Rigorous free fermion entanglement renormalization from wavelets

Transcription

1 Computational Complexity and High Energy Physics August 1st, 2017 Rigorous free fermion entanglement renormalization from wavelets arxiv: Jutho Haegeman Ghent University in collaboration with: Brian Swingle, Michael Walter, Jordan Cotler, Glen Evenbly, Volkher Scholz

2 OVERVIEW Tensor networks and quantum field theories MERA: quantum circuits, renormalization, wavelets One-dimensional Dirac fermions Fermi surface: Non-relativistic two-dimensional fermions Rigorous approximation result Outlook and extensions

3 TENSOR NETWORKS Quantum system of N spins: d H =(C d ) N = s 1,s 2,...,s s N 1 s 2 s N s n =1 d N dimensional vector rank N tensor s 1 s 2 s N approximate with a tensor network decomposition Diagrammatic notation vector: matrix: matrix product: Yang-Baxter equation: =

4 TENSOR NETWORKS Variational families of states for quantum many body systems, motivated by the structure of entanglement in low energy states (area law) = (MPS) 0i 0i 0i 0i 0i 0i 0i 0i 0i 0i 0i 0i 0i 0i classical simulation (PEPS) (MERA) Complexity scaling (MPS): Hastings; Arad, Kitaev, Landau, Vazirani, Vidick,

5 AND QUANTUM FIELD THEORY (1) Variational approach to lattice gauge theory (Hamiltonians) Very successful for (1+1)d QFT, e.g. Schwinger model TMR Byrnes et al, B Buyens et al, MC Bañuls et al, S Montangero et al, (Partial) string breaking for heavy probe charges: V Q (L)/g (m/g = 1) 25 V Q (L)/g (m/g =0.5) 25 V Q (L)/g (m/g =0.25) Q L g L g L g Boye Buyens, Jutho Haegeman, Henri Verschelde, Frank Verstraete, Karel Van Acoleyen, PRX 6, (2016)

6 AND QUANTUM FIELD THEORY (1) Variational approach to lattice gauge theory (Hamiltonians) Very successful for (1+1)d QFT, e.g. Schwinger model TMR Byrnes et al, B Buyens et al, MC Bañuls et al, S Montangero et al, (Partial) string breaking for heavy probe charges: 1.5 Ψ(z)γ 0 Ψ(z) (Q = 4.5) 2 E(z) /g (Q = 4.5) z g z g Boye Buyens, Jutho Haegeman, Henri Verschelde, Frank Verstraete, Karel Van Acoleyen, PRX 6, (2016)

7 AND QUANTUM FIELD THEORY (1) Variational approach to lattice gauge theory (Hamiltonians) Very successful for (1+1)d QFT, e.g. Schwinger model TMR Byrnes et al, B Buyens et al, MC Bañuls et al, S Montangero et al, (Partial) string breaking for heavy probe charges: 0.25 S Q (z) (L g = 0.55) x 0.25 S Q (z) (L g = 2.55) 0.25 S Q (z) (L g = 7.35) 0.25 S Q (z) (L g = 13.15) z g z g z g z g Boye Buyens, Jutho Haegeman, Henri Verschelde, Frank Verstraete, Karel Van Acoleyen, PRX 6, (2016)

8 AND QUANTUM FIELD THEORY (1) Variational approach to lattice gauge theory (Hamiltonians) Very successful for (1+1)d QFT, e.g. Schwinger model TMR Byrnes et al, B Buyens et al, MC Bañuls et al, S Montangero et al, Real time evolution: quenches, onset of thermalization? E(t)/g E(t)/g E β0 ±0.05/g t g t g t g Boye Buyens, Jutho Haegeman, Florian Hebenstreit, Frank Verstraete, Karel Van Acoleyen, arxiv: N(t) N(t) N β0 ± S(t)

