Three-dimensional topological lattice models with surface anyons

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Curt von Keyserlingk

with Fiona J. Burnell and Steven H. Simon.

University of Oxford

Nordita, August 9, 2012

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Introduction

Exactly solvable spin models in 2D

• Toric code (Kitaev 2003), Levin-Wen models (Levin & Wen(2005)).

• Often contrived, so why study them?

• Toy models for topological order.

• Fixed point hamiltonians: These models capture ‘low-energy long-wavelength’

behaviour of 2D phases of matter – tell us about the topological order that might arise in 2D strongly interacting systems.

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Three-dimensional topological lattice models with surface anyons Introduction

Exactly solvable spin models in 3D

• Here we study 3D cousins of 2D

Levin-Wen models: Exactly solvable spin models proposed by Walker and Wang [arXiv:1104.2632v2].

• Class includes 3D toric code + many novel models.

• Models capture ‘low-energy

long-wavelength’ behaviour of phases of matter.

• ‘Confined models’: Exactly solvable lattice realisation of a chiral Chern-Simons theory on boundary.

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Introduction

Outline

• Overview of Levin-Wen models, the 2D cousins of the 3D lattice models we study here.

• Review 3D toric code, the simplest Walker-Wang model.

• Introduce paradigm of confined WW models, the 3D Semion model. Understand its topological order, and the emergence of chiral anyons on its surface.

• Underlying field theories

• Generalisations

• Conclusion

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The 2D toric code

• Hilbert space: σ^z = ±1 on each edge of a trivalent lattice.

Represent configurations by colouring in σ^z = −1 edges.

Bv

s^x s^x s^x

s^x

s^z s^z s^z

s^x s^x

fHs^zL fHs^zL fHs^zL

fHs^zL

fHs^zL fHs^zL Bp

Æ s^z= -1 Æ s^z= +1

• H_TC = −P

v

Y

i ∈s(v )

σ_i^z

| {z }

Bv

−P

p

Y

i ∈∂p

σ^x_i

| {z }

Bp

s^x s^x s^x

s^xs^x s^x

B_p: p

Bp p = p

2 fig1.nb

Bv

s^x s^x s^x

s^x

s^z s^z s^z

s^x s^x

fHs^zL fHs^zL fHs^zL

fHs^zL

fHs^zL fHs^zL Bp

Æ s^z= -1 Æ s^z= +1

Bv: v

s^z s^z s^z

Bv= +1 v

Bv= -1 v

s^x s^x s^x

s^xs^x s^x

Bp: p

Bp p = p

2 fig1.nb

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Introduction

The 2D toric code: Ground states

• H_TC = −P

v

Y

i ∈s(v )

σ_i^z

| {z }

Bv

−P

p

Y

i ∈∂p

σ^x_i

| {z }

Bp

• Lower bound on the energy:

B_v = 1 ∀ v and B_p = 1 ∀ p

• Exists because

[B_v, B_v⁰] =B_p, B_p⁰ = [B_p, B_v] = 0.

• B_v = 1 → Ground state a superposition of closed loop configurations (‘Quantum loop gas’).

• B_p= 1 → states related by plaquette flips have the same coefficient in the ground state.

Bv= +1 v

B_v= -1 v

fig1.nb 3

s^x s^x s^x

s^xs^x s^x

B_p: p

B_p p = p

2 fig1.nb

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Graphical rules

=

= -1

= -

(a) Allowed vertices

(b) Deformation

(c) Loop collapsing (D1L

(d) Fusion

Semion Rules

(e) Braiding = i

Loop collapsing (D1L Allowed vertices

=

= +1

=

Deformation

Fusion

Toric Code Rules 2 tcdsem3.nb

(a)

(b)

(c)

(d)

+

+ +

+ ...

Ground state degeneracy = 2² on torus.

Topological order X.

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Introduction

Deconfined defects

• Vertex-type string operators create deconfined vertex defects (e) at their ends:

Wˆ_V(C) =Q

i ∈Cσ_i^x.

• Plaquette-type string operators create deconfined plaquette defects (m) at their ends:

Wˆ_P(C⁰) =Q

i ∈C⁰σ_i^z.

• Berry phase of −1 on exchange.

s^x

s^x s^x s^x s^x s^x

s^z s^z

s^x s^x s^x

s^x

e

m

e

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The 2D doubled semion model

The 2D doubled semion model (DSem)

• DSem is another lattice model, which is similar to the toric code except its plaquette operators are different. (Freedman et al. (2004), Levin & Wen (2005)).

• Hilbert space: σ^z = ±1 living on each edge of a trivalent lattice.

