# Topological line in frustrated Toric code models

M. H. Zarei<EMAIL_ADDRESS>Department of Physics, School of Science,
Shiraz University, Shiraz 71454, Iran  J. Abouie<EMAIL_ADDRESS>Department
of Physics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan
45137-66731, Iran

###### Abstract

Typical topological systems undergo a topological phase transition in the
presence of a strong enough perturbation. In this paper, we propose an
adjustable frustrated Toric code with a ”topological line” at which no phase
transition happens in the system and the topological order is robust against a
non-linear perturbation of arbitrary strength. This important result is a
consequence of the interplay between frustration and nonlinearity in our
system, which also causes to the emergence of other interesting phenomena such
as reentrant topological phases and survival of the topological order under
local projection operations. Our study opens a new window towards more robust
topological quantum codes which are cornerstones of large-scale quantum
computing.

###### pacs:

68.35.Rh, 3.67.-a, 03.65.Vf, 75.10.Hk

##  I Introduction

Topological phases are states of matters that are robust against local
perturbations wen1 ; wen2 ; t1 ; t2 ; t3 ; 20 ; bravyi . Designing topological
systems is of crucial importance in a wide range of practical applications
from quantum computing del ; 18 ; 19 ; 200 ; 22 ; 24  and topological
spintronics Fan ; Beenakker  to soft matter and mechanical systems Nagel ;
Ma19 , and searching for systems with rich topological properties is one of
the growing interests in both theoretical and experimental physics Hasan ;
Haldane ; Armitage ; Ozawa ; zarei2016 ; Ahmadi2020 . Toric codes (TCs) are a
kind of topological quantum models, characterizing by their robust topological
degeneracy. They were first introduced for topological quantum memory
Kitaev2003 ; 15 , and have gained significant importance in recent years in
the context of large-scale quantum computing pra ; thr ; sur ; natu . Many
attentions have been devoted to the investigation of the effects of different
types of perturbations including external magnetic fields robustness ; ala ;
schmit ; 29 ; wu ; sch ; jadid , Ising interactions karimipour ; robustness2
and local tensor perturbations nor  on the topological properties of TCs on
different lattices.

One of the important challenges is to find different properties that lead to
more robustness of the topological orders. Recently, the interplay of topology
and frustration has been studied and demonstrated that geometrical frustration
leads to the further robustness of the TC state frus1 ; frus2 . Furthermore,
it has also been shown that the perturbations that couples nonlinearly to a
topological model can lead to further robustness castel . Nevertheless, it is
believed that regardless of the kind of local perturbation, topological orders
will be eventually destroyed at a point where a topological-trivial phase
transition occurs in the system. Hence, it is raising a question of whether it
is possible to construct an adjustable system with an everlasting topological
order.

Figure 1: (Color online) An illustration of the TC on the UJ lattice (black
solid lines). The red and yellow qubits correspond to the vertex and plaquette
operators, respectively. The dual square-octagonal lattice is displayed by the
gray dashed lines where each vertex of the TC can be represented by a simple
loop in the dual lattice. The red dashed lines show a loop configuration on
the dual lattice. The qubits belonging to these loops are in the state
$|1\rangle$, an eigenstate of $Z$ with eigenvalue $-1$.

In this paper, we examine the combined effects of frustration and nonlinearity
by proposing frustrated quantum models composed of a TC coupled nonlinearly to
an external perturbation, and show that their ground states on various
lattices possess a topological phase with interesting properties, not seen in
other topological systems so far. We demonstrate that there is a topological
line (TL) at which (i) by tuning the physical parameters of the system it
remains in the topological phase in the presence of perturbation of any
strengths, (ii) the topological order of the ground state survives under local
projection operations. In order to identify this TL, and to figure out the
physical mechanism behind such a non-trivial phenomenon, we define a
topological string parameter and show that the TL is a line separating two
topological regions with different behaviors of the string parameter. Our
frustrated TC (FTC) models have also another interesting feature, in which a
reentrant topological phase transition occurs in the ground state phase
diagram of the system while the perturbation parameter increases. This
phenomenon signifies the reversibility of the topological order to the system
in the presence of strong perturbations.

