SciPost Phys. Core 4, 027 (2021)

Interrelations among frustration-free models
via Witten’s conjugation
Jurriaan Wouters1? , Hosho Katsura2,3,4† 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 Department of Physics, Graduate School of Science, The University of Tokyo,
3 Institute for Physics of Intelligence, The University of Tokyo,
4 Trans-scale Quantum Science Institute, The University of Tokyo,
7-3-1, Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan? j.j.wouters@uu.nl, † katsura@phys.s.u-tokyo.ac.jp, ‡ d.schuricht@uu.nl

Abstract
We apply Witten’s conjugation argument [1] to spin chains, where it allows us to derive
frustration-free systems and their exact ground states from known results. We particularly focus on Z p -symmetric models, with the Kitaev and Peschel–Emery line of the axial
next-nearest neighbour Ising (ANNNI) chain being the simplest examples. The approachallows us to treat two Z3 -invariant frustration-free parafermion chains, recently derived
by Iemini et al. [2] and Mahyaeh and Ardonne [3], respectively, in a unified framework.
We derive several other frustration-free models and their exact ground states, including
Z4 - and Z6 -symmetric generalisations of the frustration-free ANNNI chain.
Copyright J. Wouters et al.
This work is licensed under the Creative Commons
Attribution 4.0 International License.
Published by the SciPost Foundation.

Received 27-05-2020
Accepted 11-08-2021
Check forupdatesPublished 14-10-2021
doi:10.21468/SciPostPhysCore.4.4.027

Contents
1 Introduction

2

2 Conjugation argument

4

3 Frustration-free models revisited
3.1 XY chain in transverse magnetic field
3.2 ANNNI model
3.3 q-deformed XXZ chain
3.4 q-deformed AKLT chain

6
6
7
7
9

4 Introduction to Z p -clock models

11

5 Frustration-free Z p -generalisations of the XY chain
5.1 Z p -XY model: most general real coefficients
5.2 Z p -XY model: compact form with real coefficients
5.3 Z3 -XY model: real coefficients from complex deformation

13
13
14
14

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SciPost Phys. Core 4, 027 (2021)

6 Frustration-free Z p -generalisations of the ANNNI model
6.1 Z3 -ANNNP model: with complex coefficients
6.1.1 Z3 -ANNNP model: real coefficients reproducing Reference [3]6.1.2 Z3 -ANNNP with ground state deformation of Z3 -XY model
6.2 Z p -ANNNP model: most general real coefficients
6.3 Z2q -ANNNP model: compact form with real coefficients
6.4 Z4 -ANNNP model
6.5 Z6 -ANNNP model

15
16
16
17
17
18
19
20

7 Discussion

21

A Witten’s conjugation argument

22

B Energy gap of some Z3 -, Z4 - and Z6 -models
B.1 Knabe’s method
B.2 Gap in Z3 -models
B.3 Gap in Z4 -ANNNP model
B.4 Gap in Z6 -ANNNP model

23
23
25
27
27

References

27

1

Introduction

Strongly correlated quantum systems are notoriously hard to study. Even when restricted
to one spatial dimension the applicability of analytical methods is rather limited. Notable
exceptions are provided by systems like the quantum Ising or XY spin chain that can be mapped
to effectively non-interacting models [4], thus allowing the determination of the full spectrum
by elementary means. A second class of systems is provided by integrable models [5]. They
also allow the determination of the full energy spectrum, although more sophisticated methods
like the algebraic Bethe ansatz [6, 7] have to be employed and simple results in a closed form
are usually not available. A third class of systems is given by the so-called frustration-freemodels [8]. These are distinguished by the fact that the ground-state manifold can be given in
an exact, closed form. In this paper we will discuss such frustration-free models and present
an overarching framework connecting many of them.
One of the first frustration-free models was described by Peschel and Emery [9]. They realised that for a constrained set of couplings the ground state of the axial next-nearest neighbour Ising (ANNNI) [10] model takes the simple form of a product state, thus facilitating thestraightforward calculation of correlation functions. Along this Peschel–Emery line the model
can be viewed as a deformation of the trivial ferromagnetic Ising model. Several generalisations to other two-dimensional models including the three-state Potts model were discoveredin the following [11–14]. Recently, frustration-free models of this type have been investigatedin the context of Majorana zero modes [15–17] by employing the original results of Pescheland Emery.
Another famous example of a frustration-free model is the Affleck–Kennedy–Lieb–Tasaki(AKLT) chain [8, 18, 19], which was originally devised in the context of the Haldane conjecture for integer spin chains [20–22]. The idea to construct a parent Hamiltonian was subsequently used to construct further frustration-free models like the q-deformed AKLT model [23,2

SciPost Phys. Core 4, 027 (2021)24], valence bond solids with general Lie group symmetries [25–30], or supersymmetric systems [31, 32]. As the ground state of the AKLT model can be written as a compact matrix
product state it has served as the starting point for the development of the general theory ofmatrix product and tensor network states [33–38] and their application in numerical simulations [39, 40] as well as the classification of quantum phases and their symmetry protections [41–45].
Our investigation was motivated in particular by two recent works by Iemini et al. [2] andMahyaeh and Ardonne [3]. They constructed two different, frustration-free Z3 -clock models. The motivation for these studies was given by their relation to parafermions, thus naturally generalising Majorana zero modes to Z3 -symmetric systems [46]. Like in the case of thePeschel–Emery line discussed above, both models can be viewed as deformations of a simple
classical system, in this case the three-state zero-bias Potts chain. One of our main results is toreformulate both models in a unified framework, thus treating them on an equal footing andclarifying their relation (illustrated in Figure 3).
This will be achieved by applying Witten’s conjugation argument [1, 47], originally introduced for supersymmetric systems, to spin chains. Starting from a simple model with known
ground-state manifold, we derive interacting deformations as well as their exact ground states.
The explicit construction then allows the calculation of correlation functions and, in somecases, the proof of the existence of an energy gap. We will apply this line of argument to Z p symmetric systems, with the two specific Z3 -symmetric models mentioned above analysed indetail. Furthermore, we construct several new frustration-free models, including generalisations of the Peschel–Emery line to Z4 - and Z6 -symmetric systems.
In this context we note that a method very similar to the Witten conjugation has been applied in the field of matrix product states to construct frustration-free models from the respective parent Hamiltonians [42, 43, 48, 49]. The framework of matrix product (or generalised
valence bond solid) states also allows for the calculation of correlation functions, and provides the starting point to prove the existence of an energy gap for the corresponding parent
Hamiltonians. These proofs are based either on the martingale method [50] or finite-size criteria [33,51–53]. The latter link the energy gap of a finite-size system to a lower bound on theenergy gap in the thermodynamic limit. The first work following such an approach was doneby Knabe [54], who used exact diagonalisation on finite-size systems to obtain a lower boundfor the energy gap in the spin-1 AKLT model. We will use this approach to obtain bounds forthe energy gaps of several models considered in our manuscript. We note that our proofs canin principle be extended by using more advanced methods [50, 52], however, the obtainedbounds are physically less practical as we discuss for instance for the model in Section 6.5. Inorder to keep our discussion less abstract, we thus take a more explicit approach not relying
on matrix product states in the following but note that many of the results we present below
can be rephrased in such terms.
This article is organised as follows: In the next section we discuss Witten’s conjugationargument and tailor it to frustration-free spin chains. Section 3 recalls some known familiesof frustration-free models that are rederived using the deformation approach. In Section 4
we introduce the necessary notations to discuss Z p -symmetric clock models. In Sections 5
and 6 we analyse two types of deformations, in particular covering the models introduced
in References [2, 3] in the special case p = 3. In addition, we consider several frustrationfree Z p -models. While Witten’s conjugation argument applied here ensures the form of theground state, it does not guarantee the existence of an energy gap. Therefore, in the appendix
we apply Knabe’s method [54] to obtain lower bounds for the energy gap for some of theconsidered models.

3

SciPost Phys. Core 4, 027 (2021)

2

Conjugation argument

Originally [1] Witten introduced his conjugation argument in the context of supersymmetricquantum mechanical models. More specifically he discussed, given a supersymmetric Hamiltonian H, how to construct an inequivalent Hamiltonian H̃ with the same number of zero-energy
states. In this section we recall this argument, already tailoring the notation to the spin-chain
systems we will discuss in the following sections. For completeness we recall Witten’s original
argument in Appendix A.
We consider a lattice with a finite-dimensional Hilbert space for each of the lattice sites.
More specifically, in this work we restrict ourselves to one-dimensional chains with open
boundary conditions, and assume the local Hamiltonian to act non-trivially neighbouring sites
only. We note, however, that the argument presented here is applicable more generally, for example, to periodic boundary conditions, higher-dimensional lattices or longer-ranged models.
Coming back to our setup, we consider a Hamiltonian of the form
H=

