Disorder eects in the Z3Fock parafermion chain G. Camacho1J. Vahedi2D. Schuricht3and C. Karrasch1 1Technische Universit at Braunschweig Institut f ur Mathematische Physik

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Disorder eﬀects in the Z3Fock parafermion chain

G. Camacho,1J. Vahedi,2D. Schuricht,3and C. Karrasch1

1Technische Universit¨at Braunschweig, Institut f¨ur Mathematische Physik,

Mendelssohnstrasse 3, 38106 Braunschweig, Germany

2Department of Physics and Earth Sciences, Jacobs University Bremen, Bremen 28759, Germany

3Institute for Theoretical Physics, Center for Extreme Matter and Emergent Phenomena,

Utrecht University, Princetonplein 5, 3584 CC Utrecht, The Netherlands

(Dated: December 19, 2022)

We study the eﬀects of disorder in a one-dimensional model of Z3Fock parafermions which can

be viewed as a generalization of the prototypical Kitaev chain. Exact diagonalization is employed

to determine level statistics, participation ratios, and the dynamics of domain walls. This allows

us to identify ergodic as well as ﬁnite-size localized phases. In order to distinguish Anderson from

many-body localization, we calculate the time evolution of the entanglement entropy in random

initial states using tensor networks. We demonstrate that a purely quadratic parafermion model

does not feature Anderson but many-body localization due to the nontrivial statistics of the particles.

I. INTRODUCTION

It is well known that low-dimensional quantum

systems can host particles with statistical properties

beyond the usual boson and fermion paradigms such

as anyonic exchange [1, 2] and generalized exclusion

statistics [3, 4]. A particular type of particles with

anyonic properties is parafermions [5, 6], which can be

viewed as a generalization of the nowadays well-known

Majorana fermions [7]. The latter can be interpreted

as real and imaginary parts of spinless fermions, which

in turn possess the properties usually attributed to

quantum particles such as the existence of a Fock space

and therefore a well-deﬁned particle number. The

corresponding objects obtained from parafermions were

introduced by Cobanera and Ortiz [8, 9] and named

Fock parafermions. By construction, they possess a Fock

space with an occupation number, but, in contrast to

spinless fermions, also anyonic exchange statistics and a

generalized Pauli principle inherited from the underlying

parafermion operators.

While Fock parafermions have been utilized to

link parafermions to ordinary electrons and thereby

investigate models possessing zero-energy (edge) modes

[10], they can also be studied as particles in their own

right. The ﬁrst step in this direction was taken by

Rossini et al. [11], who studied a tight-binding chain of

Fock parafermions. A key observation was that such a

model is nonintegrable despite being quadratic in terms

of the Fock parafermion operators, and the low-energy

properties for generic ﬁlling fractions are described by a

Luttinger liquid. Due to the generalized Pauli principle,

more than one Fock parafermion can occupy a lattice site;

coherent pair-hopping processes thus become possible,

which yields further gapless phases [12]. Very recently,

the eﬀect of dissipation was also analyzed [13] which,

under suitable conditions, leads to the emergence of a

noninteracting single-particle spectrum and dark states.

Despite these eﬀorts, many open questions on Fock

parafermion systems remain, with some of the most

natural ones being related to the addition of disorder.

It is well known that in other low-dimensional systems,

the addition of a disordered potential generally leads to

the localization of quantum states and thus drastically

aﬀects the transport properties. In noninteracting

systems, Anderson localization [14, 15] manifests itself

in a complete freezing of the dynamics, detectable, for

example, in the saturation of the entanglement entropy.

The generalization of this phenomenon to interacting

systems, nowadays known as many-body localization

(MBL) [16, 17], has attracted tremendous attention

[18]. MBL provides a generic realization of a nonergodic

quantum system with potential applications in quantum

information [19, 20], and MBL phases feature interesting

properties such as an area law [21] scaling of the

entanglement in the excited states [22], unconventional

transport [23–25], or a logarithmic growth of the

entanglement entropy following quantum quenches [26–

28].

In this work, we study disorder eﬀects on Fock

parafermions. In particular, we ask whether a purely

quadratic Fock parafermion chain with random on-site

potential exhibits the phenomenology of Anderson or

many-body localization. Our results are consistent

with the MBL phenomenology, except in a special

limit where the model becomes equivalent to a free

fermionic system. Our ﬁndings hence suggest that

anyonic statistics precludes Anderson localization even in

quadratic systems. Furthermore, our results show that

the coherent pair hopping supports localization.

Many studies of MBL employ an exact diagonalization

of small systems [29, 30]. The existence of the MBL phase

in the thermodynamic limit has recently been questioned

[31–35], but no conclusive picture has emerged yet [36–

41]. A possible new viewpoint is quantum avalanches

[42–49]. In our work, we address the phenomenology of

MBL in ﬁnite-size parafermion systems in analogy to the

ﬁnite-size studies of MBL in the prototypical Heisenberg

chain [29, 30].

