{i}$ and $T_{i}$ are respectively the spin-1 and spin-2 operators and
$J=\frac{1}{2}(J_{l}+J_{d})$. Therefore the ground state is ferrimagnetically
long-range ordered. By increasing the strengths of leg and diagonal couplings,
no phase transition happens in the system and the ferrimagnetic order is
preserved.

In the case of antiferromagnetic rung couplings, the situation is completely
different and the ground state experiences various phase transitions. For
antiferromagnetic leg coupling, as seen in the previous section, in the
absence of weak diagonal interactions the system is in the RS disordered
phase, but what happens when diagonal interactions are switched on. By an
analysis of the different spin-spin correlations within unit cells, on can
shown that for $J_{l}=J_{r}=1$, in the region of $0\leq J_{d}<0.845$, we have
$\langle\tau_{i}^{(1)}\cdot\tau_{i}^{(2)}\rangle=\langle\sigma_{i}^{(1)}\cdot\sigma_{i}^{(2)}\rangle-\frac{5}{4},$
which implies that the weight of the rung-quintet states on the spin-1 rungs
is negligible in this interval.Chandra  _et al._ (2010) Therefore, one can
project out the quintet states, and map the frustrated mixed-spin-(1, 1/2)
ladder onto a frustrated uniform spin-1/2 ladder with AF leg, rung and
diagonal couplings.Chandra  _et al._ (2010) Different analytical approaches
and numerical methods, like bosonization, renormalization groups, DMRG and
exact diagonalization techniques, have been used to examine the existence of
an intermediate phase between the RS and the H phases in the ground state
phase diagram of frustrated spin-1/2 ladder with AF couplings.Vekua and
Honecker (2006); Liu  _et al._ (2008); Hikihara and Starykh (2010) It has been
demonstrated that, in the weak coupling regime, there is a columnar-dimer
phase in a very narrow neighborhood of $J_{\perp}=0.38$ (for $J_{\parallel}=1$
and $J_{\times}=0.2$) between the RS and the H phases.Liu  _et al._ (2008)
However, the spin-1/2 ladder which our mixed-spin ladder mapped onto, does not
belong to the mentioned region, and consequently it was concluded that there
is not any intermediate phase between the RS and the H phases.Chandra  _et
al._ (2010) In order to reach a definite conclusion we must carry on more
sophisticated investigations.

Here, we study the ground state phases of the frustrated mixed-spin-(1, 1/2)
ladder by examining the behavior of the EE, the ES and the string orders. We
utilize both the iTEBD and the DMRG techniques and investigate the existence
of any intermediate phase between the RS and the H phases.

The ground state energy per unit cell is plotted in Fig. 7. Different slopes
of the ground state energy curve exhibit drastic changes at different values
of $J_{d}$.

Figure 7: (Color online) The ground state energy of the frustrated mixed-
spin-(1, 1/2) ladder per unit cell versus $J_{d}$, obtained by iTEBD with
$\chi=100$ and 120. Inset: the ground state energy for the region
$0.65<J_{d}<1$.

We have also plotted in Fig. 8, the ground state EE and the Schmidt gap (the
difference between the two largest eigenvalues of the reduced density matrixDe
Chiara  _et al._ (2012)) versus $J_{d}$, obtained by iTEBD with $\chi=100$ and
120. The sudden changes on the EE and the Schmidt gap at various $J_{d}$ (see
the dashed vertical lines) imply the occurrence of different quantum phase
transitions in the system at $J_{d}=0.71,0.74,0.88$ and $1.17$.

Figure 8: (Color online) The EE (top) and the Schmidt gap (bottom) for the
frustrated mixed-spin-(1, 1/2) ladder versus $J_{d}$, obtained by iTEBD with
$\chi=100$ and 120.

By analyzing the degeneracy of the ES levels, shown in Fig. 9, and also by
investigating the behavior of the different local order parameters, plotted in
Fig. 10, we realize that in the region $0\leq J_{d}<0.71$, all local order
parameters are vanishing and the ES levels possess odd degeneracies which is a
characteristic of the RS phase. In the region $0.71\leq J_{d}<0.74$, the
dominant ES levels are not degenerate and the FiM-N local order parameter is
nonzero. For $0.74\leq J_{d}\leq 0.88$, all ES levels have even degeneracies,
and all local order parameters are vanishing. The even degeneracies in the ES
levels is a characteristic of the H phase.Pollmann  _et al._ (2010, 2012) In
the region $0.88<J_{d}\leq 1.5$, ES levels are nondegenerate, and the FiM-F
order parameter is finite, but its magnitude discontinuously changes at
$J_{d}=1.17$, where a first-order phase transition occurs in the system. For
$\left(0.88\leq J_{d}\leq 1.17\right)$ the ground state belongs to the sector
$\left[1,2,\cdots,1,2\right]$ whereas for $J_{d}>1.17$, the translational
symmetry of the ladder breaks and the ground state belongs to the sector
$\left[1,1,1,2,\cdots,1,1,1,2\right]$.