This paper is structured as follows: In Sec.(II), we introduce our FTC model
and find its phase diagram by using a mapping to the frustrated Ising model.
Then in Sec.(III), we analyze the phase diagram to find that there is a
reentrant topological phase transition. More importantly, in Sec.(IV) we
introduce the topological line which is charactrized by a topological string
parameter and accordingly we explain how the system is robust against the
perturbation along the topological line. Finally in Sec.(V), we show that the
system is also robust against local projections at the topological line.

##  II Toric code in presence of a non-linear perturbation

A TC model is described by the Hamiltonian:

$H_{\rm TC}=-\sum_{p}B_{p}-\sum_{v}A_{v},$ (1)

where $B_{p}=\prod_{i\in\partial p}Z_{i}$ and $A_{v}=\prod_{i\in v}X_{i}$ are
respectively the plaquette and vertex operators with $Z$ and $X$ being the
Pauli operators, $i\in\partial p$ refers to the qubits around the plaquette
$p$, and $i\in v$ refers to the qubits incoming to the vertex $v$ (see Fig. 1,
the yellow and red dots). This Hamiltonian is exactly solvable and its ground
state is given by $|\psi\rangle=\prod_{v}(I+A_{v})|0\rangle^{\otimes N}$,
where $|0\rangle$ is an eigenstate of $Z$ with eigenvalue 1, $I$ refers to the
identity operator and $N$ is the number of qubits. Since each vertex operator
can also be represented by a loop operator in a dual lattice (in Fig. 1, we
have illustrated the dual of a Union-Jack (UJ) lattice by the gray dashed
lines), the ground state $|\psi\rangle$ is also a loop-condensed state (each
loop is made of $|1\rangle$ states, and a loop-condensed state is a
superposition of all loop-configurations in a sea of $|0\rangle$ states). It
is important to note that, with periodic boundary condition the initial
lattice is attached to a torus with non-trivial loops which lead to different
degenerate ground states. Since these degenerate states correspond to
different topological classes, they are robust against local perturbations.

Now, let us introduce a perturbation to the system via the Hamiltonian:

$H_{e}=\sum_{v}e^{-\beta\sum_{i\in v}J_{i}Z_{i}},$ (2)

where $\beta$ and $J_{i}$ are parameters controlling the order of the system.
Using a Taylor expansion, one can see that $J_{i}$ can be served as the
magnetic moment of the qubit $i$, adjustable within the system, and $\beta$
refers to an inevitable perturbation, arising from an effective field coupled
to the moments. At small values of $\beta$, the above perturbation reduces to
a Zeeman term where the field is coupled linearly to the spins, however for
larger values of $\beta$, nonlinear effects arising from the local multi-spin
interactions around each vertex (spins that are the nearest neighbor of
vertices) become important in the topological characteristics of the system,
especially in the robustness of the topological order expan . In practice, the
perturbation $\beta$ is unavoidable, and the topological order of the system
is finally lost by strong perturbations, however our FTC system has an
additional ability that by adjusting the moments of the qubits ($J_{i}$), one
can achieve a topological order, robust against perturbation of any strengths.
The systems with adjustable magnetic moments and multi-body interactions can
be realized in experiment with current technologies by cold atoms (to find how
multi-body interactions can be implemented, see the protocol recently proposed
in Ref. Bohrdt  and references therein). Here, we set $J_{i}$ to $J_{1}$ for
all qubits on the vertical and horizontal edges and $J_{2}$ for all on the
diagonal edges.

