Ground state degeneracy
on torus in topologicallyordered phases
Haruki Watanabe (U Tokyo)
Collaborators: Yohei Fuji (U Tokyo), Meng Cheng (Yale)
Ref: HW, M. Cheng, Y. Fuji, arXiv:2211.00299

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Introduction

De ning properties of phases (textbook)
Symmetry-protected topological/trivial phases
1. Unique GS with excitation gap.
2. Stable against symmetry-preserving perturbations.
3. Can be connected to product states by local unitaries if symmetries are broken.
(Exceptions: 1D Kitaev chain, integer Quantum Hall systems, …)4. Trivial excitations in the bulk.
Anomalous states on symmetry-preserving boundaries.

Topologically-ordered phases
1. Degenerate GS with excitation gap.
Degeneracy does not originate from spontaneous breaking of symmetries.
2. Stable against any local perturbations.
3. Cannot be connected to product state by any local unitaries.
Topological entanglement entropy.
fi

4. Fractionalized (anyonic) excitations in the bulk. Anyons must appear in pairs.

Features of our model• Our model contains a parameter N ≥ 2 and a = 1,2,⋯, N.
• Here we consider the case where N is a prime and a is chosen properly.
• Ground state degeneracy (GSD) Ndeg:

N level spin

(primitive root modulo N)

2N=N
when both L1 and L2 are multiples of N − 1.
‣ deg

‣ Ndeg = 1 when L1 or L2 is not a multiple of N − 1.
• e.g. N = 11 and a = 2.

To observe Ndeg > 1, both L1 and L2 need to be multiples of 10.

Minimum system size is 10 × 10; total Hilbert space dimension = 11200 .
→ Nearly impossible to realize in any numerical study.

• Topological entanglement entropy (TEE): Stop = − log N.
•Anyons: N 2 species characterized by the electric charge qe = 1,2,⋯, N and themagnetic charge qm = 1,2,⋯, N.

De ning properties of phases (updated)
Symmetry-protected topological/trivial phases
1. Unique GS with excitation gap for any sequence of L1 and L2.
2. Stable against symmetry-preserving perturbations.
3. Can be connected to product states by local unitaries if symmetries are broken.
4. Trivial excitations in the bulk.
Anomalous states on symmetry-preserving boundaries.

Topologically-ordered phases
1. Degenerate GS with excitation gap for a sequence of L1 and L2.
Degeneracy does not originate from spontaneous breaking of symmetries.
2. Stable against any local perturbations.
3. Cannot be connected to product state by any local unitaries.
Topological entanglement entropy.

fi

4. Fractionalized (anyonic) excitations in the bulk. Anyons must appear in pairs.

De ning properties of phases (updated)
Symmetry-protected topological/trivial phases
1. Unique GS with excitation gap for any sequence of L1 and L2.
2. Stable against symmetry-preserving perturbations.
3. Can be connected to product states by local unitaries if symmetries are broken.
4. Trivial excitations in the bulk.
Anomalous states on symmetry-preserving boundaries.

Topologically-ordered phases
1. Degenerate GS with excitation gap for a sequence of L1 and L2.
Degeneracy does not originate from spontaneous breaking of symmetries.
2. Stable against any local perturbations.
3. Cannot be connected to product state by any local unitaries.
Topological entanglement entropy.

fi

4. Fractionalized (anyonic) excitations in the bulk. Anyons must appear in pairs.

fiDe nition & basic propertiesof our modelN level spins• Generalization of Pauli matrices to N ≥ 2 level spins
X=1111Z=⋱ω

1• ℤN × ℤN group with phase ω = e
ZX = ω XZ =NN

Z =X =1

•ω2

ω

ω2⋱

⋱

ω N−1

2πi
N

1

ω N−1

̂r = X̂Nr = 1.
A N level spin is placed on every bond. They satisfy Zr̂ X̂r′ = ω δr,r′ X̂r′Zr̂ and ZN• Periodic boundary condition (PBC) with system size L1 and L2 → System is put on torus.
￼￼￼•Total Hilbert space dimension is N 2L1L2.