This article is organized as follows: In the next section,

we introduce Fock parafermions, the one-dimensional

arXiv:2210.02901v2 [cond-mat.dis-nn] 16 Dec 2022

model we are considering, and recapitulate its basic

properties. Thereafter (Sec. III), we introduce our

numerical approaches, i.e., exact diagonalization (ED)

as well as the time-evolving block decimation (TEBD).

In Sec. IV, we present our results on the level

spacing statistics, the participation ratio, the imbalance

dynamics, and the entanglement entropy. In Sec. V, we

summarize our ﬁndings.

II. MODEL

A. Fock parafermions

In one dimension, the transverse-ﬁeld Ising chain is

a cornerstone model for the classiﬁcation of topological

order; it is directly linked to the Kitaev chain [6, 7]

with Majorana edge zero modes appearing in its ordered

phase. More generally, one can study p-state clock

models, and the simplest extension of the Ising chain

(p= 2) is given by the quantum Potts model,

HPotts =−JX

jσ†

jσj+1 +σ†

j+1σj

−fX

jτ†

j+τj,(1)

where the clock matrices σj, τjsatisfy the following

algebra:

σ†

j=σp−1

j, τ†

j=τp−1

σp

j=τp

j= 1, σjτk=ωδj,k τkσj,(2)

with ω= exp(2πi/p). Their explicit representation reads

[σj]kl =δk+1,l +δk,pδl,1,

[τj]kl =ωk−1δk,l, k, l ∈ {1, ..., p}.(3)

The notion of a Majorana is then generalized by

introducing two parafermion operators at each site j,

γ2j−1and γ2j, via

γ2j−1=

Y

k<j

τk

σj, γ2j=ωp−1

2γ2j−1τj,(4)

which satisfy the following algebraic relations

γjγk=ωsgn(k−j)γkγj, γp−1

j=γ†

j, γp

j= 1.(5)

For p= 2, Eq. (5) reduces to the usual anticommutation

relations for 2LMajorana fermions.

The biggest drawback of using the parafermion

operators deﬁned in Eq. (4) is that γ†

2j−1and γ†

2jcannot

be interpreted as particle creation operators. However,

it was shown that for any set of parafermion operators

governed by Eq. (5), a generalized set of annihilation

operators Fj, the so-called Fock parafermions, can be

deﬁned via [8]

Fj=p−1

pγ2j−1−1

p−1

m=1

ωm(m+p)

2γm+1

2j−1γ†m

2j.(6)

These operators feature anyonic commutators

FjFk=ωsgn(k−j)FkFj,

F†

jFk=ω−sgn(k−j)FkF†

j,(7)

and satisfy the local relations

j= 0, F †m

jFm

j+Fp−m

jF†(p−m)

j= 1,(8)

with m= 1, . . . , p −1. Note that for p= 2, this scheme

reduces to the standard representation of Majoranas

in terms of spinless fermions. The Fock parafermion

operators act on the occupation basis of a Fock space

in the usual way, thus they can be interpreted as

annihilating and creating particles that satisfy anyonic

statistics. In particular, the occupation number basis of

the Fock space is obtained by repeated application of the

creation operators over the vacuum |0i,

|n1, n2, ..., nLi=F†n1

1F†n2

2...F †nL

L|0i,(9)

with nk∈ {0,1, . . . , p −1}, and Ldenoting the total

number of lattice sites. Due to Eq. (8), each lattice

site can accommodate, at most, p−1 Fock parafermions.

One can easily show that the states (9) indeed form an

orthonormal basis,

hn1, . . . , nL|m1, ..., mLi=δn1,m1· · · δnL,mL.(10)

The number operator at a given lattice site jis given by

Nj=

p−1

m=1

F†m

jFm

j,(11)

which satisﬁes the commutation relations

Nj, F †

j=F†

j,Nj, Fj=−Fj.(12)

This entails, in particular,

Nj|n1, n2, ..., nLi=nj|n1, n2, ..., nLi.(13)

In analogy with conventional fermions, the application

of F†

jto a basis state |n1, ..., nj, ..., nLiyields

a statistical phase. In order to facilitate the

implementation of exact diagonalization and tensor

network techniques, we employ the Fradkin–Kadanoﬀ

(generalized Jordan–Wigner) transformation [5],

Fj= j−1

l=1

Ul!Bj.(14)

In the Fock basis, the operators Uj, Bjare explicitly given

Bj|n1, ..., nj, ..., nLi=|n1, ..., nj−1, ..., nLi,

Uj|n1, ..., nj, ..., nLi=ωnj|n1, ..., nj, ..., nLi,(15)

which entails that they commute on diﬀerent sites.

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DisordereectsintheZ3FockparafermionchainG.Camacho,1J.Vahedi,2D.Schuricht,3andC.Karrasch11TechnischeUniversitatBraunschweig,InstitutfurMathematischePhysik,Mendelssohnstrasse3,38106Braunschweig,Germany2DepartmentofPhysicsandEarthSciences,JacobsUniversityBremen,Bremen28759,Germany3InstituteforTheore...

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Disorder eects in the Z3Fock parafermion chain G. Camacho1J. Vahedi2D. Schuricht3and C. Karrasch1 1Technische Universit at Braunschweig Institut f ur Mathematische Physik

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