9 AND QUANTUM FIELD THEORY (1) Variational approach to lattice gauge theory (Hamiltonians) Very successful for (1+1)d QFT, e.g. Schwinger model TMR Byrnes et al, B Buyens et al, MC Bañuls et al, S Montangero et al, Can be extended to (2+1)d (and (3+1)d?): first explorations by L Tagliacozzo, E Zohar,

10 AND QUANTUM FIELD THEORY (2) Tensor networks for continuous systems: Continuous MPS: (Verstraete & Cirac, 2010) Chiral condensate in the Gross-Neveu model (N ) λ(λ)σ/ Λ exact a fit b D = 6 D = 8 D = 10 D = 16 Continuous MERA: So far: only Gaussians Interesting analytical tool to investigate relation with holography Advertisement: PhD & PostDoc positions λ -1 (Λ) = [(N - 1)g(Λ) 2 ] -1

11 MERA, RENORMALIZATION & WAVELETS Multiscale entanglement renormalization ansatz (G Vidal) 0i 0i 0i 0i 0i 0i 0i 0i 0i 0i 0i 0i 0i 0i Coarse-grain quantum state Variational ansatz Disentangle high energy degrees of freedom Quantum circuit that prepares state Captures power law decay of correlations, logarithmic violation of area law in (1+1)d, Possible relation with holography (AdS/CFT correspondence)

12 MERA, RENORMALIZATION & WAVELETS Tensor network renormalization interpretation: (G Evenbly & G Vidal; S Yang, ZC Gu, XG Wen) d imaginary time evolution

13 MERA, RENORMALIZATION & WAVELETS Tensor network renormalization interpretation: (G Evenbly & G Vidal; S Yang, ZC Gu, XG Wen) 2d imaginary time evolution

14 MERA, RENORMALIZATION & WAVELETS Tensor network renormalization interpretation: (G Evenbly & G Vidal; S Yang, ZC Gu, XG Wen) 4d imaginary time evolution

15 MERA, RENORMALIZATION & WAVELETS Tensor network renormalization interpretation: (G Evenbly & G Vidal; S Yang, ZC Gu, XG Wen) 4d imaginary time evolution For classical stat mech systems: can also be done using non-negative matrix factorization M. Bal et al, PRL 118, (2017)

16 MERA, RENORMALIZATION & WAVELETS Wavelets and renormalization: multiscale analysis

17 MERA, RENORMALIZATION & WAVELETS Wavelets and renormalization: multiscale analysis

18 MERA, RENORMALIZATION & WAVELETS Wavelets and renormalization: multiscale analysis

19 MERA, RENORMALIZATION & WAVELETS Wavelets and MERA: G Evenbly & S White, PRL 116, (2016) A free fermion MERA (unitaries generated by quadratic operators) implements a wavelet transform at the single particle level. 0i 0i 0i 0i 0i 0i 0i 1i 0i 0i 0i 0i 0i 0i scaling coefficients 0i 0i 0i 0i 0i 0i 0i 0i 1i 0i 0i 0i 0i 0i wavelet coefficients

20 MERA, RENORMALIZATION & WAVELETS Wavelets and MERA: G Evenbly & S White, PRL 116, (2016) A free fermion MERA (unitaries generated by quadratic operators) implements a wavelet transform at the single particle level. 0i 0i 0i 0i 0i 0i 0i 0i 1i 0i 0i 0i 0i 0i wavelet coefficients Free fermion ground state: fill all negative energy modes fill set of modes that span the negative energy subspace (Fermi/Dirac sea) construct wavelets that are completely supported in either positive or negative energy subspace

