Topologically ordered phases in 2 + 1 dimensions are generally characterized by three mutually related features: fractionalized (anyonic) excitations, topological entanglement entropy, and robust ground state degeneracy that does not require symmetry protection or spontaneous symmetry breaking. Such a degeneracy is known as topological degeneracy and can be usually seen under the periodic boundary condition regardless of the choice of the system sizes *L*_{1} and *L*_{2} in each direction. In this work, we introduce a family of extensions of the Kitaev toric code to *N* level spins (*N* ≥ 2). The model realizes topologically ordered phases or symmetry-protected topological phases depending on the parameters in the model. The most remarkable feature of topologically ordered phases is that the ground state may be unique, depending on *L*_{1} and *L*_{2}, despite that the translation symmetry of the model remains unbroken. Nonetheless, the topological entanglement entropy takes the nontrivial value. We argue that this behavior originates from the nontrivial action of translations permuting anyon species.

## I. INTRODUCTION

In the studies of many-body systems, one is often interested in the properties of ground states and low energy excitations. Ground state degeneracy that does not originate from spontaneous symmetry breaking or fine-tuning of parameters is called topological degeneracy.^{1–3} Such a degeneracy is robust against any local perturbations, including symmetry-breaking ones. In two dimensions, the order of topological degeneracy *N*_{deg} depends on the genus *g* of the manifold on which the system is defined.

In topologically ordered phases with U(1) symmetry (e.g., fractional quantum Hall systems), *N*_{deg} ≥ *q*^{g} when the filling is *ν* = 1/*q*. This degeneracy can be proven by a flux-threading type argument,^{4} assuming the appearance of fractional excitations with U(1) charge 1/*q*. More generally, there usually exist closed string operators that describe processes of creating a pair of anyonic excitations, dragging them apart, and pair-annihilating them again after forming a non-contractible loop. These loops commute with the Hamiltonian but not among them. Non-commutativity of loop operators implies the topological degeneracy. In particular, the topological degeneracy *N*_{deg} on a torus (*g* = 1) is often equal to the number of distinct anyonic excitations. The topological degeneracy is also tied with the topological entanglement entropy,^{5,6} which is given by $Stopo=\u2212logD$, where $D$ is the total quantum dimension and $D2$ is nothing but the number of distinct anyonic excitations for Abelian topological order. Therefore, it is often stated that the ground state degeneracy on torus, anyonic excitations, and topological entanglement entropy appear all at the same time.

In this work, we introduce a family of extensions of the Kitaev toric code^{7,8} to *N*-level spins (*N* = 2, 3, 4, …), which contains an integer parameter *a* (1 ≤ *a* ≤ *N*). The original model corresponds to the (*N*, *a*) = (2, 1) case. The model describes topologically ordered phases when *a* is not a multiple of rad(*N*) (the radical of *N*; see Sec. IV) and phases with no topological order when *a* is a multiple of rad(*N*). In particular, when *N* and *a* are coprime, these phases are characterized by the topological entanglement entropy *S*_{topo} = −log *N*, independent of system sizes *L*_{1}, *L*_{2}.

*N*, the case of

*a*= 1 is the standard $ZN$ toric code

^{8}discussed widely, for example, in Refs. 9–12, which shows topological degeneracy

*N*

_{deg}=

*N*

^{2}regardless of the choice of

*L*

_{1}and

*L*

_{2}. The case of

*a*=

*N*− 1 (

*N*≥ 3) was discussed in Refs. 13–16, although the ground state degeneracy on torus was not fully investigated. When

*N*is odd and

*a*=

*N*− 1, we find that the topological degeneracy occurs only when both

*L*

_{1}and

*L*

_{2}are even,

*N*is a prime number and

*a*is a primitive root modulo

*N*(see Sec. III A). In this case,

*N*

_{deg}is given by (see Sec. V A for the proof)

*L*

_{1}=

*L*

_{2}=

*N*− 1, for which the Hilbert space dimension is $N2(N\u22121)2$ (for example, 11

^{200}for

*N*= 11, for which

*a*= 2, 6, 7, 8 are the primitive roots). It is thus nearly impossible to see the topological degeneracy for a large

*N*in any numerical studies. Therefore, the uniqueness of the ground state for a sequence of

*L*

_{i}cannot be used as a proof of the absence of topological order, although the converse might still be the case: topological degeneracy

*N*

_{deg}> 1 in a sequence of

*L*

_{i}implies a nontrivial topological order. Note that, if an

*open*boundary condition is assumed instead of the periodic one, a unique ground state can be realized even in the original $Z2$ toric code due to the absence of any Wilson loops or constraints among stabilizers.

There is a more famous example, called Wen’s plaquette model,^{17} in which topological degeneracy depends on the system size. There are also more recent examples of this type behavior.^{18–22} However, in these examples, the ground state degeneracy on torus is at least two. Our example demonstrates that there are even cases where the ground state is unique and excitations are all gapped in a sequence of *L*_{i}, despite their nontrivial topological order. It is interesting to contrast with a known theorem about topological quantum field theory (TQFT), according to which the phase is invertible (i.e., no topological order) if *N*_{deg} = 1 on torus (and technically, on sphere as well).^{23} Our example shows that the relation between lattice models and corresponding effective field theories can be quite subtle. We will also show that the degeneracy can be understood in terms of the TQFT if the finite-size torus in the lattice system is viewed as a torus in continuum but with symmetry defect lines (or twisted boundary conditions) corresponding to the translation symmetry action in the low-energy theory.

The rest of this work is organized as follows. We summarize the definition and basic properties of our model in Sec. II. We review basic mathematical facts in number theory in Sec. III. Overall properties of our model for given integers *N* and *a* are summarized in Sec. IV. Then, the ground state degeneracy of the model in topologically ordered phases is studied in Sec. V. The relation of our model to the standard $ZN$ toric code model is clarified in Sec. VI. Topological properties such as the topological entanglement entropy and anyon statistics in our model are discussed in Sec. VII. Finally, we study the cases with no topological order in Sec. VIII. We then conclude in Sec. IX.

## II. DEFINITION OF MODEL

In this section, we explain the definition and basic properties of the $ZN$ toric code. Throughout this work, *N* is an integer greater than 1.

### A. Lattice of *N*-level spins

*N*-level spin is placed on each link of square lattice. See Fig. 1 for the illustration. The action of operators $X\u0302r$ and $Z\u0302r$ on the

*N*-level spin at

**is represented by**

*r**N*-dimensional unitary matrices,

*N*≥ 3, matrices

*X*and

*Z*are not Hermitian and $X\u0302r\u2020\u2260X\u0302r$ and $Z\u0302r\u2020\u2260Z\u0302r$.