N
−1
X

H j, j+1 =

N
−1
X

j=1

L †j, j+1 L j, j+1 ,

(1)

j=1

where each term1 H j, j+1 = L †j, j+1 L j, j+1 acts non-trivially on the neighbouring lattice sites j
and j + 1 only, and is positive semi-definite, 〈Ψ| H j, j+1 |Ψ〉 ≥ 0 for all |Ψ〉. Consequently theground-state manifold G is spanned by |Ψ1 〉 , . . . , |Ψn 〉, 1 ≤ n, with L j, j+1 |Ψi 〉T= 0 for all j; inother words, G is the intersection of the kernels of the operators L j, j+1 , G = j ker(L j, j+1 ).
The representation (1) now allows us to say something about the ground states of a deformed/conjugated Hamiltonian. Consider an invertible operator M j that acts non-trivially on
the local Hilbert space of lattice site j only,Qwith which we define an invertible operator acting
non-trivially on the whole chain via M = j M j . Using this operator we can write down the
conjugated operators as−1
M j−1 ,
L̃ j, j+1 = M L j, j+1 M −1 = M j M j+1 L j, j+1 M j+1(2)

where we used [L j, j+1 , Mk ] = 0 for k 6= j, j + 1. Now the deformed/conjugated local Hamiltonian is given by
N
−1
N
−1
X
X
H̃ =
H̃ j, j+1 =L̃ †j, j+1 C j, j+1 L̃ j, j+1 ,
(3)
j=1

j=1

where we have introduced the hermitian operator C j, j+1 as additional degrees of freedom
in the construction. The operator C j, j+1 = K †j, j+1 K j, j+1 is assumed to be positive definite,〈Ψ| C j, j+1 |Ψ〉 > 0 for all |Ψ〉, and thus invertible. The product form of M and the locality of
C j, j+1 and L j, j+1 ensure that the resulting Hamiltonian is still local. Note that in general there
is no unique annihilation operator L j, j+1 . Later in this section we will discuss the interplay
between the freedom of C j, j+1 , L j, j+1 and M j .
In this setting we can now prove the following theorem (see Reference [1] and Appendix A
for the original supersymmetric case):
Theorem 1. The ground-state manifold G̃ of the conjugated Hamiltonian H̃ is given byG̃ = span{M |Ψ1 〉 , . . . , M |Ψn 〉}, thus the ground-state degeneracies of H and H̃ are identical .
We note that the states M |Ψi 〉 do not form an orthonormal basis, but since M is invertible
the states {M |Ψ1 〉 , . . . , M |Ψn 〉} are linearly independent.
1

We use capital letters to denote operators acting on the Hilbert space of the full chain, with subindices indicatingon which lattice sites they act non-trivially.

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SciPost Phys. Core 4, 027 (2021)

T
Proof. First we show that since C j, j+1 is positive definite we have ker(H̃) = j ker( L̃ j, j+1 ).
T
The proof is simple: Note that a priori j ker( L̃ j, j+1 ) ⊆ ker(H̃). Now suppose |Ψ〉 ∈ ker(H̃),
ie, H̃ |Ψ〉 = 0, then
X
X
2
(4)
K j, j+1 L̃ j, j+1 |Ψ〉 = 0 .
〈Ψ| H̃ |Ψ〉 =〈Ψ| L̃ †j, j+1 C j, j+1 L̃ j, j+1 |Ψ〉 =jjThis implies K j, j+1 L̃ j, j+1 |Ψ〉 = 0 for all j, and consequently K †j, j+1 K j, j+1 L̃ j, j+1 |Ψ〉 = 0. Since
C j, j+1 = K †j, j+1 K j, j+1 is invertible we deduce L̃ j, j+1 |Ψ〉 = 0 for all j. Thus we have shown thatTT|Ψ〉 ∈ j ker( L̃ j, j+1 ), which implies ker(H̃) ⊆ j ker( L̃ j, j+1 ).
T
Second we have to show j ker( L̃ j, j+1 ) = G̃. Note that for all 1 ≤ j ≤ N − 1 and 1 ≤ i ≤ n
we haveL̃ j, j+1 M |Ψi 〉 = M L j, j+1 |Ψi 〉 = 0 ,(5)TTyielding G̃ ⊆ j ker( L̃ j, j+1 ). Conversely, suppose |Ψ̃〉 ∈ j ker( L̃ j, j+1 ), then for all 1 ≤ j ≤ N −1
we findL̃ j, j+1 |Ψ̃〉 = M L j, j+1 M −1 |Ψ̃〉 = 0 ,(6)Tfrom which we conclude that M −1 |Ψ̃〉 ∈ j ker(L j, j+1 ). Consequently we can expand the stateP
as M −1 |Ψ̃〉 = i ai |Ψi 〉 with suitable ai ∈ C, resulting in|Ψ̃〉 =nX

ai M |Ψi 〉 ∈ G̃ .

(7)

i=1

T
T
Therefore j ker( L̃ j, j+1 ) ⊆ G̃, which together with the above implies j ker( L̃ j, j+1 ) = G̃.
Finally, we note that since H̃ is positive semi-definite, its ground-state manifold is given byits kernel (provided it is non-zero), thus resulting in G̃ = ker(H̃) as had to be shown .
We stress that the theorem above provides a direct way to determine the ground-statedegeneracy of the deformed Hamiltonian. On the other hand, the theorem does not make any
statement about the energy gap above the ground-state manifold or the excited states of themodel. Thus, in Appendix B we will discuss a separate approach to prove the existence of afinite energy gap for some specific models.
Before applying the theorem to the construction of spin chain models, let us discuss thedegree of freedom in the choices for L j, j+1 , C j, j+1 and M j . First, assuming a local Hilbert
space of dimension p, we have the freedom to perform a local basis transformation2 v j , withv j ∈ U(p), at each lattice site j. Under this the operators M j transform as
M j → Vj M j Vj† ,Vj = 1 ⊗ . . . ⊗ 1 ⊗ v j ⊗ 1 ⊗ . . . ⊗ 1 ,(8)

and accordingly†
L j, j+1 → Vj Vj+1 L j, j+1 Vj† Vj+1,†
C j, j+1 → Vj Vj+1 C j, j+1 Vj† Vj+1.

(9)

Using this we can always choose a suitable basis in the local Hilbert spaces to simplify M j .
Second, recalling that the deformed local Hamiltonian is given byH̃ j, j+1 = (M † )−1 L †j, j+1 M † C j, j+1 M L j, j+1 M −1 ,(10)

we can also perform a transformation on the bonds between lattice sites j and j + 1 with
u j, j+1 ∈ U(p2 ). Specifically setting
L j, j+1 → U j, j+1 L j, j+1 ,2

C j, j+1 → (M † )−1 U j, j+1 M † C j, j+1 M U †j, j+1 M −1 ,We use small letters to denote operators acting on the Hilbert space of one or two lattice sites.

5

(11)

SciPost Phys. Core 4, 027 (2021)

where
U j, j+1 = 1 ⊗ . . . ⊗ 1 ⊗ u j, j+1 ⊗ 1 ⊗ . . . ⊗ 1 ,(12)

we see that the local Hamiltonian remains invariant. In the examples in the following sections
we will use these freedoms to simplify L j, j+1 .

3

Frustration-free models revisited

In this section we will revisit several known frustration-free models within the framework ofWitten’s conjugation. We first consider two spin-1/2 models: the XY model [55–57] withtransverse magnetic field and the ANNNI model [9, 15]. Then we review the q-deformed XXZ
chain [58–60], and finally we consider the q-deformed AKLT model [23, 24].

3.1

XY chain in transverse magnetic field

We rederive the frustration-free line for the XY model in a magnetic field. Our starting pointis the classical Ising chain (which is equivalent to the Kitaev/Majorana chain [61] in the decoupling limit),(13)
H j, j+1 = 2 − 2σ xj σ xj+1 ,with the exact ground states|Ψ± 〉 =

1 O
2N /2


|↑〉 j ± |↓〉 j ,

(14)

j

where |↑〉 j and |↓〉 j denote the eigenstates of σzj with eigenvalues ±1. We are looking for modQ z
els that have a Z2 -symmetry generated byj σ j . We choose M j diagonal, real and positive,
thus there is only one independent parameter in M j ,1, 0 < r < ∞.
(15)
Mj = 1 ⊗ . . . ⊗ 1 ⊗ mj ⊗ 1 ⊗ . . . ⊗ 1 , mj =r
The operator m j acts at lattice site j only, with the matrix representation given in the basis{|↑〉 j , |↓〉 j }. Hence the deformed ground states are|Ψ̃± 〉 = M |Ψ± 〉 .

(16)

Note that the states above are not orthogonal. Orthonormal ground states are instead givenby
1|Φ̃± 〉 =M |Ψ+ 〉 ± M |Ψ− 〉 ,(17)N±with suitable normalisations N± . If we take
L j, j+1 = σ xj − σ xj+1 ,C j, j+1 = 1 ,(18)

the deformation (3) gives the frustration-free line for the Kitaev chain [61], ie, the Jordan–
Wigner transform of the XY chain with magnetic field
(1)

y

y

H̃ j, j+1 = −J x σ xj σ xj+1 − J y σ j σ j+1 +B (1) z(σ j + σzj+1 ) + ε ,2(19)with the parametersJx =(r + r −1 )2
,
2

Jy =

(r − r −1 )2,
2
6

B (1) = r 2 − r −2 ,ε = r 2 + r −2 ,(20)

SciPost Phys. Core 4, 027 (2021)

which correspond to the parameters on the Barouch–McCoy circle [55]. Due to Theorem 1
P (1)the model H̃ (1) = j H̃ j, j+1 possesses a two-fold degenerate ground state. In Section 5.3 wewill discuss the Z3 -generalisation [2] of this model. Section 5 will be dedicated to generalise
the construction to arbitrary Z p -symmetry.