Figure 9: (Color online) The degeneracies of the ES levels of the frustrated
mixed-spin ladder versus $J_{d}$, obtained by iTEBD with $\chi=120$. The RS
phase is characterized by odd degeneracies in ES levels, while in the H phase
ES levels have even degeneracies. In the ferrimagnetic phases ES levels are
nondegenerate. Figure 10: (Color online) The different order parameters
defined in Eq. (IV) versus $J_{d}$, obtained by iTEBD with $\chi=120$. The
FiM-N order parameter is nonzero in a narrow interval between the RS and the H
phases, and the FiM-F order parameter is non-zero for $J_{d}\geqslant 0.88$.
The discontinuity at the H-FiM phase transition point, shows that this phase
transition is of first-order.

As seen, the iTEBD predicts an intermediate phase between the RS and the H
phases where the ES levels degeneracies do not follow the specific patterns of
the RS and the H phases. It also predicts that in this narrow interval the
ground state has long-range FiM-N order. However, as we discussed the ground
state must be a singlet state below $J_{d}<0.88$. In order to reach the true
ground state, we repeat our iTEBD computations using various initial guess
states and different bond dimensions (not shown). Although by changing the
initial state the energy of the resulted ground state changes, but its long-
range order didn’t disappear. As the former section, we employed DMRG which is
less dependent on the initial guess.

Using DMRG we achieved interesting results: consistent with the above
discussions, all local order parameters are zero
($\langle\sigma_{i}^{(1,2)}\rangle=\langle\tau_{i}^{(1,2)}\rangle=0$) below
$J_{d}<0.89$, indicating that the ground state belongs to $s^{z}_{tot}=0$ and
it is a singlet state, but as it is obviously seen from the EE and also the ES
levels degeneracies (Fig.(11, 12)), there is still a different phase between
the RS and the H phases in which the ES levels do not follow the specific
degeneracies of the RS and the H phases. In this intermediate phase the EE is
much larger than its values in the RS and the H phases. We name this
intermediate phase as ”disordered (D) phase”. This naming can be meaningful
because in this phase all the local parameters are zero, and the ES levels do
not follow a specific pattern. The very sharp and continuous increase of EE at
$J_{d}\simeq 0.71$ implies that a phase transition of second-order occurs from
the RS to the D phase in the system. Also, on the other side, i.e., at
$J_{d}\simeq 0.73$, the rapid and discontinuous growth of the EE, by
decreasing $J_{d}$, indicates the system undergoes a phase transition of
first-order from the H phase to the D phase.

Figure 11: (Color online) The EE versus $J_{d}$, for frustrated mixed-
spin-(1,1/2) ladders of different sizes $L=120,180$ and 240, obtained by DMRG
with $\chi$ gradually increased from 300 to 1000. The very sharp increases of
the EE at $J_{d}\simeq 0.71$ and $J_{d}\simeq 0.73$, indicates the existence
of a different phase between the RS and the H in the narrow interval of
$0.71\leq J_{d}\leq 0.73$. The continuous and discontinues changes of the EE
are respectively due to the occurrence of second- and first-order phase
transitions in the system.  Figure 12: (Color online) The DMRG results for the
ES levels degeneracies in different phases. In the intermediate D phase, the
ES levels do not follow a specific pattern of degeneracies.

In order to understand the nature of this D phase, we analyze the short-range
correlation functions inside a unit cell. At the RS phase for an specific
$J_{d}$, the correlations of two spins along the rungs are constant and for
both $\sigma$ and $\tau$ rungs
$\left\langle\sigma(\tau)_{i}^{(1)}\sigma(\tau)_{i}^{(2)}\right\rangle<0$. At
H phase again for an specific $J_{d}$ the correlations along the rung is
almost constant and by changing the diagonal coupling it change smoothly. In
this phase $\left\langle\sigma_{i}^{(1)}\sigma_{i}^{(2)}\right\rangle>0$ and
$\left\langle\tau_{i}^{(1)}\tau_{i}^{(2)}\right\rangle<0$. However, in the
intermediate phase, there is no similar pattern and when we are moving along
different rungs the correlations change, and in contrary to the RS and the H
phases we can not find a unique behavior for the correlations along the rungs
in the D phase.