*m*

_{i}= 0, 1, …,

*L*

_{i}− 1 (

*i*= 1, 2) and

*L*

_{i}is a positive integer,

**∈ Λ. The sets of vertices $V$ and plaquettes $P$ are given by**

*r**L*

_{1}

*L*

_{2}, and the dimension of the Hilbert space is $N2L1L2$.

### B. Hamiltonian and stabilizers

*i*= 1, 2), defined by

*a*

_{1}and

*a*

_{2}(1 ≤

*a*

_{1},

*a*

_{2}≤

*N*) are important parameters of this model. It is easy to verify that $A\u0302v$’s $(v\u2208V)$ and $B\u0302p$’s $(p\u2208P)$ all commute with each other regardless of

*a*

_{1}and

*a*

_{2}. For brevity, we set

*a*

_{1}=

*a*

_{2}=

*a*in the following, but

*a*

_{1}≠

*a*

_{2}cases can be treated in the same way.

*N*-fold, 1,

*ω*, …,

*ω*

^{N−1}.

### C. A ground state

*ϕ*

_{0}⟩ be the “ferromagnetic” product state, satisfying

_{0}⟩ is a ground state with the energy eigenvalue

*E*

_{GS}= −2

*L*

_{1}

*L*

_{2}. Here,

*N*

_{C}> 0 is the normalization factor given by

*N*

_{C}counts the number of global constraints among the vertex operators of the form

*L*

_{1}

*L*

_{2}, coincides with the total number of spins in the system. Hence, there would be no ground state degeneracy if all stabilizers were independent. Indeed, as we demonstrate in Sec. V, the number of constraints

*N*

_{C}is related to the ground state degeneracy as $Ndeg=NC2$.

_{0}⟩ is translation invariant,

*ϕ*

_{0}⟩ is translation invariant and $P\u0302$ commutes with $T\u0302i$. As we show below, the model has a nonzero excitation gap. Furthermore, all correlation functions of $X\u0302r$ and $Z\u0302r$ are short-ranged. For example,

### D. Quasiparticle excitations

*i*= 1, 2) by

*i*= 1, 2) by

*ω*and $\omega \u2212am1\u2032\u2212m1+1$, respectively. The eigenvalues of other plaquette operators remain +1. In the derivation of these relations, we used the general property of exponents $(zm)n=zmn$ for $z\u2208C$ and $m,n\u2208Z$. Similarly, the state

*ω*

^{−1}for the vertex operators $A\u0302(m1,m2)$ and $A\u0302(m1\u2032+1,m2)$, respectively. String operators along

*x*

_{2}direction also create pairs of electric or magnetic excitations at their ends [see Fig. 1(c)].

*ω*

^{q}(

*q*= 1, 2, …,

*N*− 1) costs an energy

_{pair}can thus be bounded by

These electric and magnetic excitations can be further divided into equivalence classes up to local excitations (i.e., excitations that can be created locally), which are called anyon types. They will be discussed in Sec. VII C.

## III. BASIC FACTS FROM NUMBER THEORY

In this section, we review basic mathematical facts in number theory to set up notations for Sec. IV and below.

### A. Multiplicative order and primitive root

*n*and a positive integer

*a*coprime to

*n*, the multiplicative order of

*a*modulo

*n*is defined as the smallest positive integer

*ℓ*such that

*M*

_{n}(

*a*) in this work. For example,

*M*

_{n}(

*a*) = 1 if and only if

*a*= 1 (mod

*n*). In addition, for

*n*≥ 3,

*M*

_{n}(

*a*) = 2 if

*a*= −1 (mod

*n*). Conversely, the relation in Eq. (44) implies that

*n*and

*a*are coprime. In the following applications, the integer

*n*is chosen to be

*N*itself or a divisor of

*N*that is coprime to

*a*.

The multiplicative order is related to Euler’s totient function *φ*(*n*), which is defined as the number of positive integers smaller than *n* that are relatively prime to *n*. By definition, 1 ≤ *φ*(*n*) ≤ *n* − 1. If and only if *n* is prime, *φ*(*n*) = *n* − 1.

^{25}

*a*

^{φ(n)}= 1 mod

*n*, implies that

*M*

_{n}(

*a*) is a divisor of

*φ*(

*n*). Thus,

*a*that saturate the upper bound, i.e.,

*M*

_{n}(

*a*) =

*φ*(

*n*), are called the primitive roots modulo

*n*. The primitive roots exist if and only if

*n*is 2, 4,

*p*

^{k}, or 2

*p*

^{k}, where

*p*is an odd prime number and

*k*is a positive integer. It follows that, when

*n*is a prime number, there exists an integer

*a*such that

*n*′ is also a positive integer coprime to

*a*. In this case,

*M*

_{nn′}(

*a*) is a multiple of both

*M*

_{n}(

*a*) and

*M*

_{n′}(

*a*) because $aMnn\u2032(a)=1$ (mod

*nn*′) also implies $aMnn\u2032(a)=1$ (mod

*n*) and $aMnn\u2032(a)=1$ (mod

*n*′). In particular,

*n*″ is a multiple of

*n*.

### B. Prime factorization and divisors of *N*

*N*(

*N*≥ 2) can be prime factorized into

*p*

_{j}’s (

*j*= 1, 2, …,

*n*) are prime numbers and

*r*

_{j}’s are positive integers. The radical of

*N*is defined as the product of all distinct prime factors of

*N*,

*N*by

*D*

_{N},

*N*, and rad(

*N*), for example.

*p*

_{j}’s of

*N*in Eq. (48) in such a way that

*N*that is coprime to

*a*is given by

*p*,

*q*,

*r*, …) for integers

*p*,

*q*,

*r*, … represents their greatest common divisor. By definition,

*p*/gcd(

*p*,

*q*) and

*q*/gcd(

*p*,

*q*) are positive integers. Since

*N*

_{a}is a multiple of any

*d*∈

*D*

_{N}that is coprime to

*a*, $MNa(a)$ is a multiple of

*M*

_{d}(

*a*).

## IV. CLASSIFICATION OF PHASES IN THE (*N*, *a*) MODEL

Our model describes two distinct types of phases with or without topological order depending on whether *a* (1 ≤ *a* ≤ *N*) is a multiple of rad(*N*) or not. Here, we provide a brief summary of the main features of the two phases.

