Tensor network simulation of QED on infinite lattices: learning from (1 + 1)d, and prospects for (2 + 1)d

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1 Tensor network simulation of QED on infinite lattices: learning from (1 + 1)d, and prospects for (2 + 1)d Román Orús University of Mainz (Germany) K. Zapp, RO, Phys. Rev. D 95, (2017)

2 Goal of this talk 1) Show TN results with infinite-dmrg for (1+1)d QED 2) Show that we have all the necessary ingredients for a meaningful simulation of (2+1)d QED with infinite-peps Fermions + U(1) gauge+ plaquette interactions + + improved optimization + contraction schemes +...

3 Outline 1) TN Basics 2) (1+1)d QED with idmrg 3) (2+1)d QED: roadmap for ipeps

4 Entanglement obeys area-law Entanglement 2d system E E A key resource in quantum information teleportation, quantum algorithms, quantum error correction, quantum cryptography Reduced density matrix of subsystem A Entanglement entropy (von Neumann entropy) For many ground states In d dimensions (L > ξ) topological entropy Generic state S(A) ~ L d (volume) Ground states of (most) local Hamiltonians S(A) ~ L d 1 (area) Srednicki, Plenio, Eisert, Dreißig, Cramer, Wolf Locality of interactions area-law tensor network states

5 Hilbert space is a convenient illusion Exploration time ~ O( ) sec. Most states here are not even reachable by a time evolution with a local Hamiltonian in polynomial time!!! Poulin, Qarry, Somma, Verstraete, PRL (2011) Compare to Age of the universe ~ O(10 17 ) Set of area-law states Set of product states (mean field) sec. We need a language to target the relevant corner of quantum states directly

6 Tensor Networks e.g. RO, Annals of Physics 349 (2014) Ψ = Ψ i1 i 2 i 3 i 4 i 5 i 6 i 7 i 8 i 9 i 1 i 2 i N p-level i's Ψ i1 i 2...i N 1d A B A B systems Matrix Product States (MPS) i 1 i 2 i 9 Scale-invariant Multiscale Entanglement Renormalization Ansatz (MERA) 2d, 3d... physical 1 p DMRG, PWFRG, TEBD bond 1..D (entanglement) Projected Entangled Pair States (PEPS), Tensor Product States (TPS) RG AdS/CFT, Entanglement Renormalization Tensor Product Variational Approach, PEPS & ipeps algorithms, Tensor-Entanglement Renormalization, TRG/SRG/HOTRG/HOSRG Efficient O(poly(N)), satisfy area-law, low-energy eigenstates of local Hamiltonians

7 Symmetric tensors and Schur s lemma e.g., S. Singh, R. N. C. Pfeifer, G. Vidal, PRA 82, (2010) U i k O W j symmetric tensor V = k O O 3 legs = (c,γ c ) (c)( P j O = i 2 legs (b, β b ) P (a,α a ) (c, o c ) Q (b, n b ) Q (a, m a ) degeneracy structural ( ~ identity) structural ( ~ Clebsch-Gordan) i j (a,α a ) (b, β b ) (a, m a ) (b, n b ) degeneracy Structural part depends only on the group properties (intertwiners)

8 Explosion in recent years Quantum information and computation HEP and lattice gauge theories Quantum gravity, string theory and AdS/CFT Strongly correlated systems Quantum simulations Entanglement and Tensor Networks Classical statistical mechanics Numerical tensor calculus Nuclear physics Quantum chemistry Materials science AI, Deep learning & Linguistics

9 Explosion in recent years Quantum information and computation Today s talk HEP and lattice gauge theories Quantum gravity, string theory and AdS/CFT Strongly correlated systems Quantum simulations Entanglement and Tensor Networks Classical statistical mechanics Numerical tensor calculus Nuclear physics Quantum chemistry Materials science AI, Deep learning & Linguistics

10 Outline 1) TN Basics 2) (1+1)d QED with idmrg 3) (2+1)d QED: roadmap for ipeps

11 Massive Schwinger model 1) Lagrangian density L = ψ ( i µ γ µ m)ψ 1 4 F µνf µν gψa µ γ µ ψ F µν = µ A ν ν A µ! E! = ρ Gauß law 1-flavour fermions, U(1) gauge field, coupling g 2) Lattice Hamiltonian, Kogut-Susskind staggered formulation H = i 2a n ( φ + n e iθ n φ h.c. ) n+1 + m ( 1) n φ n+ φ n + ag2 n 2 n L n 2 [ θ n, L m ] = iδ nm L n L n 1 = φ n+ φ n 1 2 ( 1 ( 1) n ) Gauß law Fermionic and bosonic variables Local interactions Invariant under translations