21 1+1 DIRAC FERMIONS Massless Dirac fermions on the lattice: staggering (Kogut-Susskind) H D = H D = X n Z + b 1,n b 2,n b 2,n b 1,n+1 + b 2,n b 1,n b 1,n+1 b 2,n dk 2 apple b1 (k) b 2 (k) apple 0 e ik 1 e ik 1 0 apple b1 (k) b 2 (k) apple 0 e ik 1 e ik 1 0 apple 0 e ik 1 e ik 1 0 apple 1 1 ie ik/2 ie ik/2 = u(k) =u(k) apple 1 1 ie ik/2 ie ik/2 apple sin(k/2) 0 0 sin(k/2) apple sin(k/2) 0 0 sin(k/2), k 2 [, + ), k 2 [, + ) u(k) = 1 p 2 apple 1 1 isign(k)e ik/2 isign(k)e ik/2 = apple isign(k)e ik/2 1 p2 apple

22 1+1 DIRAC FERMIONS u(k) = 1 p 2 apple 1 1 isign(k)e ik/2 isign(k)e ik/2 = apple isign(k)e ik/2 1 p2 apple A pair of wavelet transforms such that wavelet filters in Fourier domain (g w (k),h w (k)) isign(k)e ik/2 difference. have equal magnitude and a relative phase Scaling filters (g s (k),h s (k)) e ik/2 should have phase difference (half shift or half delay condition) same phase difference for wavelets from higher levels of the transform (scale invariance). Impossible with filters of finite support approximation?

23 1+1 DIRAC FERMIONS Problem considered by Selesnick et al: a family of solutions, h s (k) =e i (k) g s (k) satisfying, in terms of two parameters K and L, leading to filters of width 2(K+L): K = 4 L = 6

24 FERMI SURFACES Non-relativistic fermions hopping at half filling: H 1 = X n2z a na n+1 + a n+1 a n b 1,n =( 1) n a 2n,b 2,n =( 1) n a 2n+1 H D H 2 = X (m,n)2z 2 a m,na m+1,n + a m+1,n a m,n + a m,na m,n+1 + a m,n+1 a m,n b 1,x,y =( 1) x+y a x+y,x y,b 2,x,y =( 1) x+y a x+y+1,x y H = Z [, ) 2 dk x dk y apple b1 (k x,k y ) b 2 (k x,k y ) apple 0 (1 e ik x )(1 e ik y ) (1 e ik x )(1 e ik y ) 0 apple b1 (k x,k y ) b 2 (k x,k y ) u(k x,k y )= apple isign(k x )e ik x/2 apple isign(k y )e ik y/2 1 p 2 apple

25 FERMI SURFACES Branching MERA (Evenbly & Vidal) R Shankar, RG approach to interacting fermions (RMP 66, 129)

26 FERMI SURFACES S 1d MERA (R) apple c 1 + S 1d MERA (R/2) apple...apple c 1 log 2 (R)+O(1) S 2d MERA (R) apple c 2 R + S 2d MERA (R/2) apple...apple 2c 2 R + O(1) S(R) apple c 2 R + 2(R/2)S 1d MERA (R/2) + S(R/2) apple... apple 2c 1 R log 2 R + O(R) [ S(R) apple c 2 R +2 S(R/2) apple apple c 2 R log 2 (R)+O(R)]

27 RIGOROUS APPROXIMATION RESULT Given pair of scaling filters: h s (k) e ik/2 g s (k) apple, 8k 2 [, + ) B = k s k 1 Let: f 2 `2(Z) :a(f) = X n2z f[n]a n G({f i }) = h a (f 1 )...a (f N )a ( f N+1 )...a(f 2N ) i Then: G({f i }) G({f i }) MERA apple 24 p N q C2 L/2 +6 (log 2 (C/ )) 2 Where: C =2 3/2p D({f i })B(K + L) L : number of layers in the MERA

28 RIGOROUS APPROXIMATION RESULT K=L=1 K=L=3

29 OUTLOOK AND EXTENSIONS Extensions Massive theories Pairing term: fermion number fermion parity conservation Outlook Dirac cones, topological insulators,? Relevance for interacting theories? (e.g. with asymptotic freedom)

30 QUESTIONS?