In the absence of magnetic orders, the different gapped phases can be
distinguished by non-local string order parameters.den Nijs and Rommelse
(1989); Shelton  _et al._ (1996); Kim  _et al._ (2000); Haghshenas  _et al._
(2016); Pollmann and Turner (2012) Here, we define the following string order
parameters which depend on the spins located at the end sites of the string:

$\displaystyle\mathcal{O}_{Odd}^{\sigma\sigma}$ $\displaystyle=$
$\displaystyle\lim\limits_{|i-j|\rightarrow\infty}|\langle\sigma_{o,i}^{z}e^{i\pi\sum_{k=i+1}^{j-1}\sigma_{o,k}^{z}}\sigma_{o,j}^{z}\rangle|,$
$\displaystyle\mathcal{O}_{Odd}^{\tau\tau}$ $\displaystyle=$
$\displaystyle\lim\limits_{|i-j|\rightarrow\infty}|\langle\tau_{o,i}^{z}e^{i\pi\sum_{k=i+1}^{j-1}\tau_{o,k}^{z}}\tau_{o,j}^{z}\rangle|,$
$\displaystyle\mathcal{O}_{Even}^{\sigma\tau}$ $\displaystyle=$
$\displaystyle\lim\limits_{|i-j|\rightarrow\infty}|\langle\sigma_{e,i}^{z}e^{i\pi\sum_{k=i+1}^{j-1}\sigma_{e,k}^{z}}\sigma_{e,j}^{z}\rangle|,$
$\displaystyle\mathcal{O}_{Even}^{\tau\sigma}$ $\displaystyle=$
$\displaystyle\lim\limits_{|i-j|\rightarrow\infty}|\langle\tau_{e,i}^{z}e^{i\pi\sum_{k=i+1}^{j-1}\tau_{e,k}^{z}}\tau_{e,j}^{z}\rangle|,$
(12)

with

$\displaystyle\sigma_{o,i}^{z}=\sigma^{(1)z}_{i}+\sigma^{(2)z}_{i},~{}~{}~{}\tau_{o,i}^{z}=\tau^{(1)z}_{i}+\tau^{(2)z}_{i},$
$\displaystyle\sigma_{e,i}^{z}=\sigma^{(1)z}_{i}+\tau^{(2)z}_{i+1},~{}~{}~{}\tau_{e,i}^{z}=\tau^{(1)z}_{i}+\sigma^{(2)z}_{i+1}.$

As it is seen from Fig. 14, in the RS phase $\mathcal{O}_{odd}=0$ but
$\mathcal{O}_{Even}\neq 0$, while in the H phase $\mathcal{O}_{Odd}\neq 0$ and
$\mathcal{O}_{Even}=0$. Therefore, we can label these two disordered phases
with non-vanishing even or odd string order parameters. Typical configuration
of the RS and the H phases are demonstrated in Fig. 13.

Figure 13: (Color online) Typical configurations of the H and RS phases. Solid
lines represent singlet bonds. Dashed lines with green area encircling rungs:
$\sigma_{o,i}^{z}=\sigma^{(1)z}_{i}+\sigma^{(2)z}_{i}$ and
$\tau_{o,i}^{z}=\tau^{(1)z}_{i}+\tau^{(2)z}_{i}$. Notice that after we remove
all sites with $\sigma_{o,i}^{z}=\tau_{o,i}^{z}=0$, the remaining sites have
Neel order. Dashed lines with pink area encircling diagonals:
$\sigma_{e,i}^{z}=\sigma^{(1)z}_{i}+\tau^{(2)z}_{i+1}$ and
$\tau_{e,i}^{z}=\tau^{(1)z}_{i}+\sigma^{(2)z}_{i+1}$. According to the hidden
Neel order, one can see that in the RS phase $\mathcal{O}_{odd}=0$ but
$\mathcal{O}_{Even}\neq 0$, while in the H phase $\mathcal{O}_{Odd}\neq 0$ and
$\mathcal{O}_{Even}=0$.

In the D phase, however, both the odd and even string order parameters are
non-zero (based on our iTEBD calculations).

Figure 14: (Color online) Various string order parameters versus $J_{d}$,
obtained by iTEBD with $\chi=120$. This plot clearly show the difference
between the RS, the D and the H phases.

##  VI SUMMARY

In summary, we have investigated the ground state phase diagram of a
frustrated mixed-spin-(1,1/2) ladder, by studying the behavior of the EE, the
ES levels degeneracies and different order parameters. We have demonstrated
that in addition to the long-range ferrimagnetic orders, and the RS and H
phases, a quantum disordered phase also emerges in a narrow interval between
the RS and the H phases. In this D phase all local order parameters are zero,
and in contrary to the RS and the H phases where the ES levels possess
respectively odd and even degeneracies, in this region the ES levels have no
specific pattern, i.e., they have mixed odd-even degeneracies. In order to
understand the nature of this phase, we also examined the behavior of the
short-range spin correlation functions along the rungs, and showed that in
contrary to the RS and the H phases there is no a unique behavior for the
correlations in the D phase.