**Case 1**: When*a*is not a multiple of rad(*N*), our model exhibits topological degeneracy for some sequences of*L*_{1}and*L*_{2}. The ground state degeneracy on the torus is given byFor example, $Ndeg=Na2$ when both(55)$Ndeg=[gcd(aL1\u22121,aL2\u22121,Na)]2.$*L*_{1}and*L*_{2}are multiples of $MNa(a)$ and*N*_{deg}= 1 when*L*_{1}and*L*_{2}are not simultaneously multiples of*M*_{d}(*a*) for any*d*∈*D*_{N}coprime to*a*, except for*d*= 1. Correspondingly, there are $Na2$ species of anyons. This class thus falls into topologically ordered phases. It contains the important class of*a*being coprime to*N*. Examples include the cases of*a*= 1 and*a*=*N*− 1 previously discussed in the literature. The size dependence of*N*_{deg}can be understood from the translation symmetry action on the anyon excitations, which will be discussed in Sec. VII D.**Case 2**: When*a*is a multiple of rad(*N*), the ground state is unique regardless of the choice of*L*_{1}and*L*_{2}. The model thus realizes a trivial phase with regard to topological orders, but it still might be a nontrivial symmetry protected topological phase. As simplest examples, we discuss the cases of*N*=*a*and*N*=*a*^{2}.

## V. GROUND STATE DEGENERACY IN TOPOLOGICALLY ORDERED PHASES

In this section, we show that, when *a* is not a multiple of rad(*N*), the order of ground state degeneracy *N*_{deg} is greater than one for some sequences of *L*_{1} and *L*_{2}.

### A. The case of gcd(*N*, *a*) = 1

We start with the simplest case where *a* is coprime to *N*.

#### 1. When both *L*_{1} and *L*_{2} are multiples of *M*_{N}(*a*)

Suppose that both *L*_{1} and *L*_{2} are multiples of *M*_{N}(*a*) so that $aL1=aL2=1$ mod *N*. In this case, the ground state degeneracy and the low-energy excitations are basically the straightforward extension of the original toric code. For example, when *a* = 1, *M*_{N}(*a*) = 1 and the assumption automatically holds for any *L*_{1} and *L*_{2}. In contrast, when *N* ≥ 3 and *a* = *N* − 1, *M*_{N}(*a*) = 2 and both *L*_{1} and *L*_{2} need to be even.

*L*

_{1}and

*L*

_{2}are multiples of

*M*

_{N}(

*a*), there are two sets of global constraints among the stabilizers $A\u0302v$’s and $B\u0302p$’s,

*N*

_{C}in Eq. (30) is

*N*. In the derivation, we used definitions in Eqs. (16) and (17) and the periodic boundary conditions, such as $X\u0302(L1\u221212,m2)=X\u0302(\u221212,m2)$ and $X\u0302(m1,L2\u221212)=X\u0302(m1,\u221212)$. These constraints imply that not all vertex operators and plaquettes operators are independent. For example, the eigenvalues of $A\u0302v0$ [

*v*

_{0}:= (0, 0)] and $B\u0302p0$ $[p0:=(L1\u221212,L2\u221212)]$ are automatically fixed once the eigenvalues of other $A\u0302v$’s and $B\u0302p$’s are chosen.

The vertex operators $A\u0302v$ ($v\u2208V$,

*v*≠*v*_{0}) and the plaquette operators $B\u0302p$ ($p\u2208P$,*p*≠*p*_{0}). There are in total $N2(L1L2\u22121)$ different combinations of eigenvalues.Closed string operators $Z\u0302(i)$ (

*i*= 1, 2). There are*N*^{2}different combinations of eigenvalues.

_{0}⟩ in Eq. (27), which has the eigenvalue +1 for all of these 2

*L*

_{1}

*L*

_{2}operators, one can generate all $N2L1L2$ states in the Hilbert space by using the open string operators illustrated in Figs. 2(c) and 2(d) and the closed loop operators $X\u0302(i)$ (

*i*= 1, 2). They can be distinguished by $N2(L1L2\u22121)\xd7N2=N2L1L2$ distinct combinations of eigenvalues of these stabilizers. In particular, all degenerate ground states can be written as $[X\u0302(1)]j1[X\u0302(2)]j2|\Phi 0\u232a$ (

*j*

_{1},

*j*

_{2}= 0, 1, …,

*N*− 1), which has the eigenvalue $\omega ji$ of $Z\u0302(i)$. Hence, the order of topological degeneracy is

The closed loop operators in Eqs. (58) and (59) create a pair of magnetic excitations at $xi=\xb112$, dragging the one at $xi=12$ all the way to $xi=Li\u221212=\u221212$ and annihilating them in pair. The pair annihilation requires that magnetic excitations with eigenvalues *ω* and *ω*^{−1} meet. This is possible only when *L*_{i} is a multiple of *M*_{N}(*a*).

#### 2. When *L*_{1} or *L*_{2} is not a multiple of *M*_{N}(*a*)

Next, we consider the case where *L*_{1} or *L*_{2} is not a multiple of *M*_{N}(*a*). Without loss of generality, we assume that *L*_{1} is not a multiple of *M*_{N}(*a*).

*p*, respectively. The second factor creates magnetic excitations with eigenvalues

*ω*and $\omega \u2212aL1\u22121\u2212m1$ at the plaquettes

*p*and $(L1\u221212,m2+12)$, respectively. Combining these two effects, the operator $X\u0302p(1)$ create a

*single*magnetic excitation with eigenvalue $\omega 1\u2212aL1$ at the plaquette

*p*. In fact, $X\u0302p(1)$ satisfies

*L*

_{1}is not a multiple of

*M*

_{N}(

*a*), $\omega 1\u2212aL1\u22601$.

*single*electric excitation with eigenvalue $\omega aL1\u22121\u22601$ at the vertex

*v*.

*N*. In this case,

*N*. Therefore, the eigenvalue of the plaquette operator $B\u0302p$ (the vertex operator $A\u0302v$) can be freely controlled by $[X\u0302p(1)]\u2113$ $([Z\u0302v(1)]\u2113)$ without affecting others, implying the absence of global constraints involving $B\u0302p$ or $A\u0302v$, such as the ones of the form in Eq. (31) (i.e.,

*N*

_{C}= 1).

