SciPost Phys. Core 5, 008 (2022)

Phase diagram of an extended parafermion chain
Jurriaan Wouters1? , Fabian Hassler2† , Hosho Katsura3,4,5‡ and Dirk Schuricht1◦
1 Institute for Theoretical Physics, Center for Extreme Matter and Emergent Phenomena,
Utrecht University, Princetonplein 5, 3584 CE Utrecht, The Netherlands
2 JARA-Institute for Quantum Information,
RWTH Aachen University, 52056 Aachen, Germany
3 Department of Physics, Graduate School of Science, The University of Tokyo,
7-3-1, Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan
4 Institute for Physics of Intelligence, The University of Tokyo,
7-3-1, Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan
5 Trans-scale Quantum Science Institute, The University of Tokyo,
7-3-1, Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan? j.j.wouters@uu.nl† hassler@physik.rwth-aachen.de‡ katsura@phys.s.u-tokyo.ac.jp◦ d.schuricht@uu.nl

Abstract
We study the phase diagram of an extended parafermion chain, which, in addition toterms coupling parafermions on neighbouring sites, also possesses terms involving foursites. Via a Fradkin–Kadanoff transformation the parafermion chain is shown to be equivalent to the non-chiral Z3 axial next-nearest neighbour Potts model. We discuss a possible experimental realisation using hetero-nanostructures. The phase diagram contains
several gapped phases, including a topological phase where the system possesses three(nearly) degenerate ground states, and a gapless Luttinger-liquid phase.
Copyright J. Wouters et al.
This work is licensed under the Creative Commons
Attribution 4.0 International License.
Published by the SciPost Foundation.

Received 01-07-2021
Accepted 31-01-2022
Check forupdatesPublished 08-02-2022
doi:10.21468/SciPostPhysCore.5.1.008

Contents
1 Introduction

1

2 Extended parafermion chain

3

3 Proposal for experimental realisation

44 ANNNP model

5

5 Phase diagram

6

6 Upper half of the phase diagram (f ≥ 0)
6.1 Potts transition in the vicinity of C1
6.1.1 Scaling analysis

7
9
9

1

SciPost Phys. Core 5, 008 (2022)

6.2

6.1.2 Perturbation around C1
Potts transition in the vicinity of C2

1011

7 Lower half of the phase diagram (f < 0)
7.1 Limit f → −∞+ : Effective XXZ model
7.2 DMRG results: boundaries of the Luttinger-liquid phase
7.2.1 Chiral clock model at U = 0
7.3 DMRG results: nature of the gapped phases

12
12
14
16
17

8 Intricate phases for U > 1.5

18

9 Discussion

19

A Duality transformation20

B Supporting numerical results
B.1 Rough topography of the phase diagram
B.2 Finite-size scaling of energy gaps

21
21
22

C Effective XXZ chains
C.1 First-order term
C.2 Second-order term
C.3 U(1)-breaking term

23
23
24
25

References

26

1

Introduction

The properties, experimental realisations and potential applications of Majorana fermions incondensed-matter systems have been studied to a great extent in the past two decades. In aseminal work Kitaev [1] introduced, amongst other things, a one-dimensional toy model of
spinless fermions and showed that the phase diagram contained a topological phase where
Majorana zero modes are localised at the edges. The Majorana chain is equivalent to thewell-known quantum Ising chain (see, eg, Fendley [2]). The topological and trivial phasesof the Majorana chain correspond to the ferromagnetic and paramagnetic phases of the Isingmodel, separated by a transition described by a conformal field theory (CFT) [3,4] with central
charge c = 1/2. Several extensions of this toy model have been studied, like the inclusionof disorder [5–7], interactions [8–14], or both [15–18]. Without disorder, the interacting
Majorana chain is equivalent to the axial next-nearest neighbour Ising (ANNNI) model [19,20]. Besides the topological and trivial phases, already present in the absence of interactions,this model also possesses an incommensurate charge density wave phase as well as a Mott
insulating phase [10,21–24]. The Majorana zero modes were thought to find use in topologicalquantum computation, however, it turns out they are not sufficient to implement universal
quantum gates [25, 26].
The Majorana/quantum Ising chain possesses a Z2 -symmetry. An obvious path for generalisation is given by considering Z3 -symmetric1 systems, which in turn leads to parafermions [27].
1

The generalisation to arbitrary Zn -symmetry is straightforward, however, in this article we will restrict ourselves to n = 3.

2

SciPost Phys. Core 5, 008 (2022)

In the corresponding parafermion chain the Z3 -symmetry turns out to be less restrictive thanthe Z2 -symmetry of its Majorana cousin. For example, the breaking of time-reversal and spatial
parity symmetry via chiral interactions is allowed. The parafermion chain is equivalent [2] tothe Z3 -clock model, which, in the non-chiral case, simplifies to the three-state quantum Potts
chain [28]. The latter possesses an ordered phase with three-fold degenerate ground state,
which is separated from a paramagnetic phase by a quantum phase transition described by a
CFT with central charge c = 4/5. In addition, the chiral model possesses an incommensurate
phase [29–31]. Interestingly, the transition between the ordered and paramagnetic phases in
the non-chiral model is no longer described by a CFT [31]. In the parafermion description theordered phase is topological, possessing zero-energy modes linked to the degeneracy of theground state [2, 32–34]. As for the Majorana fermions, this degeneracy alone is insufficientfor universal quantum computation. However, through the Read–Rezayi state [35,36] or withthe help of the Aharanov–Casher effect [37] universal gates can be realised from parafermion
modes. First steps towards the experimental realisation of parafermion excitations have beentaken recently [38–40].
In addition to the chiral interactions, the Z3 -symmetry allows several extensions of theparafermion chain, which correspond to the terms coupling parafermions beyond neighbouring sites [41, 42, 44, 45]. The equivalent clock models can be viewed as Z3 -generalisations of
the ANNNI model. It is interesting to note that for specific parameters these clock models become frustration free [42, 46], implying that the degenerate ground states can be constructed
explicitly. This behaviour generalises the well-known frustration-free Peschel–Emery line [19]of the ANNNI model.
In this work we focus on a specific extension of the parafermion chain, which, in addition
to terms coupling parafermions on neighbouring sites, also possesses terms involving four sites
next to each other. In terms of clock variables our model becomes the non-chiral Z3 axial nextnearest neighbour Potts (ANNNP) model [47]. Our specific choice is motivated by a possible
experimental realisation of this extended parafermion chain using heterostructures containing
ferromagnets, superconductors and fractional quantum Hall states. We provide a detailed
characterisation of the phase diagram of our model (shown in Figure 2), which, for moderate
strengths of the extension, contains four gapped phases: the topological and trivial phases
already present in the pure parafermion chain, and two phases showing antiferromagnetic
and ferromagnetic Ising-type order. In addition, we identify a critical Luttinger-liquid phasewith central charge c = 1. The latter as well as the two Ising-type phases can be linked tothe physics of the spin-1/2 XXZ Heisenberg chain. Furthermore, we provide evidence that the
topological phase is pinched between the Luttinger-liquid phase and the ferromagnetic Ising
phase.
This article is organised as follows: In the next section we define the extended parafermion
chain. In Section 3 we discuss a proposal to experimentally realise it in heteronanostructures,
thus motivating our specific choice of the considered extension. We then link the extended
parafermion chain to the non-chiral ANNNP model, which provides the starting point for ourfurther analysis. In Section 5 we give a qualitative discussion of the phase diagram, whose
details are elaborated on in Sections 6 and 7. We then give a brief outlook on the phase
diagram at stronger extension parameters, followed by a concluding discussion of our resultsin Section 9. The appendix contains further details of our analysis, including a discussion ofduality transformations, additional supporting numerical results, and details of the mappingto the effective XXZ chain.

3

SciPost Phys. Core 5, 008 (2022)

2

Extended parafermion chain

In this article we are investigating the phase diagram of a one-dimensional parafermionicsystem which can be viewed as an extension of the parafermion chain [2,32] by terms coupling
parafermions on four neighbouring sites. Specifically, we consider an open chain of length 2L.
At each lattice site we define parafermion operators χl , l = 1, . . . , 2L, satisfying (we recall that
we consider Z3 -symmetric systems only)χl3 = 1,χl† = χl2 ,χl χm = ωsgn(m−l) χm χlfor m 6= l,

ω = e2πi/3 ,(1)which can be regarded as a direct generalisation of Majorana fermions. Using this the Hamiltonian of the extended parafermion chain can be written as
H = −J

L−1
X
j=1

χ2 j χ2† j+1 − f

LXχ2† j−1 χ2 j + U

j=1

L−1
X

χ2† j−1 χ2 j χ2† j+1 χ2 j+2 + h.c. .

(2)

j=1

The parameters J, f and U are assumed to be real2 , making the model non-chiral. Unless it isstated otherwise, we set J = 1. In the absence of the last term, ie, U = 0, this model is known
as the parafermion chain [2, 32]. The term ∝ U corresponds to an extension involving fourneighbouring sites. One thus might be tempted to call the model (2) “interacting parafermion
chain", however, due to the non-trivial relations (1) the model is not quadratically solvableeven for U = 0. We note that a similar extension to the parafermion chain has been studied by
Milsted et al. [43] and Zhang et al. [44]. The former focused on the Z6 -variant of Equation (2),while the latter discussed the Z3 -model in a different parameter regime3 . For f = U = 0 we
recognise that χ1 and χ2L decouple from the system and form a non-local zero-energy edge
mode that generates a three-fold degeneracy throughout
whole spectrum. This degeneracy
Q the†is protected by the non-local Z3 -symmetry ω P =(χχ ). Contrary to the Majoranaj2 j−1 2 jchain, these exact modes disappear when going away from the classical point. While theground state might retain its degeneracy, the degenerate excited states hybridise and thus
split in energy [2,32–34], ie, the zero modes cease to commute with the full Hamiltonian. The
region around the classical point where the ground state remains (approximately) degenerate
is called the topological phase.
Before analysing the phase diagram of the extended parafermion chain (2), in the next
section we present a proposal to experimentally realise the model using heterostructures containing ferromagnets, superconductors and fractional quantum Hall states.

3

Proposal for experimental realisation

Recently, there have been several proposals put forward that allow to realise parafermionicbound states by cleverly constraining the fractionalised edge states of two-dimensional interacting systems [25, 36, 48]. To fix the ideas, we discuss the set-up described by Ref. [48] inmore detail. As our starting point we consider helical edge states of a fractional quantum spin
Hall state at filling factor ν = 1/m, experimentally observed in [38,39]. Such an edge configuration can also be realised at the interface of two fractional quantum Hall states with g-factorsof opposite signs [25]. Independent of the realisation, the low-energy degrees of freedom arecounter-propagating modes of fractionalised electrons with charge e∗ = e/m and spin 1/m (inunits of the electron spin), see Figure 1.
2

Complex parameters would lead to chiral interactions, which in turn break spatial parity and time-reversal
symmetry.
3
The Hamiltonian in Reference [44] is related to Equation (2) via a duality transformation as discussed inAppendix A.