###### Acknowledgements.

The authors would like to thank Prof. D. Baeriswyl for insightful comments and
discussion. RH acknowledges support for this work from the US Department of
Energy, Office of Science, via award no DE-SC0019374. We have used _Uni10_ Kao
_et al._ (2015) and ITensor Fishman  _et al._ (2020) as middleware libraries
to build up variational gauging ansatz.

##  Appendix A Second order correction of ground state

In this Appendix we will obtain the ground state of the unfrustrated mixed-
spin-(1,1/2) ladder up to second order in $J_{l}/J_{r}$. For a spin-s system
the quintet states can be represented as:

$\displaystyle\left|q^{+2}\right\rangle_{i}$ $\displaystyle=$ $\displaystyle
Q(s)\sum_{m=-s+1}^{s-1}(-1)^{s-m}\sqrt{s(s+1)-m(m-1)}$ $\displaystyle\times$
$\displaystyle\sqrt{s(s+1)-m(m+1)}\left|m+1\right\rangle_{i}^{(1)}\left|-m+1\right\rangle_{i}^{(2)},$
$\displaystyle\left|q^{+1}\right\rangle_{i}$ $\displaystyle=$ $\displaystyle
Q(s)\sum_{m=-s}^{s-1}(-1)^{s-m}\sqrt{s(s+1)-m(m+1)}$ $\displaystyle\times$
$\displaystyle(2m+1)\left|m+1\right\rangle_{i}^{(1)}\left|-m\right\rangle_{i}^{(2)},$
$\displaystyle\left|q^{0}\right\rangle_{i}$ $\displaystyle=$ $\displaystyle
Q(s)\sqrt{\frac{2}{3}}\sum_{m=-s}^{s}(-1)^{s-m}(s(s+1)-3m^{2})$
$\displaystyle\times$
$\displaystyle\left|m\right\rangle_{i}^{(1)}\left|-m\right\rangle_{i}^{(2)},$
$\displaystyle\left|q^{-1}\right\rangle_{i}$ $\displaystyle=$ $\displaystyle
Q(s)\sum_{m=-s}^{s-1}(-1)^{s-m}\sqrt{s(s+1)-m(m+1)}$ $\displaystyle\times$
$\displaystyle(2m+1)\left|m\right\rangle_{i}^{(1)}\left|-m-1\right\rangle_{i}^{(2)},$
$\displaystyle\left|q^{-2}\right\rangle_{i}$ $\displaystyle=$ $\displaystyle
Q(s)\sum_{m=-s+1}^{s-1}(-1)^{s-m}\sqrt{s(s+1)-m(m-1)}$ $\displaystyle\times$
$\displaystyle\sqrt{s(s+1)-m(m+1)}\left|m-1\right\rangle_{i}^{(1)}\left|-m-1\right\rangle_{i}^{(2)}.$

where

$Q(s)=\frac{-\sqrt{\frac{15}{2}}}{\sqrt{s(s+1)(2s+1)(2s+3)(2s-1)}}.$

The second order correction of the ground state is readily obtained as
follows:

$\displaystyle\left|2\right\rangle$ $\displaystyle=$
$\displaystyle-\left(\frac{J_{l}}{J_{r}}\right)^{2}\frac{2a}{3}L\left|0\right\rangle+F(j,j)$
$\displaystyle+$
$\displaystyle\left(\frac{J_{l}}{J_{r}}\right)^{2}\frac{\sqrt{a}}{2\sqrt{3}}\sum_{i}\left(\cdots\gamma_{i,i+1}^{1,\sigma,\tau}\cdots+\cdots\gamma_{i-1,i}^{1,\tau,\sigma}\cdots\right)$
$\displaystyle-$
$\displaystyle\left(\frac{J_{l}}{J_{r}}\right)^{2}\frac{2a}{3\sqrt{3}}\sum_{u}\left(\cdots\gamma_{i,i+2}^{1,\sigma,\sigma}\cdots+\cdots\gamma_{i-1,i+1}^{1,\tau,\tau}\cdots\right)$
$\displaystyle+$
$\displaystyle\left(\frac{J_{l}}{J_{r}}\right)^{2}\frac{b\sqrt{a}}{18\sqrt{5}}\sum_{i}\left(\cdots\gamma_{i,i+1}^{2,\sigma,\tau}\cdots+\cdots\gamma_{i-1,i}^{2,\tau,\sigma}\cdots\right)$
$\displaystyle+$
$\displaystyle\left(\frac{J_{l}}{J_{r}}\right)^{2}\frac{2\sqrt{2}}{15\sqrt{3}}\tau(\tau+1)c\sum_{i}\left(\cdots\gamma_{i-1,i,i+1}^{\tau,\sigma,\tau}\cdots\right)$
$\displaystyle+$
$\displaystyle\left(\frac{J_{l}}{J_{r}}\right)^{2}\frac{2\sqrt{2}}{15\sqrt{3}}\sigma(\sigma+1)d\sum_{i}\left(\cdots\gamma_{i,i+1,i+2}^{\sigma,\tau,\sigma}\cdots\right),$