3.2

ANNNI model

For the second example we obtain an interacting parent Hamiltonian of (16) by choosing
C j, j+1 =r 2 −2 −2
M M j+1 ,
2 j

(21)

which acts non-trivially on the neighbouring lattice sites j and j + 1, with M j and L j as in
Section 3.1. The resulting deformed local Hamiltonian is the ANNNI model(2)H̃ j, j+1 = −σ xj σ xj+1 + Jz σzj σzj+1 +B (2) zσ j + σzj+1 + ε ,
2

(22)

with

r 2 − r −2
(r + r −1 )2(r − r −1 )2, B (2) =
, ε=
.
(23)
4
2
4
The frustration-free line rediscovered
here is the well-known Peschel–Emery line [9, 15] dep(2)
fined by the relation B = 2 Jz (1 + Jz ). The exact two-fold ground-state degeneracy of
P (2)H̃ (2) = j H̃ j, j+1 is assured by Theorem 1. In Section 6.1.1 we discuss the Z3 -generalisation [3]of this setup, while in Section 6.4 we extend the construction to Z4 -symmetry.
By construction the models (19) and (22) share the same ground states. Thus their combination is also a parent Hamiltonian,
Jz =

(1)

(2)

H̃ j, j+1 = α1 H̃ j, j+1 + α2 H̃ j, j+1 ,(24)

as long as αi ≥ 0. The parameters in the resulting spin model reproduce the frustration-free
condition for the XYZ model [56, 57, 62]. Furthermore, the existence of an energy gap above
the ground states for (24) has been proven [15]. We also note that the construction above canbe extended to inhomogeneous magnetic fields, in particular with an alternating bias [17], or
higher-dimensional systems [63, 64].
Finally, we note that the states (16) allow a straightforward calculation of correlation functions. For example, the two-point function of the Ising order parameter is independently of
the separation j − j 0 given by [55]〈Ψ̃± |σ xj σ xj0 |Ψ̃± 〉〈Ψ̃± |Ψ̃± 〉=4,(r + r −1 )2(25)

which simplifies to unity at the Ising point (r = 1) as expected.

3.3

q-deformed XXZ chain

As a third example we show that the XXX chain and the q-deformed XXZ chain are related
via Witten’s conjugation. We start with the local Hamiltonian of the spin-1/2 XXX Heisenberg
chain

y y
H j, j+1 = 1 − σ xj σ xj+1 + σ j σ j+1 + σzj σzj+1 .
(26)
We first note that the local Hamiltonian satisfies (H j, j+1 )2 = 4H j, j+1 , which means that theoperators H j, j+1 /4 act as projectors. Thus we can write1
H j, j+1 = |sing〉 j, j+1 〈sing| j, j+1 ,
4
7

(27)

SciPost Phys. Core 4, 027 (2021)pwith3 |sing〉 j, j+1 = (|↑〉 j |↓〉 j+1 − |↓〉 j |↑〉 j+1 )/ 2 denoting the singlet state on the lattice sites
j and j + 1. On all other lattice sites H j, j+1 acts trivially. To make the link to the notion
introduced above we write1
H j, j+1 = L †j, j+1 L j, j+1 ,4

L j, j+1 = |↑〉 j |↓〉 j+1 〈sing| j, j+1 .

(28)

Next we consider the generators of Uq (sl2 ) [65]
z

Sq± =

qS ,

N
X

σz

qσ1 /2 · · · q j−1
z

/2

σ±j q

−σzj+1 /2· · · q−σN /2 ,

(29)

.

(30)

z

j=1

where we assume q ∈ R, q > 0, and
y

N

1X z
σ ,
S =
2 j=1 jσ±j =z

σ xj ± iσ j
2These generators satisfy the algebrazzzqS Sq± q−S = q±1 Sq± ,[Sq+ , Sq− ] =z

q2S − q−2S,
q − q−1(31)

which reduce to the standard relations among the generators of SU(2) in the limit q → 1.
In order to proceed, we next define the operator M via
M (q) = q−σ1 /2 · · · qz− jσzj /2· · · q−N σN /2 ,z

(32)with the inverse given by
M (q)−1 = qσ1 /2 · · · q

jσzj /2· · · q N σN /2 .

(33)|↑〉 j |↓〉 j+1 〈sing(q)| j, j+1 ,(34)(q−1/2 |↑〉 j |↓〉 j+1 − q1/2 |↓〉 j |↑〉 j+1 ) .

(35)

z

z

With this one gets
−1L̃ j, j+1 = M (q)L j, j+1 M (q)=v
t 1 + q22

where the q-deformed singlet state is given by|sing(q)〉 j, j+1 = p

1q + q−1

Thus we obtainL̃ †j, j+1 L̃ j, j+1 =1 + q2|sing(q)〉 j, j+1 〈sing(q)| j, j+1 ,2(36)

which is manifestly Uq (sl2 ) invariant as it is the projection onto the q-deformed singlet stateon the bond ( j, j + 1). A straightforward calculation choosing C j, j+1 = 1 shows that
1H̃ j, j+1 = L̃ †j, j+1 L̃ j, j+14 q − q−1 z
q
q + q−1 z z
y y
x x
z
= − σ j σ j+1 + σ j σ j+1 +σ j σ j+1 − 1 +σ j − σ j+1 ,
4
2
2
3

(37)
(38)

For the tensor product of states on neighbouring lattice sites we use the short-hand notation|↑〉 j |↓〉 j+1 = |↑〉 j ⊗ |↓〉 j+1 and so on.

8SciPost Phys. Core 4, 027 (2021)

which, up to the prefactor q, is the local Hamiltonian of the q-deformed XXZ chain [58–60,65].
After deriving the Hamiltonian, let us consider the ground states in more detail. The
ground states of the Heisenberg chain (26) are given by4(S1− )i |⇑〉 ,i = 0, 1, . . . , N ,with|⇑〉 = |↑ · · · ↑〉 .

(39)Consequently, according to Theorem 1 the ground states of the q-deformed model readM (q) (S1− )i |⇑〉 ∝ (S̃1− )i |⇑〉 ,S̃1− = M (q)S1− M (q)−1 =

with

N
X

q j σ−j .

(40)

j=1

However, the Uq (sl2 ) algebra dictates that the ground-state manifold is spanned by(Sq− )i |⇑〉 .

(41)

By induction we will show that there is a correspondence (up to normalisation) between thesesets of states, ie,(Sq− )i |⇑〉 ∝ (S̃1− )i |⇑〉 .
(42)
Obviously this relation holds for i = 0. Now suppose that (42) is true up to i − 1. If we writez 1
Sq± = q−S ± 2

N
X

σz

qσ1 · · · q j−1 σ±j ,
z

(43)

j=1

then
(Sq− )i |⇑〉 ∝ Sq− (S̃1− )i−1 |⇑〉 = (i − 1)!Sq−
N +1= (i − 1)!q− 2
=

q

i

i

j1 <···< ji−1

q j1 +···+ ji−1 σ−j · · · σ−j |⇑〉1

q i − q−i X
q j1 +···+ ji σ−j · · · σ−j |⇑〉1iq − q−1 j <···< j1− N 2+1

X

i−1

(44)
(45)

i

−i

q −q
(S̃ − )i |⇑〉 ,q − q−1 1(46)

where the precise prefactor is in fact irrelevant for our purpose. This shows that the relation(42) is indeed fulfilled, and thus that the ground states of the q-deformed model are given by
M (q) (S1− )i |⇑〉.

3.4

q-deformed AKLT chain

Arguably one of the most prominent frustration-free models is the AKLT chain [8,18,19]. Even
though the ground state of this system is a matrix product state, we will see that we can still
employ the tools outlined above to derive its q-deformed generalisation [23, 24, 66].
We start with the original AKLT chain written as
H=X

H j, j+1 ,H j, j+1 ≡2X|ψm 〉 j, j+1 〈ψm | j, j+1 ,

(47)

m=−2

j

where H j, j+1 is the projector onto the subspace of total spin-2 on the neighbouring lattice sites
j and j + 1. It can be written in terms of the corresponding eigenstates |ψm 〉 j, j+1 and acts
4−
We note that the subscript refers to the deformation parameter, ie, S1− ≡ Sq=1.

9

SciPost Phys. Core 4, 027 (2021)

trivially on all other lattice sites. The eigenstates are explicitly given by
1|ψ1 〉 j, j+1 = p |+〉 j |0〉 j+1 + |0〉 j |+〉 j+1 ,2|+〉 j |−〉 j+1 + |−〉 j |+〉 j+1 + 2 |0〉 j |0〉 j+1 ,|ψ2 〉 j, j+1 = |+〉 j |+〉 j+1 ,1|ψ0 〉 j, j+1 = p61|ψ−1 〉 j, j+1 = p |0〉 j |−〉 j+1 + |−〉 j |0〉 j+1 ,2|ψ−2 〉 j, j+1 = |−〉 j |−〉 j+1 ,(48)

with |±〉 j , |0〉 j denoting the eigenstates of the spin-1 operator S zj at a given lattice site j. Notethat since H j, j+1 is a projector, we can match our convention by simply setting L j, j+1 = H j, j+1 .
For the deformation we choose (q ∈ R, q > 0)
M (q) =Y

M j (q),M j (q) = q−2 jS zjj

q + q−1
2

(S z )2 /2
j

,

(49)

and we define q-deformed statesq|ψ̃2 〉 j, j+1 = |+〉 j |+〉 j+1 ,1

q|ψ̃1 〉 j, j+1 = p

1 + q4|+〉 j |0〉 j+1 + q2 |0〉 j |+〉 j+1 ,|+〉 j |−〉 j+1 + q2 |−〉 j |+〉 j+1 + (q + q−1 ) |0〉 j |0〉 j+1,p
q4 + q−4 + (q + q−1 )21qq|ψ̃−1 〉 j, j+1 = p|0〉 j |−〉 j+1 + q2 |−〉 j |0〉 j+1 , |ψ̃−2 〉 j, j+1 = |−〉 j |−〉 j+1 .
4
1+q

q
q
|ψ̃0 〉 j, j+1 =

−2

(50)