𝒫𝒱ℤN toric code
• Vertex and plaquette terms (all commute regardless of a)̂ ,m ) = X̂−a 1 X̂−â(m − 1 ,m ) X̂(m ,m − 1 )
A(mX
11 2
1 22
1 2 2
(m + ,m ) (m ,m + )122

122−aâ̂ +1,m + 1 )Z−1̂̂B̂(m1+ 12 ,m2+ 12 ) = Z(mZZ
1112 2(m + ,m +1) (m ,m + ) (m + 1 ,m )122122122• a = 1,2,⋯, N is a very important parameter.
• Hamiltonian is sum of stabilizers:Ĥ = −1 ̂1 ̂(A + h.c.) −(B + h.c.)∑2 v∑2 pv∈
p∈= {(m1, m2) | mi = 0,1,⋯, Li − 1}
1
1= {(m1+ 2 , m2+ 2 ) | mi = 0,1,⋯, Li − 1}
𝒫𝒱

•

̂ .
Translation symmetry: Tî X̂rT ̂†i = X̂r+ei and Tî Zr̂ T ̂†i = Zr+eiA ground statêv = B̂Np = 1, the eigenvalues of Av̂ and B̂p are N-fold: 1,ω, ⋯, ω N−1.
Since AN•• Any state with eigenvalues +1 for all vertices and plaquettes is a ground state.
• A ground state can be constructed by(i) starting from the ferromagnetic state Zr̂ | ϕ0⟩ = | ϕ0⟩(ii) applying the projector P̂ =

1N−1

∏∑
N L1L2 v∈ℓ=0( ∀r ∈ Λ)

Aℓv̂ with the following properties.

Then | Φ0⟩ ∝ P̂ | ϕ0⟩ has eigenvalues +1 for all vertices and plaquettes.
2̂
P= P̂
‣̂
̂
̂̂
̂
A
P
=PA=P‣ vv
̂̂̂̂BP=PBp‣ p𝒱‣ Tî P̂ = PT̂ îSimple case 1: a = 1• This case is basically the same as the original toric code.
Global constraints•∏v∈

X loops: X̂ (1) =

•L1−1

∏
ℓ=0Z loops: Z ̂(1) =

•

L1−1

∏
ℓ=0Av̂ =∏p∈B̂p = 1.

X̂(ℓ,L2− 12 ), X̂ (2) =̂ 1 ,0), Z ̂(2) =Z(ℓ+2L2−1∏ℓ=0L2−1∏ℓ=0X̂(L1− 12 ,ℓ).

̂1 .
Z(0,ℓ+2)(1) ̂ (2)(2) ̂(1)
̂̂
ZX=
ωXZ .
‣(2) ̂ (1)(1) ̂(2)̂̂
ZX=ωXZ .
‣•(1) ̂ (1)(2) ̂(2)̂̂
[Z,X]=[X, Z ] = 0.
‣𝒫𝒱(1) ̂(2)(1) ̂ (2)̂̂
[Z,Z]=[X, X ] = 0.
‣GSD: Ndeg = N 2. TEE: Stop = − log N. N 2 species of anyons.

Open string operators
Control the eigenvalues of• Av̂ (v ∈ , v ≠ v0)p0• B̂p (p ∈ , p ≠ p0)
Zr̂ X̂r′ = ω δr,r′ X̂r′Zr̂

v0Av̂ 0 and B̂p0 are xedby global constraints.
p0
Closed string operatorsX̂ (2) and Z ̂(2)Control the eigenvalues of• X̂ (1) and Z ̂(1)￼fi￼𝒫𝒱￼Z ̂(2)X̂ (1) = ω X̂ (1)Z ̂(2)Z ̂(1)X̂ (2) = ω X̂ (2)Z ̂(1)v0

Simple case 2: a = N•

Ĥ =∑r∈Λĥ r with11
̂ ,m − 1 )Z−1̂ĥ (m1,m2) = (X̂(m1− 12 ,m2)X̂(m1,m2− 12 ) + h.c.) − (Z(m+ h.c.)1
1 2 2(m−,m)1 2222• Completely decoupled → product state.
• GSD: Ndeg = 1. TEE: Stop = 0. No anyons.

→More general cases

Excited states• Applying open strings of X̂ and Z ̂ creates a pair of electric and magnetic particles.
• Translation permutes anyons (i.e. changes the charges of e and m particles).

X̂ (1)

￼￼

￼

￼

￼

(m1− 12 ,m2+ 12 ),(m′1+ 12 ,m2+ 12 )=m′1−m1∏ℓ=0aℓ
̂X(m +ℓ,m + 1 )122Z ̂(2)(m1,m2),(m1,m′2+1)=

m′2−m2′â m2−m2−ℓZ∏ (m1,m2+ℓ+ 12 )ℓ=0

Global constraints
Then GSD is given by Ndeg = NC2 (proof is given later),•‣‣ℓwhere NC is the number of global constraints:
Av̂ v = 1 (0 ≤ ℓv ≤ N − 1)).
∏v∈

There will be an equal number of constraints:∏p∈ℓp̂Bp = 1 (0 ≤ ℓp ≤ N − 1).