The frustrated Hamiltonian, $H_{\rm FTC}=H_{\rm TC}+H_{e}$, is indeed a type
of stochastic matrix form Hamiltonians castel2005 , and its ground state can
be exactly found castel ; zarei20 ; zarei18  as:

$|G(\beta,\\{J_{i}\\})\rangle=\frac{1}{\sqrt{\mathcal{Z}(\beta)}}e^{\frac{\beta}{2}\sum_{i}J_{i}Z_{i}}|\psi\rangle,$
(3)

where $|\psi\rangle$ is the ground state of the Hamiltonian $H_{\rm TC}$. By
applying the operator $e^{\frac{\beta}{2}\sum_{i}J_{i}Z_{i}}$ on the state
$|\psi\rangle$, we will have a superposition of loop-configurations with
amplitudes $e^{\frac{\beta}{2}\sum_{i}J_{i}\sigma_{i}}$, where
$\sigma_{i}=-1(+1)$ for links with qubits in the state $|1\rangle$
($|0\rangle$). The normalization factor in Eq. (3) is thus obtained as
$\mathcal{Z(\beta)}=\sum_{lc}e^{\beta\sum_{i}J_{i}\sigma_{i}}$, where the
summation runs over all loop-configurations. This function is nothing but the
partition function of a classical Ising model on the UJ lattice, where the
parameter $\beta$ plays the role of the inverse of the thermal energy
$k_{B}T$, $J_{i}$ is the local exchange interaction between the two nearest
neighbor Ising spins, $S_{i}$ and $S_{i+1}$, located at the UJ lattice points
$i$ and $i+1$, and $\sigma_{i}=S_{i}S_{i+1}$ is equal to +1 (-1) when the
nearest neighbor spins are parallel (antiparallel) Steph1 . In the low
temperature expansion of this partition function, we can see that each spin-
configuration is also represented by a loop-configuration in the dual square-
octagonal lattice.

Figure 2: (Color online) The ground state phase diagram of our FTC on the UJ
lattice. The vertical and horizontal axes are respectively $\beta^{-1}$ and
$J_{2}$ (we set $J_{1}=1$). The green and the pink regions are respectively
the topological and the trivial phases which are seperated by the phase
boundaries with equations: $\cosh(4\beta J_{1})=e^{-4\beta J_{2}}\pm
2e^{-2\beta J_{2}}$, where the signs $+$ and $-$ are respectively for the blue
curves at the right and the left sides of the red dashed line. The dashed red
line is the TL. This line divides the green region into two parts where the
topological string parameter decays oscillatory and non-oscillatory with
respect to the string length. In the interval $J_{2}^{i}<J_{2}<J_{2}^{f}$ a
reentrant topological phase appears at small values of $\beta^{-1}$ (the green
region between $\beta_{1}^{-1}$ and $\beta_{2}^{-1}$). For the case of
$J_{1}=1$, the parameters $J_{2}^{i}$ and $J_{2}^{f}$ are approximately equal
to $-1$ and $-0.9$.