We can then work out that the conjugated annihilation operator L̃ j, j+1 is given by
q−1

q−1

q

q

L̃ j, j+1 ≡ |ψ̃2 〉 j, j+1 〈ψ̃2 | j, j+1 + |ψ̃−2 〉 j, j+1 〈ψ̃−2 | j, j+1


q−1
q
q−1
q
q
+ a(q) |ψ̃1 〉 j, j+1 〈ψ̃1 | j, j+1 + |ψ̃−1 〉 j, j+1 〈ψ̃−1 | j, j+1 + b(q) |φ̃0 〉 j, j+1 〈ψ̃0 | j, j+1 , (51)with the auxiliary state|φ̃0 〉 j, j+1 = q2 |+〉 j |−〉 j+1 + q−2 |−〉 j |+〉 j+1 +4|0〉 j |0〉 j+1
q + q−1(52)

and the parametersa(q) =

q2 + q−2,2b(q) =(q2 + q−2 )(q2 + q−2 + 1)
.
6

(53)

Now we choose C j, j+1 as
q−1

q−1

q−1

q−1

C j, j+1 = |ψ̃2 〉 j, j+1 〈ψ̃2 | j, j+1 + |ψ̃−2 〉 j, j+1 〈ψ̃−2 | j, j+11  q−1
q−1
q−1
q−1
|
ψ̃
〉
〈
ψ̃
|
+
|
ψ̃
〉
〈
ψ̃
|
+
1
1
−1
−1
j,
j+1
j,
j+1
j,
j+1
j,
j+1
a(q)2
1
+
|φ̃0 〉 j, j+1 〈φ̃0 | j, j+1 ,
b(q)2

(54)

such that the deformed local Hamiltonian becomes the projectorH̃ j, j+1 = L̃ †j, j+1 C j, j+1 L̃ j, j+1 ≡

2
X
m=−2

10

|ψ̃qm 〉 j, j+1 〈ψ̃qm | j, j+1 .

(55)

SciPost Phys. Core 4, 027 (2021)

Hence we obtain the q-deformed AKLT model [23, 24, 66].
The above result shows the deformation at the level of the Hamiltonian. Let us also look
explicitly at the ground state. The four ground states of the undeformed AKLT chain can be
written in the matrix product state representation as
 1,1

p


1,2
|ΨAKLT 〉 |ΨAKLT 〉|0〉 j− 2 |+〉 j= A1 · · · A L , with A j = p
.
(56)
2,1
2,2
2 |−〉 j− |0〉 j|ΨAKLT 〉 |ΨAKLT 〉According to Theorem 1, the ground state of the q-deformed model is generated by the matrix


p
|0〉 j
−q−2 j q + q−1 |+〉 j
p.
(57)
Ã j = 2 j
q + q−1 |−〉 j− |0〉 jq
Generically a matrix product state is defined up to a gauge freedom. If we take jq
f j−1, j =,q−( j−1)
we can redefine the matrix representation as
q−1 |0〉 j
tr−1Ã j = f j−1, j Ã j f j, j+1 = p
q + q−1 |−〉 j−pq + q−1 |+〉 j−q |0〉 j

(58)


,

(59)

which is identical to the one given in References [24,66] for the ground state of the q-deformed
AKLT chain.
Finally we note that a similar derivation to the one presented in this section can be used
to relate the AKLT chain (47) to a frustration-free point in the (representation) symmetry
protected phase of S3 -invariant chains recently studied by O’Brien et al. [67].

4Introduction to Z p -clock models

The rest of the paper considers Z p -clock models and frustration-free systems of this type.
Therefore, let us first briefly review the Z p -clock algebra. Consider a local p-dimensional
Hilbert space and two local operators σ and τ satisfying
σ p = τp = 1 ,σ p−1 = σ† ,

τ p−1 = τ† ,στ = ωτσ ,(60)

where ω = exp(2πi/p) is the pth root of unity. Denoting the eigenstates of σ and τ by |σ, i〉and |τ, i〉 with i = 0, . . . , p − 1 respectively, the action of the operators is given by
σ |σ, i〉 = ωi |σ, i〉 ,i

τ |τ, i〉 = ω |τ, i〉 ,

τ |σ, i〉 = |σ, i + 1〉 ,(61)

σ |τ, i〉 = |τ, i − 1〉 ,(62)

where i ± 1 has to be taken modulo p. The states |σ, i〉 can be represented in terms of thestates |τ, i〉 as
1|σ, i〉 = p |τ, 0〉 + ωi |τ, 1〉 + · · · + ω(p−1)i |τ, p − 1〉 .
p

(63)

The Potts/clock model is a generalisation of the Ising model. Here we start with the counterpart of the classical Ising chain (13), namely the classical Potts model, whose local Hamiltonian is given by
H j, j+1 = 2 − σ j σ†j+1 − σ†j σ j+1 ,(64)
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SciPost Phys. Core 4, 027 (2021)

where σ j and τ j denote the operators σ and τ introduced above, now acting non-trivially on
the local Hilbert space of site j. The classical Potts model possesses a p-fold degenerate groundstateO|Ψi 〉 =|σ, i〉 j ,(65)
jwith |σ, i〉 j denoting the eigenstates of σ j . Furthermore, the system has an energy gap above
the ground states.
Finally, we note that the clock operators σ j and τ j have a parafermionic dual by virtue ofthe Fradkin–Kadanoff transformation [68], which is the generalisation of the Jordan–Wignertransformation to Z p -symmetry. The resulting parafermions can be regarded as a generalisation of Majorana fermions [46].
We can already discuss the most general form of deformation that we consider in the restof the paper. So far the only requirement for M j is the invertibility. In this work we restrict
ourselves to models that preserve Z p -symmetry generated by
ωP =

Y

τj .

(66)

j

Since M has to commute with ω P , the local operator m j has to be diagonal in the τ-basis, ie,
mj = 


α0


α1
..

p−1
 1 X
=
αk ω−kl τl ,
 p

.

(67)

k,l=0

α p−1

for αk ∈ C. Note that we can take out an overall scaling factor, so we end up with p − 1
independent complex coefficients αk /α0 , k = 1, . . . , p − 1. For now we will leave it in themost general form. In line with the cyclicity of the algebra, the coefficients αk are also definedmodulo p,
αk = αk mod p ,(68)
for instance α−k = α p−k . Later we will see that in specific examples we get more constraints
on the coefficients αk .
Starting withstates (65) we obtain the deformed states by acting with the
Q the groundN
operator M = j M j = j m j , ie,|Ψ̃i 〉 = M |Ψi 〉 =Om j |σ, i〉 j .

(69)

j

This form immediately allows us to calculate correlation functions. For example, the two-point
function of the order parameter σ becomes〈Ψ̃i |σ j σ†j 0 |Ψ̃i 〉〈Ψ̃i |Ψ̃i 〉=|α∗ αk+1 |2
Pk k
≤ 1,
( k |αk |2 )2

P

(70)

where the upper bound is obtained by virtue of the Schwarz inequality. Other correlation
functions can be obtained in a similar way. In the following sections we will derive the parent
Hamiltonian for the deformed ground states.

12

SciPost Phys. Core 4, 027 (2021)

5

Frustration-free Z p -generalisations of the XY chain

In this section we generalise the Z2 -XY chain discussed in Section 3.1 to arbitrary Z p -symmetry.
Specifically we use the term XY in the sense that we take L j, j+1 and C j, j+1 of the following form
L j, j+1 = σ j − σ j+1 ,C j, j+1 = 1 .

(71)

Furthermore we require the resulting model to possess ω P -symmetry, which fixes m j to begiven by (67). In the case p = 3 we recover a model recently studied by Iemini et al. [2], see
Section 5.2.
For the choices (67) and (71) it is straightforward to work out the conjugated Hamiltonian(we set α−1 ≡ α p−1 to lighten the notation)p−1

L̃ j, j+1 =
1 X αk−1 −kl ωσ j τlj − σ j+1 τlj+1 ,p k,l=0 αk(72)where we used [see Equation (60)]M j σ j M j−1 =p−1
1 X αk−1 −kl
1 X αk −(k+1)l−k0 l 0
l+l 0ωσjτj =
ω σ j τlj .
p2 k,k0 ,l,l 0 αk0
p k,l=0 αk(73)With (72) the conjugated local Hamiltonian then becomesH̃ j, j+1 =

∗

 
1 X αk−1 αk0 −1 kl−k0 l 0  l 0 −ll0−l †l 0 −l
+
h.c.
σ
σ
τ
−
τ
+
τ
ω
τ
j+1
j+1
j
j+1
j
∗
j
p2 k,k0 ,l,l 0 αk αk0

p−1


 X

= − B †j σ†j σ j+1 B j+1 + h.c. +γl τlj + τlj+1 ,

(74)

l=0

where
Bj =

p−1
X
l=0

p−1

βl τlj ,1 X αk−1 −kl
βl =ω ,p k=0 αkp−1

1 X αk−1
γl =p k=0 αk2

ω−kl .

(75)

Admittedly this form is not yet very insightful. Thus in the following sections we will consider
specific cases for which the Hamiltonian simplifies.

5.1

Z p -XY model: most general real coefficients

One simplification occurs with the requirement that the coefficients βl and γl are real. For odd
p this implies the following conditions (we set α0 = r0 = 1 due to the freedom in the overallscaling of m j )( iθ
p−1
e k rk ,k = 1, . . . , 2 ,
αk =
(76)
r2
p+1
eiθp−k−1 r(p−1)/2, k = 2 ,..., p − 1,
p−k−1
for r1 , . . . , r(p−1)/2 > 0 and θ1 , . . . , θ(p−1)/2 ∈ [0, 2π). Similarly, for p even βl and γl are real
provided¨peiθk rk ,k = 1, . . . , 2 − 1 ,
αk =
(77)
p
s
, k = 2,..., p − 1,±eiθp−k−1 rp−k−1
for r1 , . . . , r p/2−1 , s > 0 and θ1 , . . . , θ p/2−1 ∈ [0, 2π).