• H_DSEM = −P

v

Y

s(v )

σ_i^z

| {z }

Bv

+P

p(Y

i ∈∂p

σ_i^x) Y

j ∈s(p)

i^(1−σ^j^z^)/2

| {z }

Bp

• Different B_p operator!

s^x s^x s^x

s^xs^x s^x

Bp: p

B_p p = p

s^x s^x i^H1-s^z^Lê2

i^H1-s^z^Lê2

i^H1-s^z^Lê2 i^H1-s^z^Lê2 i^H1-s^z^Lê2 i^H1-s^z^Lê2

s^x s^xs^x

s^x

Bp: p

2 fig1.nb

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Introduction

The 2D doubled semion model

• H_DSEM =

−P

v

Y

s(v )

σ_i^z

| {z }

Bv

+P

p(Y

i ∈∂p

σ_i^x) Y

j ∈s(p)

i^(1−σ^j^z^)/2

| {z }

Bp

• Lower bound on the energy:

Bv = 1 ∀ v and Bp= −1 ∀ p

• B_v = 1 → Ground state is superposition of closed loops (‘Quantum loop gas’).

• B_p= −1 → New graphical rules.

Bv= +1 v

B_v= -1 v

fig1.nb 3

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Graphical rules

=

= -1

= -

(b) Deformation

(d) Fusion

Semion Rules

(e) Braiding = i

=

= -1

= -

Allowed vertices

Deformation

Loop collapsing (D₁L

Fusion

Semion Rules

Braiding = i

2 tcdsem3.nb

(a)

(b)

(c)

(d)

-

- -

+ +

+ ...

Ground state degeneracy = 2² on torus.

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Introduction

String operators

• Two chiralities of vertex-type string operators:

Wˆ_V^±(C) =Q

i ∈Cσ^x_i Q

k∈L vertices(−1)¹⁴^(1−σ^zⁱ^)(1+σ^j^z⁾Q

l ∈R legs(±i )^(1−σ^l^z^)/2.

Σ^x

Σ^x Σ^x Σ^x Σ^x

Σ^x

Σ^z

Σ^z Σ^z

Σ^x Σ^x Σ^x

Σ^x

L L

R R

R

R R

• Plaquette-type string operator (achiral bound state): ˆW_P(C⁰) =Q

i ∈C⁰σ^z_i.

• Vertex defects of same chirality are relative semions. Topological order X.

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Summary of 2D models

Topological order in the 2D models

The 2D toric code

• GS degeneracy of 2² on torus X

• Relative statistics between point and vortex defectsX

• Topological entanglement entropy of log 2X

• Fixed point HamiltonianX

• → Topological order

• GS degeneracy of 2² on torus X

• Semionic statistics between vertex defects of same chiralityX

• → Different topological order

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Introduction

The 2D toric code

m m

e e

p vortex - p vortex - e qp e qp

Toric code "Superconductor"

2 DSemTODSem.nb

• Superconductors are

topologically ordered (Hansson et al. (2004)).

• Vertex defects

⇔ charge e quasi-particles.

• Plaquette defects

⇔ π-flux vortices.

eê 2 - eê 2

n = -1ê 2

- -

+ + ^{eê 2} ^{- eê 2}

DSem n = +1ê 2

• Two copies of a ν = ±1/2 bosonic Laughlin.

• Vertex defects, chirality ±

⇔ charge e/2 quasi-particles in ν = ±1/2 layer.

• Plaquette defects

⇔ Bound state of ± chirality quasi-particles.

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Generalising the 2D models to 3D

• H = −P

vB_v −P

pB_p

• The first of these will be the familiar 3D toric code (Hamma et al. 2005).

• The second will the the ‘3D semion model’. It is qualitatively very different from the 3D toric code.

Point Split

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The 3D toric code

• Hilbert space: σ^z = ±1 on each edge.

• H_TC =

−P

v

Y

i ∈s(v )

σ_i^z

| {z }

Bv

−P

p

Y

i ∈∂p

σ_i^x

| {z }

Bp

(a) (b) (c)

(d) (e) (f)

p

I-BpM = -i

Px y

s^x s^x

s^x s^x s^x

s^x s^x s^x s^x s^x s^x s^x s^x

s^x s^x s^x s^x s^x s^x s^x

s^x s^x s^x

s^x s^x

Bp:

Py z Pz x

p

Bp =

Bv

s^x s^x s^x

s^x

s^z s^z s^z

s^x s^x

fHs^zL fHs^zL fHs^zL

fHs^zL

fHs^zL fHs^zL Bp

Æ s^z= -1 Æ s^z= +1

Bv: v

s^z s^z s^z

Bv= +1 v

Bv= -1 v

(a) (b) (c)

(d) (e) (f)

p

I-BpM = -i

Px y

s^x s^x

s^x s^x s^x

s^x s^x s^x s^x s^x s^x s^x s^x

s^x s^x s^x

s^x s^x

Bp:

Py z Pz x

p

Bp =

2 3DSemplaquettes.nb

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The 3D toric code

• H_TC = −P

v

Y

i ∈s(v )

σ_i^z

| {z }

Bv

−P

p

Y

i ∈∂p

σ^x_i

| {z }

Bp

• [B_v, B_v⁰] =B_p, B_p⁰ = [B_p, B_v] = 0.