It is intuitive to compare the ground state of the TC on a square lattice in
the presence of a uniform magnetic field with the ground state of our FTC
model. In both cases they are a superposition of loop-configurations, but the
effects of perturbations are different. In the former case, the perturbation
causes the generation of open strings robustness ; frus1 , while in ours,
owing to nonlinearities arising from the local multi-body interactions, the
perturbation only changes the amplitudes of the loop-configurations. For
example, in the case of $J_{1}=J_{2}=+1$, the amplitudes in Eq. (3) are in the
form of $\sim e^{-\beta l}$, where $l$ is the total perimeter of all loops.
This term shows that the perturbation acts as a ”tension”, and an increase of
$\beta$ decreases the amplitude of large loops. In particular, in the limit of
$\beta\rightarrow\infty$, the amplitude of all loop-configurations goes to
zero and the final state will be the product state $|00...0\rangle$. Since the
initial state at $\beta=0$ is the topological loop-condensed state
$|\psi\rangle$, it is concluded that there must be occurred a topological-
trivial phase transition by the increase of $\beta$ from zero to $\infty$
(more details for the ground state phase diagram of the TC on a simple
unfrustrated square lattice has been addressed in castel ; zarei20 ). On the
other hand, the situation is different if $J_{1}=J_{2}=-1$. In this case, the
amplitudes in Eq. (3) are simplified as $\sim e^{\beta l}$. In contrast to the
previous case, here the perturbation $e^{\beta l}$ plays the role of a
”pressure”, and an increase of $\beta$ leads to the generation of loops with
larger total perimeter. In the limit of $\beta\rightarrow\infty$, the loop-
configuration with maximum total perimeter is dominant and the ground state is
the product state $|11...1\rangle$. Finally, in the case of $J_{1}\geq 0$ and
$J_{2}\leq 0$, both the string tension and pressure are present in the system.
In this case the interplay of the string tension and pressure causes the
system to be frustrated. The simultaneous presence of the frustration and
nonlinearities arising from the local multi-body interactions is in the favor
of topological robustness, and leads to the interesting phenomena discussed in
the following sections.

##  III Reentrant topological phase

In order to obtain the topological phase transition points in our FTC model,
we investigate the behavior of the ground state fidelity; $F=\langle
G(\beta,\\{J_{1},J_{2}\\})|G(\beta+d\beta,\\{J_{1},J_{2}\\})\rangle$.
Employing a Taylor expansion, the ground state fidelity zanardi  is readily
obtained in terms of the specific heat ($C_{V}$) of the Ising model as:

$F\simeq 1-\frac{C_{V}}{8\beta^{2}}d\beta^{2},$ (4)

where we have used the equality
$C_{V}=\beta^{2}\frac{\partial^{2}\ln\mathcal{Z}}{\partial\beta^{2}}$. The
above relation indicates that corresponding to a phase transition temperature
in the classical model where the specific heat shows a singularity, there must
be a topological phase transition point, where the ground state fidelity
becomes singular.

Fortunately, the phase diagram of the Ising model on different lattices such
as the UJ and the triangular lattices is exactly known Steph1 ; diep . Here,
as an interlude we explain the classical phase diagram of the Ising model on
the UJ lattice with $J_{1}>0$ and $J_{2}$ couplings. This model possesses two
different phases with ferromagnetic and antiferromagnetic long range orders at
low temperatures and a paramagnetic disorder phase at high temperatures. For
large positive (negative) values of $J_{2}/J_{1}$, strong ferromagnetic
(antiferromagnetic) couplings cause the system to be in the ferromagnetic
(aniferromagnetic) phase at low temperatures. However, at intermediate
strengths of $J_{2}/J_{1}<0$, the system behaves exotically, it is in the
paramagnetic phase at high temperatures and goes to the antiferromagnetic
phase by decreasing temperature. By further decreasing of temperature, we
expect the staggered magnetization to increase and the true long range Neel
order to form in the classical phase diagram, but this will not happen and
amazingly a phase transition occurs to the paramagnetic phase. This phenomena
which is a consequences of frustration is called reentrant phase transition.

Now we come back to our frustrated TC model on the UJ lattice, where $\beta$
is the perturbation parameter. Since $\beta^{-1}$ plays the role of
temperature in the classical Ising model, the paramagnetic phase corresponds
to a topological order at small values of $\beta$, and the ordered
ferromagnetic and antiferromagnetic phases at low temperatures correspond to a
topologically trivial phase at large values of $\beta$. In Fig. 2, we have
presented the ground state phase diagram of our FTC model on the UJ lattice.
Let us focus on the interesting region of $J_{2}^{i}<J_{2}<J_{2}^{f}$. At
large values of $\beta^{-1}$ (or small strengths of perturbation) the ground
state possesses a topological order, however this order can not persist in the
presence of stronger perturbations (or smaller $\beta^{-1}$) and disappears
eventually at a transition point ($\beta_{3}^{-1}$) where the system enters to
a trivial phase. Now, we expect the system to be locked in the trivial phase,
and perturbations wash out the topological order completely, however we see
that amazingly a ” reentrant topological phase transition ” occurs in the
system and the topological order revives at the second phase transition point
($\beta_{2}^{-1}$). We have also examined our FTC model on other geometrically
frustrated lattices such as Kagome and triangular lattices (the results are
not shown here), and observed the above mentioned phenomenon also in these
systems. Actually, the emergence of a reentrant topological phase is a
dramatic impact of frustration and nonlinearities arising from the multi-body
interactions introduced by the Hamiltonian in Eq. (2) to the system.