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SciPost Phys. Core 4, 027 (2021)

5.2

Z p -XY model: compact form with real coefficients

In order to obtain a compact form for the Hamiltonian (74) the results from the previous
section can be further specified. Taking αk = r k with r ∈ R\{0} such that the ratio between
consecutive αk is constant, we obtain for the coefficients
βl =


1
r p + pδl,0 − 1 ,
pr

γl =


1
2p
r
+
pδ
−
1
.
l,0
pr 2

(78)

Thus the local Hamiltonian simplifies toH̃ j, j+1 = ε −

1+ b

p−1
X


τlj


σ†j σ j+1

1+ b

l=1

with
b=

rp − 1,
rp + p − 1

p−1
X


τlj+1

l=1

f =

2p(1 − r 2p ),(r p + p − 1)2



p−1


f X lτ j + τlj+1 ,+ h.c. −2 l=1

ε=p(r 2p + p − 1),(r p + p − 1)2

(79)

(80)

where we have done a multiplicative rescaling to set the coupling of σ†j σ j+1 to −1. For p = 2
the model simplifies to the XY model (19) discussed in Section 3.1.
We note that for odd p the model parameters depend on the sign of r, while for even p
the coefficients only contain even powers of r. The latter suggests that there are two sets ofground states for the same Hamiltonian,|Ψ̃i+ 〉 = M (r) |Ψi 〉 ,|Ψ̃i− 〉 = M (−r) |Ψi 〉 .

(81)However, from the expansion we recognise+〉,|Ψ̃i− 〉 = |Ψ̃i+p/2(82)

so both M (r) and M (−r) provide the same set of ground states. Moreover, the physical properties do not change under r→
1/r.
We can see this from
n
−1
−n
n n−1
m j = diag(1, r, . . . , r ) → diag(1, r , . . . , r ) ∝ diag(r , r , . . . , 1). The latter is related
to the original m j by a conjugation and cyclic rotation of the basis, hence the physical properties remain invariant.
Finally we note that for p = 3 and r > 0 we reproduce the model introduced by Iemini et al. [2]. There the authors also derive the positive-definite form (3) by the use of Fockparafermions [69]. Using elementary methods, in Appendix B.2 we show that the model possesses a finite energy gap for 0.5695 ® r ® 1/0.5695 ≈ 1.7560, thus confirming the corresponding numerical results [2]. We note that our proof does not exclude the existence of anenergy gap outside this interval, which can be extended by improving our analysis or using
alternative methods [33,50,52]. We note, however, that special care has to be taken regarding
the treatment of the boundary conditions.

5.3

Z3 -XY model: real coefficients from complex deformation

Our construction allows us to directly generalise the model discussed above. From Section 5.1
we see that for p = 3 there is an additional freedom in the choice of m j in the form of acomplex phase, ie, we can choose


1
,
eiθ r
mj = 
(83)
2
r

14

SciPost Phys. Core 4, 027 (2021)

which results in




 f

H̃ j, j+1 = ε − 1 + b+ τ j + b− τ†j σ†j σ j+1 1 + b− τ j+1 + b+ τ†j+1 +τ j + τ j+1 + h.c. ,2(84)with the parameterspr 3 − cos θ ± 3 sin θ6(r 6 + 2)
6(1 − r 6 )
±
,
b
=
,
ε
=
.
(85)
f = 3
(r + 2 cos θ )2
r 3 + 2 cos θ(r 3 + 2 cos θ )2
For θ = 0 we recover the modelstudied in Reference [2]. We note that the parameters (85)p3
possess a divergence at r = −2 cos θ provided θ ∈ [ π2 , 3π2 ]. This divergence is an artefact offixing the prefactor of the −σ†j σ j+1 -term to unity, it can be removed by rescaling the Hamiltonian by (r 3 + 2 cos θ )2 .

6

Frustration-free Z p -generalisations of the ANNNI model

In this section we construct Z p -invariant generalisations of the ANNNI model (see Section 3.2),
for which we will use the term5 axial next-nearest neighbour Potts (ANNNP) model [14].
More specifically we consider Z p -invariant Hamiltonians where besides the classical Potts term0σ j σ†j+1 + σ†j σ j+1 only terms of the form τlj τlj+1 with l, l 0 = 0, . . . , p − 1 appear. In particular,
there are no terms containing products of σ- and τ-operators.
First we will derive some general results following from this simple set of rules. Then we
discuss several specific examples. We take m j to be defined by (67) and L j, j+1 = σ j − σ j+1 as
before. Furthermore, generalising (21) we set C j, j+1 = K j K j+1 , where K j acts non-trivially at
lattice site j with the matrix k j . Now making the ansatz (in the τ-basis)
α p−1 α0
α1 α2k j = diag
, ,...,,
α0 α1
α p−2 α p−1



,

(86)

and recalling that C j, j+1 has to be hermitian and positive definite, we deduce that the αk haveto be real and positive (we set α0 = 1). From the form above we also deduce that the following
identity holds, K j M j σ j M j−1 = σ j . Hence we find for the deformed local HamiltonianH̃ j, j+1 = L̃ †j, j+1 K j K j+1 L̃ j, j+1 = L̃ †j, j+1 σ j K j+1 − K j σ j+1



−1 †
= M j−1 σ†j M j − M j+1
σ j+1 M j+1 σ j K j+1 − K j σ j+1

 
−1 †
= − σ j σ†j+1 + σ†j σ j+1 + M j−1 σ†j M j σ j K j+1 + K j M j+1
σ j+1 M j+1 σ j+1 .

(87)

Here the first two terms represent the classical Potts model. Note that both M j and K j are
diagonal in the τ-basis and can therefore be expanded in powers of τ j .
M j−1 σ†j M j σ j =X∆l τlj ,

l

Kj =

X

Γl τlj ,(88)lwhere we introduced the abbreviations
1 X αk−1 −klω ,∆l =p k αk5Γl =1 X αk+1 −klω .
p k αk(89)

Alternatively, since the models will be written in terms of the clock operators, we could use the term axial
next-nearest neighbour clock (ANNNC) model [70].

15SciPost Phys. Core 4, 027 (2021)

0

Hence, the last two terms in (87) only produce contributions of the form τlj τlj+1 , as was intended. We will not write down the explicit expansion, since it is tedious and not insightful.
Instead, in the next sections we will discuss several explicit examples. Doing so we obtain ageneral complex Z3 -ANNNP model. Furthermore, we rediscover the known frustration-freeline [3, 14] in the Z3 case, with purely real coefficients. Finally, we discuss a frustration-freeline for p = 2q even, of which the original ANNNI model (22) is the simplest representativeand Z4 - and Z6 -ANNNP examples are given below.

6.1

Z3 -ANNNP model: with complex coefficients

The simplest non-trivial example (besides ANNNI) we can derive with this construction is theZ3 -ANNNP. The most general deformation for Z3 is


1
r ,
(90)
mj = 
swith the corresponding C j, j+1 determined by (86). The deformed Hamiltonian takes the simple
form

X
f
†
†
σ j σ j+1 + (τ j + τ j+1 ) + g1 τ j τ j+1 + g2 τ j τ j+1 + h.c. + ε ,H̃ = −(91)2
jwhich is also the quantum limit of the axial next-nearest neighbour Potts model [14]. Since theoperators σ j and τ j are not self-adjoint, more terms and coefficients than in the original ANNNI
model (22) appear. The similarity with the ANNNI model is exemplified by the following
identifications:
ANNNI modelσ xj σ xj+1
σzj

→
→

Z3 -ANNNP modelσ j σ†j+1
τjσzj σzj+1→

τ j τ j+1 , τ j τ†j+1

The coefficients generated by the deformation (90) are in general complex


 1
2
r
s2
2 
∗ s
∗r
2 rs + ω 2 + ω 2 −
+ω
+ω
,
f =
9
s
r
rs
s
r
  2

 2


r
s
r
1
2
∗ s
+ 2 +ω
+ 2 + rs +,g1 = − ω
9
s
r
r
s
rs





1
1
s2 r 2
s
r
g2 =
3+
+ +
− 2 rs + 2 + 2 .
9
rs
r
s
r
s

(92)
(93)
(94)

Even though there is some elegance in the generality of this model, these complex coefficientsare not very practical. Therefore in the next sub-sections we discuss two specific cases.
6.1.1 Z3 -ANNNP model: real coefficients reproducing Reference [3]The first example features purely real coefficients. This model was originally obtained by direct
calculation by Mahyaeh and Ardonne [3]. We rediscover it by considering the deformation(90) with s = r, ie,




1
r
 , L j, j+1 = σ j − σ j+1 , C j, j+1 = K j K j+1 , k j =  1
 , (95)r
mj = 
−1r
r
16

SciPost Phys. Core 4, 027 (2021)

such that the coefficients become2(1 + 2r)(1 − r 3 )2(1 + r + r 2 )2
,
ε
=
,
9r 2
9r 2
2(1 − r)2 (1 + r + r 2 )(1 − r)2 (1 − 2r − 2r 2 )g1 = −
,
g
=
.
2
9r 2
9r 2
f =

(96)
(97)

The exact ground states originally constructed in Reference [3] follow by direct applicationof Theorem 1. Furthermore, in Appendix B.2 we proveqthat the model possesses an energy
3p
gap above these ground states at least in the interval
2 2 − 2 ≈ 0.3483 ® r ® 3.9912.
p
Finally we note that for r = ( 3 − 1)/2 ≈ 0.366 the model (91) simplifies as the parameterg2 vanishes.
6.1.2 Z3 -ANNNP with ground state deformation of Z3 -XY model
For the second example we consider the deformation that we encountered before for Z3 -XY,
namely s = r 2 ,


1
.
r
mj = 
(98)
r2
Thus the deformed ground states are identical to the ones for θ = 0 discussed in Section 5.3.
However, due to the non-trivial choice for C j, j+1 the Hamiltonian will differ, specifically weobtain (91) with the coefficientsp(1 − r 3 ) (1 − r 3 ) + 3 3i(1 + r 3 )(1 − r 3 )2
∗
,
g
=
−2ωg
,
g
=
−
.
(99)
f =ω
1
2
2
9r 3
9r 3
The coefficient g1 can be chosen to be real via a gauge transformation, ie, a permutation of
diagonal elements of m j .