4

SciPost Phys. Core 5, 008 (2022)

Figure 1: Schematic display of a fractional quantum Hall system with appearing effective parafermion degrees of freedom. The alternating placement of superconductors(SC) and ferromagnets (FM) traps the edge modes. These trapped modes obey theZ6 -parafermion algebra.
There are two (dual) ways of opening a gap in these edge states. Coupling them to a (swave) superconductor (SC) allows a transfer of charge 2e to and from the superconducting
condensate. The electric charge eQ j on the j-th superconducting island can thus assume thevaluese 2e(2m − 1)e
eQ j = 0, , , . . . ,(mod 2e) .
(3)
m m
m
The (clock) operator describing the charge is thus given by eiπQ j and commutes with the Hamiltonian [48]. First indications for induced superconductivity in fractional quantum Hall edge
states have been reported in Reference [40].
The second way to open a gap is via backscattering. This involves a change of the spin
which can be achieved by coupling the edge state to a ferromagnetic (FM) insulator. The spin
S j in the j-th ferromagnetic region may assume the values
S j = 0,1 2(2m − 1)
, ,...,(mod 2) ,
m m
m

(4)

due to the fact that the ferromagnet serves as a reservoir of spins in units of 2. Note that thebackscattering leads to the formation of an insulating phase and correspondingly the charge
vanishes in the FM segments. The corresponding clock operators satisfyπeiπS j eiπQ k = ei m (δ j,k+1 −δ j,k ) eiπQ k eiπS j ,(5)displaying the fractional statistics. Using the algebra in Equation (5) the SC and FM operators
can be represented by the parafermion modes on the interfaces,
χ2 j χ2† j−1 = eiπQ j ,χ2 j+1 χ2† j = eiπS j .

(6)With this procedure only Z2m -parafermions can be realised natively while we concentrate on
the case Z3 in this work. Note however that starting from Z6 (m = 3), Z3 -parafermions
naturally emerge by allowing for fluctuations of the gauge field with restricted dynamics [51].
An alternative experimental avenue to the Z3 -parafermions is the spin-unpolarised ν = 2/3state [36].
The entrapment of the parafermions is not perfect and exchange processes through theFMs and SCs couple the parafermions. Tunnelling of a fractional charge e∗ through the FM
segments is described by the operator eiπS j and yields the termX
H J = −Jeiθ eiπS j + h.c. ,
(7)
j

5

SciPost Phys. Core 5, 008 (2022)with some coupling J eiθ . As the coupling is due to tunneling of quasiparticles, it is given byJ ∝ exp[−cB ∆B L B /(h̄v)] and depends exponentially on the length L B and the gap ∆B of themagnetic island, with v the velocity of the edge modes and cB ∼ 1 is some constant. A priori,
the coupling is complex, but using an instanton approach it can be shown [49] that θ = 0.
Moreover, we set J = 1, fixing the overall energy scale.
Charging effects on the small mesoscopic islands perturbatively can only involve the operator eiπQ j . The charging effects are due to the Aharanov–Casher phase of a superconducting
vortex encircling the island [10]. The charging energy assumes the formX
Hf = −
f eiπQ j + h.c. ,(8)jwhere f ∝ exp[−cS ∆S LS /(h̄v)], with the length LS and the gap ∆S of the superconducting
island, can be made real by an appropriate gate voltage [49, 50]. This term is due to the selfcapacitance of the island. The terms (7) and (8) realise the (dual of) Z3 -Potts model studied
in Reference [2]. This is the parafermionic analog of the Kitaev chain [1].
Following Reference [10], we argue that the charging effects due to cross-capacitances
between adjacent islands are important. They are described by the termX
HU = U
eiπ(Q j +Q j+1 ) + h.c. ,(9)jwith U ∈ R due to the Aharonov–Casher effect encircling two adjacent islands. We note thattherealisation
proposed
here
will
generically
lead
to
the
regime|U|∝ exp[−2cS ∆S LS /(h̄v)] ® | f |. With the relation (6), the effective Hamiltonian H J +H f +H U
maps to (2) whose phase diagram we will investigate in the following.

4

ANNNP model

The analysis of the phase diagram of the extended parafermion chain will be fostered by mapping it to the equivalent non-chiral Z3 -ANNNP model [47]. The latter generalises the quantum Potts chain by including an additional coupling term, which is reminiscent of the additionof a transverse interaction term when generalising the quantum Ising chain to the ANNNI
model [19, 20].
We begin with the Fradkin–Kadanoff transformation [27]
!
!
j−1
j−1
Y
Y
χ2 j−1 =(10)
τk σ j , χ2 j =τk σ j τ j = χ2 j−1 τ j ,
k=1

k=1

which relates the 2L parafermion operators χl to clock operators σ j and τ j , j = 1, . . . , L. Theseclock operators commute off-site,[τi , τ j ] = [σi , σ j ] = [τi , σ j ] = 0,i 6= j ,(11)while on the same lattice site they satisfy
σ3j = τ3j = 1,σ†j = σ2j ,τ†j = τ2j ,σ j τ j = ωτ j σ j ,ω = e2πi/3 .

(12)

An explicit matrix representation for the clock operators on an individual lattice site is given
by




1
1
, σ = 
1 .
ω
(13)
τ=
2
ω
1
6

SciPost Phys. Core 5, 008 (2022)

In terms of the clock operators the extended parafermion chain (2) becomes4
H = −J

L−1
X
j=1

σ j σ†j+1 − f

L
X
j=1

τj + UL−1
Xτ j τ j+1 + H.c. ,

(14)

j=1

with J = 1. We note that the ANNNP model resides on a chain of length L, ie, there has
5been an effective halving
Qof the system size. The non-local Z3 -symmetry of the HamiltonianP
is generated by ω =j τ j . On the clock variables the spatial parity transformation acts
as [30] Πσ j Π = σ L− j+1 , Πτ j Π = τ L− j+1 , while time reversal is implemented via T σ j T = σ j ,T τ j T = τ†j together with complex conjugation of scalars. This shows that indeed for realparameters J, f and U the system is time-reversal and parity invariant. At U = 0 the model
reduces to the quantum Potts chain [28], which possesses a critical point at f = 1 describedby a CFT with central charge c = 4/5 [3, 4]. At f = 0 we obtain the classical (ferromagnetic)
Potts model, which has a three-fold degenerate ground state.

5

Phase diagram

The phase diagram of the extended parafermion chain/Z3 -ANNNP model for weak to moderatevalues of U is shown in Figure 2. The phases and transitions were studied using a combinationof numerical simulations, conformal field theory [3, 4] and perturbative arguments. For thenumerics we used the TeNPy implementation [52] of the density matrix renormalisation group(DMRG) algorithm [53, 54], with a typical bond dimension of 500–1000, unless otherwise
stated. First the rough topography of the phase diagram was obtained from an inexpensive
DMRG calculation, see Figure 12 of the supporting numerical results in Appendix B. Then thedetailed properties of the phases and transitions were investigated, as is discussed in Sections 6
and 7.
We see that the model displays a variety of phases. The top half of the phase diagram( f ≥ 0) resembles the picture for the ANNNI model [10,12], with two gapped phases separated
by a critical line. The ground state of the paramagnetic phase is singly degenerate, while theZ3 -ordered phase has a three-fold degenerate ground state. The latter is due to approximate
zero-energy parafermion modes, which explains the term “topological phase". The top half ofthe phase diagram is discussed in detail in Section 6.
In contrast to the ANNNI model, the Z3 -ANNNP model is not invariant under f → − f(which is a consequence of the Z2 -symmetry of the ANNNI model). The lack of this invarianceis manifest in the phase diagram, which shows four phases for f < 0: the topological phase, agapped antiferromagnetic phase, a critical XXZ phase, and a ferromagnetic phase. The latter
three can be related to the physics of the XXZ chain in the limit f → −∞, which predicts
the transitions to be at U = ±1/3. The detailed description of these phases is presented in
Section 7.

6

Upper half of the phase diagram (f ≥ 0)

Given that the Z3 -ANNNP model is not integrable, the applicability of analytical methods islimited. Still, the quantum Potts model (U = 0) is well understood due to its relation to the4
We note that Z3 -symmetry also allows terms like ∝ τ j τ†j+1 , which we do not consider here. For generalZn -symmetric models the complexity increases accordingly.
5
We note in passing that the model (14) possesses an additional Z2 -symmetry σ j → σ†j , τ j → τ†j which enlarges
the Z3 -symmetry to a full S3 -symmetry [34].

7

SciPost Phys. Core 5, 008 (2022)

4
3

C1

1

?

C2
G1

f

0

G2
Z3-ordered(topological)

paramagnetic(trivial)

2

−2
−4
−6

Z2-orderIsing antiferromagnetic

−10
−1.5

−1.0

−8

−∞+

Z2-orderIsing
ferromagnetic

critical
XXZ
c=1
−0.5

0.0
U
− 13

≈ 0.058

0.5

1.0

1.5

1
3

Figure 2: Phase diagram of the extended parafermion chain/Z3 -ANNNP model. We
distinguish the following four gapped phases: a paramagnetic phase (red), a topological phase (yellow), an Ising antiferromagnetic (purple) and an Ising ferromagnetic(green) phase. Furthermore, we identify a critical XXZ like phase (violet) with central charge c = 1. We also indicate the transition points C1 and C2 corresponding
to specific conformal field theories, and the points G1 and G2 at which the model
becomes frustration-free and thus allows an exact description of the ground state. At
the bottom we show the phase diagram in the limit f → −∞+ obtained analyticallyin Section 7.1. The dashed lines indicate cuts along which detailed results are shownin Figures 3, 4, and 8(b) (with the corresponding symbols for marked points).
two-dimensional classical Potts model. Two topologically distinct phases are separated by aquantum phase transition at f = 1 (C1 in Figure 2) described by a CFT with central chargec = 4/5. The two distinct phases can be characterised by analysing the limiting cases f → ∞and f = 0, respectively.
In the limit f → ∞ the ground state is unique and given by a product state|Ψ0 〉 = |0〉⊗L
τ ,

(15)

where |i〉τ , i = 0, 1, 2, span the space of eigenstates of τ,τ |i〉τ = ωi |i〉τ .

