Toric-code model Z 2 topological order, Z 2 gauge theory

Transcription

1 Toric-code model 2 tooloical order, 2 aue theory Dancin rules: Φ str = Φ str, Φ str = Φ str The Hamiltonian to enforce the dancin rules: I le i ede H = U Q I I F, Q I = σi z, F = Ground state wave function Φ = const. les of I edes of σ x i

2 Stable round state deeneracy and too. order The hamiltonian is a sum of commutin oerators [F, F ] = 0, [Q I, Q I ] = 0, [F, Q I ] = 0. F 2 = Q 2 I = 1 Ground state Ψ rnd : F Ψ rnd = Q I Ψ rnd = Ψ rnd, E rnd = 2UN cell N cell e e Quasiarticle excitation enery a Δ Q = 2U, Δ F e = 2 Ground state deeneracy Identities I Q I = 1, F = 1. Number of indeendent quantum numbers F = ±1, Q I = ±1 on torus: N label = 2 2N cell2 N cell/4 Number of states on torus: N label = 2 2N cell2 N cell e o o o o D tor =4 H is a function of F, Q I. The deeneracy of any eienstates is 4. On enus surface, round state deeneracy D = 4 The above deenerate round states form a code, which has a lare code distance of order L (the linear size of the system).

3 Stable aed round state aed quantum hase E E E E - Second-order transition oint = aless state - First-order transition oint = unstable aed state aed quantum hase = stable aed round state Stable round state deeneracy Gaed quantum hase However, for a lon time, we thouht that without symmetry, the stable round state deeneracy always = 1 with symmetry, the stable round state deeneracy 1, symmetry breakin = emerence of round state deeneracy which is stable aainst any erturbation that reect the symmetry. Δ Δ ε

4 Stable aed round state aed quantum hase E E E E - Second-order transition oint = aless state - First-order transition oint = unstable aed state aed quantum hase = stable aed round state Stable round state deeneracy Gaed quantum hase However, for a lon time, we thouht that without symmetry, the stable round state deeneracy always = 1 with symmetry, the stable round state deeneracy 1, symmetry breakin = emerence of round state deeneracy which is stable aainst any erturbation that reect the symmetry. The above tooloy-deendent round state deeneracy D, is stable aainst any erturbations: =1 =0 =2 a new kind of order too. order Wen 89; Wen-Niu 90 De.=1 De.=D De.=D Δ Δ 1 2 ε

5 Double-semion theory Dancin rules: Φ str = Φ str, Φ str = Φ str The Hamiltonian to enforce the dancin rules: I le i ede le H = U I Q I 2 Q I = les of I σz i, (F + h.c.), F = ( edes of σx j )( les of i 1 σ z j 2 ) Ground state wave function Φ = ( ) c, where c is the number of loos in the strin confiuration.

6 More eneral atterns of lon-rane entanlement Generslize the 2 /double-semion ( dancin ) rule: Φ str = Φ str, Φ str = ±Φ str Grahic state: m More eneral wave functions are defined k on rahs, with N + 1 states on links i=0,...,n and N v = N ij k states on vertices: γ α= j More eneral local rule: F-move Levin-Wen, 2005; Chen-Gu-Wen, 2010 i j k N α N kjn ( N nil F-move: Φ β m = F ijk;mαβ χ l;nχδ Φ δ n l The matrix F ijk l (F ijk l ) mαβ nχδ n=0 χ=1 δ=1 n i j k = local unitary transformation l β 1,...,N v λ ) l