_{0}⟩ satisfying Eqs. (28) and (29), one can generate all $N2L1L2$ states in the Hilbert space by successively applying $Z\u0302v(1)$’s and $X\u0302p(1)$’s. In particular, there is no state other than |Φ

_{0}⟩ that has eigenvalue +1 for all vertex operators and plaquette operators. This proves the uniqueness of the ground state

*N*. This condition is satisfied, for example, (i) when

*N*is prime and

*L*

_{1}is not a multiple of

*M*

_{N}(

*a*) (in this case, $aL1\u22121\u22600$ mod

*N*) and (ii) when

*N*is odd,

*a*=

*N*− 1, and

*L*

_{1}is not a multiple of

*M*

_{N}(

*a*) = 2 (in this case, $aL1\u22121=\u22122$ mod

*N*). This completes the proof of Eqs. (1) and (2).

The gap to the first excited states is given by Δ_{q} in Eq. (41), although these states are created by nonlocal operators $X\u0302p(1)$ and $Z\u0302v(1)$. Local excitations are still given by pairs of magnetic excitations and electric excitations, for which the exaction gap is bounded by Eq. (43).

### B. General case

Next, we discuss the most general case where $a/rad(N)\u2209Z$ but *a* is not necessarily coprime to *N*. In this case, we will see that *N*_{a} in Eq. (52) plays the role of *N* in the above discussion.

*n*∈

*D*

_{N}is a parameter specified shortly. In order to set the products in Eqs. (75) and (76) to be 1, we need

*d*

_{i,a}∈

*D*

_{N}(

*i*= 1, 2) by

*d*of

*N*such that (i)

*d*is coprime to

*a*and (ii)

*L*

_{i}is a multiple of

*M*

_{d}(

*a*). For example,

*d*

_{i,a}=

*N*when

*a*= 1 and

*d*

_{i,a}= 1 when $aLi\u22121$ is coprime to

*N*. The smallest positive integer

*n*∈

*D*

_{N}satisfying Eq. (77) is given by

*n*′ = 0, 1, 2, …,

*d*

_{a}− 1, suggesting that

*N*

_{C}in Eq. (30) is given by

*d*

_{a}. These constraints imply that not all vertex operators and plaquettes operators are independent. For example, the eigenvalues of $A\u0302v0na$ and $B\u0302p0na$ can be automatically fixed once the eigenvalues of other $A\u0302v$’s and $B\u0302p$’s are chosen. Then, as the set of independent stabilizers commuting with $H\u0302$, one can choose the following set of operators:

The vertex operators $A\u0302v$ ($v\u2208V$,

*v*≠*v*_{0}) and the plaquette operators $B\u0302p$ ($p\u2208P$,*p*≠*p*_{0}). There are in total $N2(L1L2\u22121)$ different combinations of eigenvalues.The residual free parts of $A\u0302v0$ and $B\u0302p0$. The eigenvalues of these operators can be written as $\omega x+da\u2113$ (

*ℓ*= 0, 1, …,*n*_{a}− 1), where the value of*x*(*x*= 0, 1, …,*d*_{a}) is automatically determined by the constraints in Eqs. (81) and (82). Hence, there are effectively $na2$ different combinations of eigenvalues.Loop operators $[X\u0302(1)]n1,a$ and $[Z\u0302(1)]n1,a$ (or $[X\u0302(2)]n2,a$ and $[Z\u0302(2)]n2,a$), where

*n*_{i,a}:=*N*/*d*_{i,a}(*i*= 1, 2). Their eigenvalues are*d*_{i,a}-fold: $\omega ni,aj$ (*j*= 0, 1, …,*d*_{i,a}− 1), which include $\omega naj\u2032(j\u2032=0,1,\u2026,da\u22121)$ as a subset. As detailed below, only $da2$ different eigenvalues of these operators can be manipulated without affecting the eigenvalues of other stabilizers.

Hence, starting from the ground state |Φ_{0}⟩ in Eq. (27), one can generate all $N2L1L2$ states in the Hilbert space, which can be distinguished by $N2(L1L2\u22121)\xd7na2\xd7da2=N2L1L2$ distinct combinations of eigenvalues of these stabilizers. This implies that the order of the ground state degeneracy is given by Eq. (55).

*v*≠

*v*

_{0}) and $B\u0302p$ ($p\u2208P$,

*p*≠

*p*

_{0}). The remaining operators satisfy the following algebra:

*α*

_{i}(0 ≤

*α*

_{i}≤

*N*− 1) is defined by

*a*and a multiple of

*d*

_{i,a}= gcd(

*α*

_{i},

*N*

_{a}). All these operators commute with $A\u0302v$ ($v\u2208V$,

*v*≠

*v*

_{0}) and $B\u0302p$ ($p\u2208P$,

*p*≠

*p*

_{0}) and thus do not change their eigenvalues.

#### 1. Case 1: *α*_{1} = *α*_{2} = *0*

When *α*_{1} = *α*_{2} = 0, *N* is coprime to *a* and both *L*_{1} and *L*_{2} are multiples of *M*_{N}(*a*). This case was covered in Sec. V A 1.

#### 2. Case 2: Either *α*_{1} = *0* or *α*_{2} = *0*

Next, we discuss the case when either *α*_{1} = 0 or *α*_{2} = 0. Without loss of the generality, here, we assume *α*_{1} ≠ 0 and *α*_{2} = 0. In this case, *N* is again coprime to *a*, and we have *d*_{1,a} = *d*_{a} = gcd(*α*_{1}, *N*) and *d*_{2,a} = *N*.

*α*

_{1}/

*d*

_{1,a}is coprime to

*n*

_{1,a}=

*N*/

*d*

_{1,a}, there exists an integer

*ℓ*

_{1}(1 ≤

*ℓ*

_{1}≤

*n*

_{1,a}− 1) such that

*n*

_{1,a}=

*n*

_{a}=

*N*/

*d*

_{a}, this is what we needed.

#### 3. Case 3: *α*_{1} ≠ *0* and *α*_{2} ≠ *0*

Finally, we discuss the case when *α*_{1} ≠ 0 and *α*_{2} ≠ 0. We define operators $X\u0302(\u21131,\u21132):=[X\u0302(1)]\u21131[X\u0302(2)]\u21132$ and $Z\u0302(\u21131,\u21132):=[Z\u0302(1)]\u21131[Z\u0302(2)]\u21132$.