6.2

Z p -ANNNP model: most general real coefficients

For general Z p we discuss the case when all coefficients take real values. From (88) and (89)0
we recognise that the coefficient of τlj τlj+1 is ∆l Γl 0 + Γl ∆l 0 . This is real for example if Γl∗ = ∆l ,which yields the constraints (recall that αk > 0)α−k ≡ α p−k = αk ,(100)

for all k. Thus there are (p − 1)/2 real degrees of freedom for p odd and p/2 for p even. The
expansion is still not in a compact form. In Section 6.3 we will discuss a Hamiltonian with acompact form for p even. For p odd we did not obtain a simple compact form, except for thecase p = 3 discussed in the next section.
The condition (100) has another consequence. Under charge conjugation
σ j → σ†j ,τ j → τ†j ,(101)

K j → M j−1 σ†j M j σ j .

(102)

we see that
M j−1 σ†j M j σ j → K j ,In this particular case the Hamiltonian (87) is invariant under charge conjugation, and together with the Z p -symmetry generated by ω P , the full symmetry group is the dihedral group
D p [71, 72]. Note that for p = 3 the dihedral group is isomorphic to the symmetric group S3of all permutations.
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SciPost Phys. Core 4, 027 (2021)

6.3

Z2q -ANNNP model: compact form with real coefficients

For even p = 2q it is possible to construct a model depending on a single parameter which
possesses real coefficients and a simple closed form. We start with
m j = diag(1, r, . . . , r q−1 , r q , r q−1 , . . . , r) ,k j = diag(r, . . . , r, r −1 , . . . , r −1 ) .

(103)

Using Equation (67) we see that
M j−1 σ†j M j σ j
Kj

=
=

p−1
X



r −1 + (−1)l r θ q (l)τlj ,

(104)



(−ω)l r −1 + (−1)l r θ q (l)τlj ,

(105)

l=0
p−1
X
l=0

where


1

2 ,
1
θ q (l) =ω−kl = 0 ,
2q k=1 1 Pqif l = 0 ,

q
X

2q

k=1

if l even 6= 0 ,
ω

−kl

,

(106)

if l odd .

Note that both (104) and (105) only contribute odd powers of τ (or the identity), hence thelast two terms in (87) can only give odd powers of τ j -operators. The full expression becomes−1 †σ j+1 M j+1 σ j+1
M j−1 σ†j M j σ j k j+1 + k j M j+1
X




0
0
0
=
(−ω)l + (−ω)l r −1 + (−1)l r r −1 + (−1)l r θ q (l)θ q (l 0 )τlj τlj+1 .

(107)

l,l 0

Let us consider the different terms individually. First, the term with l = l 0 = 0 turns into anenergy shift given by(r + r −1 )2
ε=
.
(108)
2
f

Second, the terms with l = 0 or l 0 = 0 turn into a magnetic-field term of the form − 2 (τlj +τlj+1 )for odd l, with the prefactor given byq

f = −(r −2 − r 2 )(1 − ωl )θ q (l) = r −2 − r 2
r 2 − r −2 X −kl
ω − ω−(k−1)l =
.
2q
q
k=1

(109)

0

Finally, the remaining terms with l, l 0 6= 00 yield the terms Ull 0 τlj τlj+1 with20Ul l 0 = − ωl + ωl r − r −1 θ q (l)θ q (l 0 ) .

(110)


∗
0
Note that ωl θ q (l)θ q (l 0 ) = ωl θ q (l)θ q (l 0 ) and therefore Ull∗ 0 = Ull 0 = Ul 0 l , such that the full
local Hamiltonian becomes
p−1 
p−1
X

 f X
0
H̃ j, j+1 = − σ j σ†j+1 + σ†j σ j+1 −τlj + τlj+1 +Ull 0 τlj τlj+1 + ε .
2 l=1
0
l odd

l,l =1l,l 0 odd

We note that the Hamiltonian for even p is invariant under r → 1/r and τ → −τ.

18

(111)

SciPost Phys. Core 4, 027 (2021)

˜
∆

2
10−1

1.5

10−2
0.2

0.3

0.4

0.5

1
0.5

DMRG
4r 2
r 2 +r −2

0
0.2

0.4

0.6

0.8

1

r
˜ above the four-fold degenerate ground states |Ψ̃i 〉 of the
Figure 1: Energy gap ∆frustration-free ANNNP model (112). The dots show the energy gap obtained byextrapolating the finite-size data for L = 64, 76, 88, 100 to the thermodynamic limit.
The dashed line is the lower bound for the energy gap proven to exist in Appendix B.3.
Inset: Zoom in on the small-r region, logarithmic scale.

6.4

Z4 -ANNNP model

The first new non-trivial example originating from the construction of the previous section is
obtained for p = 4. In this case the local Hamiltonian becomes remarkably simple
f†H̃ j, j+1 = − σ j σ j+1 + (τ j + τ j+1 ) − Uτ j τ j+1 + h.c. + ε ,(112)
2with the parametersf =r −2 − r 2,
2

U=(r − r −1 )2,4

ε=(r + r −1 )2
.
2

(113)

Note the absence of terms like τ2j , τ j τ2j+1 and τ j τ†j+1 , in contrast to the frustration-free Z3 ANNNP model (91). The correlation functions in the four-fold degenerate ground states |Ψ̃i 〉are identical to the ones in the ANNNI model, see Equation (25),〈Ψ̃i |σ j σ†j 0 |Ψ̃i 〉〈Ψ̃i |Ψ̃i 〉=4.
(r + r −1 )2(114)

˜ above the ground
In Appendix B.3 we prove that the model (112) possesses an energy gap ∆states. More specifically, we show that the lower bound for the gap in the thermodynamic limit
4 min(r 2 ,r −2 )˜ For completeness in Figure 1 we compare this to numerical results
is given by r 2 +r −2 ≤ ∆.
for the energy gap. The latter were obtained by extrapolating finite-size data from system sizes
L = 64, 76, 88, 100 to L → ∞, with the finite-size results being calculated by employing thedensity matrix renormalisation group (DMRG) method [39, 73] using the TeNPy [74] library.
Closerinspection of the parameters (113) shows that they satisfy the relationpf = 2 U(1 + U), which is identical to the relation along the Peschel–Emery line in the ANNNI
model. This points towards a closer relation between the models (112) and (22), which we
discuss in the following. In fact, even away from the frustration-free line one can map the Z4 ANNNP chain to two decoupled ANNNI chains. For simplicity we consider an infinitely long
19

SciPost Phys. Core 4, 027 (2021)

system (ie, we ignore the boundary conditions) and drop the constant energy shift ε; thus(112) turns into the HamiltonianX
HANNNP = −σ j σ†j+1 + f τ j − Uτ j τ j+1 + h.c. .
(115)
jIntroducing the dual operators viaσ†j σ j+1 → τ̃ j ,τ j → σ̃ j−1 σ̃†j ,(116)

which satisfy the clock algebra (60) with p = 4, we can rewrite this as
X

dual
HANNNP=−τ̃ j + f σ̃ j σ̃†j+1 − U σ̃ j σ̃†j+2 + h.c. .

(117)

j
x/z

Next we introduce two sets of Pauli matrices σi, j , i = 1, 2, per lattice site j, and consider the
mapping [72, 75, 76] xx σ1, j − iσ2,
j
i π4
z
z
σ̃ j = e, τ̃ j + τ̃†j = σ1,(118)
pj + σ2, j .
2
From the second relation in (118) we can already infer that the τ̃ j -terms are mapped to atransverse magnetic field on the Ising ladder. For the other terms, we use the following simple
identityxxxxσ̃ j σ̃†j+ j 0 + h.c. = σ1,
j σ1, j+ j 0 + σ2, j σ2, j+ j 0 .

(119)Thus the dual of the Z4 -ANNNP model can be written as the sum of two decoupled ANNNI
chains
dual
dual
dual
,
(120)
+ HANNNI,2= HANNNI,1
HANNNP
withdual=−
HANNNI,iXσzi, j + f σi,x j σi,x j+1 − Uσi,x j σi,x j+2 .

(121)

j

Performing another duality transformation (121) can be brought into the form (22) discussed
in Section 3.2. The condition for the parameters f and U to be on the frustration-free line
directly turns into the Peschel–Emery line for the two ANNNI models.