7 Consistent conditions for F ijk;mαβ l;nχδ : the entaon identity i j k l α β m χ n Φ Φ ( ( i j k l α β m χ n i j k l α β m χ n can be trans. to ) ) = q,δ,ɛ F mkl;nβχ ;qδɛ Φ = t,η,ϕ F ijk;mαβ n;tηϕ Φ i j k l δ φ γ q s ( ( i j α k l δ m ε q i j k l η ϕ t n χ = t,η,κ;ϕ;s,κ,γ;q,δ,φ F ijk;mαβ n;tηϕ throuh two different aths: ) ) = ( q,δ,ɛ;s,φ,γ F mkl;nβχ ;qδɛ F ijq;mαɛ ;sφγ Φ ( = t,η,ϕ;s,κ,γ F ijk;mαβ n;tηϕ F;sκγ itl;nϕχ Φ F itl;nϕχ ;sκγ F jkl;tηκ s;qδφ Φ i j k l δ φ γ q s The two aths should lead to the same LU trans.: Fn;tηϕ ijk;mαβ F;sκγ itl;nϕχ F jkl;tηκ s;qδφ = F mkl;nβχ ;qδɛ F ijq;mαɛ ;sφγ t,η,ϕ,κ ɛ Such a set of non-linear alebraic equations is the famous entaon identity. Thier solution N ij k, F ijk;mαβ l;nχδ Unitary fusion cateory (UFC) strin-net states. i j k l δ φ γ q s i j η k l t κ γ s ) ),

8 A comlete characterization of 2+1D tooloical order Both 2 tooloical order and double-semion tooloical order have the same deeneracy 4 on enus surfaces. Do they have the same tooloical order and belon to the same hase?

9 A comlete characterization of 2+1D tooloical order Both 2 tooloical order and double-semion tooloical order have the same deeneracy 4 on enus surfaces. Do they have the same tooloical order and belon to the same hase? Non-Abelian eometric hases Wilczek-ee 84 of the deenerate round state by deformin the torus: (1) Ψ α Ψ α = T αβ Ψ β ˆT : (2) 90 rotation Ŝ: Ψ α Ψ α = S αβ Ψ β ˆT, Ŝ enerate the MCG SL(2, ) of torus T αβ, S αβ, c comlete characterization of tooloical order - D =1 number quasiarticle tyes - Eienvalues of T αβ quasiarticle fractional statistics

10 Measure too. order: Universal wavefunction overla Consider the round states Ψ α on torus T 2, and two mas, Ŝ = 90 rotation and ˆT = Dehn twist. The non-abelian eometric hases S, T via overla S αβ e f S L 2 +o(l 1) = Ψ α Ŝ Ψ β T αβ e f T L 2 +o(l 1) = Ψ α ˆT Ψ β S T

11 Measure too. order: Universal wavefunction overla Consider the round states Ψ α on torus T 2, and two mas, Ŝ = 90 rotation and ˆT = Dehn twist. The non-abelian eometric hases S, T via overla S αβ e f S L 2 +o(l 1) = Ψ α Ŝ Ψ β T αβ e f T L 2 +o(l 1) = Ψ α ˆT Ψ β For the first too. order: Ψ 1 = strin-lenth Ψ 2 = ( ) W x str-len Ψ 3 = ( ) W y str-len Ψ 4 = ( ) W x +W y str-len S T

12 Measure too. order: Universal wavefunction overla Consider the round states Ψ α on torus T 2, and two mas, Ŝ = 90 rotation and ˆT = Dehn twist. The non-abelian eometric hases S, T via overla S αβ e f S L 2 +o(l 1) = Ψ α Ŝ Ψ β T αβ e f T L 2 +o(l 1) = Ψ α ˆT Ψ β For the first too. order: Ψ 1 = strin-lenth Ψ 2 = ( ) W x str-len Ψ 3 = ( ) W y str-len Ψ 4 = ( ) W x +W y str-len < 0.8 small-loo hase Ψ α are the same state > 0.8 lare-loo hase Ψ α are four diff. states (a) S T =0.802 (c) (b)

13 Measure too. order: Universal wavefunction overla Consider the round states Ψ α on torus T 2, and two mas, Ŝ = 90 rotation and ˆT = Dehn twist. The non-abelian eometric hases S, T via overla S αβ e f S L 2 +o(l 1) = Ψ α Ŝ Ψ β T αβ e f T L 2 +o(l 1) = Ψ α ˆT Ψ β For the first too. order: Ψ 1 = strin-lenth Ψ 2 = ( ) W x str-len Ψ 3 = ( ) W y str-len Ψ 4 = ( ) W x +W y str-len < 0.8 small-loo hase Ψ α are the same state > 0.8 lare-loo hase Ψ α are four diff. states For the second too. order: S = (a) S T =0.802 (c) (b) , T =