*a*and

*α*

_{i}are coprime,

*d*

_{a}in Eq. (80) can also be written as gcd(

*α*

_{1},

*α*

_{2},

*N*). It follows that gcd(

*α*

_{1},

*α*

_{2})/

*d*

_{a}is coprime to

*n*

_{a}=

*N*/

*d*

_{a}. Thus, there exists an integer

*b*

_{0}such that

*b*

_{1}and

*b*

_{2}such that

*ℓ*

_{i}=

*b*

_{0}

*b*

_{i}mod

*n*

_{a}(0 ≤

*ℓ*

_{i}≤

*n*

_{a}− 1). The eigenvalues of $A\u0302v0$ and $B\u0302p0$ can be controlled by $X\u0302(\u21131,\u21132)$ and $Z\u0302(\u21131,\u21132)$,

## VI. RELATION TO THE STANDARD $ZN$ TORIC CODE

In this section, we clarify the relation of our model in Eq. (14) to the $a=1ZN$ toric code with a twisted boundary condition. This connection for a prime *N* is implied by the result in Ref. 26, but our discussion goes more generally whenever *N* and *a* are coprime.

*a*

_{1}=

*a*

_{2}=

*a*. The ground states are still given by those who have eigenvalue +1 for all $A\u0302v$ $(v\u2208V)$ and $B\u0302p$ $(p\u2208P)$, and the ground state degeneracy remains unchanged.

**∈ Λ), whose action on the local spin is given by a unitary matrix**

*r**U*

_{i,j}:=

*δ*

_{j,1+mod[(i−1)a,N]}. This operator satisfies

*ℓ*= 1, 2, …,

*M*

_{N}(

*a*)] should be understood as $X\u0302r\xb1aMN(a)\u2212\u2113$ (recall that $aMN(a)=1$ mod

*N*). The global operator $\u220fr\u2208\Lambda U\u0302r$ is a symmetry of $H\u0302\u2032$ as it commutes with $H\u0302\u2032$. When gcd(

*N*,

*a*) ≠ 1, such a unitary operator does not exist.

*m*

_{1}≤

*L*

_{1}− 2 and 1 ≤

*m*

_{2}≤

*L*

_{2}− 2) to those for

*a*= 1,

*a*

_{1}=

*a*

_{2}= 1, respectively. Therefore, except for boundary terms,

*a*= 1). Boundary terms are given by

## VII. TOPOLOGICAL PROPERTIES IN TOPOLOGICALLY ORDERED PHASES

In Sec. V, we showed that the order of ground state degeneracy under the periodic boundary condition can be 1 depending on the system size. Then, one might suspect that the system is not in a topologically ordered phase. In this section, we show this is not the case by demonstrating nontrivial topological entanglement entropy and anyonic excitations in the system. In addition, we discuss how the size dependence of the ground state degeneracy can be understood by viewing the lattice system as a continuum torus but with lattice translation symmetry defects.

### A. Topological entanglement entropy

*S*

_{topo}of the ground state of our model. We use the Kitaev–Preskill prescription,

^{5}

_{0}⟩. The von Neumann entropy shows the area law behavior

*S*

_{R}=

*α∂*R +

*S*

_{topo}(

*∂*R is the length of the boundary of the region R). The formula in Eq. (128) is designed in such a way that contributions from the area law term cancel.

*S*

_{R}for a stabilizer Hamiltonian can be computed easily.

^{10,27}Let

*G*be the multiplicative group generated by all $A\u0302v$’s $(v\u2208V)$, $B\u0302p$’s $(p\u2208P)$ and possible closed string operators for which |Φ

_{0}⟩ has the eigenvalue +1. Suppose that |Φ

_{0}⟩ is the unique state that has the eigenvalue +1 for all operators in

*G*. Then, the projector onto |Φ

_{0}⟩ can be written as

*λ*

_{*}≠ 1. The former is simply the definition of |Φ

_{0}⟩. The latter follows by applying $P\u0302G=P\u0302Gg\u0302*$ to the state |Ψ⟩,

*G*is given by $|G|=N2L1L2$. Similarly, $trR\u0304[g\u0302]$ can be nonzero only when $g\u0302$ is identity over $R\u0304$. Thus,

*n*

_{R}is the number of

*N*-level spins in R and

*G*

_{R}is the subgroup of

*G*supported in R. In the last step, we introduced the projector

^{10,27}

*a*.

*a*is coprime to

*N*, |

*G*

_{R}| is given by $NmR$, where

*m*

_{R}is the number of generators of

*G*supported in R.

^{10}Therefore, the formula in Eq. (134) reduces to

*L*

_{1}and

*L*

_{2}, as far as

*a*is coprime to

*N*. For example, for the subregions A, B, and C illustrated in Fig. 4(a), we have

*L*

_{1}=

*L*

_{2}= 3 and

*N*= 3.

*N*and

*a*have a common divisor, one needs to directly use the formula in Eq. (134). For example, let us take positive, mutually coprime integers

*N*

_{1},

*N*

_{2},

*a*′ and set

*N*=

*N*

_{1}

*N*

_{2}and

*a*=

*N*

_{1}

*a*′. For the subregions A, B, and C illustrated in Fig. 4(a), we find

*G*

_{R}used in the calculation are shown in Figs. 4(b)–4(g) using region C as an example. This result is what one would expect from the $ZNa$ topological order. However, more generally, we have

*N*/[

*N*

_{a}gcd(

*N*,

*a*)] is a positive integer. When it is larger than one,

*S*

_{topo}is shifted from the expected value −log

*N*

_{a}. We examine this additional contribution to

*S*

_{topo}in detail below.

### B. Spurious contributions

It is known that the topological entanglement entropy may suffer from spurious contributions and may become nonzero even when the ground state does not have a topological order.^{10,28,29} Thus, we need to verify that the nonzero topological entanglement entropy found in Sec. VII A is the legitimate one.

*a*is coprime to

*N*, we find

*S*

_{topo}in Eq. (136) is physical. This remains true more generally when

*N*/[

*N*

_{a}gcd(

*N*,

*a*)] = 1.

*N*/[

*N*

_{a}gcd(

*N*,

*a*)] > 1. For example, when

*N*=

*a*

^{2}(

*a*> 1), there are no anyons (

*N*

_{a}= 1) and the phase must be topologically trivial as we will discuss in Sec. VIII. However, in this case,

*N*/[

*N*

_{a}gcd(

*N*,

*a*)] =

*a*> 1 and

*S*

_{topo}in Eq. (139) becomes

*N*

_{a}≠

*N*. They have nontrivial contribution to

*S*

_{topo}and

*S*

_{dumb}when their ends have a shape illustrated by dashed lines in Figs. 4(b) and 4(c), which occurs when

*N*/[

*N*

_{a}gcd(

*N*,

*a*)] > 1. Indeed, when

*N*=

*a*

^{2}(

*a*> 1), we find

*G*

_{ABC}used in the calculation in Figs. 5(b)–5(f). These behaviors imply that

*N*=

*a*

^{2}cases realize subsystem symmetry-protected topological (SSPT) phases, and we will come back to this point in Sec. VIII C.