where

$\displaystyle F(i,j)$ $\displaystyle=$
$\displaystyle\left(\frac{J_{l}}{J_{r}}\right)^{2}\frac{a}{6}\sum_{\left|i-j\right|>1}\left(\cdots\gamma_{i-1,i}^{1,\tau,\sigma}\cdots\gamma_{j-1,j}^{1,\tau,\sigma}\right)$
$\displaystyle+$
$\displaystyle\left(\cdots\gamma_{i-1,i}^{1,\tau,\sigma}\cdots\gamma_{j,j+1}^{1,\sigma,\tau}\right)+\left(\cdots\gamma_{i,i+1}^{1,\sigma,\tau}\cdots\gamma_{j-1,j}^{1,\tau,\sigma}\right)$
$\displaystyle+$
$\displaystyle\left(\cdots\gamma_{i,i+1}^{1,\sigma,\tau}\cdots\gamma_{j,j+1}^{1,\sigma,\tau}\right).$

Here,

$\displaystyle a$ $\displaystyle=$
$\displaystyle\sigma\tau(\sigma+1)(\tau+1),$ $\displaystyle b$
$\displaystyle=$
$\displaystyle\sqrt{(2\sigma+3)(2\sigma-1)(2\tau+3)(2\tau-1)},$ $\displaystyle
c$ $\displaystyle=$
$\displaystyle\sqrt{\sigma(\sigma+1)(2\sigma+3)(2\sigma-1)},$ $\displaystyle
d$ $\displaystyle=$ $\displaystyle\sqrt{\tau(\tau+1)(2\tau+3)(2\tau-1)},$

and

$\displaystyle\gamma_{i,j}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{5}}(\left|q^{+2}\right\rangle_{i}\left|q^{-2}\right\rangle_{j}-\left|q^{+1}\right\rangle_{i}\left|q^{-1}\right\rangle_{j}$
$\displaystyle+$
$\displaystyle\left|q^{0}\right\rangle_{i}\left|q^{0}\right\rangle_{j}-\left|q^{-1}\right\rangle_{i}\left|q^{+1}\right\rangle_{j}+\left|q^{-2}\right\rangle_{i}\left|q^{+2}\right\rangle_{j}),$
$\displaystyle\gamma_{i,j,k}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}(\left|\alpha^{+1}\right\rangle_{i,j}\left|t^{-1}\right\rangle_{k}-\left|\alpha^{0}\right\rangle_{i,j}\left|t^{0}\right\rangle_{k}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+$
$\displaystyle\left|\alpha^{-1}\right\rangle_{i,j}\left|t^{+1}\right\rangle_{k})$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}(\left|t^{+1}\right\rangle_{i}\left|\alpha^{-1}\right\rangle_{j,k}-\left|t^{0}\right\rangle_{i}\left|\alpha^{0}\right\rangle_{j,k}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+$
$\displaystyle\left|t^{-1}\right\rangle_{i}\left|\alpha^{+1}\right\rangle_{j,k}),$

where

$\displaystyle\left|\alpha^{+1}\right\rangle_{i,j}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{10}}\left|t^{+1}\right\rangle_{i}\left|q^{0}\right\rangle_{j}-\sqrt{\frac{3}{10}}\left|t^{0}\right\rangle_{i}\left|q^{+1}\right\rangle_{j}$
$\displaystyle+$
$\displaystyle\sqrt{\frac{3}{5}}\left|t^{-1}\right\rangle_{i}\left|q^{+2}\right\rangle_{j},$
$\displaystyle\left|\alpha^{0}\right\rangle_{i,j}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{3}{10}}\left|t^{+1}\right\rangle_{i}\left|q^{-1}\right\rangle_{j}-\sqrt{\frac{2}{5}}\left|t^{0}\right\rangle_{i}\left|q^{0}\right\rangle_{j}$
$\displaystyle+$
$\displaystyle\sqrt{\frac{3}{10}}\left|t^{-1}\right\rangle_{i}\left|q^{+1}\right\rangle_{j},$
$\displaystyle\left|\alpha^{-1}\right\rangle_{i,j}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{3}{5}}\left|t^{+1}\right\rangle_{i}\left|q^{-2}\right\rangle_{j}-\sqrt{\frac{3}{10}}\left|t^{0}\right\rangle_{i}\left|q^{-1}\right\rangle_{j}$
$\displaystyle+$
$\displaystyle\sqrt{\frac{1}{10}}\left|t^{-1}\right\rangle_{i}\left|q^{0}\right\rangle_{j}.$

