Universal phase transitions in Topological lattice models

Transcription

1 Universal phase transitions in Topological lattice models F. J. Burnell Collaborators: J. Slingerland S. H. Simon September 2, 2010

2 Overview Matter: classified by orders Symmetry Breaking (Ferromagnet) Topological (Quantum Hall) Phase transitions between different orders Landau-Ginzburg Symmetry dictates: initial and final phase; Critical behavior

3 Symmetry breaking: Symmetry groups G 1 and G 2 tell us: Particles: representations of G 1 (G 2 ) G 1 = Z 2 G 2 = 1 2 nd order phase transition? Yes, if G 2 G 1 Condensation transition: G 1 X =0 G 2 X = S i Field theory of the transition: (Ginzburg-Landau theory ) ( ) 2 L = ( t φ) 2 φ m 2 φ 2 λφ 4

4 Topological Order (TQFT) Characterize by Topological ground state degeneracy (1 on the sphere, multiple on the torus) Excitations with anyonic statistics gapless edge modes

5 Topological Order (TQFT) Characterize by Topological ground state degeneracy (1 on the sphere, multiple on the torus) Excitations with anyonic statistics gapless edge modes effective topological field theory (TQFT) gauge theory Fixes GS degeneracy, edge modes, and quasi-particle statistics σ model of topological phases

6 Why Topological Order? e.g. Quantum Hall effect Wave functions at filling 5/2 Moore-Read Halperin 331 σ xy = 5 e 2 2 h σ xy = 5 e 2 2 h Non-Abelian anyons Abelian anyons Same Hall conductivity Different edge theories and excitations

7 Topological symmetry breaking: condensation in TQFTs When can I condense a boson to get from TQFT 1 to TQFT 2? Symmetry breaking Topological order G 1, G 2 T 1, T 2 local order parameter X Higgs mechanism

8 Superconductor Hansson, Oganesyan, and Sondhi Ann. Phys. 313, 497 Start with electrons and photons TQFT 1: U(1) gauge theory with matter

9 Superconductor Hansson, Oganesyan, and Sondhi Ann. Phys. 313, 497 Start with electrons and photons TQFT 1: U(1) gauge theory with matter Condense c k c k (charge 2e ) Cooper pair

10 Superconductor Hansson, Oganesyan, and Sondhi Ann. Phys. 313, 497 Start with electrons and photons TQFT 1: U(1) gauge theory with matter Condense c k c k (charge 2e ) Cooper pair Vortices: φ = n 2e Vortex

11 Superconductor Hansson, Oganesyan, and Sondhi Ann. Phys. 313, 497 Start with electrons and photons TQFT 1: U(1) gauge theory with matter Condense c k c k (charge 2e ) Cooper pair Vortices: φ = n 2e Charge picks up Berry s phase of π when traveling around the vortex Vortex electron

12 Superconductor Hansson, Oganesyan, and Sondhi Ann. Phys. 313, 497 Start with electrons and photons TQFT 1: U(1) gauge theory with matter Condense c k c k (charge 2e ) Cooper pair Vortices: φ = n 2e Charge picks up Berry s phase of π when traveling around the vortex electron Vortex

13 Superconductor Hansson, Oganesyan, and Sondhi Ann. Phys. 313, 497 Start with electrons and photons TQFT 1: U(1) gauge theory with matter Condense c k c k (charge 2e ) Cooper pair Vortices: φ = n 2e Charge picks up Berry s phase of π when traveling around the vortex Vortex electron Ψ Ψ

14 Superconductor Hansson, Oganesyan, and Sondhi Ann. Phys. 313, 497 Start with electrons and photons TQFT 1: U(1) gauge theory with matter Condense c k c k (charge 2e ) Cooper pair Vortices: φ = n 2e Charge picks up Berry s phase of π when traveling around the vortex TQFT 2: Z 2 gauge theory Vortex electron Ψ Ψ

15 Effects of Topological Symmetry Breaking Superconductor TSB Condense a boson: Condense a boson: = c c φ 0

16 Effects of Topological Symmetry Breaking Superconductor TSB Condense a boson: Condense a boson: = c c φ 0 Vacuum: Vacuum: Ω = (u + vc k c k ) 0 Ω = (1 + φ ) 0

