Lecture on quantum entanglement in condensed matter systems

Transcription

1 Lecture on quantum entanglement in condensed matter systems Shinsei Ryu University of Chicago January 12, 2018

2 Overview Quantum entanglement as an order parameter SPT phases (free systems) (1+1)d CFTs Perturbed CFTs (2+1)d topologically ordered phases... Developing theoretical/computational tools: DMRG, MPS, PEPS, MERA, and other tensor networks Other applications ETH and many-body localization, thermalization and chaos in dynamical systems, etc. Applications to physics of spacetime

3 Phases of matter Spontaneous symmetry breaking Phases of matter No spontaneous symmetry breaking Ordered phases Quantum disordered phases Gapless Gapped Gapless Gapped Continuous symmetry-broken phases Discrete symmetry-broken phases Quantum critical disordered phases and critical points Gapped quantum disordered phases No topological order Topological order Other phases Symmetry breaking coexists with topological order... Trivial phases Short-range entangled states (a.k.a "invertible" states) Symmetry Topologically ordered phases Long-range entangled states Symmetry Symmetry protected topological phases (SPT phases) Symmetry enriched topological phases (SET phases)

4 Entanglement and entropy of entanglement (0) States of your interest, e.g., ρ tot = Ψ Ψ. (i) Bipartition Hilbert space H = H A H B. (ii) Partial trace: ρ A = Tr B Ψ Ψ = j p j ψ j A ψ j A ( j p j = 1) (1) (iii) von Neumann Entanglement entropy: S A = Tr A [ρ A ln ρ A] = j p j ln p j (2) (iv) Entanglement spectrum ρ A exp( H e)/z: {ξ i} where p i =: exp( ξ i)/z (3)

5 Mutual information: I A:B S A + S B S A B (4) Rényi entropy: Note that S A = lim q 1 R (q) The Rényi mutual information: R (q) A = 1 1 q ln(tr ρq A ). (5) A. {R(q) A }q = entanglement spectrum. I (q) A:B R(q) A + R(q) B R(q) A B (6) Other entanglement measures, e.g., entanglement negativity.

6 Some key properties If ρ tot is a pure state and B = Ā, SA = SB. If ρ tot is a mixed state (e.g., ρ tot = e βh ), S A S B even when B = Ā, If B =, S A = S thermal. Subadditivity: i.e., the positivity of the mutual information: I A:B = S A + S B S A+B 0. Strong subadditivity S A+B S A + S B. (7) S B + S ABC S AB + S BC (8) By setting C =, we obtain the subadditivity relation.

7 ES in non-interacting systems Consider the ground states GS of free (non-interacting) systems, and bipartitioning H = H L H R. When ρ tot = GS GS is a Gaussian state, H e is quadratic [Pesche (02)]. H e = ψ I K IJψ J, I = r, σ, i,... (9) I,J L H e can be reconstructed from 2pt functions: C IJ := GS ψ I ψ J GS. ( ) CL C LR C =, C RL = C LR. (10) C RL Correlation matrix is a projector: C R C 2 = C, Q 2 = 1 (Q IJ := 1 2C IJ). (11) Entanglement Hamiltonian: H e = ψ I K IJψ J, K = ln[(1 C L)/C L]. (12) I,J L

8 E.g. the integer quantum Hall eect A prototype of topological phases Characterized by quantized Hall conductance σ xy = (e 2 /h) (integer). Gapped bulk, gapless edge Robust against disorder and interactions Chiral edge states in ES Figure: Physical v.s. entanglement spectra of a Chern insulator [SR-Hatsugai (06)]

9 1d lattice fermion model: E.g. the SSH model H = t i ( a i b i + h.c. ) + t i ( b i a i+1 + h.c. ) (13) Phase diagram: Physical spectrum, entanglement spectrum, entanglement entropy. Figure: [SR-Hatsugai (06)]

10 Symmetry-protected degeneracy in ES Robust zero mode in ES; 2-fold degeneracy for each level. S A = A log ξ/a 0 + log 2 Degeneracy is symmetry-protected; Symmetry: a i a i, bi b i. (Class D or AIII/BDI topological insulator) Symmetry-protected degeneracy is an indicator of symmetry-protected topological (SPT) phases. [Pollmann-Berg-Turner-Oshikawa (10)]