Finally the ground state of the ladder up to second order in $J_{l}/J_{r}$ is
given by:

$\left|GS^{(2)}\right\rangle=\left|GS^{(0)}\right\rangle+\left|1\right\rangle+\left|2\right\rangle.$
(17)

##  Appendix B Different initial guesses in iTEBD algorithm

Since the iTEBD algorithm is very sensitive to the initial guess for the
ground state, in the vicinity of phase transition points it may lead to
results which are not compatible with fundamental theories in condensed matter
physics such as Mermin-Wagner theorem. For example, in our unfrustrated mixed-
spin ladder, in the region $0<R<0.08$, the iTEBD results predict the presence
of a long-range ferrimagnetic order. We examined the ground state of the
system using various initial states and the result has been summarized in the
following table. As it is obvious the state with minimum energy shows a
magnetic long-range order.

Table 1: Ground state energy for different initial guesses obtained by iTEBD at $R=0.01$. initial guess | Ground state energy | $\mathcal{O_{FiM-N}}$ | $\chi$
---|---|---|---
ًRandom | $-288.663089927308$ | $-0.871032827742$ | $120$
Random | $-288.663684562826$ | $-0.089252552437$ | $140$
Random | $-288.663978011657$ | $-0.86637333846$ | $160$
Random | $-288.664158723753$ | $-0.715664129962$ | $180$
Random | $-2.88662408726905$ | $-0.322314791619$ | $200$
Random | $-288.664458015598$ | $-0.611408290830$ | $220$
FM | $-288.603278537842$ | $0.532418953609$ | $120$
Haldane | $-288.692834152507$ | $0.903095324450$ | $120$
$RS$ | $-288.603278538337$ | $-0.706106745175$ | $120$
Columnar dimer | $-288.692834152497$ | $0.29916032904$ | $120$
Staggered dimer | $-288.692834152576$ | $-1.015885788427$ | $120$
برهم‌نهی هالدین و $RS$ | $-288.692834151684$ | $-0.189865736778$ | $120$
حالتی با تقارن $SU(2)$ | $-288.687750446529$ | $-0.000413096230$ | $120$