• Trial global constraints:
L1−1 L2−1

‣

∏
‣ m∏
=0 m =0𝒫1

𝒱

L2−1

m1 L 2m2 L1−a(a−1)−a(a −1)
â m1+m2̂A(m1,m2) =X̂
X.
11∏∏∏ (m1,− 2 )∏ (− 2 ,m2)
m1=0 m2=0m1=0m2=0L1−1 L2−1

•

L1−12

a (L1−1−m1)+(L2−1−m2)
̂B(m + 1 ,m + 1 )=1222L1−1L2−1(L 2−1−m2) L1â (L1−1−m1)(a L2−1)−a(a −1)̂.
ZZ
11∏ (m1+ 2 ,0)∏ (0,m2+ 2 )m1=0
m2=0Consider their n-th power. We need (a L1 − 1)n = (a L2 − 1)n = 0mod N.

Loops(i) ̂a Li−1 ̂(i)(i) ̂1−a Li ̂(i)
̂̂̂̂
X
B=
ωBX
ZA=
ωAZand.
p0p0v0v0•• Closed loops are not really closed. Create an anyon without forming a pair.
X̂ (1) =L1−1

∏
ℓ=0aℓ
̂
X(ℓ,L − 1 )2

Z ̂(2) =2X̂ (2) =L2−1∏ℓ=0X̂a(L − 1 ,ℓ)ℓ

1

2L2−1â L 2−1−ℓ
Z(0,ℓ+ 1 )∏2ℓ=0Z ̂(1) =L1−1â L1−1−ℓZ(ℓ+ 1 ,0)∏2ℓ=0Math de nitions• Multiplicative order of a modulo n‣ Suppose n and a are coprime.
ℓisthesmallest
positive
integersuchthatM(a)ℓa= 1 (mod n).
‣ n‣ Mn(a) = 1 if and only if a = 1 (mod n).
‣ When n ≥ 3, Mn(a) = 2 if a = n − 1 (mod n).
• Euler's totient function φ(n)‣ The number of positive integers smaller than n that are relatively prime to n.
e.g., φ(n) = n − 1 when n is a prime.

• Primitive root modulo n‣ a is a primitive root modulo n if Mn(a) = φ(n).

fi‣

The primitive roots exist if and only if n is either 2, 4, p k, or 2p k (p is an odd prime &k is a positive integer).

Case 3: N is a prime•When N is a prime, (a Li − 1)n = 0 (mod N) ⇔ a Li = 1 (mod N) ⇔ Li = 0 (mod MN (a)).

•If Li ≠ 0 (mod MN (a)), a Li − 1 is coprime to N. Eigenvalues of Av̂ 0 and B̂p0 can be freely(i) ̂a Li−1 ̂(i)(i) ̂1−a Li ̂̂̂̂
controlled by using X Bp0 = ω
Bp0 X and X Ap0 = ωAp0 X̂ (i).
This argument can be extended to all v ∈and p ∈→ Absence of global

constraints.

• GSD Ndeg depends on L1 and L2:

Example: N = 11ℓ(mod N)
aa2Ndeg = N when both L1 and L2 are multiples of MN(a). 1 1
22, 4, 8, 5, 10, 9, 7, 3, 6, 1
33, 9, 5, 4, 1
Ndeg = 1 when L1 or L2 is not a multiple of MN(a).
44, 5, 9, 3, 1
5
5, 3, 4, 9, 1
a is a primitive root if the smallest integer ℓ
6
6, 3, 7, 9, 10, 5, 8, 4, 2, 1ℓsuch that a = 1 (mod N) is N − 1.
7
7, 5, 2, 3, 10, 4, 6, 9, 8, 1
N−1
c.f. Fermat’s little theorem: a= 1 (mod N)
88, 9, 6, 4, 10, 3, 2, 5, 7, 1
99, 4, 3, 5, 1
TEE: Stop = − log N. N 2 species of anyons.
10 10, 1
Primitive roots are a = 2,6,7,8.