11 Symmetry-protected topological phases (SPT phases) "Deformable" to a trivial phase (state w/o entanglement) in the absence of symmetries. (Unique ground state on any spatial manifold "invertible") But sharply distinct from trivial state, once symmetries are enforced. Example: SSH model, time-reversal symmetric topological insulators, the Haldane phase Symmetry-breaking paradigm does not apply: no local order parameter

12 Entanglement spec. and non-spatial symmetry How about symmetry? Corr. matrix inherits symmetries of the Hamiltonian ψ I U IJψ J, H phys U H phys U = H phys, Q U QU = Q (14) Non-spatial symmetry, the sub block of corr. matrix inherits symmetries: Q L U Q LU = Q L (15) So does the entanglement Hamiltonian. This may result in degeneracy in the ES.

13 Another example Spin-1 Antiferromagnetic spin chain H = S j S j+1 + U zz (Sj z ) 2 (16) j j Figure: [Pollmann-Berg-Turner-Oshikawa (10)]

14 View from Matrix product states Matrix product state representation: Ψ(s 1, s 2, ) = A s 1 i 1 i 2 A s 2 i 2 i 3 A s 3 i 3 i 4 s a = 1, 0, 1 {i n=1, } Symmetry action: for g, h Symmetry group, we have U(g) acting on physical Hilbert space: U(g)U(h) = U(gh) U(g) s s A s = V 1 (g)a s V (g)e iθg (17) Symmetry acts on the internal space projectively: V (g)v (h) = e iα(g,h) V (gh) (18) [Chen et al (11), Pollmann et al (10-12), Schuch et al (11)]

15 (Entanglement spec) 2 and SUSY QM From C 2 = C: CL 2 C L = C LRC RL, Q LC LR = C LRQ R, C RLQ L = Q RC ReL, CR 2 C R = C RLC LR (19) Introduce: ( Q 2 S = 1 L 0 0 Q 2 R ) ( 0 2CLR, Q = 0 0 ) (, Q = ) C RL 0 (20) SUSY algebra [S, Q] = [S, Q ] = 0, {Q, Q } = S, {Q, Q} = {Q, Q } = 0 (21)

16 Entanglement spec. and spatial symmetries L/R = fermionic/bosonic sector; C L,R intertwines the two sectors: H L Spatial symmetry O: choose bipartitioning s.t. O = ( 0 OLR O RL 0 Symmetry of entanglement Hamiltonian: C LR C RL H R (22) O : H L H R (23) ), O LRO LR = ORLO RL = 1 (24) Q LC LRO LR = CLRO LR Q L (25) [Turner-Zhang-Vishwanath (10), Hughes-Prodan-Bernevig (11), Fang-Gilbert-Bernevig (12-13), Chang-Mudry-Ryu (14)]

17 Graphene with Kekule order Kekule distortion in graphene Degeneracy protected by inversion Entanglement spec. is more useful than physical spec.

18 Short notes: Conformal eld theory in (1+1)d Scale invariance in (1+1)d conformal symmetry (Polchinski) Conformal symmetry is innite dimensional. Holomorphi-anti-holomorphic factorization Innite symmetry generated by stress energy tensor T (z) = Virasoro algebra + n= L nz n 2, T ( z) = + n= L n z n 2, (26) [L m, L n] = (m n)l m+n + c 12 (m3 m)δ m, n (27) Characterized by a number c central charge (among others)

19 Short notes: CFT in (1+1)d Structure of the spectrum: tower of states: h, N; j h, N; j, L 0 h, N; j = (h + N) h, N; j. L 0 h, N; j = ( h + N) h, N; j. (28) In other words: H = h, h n h, hv h V h, (29) n h, h: the number of distinct primary elds with conformal weight (h, h). (For simplicity, we only consider the diagonal CFTs with n h, h = δ h, h.)