### C. Anyons

When *a* is coprime to *N*, all magnetic and electric excitations can be understood as anyons with nontrivial mutual braiding statistics. They are created in pairs by open string operators, as in Sec. II D, or by extended string operators in Eqs. (67) and (70) without forming a pair. The appearance of anyonic excitations is another hallmark of topologically ordered phases.

*a*is not coprime to

*N*, some of magnetic and electric excitations are trivial in the sense they can be created locally without forming a pair. To see this, let us focus on divisors of

*N*given by

*k*<

*k*′,

*d*

_{k}/

*d*

_{k′}is a positive integer because

*d*

_{k}=

*N*

_{a}for every $k\u2265max{rj}j=m+1n$, where

*N*

_{a}is defined in Eq. (52) and

*r*

_{j}’s are powers appearing in the prime factorization in Eq. (48). Therefore, all

*d*

_{k}’s are multiples of

*N*

_{a}.

*q*is not a multiple of

*N*

_{a}, excitations with eigenvalue

*ω*

^{q}need to be created in pairs. Therefore, only excitations with the charge

*q*= 1, 2, …,

*N*

_{a}− 1 are nontrivial.

*q*

_{e}and

*q*

_{m}, where

*q*

_{e},

*q*

_{m}∈ {0, 1, …,

*N*

_{a}− 1}. The topological order of this model is thus identical to that of the standard $ZNa$ toric code model, i.e., the same anyon types, fusion rules, and braiding statistics. In particular, they satisfy the following fusion rule:

*x*mod

*N*

_{a}. Thus, we may view the anyons as an Abelian group $A=ZNa\xd7ZNa$, with the multiplication given by fusion.

^{30}as the standard $ZNa$ toric code. More specifically, under a unit translation in

*x*

_{1}or

*x*

_{2}, an anyon (

*q*

_{e},

*q*

_{m}) becomes

*q*= 1, 2, …,

*N*

_{a}− 1, there exists

*ℓ*(1 ≤

*ℓ*≤

*N*

_{a}− 1) such that

*q*=

*a*

^{ℓ}mod

*N*

_{a}. Then,

*aq*:=

*a*

^{ℓ+1}and

*a*

^{−1}

*q*:=

*a*

^{ℓ−1}mod

*N*

_{a}. When

*q*= 0,

*aq*=

*a*

^{−1}

*q*= 0.

We should mention that to completely describe the symmetry-enriched topological order, there are further information beyond the permutation action.^{30} However, they are not relevant for our purpose, so we will not consider them in more details.

### D. Symmetry defects

*a*≠ 1, the

*T*

_{i}action generally changes anyon types. We can also see that $TiMNa(a)$ keeps all anyon types invariant, so effectively

*T*

_{i}generates a $ZMNa(a)$ symmetry group of the low-energy topological theory. In this section, we will use $\rho ak$ to denote the permutation

Before we continue, it will be very useful to understand the properties of (point-like) symmetry defects, i.e., dislocations in this case.^{30–36} Generally, each symmetry defect is uniquely associated with a group element, which determines the symmetry action that takes place when moving around the defect. We denote the set of all defects associated with symmetry group element *g* by $Cg$. Note that for *g* = 1, trivial defects are nothing but the anyons. Symmetry defects are always at the end points of defect lines, which can be intuitively thought of as branch cuts where the symmetry action takes place. Just like anyons, defects can fuse with each other to new defects, and the fusion rules must respect the group multiplication structure. Defects can also fuse with anyons, which do not change the associated group element. See Ref. 30 for a more systematic discussion of defect fusion rules.

*q*

_{e},

*q*

_{m}. To see this, one can locally create a pair of anyons (

*q*

_{e},

*q*

_{m}) and (−

*q*

_{e}, −

*q*

_{m}) near the defect, move (

*q*

_{e},

*q*

_{m}) around the defect so it becomes (

*a*

^{k}

*q*

_{e},

*a*

^{−k}

*q*

_{m}), and then fuse it again with (−

*q*

_{e}, −

*q*

_{m}) to give [(

*a*

^{k}− 1)

*q*

_{e}, (

*a*

^{−k}− 1)

*q*

_{m}]. In other words, $\sigma ak,0$ and $\sigma ak,0\xd7((ak\u22121)qe,(a\u2212k\u22121)qm)$ are related by a local operation, so must be the same type of defect.

*a*

^{k}− 1, 0) and (0,

*a*

^{−k}− 1).

^{30,35}We will denote by [

*q*

_{e},

*q*

_{m}] the equivalence classes of anyons under this quotient. Define $tak=gcd(ak\u22121,Na)=gcd(a\u2212k\u22121,Na)$ [the second equality follows from gcd(

*a*

^{k},

*N*

_{a}) = 1]; then, we can label the defects by $\sigma ak,[qe,qm]$ with $qe,m=0,1,\u2026,tak$ as representatives of the equivalence classes,

As an example, if *N*_{a} is a prime and *a* ≠ 1 mod *N*_{a}, then the subgroup generated by *a*^{k} − 1 for 0 < *k* < *N*_{a} − 1 is basically the entire group $ZNa$. Hence, the quotient group has a single element, and there is only a unique type of defect.

Let us now consider the ground state degeneracy on a torus, with a $\rho ak1$ defect line in one direction and a $\rho ak2$ defect line in the other direction. According to the general theory in Ref. 30, the ground state degeneracy is equal to the number of $\rho ak$ defect types invariant under $\rho ak\u2032$ action given in Eq. (152).

*q*

_{e},

*q*

_{m}satisfy

*e*. To shorten notations, define

*b*

_{1}=

*a*

^{k}− 1,

*b*

_{2}=

*a*

^{k}′ − 1, and

*t*

_{i}= gcd(

*b*

_{i},

*N*

_{a}). In the electric sector, Eq. (154) means that there exists an integer

*r*such that

*q*, this is possible if and only if

*t*

_{1}= gcd(

*b*

_{1},

*N*

_{a}) divides

*b*

_{2}

*q*. In other words, there exists an integer

*r*′ such that

*q*that makes it solvable is $t1gcd(t1,b2)$. Note that gcd(

*t*

_{1},

*b*

_{2}) = gcd(gcd(

*b*

_{1},

*N*

_{a}),

*b*

_{2}) = gcd(

*b*

_{1},

*b*

_{2},

*N*

_{a}). Therefore, the number of solutions is precisely gcd(

*b*

_{1},

*b*

_{2},

*N*

_{a}). The same argument works for the magnetic sector, so together, we find that the total number of solutions to Eq. (154) is given by $gcd(b1,b2,Na)2=gcd(ak\u22121,ak\u2032\u22121,Na)2$.