‣•

𝒫•

𝒱‣

Calculation of TEE• Kitaev-Preskill prescription‣ Stopo = (SA + SB + SC) − (SAB + SBC + SCA) + SABC
‣ SR = − tr[ρR̂ log ρR̂ ]• Useful formula: SR = nR log N − log | GR |‣ nR is the number of N-level spins in R.
‣ GR is the subgroup of G supported in R.
̂̂
Gisthemultiplicativegroup
generatedbyallA’s(v∈),B), andv
p’s (p ∈‣possible closed string operators for which | Φ0⟩ has the eigenvalue +1.

• When N and a are coprime: SR = (nR − mR)log N𝒫𝒱‣ mR is the number of generators of G supported in R.

General casen•
•

rjrn
r1 r2
Prime factorization N =pj = p1 p2 ⋯pn .
∏j=1

Radical of N: rad(N) =n∏
j=1

pj = p1 p2⋯pn.

• Na = the largest divisor of N that is coprime to a.
‣ Na ≠ 1 if a is not a multiple of rad(N) → Topologically-ordered phases‣ Na = 1 if a is a multiple of rad(N) → SPT phases•GSD Ndeg = da2 with da = gcd(a L1 − 1,a L2 − 1,Na).

2
N• a anyon species characterized by the electric charge qe = 1,2,⋯, Na and themagnetic charge qm = 1,2,⋯, Na.

Justi cation• Trial global constraints:
L1−1 L2−1

‣

L1−1

L2−1

−a m1(a L2−1)−a m2(a L1−1)
â m1+m2̂̂
A(m1,m2) =X
X 1.
1
∏∏
∏ (m1,− 2 )∏ (− 2 ,m2)
m1=0 m2=0m1=0m2=0L1−1 L2−1∏‣ m∏=0 m =012

a (L1−1−m1)+(L2−1−m2)
̂B(m + 1 ,m + 1 )=1222L1−1

L2−1(L 2−1−m2) L1â (L1−1−m1)(a L2−1)−a(a −1)
̂ZZ.
11∏ (m1+ 2 ,0)∏ (0,m2+ 2 )m1=0m2=0•

Consider their n-th power. We need (a L1 − 1)n = (a L2 − 1)n = 0•

Similarly, [X̂ (i)]ni,a commutes with B̂p0 if (a Li − 1)ni,a = 0 (mod N).

N
Solution: n =with da = gcd(a L1 − 1,a L2 − 1,Na).
dafi

N
Solution: ni,a =.
Lgcd(a i − 1,Na)

mod N.

Open string operators
Control the eigenvalues of• Av̂ (v ∈ , v ≠ v0)p0• B̂p (p ∈ , p ≠ p0)[Av̂ 0] and [B̂p0] are xed
N
daN
dav0

by global constraints.

p0
Closed string operators X̂ (i) and Z ̂(i)Control the eigenvalues of̂̂ABResidual
partofandv0p0.
•fi𝒫𝒱(1) n1,a(1) n1,â̂
Closedstrings:and[X]
[Z] .
•v0

Case 4: a2• Rotate the model by 45 degree.
• Subsystem symmetries for each row‣‣X̂m̄2 =Zm̄̂ =2L̄1−1∏m̄1=0L̄1−1∏m̄1=0X̂(m̄1,m̄2)(−1)m̄1̂Z(m̄1,m̄2)• Subsystem-Protected Topological (SSPT) Phase‣ Charge pumping‣ Zero energy edge states=NSummary 1• We introduced a family of ℤN toric code with an integer parameter a.
• Na = the largest divisor of N that is coprime to a.
‣ Na ≠ 1 if a is not a multiple of rad(N) → Topologically-ordered phases‣ Na = 1 if a is a multiple of rad(N) → (S)SPT phases•GSD Ndeg = da2 with da = gcd(a L1 − 1,a L2 − 1,Na).

2N• a anyon species characterized by the electric charge qe = 1,2,⋯, Na and themagnetic charge qm = 1,2,⋯, Na.

Ref: HW, M. Cheng, Y. Fuji, arXiv:2211.00299

Summary 2
Symmetry-protected topological/trivial phases
1. Unique GS with excitation gap for any sequence of L1 and L2.
2. Stable against symmetry-preserving perturbations.
3. Can be connected to product states by local unitaries if symmetries are broken.
4. Trivial excitations in the bulk.
Anomalous states on symmetry-preserving boundaries.

Topologically-ordered phases
1. Degenerate GS with excitation gap for a sequence of L1 and L2.
Degeneracy does not originate from spontaneous breaking of symmetries.
2. Stable against any local perturbations.
3. Cannot be connected to product state by any local unitaries.
Topological entanglement entropy.
4. Fractionalized (anyonic) excitations in the bulk. Anyons must appear in pairs.