20 Central charge c = Weyl anomaly; at critical points, there are emergent scale invariance, but this emergent symmetry is broken by an anomaly. c (number of degrees of freedom) c shows up in free energy and specic heat, etc: Note: v is non-universal. c V = πc 3vβ Can be extracted from the entanglement entropy scaling: (30) S A = c log R + (31) 3 RG monotone. (Zamolodchikov c-function; entropic c-function)

21 Radial and angular quantization w(z) = log z z = x + iy w = u + iv w(z) = log z CFT on a plane CFT on a cylinder Radial evolution Hamiltonian Angular evolution (Entanglement or Rindler Hamiltonian) Hamiltonian with boundary

22 Radial ow Finite size scaling CFT on a cylinder of circumference L L H = 1 dv T uu(u 0, v) 2π 0 = 1 dw T (w) + (anti-hol) (32) 2π C w Conformal map: cylinder plane w = L 2π log z dw T (w) = dz dw ( 2π C w C z dz L ( L = dz C z 2π ) 2 [ z 2 T (z) c 24 ) [ zt (z) c 24 ] 1 z CFT Hamiltonian on a cylinder can be written in terms of dilatation operator L 0 + L 0 on a plane: H = 2π ( L 0 + L L 0 c ) 24 ] (33) (34)

23 Gives relation between stress tensor (on z-plane) to a physical Hamiltonian on a nite cylinder. Level spacing scales as 1/L. Levels are equally spaced (within a tower) The c/24 1/L part allows us to determine c (numerically). (the extensive part A L has to be subtracted.) Degeneracy full identication of the theory

24 Radial ow Numerics XX model: H = j ( S x j S x j+1 + S y j Sy j+1) For a given tower, all levels are equally spaced. Level spacing scales as 1/L.

25 Angular ow

26 Angular ow Corner transfer matrix Corner transfer matrix A σ σ and partition function Z = Tr A 4 [Baxter (80's); Figures:Wikipedia]

27 Angular ow = Entanglement (Rindler) Hamiltonian In Euclidean signature, z = x + iy = e w = e u+iv maps the complex z-plane to a cylinder. In Minkowski signature: (t, x) (u, v) (Rindler coordinate): x = e u cosh v, t = e u sinh v. In the Rindler coord., the half of the 2d spacetime is inaccessible (traced out). Radial evolution in the complex z-plane u-evolution in the cylinder Angular evolution in the complex z-plane v-evolution in the cylinder = entanglement (or Rindler) Hamiltonian [Figures: Wikipedia]

28 Rindler Hamiltonian Constant u trajectories = World-lines of observer with constant acceleration a where a = 1 in our case. Accelerated observer in Minkowski space = Static observer in Rindler space Unruh eect: Vacuum is observer dependent. Observer in an accelerated frame (Rindler observer) sees the vacuum of the Minkowski vacuum as a thermal bath with Unruh temperature T = a 2π = 1 2π (35) This is due to a Rindler horizon and inability to access the other part of spacetime. Rinder coordinates covers with metric only covers x > t (the right Rindler wedge). Left Rindler wedge is dened by ds 2 = e 2au ( dv 2 + du 2 ) (36) x = e u cosh v, t = e u sinh v.

29 Entanglement Hamiltonian for nite interval w(z) = ln(z + R)/(z R) Entanglement hamiltonian on nite interval [ R, +R] Hamiltonian with boundaries Transforming from strip to plane: H = du T vv v0 =π = Entanglement spec: 1/ log(r) scaling +R R E.g., Casini-Huerta-Myers (11), Cardy-Tonni (16) dx (x2 R 2 ) T yy y=0 (37) 2R

30 SSH chain Entanglement spectrum of CFT GS: H E = const. H = t i ( ) a i b i + h.c. + t i L 0 log(r/a) (b i a i+1 + h.c. ) (38) with t = t 20 Entanglement spectrum / log(na/a0) Figure: [Cho-Ludwig-Ryu (16)]

31 Numerics Figure: [Lauchli (13)]

32 Remarks: What is an analogue of the radial direction? It is related to the so-called sine-square deformation (SSD). [Gendiar-Krcmar-Nishino (09), Hikihara-Nishino (11),...] Evolution operator: H = π In the limit R 0, 0 dv T uu(u 0, v) = r 2 0 H L 0 2π 0 dθ cos θ + cosh u0 sinh u 0 T rr(r, θ) (39) ( ds sin 2 πs ) ( L T rr L 2π, 2πs ) L [Ishibashi-Tada (15-16); Okunishi (16); Wen-Ryu-Ludwig (16)] (40)

33 Perturbed CFT Add a relevant perturbation S = S + g d 2 z φ(z, z) (41) and go into a massive phase; Consider the entanglement Hamiltonian for half space. The above conformal map leads to an exponentially growing potential u2 2π S + g du dv e yu Φ(w, w) (42) u 1 0 with length scale log(ξ/a).