17 Effects of Topological Symmetry Breaking Superconductor TSB Condense a boson: Condense a boson: = c c φ 0 Vacuum: Vacuum: Ω = (u + vc k c k ) 0 Ω = (1 + φ ) 0 Mix particles and holes: c c Identify particles that mix X φ Y

18 Effects of Topological Symmetry Breaking Superconductor TSB Condense a boson: Condense a boson: = c c φ 0 Vacuum: Vacuum: Ω = (u + vc k c k ) 0 Ω = (1 + φ ) 0 Mix particles and holes: c c Confine magnetic field (quantized vortices 2e ) Identify particles that mix X φ Y Confine excitations which braid non-trivially with condensate

19 Effects of Topological Symmetry Breaking Superconductor TSB Condense a boson: Condense a boson: = c c φ 0 Vacuum: Vacuum: Ω = (u + vc k c k ) 0 Ω = (1 + φ ) 0 Mix particles and holes: c c Confine magnetic field (quantized vortices 2e ) Identify particles that mix X φ Y Confine excitations which braid non-trivially with condensate Split particles related by broken symmetry

20 Topological symmetry breaking: condensation in TQFTs Condense a boson to get from TQFT 1 to TQFT 2? Non-Abelian Yang-Mills theories More general topological orders De Wild Propitius PhD thesis (F. A. Bais) Slingerland and Bais, PRB 79,

21 Questions about Topological Symmetry breaking Physical models that realize these transitions? lattice gauge theories Other TQFT s? How are excitations in TQFT 1 and TQFT 2 related? How do some excitations split? What is the field theory of the phase transition? Universality?

22 What follows 1. Condensation and confinement in lattice Ising gauge theory (Toric code) 2. SU(2) 2 SU(2) 2 Chern-Simons theory Ising gauge theory 3. Phase transitions between more general topological phases SU(2)k SU(2) k Chern-Simons theory and the Ising phase transition

23 I. Toric Code to Vacuum Toric code (Ising gauge theory with matter) A. Kitaev, Ann. Phys. 303 ( 03) 2 Edges σ z = ±1 (electric flux) E = 0 : σ z (i) = 1 σz =1 σz = 1 Magnetic flux: σ x flips all electric fluxes from + to

24 I. Toric Code to Vacuum Toric code (Ising gauge theory with matter) A. Kitaev, Ann. Phys. 303 ( 03) 2 Edges σ z = ±1 (electric flux) E = 0 : σ z (i) = 1 σz =1 σz = 1 Magnetic flux: σ x flips all electric fluxes from + to Hamiltonian : H = V ɛ e σz (i) P ɛ m σx (i) Plaquette and vertex projectors commute Exactly solvable

25 I. Toric Code to Vacuum Ground State Loop gas Excitations Boson e (Electric source) Boson m (Vortex) Fermion em (Both!) 1 Statistics : Z 2 topological order e with m, em : 1 m with e, em : 1

26 Phase transition and confinement Confining transition : Decrease gap to creating a vortex ɛ m ɛ m (1 α) Add an E 2 term that spontaneously creates and hops vortices: H = α σ z (i) edges i References: Fradkin and Shenker PRD 19, 3682 ( 79) Trebst et. al. PRL 98, ( 07) C. Castelnovo and C. Chamon PRB ( 08 ) Vidal PRB 79, ( 09) etc.

27 Phase transition and confinement H = V ɛ e σz (i) P (1 α)ɛ m σx (i) α σ z (i) edges i Small α :deconfined ( Toric Code ) QCP α 1:confined ( Trivial )

28 Effects of Topological Symmetry Breaking Superconductor Mix particles and holes: c c Confine magnetic field (quantized vortices 2e ) TSB Identify particles that mix X φ Y Confine excitations which braid non-trivially with condensate Split particles related by broken symmetry