6.5 Z6 -ANNNP modelInterestingly, in the case p = 6 the deformation (103) leads to another rather simple modelwith the local Hamiltonian



f
1
1
H̃ j, j+1 = − σ j σ†j+1 +τ j + τ3j + τ j+1 + τ3j+1
2
2
2



1
1
1
1
−U τ j τ j+1 + τ j τ3j+1 + τ3j τ j+1 − τ j τ†j+1 + τ3j τ3j+1 + h.c. + ε,
4
4
2
8
(122)
where the parameters are given byf =r −2 − r 2
,
3

U=

2
(r − r −1 )2 ,9ε=(r + r −1 )2
.
2

(123)
0

We note that even though the Z6 -symmetry allows a wealth of terms of the form τlj τlj+1 , along
the frustration-free line the relative prefactors of them are fixed to fairly simple values. In
20

SciPost Phys. Core 4, 027 (2021)

˜
∆

1
10−2

0.8
0.6

10−3
10−4

0.2

0.3

0.4

0.4
0.2
DMRG
lower bound

0
0.2

0.4

0.6

0.8

1

r

˜ above the six-fold degenerate ground states of the frustrationFigure 2: Energy gap ∆free ANNNP model (122). The dots show the energy gap obtained by extrapolating
the finite-size data for L = 64, 76, 88, 100 to the thermodynamic limit. The dashedline is the lower bound for the energy gap obtained in Appendix B.4.
Figure 2 we show the energy gap above the six-fold degenerate ground state. The numericalresults were obtained by extrapolation from finite-size data, they clearly indicate the existenceof a finite energy gap along the frustration-free line. In addition, in Appendix B.4 we provethat the model is gapped at least in the interval 0.5754 ® r ® 1/0.5754 ≈ 1.7379. We note inpassing that using more advanced methods for open boundary conditions [52] it is possible toenlarge the region for which the existence of a finite energy gap can be proven. However, theobtained lower bounds are found to be quite small (< 10−5 ).

7Discussion

We have presented a constructive approach to understand and derive one-dimensional
frustration-free spin models. Starting from a simple point, for example a classical system,
we derived the corresponding frustration-free quantum models and their exact ground states.
We have shown that many known frustration-free spin-1/2, spin-1 and Z p -clock models canbe understood in this framework on an equal footing. Hence our approach provides an overarching framework for many frustration-free systems.
More specifically, the approach allowed us to connect two distinct frustration-free Z3 -clock
models recently introduced by Iemini et al. [2] and Mahyaeh and Ardonne [3]. As we haveshown, both models can be interpreted as different deformations of the classical three-state
Potts chain, see Figure 3 for an illustration of their relation. As a side remark, we analyticallyshowed that the energy gap remains finite in a finite region around the classical point for bothmodels. This in particular implies that both models (or their parafermion analogues) are in thesame (topological) phase. Furthermore, we have constructed several new frustration-free Z p clock models, including Z4 - and Z6 -generalisations of the Peschel–Emery line of the original
ANNNI chain.
We stress that the list of frustration-free clock models considered above is by no means
extensive. On the contrary, the examples discussed here should be regarded as a proof ofprinciple on how to apply the general construction. Several generalisations come to mind:
First, one may consider chiral classical models [77, 78] as starting points in the deformation
construction. However, since in this case the local Hamiltonians are no longer given by simple
projectors, the deformed Hamiltonians so obtained may become quite complicated. Second,
21

SciPost Phys. Core 4, 027 (2021)

ANNNP-deformation [3], Sec. 6.1.1
0
1
0
1
r
A , kj = @mj = @ rr<latexit sha1_base64="sPW4aYdRgr+smlT3rbZYzwQHNg0=">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</latexit>

gapped regions(see App. B.2)

1
r 1

1
A

r ⇡ 3.9912<latexit sha1_base64="aSc8KqjYIdOkWj2hEyDaLDvZS8M=">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</latexit>r ⇡ 1.7560<latexit sha1_base64="YdxT+ykL+Sc/SFPAkFyv3ed1uRY=">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</latexit>r ⇡ 0.5695<latexit sha1_base64="+mancnyVeW6biFm71TiqXFCXL54=">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</latexit>

XY-deformation [2], Secs. 5.2 and 5.3
0
1
0
1
1
1
@
A
@
r
1 A
mj =
, kj =
r2
1<latexit sha1_base64="Nh1XTt05tVJEioGvIxqHJ/mNpSs=">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</latexit>

classical Potts model (64)Lj,j+1 = j
j+1<latexit sha1_base64="YZwp/H/rMIuBzboWl//O0osuarY=">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</latexit><latexit sha1_base64="opMb7LTJCdSALGA9/0WlfNdvWsQ=">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</latexit>

r=1

r ⇡ 0.3483<latexit sha1_base64="W+o1Zhdl72GW6+Luv14QPqxcN9w=">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</latexit>

Figure 3: Schematic sketch of the relation between the two frustration-free Z3 -clock
models introduced by Iemini et al. [2] (see Sections 5.2 and 5.3) and Mahyaeh andArdonne [3] (see Section 6.1.1). Both models can be obtained as deformations of theclassical three-state Potts chain (red dot) using the local deformations m j and central
term k j depending on the parameter r. The green lines indicate the regions in which
the systems are proven to be gapped in Appendix B.2. In particular, within this region
the two models can be connected without closing the energy gap, implying that they
are in the same phase.
in this paper we have kept the considered deformations to be homogeneous, a restriction thatis not required by Theorem 1. Thus our results can be extended to inhomogeneous systems.
Third, another generalisation would be to relax the requirement for the operator C j, j+1 to
be positive definite. In such a case, the ground states of the undeformed model are no longer
transformed into ground states of the new model. However, they will still be exact eigenstates,
potentially in the middle of the spectrum, and thus may be relevant in the context of quantum
many-body scars [79–84].

Acknowledgements
We thank Floris Elzinga, Vladimir Korepin, Marius Lemm and Iman Mahyaeh for useful discussions and correspondence. H. K. was supported in part by JSPS Grant-in-Aid for Scientific
Research on Innovative Areas No. JP18H04478 and JP20H04630, and JSPS KAKENHI Grant
No. JP18K03445. This work is part of the D-ITP consortium, a program of the Netherlands
Organisation for Scientific Research (NWO) that is funded by the Dutch Ministry of Education,
Culture and Science (OCW).

A

Witten’s conjugation argument

In this appendix we recall Witten’s original conjugation argument on the ground-state degeneracy of supersymmetric Hamiltonians. Consider two supercharges Q and Q† as well as a

22

SciPost Phys. Core 4, 027 (2021)

Hamiltonian H satisfying
Q2 = (Q† )2 = 0 ,H = Q†Q + QQ† .

(124)

First we note that any zero-energy ground state |ψ〉 of H is annihilated by both Q and Q† .
Furthermore, it is not possible to obtain |ψ〉 by action of Q, ie, |ψ〉 6= Q |φ〉 for any state|φ〉. (To see this assume |ψ〉 = Q |φ〉. But since |ψ〉 is a zero-energy ground state we have
0 = Q† |ψ〉 = Q†Q |φ〉 which implies 〈φ| Q†Q |φ〉 = kQ |φ〉 k2 = 0 and thus Q |φ〉 = 0 in
contradiction with the assumption that |ψ〉 is a ground state.)
Now let us consider the deformed/conjugated operators Q̃ = MQM −1 , Q̃† = (Q̃)† andH̃ = Q̃†Q̃ + Q̃Q̃† with M being invertible. Obviously, if |ψ〉 is a ground state of H, the deformed
state |ψ̃〉 = M |ψ〉 is annihilated by Q̃. Furthermore, |ψ̃〉 cannot be written as |ψ̃〉 = Q̃ |φ̃〉for any |φ̃〉, since this would imply that |ψ〉 = QM −1 |φ̃〉 in contradiction with the assumptionthat |ψ〉 was a ground state of H. Thus |ψ̃〉 is a ground state of H̃ establishing a one-to-one
correspondence between the ground-state manifolds of H and H̃.

B

Energy gap of some Z3 -, Z4 - and Z6 -models

The conjugation argument does only provide information about the ground-state manifold. Inorder to obtain information about the energy gap above it, additional techniques have to be
employed. In Appendix B.1 we recall Knabe’s method [54], which was originally applied to
the AKLT model with periodic boundary conditions. This is then applied in Appendices B.2,
B.3 and B.4 to prove the existence of an energy gap in specific Z3 -, Z4 -, and Z6 -models.

B.1

Knabe’s method

We consider a system with N sites, open boundary conditions and the Hamiltonian
HN =

N
−1
X

P j, j+1 ,

(125)

j=1

T
with the P j, j+1 being two-site projection operators. We assume j ker(P j, j+1 ) 6= {0}, ie, theground state is at zero energy, and denote the energy gap of H N by ∆N . Then we have
Theorem 2 (Knabe’s method [54]). For the projector Hamiltonian H N the gap above the groundstate (∆N ) is bounded from below by


m−1
1
∆N ≥
min
{∆
}
−
,
(126)
0
m
m − 2 m0 =2,...,mm−1where ∆m0 denotes the gap of the m0 -site, sub-system Hamiltonian
h j,m0 =

0
j+m
X−2

Pk,k+1 .

(127)

k= j

Proof. Note that H N is positive semi-definite, therefore H N2 ≥ ∆N H N . In other words, if we
obtain the above inequality with ∆N , the Theorem is proven. We have the analogous statement
for h j,m0 , h2j,m0 ≥ ∆m h j,m0 , and moreover realise that P j,2 j+1 = P j, j+1 and [P j, j+1 , Pk,k+1 ] = 0 for| j − k| > 1.