(16)

In the parafermionic language this is identified as the trivial phase due to the absence of boundary modes. The whole phase denoted as paramagnetic in Figure 2 is adiabatically connectedto this limit, in particular, it possesses a unique ground state with an energy gap above it.
Explicit numerical evidence for the gap at a representative point (U = −1, f = 1) is shown in
Figure 13(a) of Appendix B.2.
The nature of the Z3 -ordered phase is obvious from studying the f = 0 point (G1 ). Herethe three-fold degenerate ground state is given by|Φ0i 〉 = |i〉⊗Lσfor i = 0, 1, 2 ,(17)where |i〉σ span the space of eigenstates of σ,
σ |i〉σ = ωi |i〉σ .
8(18)

SciPost Phys. Core 5, 008 (2022)

Z3

paramagnetic

G2

∆

0.2
∆1
∆3

0.1
0
0.95

0.96

0.97

0.98

0.99

1

U

Figure 3: Energy gaps ∆n between the ground state and p
the nth eigenstate obtained
from finite-size scaling as a function of U for fixed f = 1+ 3 (see dashed line close to
G2 in Figure 2). In the Z3 -ordered phase we observe the three-fold degeneracy of theground state (∆1 = ∆2 = 0) with a finite gap above it (∆3 > 0). In contrast, in theparamagnetic phase the ground state is unique (∆1 > 0). The transition (determinedwith the methods discussed in Section 6.1) is located at Uc ≈ 0.97. The gap ∆3 isvery small close to the transition.
The parafermion dual of this system is topological, with edge states χ1 and χ2L .
Recent progress on frustration-free models allows us to analytically discusspone additional
point in the topological phase. In Ref. [42] it was shown that at U = 1, f = 1+ 3 the model isfrustration free (point G2 ), enabling the construction of the exact ground states. Furthermore,this point is adiabatically (ie, without closing the energy gap) connected to the classical Potts
model (G1 ) [46]. In fact, the points G1 and G2 lie on a frustration free line of a more general
Hamiltonian, obtained from (14) by adding a term ∝ (τ j τ†j+1 +H.c.) with a suitable prefactor.
The situation is reminiscent to the Peschel–Emery line in the ANNNI model [12, 19]. The
numerically calculated energy gaps shown in Figure 3 confirm that at G2 the model indeed
possesses a three-fold degenerate ground state. The model is gapped down to the transition
to the paramagnetic phase at Uc ≈ 0.97. Further numerical results presented in Appendix B.2[see Figure 13(b) for the point U = f = 1] show that the model is gapped with a three-folddegenerate ground state throughout the topological phase.
Finally, a simplification occurs along the line f = 0. Performing two duality transformations (see Appendix A for the details) we can bring the Hamiltonian in the following form


L/2−1
X
X L/2−2Xτaj  + H.c. ,
(19)
H =−
σ aj (σ aj+1 )† − U
a=o,e

j=1

j=1

where we omitted the boundary terms6 . The result (19) represents two decoupled (o/e) quantum Potts chains. Consequently, at U = −1 the model possesses a second-order phase transition corresponding to a CFT with c = 4/5 + 4/5 = 8/5 (see also Reference [55]) depicted by
C2 in Figure 2, separating a trivial from a topological phase.

6.1

Potts transition in the vicinity of C1

In this subsection we perform a more detailed analysis of the Potts transition. We begin witha scaling analysis of finite-size data, followed by a inspection of the vicinity of the point C1 .
6
Here we are only interested in bulk criticality, for which boundary terms can be disregarded. The boundary
terms are given in Appendix A.

9SciPost Phys. Core 5, 008 (2022)

fc = 0.3394, c = 0.813
L0.954 ∆

c(L)

1

0.85

0.5

L
64

8

70
76
82
88
94

6

0.8

6.5

100

0.75

0

0.339

0.336

0.3395

0.338

6

0.340

0.340

0.342

0.336

0.344

0.338

0.340

f

f

(a)

(b)

0.339

0.3395

0.342

0.340

0.344

L−1.691 S

L1.179 β

1

−4

−5

0.8
−4.1

0.82

−4.2

0.8

0.6 0.78

−4.3
0.339

0.336

0.3395

0.338

0.339

0.340

0.340

0.342

0.336

0.344

0.3395

0.338

0.340

0.340

f

f

(c)

(d)

0.342

0.344

Figure 4: Finite-size results for U = −0.5, locating the transition at f c = 0.3394
from the central charge (a) with c ≈ 0.813 (Lmax = 100). From (b) we confirm the
dynamical exponent is close to 1. From the Callan–Symanzik β function in (c) wederive 1/ν ≈ 1.179 and the structure factor S gives 2 − η = 1.691. The exponents in(b), (c) and (d) are obtained by requiring that the finite-size data are independentof L at the transition f c .
6.1.1

Scaling analysis

In Figure 4 we show several observables along a cut at U = −0.5 (indicated by a dashed line
in Figure 2). The numerical data were obtained for system sizes L = 64, 70, . . . , 100.
First, we consider the entanglement entropy S. In a conformally invariant system this ispredicted by the Calabrese–Cardy formula [56, 57]

 
c
L
πl
S(L, l) = S0 + log
sin
,
(20)
6
π
L
with c being the central charge, l the bipartition length, and S0 being a model-dependent
constant. Setting l = L/2 we obtain the central cut entanglement entropy, for which werealise that
S(L, L/2) − S(Lmax , Lmax /2)
c=6
.
(21)
log(L/Lmax )For a critical system the right-hand side of (21) is length (L) independent. Thus we canlocate the transition as the point where the finite-size data collapse, obtaining the central
charge in the process. From the entanglement-entropy results in Figure 4(a) we can infer thatf c = 0.3394 with c ≈ 0.813, which is in good agreement7 with the predicted value of c = 4/5.
7

The numerical results typically improve with an increase in system size and bond dimension.

10

SciPost Phys. Core 5, 008 (2022)

Second, we consider the energy gap whose scaling behaviour is given by [31, 58]
˜ 1/ν | f − f c |) ,∆(L) = L −z ∆(L(22)

with z being the dynamical exponent. The critical exponent ν governs the divergence of thecorrelation length ξ ∝ | f − f c |−ν . At the f = f c we find z by requiring L z ∆(L) to be independent of L. From this ansatz we obtain z ≈ 0.954 [see Figure 4(b)], in good agreement withthe value z = 1 expected for a CFT.
Third, we consider the Callan–Symanzik function β [59]β=∆∆ − 2 ∂∂ln∆f∝ | f − fc | ,(23)which allows us to determine the critical exponent ν. The finite-size ansatz implies that β(L)scales as L −1/ν . The CFT prediction for the Potts transition is determined from the scaling
dimension of the perturbing field, in this case the energy operator E, to be
ν=

1
5
= ,
2 − ∆E
6(24)with ∆ E = 4/5 for the critical Potts model [4]. From Figure 4(c) we get the numerical value
1/ν = 1.179, again close to the prediction.
Finally, the last critical exponent we can easily study is the scaling of the two-point correlation function Γ(r) = 〈σ†i+r σi 〉 ∝ r −η with 〈.〉 denoting the ground-state expectation value.
From the finite-size scaling ansatz we see that the structure factor behaves as
X
S (L) =〈σi σ†j 〉 ∝ L 2−η .
(25)
i, j

From the CFT description we recognise that η relates to the scaling dimension of the σ-field [4]η = 4∆σ = 4/15. Consequently, the theoretical prediction is 2 − η = 26/15 ≈ 1.7333, with
the numerical data in Figure 4(d) yielding the estimate 2 − η = 1.691.
We obtained similar results for several points along the transition line depicted in Figure 2,
indicating that the transition along the whole line is described8 by the Potts CFT with c = 4/5.
6.1.2

Perturbation around C1

In general it is possible to link the lattice operators in the quantum Potts chain to scaling fields
in the Potts CFT [60]. Unfortunately, for the τ j τ j+1 -perturbation coupled to U, which is ofinterest here, the corresponding field expansion was not derived in Reference [60]. However,from numerical analysis we can obtain its scaling dimension ∆U . The Callan–Symanzik function (23) in Figure 5 shows that the τ j τ j+1 perturbation at C1 scales with 1/ν = 0, ie, isindependent of the system size at the transition. From Equation (24) we conclude that thecorresponding field has scaling dimension ∆U = 2 and is thus marginal.
The qualitative behaviour of the transition line close to C1 is consistent with a simple
mean-field argument. Decoupling the U(τ j τ j+1 + H.c.) perturbation is tantamount to a shift
in the on-site field term, f → f ∗ = f − 2 〈τ j 〉 U; implying that the transition is shifted to
f c = 1 + 2 〈τ j 〉 U. Numerically we obtain 〈τ j 〉 = 0.609 > 0 at U = 0, in qualitative agreementwith the positive slope of the transition between the trivial and topological phase.
8

The scaling behaviour at the transition will be different in the Zn -symmetric model.

11

SciPost Phys. Core 5, 008 (2022)

L = 64
L = 76
L = 88
L = 100

L0 β

1.2

1
1.01
1

0.8

0.99

0.6
-0.001-0.000100.0001-0.000500.00050.001

UFigure 5: The Callan–Symanzik β function for the U(τ j τ j+1 + H.c.) perturbation at
C1 . The scaling at the transition is independent of the system size, ∝ L 0 , indicatingthat the perturbation is marginal.

6.2

Potts transition in the vicinity of C2

Finally, let us look more closely at the phase transition in the vicinity of U = −1. As alreadydiscussed in relation to (19), using a duality transformation the model with f = 0 can be
written as two copies of a quantum Potts chain, implying that the transition at U = −1, f = 0
possesses central charge c = 8/5. Now let us reinstate the f -term within the dual description, which results in the Hamiltonian (again dropping the boundary terms; for details see
Appendix A)


L/2−1
L/2−1
X
X L/2−2
X
X
†
a
a
a

(26)
τj − f(µoj µej + µej µoj+1 ) + H.c.
H =−σ j (σ j+1 ) − U
a=o,e

j=1

j=1

j=1

Starting from the U = −1, f = 0, the perturbing fields related to the lattice operators are
known to be [60](U + 1)(E o + E e ), 2 f µo µe ,(27)
which are the energy density and disorder fields respectively for each copies of the Potts chain.
Both terms independently open up a gap, as can be seen in the phase diagram Figure 2. However, a proper combination of the perturbations will leave the system gapless, ie, there will
be a gapless line f c (U). At first order in the couplings the renormalisation-group equations
contain the scaling dimensions of the relevant fields E o,e and µo,e∂l (U + 1) = (2 − ∆ E )(U + 1),∂l f = (2 − ∆µµ ) f = (2 − 2∆µ ) f ,(28)

with ∆ E = 4/5 and ∆µ = 2/15 [4]. At the phase transition neither flows to strong coupling,11

thus the scalings are necessarily proportional: | f c | 2−2∆µ ∝ |Uc + 1| 2−∆E . Therefore, the transition follows a power law in the vicinity of Uc = −1 (see, eg, References [61, 62] for a similarline of argument),| f c |∝ |U + 1|13/9 .
(29)
In Figure 6 we see that the numerically obtained transition points (black dots) are in very good
agreement with the scaling prediction (red line). Thus the emerging picture is that under the
perturbations (27) the c = 8/5 fixed point is unstable, with the flow along the line (29) being
described by the Potts CFT with c = 4/5. This is also consistent with the fact that due to the
c-theorem [4, 63] the central charge cannot increase under the renormalisation-group flow.
We note in passing that such an analysis for the Ising transition in the ANNNI model shows
similar behaviour, with the scaling exponent replaced by 7/4 [62].
12

SciPost Phys. Core 5, 008 (2022)

0.20
DMRG

0.15

13

f

fc = 0.876(U + 1) 9

0.10
0.05
0
−1

−0.95

−0.9

−0.85

U

−0.8

−0.75

−0.7

−0.65

Figure 6: The phase boundary between the paramagnetic and topological phase. The
dots are obtained with finite-size scaling from the DMRG calculation. The red line is
the CFT prediction (29), with the prefactor obtained from a fit to the numerical data.

Lower half of the phase diagram (f < 0)

7

The phase diagram of ANNNI model is symmetric around the f -axis due to the underlying
Z2 -symmetry of the model. In contrast, the ANNNP model possesses a Z3 -symmetry, which
in turn breaks the symmetry of the phase diagram under f → − f . While we have discussed
above the phase digram in Figure 2 for f > 0, and seen that it looks very similar to the one of
the ANNNI model, for f < 0 a completely different topography appears. It is the aim of thissection to discuss the lower half of the phase diagram in detail.

