Topologically protected boundary discrete time crystal for a solvable model Peng Xu1and Tian-Shu Deng2 1School of Physics Zhengzhou University Zhengzhou 450001 China

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Topologically protected boundary discrete time crystal for a solvable model

Peng Xu1and Tian-Shu Deng2, ∗

1School of Physics, Zhengzhou University, Zhengzhou 450001, China

2Institute for Advanced Study, Tsinghua University, Beijing,100084, China

Floquet time crystal, which breaks discrete time-translation symmetry, is an intriguing phe-

nomenon in non-equilibrium systems. It is crucial to understand the rigidity and robustness of

discrete time crystal (DTC) phases in a many-body system, and ﬁnding a precisely solvable model

can pave a way for understanding of the DTC phase. Here, we propose and study a solvable spin

chain model by mapping it to a Floquet superconductor through the Jordan-Wigner transformation.

The phase diagrams of Floquet topological systems are characterized by topological invariants and

tell the existence of anomalous edge states. The sub-harmonic oscillation, which is the typical signal

of the DTC, can be generated from such edge states and protected by topology. We also examine

the robustness of the DTC by adding symmetry-preserving and symmetry-breaking perturbations.

Our results on topologically protected DTC can provide a deep understanding of the DTC when

generalized to other interacting or dissipative systems.

I. INTRODUCTION

Periodic driving is a powerful tool to engineer vari-

ety of unique and fascinating phenomena in many-body

systems, such as Mott-insulator-superﬂuid transition1,

dynamical gauge ﬁeld2,3, many-body echo4–6, the re-

alization of topological band structures7,8. Recently,

Floquet systems also bring new possibilities to simu-

late time-translation symmetry broken phases, which are

called discrete time crystals (DTC), and this has at-

tracted much attention from both experimentalists9–20

and theorists21–28. The concept of the time crystal

was originally proposed by Wilczek29. However, it was

ruled out by the no-go theorem in thermal equilib-

rium systems30,31. Then Shivaji Sondhi’s32 and Chetan

Nayak’s33 groups generalized the concept of time crystal

and proposed the DTC, which exhibits a unique prop-

erty that the expectation values of generic observables

manifest a sub-harmonic oscillation. For example, the

kicked Ising chain model with disordered interaction,

where spins collectively ﬂip after one period and back

to their initial state after two periods, is a canonical re-

alization of the DTC.

In a non-interacting spin chain system, taking ˆ

exp (−iθ Pjˆ

Xj) with θ=π/2 as a Floquet evolution

operator is a straightforward method to ﬂip all spins in

one period. Here, ˆ

Xjis a spin operator acting on site

j. However, when θdeviates slightly from π/2, the pe-

riod of observables also deviates from twice of the Flo-

quet period. It means the sub-harmonic response in-

duced by ˆ

U= exp (−iθ Pjˆ

Xj) is a ﬁne-tuned result

and easily destroyed by perturbations. This simple ex-

ample implies that the robustness of sub-harmonic re-

sponse is a crucial property of the DTC. According to

the previous results, many-body localization and pre-

thermalization may provide two mechanisms to stabilize

the sub-harmonic response. Besides, topologically pro-

tected anomalous edge states34–36 also suggest another

mechanism of generating the robust DTC and the rea-

sons are listed in the following. Firstly, edge states in

topological insulators (superconductors) are protected by

symmetries. As long as the symmetries are not broken

and the gap does not close, topological edge states are

stable and robust. Secondly, Floquet topological insula-

tors (superconductors) host anomalous edge states with

quasi-energy π/T which can generate a sub-harmonic re-

sponse to driving frequency 2π/T . Although the rela-

tion between π-mode edge states in Floquet topologi-

cal system and the DTC has been discussed in previ-

ous research37–42, detailed and systematic of topologi-

cally protected DTC in a more general spin chain model

are still lacking. Bridging Floquet topological supercon-

ductors to the topologically protected DTC in a general

periodic driven spin chain model and explicitly analyzing

the dynamics of observables are signiﬁcantly helpful for

deeply understanding the existence and robustness of the

DTC.

In this work, we investigate the existence of the DTC

phase in a general Floquet spin chain model. Such a

periodically driven spin chain can be mapped to a Flo-

quet superconductor through the Jordan-Wigner trans-

formation, after which the model becomes the form of

Majorana fermion. This model is intrinsic with particle-

hole symmetry. Furthermore, this system can be clas-

siﬁed into D class or BDI class, which is dependent on

whether chiral symmetry is preserved. The topological

Floquet superconductor exhibit a special kind of topo-

logically non-trivial phase, where anomalous edge states

with quasi-energy π/T exist. In order to observe a robust

DTC, the observable should be selected as the anomalous

edge mode or the end spin. We numerically demonstrate

that both observables manifest a sub-harmonic response

with a generic initial product state. Finally, we also con-

ﬁrm the robustness of the DTC by adding symmetry-

preserving and symmetry-breaking perturbations.