14 Local and tooloical quasiarticle excitations In a system: H = x H x a article-like excitation: enery density = Ψ exc H x Ψ exc excitation enery density Ψ exc is the aed round state of H + δh tra ( x). Local quasiarticle excitation: Ψ exc = Ô( x) Ψ rnd round state enery density Tooloical quasiarticle excitations Ψ exc Ô( x) Ψ rnd for any local oerators Ô( x) Tooloical quasiarticle tyes: if Ψ exc = Ô( x) Ψ exc, then Ψ exc and Ψ exc belon to the same tye. Number of tooloical quasiarticle tyes is an imortant tooloical invariant that characterizes the tooloical order. Only tooloical quasiarticles can carry fractional statistics and fractional quantum numbers.

15 The strin oerators in the 2 tooloically ordered state: the creation oerator of tooloical quasiarticle Toric code model: H = U I Q I F Q I = les of I σz i, F = edes of σx i Tooloical excitations: Q I = 1 Q I = 1 F = 1 F = 1 Tye-I strin oerator W I = strin σx i σ z σ σ z z σ x I i σ x σ x Tye-II strin oerator W II = strin* σz i Tye-III strin o. W III = strin σx i les σz i [H, WI closed ] = [H, WII closed ] = [H, WIII closed ] = 0. WI closed Ψ rnd = WII closed Ψ rnd = WIII closed Ψ rnd = Ψ rnd F : closed tye-i strin oertors. Q I : closed tye-ii strin oertors.

16 The strin oerators in the 2 tooloically ordered state: the creation oerator of tooloical quasiarticle Toric code model: H = U I Q I F Q I = les of I σz i, F = edes of σx i Tooloical excitations: Q I = 1 Q I = 1 F = 1 F = 1 σ z σ σ z z σ x I i σ x σ x Tye-I strin oerator W I = strin σx i e-tye. e e = 1 Tye-II strin oerator W II = strin* σz i m-tye. m m = 1 Tye-III strin o. W III = strin σx i les σz i ɛ-tye = e m [H, WI closed ] = [H, WII closed ] = [H, WIII closed ] = 0. WI closed Ψ rnd = WII closed Ψ rnd = WIII closed Ψ rnd = Ψ rnd F : closed tye-i strin oertors. Q I : closed tye-ii strin oertors. Oen strin oerators create tooloical excitations. Oen strin oerators are hoin oerators of too. excitations

17 Emerence of fractional sin/statistics Why electron carry sin-1/2 and Fermi statistics? Ends of strins (tye-i) are oint-like excitations, which can carry sin-1/2 and Fermi statistics? Fidkowski-Freedman-Nayak-Walker-Wan 06 Φ str = 1 strin liquid Φ str = Φ str rotation: and = : R 360 = has a sin 0 mod 1. has a sin 1/2 mod 1. Φ str = ( ) # of loos strin liquid Φ str 360 rotation: and = : R 360 = = Φ str ( i has a sin 1/4 mod 1. i has a sin 1/4 mod 1. )

18 Sin-statistics theorem (a) (b) (c) (d) (e) (a) (b) = exchane two strin-ends. (d) (e) = 360 rotation of a strin-end. Amlitude (a) = Amlitude (e) Exchane two strin-ends lus a 360 rotation of one of the strin-end enerate no hase. Sin-statistics theorem

19 Statistics of ends of strins The statistics is determined by article hoin oerators Levin-Wen 03: c a c t cb b t ba t bd d t bd t cb t ba t ba t cb t bd a a b c b d d a 4 c 3 1 b 2 d An oen strin oerator is a hoin oerator of the ends. The alebra of the oen strin oerator determine the statistics. For tye-i strin: t ba = σ x 1, t cb = σ x 3, t bd = σ x 2 We find t bd t cb t ba = t ba t cb t bd The ends of tye-i strin are bosons For tye-iii strins: t ba = σ x 1, t cb = σ x 3 σz 4, t bd = σ x 2 σz 3 We find t bd t cb t ba = t ba t cb t bd The ends of tye-iii strins are fermions