29 II. SU(2) 2 SU(2) 2 Toric code String net lattice Hamiltonian Levin and Wen PRB 71, ( 05) Hilbert space : Edge labels 1, σ, ψ Allowed vertices ( E = 0 ): (1, 1, 1) (1, ψ, ψ) (σ, σ, ψ) (σ, σ, 1) Allowed label flips (construct B 2 term): = Add flux (gauge theory) fusion (CFT)

30 Lattice Hamiltonian: H = V ɛ V δ E1,E 2,E 3 P ɛ m P (0) P Vertex term P V : energy penalty for vertices that are not allowed! Plaquette projector P P : flips strings (changes their electric flux) Projects onto states with no magnetic flux Commuting projectors: the model is exactly solvable Topological order: SU(2) 2 SU(2) 2 Chern-Simons theory

31 Excitations Two sectors (R and L): particles can carry both electric and magnetic flux electric : R and L magnetic : L only σ R ψ R ψ ψ Particle Type ψ R, ψ L σ R, σ L, ψ R σ L, σ R ψ L σ R σ L ψ R ψ L Statistics Fermion Chiral anyon Boson Boson

32 Condensation Boson to condense: ψ vortex product state ψ R ψ L violates plaquettes has no electric flux Z 2 flux addition rules ψ ψ

33 Phase transition Condense ψ vortices: decrease mass gap for ψ vortex Pair-create ψ vortices: H = J x i=edges ( 1)nσ(i) deconfined ( SU(2) 2 SU(2) 2 ) QCP confined ( Toric Code )

34 Questions about Topological Symmetry breaking Physical models that realize these transitions? lattice gauge theories Other TQFT s? Levin-Wen +... How are excitations in TQFT 1 and TQFT 2 related? How do some excitations split? What is the field theory of the phase transition? Universality?

35 The confined phase Identified: ψ R ψ L (ψ R ψ L ) ψ R = ψ L Confined: any excitation with an odd number of σ R,L strings σ R, σ L, ψ R σ L, σ R ψ L Split: σ R σ L σ R σ L (σ, σ) 1 (σ, σ) ψ (σ, σ)1 : No electric flux (σ, σ) ψ : electric flux ψ Flux of ψ conserved once σ eliminated!

36 The confined phase Deconfined Particle Confined particle 1 1 ψ R ψ L Condensed: ψ R ψ L 1 ψ R, ψ L Identified: ψ R ψ L σ R,L Confined σ R ψ L, ψ R σ L Confined σ R σ L Split : (σ, σ) 1, (σ, σ) ψ Net spectrum: 1, ψ, (σ, σ) 1, (σ, σ) ψ Bosons (σ, σ)1, (σ, σ) ψ with a mutual braiding phase of 1 Fermion ψ which has mutual braiding statistics of 1 with both (σ, σ) s. Final theory: Toric code! (Z 2 Topological order) e (σ, σ) ψ m (σ, σ) 1 e m ψ

37 Questions about Topological Symmetry breaking Physical models that realize these transitions? lattice gauge theories Other TQFT s? Levin-Wen +... How are excitations in TQFT 1 and TQFT 2 related? How do some excitations split? What is the field theory of the phase transition? Universality

38 Toric code and TFIM T = 0: no electric sources are created. Dual to the transverse field Ising model: +1 i i no vortex spin up σ z (i) S x (i)s x 1 vortex spin down Electric flux Domain σ x S z (i) = ±1

39 Toric Code and TFIM T = 0: no electric sources are created. Dual to the transverse field Ising model: i i j x spin up σ z (i) S x (i)s x (j) spin down Electric flux Domain wall Vortex pair creation: Ising interaction

40 Toric Code and TFIM Another picture of the phase diagram: Small α :ToricCode ( Paramagnetic ) QCP (3D Ising) α 1:Trivial ( Ferromagnetic )

41 SU(2) 2 SU(2) 2 and TFIM Objective: SU(2) 2 SU(2) 2 ( Paramagnetic ) QCP 3D Ising Toric Code (Ferromagnetic )

42 SU(2) 2 SU(2) 2 and TFIM Effective Hamiltonian for the phase transition Ising! H = H 0 + J z 2 H 0 = V P ( P (0) P m e δ(e 1, E 2, E 3 ) + P ) P(ψ) P J x ( P (0) P i=edges + P(ψ) P ( 1) nσ(i) ) Effective TFIM!