7.1

Limit f → −∞+ : Effective XXZ model

We start the discussion by considering the limit f → −∞, in which the field term − f (τ j + τ†j )in (14) becomes dominant. As the local eigenstates |0〉 j , |1〉 j , |2〉 j have energies −2 f , f , f , thislimit projects onto the two local states |1〉 j , |2〉 j . This allows us to derive9 an effective spin-1/2
model, with the third state, |0〉 j , only appearing in virtual processes.
The remaining terms in (14) are treated perturbatively. The first-order contributions to
the effective Hamiltonian are (see Appendix C for the derivation)
X
3U z z(1)− ++ −Heff = −σ j σ j+1 + σ j σ j+1 +
σ σ
,
(30)
2 j j+1jywhere σ±j = (σ xj ±iσ j )/2 with σ aj , a = x, y, z, denoting the Pauli matrices acting on lattice site
j. Thus at
P leading order we recognise the spin-1/2 XXZ model with an U(1) symmetry generated by j σzj . [We note that a similar argument was used in References [61, 64] to explain
the appearance of critical c = 1 phases in parafermion chains to the XY phase of (30).] For thisintegrable model, the phase diagram is well-known [65] and consists of an antiferromagnetic
Ising phase for 3U < −1, a ferromagnetic Ising phase 3U > 1, and a Luttinger-liquid phasewith c = 1 in between. The Luttinger parameter of the critical phase is given by (at f = −∞)K=

π.
2 arccos(3U)(31)

Note that the ferromagnetic Heisenberg point (U = 1/3) is not described by a CFT, as thedispersion becomes quadratic, or equivalently Luttinger parameter diverges. The transition9

We note in passing that the mapping to an effective two-state system seems possible for Zn -symmetricparafermion chains with n being odd. However, we have not analysed the phase diagram in this case.

13

SciPost Phys. Core 5, 008 (2022)

to the antiferromagnetic (AFM) phase at U = −1/3 appears at K = 1/2, where a gap opens
due to the relevance of perturbations to the Luttinger-liquid field theory. However, the phase
diagram obtained in this way, showing these three XXZ-phases, is not the complete picture.
When we go slightly away from f → −∞, denoted as f = −∞+ in Figure 2, an additional
Z3 -phase emerges. This phase originates from the second-order contributions (see Appendix C
for the details),
X 1
1 z z
(2)
Heff =(σ+j σ−j+1 + σ−j σ+j+1 ) +
σ σ
(32)
6f
4 f j j+1
j

1
2
+ −
− +
+ +
+
− −
−
+
(σ j σ j+2 + σ j σ j+2 ) +(σ j σ j+1 σ j+2 + σ j σ j+1 σ j+2 ) .
(33)
3f
3f
In the absence of U, a similar expansion has been obtained in Reference [64] in the analysis
of S3 -invariant spin chains [66]. In Reference [67], a similar effective description was found,discussing edge effects in fractional quantum Hall systems. The final term in (33) was also
found in the effective study of Rydberg atoms in Reference [68].
The effect of the second-order contributions (32) is as follows: The first two terms only
cause a redefinition of the XXZ parameters, which for example shifts the AFM transition to12
U =− +
.
3 9f

(34)

We note that the transition point is shifted to the left, in qualitative agreement with the numerical results leading to the phase diagram. In addition, the Luttinger parameter will also acquire
corrections to the leading result (31). The two terms (33) need a more careful consideration:
The first term is a next-nearest neighbour spin-flip term, conserving the U(1) symmetry. It
has been shown that, for small perturbations, this terms only renormalises the XXZ parameters [69], leading to a further shift of the transition points on top of (34). The second termof (33) is more involved. It breaks the U(1) symmetry down to Z3 . Within the bonsonisation
formalism this perturbation corresponds to a field with scaling dimension ∆+++ = 9/(4K) (seeAppendix C.3). Whenever ∆+++ < 2 this U(1)-breaking term is relevant, thus with (31) we
see that at f = −∞+ a Z3 gapped phase should appear for
 
1
4π
1
0.058 ≈ cos
<U< .
(35)
3
9
3
These transition points will be shifted by 1/ f -corrections due to the corrections of the XXZ
parameters originating from (32). We note in passing that a similar U(1)-breaking term has
been shown to lead to Z3 -order in a dilute Bose gas [70], although the precise relation to our
setup remains unclear.
Finally we note that the analysis presented above critically depends on the absence ofchirality breaking in the original model (14). Introducing a chirality-breaking
P term would, inthe limit f → −∞, result in an additional, strong magnetic field term ∝ f j σzj to be added
to the effective Hamiltonian (30). This in turn would destroy the Luttinger-liquid phase as
well as the antiferromagnetic and ferromagnetic Ising phases of the XXZ model and transformthem into a trivial, paramagnetic phase.
In the following section we provide numerical evidence that the qualitative phase diagram
deduced from bosonisation at f = −∞+ is also valid in the perturbative regime at f = −30.
Furthermore, in Section 7.3 we show that the effective spin-1/2 description can be linked to
the full ANNNP model even in the non-perturbative region at f = −3.

14

SciPost Phys. Core 5, 008 (2022)

100
b = 0.55

G(l)

S(l)

1

10−1

c = 0.98
0.5

10−2
0

0

50

100
l
(a)

150

200

100

101
l
(b)

102

Figure 7: DMRG results (orange dots) for the Z3 -ANNNP model for U = −0.25,
f = −3 and L = 200. (a) Entanglement entropy together with the fitted prediction(36) (solid line). The alternation is the result of edge effects as was shown in Reference [71], see Equation (36). (b) Correlation function G(l) with the corresponding
power scaling l −b (solid line).

7.2

DMRG results: boundaries of the Luttinger-liquid phase

In principle, the DMRG simulations allow for a straightforward calculation of the central charge
from the entanglement entropy via the Calabrese–Cardy formula (20). However, as the entanglement entropy of the XXZ model is sensitive to finite-size effects, a modified relation was
proposed [71] taking the finite-size oscillations into account,a cos(πl)
Smod (L, l) = S(L, l) + L.
πlπ sin L(36)

Furthermore, we study the correlation function
G(l) = 〈σ†j σ j+l 〉 ∝ l −b ,

b=

1
,
2K

(37)

for which we obtain the scaling exponent b from the XXZ description. With the spin-1/2
projection we recognise〈σ†j σ j+l 〉 = 〈σ̃+j σ̃−j+l 〉withσ̃± = diag(0, σ± ) .

(38)

The scaling behaviour then follows from standard bonsonisation [65].
As an example Figure 7 shows fits of these predictions to numerical results for U = −0.25
and f = −3, confirming c ≈ 1 in the critical XXZ region as well as determining the Luttinger
parameter to be K = 0.9.
To study the perturbative regime first, we take a cut along f = −30. For this cut, Figure 8(a) shows the central charge (c) and the scaling exponent (b). The differently coloured
circles correspond to the full Z3 -ANNNP model (red) and the various XXZ perturbative approximations (first order in dark blue, second order without the U(1)-breaking term in light blue,
second-order with U(1)-breaking term in yellow). First of all, we note that the agreement of
the results for the different models is remarkable except for the Z3 -phase. This tells us that (i)the XXZ picture is a good approximation, (ii) the next nearest-neighbour spin-flip term, (33),
denoted by “2nd no U(1)-br” is irrelevant as the results are indistinguishable from the “1st ”
order plain XXZ results, (iii) for 0 ® U ® 1/3 the U(1)-breaking term is important.
The central charge for the Z3 -ANNNP model in the top panel of Figure 8(a) shows that
there is a critical phase between gapped phases. However, the gapless region with c > 0
15

SciPost Phys. Core 5, 008 (2022)cI
anti-ferroIIcrit XXZ

III

Z3

I

ferro

f = −3IIZ3anti-ferro crit XXZ

III

ferro

1st
2nd no U(1)-br
2nd with U(1)-br

b

1.5
1
0.5
0
2
1.5
1
0.5
0
1

f = −30

Z3 ANNNP49⟨Σj (−1)j σ̃ z ⟩j†|⟨σ σj+L/2 ⟩|j⟨Σj σ̃ z ⟩
j

0.5
0
−0.5

0

0.5

−0.5

U
(a)

0

0.5

U
(b)

Figure 8: Numerical results for the different XXZ approximations (with and without
the U(1)-breaking term) and the full Z3 -ANNNP model for L = 200 for (a) f = −30
and (b) f = −3. The top and middle panels show the central charge c and scaling
exponent b, respectively; the solid horizontal lines at b = 1 and b = 4/9 indicate thevalues of the exponent belonging to transitions I and II. The bottom panels show thethree order parameters introduced in (41), it will be discussed in Section 7.3. The
black circles highlight the points for which further results are shown in Figure 11.
exceeds the regime denoted by “critical XXZ”, which is estimated from values of the exponent
b shown in the middle panel. This deviation is the result of finite-size effects around the twotransitions (I,II). Since both are described by a sine-Gordon term opening a gap in a Luttinger
liquid, they are Kosterlitz–Thouless transitions (KTT) [65, 72]. The respective sine-Gordon
terms responsible for transitions I and II are relevant for K < 1/2 and K > 9/8 respectively.
Since the scaling of the correlation function (37) is related to K, we locate I at b = 1 and II at
b = 4/9. As we see in the middle panel, the obtained transitions correspond accurately to thebosonisation predictions10 (34) and (35) of UcI ≈ −0.341 and UcII ≈ 0.0526 respectively.
Let us have a closer look at transition II between the Luttinger-liquid phase and the Z3 phase. In contrast to the Potts transition discussed in Section 6, which was a second-ordertransition between to gapped phases, here we have to analyse a KTT between a gapped andgapless phase. For this it is known [65] that the gap closes extremely slowly and the observing the true transition point requires very large systems sizes. There are several studies [68, 73–79] addressing finite-size scaling for the KTT with the help of correlation length,10Note that the value for UcII is slightly shifted from the prediction (35) because of the renormalisation of the
Luttinger parameter K due to (32).

16

SciPost Phys. Core 5, 008 (2022)

S(L)

1.4

′
Smin

0.25
0.2

U

1.6

L = 100
L = 140
L = 200
L = 280
L = 380
L = 500

0.15
0.1
0.05
0

0.004 0.008 0.012
1log(bL)2

1.2
0.1

0.125 0.15 0.175

0.2

0.225 0.25 0.275

0.3

UFigure 9: Central-cut entanglement entropy for L = 100 − 500 at f = −30, displaying the finite-size features close to the KTT between the Z3 - and Luttinger-liquid
phases. The inflection points are highlighted. The inset shows the finite-size scalingof the position of the inflection points, confirming the thermodynamic transition at
UcII ≈ 0.0526 using (40).
fidelity and entanglement entropy. In our experience certain features of the central-cut entanglement entropy (CCEE) proved most useful in this case. Starting with (20) we define the
CCEE as S(L) ≡ S(L, L/2). Deep in the gapped Z3 -phase the CCEE is independent of system
log(L)
size, S(L) = log(3). In contrast, in the critical phase it follows (20), S(L) ≈ 6 . As thesystem approaches the critical region, there is a bump due to finite-size effects. This is shownin Figure 9, where CCEE is plotted for several system sizes for f = −30 as a function of U.
We are not interested in the bump, but rather in the inflection point to the right of this bump(highlighted in Figure 9), in particular the position Uinfl (L). Since the CCEE diverges with L
in the critical phase, we assume Uinfl (L) to approach the true thermodynamic transition (Uc ).
The finite-size scaling follows from the observation that ÆL ≈ ξ ∝ exp C/ |U − Uc | ,(39)
for a KTT, with ξ being the correlation length which is cut off by the system size L, and C someconstant. We can rewrite (39) as
U(L) = Uc +a,log(b L)2

(40)

with a and b being some constants, and assume that features like the inflection point of the
CCEE follow this scaling. The inset of Figure 9 shows that this assumption together with thepredicted value UcII ≈ 0.0526 is indeed satisfied.
Applying the reasoning above at f = −3, ie, outside the perturbative region, we obtainfrom the scaling of the correlation function that UcI ≈ 0.475 and UcII ≈ 0.185 as shown inthe middle panel of Figure 8(b). The latter is in agreement with the results from finite-size
scaling of the CCEE, not shown in this paper. Moreover, we note that the effective spin-1/2
description deviates qualitatively around transition II. This is not surprising, since we haveleft the perturbative regime11 . Nonetheless, even at f = −3 the local state |0〉 j is (almost)projected out, hence the spin-1/2 interpretation is still reasonable.
11
The numerics for the spin-1/2 model does show a (narrower) Z3 -phase in between the Luttinger liquid andthe ferromagnetic phase.