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SciPost Phys. Core 4, 027 (2021)

To prove the bound, we first expand H N2
H N2 =

N−1
X

h2j,2 +m−2
X N −m
X0 −1
m0 =1

j=1

j=1

m−2
X

X


h j,2 h j+m0 ,2 + h.c. +h j,2 hk,2(128)

| j−k|>m−2

m − m0 − 1≥ HN +
m−2
m0 =1

N −m
X0 −1


h j,2 h j+m0 ,2 + h.c. ,

(129)

j=1

with the second step following from the fact that h j,2 hk,2 is positive semi-definite for |k− j| > 1.
This can be further reduced to
N −m+1
m−1
X
X


1 
h2j,m +
h21,m0 + h2N −m0 +1,m0 − (m − 1)H N (130)
H N2 ≥ H N +
m−2
j=1
m0 =2




N −m+1
m−1
X
X

m−1
1 
≥ 1−
HN +
∆m
h j,m +
∆m0 h1,m0 + hN −m0 +1,m0  . (131)
m−2
m−2
0
j=1
m =2
Because we have the expansion

HN =

1 
m−1

N −m+1Xj=1

h j,m +m−1Xh1,m0 + hN −m0 +1,m0 

(132)

m0 =2

and ∆m0 ≤ 1, the last term of (131) can be simplified to obtain
m−1
1
HN +
min {∆m0 } H N
m−2
m − 2 m0 =2,...,m


1
m−1
min {∆m0 } −HN .
=m − 2 m0 =2,...,mm−1

H N2 ≥ −

(133)
(134)


1
This proves the Theorem with the lower bound ∆N ≥ m−1
min
0 =2,...,m {∆ m0 } −
m
m−2
m−1 .
Remark 3. Given that the models considered here can be viewed as parent Hamiltonians formatrix product ground states, one can apply more powerful tools [33,50,52] to prove the existenceof energy gaps.6 These methods allow one to extend the parameter regions with proven energy
gaps. However, in some cases, Knabe’s method gives us a better lower bound for an energy gap for
fixed parameters. We also note that when analysing the energy gap, special care has to be taken
regarding the treatment of different boundary conditions .
Note that the argument above assumes the Hamiltonian to be the sum of projectors. The
systems studied in our paper do not fit that picture. However, since they are frustration-free
we can still obtain a bound using the following observation:
Corollary 4. For a frustration-free model with an n-fold degenerate zero-energy ground state and˜ k anda p-dimensional local Hilbert space, with p2 > n, we can arrange the two-site eigenvalues ∆2
6

For example, for a so-called injective matrix product state one can prove that the corresponding parent Hamiltonian has a unique ground state and a finite energy gap [33, 34, 50]. However, most of the ground states we havelooked at in this article do not qualify as injective (see, eg, Reference [16] for the ANNNI model), because of thedegeneracy.

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SciPost Phys. Core 4, 027 (2021)

˜k ≤ ∆
˜ l for k < l and ∆˜ 1, . . . , ∆˜ n = 0. Then two-sitenormalised eigenstates |ψ̃k 〉 such that ∆2222
Hamiltonian can be bounded from below as follows,2H̃ j, j+1 =pX2

˜ k |ψ̃k 〉 〈ψ̃k | = ∆
˜ n+1
∆
2
2

k=n+1
˜ n+1 P j, j+1 = ∆˜ 2 P j, j+1 ,
≥∆
2

p
X

2

|ψ̃k 〉 〈ψ̃k | +

k=n+1

p
X


˜k −∆
˜ n+1 |ψ̃k 〉 〈ψ̃k |
∆
2
2

(135)

k=n+1

(136)

˜2 = ∆
˜ n+1 of the frustration-free Hamiltonian H̃ j, j+1 and P j, j+1 denoting the prowith the gap ∆2
jector onto the space orthogonal to its ground-state manifold. The min-max theorem [85] then˜ N of the frustration-free model on N sites
implies for the gap ∆˜N ≥∆˜ 2 ∆N .
∆(137)

˜ aboveThus in order to prove that a frustration-free Hamiltonian possesses an energy gap ∆its ground states in the thermodynamic limit, we proceed as follows: (i) We consider projectors
P j, j+1 onto the space orthogonal to the local ground states on the lattice sites j and j + 1 anddetermine the gap ∆2 above
Pm these ground states. (ii) From that we construct the auxiliary msite Hamiltonian h1,m = j=1 P j, j+1 and determine its energy gap ∆m . (iii) If this gap satisfies
minm0 =2,...,m {∆m0 } > 1/(m − 1), then the auxiliary N -site Hamiltonian H N will have a gap ∆N
˜ of the original frustration-free Hamiltonian issatisfying (126). (iv) Due to (137) the gap ∆bounded from below by


m−1
1
˜
˜
˜
˜
∆ = lim ∆N ≥ lim ∆2 ∆N ≥ ∆2
min {∆m0 } −
.
(138)
N →∞
N →∞
m − 2 m0 =2,...,mm−1
Every m > 2 gives a lower bound on the gap, so the supremum over subsysten sizes is also
a lower bound. Usually, the bound increases for increasing m. Since the computation of ∆mrequires exact diagonalization of a p m × p m matrix, the maximal feasible m is constrained bycomputational resources. In the following appendices we apply this line of argument to several
models.

B.2

Gap in Z3 -models

In order to treat both Z3 -models (84) (for θ = 0) and the models discussed in Section 6.1
within the same framework, we consider the general, diagonal deformation with


1
r ,
mj = 
(139)s
where r, s > 0. For each point in the (r, s)-plane we get a lower bound on the thermodynamic
gap by means of (126), provided that for some feasible m the relevant energy gap of the auxiliary m-site Hamiltonian satisfies minm0 =2,...,m {∆m0 } > 1/(m − 1). Computational resources
allow us to go up to m = 7. In Figure 4 we have depicted the maximal lower bound form = 3, . . . , 7 in the (r, s)-plane obtained from this. Note that this is a lower bound for thegap of the auxiliary projector Hamiltonian. For a particular parent Hamiltonian like (84) and˜ 2 . As long as the local parent Hamiltonian has(91), the true gap depends on the local gap ∆the same degeneracy as the local auxiliary Hamiltonian it is gapped for the same parameter
regime, by virtue of (138). We only consider the triangle s ≤ r ≤ 1, since due to the dihedral
symmetry of the model there is a six-fold symmetry in the (r, s)-plane. The red line denotes
the boundary of the region that is definitively gapped, ie, for all points above this line in the(r, s)-plane it is assured that the full system is gapped in the thermodynamic limit. The blue
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SciPost Phys. Core 4, 027 (2021)

1.0

0.8
0.6
0.4
0.2
0

0.8
0.6

0.4

Gapped

0.2
0.2

0.4

0.6

0.8

1.0

Figure4:Log-log
contour  plot
of
the
lower
bound
1
}
−
maxm=3,...,7 m−1
min
{∆
for
the
deformationinEquation(139).
00m =2,...,m
m
m−2
m−1
For a finite lower bound the system is gapped in the thermodynamic limit N → ∞,ie, all points above the red line yield gapped systems. The blue and green lines
correspond to the ground states of (84) (for θ = 0) and (91), respectively. The staris the special point with g2 = 0.
and green lines correspond to the ground states of (84) (for θ = 0) and (91), respectively,
with the black star indicating the model (91) at the special point g2 = 0. Given the six-foldsymmetry in the (r, s)-plane, we have to be careful how to display the green (1, r, r) and blue
(1, r, r 2 ) lines. For the blue line, note that (1, r, r 2 ) ' (r −2 , r −1 , 1), since the Hamiltonian isinvariant under rescaling of M . Also the freedom in the form of the dihedral symmetry lets us
write (1, r, r 2 ) ' (1, r −1 , r −2 ), permuting the entries. Hence the blue line for r > 1, maps tothe blue line for r < 1 under the symmetry. Using the same reasoning for the green line we
obtain (1, r, r) ' (r −1 , 1, 1) ' (1, 1, r −1 ), mapping (1, r, r) for r > 1 to (1, 1, s) for s = r −1 .
Let us zoom in on the two lines s = r 2 and s = r that correspond to the ground states of(84) and (91) respectively. In Table 1 we list the lower and upper limit r low,up for the gapped
region for different sub-system sizes m. For s = r 2 the upper limit is simply r up = 1/r low , asfollows from the symmetry discussed above. As m increases we see that the region increases
in both directions.
On the other hand, for s = r something peculiar occurs. The lower limit r low is significantly better for mq= 3 than for m = 4, . . . , 7. This lower limit has the exact value ofpr low = 21/4 − 2−1/4 = 32 2 − 2 ≈ 0.3483 The upper limit, on the other hand, does become
more informative as m increases.
In total we deduce that the full system (84) (for θ = 0) is gapped in the thermodynamic
limit for 0.5695 ® r ® 1/0.5695 and (91) for 0.3483 ® r ® 3.9912. In particular this implies
that in this parameter regime the models can be adiabatically connected to the classical modelobtained for r = s = 1 as sketched in Figure 3.

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SciPost Phys. Core 4, 027 (2021)

Table 1: Lower and upper limit r low,up for the gapped regions of the Z3 -XY model(84) and Z3 -ANNNP model (91) as deduced from different sub-system sizes m. The
bold values indicate the extremal values which are stated in the main text.

m
3
4
5
6
7

B.3

s = r 2 (Z3 -XY model)
r low
r up
0.63371.57790.6204
1.61190.60261.65950.58531.7086
0.56951.7560s = r (Z3 -ANNNP model)
r low
r up
0.34832
0.42162.67960.42593.01460.42003.62330.4116
3.9912

Gap in Z4 -ANNNP model

We can apply the same method to analyse the gap of the Z4 -ANNNP model (112). For thismodel it is sufficient to consider m = 3, since∆3 =1 min(r 2 , r −2 )+,2
r 2 + r −2(140)which is strictly larger than 1/2 for 0 < r < ∞. Thus we deduce for the gap in the thermodynamic limit
4 min(r 2 , r −2 )
˜ ≥
∆
.
(141)
r 2 + r −2
Instead of using Corollary 4, the lower bound (141) can also be obtained from the mapping
to two decoupled ANNNI chains, together with the lower bound for the energy gap along thePeschel–Emery line of the ANNNI chain obtained in Reference [15].

B.4

Gap in Z6 -ANNNP model

For the Z6 -ANNNP model (122) the condition of ∆3 > 1/2 shows that the model is gapped atleast in the interval 0.5754 ® r ® 1/0.5754, where we have used the invariance of the model
under r → 1/r. The region does not improve for m = 4, 5 and higher sub-systems sizes are
not accessible with our current resources.

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