43 SU(2) 2 SU(2) 2 and TFIM ψ-vortex Ising spin on the plaquette (Exact mapping = non-local) Plaquette terms: ( ) P (0) P + P(ψ) P 1 ( ) P (0) P P(ψ) P S z ψ ψψ Creation terms : Ising interaction S x (i)s x (j) σ domain wall (Ising gauge field) =( 1) n σ = i j S x (i)s x (j)

44 Questions about Topological Symmetry breaking Physical models that realize these transitions? lattice gauge theories Other TQFT s? Levin-Wen +... How are excitations in TQFT 1 and TQFT 2 related? How do some excitations split? What is the field theory of the phase transition? Universality Ising

45 Universality Why Ising transition? ψ L ψ R Ising spin (ψl ψ R ) 2 1 ψl ψ R carries no electric flux Pair-create ψ vortices without creating other excitations H commutes with n σ, n ψ Other theories with similar transitions? TFIM SU(2)k SU(2) k Transverse-field Potts Order q simple current

46 Conclusions Phase transitions between different topological orders Mechanism: Yang-Mills theory: Higgs General TQFT: Topological Symmetry Breaking Toy models realizing these Topological lattice Hamiltonians + perturbation Field Theory & Universality Condense Ising-like particle: TFIM transition +1 no vortex spin up i σz(i) i Sx Condense Potts-like particle: Transverse field Potts transition 1 vortex spin down Electric flux Do

47 Universality and Generalizations What is the field theory describing topological phase transitions? Symmetry breaking: Critical theory insensitive to microscopics; depends only on symmetry breaking Here: Critical theory is indifferent to many features of the topological order Condense a boson which behaves like an Ising spin Transition: TFIM Generalizations: Condense a boson which behaves like a q-state Potts spin (order q simple current) Effective theory of phase transitions: transverse field Potts model

48 Condensation in SU(2) 3 SU(2) 3 Lattice Hamiltonian (Levin-Wen type) Excitations: spins 0, 1/2 L,R, 1 L,R, 3/2 L,R Condense φ 3/2 L 3/2 R Boson with 2 states per site Carries no electric flux P (0) + P (φ) = Preserves spin on each edge mod 1 P (0) P (φ) = Interchanges 1/2-int int. spins Modify the Hamiltonian: H = ɛ V P V 1 [ɛ m (P (0) + P (φ)) 2 V P ( +J z P (0) P (φ))] + J x ( 1) 2s Effective Ising model of phase transition! e

49 Condensation in SU(2) 3 SU(2) 3 Uncondensed Condensed (0, 0), (0, 1), (1, 0), (1, 1) (0, 0), (0, 1), (1, 0), (1, 1) (3/2, 3/2), (3/2, 1/2), (1/2, 3/2), (1/2, 1/2) Identified with above (0, 1/2), (0, 3/2), (1, 1/2), (1, 3/2) Confined (1/2, 0, (3/2, 0), (1/2, 1), 3/2, 1) Confined Final theory:doubled Fibbonacci Generalizes to any odd k ( SO(3) k SO(3) k )

50 Condensation in SU(2) k SU(2) k (k even) Start: doubled Chern-Simons theory particles spin 0, 1/2,...k/2 in non-interacting R and L sectors Realizeable via a lattice Hamiltonian of commuting projectors (string net) R L symmetric boson: k/2r k/2 L, which violates only plaquettes in the lattice model Phase transition: condense k/2 R k/2 L Effective description of ground-states + condensing boson: TFIM Dual effective description: Ising gauge theory Phase transition: Ising type

51 Condensation in SU(2) k SU(2) k (k even) End: a new topological theory... Confinement of particles of net 1/2-integer spin Particle types: spin j L i R, j/2 L i/2 R, j = 0, 1, k/2 1, i = 0, 1,...k/2, i > j Splitting of k/4l k/4 R due to an emergent Z 2 symmetry in confined phase Features of the final theory k = 4: D 3 gauge theory k > 4...? In general, it is not a gauge theory...