17

cSciPost Phys. Core 5, 008 (2022)

2
1
0crit XXZ

II

U =0
Z3

ferro

b

4
9

0
−10

−8

f

−6

−4

Figure 10: Numerical results for the Z3 -ANNNP model for L = 200 at U = 0. The top
and bottom panels show the central charge c and scaling exponent b, respectively.
The solid horizontal line at b = 4/9 indicates the value of the exponent belonging totransition II.
7.2.1

Chiral clock model at U = 0

In the absence of the U-term the model (14) becomes a special case of the chiral Z3 -clock
model [2], whose phase diagram as a function of the chiral angles (φ, θ ) was studied by
Zhuang et al. [30]. More specifically, our model (14) at U = 0, f < 0 is equivalent to the chiralmodel at positive field strength and φ = π/3, θ = 0. The phase diagram for the latter showsa transition between the topological phase and a gapless, incommensurate phase with central
charge c = 1. Using our conventions this translates into a transition from the topologicalphase to a gapless phase with c = 1 at f ≈ −4. However, along the line U = 0 our results
show a transition at f cII ≈ −7.87. Both finite-size scaling of the CCEE and scaling of correlation
function confirm this value. The latter can be seen in the bottom panel of Figure 10, whereb = 4/9 signals the Z3 -phase. We attribute the discrepancy with the estimated value f ≈ −4
of Reference [30] to finite-size effects.

7.3

DMRG results: nature of the gapped phases

In order to further characterise the gapped phases, we have calculated three order parametersin the full ANNNP model. The results are shown in the bottom panels of Figure 8. Specifically,
we determined the Z3 -embedded antiferromagnetic and ferromagnetic order parameters aswell as the long-range Z3 -order defined as
¬P
¶
¬P
¶
j z
z
(−1)
σ̃
,
σ̃
G(L/2) = 〈σ†j σ j+L/2 〉 .
(41)
j
j j ,
j
Here σ̃zj = diag(0, 1, −1)τ in the local eigenbasis of τ j . In Figure 11 we show the order parameters for representative points in the different gapped phases, clearly confirming the natureof these phases as antiferromagnetic, ferromagnetic and Z3 -long-range ordered, respectively.
We note that the study of G(L/2) is preferable over the short-range correlations |〈σ†j σ j+1 〉 |,because the latter can be potentially close to 1 in the critical phase, while the long-range correlation decays with the system size (although with a power law). In contrast, G(L/2) will
become constant in the Z3 -ordered regime, as can be seen exemplarily in Figure 11(b). We
also note that the antiferromagnetic and ferromagnetic Ising phases have a two-fold degenerate ground state with a finite energy gap above; see Figure 14 in Appendix B.2. On the other
hand, in the critical XXZ phase shows an even-odd effect (Figure 15, Appendix B.2).
18

SciPost Phys. Core 5, 008 (2022)

1
G(l)

0.6

0.5
hσ̃jz iU = −0.75
U = 0.75

0

0.4
U = 0.25
0.2

−0.5
0

50

100

150

0

200

0

50

100

150

200

l

j
(b)

(a)

Figure 11: Example points displaying the respective order for the various gapped
phases for f = −3. (a) The antiferromagnetic (U = −0.75) and ferromagnetic(U = 0.75) behaviour of the local magnetisation. (b) The correlation function (37)signalling Z3 /topological order. All data for L = 200.
Coming back to the bottom panels of Figure 8 we see that the three gapped phases can bewell distinguished by the order parameters (41). This reveals the nature of the phases and canbe used to locate the phase transitions. In particular, the transition between the Z3 -ordered
and ferromagnetic phases becomes clear from the crossover in the respective order parameters.
Finally, from the topography of the phase diagram shown in Figure 12 in Appendix B.1 wededuce that the antiferromagnetic region extends up to vanishing f , while the transition toZ3 -ordered phase keeps the ferromagnetic region from touching the f = 0 axis (in the studied
range for U). We also infer, in combination with the discussion above, that XXZ critical region
extends up to U = −1, f = 0 where the critical point with c = 8/5 is located. We would liketo stress that the vicinity of this point is very difficult to study numerically, since both the left
and right transitions are rather soft. Similar difficulties were experienced in the vicinity of amulti-critical point in the ANNNI model [62].

8Intricate phases for U > 1.5

We have focused on the phase diagram of the model (14) for small to moderate values of U. In
particular we identified a topological phase as well as gapped and gapless trivial ones. For the
ANNNI model it is well known [10, 20–24] that at strong interaction strengths U other phases(like a Mott insulating phase) exist. In analogy we expect the existence of intricate phases atstrong values of U in the ANNNP model as well.
A first idea can be obtained from the dual description (26) of the model. At f = 0 themodel is equivalent (up to boundary terms) to two decoupled quantum Potts chains. Thus we
expect a transition form the topological phase to a gapless phase with c = 2 at U ≈ 8, whichis consistent with preliminary numerical data. The coupling of the two Potts chains for f 6= 0
involves non-local terms in the dual description, thus intricate behaviour can be expected. The
preliminary numerical data indicate the existence of several phases, including a critical XXZ
phase showing even-odd effects in the system length. However, as the detailed analysis of thispart of the phase diagram lies outside the scope of the present manuscript, we will leave it for
future investigation [80].

19SciPost Phys. Core 5, 008 (2022)

9

Discussion

In this article we have studied an extended parafermion chain, which possessed terms coupling
parafermions on four neighbouring sites. We mapped the model to the non-chiral Z3 -ANNNP
model via a Fradkin–Kadanoff transformation and analysed the phase diagram for weak tomoderate couplings of the four-site term. By applying a combination of DMRG simulations,scaling arguments and analytical results in special limiting cases we identified four gapped
phases: a topological phase possessing a three-fold degenerate ground state, a trivial (paramagnetic) phase as well as an antiferromagnetic and ferromagnetic Ising ordered phase. The
latter two as well as an additional critical Luttinger-liquid phase can be connected to the wellknown phase diagram of the XXZ Heisenberg chain. We provided evidence that the topologicalphase appears in between the Luttinger-liquid phase and the ferromagnetic Ising phase, and isdue to the U(1)-breaking nature of the Z3 -ANNNP model. Furthermore, we discussed a possible experimental realisation of the extended parafermion chain using hetero-nanostructuresconsisting of ferromagnets, superconductors and fractional quantum Hall states.
There are several directions for future studies: (i) Obviously the phase diagram for strong
couplings U of the four-site term could be analysed. For the interacting Majorana chain it
is known [10, 22–24] that in the limit of strong interactions two additional phases exist, aMott insulator and an incommensurate charge density wave. Our preliminary numerical data
indicate that around U ≈ 8 additional phases appear in the extended parafermion chain, so
it would be interesting to analyse their properties and link them to the known results for the
Majorana chain.
the inclusion of terms in
P (ii) The Z3 -symmetry of the model (14) allows P
addition to U j (τ j τ j+1 + H.c.). For example, including a term ∼ j (τ j τ†j+1 + H.c.) allows
the construction [42, 46] of a family of frustration-free models, of which the points G1 and G2
in Figure 2 are just special cases. These frustration-free models could serve as starting point
for an analytic study of the topological phase. We note that the addition of such a term is also
feasible within the framework of the hetero-nanostructures discussed in Section 3. (iii) The
properties of the parafermion chain critically depend on the chirality breaking in the model,see, eg, References [30, 31, 44] for studies of the phase diagram in chiral parafermion chains.
Thus it would be natural to extend the model (14) by including chirality breaking, which, as
indicated in Section 7.1, is expected to have a drastic effect on the phase diagram.

Acknowledgements
We thank Philippe Corboz and Paul Fendley for useful discussions. F.H. acknowledges support
by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – Cluster of Excellence Matter and Light for Quantum Computing(ML4Q) EXC 2004/1 390534769. H.K. was supported in part by JSPS Grant-in-Aid for Scientific Research on Innovative Areas No. JP20H04630, JSPS KAKENHI Grant No. JP18K03445,
and the Inamori Foundation. This work is part of the D-ITP consortium, a program of theNetherlands Organisation for Scientific Research (NWO) that is funded by the Dutch Ministryof Education, Culture and Science (OCW).

A

Duality transformation

In this appendix we discuss the duality transformation of the Potts model (see, eg, Reference [81]), with extra care to treat the boundary terms. We start with the Hamiltonian (14)20

SciPost Phys. Core 5, 008 (2022)

H =−

L−1
X

σ j σ†j+1 − f

j=1

L
X

τj + UL−1
Xτ j τ j+1 + H.c. .

(42)

Y

(43)

j=1

j=1

Let us now apply the following transformation,Yν j = σ j σ†j+1 , µ j =
τ†i
⇔

σj =

i≤ j

ν†i ,

τ j = µ j−1 µ†j ,i< jwith auxiliary operator σ1 = ν†0 and the exception τ1 = µ†1 . Note that ν L is not defined, which
is not a problem for the moment. Applying this, the dual Hamiltonian reads
H =−

L−2
X

νj − fL−1
Xj=1µ j µ†j+1 + U

j=1

L−2
X

µ j µ†j+2 + B + H.c. ,

(44)

j=1

where B = −ν L−1 − f µ†1 + Uµ†2 . Up to boundary terms, for U = 0 we recognise that (42)and (44) are physically equivalent at f = 1, ie, the model is self-dual at this point in thethermodynamic limit. We note that the model (44) has been studied by12 Zhang et al. [44]with a focus on the phase diagram in the presence of chirality breaking.
Next, we turn off the perpendicular field, ie, we consider f = 0. The operators ν and µ
can be split on the odd/even (o/e) sites to obtain


L/2−1
X
X L/2−1X(45)µaj (µaj+1 )†  + B + H.c. ,νaj − U
H =−
a=o,e

j=1

j=1

where B = −νoL/2 + U(µe1 )† contains the boundary terms. We recognise two decoupled Potts
chains in their dual representation: For each chain we can do another duality transformation
YY(46)(τai )† , νaj = σ aj−1 (σ aj )† ,⇔µaj =(νai )†
τaj = µaj (µaj+1 )† , σ aj =
i< j

i≤ j

with auxiliary operator µ1 = τ†0 and the exception ν1a = (σ1a )† . This gives


L/2−1
X
X L/2−2Xτaj  + B + H.c. ,H =−σ aj (σ aj+1 )† − U
a=o,e

with

(47)

j=1

j=1



B = − νoL/2 − νe1 − νo1 + U(µe1 )† .