The paper is organized as follows. In section II, we

describe a general periodically driven spin chain model

and map it to a Floquet superconductor. In section III,

we discuss the topological classiﬁcation of this model,

calculate the topological invariants and obtain the phase

diagrams. In section IV, we demonstrate the existence

arXiv:2210.15222v1 [cond-mat.quant-gas] 27 Oct 2022

of the DTC resulting from the Floquet superconduc-

tors, by selecting the observable as the anomalous edge

mode or the end spin. Finally, in section V, we exam-

ine the robustness of the DTC is protected by topolog-

ically non-trivial phase by adding symmetry-preserving

and symmetry-breaking perturbations.

II. MODEL: PERIODICALLY DRIVEN SPIN

CHAIN

We consider a periodically driven spin-1/2 chain as il-

lustrated in Fig. 1(a),

H(t) = (ˆ

H1,for nT 6t < nT +t1

H2,for nT +t16t < (n+ 1)T,(1)

where

Hm=Jxx

Xjˆ

Xj+1 +Jyy

Yjˆ

Yj+1

+Jxy

Xjˆ

Yj+1 +Jyx

Yjˆ

Xj+1 +hz

Zj.(2)

Xj,ˆ

Yjand ˆ

Zjare spin operators (with the form of Pauli

matrices) acting on the j-th site and the chain contains

Nspins. Jxx

m,Jyy

m,Jxy

m,Jyx

mrepresent the strengths of

nearest spin interactions and hz

mthe transverse ﬁeld dur-

ing m-th time interval. By employing the Jordan-Wigner

transformation

Xj= (ˆc†

j+ ˆcj)eiπ Pl<j ˆc†

lˆcl,

Yj=−i(ˆc†

j−ˆcj)eiπ Pl<j ˆc†

lˆcl,

Zj= 2ˆc†

jˆcj−1.

(3)

where ˆc†

jand ˆcjare the creation and annihilation fermion

operators on j-th site. Then the Hamiltonian in Eq. (2)

can be mapped to a fermionic system,

Hm=X

(Jxx

m−Jyy

m−iJxy

m−iJyx

m) ˆc†

jˆc†

j+1

(Jxx

m+Jyy

m+iJxy

m−iJyx

m) ˆc†

jˆcj+1

+hz

(2ˆc†

jˆcj−1) + h.c..

(4)

Furthermore, by deﬁning ˆγj,1=1

√2ˆc†

j+ ˆcjand

ˆγj,2=i1

√2ˆcj−ˆc†

j, we obtain a Hamiltonian in Ma-

jorana representaion, as illustrated in Fig. 1(b),

Hm= 2Jxx

iˆγj,2ˆγj+1,1−2Jyy

iˆγj,1ˆγj+1,2

+ 2Jxy

iˆγj,2ˆγj+1,2−2Jyx

iˆγj,1ˆγj+1,1

+ 2Jz

iˆγj,2ˆγj,1.

(5)

ˆγ1

ˆγ2

π/T

bulk

edge

E+π/T

β†

(a)

(b)

FIG. 1. (a) Schematic of spin chain. (b) Schematic of Ma-

jorana chain. (c) Quasi-energy spectrum exp(−iµT) with

µthe quasi-energy excitation. 0 and π/T represent two edge

states quasi-energy excitation. (d) Eigenvalues of the Floquet

evolution operator ˆ

UT. Quasi-particle operator ˆ

β†

πexcites a

state |Eiwith eigenenergy Eto another state |E+π/T i.

Both Hamiltonians ˆ

H1and ˆ

H2in Eq. (5) are quadratic,

so they can be rewritten as

Hm=1

2ˆ

Ψ†Hmˆ

Ψ,(6)

with

Ψ†=ˆγ1,1ˆγ1,2ˆγ2,1ˆγ2,2... ,(7)

where Hmis a 2N∗2Nanti-symmetric matrix. Here

and after, ˆ

H(ˆ

U) represents an operator and H(U)

represents a matrix. Applying Fourier transformation

ˆγj,m =1

√NPkˆγk,meikj ,ˆ

Hm(k) has such a form

Hm=X

kˆγ†

k,1ˆγ†

k,2d0

m(k)I+dm(k)·σˆγk,1

ˆγk,2,

(8)

where σ= (σx, σy, σz) denote Pauli matrices. d0

m(k) and

dm(k) are given by

m(k)=(Jyx

m−Jxy

m) sin(k),

m(k)=(Jyy

m−Jxx

m) sin(k),

m(k) = Jz

m+ (Jxx

m+Jyy

m) cos(k),

m(k)=(Jxy

m+Jyx

m) sin(k).

(9)

For the periodically driven Hamiltonian in Eq. (1), the

Floquet evolution operator ˆ

UTand the eﬀective Hamil-

tonian ˆ

Heﬀ are deﬁned by

UT=ˆ

Texp −iZT

H(t)dt!= exp(−iˆ

H2t2) exp(−iˆ

H1t1)

= exp(−iˆ

Heﬀ T),(10)

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TopologicallyprotectedboundarydiscretetimecrystalforasolvablemodelPengXu1andTian-ShuDeng2,1SchoolofPhysics,ZhengzhouUniversity,Zhengzhou450001,China2InstituteforAdvancedStudy,TsinghuaUniversity,Beijing,100084,ChinaFloquettimecrystal,whichbreaksdiscretetime-translationsymmetry,isanintriguingphe-nome...

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Topologically protected boundary discrete time crystal for a solvable model Peng Xu1and Tian-Shu Deng2 1School of Physics Zhengzhou University Zhengzhou 450001 China

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