We now show that knowing the permutation action of *T*_{i} on anyons is enough to derive the topological degeneracy. Here, the key is to think of a *L*_{1} × *L*_{2} torus as a torus in continuum, but with a $T1L1$ defect line along *x*_{2} and a $T2L2$ defect line along *x*_{1}. Intuitively, this is because traveling across the torus in the *x*_{i} direction is the same as translating by *L*_{i}. With this picture, the ground state degeneracy is obtained by substituting *k* = *L*_{1} and *k*′ = *L*_{2}, which reproduces the result in Eq. (55). In our model, as shown in Sec. VI, when *a* and *N* are coprime, we can indeed explicitly map the Hamiltonian on a torus to the standard toric code (where translation symmetry acts trivially) with a twisted boundary condition or equivalently with symmetry defect lines wrapping around the two non-contractible cycles, in full agreement with the argument in this section. The standard toric code has a smooth continuum limit; thus, the finite-lattice effect is completely captured by the defect lines, establishing the continuum picture at the microscopic level.

## VIII. PHASES WITH NO TOPOLOGICAL ORDER

In this section, we consider the case when *a* is a multiple of rad(*N*).

### A. Uniqueness of the ground state

Let us demonstrate the uniqueness of the ground state regardless of the choice of the system size *L*_{1} and *L*_{2} although it is already implied by our general formula in Eq. (55) with *N*_{a} = 1.

*ℓ*

_{0}be the smallest positive integer such that

*ℓ*

_{0}indeed exists, let us write $rM:=max{ri}i=1n$, where

*r*

_{i}’s are powers appearing in the prime factorization of

*N*in Eq. (48). Because $rad(N)rM=\u220fj=1npjrM$ is a multiple of

*N*and also because $arM$ is a multiple of $rad(N)rM$, we have

*ℓ*

_{0}is in the range 1 ≤

*ℓ*

_{0}≤

*r*

_{M}.

*ω*at the plaquette $p=(m1+12,m2+12)$. The eigenvalue of the plaquette operator $B\u0302(m1+\u21130+12,m2+12)$ remains 1. Most importantly, the string operator in Eq. (159) is local in the sense that its length

*ℓ*

_{0}does not depend on the system size. Hence, a

*single elementally*magnetic excitation can be created locally. Similarly, the state

*ω*

^{−1}at the vertex

*v*= (

*m*

_{1},

*m*

_{2}). The rest of the discussion proceeds exactly the same as in Sec. V A 2. Therefore,

*L*

_{1}and

*L*

_{2}.

### B. Example 1: *N* = *a*

*N*=

*a*. In this case, the Hamiltonian is completely decoupled,

_{1}. We denote the ground state by $|\varphi 0\u232ar$. Then, the unique ground state of $H\u0302$ is given by the product state $\u2297r\u2208\Lambda |\varphi 0\u232ar$. Therefore, this phase is completely trivial. Indeed, the topological entablement entropy in Eq. (128) vanishes

### C. Example 2: *N* = *a*^{2}

Next, we discuss the case of *N* = *a*^{2}. We argue that this case realizes a SSPT phase.^{37,38}

#### 1. Charge pumping

Let us work with the periodic boundary condition first. We identify $r+(n1L\u03041,n2L\u03042)$ with ** r** for $n1,n2\u2208Z$. Both $L\u03041$ and $L\u03042$ are assumed to be even. Unit translation symmetries of the model shift

**by either (1, 1) or (1, −1).**

*r**N*=

*a*

^{2}, these operators can be rewritten in terms of stabilizers as

_{0}⟩ where all vertex operators and plaquette operators take the value +1, we have

*ω*

^{−a}to the vertex term $A\u0302(L\u03041\u22121,2j2)$ [the red shaded vertex in Fig. 6(b)],

*ω*

^{a}. Therefore, using Eqs. (172) and (174), we find

*ω*

^{a}is pumped for the subsystem symmetry $X\u03022j2$ upon inserting the symmetry flux associated with the subsystem symmetry $Z\u03022j2+1$. This pumped charge is a topological invariant that distinguishes this phase from product states.

#### 2. Zero energy edge states

Next, let us consider the open boundary condition. We impose the subsystem symmetries $X\u0302m\u03042$ and $Z\u0302m\u03042$ in Eqs. (170) and (171) for every $m\u03042$, including the edges.

*s*,

*s*′ =

*L*,

*R*and $j,j\u2032=1,2,\u2026,L\u030422\u22121$. A pair of $\sigma \u03022j,sx$ and $\sigma \u03022j,sz$ generates a $ZN\xd7ZN$ symmetry, implying

*N*-fold degeneracy, and there are $L\u03042\u22122$ such pairs. This $NL\u03042\u22122$-fold degeneracy cannot be lifted by perturbations on the edges, as long as the subsystem symmetries are maintained. In contrast, the two edges at $m\u03042=0$ and $m\u03042=L\u03042\u22121$ can be gapped by edge perturbations.

## IX. DISCUSSIONS

As a concluding remark, let us discuss implications of our example on the Lieb–Schultz–Mattis (LSM)-type theorems,^{39–53} which formulate necessary conditions for the unique ground state with a nonzero excitation gap under the periodic boundary condition. When one of these conditions are not satisfied, the appearance of ground state degeneracy or gapless excitations is guaranteed. The ground state degeneracy originates from either spontaneous symmetry breaking or topological degeneracy. Hence, violation of LSM-type conditions in symmetric and gapped phases can be used as a sufficient condition for a nontrivial topological order.^{45,47}

There are a variety of such theorems applicable to quantum many-body systems in different settings. For example, in one dimension, an early version of LSM theorems for quantum spin chains with spin-rotation symmetry states that *S* needs to be an integer in the presence of the time-reversal symmetry.^{39,40} More generally, *S* − *m* (*m* is the magnetization per unit cell) must be an integer to realize a unique gapped ground state.^{41} Similarly, in fermionic systems with *U*(1) symmetry, the filling *ν* (the average number of fermions per unit cell) must be an integer.^{42} These results apply to any sequence of *L*_{1}. One can even start with the infinite system from the beginning.^{52,53}

In contrast, there is usually a restriction on the choice of the sequence of *L*_{i}’s in higher-dimensional extensions of these theorems. In the formulation, one usually starts with a finite size system with the length *L*_{i} in *x*_{i} direction (*i* = 1, …, *d*) and considers the limit *L*_{1}, …, *L*_{d} → +∞. For example, for spin systems, the arguments in Refs. 44 and 46 are effective only when *L*_{2}, …, *L*_{d} are all odd. For particle systems, the discussions in Refs. 43 and 51 assume that *L*_{2}, …, *L*_{d} are coprime to *q* when *ν* = *p*/*q*. There is a way to remove such a restriction by modifying the boundary condition to a tilted one,^{54} but this argument is not about the original periodic boundary condition. Namely, changing the boundary condition from the periodic one to the tilted one might affect the degeneracy or excitation gap.