(48)

It is interesting to relate the original order parameter σ j to the new operators σ aj ,(Qσj =(νo )† (νei )†Qi<( j−1)/2 io † e †o†i< j/2 (νi ) (νi ) (ν j/2 )with the inverse relation given by¨Q
σ aj == σ(oj−1)/2 σ(ej−1)/2 ,j odd,= σej/2−1 σoj/2 ,j even ,†(−1)i,
i< j ν2i−1 =i<2 j (σi )QQ†−(−1)i,
i< j ν2i =2<i<2 j+1 (σi )Qa = o,
a = e.

(49)(50)

The relation between the parameters in Equation (4) of Reference [44] and the ones in (44) is given by
h → J ≡ 1, J → f and J 0 → U. In particular, the supercritical point corresponds in our convention to the limit
f = 2U with U → ∞, indicating that the Potts transition between the trivial and topological phases extends toarbitrary large U.
12

21

SciPost Phys. Core 5, 008 (2022)

Central-cut entanglement entropy

3

1.8

3

2

1.6

2

1

1.4

1

0

1.2

0

−1

1.0

−1

−2

0.8

−2

−3

0.6

−3

−4

0.4

−4

−5

0.2

−5

0.0

−6
−1.5

Central charge

1.6
1.4
1.2

f

f

1.0

−6
−1.5

−1.0

−0.5

0.0
U

0.5

1.0

1.5

(a)

0.8
0.6
0.4
0.2

−1.0

−0.5

0.0
U

0.5

1.0

1.5

0.0

(b)

Figure 12: Rough topography of the phase diagram for the Z3 -ANNNP model. The
central-cut entanglement entropy and central charge results were obtained for small
system sizes L =50–100 and low bond dimension in the parity sector 0. Even though
the nature of the phases and transitions cannot be conclusively derived from theseplots, it gives a good visual guide for the features to be studied in more detail.
Thus we see that the relation is non-local involving string
Qoperators.
Q e
†
e
†
We can also rewrite the symmetry operator, ω P =j τ j = µ L = (µ L/2 ) =j τ j . For
Q
Q
P
a †
f = 0, the original Hamiltonian has another symmetry ω̃ = j σ j = j,a (σ j ) .
Finally, for completeness we can reinstate the f -term for the second transformation. Even
though the resulting lattice model is non-local in terms of the original operators σ and τ, thefollowing expression will be useful in Section 6.2


L/2−1
L/2−1
X
X
X L/2−1Xf µej (µoj )† + (µoj+1 )† + B + H.c. , (51)µaj (µaj+1 )†  −H =−νaj − U
a=o,e

j=1

j=1

j=1



with B = −νoL/2 − f (µo1 )† + µoL/2 (µeL/2 )† + U(µe1 )† .

B

Supporting numerical results

In this appendix we present additional numerical material to support certain points in the main
text.

B.1

Rough topography of the phase diagram

The overall structure of the phase diagram presented in Section 5 was determined largely
based on an inexpensive DMRG calculation, ie, for small systems (L = 50 − 100). The resultsof these calculations are shown in Figure 12. It displays the central-cut entanglement entropy
and central charge. The entanglement entropy follows naturally from the DMRG calculation.

22

SciPost Phys. Core 5, 008 (2022)

With Schmidt decomposition we can write the ground state asX|Ψ〉 =sa |ΨAa〉 |ΨBa 〉 ,

(52)

a

P
where A, B are the left and right subsystems, such that LA + L B = L and a sa2 = 1. Also |ΨAa〉
and |ΨBa 〉 form an orthonormal basis in their respective subspace. The reduced density matrix
becomesX
ρA = TrB ρ =sa2 |ΨAa〉 〈ΨAa| ,(53)awith the entanglement entropy given by [54]S(L, LA) = −TrρA log(ρA) = −X

sa2 log(sa2 ) .

(54)

a

The area law predicts that the entanglement entropy should be constant with respect to systemsize for gapped systems, which can be used as a first tool to identify gapped phases studying
the central-cut entanglement entropy S(L, L/2).
As an example, consider the unique product state (15) the central-cut entanglement entropy is simply given by S = −1 log(1) = 0. In the top left of the phase diagram in Figure 12(a)we find S ≈ 0, indicating that this region is indeed connected to the trivial product state. On
the other hand, for the Z3 -ordered phase at U = f = 0, the ground state for each parity sector is a linear combination of the three degenerate
ground states (17). Hence the central-cut
P2
entanglement entropy is given by S = − a=0 13 log(1/3) = log(3) ≈ 1.09, which we observethroughout the topological phase. We note that the central-cut entanglement entropy can also
be deceiving. For example, the ground states for the antiferromagnetic and ferromagnetic
phases for f < 0 seem to be singly degenerate (S = 0), while they are in fact doubly degenerate with the two degenerate ground states lying in different symmetry sectors and thecentral-cut entanglement entropy vanishing in each of them.
In the critical regions the central-cut entanglement entropy is not a good indicator, since
it diverges logarithmically with the system size. Instead, here we employ the central chargec obtained by fitting the entanglement entropy (54) to the Calabrese–Cardy formula [56, 57](20). It is important to note that this fit only give a qualitative view. The central charge in
Figure 12(b) is often overestimated at points close to transitions, because at finite sizes thecorrelation lengths exceed the system size. Nevertheless, it shows the presence of a transition in the top left and bottom right. Moreover, there are several critical regions that can be
identified, in particular the critical XXZ phase in the bottom left (see Section 7).

B.2

Finite-size scaling of energy gaps

Here we present data for the finite-size scaling in the gapped regions discussed in Section 6
and 7. The two plots in Figure 13 show the gap for the paramagnetic/trivial phase and theZ3 -ordered/topological phase. These confirm both the thermodynamic gaps as well as therespective degeneracies of the ground states.
In Figure 14 we show the finite-size scaling for the energy gap in the antiferromagnetic
and ferromagnetic phase described in Section 7, showing that both are indeed gapped with a
two-fold degeneracy.
On the other hand, in Figure 15 we see that both the gap to the first, second and third
excited states vanish at U = −0.25, f = −3, thus this point indeed belongs to a critical region
and not the three-fold degenerate topological phase. There is even-odd effect in the finite-size
gap that we can explain from the effective XXZ description; for an extensive discussion see
Reference [71]. For even chain lengths (and in the absence of a magnetic field) the ground23

SciPost Phys. Core 5, 008 (2022)

1
∆(L)

∆(L)

4
∆1

0.5

2

0

∆3
∆1 = ∆ 2

0

0.02

0.04

0.06

0.08

0

0.1

0

0.02

0.04

0.06

0.08

0.1

1/L

1/L

(b) Z3 -ordered/topological ( f = 1, U = 1)(a) Paramagnetic/trivial ( f = 1, U = −1)

Figure 13: Energy gaps ∆n between the nth energy eigenstate and the groundstate, obtained from finite-size scaling for system sizes L = 10, 20, . . . , 60: (a) at
f = 1, U = −1 in the paramagnetic/trivial phase, (b) at f = U = 1 in the Z3 ordered/topological phase.
Pstate is unique with total spin 〈S z 〉 = 〈 j σzj 〉 = 0. The first excited state is two-fold degeneratewith 〈S z 〉 = ±1, with the two states related by a global spin flip. On the other hand, for odd
lengths the smallest magnetisation commensurate with the system is 〈S z 〉 = ± 12 , hence there
is a double degeneracy of the ground states. We recognise this pattern in the finite-size scalingin Figure 15.

C

Effective XXZ chains

In this appendix we derive the effective XXZ chain describing the limit f → −∞, which waspresented in Section 7.1. We note that a similar expansion has been obtained in Reference [64].

C.1First-order term

The eigenvalues of the local field term − f (τ j +τ†j ) are −2 f , f , f for the eigenstates |0〉 j , |1〉 j , |2〉 j
respectively. Thus for f → −∞ there will be a large energy gap between the state |0〉 j and thestates |1〉 j , |2〉 j , which allows us to project onto a local, two-dimensional Hilbert space. Let
us denote the resulting projected many-body Hilbert space by G , with the notation |Ψi 〉 ∈ G
and |Φi 〉 ∈/ G , and the respective energies due to this leading term by EΨi and EΦi . For theremaining
write down an effective first-order Hamiltonian describing the action
Pterms we canσof V σ = j v σ
,
with
v
=
−σ j σ†j+1 − σ†j σ j+1 , ie, terms that represent 〈Ψi |V σ |Ψk 〉. If we viewjj
the operators as tensor products of 2 × 2 matrices acting on the local states |1〉 j , |2〉 j , we canwrite−σ j σ†j+1 − σ†j σ j+1 = −σ+j σ−j+1 − σ−j σ+j+1 ,(55)
with σ±j being the effective spin-1/2 raising and lowering operators acting at site j,ie, σ+j = |1〉 j 〈2| j and σ−j = |2〉 j 〈1| j . Similarly, we have σzj = |1〉 j 〈1| j − |2〉 j 〈2| j . Using
this we recognise that
p
3 z
1
σzj
τj = ω = − + i
σ .
(56)
2
2 j
This allows us to rewrite the term Uτ j τ j+1 + H.c. asUτ j τ j+1 + H.c. = −24

3U z zUσ σ + .
2 j j+1 2(57)

SciPost Phys. Core 5, 008 (2022)

1.5

∆(L)

∆(L)

0.4

∆2
∆1

0.2

1

∆2
∆1

0.5
0

0
0

0.005

0.01

0.015

0

0.005

0.01

1/L

0.015

1/L

(a) Ising antiferromagnetic ( f = −3, U = −0.75)(b) Ising ferromagnetic ( f = −3, U = 0.75)

Figure 14: Finite-size scaling of the energy gaps ∆n for representative points in the
Ising antiferromagnetic (a) as well as the Ising ferromagnetic (b) phase. Both DMRG
results are for system sizes L = 64, 65, . . . , 100. In both cases we find ∆1 = 0, showing that the ground states are two-fold degenerate, while ∆2 > 0 in the thermodynamic limit.
Taken together we thus deduce that at leading order the effective Hamiltonian describing thelarge − f limit of the ANNNP model becomesX
3U z z(1)σ+j σ−j+1 + σ−j σ+j+1 +(58)
Heff = −σ j σ j+1 .
2jHence the behaviour of the ANNNP model in this limit is governed by the XXZ Heisenberg
chain, which is known to be critical for |U|≤ 1/3 with central charge c = 1 [65].