## References

  * Diep (2005) H. T. Diep,  _Frustrated Spin Systems_ (WORLD SCIENTIFIC, 2005) https://www.worldscientific.com/doi/pdf/10.1142/5697 .
  * Zhou  _et al._ (2017) Y. Zhou, K. Kanoda, and T.-K. Ng, Rev. Mod. Phys. 89, 025003 (2017).
  * Nisoli  _et al._ (2013) C. Nisoli, R. Moessner, and P. Schiffer, Rev. Mod. Phys. 85, 1473 (2013).
  * Ortiz-Ambriz  _et al._ (2019) A. Ortiz-Ambriz, C. Nisoli, C. Reichhardt, C. J. O. Reichhardt, and P. Tierno, Rev. Mod. Phys. 91, 041003 (2019).
  * Zapf  _et al._ (2014a) V. Zapf, M. Jaime, and C. D. Batista, Rev. Mod. Phys. 86, 563 (2014a).
  * Zapf  _et al._ (2014b) V. Zapf, M. Jaime, and C. D. Batista, Rev. Mod. Phys. 86, 1453 (2014b).
  * Fáth  _et al._ (2001) G. Fáth, O. Legeza, and J. Sólyom, Phys. Rev. B 63, 134403 (2001).
  * Kim  _et al._ (2000) E. H. Kim, G. Fáth, J. Sólyom, and D. J. Scalapino, Phys. Rev. B 62, 14965 (2000).
  * Hijii  _et al._ (2005) K. Hijii, A. Kitazawa, and K. Nomura, Phys. Rev. B 72, 014449 (2005).
  * Karrasch  _et al._ (2015) C. Karrasch, D. M. Kennes, and F. Heidrich-Meisner, Phys. Rev. B 91, 115130 (2015).
  * Ramos and Xavier (2014) F. B. Ramos and J. C. Xavier, Phys. Rev. B 89, 094424 (2014).
  * Aristov  _et al._ (2010) D. N. Aristov, C. Brünger, F. F. Assaad, M. N. Kiselev, A. Weichselbaum, S. Capponi, and F. Alet, Phys. Rev. B 82, 174410 (2010).
  * Batchelor  _et al._ (2007) M. T. Batchelor, X. W. Guan, N. Oelkers, and Z. Tsuboi, Advances in Physics 56, 465 (2007), https://doi.org/10.1080/00018730701265383 .
  * Hikihara and Starykh (2010) T. Hikihara and O. A. Starykh, Phys. Rev. B 81, 064432 (2010).
  * Mishra  _et al._ (2015) T. Mishra, S. Greschner, and L. Santos, Phys. Rev. A 91, 043614 (2015).
  * Bhaseen and Tsvelik (2003) M. J. Bhaseen and A. M. Tsvelik, Phys. Rev. B 68, 094405 (2003).
  * Sugimoto  _et al._ (2018) T. Sugimoto, M. Mori, T. Tohyama, and S. Maekawa, Phys. Rev. B 97, 144424 (2018).
  * Sasaki  _et al._ (2020) K. Sasaki, T. Sugimoto, T. Tohyama, and S. Sota, Phys. Rev. B 101, 144407 (2020).
  * Hung  _et al._ (2006) H.-H. Hung, C.-D. Gong, Y.-C. Chen, and M.-F. Yang, Phys. Rev. B 73, 224433 (2006).
  * Barbarino  _et al._ (2018) S. Barbarino, M. Dalmonte, R. Fazio, and G. E. Santoro, Phys. Rev. A 97, 013634 (2018).
  * Vekua and Honecker (2006) T. Vekua and A. Honecker, Phys. Rev. B 73, 214427 (2006).
  * Liu  _et al._ (2008) G.-H. Liu, H.-L. Wang, and G.-S. Tian, Phys. Rev. B 77, 214418 (2008).
  * Dagotto and Rice (1996) E. Dagotto and T. M. Rice, Science 271, 618 (1996), https://science.sciencemag.org/content/271/5249/618.full.pdf .
  * Allen  _et al._ (2000) D. Allen, F. H. L. Essler, and A. A. Nersesyan, Phys. Rev. B 61, 8871 (2000).
  * Fouet  _et al._ (2006) J.-B. Fouet, F. Mila, D. Clarke, H. Youk, O. Tchernyshyov, P. Fendley, and R. M. Noack, Phys. Rev. B 73, 214405 (2006).
  * Abouie and Langari (2004) J. Abouie and A. Langari, Phys. Rev. B 70, 184416 (2004).
  * Abouie  _et al._ (2006) J. Abouie, S. A. Ghasemi, and A. Langari, Phys. Rev. B 73, 014411 (2006).
  * Heydarinasab and Abouie (2017) F. Heydarinasab and J. Abouie, Phys. Rev. B 96, 104406 (2017).
  * Rezai  _et al._ (2010) M. Rezai, A. Langari, and J. Abouie, Phys. Rev. B 81, 060401 (2010).
  * Heydarinasab and Abouie (2020) F. Heydarinasab and J. Abouie, Journal of Physics: Condensed Matter 32, 165804 (2020).
  * Langari  _et al._ (2011) A. Langari, J. Abouie, M. Z. Asadzadeh, and M. Rezai, Journal of Statistical Mechanics: Theory and Experiment 2011, P08001 (2011).
  * Chandra  _et al._ (2010) V. R. Chandra, N. B. Ivanov, and J. Richter, Phys. Rev. B 81, 024409 (2010).
  * Kim  _et al._ (2008) E. H. Kim, O. Legeza, and J. Sólyom, Phys. Rev. B 77, 205121 (2008).