##  IV Topological line

As we discussed, there is a topological order in the ground state phase
diagram of our FTC model which is robust against small perturbations, $\beta$,
but disappears in the presence of strong perturbations when a topological-
trivial phase transition occurs in the system. However, surprisingly we see
that exactly at a line in the topological phase, the system remains in the
topological phase in the presence of any strength of the parameter $\beta$. We
call this line as ” Topological line (TL)”, because the topological order
survives at this line regardless of the strength of $\beta$ (see the dashed
line in Fig. 2).

Figure 3: (Color online) A UJ lattice with Ising spins (white dots). The solid
lines are strings that connect the two spins $S_{i}$ and $S_{j}$ located at
the lattice points $i$ and $j$. Red and white dotes are respectively spin $+1$
and $-1$. The blue (green) string crosses a loop for even (odd) times.

In order to obtain the equation of this TL in the ground state phase diagram,
we utilize the TC-Ising mapping explained in the previous section. In the
language of the classical Ising model on the UJ lattice, the TL is a disorder
line in the paramagnetic phase which separates two regions with different two-
point correlation functions. In the region next to the ferromagnetic phase the
two-point correlations decay nonoscillatory by increasing the separation
distance of the Ising spins, while oscillatory in the region next to the
antiferromagnetic phase. The two-point correlation functions in the Ising
model are equal to the signed summations of the Boltzmann weights where the
sign behind each Boltzmann weight is determined by the sign of $S_{i}S_{j}$
($S_{i}$ is an Ising spin located at UJ lattice point $i$). Since, each spin-
configuration on the UJ lattice is represented by a loop-configuration on its
dual lattice, we can express the two-point correlation functions as signed
summations of the Boltzmann weights corresponding to different loop-
configurations. In order to determine the sign of the mentioned Boltzmann
weights, we pull a string $\gamma$ between the two spins $S_{i}$ and $S_{j}$
on the UJ lattice (see Fig. 3). For spin-configurations with parallel
(antiparallel) $S_{i}$ and $S_{j}$, the corresponding loop-configurations on
the dual lattice will cross the string $\gamma$ for even (odd) times (see Fig.
3), and the sign of the Boltzmann weight becomes $+1~{}(-1)$. Now, we define a
string operator as $\mathcal{S}_{\gamma}=\prod_{i\in\gamma}Z_{i}$ where
$i\in\gamma$ refers to all qubits belonging to the string $\gamma$ string .
The expectation value of this operator in the ground state of our FTC model is
equal to the two-point correlation function $\langle S_{i}S_{j}\rangle$ in the
Ising model, i.e. $\langle S_{i}S_{j}\rangle=\langle
G(\beta)|\mathcal{S}_{\gamma}|G(\beta)\rangle$. By using this relation, we can
obtain the equation of the TL as:

$\cosh(4\beta J_{1})=\exp(-4\beta J_{2}).$ (5)