C.2

Second-order term

The second-order terms originate from perturbations of the form
X 〈Ψ |V σ |Φ 〉 〈Φ |V σ |Ψ 〉
i

k

k

l

EΨ − EΦk

k

,

(59)

where EΨ = EΨi = EΨl the unperturbed energies of the ground states. Let us start with thecontributions to effective two-site terms. The diagonal terms readσ〈12|v σj |00〉 〈00|v j |12〉

EΨ − EΦkσ〈11|v σj |20〉 〈20|v j |11〉EΨ − EΦk==+σ〈21|v σj |00〉 〈00|v j |21〉EΨ − EΦk

1
,
6f

(60)

σ
〈11|v σj |02〉 〈02|v j |11〉σ
〈22|v σj |10〉 〈10|v j |22〉

EΨ − EΦk=(61)

EΨ − EΦk+σ〈22|v σj |01〉 〈01|v j |22〉

EΨ − EΦk

=

2
,
3f

(62)

which can be summarised as 41f σzj σzj+1 + 21f . Similarly, the off-diagonal two-site contribution
is given byσ〈12|v σ1
j |00〉 〈00|v j |21〉=,(63)
EΨ − EΦk
6f

25

SciPost Phys. Core 5, 008 (2022)

δ2 (L odd) = δ3 (L)δ2 (L even)δ1 (L even)δ1 (L odd)

δ(L)

0.15

0.1

0.05

0
0

0.005

0.01

0.015

1/L

Figure 15: Finite-size scaling of the gap δn to the n-th excited states from system sizes
L = 64, 65, . . . , 100 for the system in the critical XXZ phase (U = −0.25, f = −3).
The gap is depicted with a small δ, to signal that it is zero in the thermodynamic
limit.
which becomes the spin-flip term 61f (σ+j σ−j+1 + σ−j σ+j+1 ). Furthermore, there are three-site
contributions such asσ〈111|v σ
j+1 |102〉 〈102|v j |222〉EΨ − EΦ k+σ〈111|v σ
j |201〉 〈201|v j+1 |222〉EΨ − EΦk

=

2
,
3f

(64)

and its hermitian conjugate, which taken together become 32f (σ+j σ+j+1 σ+j+2 + σ−j σ−j+1 σ−j+2 ).
This term breaks the U(1) symmetry of the XXZ chain, but preserves the Z3 -symmetry. Finally,
there is a next-nearest neighbour hopping term,σ〈112|v σ
j |202〉 〈202|v j+1 |211〉

EΨ − EΦk=σ〈122|v σ
j+1 |101〉 〈101|v j |221〉EΨ − EΦk

=

1
,
3f

(65)

which can be written as 31f (σ+j σ−j+2 + σ−j σ+j+2 ). Taken together we arrive at the second-order
Hamiltonian
X 1
1 z z
(2)
Heff =(σ+j σ−j+1 + σ−j σ+j+1 ) +σ σ
6f
4 f j j+1j12+
(σ+j σ−j+2 + σ−j σ+j+2 ) +(σ+j σ+j+1 σ+j+2 + σ−j σ−j+1 σ−j+2 ) .
(66)
3f
3f

C.3

U(1)-breaking term

In order to analyse the effect of the U(1)-breaking term within the bosonisation framework,
we first bring the Hamiltonian (30) to its standard form. This is achieved by flipping the signof the first two terms using the transformation σ±j → (−1) j σ±j . Now the bosonisation dictionary [65] shows that the low-energy behaviour of the XXZ model is governed by a Luttingerliquid Hamiltonian
Z


u
1
2
2
d x K (∇θ (x)) + (∇φ(x)) ,
(67)
H=
2π
K
with φ and θ being a bosonic field and its dual, and (at f = −∞)
 

π
1
1
K=
, u=
sin π 1 −2 arccos(3U)
2 − 1/K
2K
26

(68)

SciPost Phys. Core 5, 008 (2022)

denoting the Luttinger parameter and velocity respectively. The local spin-flip operators σ+j
are related to the bosonic field via [the Fermi momentum is given by kF = π/(2a)]
σ+j =

p


e−iθ (x) aS + (x) = p(−1) x + cos(2φ(x)) ,
2π

(69)

where x = ja with a being the lattice constant (which we set to one), and S + (x) the continuum
operator related to σ+j . Using (69) with the transformation σ±j → (−1) j σ±j discussed above(2)

in mind, the U(1)-breaking term in Heff becomes(−1) j2(σ+j σ+j+1 σ+j+2 + σ−j σ−j+1 σ−j+2 ) ∝ e−i3θ (x) + . . . ,
3f

(70)

with the dots representing terms that are either less relevant or contain rapidly oscillating
factors (−1) x , which will thus not contribute in the continuum limit. Using the individual2
a2
scaling dimensions ∆a,b = 4K
+ b 4K of general vertex operators e−iaθ (x)−ibφ(x) , we deduce thatthe scaling dimension of the U(1)-breaking term is given by∆+++ =

9
.
4K

(71)

This shows that the U(1)-breaking term is relevant whenever K > 9/8.

References[1] A. Yu Kitaev, Unpaired Majorana fermions in quantum wires, Phys.-Usp. 44, 131 (2001),
doi:10.1070/1063-7869/44/10S/S29.
[2] P. Fendley, Parafermionic edge zero modes in Zn -invariant spin chains, J. Stat. Mech.
P11020 (2012), doi:10.1088/1742-5468/2012/11/P11020.
[3] P. Di Francesco, P. Mathieu and D. Sénéchal, Conformal field theory, Springer, New
York,(1997).
[4] G. Mussardo, Statistical field theory, Oxford University Press, Oxford (2010).
[5] W. DeGottardi, D. Sen and S. Vishveshwara, Majorana fermions in superconducting 1D
systems having periodic, quasiperiodic, and disordered potentials, Phys. Rev. Lett. 110,146404 (2013), doi:10.1103/PhysRevLett.110.146404.
[6] W. DeGottardi, M. Thakurathi, S. Vishveshwara and D. Sen, Majorana fermions in superconducting wires: Effects of long-range hopping, broken time-reversal symmetry, and potential landscapes, Phys. Rev. B 88, 165111 (2013), doi:10.1103/PhysRevB.88.165111.
[7] A. Altland, D. Bagrets, L. Fritz, A. Kamenev and H. Schmiedt, Quantum criticality of quasione-dimensional topological anderson insulators, Phys. Rev. Lett. 112, 206602 (2014),
doi:10.1103/PhysRevLett.112.206602.
[8] S. Gangadharaiah, B. Braunecker, P. Simon and D. Loss, Majorana edge statesin interacting one-dimensional systems, Phys. Rev. Lett. 107, 036801 (2011),
doi:10.1103/PhysRevLett.107.036801.
[9] E. Sela, A. Altland and A. Rosch, Majorana fermions in strongly interacting helical liquids,
Phys. Rev. B 84, 085114 (2011), doi:10.1103/PhysRevB.84.085114.
27

SciPost Phys. Core 5, 008 (2022)[10] F. Hassler and D. Schuricht, Strongly interacting Majorana modes in an array of Josephson
junctions, New J. Phys. 14, 125018 (2012), doi:10.1088/1367-2630/14/12/125018.
[11] R. Thomale, S. Rachel and P. Schmitteckert, Tunneling spectra simulation of interacting
Majorana wires, Phys. Rev. B 88, 161103 (2013), doi:10.1103/PhysRevB.88.161103.
[12] H. Katsura, D. Schuricht and M. Takahashi, Exact ground states and topological order in interacting Kitaev/Majorana chains, Phys. Rev. B 92, 115137 (2015),
doi:10.1103/PhysRevB.92.115137.
[13] A. Rahmani, X. Zhu, M. Franz and I. Affleck, Emergent supersymmetry fromstrongly interacting Majorana zero modes, Phys. Rev. Lett. 115, 166401 (2015),
doi:10.1103/PhysRevLett.115.166401/[14] A. Rahmani, X. Zhu, M. Franz and I. Affleck, Phase diagram of the interacting Majorana
chain model, Phys. Rev. B 92, 235123 (2015), doi:10.1103/PhysRevB.92.235123.
[15] N. M. Gergs, L. Fritz and D. Schuricht, Topological order in the Kitaev/Majorana
chain in the presence of disorder and interactions, Phys. Rev. B 92, 235123 (2015),
doi:10.1103/PhysRevB.93.075129.
[16] M. McGinley, J. Knolle and A. Nunnenkamp, Robustness of Majorana edge modes andtopological order: Exact results for the symmetric interacting Kitaev chain with disorder,
Phys. Rev. B 96, 241113 (2017), doi:10.1103/PhysRevB.96.241113.
[17] G. Kells, N. Moran and D. Meidan, Localization enhanced and degraded topological order in interacting p-wave wires, Phys. Rev. B 97, 085425 (2018),
doi:10.1103/PhysRevB.97.085425.
[18] J. Wouters, H. Katsura and D. Schuricht, Exact ground states for interacting Kitaev chains,
Phys. Rev. B 98, 155119 (2018), doi:10.1103/PhysRevB.98.155119.
[19] I. Peschel and V. J. Emery, Calculation of spin correlations in two-dimensional Ising systems from one-dimensional kinetic models, Z. Physik B - Condens. Matter 43, 241 (1981),
doi:10.1007/BF01297524.
[20] W. Selke, The ANNNI model — Theoretical analysis and experimental application, Phys.
Rep. 170, 213 (1988), doi:10.1016/0370-1573(88)90140-8.
[21] D. Allen, P. Azaria and P. Lecheminant, A two-leg quantum Ising ladder: A bosonization
study of the ANNNI model, J. Phys. A: Math. Gen. 34, L305 (2001), doi:10.1088/03054470/34/21/101.
[22] M. Beccaria, M. Campostrini and A. Feo, Density-matrix renormalization-group study ofthe disorder line in the quantum axial next-nearest-neighbor Ising model, Phys. Rev. B 73,
052402 (2006), doi:10.1103/PhysRevB.73.052402.
[23] M. Beccaria, M. Campostrini and A. Feo, Evidence for a floating phase of thetransverse ANNNI model at high frustration, Phys. Rev. B 76, 094410 (2007),
doi:10.1103/PhysRevB.76.094410.
[24] E. Sela and R. G. Pereira, Orbital multicriticality in spin-gapped quasi-one-dimensional
antiferromagnets, Phys. Rev. B 84, 014407 (2011), doi:10.1103/PhysRevB.84.014407.
[25] D. J. Clarke, J. Alicea and K. Shtengel, Exotic non-Abelian anyons from conventional fractional quantum Hall states, Nat. Commun. 4, 1348 (2013), doi:10.1038/ncomms2340.
28