34 Entanglement Spectrum Entanglement spectrum for gapped phases is given by a CFT with boundaries (Boundary CFT in short) of a nearby CFT Partition function: Z AB = Tr AB e He (43) Here, A = vacuum and B = SP T. [RG domain wall idea:] Spectrum is given by half of the full CFT: L 0 H e = const. log(ξ/a)

35 Numerics: SSH model Spectrum depends on type of boundaries (type of SPTs): There is symmetry-protected degeneracy in the topological phase Entanglement spectrum Entanglement spectrum Subsystem size (NA) Subsystem size (NA)

36 BCFT and SPT Entanglement spectrum for gapped phases is given by BCFT When the gapped phase is an SPT, the topological invariant can also be computed from BCFT. [Cho-Shiozaki-Ryu-Ludwig (16)] Switching space and time, Z = Tr e β/ll 0 = A e l/β(l 0+ L 0 ) B (44) we introduce boundary states A and B : (L n L n) B = 0, n Z (45) From B, the corresponding SPT phase can be identied by the phase g B h = ε B (g h) B h, g, h G where B h is the boundary state in h-twisted sector. discrete torsion phase ε B(g h) H 2 (G, U(1)). This phase is called the

37 Boundary states as gapped states Conformally invariant boundary states, (L n L n) B = 0. Boundary states B do not have real-space correlations: B e ɛh O 1(x 1) O n(x n)e ɛh B / B e 2ɛH B where x 1,, x n refer n dierent spacial positions. In the limit ɛ 0 with x i x j the correlation function factorizes and does not depend on x i x j. Boundary states represent a highly excited state within the Hilbert space of a gapless conformal eld theory and can be viewed as gapped ground states. [Miyaji-Ryu-Takayanagi-Wen (14), Cardy (17), Konechny (17)]

38 Free fermion example A massive free massive Dirac fermion in (1+1)d: [ ] H = dx iψ σ z xψ + mψ σ xψ, ψ = (ψ L, ψ R ) T The ground state of this Hamiltonian is given by GS = exp m ( L k) ψ k>0 m 2 + k 2 Lk + k ψ Rk + ψ R k ψ G L G R where ψ L,Rk is the Fourier component of ψ L,R(x), and G L,R is the Fock vacuum of the left- and right-moving sector. In the limit m (m/(v F k) ), GS reduces to the boundary states of the free massless fermion theory.

39 More details SPT phases in (1+1)d are classied by group cohomology H 2 (G, U(1)). [Chen-Gu-Liu-Wen (02)] Recall: V (g)v (h) = e iα(g,h) V (gh) (46) CFT context: Discrete torsion phases in CFT [Vafa (86)...] and in BCFT [Douglas (98)...]. Discrete torsion phases and entanglement spectrum (symmetry-protected degeneracy): Twisted partition function: ZAB h βh = Tr HAB [ĥe open AB vanishes when A B. (symmetry-enforced vanishing of partition function). Exchange time and space, Z h AB = h A e l 2 Hclosed B h and insert g to show [ε B(g h) ε A(g h)]z h AB = 0 ]

40 RG and entanglement: entropic c-theorem Entropic c-function [Casini (04)] : c E(R) := 3R ds(r) dr At critical points, c E = c (central charge). From strong subadditivity: can argue that S is concave w.r.t. log R: t (47) S A + S B S A B + S A B (48) A B t = x A B t = x A B x Taking the limit:r R: 2S( rr) S(R) + S(r) (49) c E(R) 3 = S (R) + RS (R) 0 (50)

41 Remark: F-theorem Is there an analogue of c and c-theorem in (2+1)d? (No weyl anomaly in (2+1)d) EE of a disc D of radius R [Ryu-Takayanagi (06), Myers-Sinha (10)]: S D(R) = α 2πR ɛ F (R) (51) F at the critical point is a universal constant. C.f. topological entanglement entropy. F is related to the partition functions on a sphere S 3, F = log Z(S 3 ) [Casini-Huerta-Myers (11)]. F-theorem: [Jaeris et al (11), Klebanov et al (11)]: F UV F IR Entropic F function: [Liu-Mezei (13)] ( F(R) = R ) R 1 S D(R) (52) F(R) CF T = F and F (R) 0 [Casini-Huerta (12)] Applications [Grover (12),...] Stationarity?