• B_v = 1 → Ground state is superposition of closed loops.

• B_p= 1 → states that are related by plaquette flips have the same coefficient in the ground state.

Bv= +1 v

B_v= -1 v

fig1.nb 3

(a) (b) (c)

(d) (e) (f)

p

I-BpM = -i

Px y

s^x s^x

s^xs^x

s^x s^x s^x s^x s^x s^x s^x s^x s^x s^x s^x

s^x s^x s^x s^x s^x s^x

s^x s^x Bp:

Py z Pz x

p

Bp =

2 3DSemplaquettes.nb

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Introduction The 3D toric code

Graphical rules

• 2³ ground states on the 3-torus.

• Topological order X.

+

+ +

+ ...

HaL

HbL

+

+ +

+ ...

HaL

HbL

6 more

=

= +1

=

= -1

= -

(b) Deformation

(d) Fusion

Toric Code Rules Semion Rules

(e) Braiding = = i

=

= +1

=

= -1

= -

(a) Vertex constraints

(b) Smooth deformation

(c) Quantum dimension (D1L

(d) Fusion

Toric Code Doubled Semion

(e) Braiding

=

= = -

= i

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The 3D toric code

Defects

• Vertex-type string operators:

Wˆ_V(C) =Q

i ∈Cσ^x_i.

• Lines of plaquette defects (‘vortex rings’): ˆWP(S) =Q

i ∈Sσ_i^z.

• Berry phase of −1 on braiding.

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Introduction The 3D toric code

3D Toric code ⇔ ‘3D superconductor’

• Superconductors are topologically ordered (Hansson et al. (2004)).

• Vertex defects

⇔ charge e quasi-particles.

• Lines of plaquette defects

⇔ π-flux vortex rings.

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The 3D toric code

The 3D semion model

• Hilbert space: σ^z = ±1 on each edge. Again represent configurations by colouring in σ^z = −1 edges.

• H_3DSem = −P

vBv

+P

p(Y

i ∈∂p

σ^x_i)( Y

j ∈s(p)

iⁿ^j) i^P^{j red}ⁿ^j⁻^P^{j blue}ⁿ^j

| {z }

Bp

(a) R

R R R B R

B

B B

R

(b) (c)

(d) (e) (f)

p

I-BpM = -i

p

Bp = i

2 3DSemplaquettesflip.nb

(a) R

R R R B R

B

B B

R

(b) (c)

(d) (e) (f)

p

I-BpM = -i

(a) R

R

R R

R

B B

B

B B

R

(b)

(c)

(d)

(e)

(f)

2 3DSemplaquettes2.nb

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Introduction The 3D semion model

The 3D semion model

• H_3DSem = −P

vB_v +P

pB_p

• B_v = 1 → Ground state is a superposition of closed loops.

• B_p= −1 → Semion graphical rules determine relative amplitudes of loop configurations.

Bv= +1 v

Bv= -1 v

fig1.nb 3

(a) R

R R R B R

B B

B

B B

R

(b) (c)

(d) (e) (f)

p

I-BpM = -i

p

Bp = i

2 3DSemplaquettesflip.nb

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The 3D semion model

Graphical rules

• Unique ground state on the 3-torus.

� � � �i � ...

• NOT a ground state:

- + + -i + ...

+ - - - + ...

=

= -1

= -

(b) Deformation

(d) Fusion

Semion Rules

(e) Braiding = i

=

= -1

= -

Allowed vertices

Deformation

Loop collapsing (D1L

Fusion

Semion Rules

Braiding = i

2 tcdsem3.nb

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Defects

Flipped edges

Vertex defect

Plaquette defects

• Vertex-type string operators are confined.

Any operator ˆWV(C) =Q

i ∈Cσ_i^x× phases necessarily produces plaquette defects along its length.

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The 3D semion model

Unique ground state ⇔ Confined excitations

• ˆW_V(C) =Q

i ∈Cσ_i^x toggles between two ground state sectors.

+

+ +

+ ...

HaL

HbL

+

+ +

+ ...

HaL

HbL

6 more

• In 3D semion model, applying such an operator to GS leaves an energetic string behind.