SciPost Phys. Core 5, 008 (2022)[26] M. B. Hastings, C. Nayak and Z. Wang, Metaplectic anyons, Majorana zeromodes, and their computational power, Phys. Rev. B 87, 165421 (2013),
doi:10.1103/PhysRevB.87.165421.
[27] E. Fradkin and L. P. Kadanoff, Disorder variables and para-fermions in two-dimensional
statistical mechanics, Nucl. Phys. B 170, 1 (1980), doi:10.1016/0550-3213(80)90472-1.
[28] G. von Gehlen and V. Rittenberg, Operator content of the three-state Potts quantum chain,
J. Phys. A: Math. Gen. 19, L625 (1986), doi:10.1088/0305-4470/19/10/013.
[29] S. Ostlund, Incommensurate and commensurate phases in asymmetric clock models, Phys.
Rev. B 24, 398 (1981), doi:10.1103/PhysRevB.24.398.
[30] Y. Zhuang, H. J. Changlani, N. M. Tubman and T. L. Hughes, Phase diagram oftheZ3parafermionic chain with chiral interactions, Phys. Rev. B 92, 035154 (2015),
doi:10.1103/PhysRevB.92.035154.
[31] R. Samajdar, S. Choi, H. Pichler, M. D. Lukin and S. Sachdev, Numerical study of the chiralZ3 quantum phase transition in one spatial dimension, Phys. Rev. A 98, 023614 (2018),
doi:10.1103/PhysRevA.98.023614.
[32] A. S. Jermyn, R. S. K. Mong, J. Alicea and P. Fendley, Stability of zero modes in parafermion
chains, Phys. Rev. B 90, 165106 (2014), doi:10.1103/PhysRevB.90.165106.
[33] A. Alexandradinata, N. Regnault, C. Fang, M. J. Gilbert and B. Andrei Bernevig,
Parafermionic phases with symmetry breaking and topological order, Phys. Rev. B 94,
125103 (2016), doi:10.1103/PhysRevB.94.125103.
[34] J. Motruk, E. Berg, A. M. Turner and F. Pollmann, Topological phases ingapped edges of fractionalized systems, Phys. Rev. B 88, 085115 (2013),
doi:10.1103/PhysRevB.88.085115.
[35] C. Nayak, S. H. Simon, A. Stern, M. Freedman and S. D. Sarma, Non-Abelian
anyons and topological quantum computation, Rev. Mod. Phys. 80, 1083 (2008),
doi:10.1103/RevModPhys.80.1083.
[36] R. S. K. Mong et al., Universal topological quantum computation from asuperconductor-abelian quantum hall heterostructure, Phys. Rev. X 4, 011036 (2014),
doi:10.1103/PhysRevX.4.011036.
[37] A. Dua, B. Malomed, M. Cheng and L. Jiang, Universal quantum computing
with parafermions assisted by a half-fluxon, Phys. Rev. B 100, 144508 (2019),
doi:10.1103/PhysRevB.100.144508.
[38] T. Wu et al., Formation of helical domain walls in the fractional quantum Hall regime asa step toward realization of high-order non-Abelian excitations, Phys. Rev. B 97, 245304(2018), doi:10.1103/PhysRevB.97.245304.
[39] Y. Wang, V. Ponomarenko, Z. Wan, K. W. West, K. W. Baldwin, L. N. Pfeiffer, Y. LyandaGeller and L. P. Rokhinson, Transport in helical Luttinger liquids in the fractional quantum
Hall regime, Nat. Commun. 12, 5312 (2021), doi:10.1038/s41467-021-25631-2.
[40] Ö. Gül et al., Andreev reflection in the fractional quantum Hall state, arXiv:2009.07836.
[41] W. Li, S. Yang, H.-H. Tu and M. Cheng, Criticality in translation-invariant parafermion
chains, Phys. Rev. B 91, 115133 (2015), doi:10.1103/PhysRevB.91.115133.
29

SciPost Phys. Core 5, 008 (2022)[42] I. Mahyaeh and E. Ardonne, Exact results for a Z3 -clock-type model and some close relatives,
Phys. Rev. B 98, 245104 (2018), doi:10.1103/PhysRevB.98.245104.
[43] A. Milsted, E. Cobanera, M. Burrello and G. Ortiz, Commensurate and incommensurate states of topological quantum matter, Phys. Rev. B 90, 195101 (2014),
doi:10.1103/PhysRevB.90.195101.
[44] S.-Y. Zhang, H.-Z. Xu, Y.-X. Huang, G.-C. Guo, Z.-W. Zhou and M. Gong, Topological phase,supercritical point, and emergent phenomena in an extended parafermion chain, Phys. Rev.
B 100, 125101 (2019), doi:10.1103/PhysRevB.100.125101.
[45] V. Kaskela and J. L. Lado, Dynamical topological excitations in parafermion chains, Phys.
Rev. Research 3, 013095 (2021), doi:10.1103/PhysRevResearch.3.013095.
[46] J. Wouters, H. Katsura and D. Schuricht„ Interrelations among frustrationfree models via Witten’s conjugation, SciPost Phys. Core 4, 027 (2021),
doi:10.21468/SciPostPhysCore.4.4.027.
[47] I. Peschel and T. T. Truong, The kinetic Potts chain and related potts models with competinginteractions, J. Stat. Phys. 45, 233 (1986), doi:10.1007/BF01033089.
[48] N. H. Lindner, E. Berg, G. Refael and A. Stern, Fractionalizing Majorana fermions: NonAbelian statistics on the edges of Abelian quantum hall states, Phys. Rev. X 2, 041002(2012), doi:10.1103/PhysRevX.2.041002.
[49] S. Groenendijk, A. Calzona, H. Tschirhart, E. G. Idrisov and T. L. Schmidt, Parafermion
braiding in fractional quantum Hall edge states with a finite chemical potential, Phys. Rev.
B 100, 205424 (2019), doi:10.1103/PhysRevB.100.205424.
[50] C. Chen and F. J. Burnell, Tunable splitting of the ground-state degeneracy inquasi-one-dimensional parafermion systems, Phys. Rev. Lett. 116, 106405 (2016),
doi:10.1103/PhysRevLett.116.106405.
[51] E. Cobanera, J. Ulrich and F. Hassler, Changing anyonic ground degeneracy with engineered
gauge fields, Phys. Rev. B 94, 125434 (2016), doi:10.1103/PhysRevB.94.125434.
[52] J. Hauschild and F. Pollmann, Efficient numerical simulations with Tensor Networks: Tensor Network Python (TeNPy), SciPost Phys. Lect. Notes, 5 (2018),
doi:10.21468/SciPostPhysLectNotes.5.
[53] S. R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev.
Lett. 69, 2863 (1992), doi:10.1103/PhysRevLett.69.2863.
[54] U. Schollwöck, The density-matrix renormalization group in the age of matrix productstates, Ann. Phys. 326, 96 (2011), doi:10.1016/j.aop.2010.09.012.
[55] V. Lahtinen, T. Mansson and E. Ardonne, Quantum criticality in many-body parafermion
chains, SciPost Phys. Core 4, 014 (2021), doi:10.21468/SciPostPhysCore.4.2.014.
[56] P. Calabrese and J. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech.:
Theor. Exp. P06002 (2004), doi:10.1088/1742-5468/2004/06/P06002.
[57] P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A:
Math. Theor. 42, 504005 (2009), doi:10.1088/1751-8113/42/50/504005.
[58] M. E. J. Newman and G. T. Barkema, Monte Carlo methods in statistical physics, Oxford
University Press, New York (1999).
30

SciPost Phys. Core 5, 008 (2022)[59] C. J. Hamer, J. B. Kogut and L. Susskind, Strong-coupling expansions and phase diagramsfor the O(2), O(3), and O(4) Heisenberg spin systems in two dimensions, Phys. Rev. D 19,3091 (1979), doi:10.1103/PhysRevD.19.3091.
[60] R. S. K. Mong, D. J. Clarke, J. Alicea, N. H. Lindner and P. Fendley, Parafermionicconformal field theory on the lattice, J. Phys. A: Math. Theor. 47, 452001 (2014),
doi:10.1088/1751-8113/47/45/452001.
[61] E. M. Stoudenmire, D. J. Clarke, R. S. K. Mong and J. Alicea, Assembling Fibonacci anyons from a Z3 parafermion lattice model, Phys. Rev. B 91, 235112 (2015),
doi:10.1103/PhysRevB.91.235112.
[62] I. Mahyaeh and E. Ardonne, Study of the phase diagram of the Kitaev-Hubbard chain,
Phys. Rev. B 101, 085125 (2020), doi:10.1103/PhysRevB.101.085125.
[63] A. B. Zamilochikov, “Irreversibility" of the flux of the renormalization group in a 2D field
theory, JETP Lett. 43, 730 (1986).
[64] E. O’Brien, Exotic phases of interacting Majorana fermions and parafermions, Ph.D. thesis,
University of Oxford (2020).
[65] T. Giamarchi, Quantum Physics in One Dimension, Oxford University Press, Oxford(2004).
T. Giamarchi, Quantum physics in one dimension, Oxford University Press, Oxford (2004).
[66] E. O’Brien, E. Vernier and P. Fendley, “Not-A”, representation symmetry-protected topological, and Potts phases in an S3 -invariant chain, Phys. Rev. B 101, 235108 (2020),
doi:10.1103/PhysRevB.101.235108.
[67] Y. Alavirad, D. Clarke, A. Nag and J. D. Sau, Z3 - Parafermionic zero modes without Andreev
backscattering from the 2/3 fractional quantum Hall state, Phys. Rev. Lett. 119, 217701(2017), doi:10.1103/PhysRevLett.119.217701.
[68] N. Chepiga and F. Mila, Lifshitz point at commensurate melting of chains of Rydberg atoms,
Phys. Rev. Research 3, 023049 (2021), doi:10.1103/PhysRevResearch.3.023049.
[69] R. Jafari and A. Langari, Second order quantum renormalisation group of XXZ chain
with next-nearest neighbour interactions, Phys. A: Stat. Mech. Appl. 364, 213 (2006),
doi:10.1016/j.physa.2005.09.048.
[70] S. Whitsitt, R. Samajdar and S. Sachdev, Quantum field theory for the chiral clock transition in one spatial dimension, Phys. Rev. B 98, 205118 (2018),
doi:10.1103/PhysRevB.98.205118.
[71] N. Laflorencie, E. S. Sørensen, M.-S. Chang and I. Affleck, Boundary effects in the critical scaling of entanglement entropy in 1D systems, Phys. Rev. Lett. 96, 100603 (2006),
doi:10.1103/PhysRevLett.96.100603.
[72] J. M. Kosterlitz and D. J. Thouless, Ordering, metastability and phase transitions in twodimensional systems, J. Phys. C: Solid State Phys. 6, 1181 (1973), doi:10.1088/00223719/6/7/010.
[73] C. C. Rulli and M. S. Sarandy, Entanglement and local extremes at an
infinite-order quantum phase transition, Phys. Rev. A 81, 032334 (2010),
doi:10.1103/PhysRevA.81.032334.
31

SciPost Phys. Core 5, 008 (2022)[74] Y.-D. Hsieh, Y.-J. Kao and A. W. Sandvik, Finite-size scaling method for the BerezinskiiKosterlitz-Thouless transition, J. Stat. Mech. P09001 (2013), doi:10.1088/17425468/2013/09/P09001.
[75] G. Sun, A. K. Kolezhuk and T. Vekua, Fidelity at Berezinskii-Kosterlitz-Thouless quantum
phase transitions, Phys. Rev. B 91, 014418 (2015), doi:10.1103/PhysRevB.91.014418.
[76] G. Sun, T. Vekua, E. Cobanera and G. Ortiz, Phase transitions in the Z p and U(1) clock
models, Phys. Rev. B 100, 094428 (2019), doi:10.1103/PhysRevB.100.094428.
[77] N. Chepiga and F. Mila, Floating Phase versus Chiral Transition in a 1D Hard-Boson Model,
Phys. Rev. Lett. 122, 017205 (2019), doi:10.1103/PhysRevLett.122.017205.
[78] J. Zhang, Y. Meurice and S.-W. Tsai, Truncation effects in the charge representation of theO(2) model, Phys. Rev. B 103, 245137 (2021), doi:10.1103/PhysRevB.103.245137.
[79] J. Zhang, Fidelity and entanglement entropy for infinite-order phase transitions, Phys. Rev.
B 104, 205112 (2021), doi:10.1103/PhysRevB.104.205112.
[80] In preparation.
[81] L. Turban, Exact results for quantum chains with multisite interactions, J. Phys. A: Math.
Gen. 18, 2313 (1985), doi:10.1088/0305-4470/18/12/029.

32