42 Topological phases of matter Topologically ordered phases: phases which support anyons ( support topology dependent ground state degeneracy) E.g., fractional quantum Hall states, Z 2 quantum spin liquid, etc. Quantum phases which are not described by the symmetry-breaking paradigm. (I.e., Landau-Ginzburg type of theories) Instead, characterized by properties of anyons (fusion, braiding, etc.) (I.e., topological quantum eld theories)

43 Algebraic theory of anyons (Bosonic) topological orders are believed to be fully characterized by a unitary modular tensor category (UMTC). (i) Finite set of anyons {1, a, b,...} equipped with quantum dimensions {1, d a, d b,...} (d a 1). Total quantum dimension D: D = d 2 a (53) (ii) Fusion a b = c N c abc. (iii) The modular T matrix, T = diag (1, θ a, θ b,...) where θ a = exp 2πih a is the self-statistical angle of a with h a the topological spin of a. (iv) The modular S matrix encodes the braiding between anyons, and given by (dened by) S ab = 1 D c a N c ab θ c θ aθ b d c. (54)

44 Chiral central charge There may be topologically ordered phases with the same braiding properties, but dierent values of c, the chiral central charge of the edge modes. Albeit the same braiding properties, they cannot be smoothly deformed to each other without closing the energy gap. Topological order is conjectured to be fully characterized by (S, T, c)

45 Ground states and S and T Ground state degeneracy depending on the topology of the space (topological ground state degeneracy), related to the presence of anyons [Wen (90)] Ground state degeneracy on a spatial torus, { Ψ i }. S and T are extracted from the transformation law of { Ψ i } [Wen (92)]

46 Topologically ordered phases and quantum entanglement Consider: the reduced density matrix ρ A obtained from a ground state GS of a topologically-ordered phase by tracing out half-space. ρ A Tr R e ɛh B.S. B.S. e ɛh [Qi-Katsura-Ludwig (12), Fliss et al. (17), Wong (17)] Dierent (Ishibashi) boundary states correspond to dierent ground states With this explicit form of the reduced density matrix, various entanglement measures can be computed: [Wen-Matsuura-SR] the entanglement entropy the mutual information the entanglement negativity

47 Bulk-boundary correspondence Bulk anyon twisted boundary conditions at edge: Bulk wfn Ψ i boundary partition function χ i Bulk S and T matrices acting on Ψ i on spatial torus S and T matrices acting on boundary partition function χ i on spacetime torus [Cappelli (96),...] ( χ a e 4πβ ) l = a S aa χ a ( e πl ) β (55)

48 Conformal BC: L n b = L n b ( n Z) Ishibashi boundary state: h a N=0 d ha (N) j=1 h a, N; j h a, N; j (56) Topological sector dependent normalization (regularization): h a = e ɛh na h a so that h a h b = δ ab. (57) More generically, one can consider a superposition ψ = a ψa ha Reduced density matrix: ρ L,a = Tr R( h a h a ) = 1 e 8πɛ (h l a+n 24 c ) h a, N; j h a, N; j. (58) n a N,j

49 Some details Trance of the reduced density matrix: Tr L (ρ L,a) n = 1 n n a Modular transformation ( χ a e 8πnɛ l ) = ( χ a e 8πnɛ l a S aa χ a ( ) ) χ a e 8πnɛ l = ( e 2nɛ πl ) ( χ a e 8πɛ l ) n (59) S a0 e πcl 48nɛ (l/ɛ ), (60) i.e., only the identity eld I, labeled by 0 here, survives the limit. Hence, in the thermodynamic limit l/ɛ : Tr L (ρ L,a) n a S = aa χ a [ a S aa χ a ( ) e 2nɛ πl ( e πl 2ɛ ) ] n e πcl 48ɛ ( n 1 n) (S a0) 1 n, (61)

50 Final result: where S (n) L S vn L = 1 + n n πc 48 l ɛ ln D + 1 ln d1 n a 1 n = πc 24 l ln D + ln da (62) ɛ S a0 = d a/d (63) is the quantum dimension.

51 Lessons Entanglement cut may be more useful than physical cut. Entanglement and universal information of many-body systems. Entanglement can tell the direction of the RG ow. Entanglement and spacetime physics Entanglement has a topological interpretation in particular in topological eld theories.