� � � �i � ...

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Is the 3D semion model even topologically ordered?

3D toric code: 3D semion model:

• GS degeneracy of 2³ X

• Universal statistics between point defects and vortex linesX

• → Topological order

• Unique ground state 7

• All excitations confined in the bulk7

• Topological entanglement entropy of 07

• → Topological order ??? Not in the traditional sense . . .at least in the bulk!

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The 3D semion model

3D semion model with boundary

• Cut off the lattice in a ‘smooth’ manner.

Bp

Bv

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3D semion model on the solid donut

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The 3D semion model

Ground states on the solid donut

• Ground state degeneracy of 3D semion model on solid donut is 2. Topological order X.

- + + -i + ...

+ - - - + ...

- + + -i

-i +i + -i

+ ...

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The 3D semion model

Deconfined surface anyons

�a� Ψ� �b� W�

V��z� Ψ�

W�

V��AB� Ψ�

�c� W�

V��Φ� Ψ �

�c� �d�

A

B

Ψ� W�

V��AB� Ψ�

A

B

Fuse using graphical rules

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The 3D semion model

What is the topological order of the 3D semion model?

• Not topologically ordered in the bulk, at least according to Wen’s criteria.

• However, a sample with a surface has topological order.

• Topological order: Bosonic ν = 1/2 laughlin state

Chiral semion

n = 1/2 bosonic Laughlin

-

n = -1ê 2

-

+

^{n = +1}^{ê 2}

+

bosonic Laughlin

4 thickeningDSem.nb

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Field Theory

Introduction

The 2D toric code

The 2D doubled semion model Summary of 2D models The 3D toric code The 3D semion model Field Theory

Generalisations

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Three-dimensional topological lattice models with surface anyons Field Theory

Field theory

• ν = 1/2 bosonic hall effect → k=2 Chern-Simons theory.

• Ground-space of 3D semion model on solid donut ⇔ Hilbert space of compact k = 2 Chern-Simons theory on surface.

Using ^µνρσF_µνF_ρσ = ^µνρσ∂_µ(A_ν∂_ρA_σ) suggests:

SFF[A] = Z

d⁴x

k

16π^µνρσFµνFρσ+ AµJ^µ

,

n = -1ê 2

- -

+

^{n = +1}^{ê 2}

+

k = 2 Chern - Simons

Bulk k = 2 F ^ F theory

k = -2 Chern - Simons

thickeningDSem.nb 3

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Field Theory

Field theory

SFF[A] = Z

d⁴x

k

16π^µνρσFµνFρσ+ AµJ^µ

,

• Correct GS degeneracy, and anyonic statistics between surface particles (J).

• No deconfined point particles in the bulk: If J is

non-vanishing in the bulk, the partition function becomes:

Z

DA_Se^4π^ik^S^CS^[A^S^]+i^{R J}^S^·A^Sδ

J⁰ = − k

4π^ijk∂i∂jA_kin bulk

• Pairs of bulk point particles connected by a line defect in A (corresponds to the presence of a energetic line defect in lattice model).

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Three-dimensional topological lattice models with surface anyons Generalisations

Generalisations

• 3D semion model

⇔ k = 2 F ∧ F theory ⇔ k = 2 surface CS theory.

• Can realise many compact chiral (non-abelian) Chern-Simons theory on the surface of a Walker-Wang model.

Walker-Wang model based on category SUH2Lk

Chiral SUH2Lk

Chern-Simons anyon

theory on surface

thickeningCS.nb 5

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Generalisations

• e.g. To realise an SU(2)₂ Chern-Simons theory (Ising anyon theory), construct a Walker-Wang model based on the Ising category.

• Hilbert space: 3-state system on each edge of the lattice.

Labels {0,¹₂, 1} (or {I , σ, ψ, })

• Allowed vertices:

I s s y

• ...more complicated graphical rules determining ground states.

• Leads to non-abelian anyons on the surface of the model.

s s

y

y y

Walker-Wang model based on category SUH2L2

Chiral SUH2L2

Chern-Simons theory on surface

6 thickeningCS.nb

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Conclusion

• We described the ‘surface topological order’ of the 3D semion model.

• Explicitly constructed the chiral surface semions.

• Made more concrete the correspondence:

Lattice model ⇔ FF -theory.

• Works for any Chern-Simons theory!

• Known that

3D toric code ⇔ 3D superconductor ⇔ bF -theory (Hansson et al. (2004))

• Speculate:

3D semion model ⇔ Novel phase?! ⇔ F ∧ F -theory

• Putative novel phase: Gapped bulk with confined excitations, deconfined anyons on boundary. P and T odd low-energy Hamiltonian.

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Generalisations

Thanks for listening!