As we demonstrated through an example, a topologically ordered phase may not show topological degeneracy on torus depending on the sequence of system size. Hence, even when all the LSM-type conditions are fulfilled and the ground state is indeed unique in some sequences of the system size, it still might be the case that the ground state is actually topologically ordered.

## ACKNOWLEDGMENTS

H.W. thanks Hiroki Hamaguchi for informing us of the proof of Eq. (A4) in Appendix A. The work of H.W. was supported by JSPS KAKENHI under Grant Nos. JP20H01825 and JP21H01789. M.C. acknowledges support from NSF under Award No. DMR-1846109. The work of Y.F. was supported by JSPS KAKENHI under Grant No. JP20K14402 and JST CREST under Grant No. JPMJCR19T2. H.W. acknowledges the hospitality and fruitful discussions at the Institute of Basic Science, Daejeon, Korea, during the week of Conference on Advances in The Physics of Topological and Correlated Matter.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Haruki Watanabe**: Formal analysis (lead); Investigation (lead); Writing – original draft (lead); Writing – review & editing (lead). **Meng Cheng**: Formal analysis (equal); Investigation (equal); Supervision (equal); Validation (lead); Writing – original draft (equal); Writing – review & editing (equal). **Yohei Fuji**: Formal analysis (equal); Investigation (equal); Supervision (lead); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

### APPENDIX A: PROOF OF EQ. (105)

*N*/

*d*

_{a}is coprime to gcd(

*α*

_{1}+

*b*

_{1}

*N*,

*α*

_{2}+

*b*

_{2}

*N*)/

*d*

_{a}. Hence, Eq. (105) can be rewritten as

Below, we prove the following statement: for any integer *N* ≥ 2 and integers *α*_{1} and *α*_{2} in the range 1 ≤ *α*_{1}, *α*_{2} ≤ *N* − 1, there always exist integers *b*_{1} and *b*_{2} such that Eq. (A4) holds.^{55} In particular, *b*_{2} can be set 0. Since this is trivially the case when *d*_{a} = 1, in the following, we assume *d*_{a} ≠ 1. We introduce shorthands $\alpha 1\u2032:=\alpha 1/da$, $\alpha 2\u2032:=\alpha 2/da$, and *N*′ := *N*/*d*_{a}.

*m*and a prime

*p*, let us denote by

*ν*

_{p}(

*m*) the largest non-negative integer

*ν*such that

*p*

^{ν}divides

*m*. Suppose that $ej:=\nu pj(da)\u22651$ for

*j*= 1, 2, …,

*J*. In other words,

*d*

_{a}can be prime-factorized as $da=\u220fj=1Jpjej$. Then, Eq. (A4) holds if and only if

*j*= 1, 2, …,

*J*. In addition, by definition,

*j*= 1, 2, …,

*J*, Eq. (A5) is fulfilled. In the following, we write $nj:=\nu pj(N\u2032)$ and $mj:=\nu pj(\alpha 1\u2032)$.

*n*

_{j}≥

*m*

_{j}, we need

*p*

_{j}. In this case, we can set

*b*

_{1}= 0 mod

*p*

_{j}. On the other hand, when

*m*

_{j}>

*n*

_{j}, we need

*p*

_{j}and $N\u2032/pjnj\u22600$ mod

*p*

_{j}. In this case, we can set

*b*

_{1}= 1 mod

*p*

_{j}. After all, we found a condition of the form

*b*

_{1}=

*x*

_{j}mod

*p*

_{j}for each

*j*= 1, 2, …,

*J*. The Chinese remainder theorem guarantees the existence

*b*

_{1}in the range 0 to $\u22121+\u220fj=1Jpj$ such that these conditions are simultaneously satisfied.

### APPENDIX B: REDUCTION OF GENERALIZED PAULI MATRICES

*N*

_{1}and

*N*

_{2}are coprime, the

*N*=

*N*

_{1}

*N*

_{2}-level spin can be decomposed into the tensor product of

*N*

_{1}- and

*N*

_{2}-level spins. To see this, let us write the matrices in Eqs. (3) and (4) as

*X*(

*N*) and

*Z*(

*N*), respectively. We have

*i*

_{1}= 1, …,

*N*

_{1},

*i*

_{2}= 1, …,

*N*

_{2}, and

*i*= 1, …,

*N*. These reduction formulas can be readily shown by using the representations in Eqs. (3) and (4).

*N*=

*N*

_{1}

*N*

_{2}and

*N*

_{1}and

*N*

_{2}are coprime. We introduce another modified Hamiltonian,

*a*

_{1}=

*a*

_{2}=

*a*. The eigenstates of this Hamiltonian are also identical to those for $H\u0302$ in Eq. (14), and the ground state degeneracy remains unchanged.

*N*level spin is given by the unitary matrix

*V*above. Using the reduction formulas, we find

*N*

_{i}-level spins. Ground states have the eigenvalue +1 for all $A\u0302v(Ni)$’s and $B\u0302p(Ni)$’s. This result indicates that, if we denote the ground state degeneracy of the our model for

*N*-level spin by

*N*

_{deg}(

*N*),

*N*

_{1}and

*N*

_{2}are coprime. More generally, for the form of

*N*in Eq. (48), we have

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*Physics and Mathematics of Quantum Many-Body Systems*

*Elementary Number Theory: Primes, Congruences, and Secrets: A Computational Approach*

*The Physics and Mathematics of Elliott Lieb*, edited by R. L. Frank, A. Laptev, M. Lewin, and R. Seiringer (EMS Press, 2022), Vol. 2, pp. 405-446.

The proof is due to Hiroki Hamaguchi.