This TL divides the topological phase into two parts as shown in Fig. 2. In
these regions the topological string parameter decays oscillatory and
nonoscillatory by increasing the length of the string, respectively. These
different behaviors can be well interpreted by comparing the strengths of the
string tension and pressure in the FTC model. Actually, in the topological
phase at the left of the TL, the pressure causes several small loops to be
generated in the system, and hence loop-configurations with larger total
perimeters play the dominant role in determining the ground state of the
system. It should be noted that loop-configurations with large perimeters are
those which are generated from several small loops. Accordingly, a typical
string $\gamma$ with the length of $r$ successively crosses the small loops,
and consequently the sign of the string parameter
$\langle\mathcal{S}_{\gamma}(r)\rangle$ oscillates by increasing $r$. On the
other hand, in the topological phase at the right of the TL, the string
tension plays the dominant role, and the loop-configurations with very few
small loops are crucial in determining the ground state. In this region,
unlike the topological phase at the left of the TL, the sign of the string
parameter does not change by $r$ (see the inset plots in Fig. 2). Finally, at
the TL, the reciprocal effects of tension and pressure are balanced, resulting
in the formation of closed loops with different sizes. The stability of
various loops leads to robustness of the topological order at the TL.

##  V Topological line and robustness against local projections

The TL has also another characteristic at which local projection operations
are not able to destroy the topological order of the ground state. To explain
this phenomenon we start with an important property of the TC state
$|G(\beta=0)\rangle=|\psi\rangle$, in which if we apply a projection operator
like $|+\rangle\langle+|$, with $|+\rangle$ being an eigenstate of the Pauli
operator $X$, on a single qubit of the state $|\psi\rangle$, it removes the
corresponding edge from the lattice and the quantum state of the rest will be
again a TC state ras ; zar . Now, we consider our FTC model on a triangular
lattice with three tuning parameter $J_{1}$, $J_{2}$, and $J_{3}$
corresponding to three qubits of each triangle, see Fig. 4. If we project out
the three spins of a triangle, the quantum state of the rest is given by:

$|\tilde{G}(\beta)\rangle=\langle+_{1}+_{2}+_{3}|G(\beta)\rangle,$ (6)

where $|G(\beta)\rangle$ is the ground state of the FTC on a triangular
lattice. By applying the mentioned projection operators sequentially on the
ground state, different excitations will be created in the system. They move
in the lattice and destroy the topological order of the ground state. In
general, the reduced ground state $|\tilde{G}(\beta)\rangle$ is not the same
as the initial state $|G(\beta)\rangle$. However, at the TL they are exactly
identical. In order to prove this important property of the TL, let us project
out the three spins of a triangle, for example the triangle denoted by $p$
shown in Fig. 4. The ground state of our FTC is written as
$|G(\beta)\rangle=\frac{1}{\mathcal{Z}}e^{\sum_{i}\frac{K_{i}}{2}Z_{i}}|\psi\rangle$
where $K_{i}=\beta J_{i}$ and $|\psi\rangle$ is the pure TC state on the
triangular lattice which can be written in the form of
$\prod_{\Delta}(1+B_{\Delta})|+\rangle^{N}$, where $\Delta$ refers to
triangular plaquettes, $N$ is the number of edges and
$B_{\Delta}=\prod_{i\in\Delta}Z_{i}$ is plaquette operator. Substituting
$|G(\beta)\rangle$ in Eq. (6), we obtain

$\displaystyle|\tilde{G}(\beta)\rangle=e^{\sum{i\neq
1,2,3}\frac{K_{i}}{2}Z_{i}}\prod_{\Delta\neq
p,q}(1+B_{\Delta})|+\rangle^{\otimes(N-3)}\times$
$\displaystyle\langle+_{1}+_{2}+_{3}|e^{\sum_{i=1}^{3}\frac{K_{i}Z_{i}}{2}}(1+B_{p})(1+B_{q})|+_{1}+_{2}+_{3}\rangle,$

where $B_{p}=Z_{1}Z_{2}Z_{3}$ and $B_{q}=Z_{3}Z_{4}Z_{5}$. Here, $p$ and $q$
are the label of triangles which share the qubit 3 (see Fig. 4). By doing
straightforward calculations we achieve the ground state of the FTC as:

$\displaystyle|\tilde{G}(\beta)\rangle=e^{\sum{i\neq
1,2,3}\frac{K_{i}}{2}Z_{i}}\prod_{\Delta\neq
p,q}(1+B_{\Delta})|+\rangle^{\otimes(N-3)}\times$
$\displaystyle[A(K_{1},K_{2},K_{3})+B(K_{1},K_{2},K_{3})Z_{1}Z_{2}],$ (7)

with

$\displaystyle
A(K_{1},K_{2},K_{3})=\prod_{i=1}^{3}\cosh(K_{i}/2)+\prod_{i=1}^{3}\sinh(K_{i}/2),$
$\displaystyle
B(K_{1},K_{2},K_{3})=\prod_{i=1}^{2}\cosh(K_{i}/2)\sinh(K_{3}/2)$
$\displaystyle~{}~{}~{}~{}~{}+\prod_{i=1}^{2}\sinh(K_{i}/2)\cosh(K_{3}/2).$
(8)

Figure 4: (Color online) Schematic illustration of our FTC model on a
triangular lattice with $J_{1}$, $J_{2}$, and $J_{3}$ tuning parameters. The
plaquettes $p$ and $q$ share the qubit 3. At the TL, by projecting out the red
qubits denoted by 1, 2, and 3 by the state $\langle+|$, leads to the same
frustrated toric code model on a reduced triangular lattice where the
plaquette $p$ is removed.

Now, if we set $B(K_{1},K_{2},K_{3})$ to zero, according to Eq. (7), the
ground state $|\tilde{G}(\beta)\rangle$ is identical to $|G(\beta)\rangle$ on
the reduced triangular lattice where the triangle $p$ is removed.
Consequently, for those $K_{1}$, $K_{2}$ and $K_{3}$ which satisfy the
constraint $B(K_{1},K_{2},K_{3})=0$, the final state will be again the FTC
state. This constraint is nothing but the equation of the TL. Finally, we
point out that regarding to the classical-quantum mapping, the above process
for our FTC model is similar to the ” dimensional reduction ” in frustrated
Ising model where one can obtain the partition function of the classical Ising
model at the disorder line by tracing over spin degrees of freedom row-by-row
diep .

The invariance of our FTC ground state under the local projections implies
that independent of the strength of perturbations, the excitations are
suppressed at the TL by frustration and nonlinearity, and the topological
order survives under the above local projections. The existence of such a TL
in our FTC models is crucial in practical applications. In particular, one can
imagine that in the presence of a perturbation with arbitrary strength, we can
tune the moment $J_{2}$ related to physical qubits living in the diagonal
edges of the lattice so that the system remains permanently in the topological
phase. Actually, having two types of physical qubit with different moments,
$J_{1}$ and $J_{2}$, is an additional ability of our FTC which is absent in
the TC.

##  VI Summary and outlook

The robustness of topological orders against local perturbations is of
crucially importance for modern practical applications, and constructing
highly-robust topological systems is one of the most important current
challenges. In this paper we have taken an important step forward by
introducing realizable models with a robust topological order. We demonstrated
that the interplay of frustration and nonlinearity in our system leads to the
formation of a TL at which no phase transition occurs in the system, and the
topological order is robust even against local projection operations. We also
found another interesting phenomena, not seen in other frustrated systems,
that the ground state of our frustrated Toric code model experiences a
reentrant topological phase transition. This phenomenon signifies the
reversibility of the topological order to the system in the presence of
frustration.

Our paper paves the way for future studies for implementing more robust
topological quantum codes. In particular, it is interesting to explore the
existence of such a TL in different quantum codes including color codes and
fracton codes, employing proper quantum-classical mappings.

## Acknowledgement

The authors would like to thank S. S. Jahromi, A. Ramezanpour, A. Montakhab
and L. Memarzadeh for fruitful discussions.

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