  * Weihong  _et al._ (1998) Z. Weihong, V. Kotov, and J. Oitmaa, Phys. Rev. B 57, 11439 (1998).
  * Starykh and Balents (2004) O. A. Starykh and L. Balents, Phys. Rev. Lett. 93, 127202 (2004).
  * White (1992) S. R. White, Phys. Rev. Lett. 69, 2863 (1992).
  * Vidal (2003) G. Vidal, Phys. Rev. Lett. 91, 147902 (2003).
  * Schollwöck (2005) U. Schollwöck, Rev. Mod. Phys. 77, 259 (2005).
  * Vidal (2004) G. Vidal, Phys. Rev. Lett. 93, 040502 (2004).
  * Jordan  _et al._ (2008) J. Jordan, R. Orús, G. Vidal, F. Verstraete, and J. I. Cirac, Phys. Rev. Lett. 101, 250602 (2008).
  * Haghshenas and Sheng (2018) R. Haghshenas and D. N. Sheng, Phys. Rev. B 97, 174408 (2018).
  * Haghshenas  _et al._ (2019a) R. Haghshenas, S.-S. Gong, and D. N. Sheng, Phys. Rev. B 99, 174423 (2019a).
  * Haghshenas  _et al._ (2019b) R. Haghshenas, M. J. O’Rourke, and G. K.-L. Chan, Phys. Rev. B 100, 054404 (2019b).
  * Osterloh  _et al._ (2002) A. Osterloh, L. Amico, G. Falci, and R. Fazio, Nature 416, 608 (2002).
  * Levin and Wen (2006) M. Levin and X.-G. Wen, Phys. Rev. Lett. 96, 110405 (2006).
  * Kitaev and Preskill (2006) A. Kitaev and J. Preskill, Phys. Rev. Lett. 96, 110404 (2006).
  * Li and Haldane (2008) H. Li and F. D. M. Haldane, Phys. Rev. Lett. 101, 010504 (2008).
  * Pollmann  _et al._ (2010) F. Pollmann, A. M. Turner, E. Berg, and M. Oshikawa, Phys. Rev. B 81, 064439 (2010).
  * Haghshenas  _et al._ (2014) R. Haghshenas, A. Langari, and A. T. Rezakhani, Journal of Physics: Condensed Matter 26, 456001 (2014).
  * Brehmer  _et al._ (1997) S. Brehmer, H.-J. Mikeska, and S. Yamamoto, Journal of Physics: Condensed Matter 9, 3921 (1997).
  * Yamamoto  _et al._ (1998) S. Yamamoto, S. Brehmer, and H.-J. Mikeska, Phys. Rev. B 57, 13610 (1998).
  * Yamamoto  _et al._ (2000) S. Yamamoto, T. Fukui, and T. Sakai, The European Physical Journal B - Condensed Matter and Complex Systems 15, 211 (2000).
  * Trumper and Gazza (2001) A. E. Trumper and C. Gazza, Phys. Rev. B 64, 134408 (2001).
  * Östlund and Rommer (1995) S. Östlund and S. Rommer, Phys. Rev. Lett. 75, 3537 (1995).
  * Eisert  _et al._ (2010) J. Eisert, M. Cramer, and M. B. Plenio, Rev. Mod. Phys. 82, 277 (2010).
  * White and Feiguin (2004) S. R. White and A. E. Feiguin, Phys. Rev. Lett. 93, 076401 (2004).
  * Daley  _et al._ (2004) A. J. Daley, C. Kollath, U. Schollwöck, and G. Vidal, Journal of Statistical Mechanics: Theory and Experiment 2004, P04005 (2004).
  * Orús and Vidal (2008) R. Orús and G. Vidal, Phys. Rev. B 78, 155117 (2008).
  * Hohenberg (1967) P. C. Hohenberg, Phys. Rev. 158, 383 (1967).
  * Mermin and Wagner (1966) N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 (1966).
  * Pollmann  _et al._ (2012) F. Pollmann, E. Berg, A. M. Turner, and M. Oshikawa, Phys. Rev. B 85, 075125 (2012).
  * Huang and Lin (2011) C.-Y. Huang and F.-L. Lin, Phys. Rev. B 84, 125110 (2011).
  * Ye  _et al._ (2016) B.-T. Ye, L.-Z. Mu, and H. Fan, Phys. Rev. B 94, 165167 (2016).
  * De Chiara  _et al._ (2012) G. De Chiara, L. Lepori, M. Lewenstein, and A. Sanpera, Phys. Rev. Lett. 109, 237208 (2012).
  * den Nijs and Rommelse (1989) M. den Nijs and K. Rommelse, Phys. Rev. B 40, 4709 (1989).
  * Shelton  _et al._ (1996) D. G. Shelton, A. A. Nersesyan, and A. M. Tsvelik, Phys. Rev. B 53, 8521 (1996).
  * Haghshenas  _et al._ (2016) R. Haghshenas, A. Langari, and A. T. Rezakhani, Journal of Physics: Condensed Matter 28, 176001 (2016).
  * Pollmann and Turner (2012) F. Pollmann and A. M. Turner, Phys. Rev. B 86, 125441 (2012).
  * Kao  _et al._ (2015) Y. Kao, Y.-D. Hsieh, and P. Chen (2015).
  * Fishman  _et al._ (2020) M. Fishman, S. R. White, and E. M. Stoudenmire, “The ITensor software library for tensor network calculations,” (2020), arXiv:2007.14822 .