12 Massive Schwinger model 3) After Jordan-Wigner transformation H = g 2 x n L n 2 + µ 2 n ( ( ) n ) + x σ n ( 1) n σ z n + 1 n ( + e iθ n σ ) n+1 + h.c. ( ( ) n ) x 1/ (g 2 a 2 ) µ 2m x / g L n L n 1 = 1 2 σ z n + 1 Gauß law 4) Integrating out Gauss law (only in 1+1 dimensions) H = x ( σ n+ σ n+1 +σ n σ + ) n+1 + µ 2 n Good formulation for infinite-mps (TDVP, idmrg...) B. Buyens, K. Van Acoleyen, J. Haegeman, F. Verstraete, PoS(LATTICE2014)308. n ( 1+ ( 1) n z σ ) + l + 1 n 2 n n k=0 (( 1) k z +σ ) k 2 Non-local interactions Good formulation for finite-mps (DMRG) M. C. Bañuls, K. Cichy, J. I. Cirac, K. Jansen, H. Saito, PoS(LATTICE2013)332.

13 Gauge-invariant idmrg Gauge-invariant idmrg simulations e + B + C A e - B - C fermion gauge boson B. Buyens, K. Van Acoleyen, J. Haegeman, F. Verstraete, PoS(LATTICE2014)308. degeneracy structural q,r indices for U(1) gauge symmetry sector (structural) Greek degeneracy indices H = translation invariant MPO, bond dimension = 4

14 1-site idmrg crash-course (technical) 1) Get L H and R H environment tensors 4) Odd step: absorb to the left 2) Define effective Hamiltonian 5) Even step: absorb to the right 3) Solve eigenvalue problem Mixed canonical form, and 2-site idmrg also possible

15 Chiral condensate m/g=0.25

16 Chiral condensate m/g=0.25 DMRG itdvp Similar results with 2-site idmrg A similar approach should be possible in (2+1)d

17 Outline 1) TN Basics 2) (1+1)d QED with idmrg 3) (2+1)d QED: roadmap for ipeps

18 Lattice (2+1)d QED with PEPS Why interesting? True fermions, chiral symmetry breaking, confinement, higher dimensions,... H = i 2a + ag2 2 i, j Staggered fermions in (2+1)d i, j ( φ + i e iθ ij φ j h.c. ) + m 1 i ( ) s(i) φ i+ φ i L 2 ij 1 cos θ ag 2 1 +θ 2 θ 3 θ 4 p ( ) electric energy magnetic energy L ij L i( j+1) = φ i+ φ i 1 2 Jordan-Wigner no longer useful Plaquette interaction Integrating out Gauß law no longer useful Truncation of gauge-boson Hilbert space (quantum link model) Gauge U(1), global fermionic parity Z 2 ( 1 ( 1) s(i) ) Gauß law

19 Lattice (2+1)d QED with PEPS Ok, so what do we need for this simulation? Fermions U(1) symmetry e.g., P. Corboz, PRB (2016) Plaquette interactions e.g., B. Bauer, P. Corboz, RO, M. Troyer, PRB (2011) Efficient accurate schemes Simple update, full update, fast full update, CTMs, TRG, TERG, boundary-mps, TDVP, variational, imag.-time evolution, large unit cells, finite-d scaling... e.g., S. Dusuel, M. Kamfor, RO, K. P. Schmidt, J. Vidal, PRL 106, (2011) We have all the ingredients to do this simulation (a priori)

20 An option:

21 Lattice (2+1)d QED with PEPS fermion (site) Symmetric tensors X gauge boson (link) free param. fermionic Z 2 gauge U(1) fermionic and bosonic indices U(1) indices oriented (2+1)d variational ansatz Global fermionic Z 2 Gauge U(1) See also E. Zohar, M. Burrello, T. B. Wahl, J. I. Cirac, Ann. Phys (2015); E. Zohar, M. Burrello, NJP 18, (2016); E. Zohar, T. B. Wahl, M. Burrello, J. I. Cirac, Ann. Phys (2016)

22 Lattice (2+1)d QED with PEPS H = i 2a + ag2 2 i, j i, j ( φ + i e iθ ij φ j h.c. ) + m 1 i ( ) s(i) φ i+ φ i L 2 ij 1 cos θ ag 2 1 +θ 2 θ 3 θ 4 p ( ) e Ht Ψ Trotterize, etc Fermion-gauge boson interaction: 3-body gate

23 Lattice (2+1)d QED with PEPS H = i 2a + ag2 2 i, j i, j ( φ + i e iθ ij φ j h.c. ) + m 1 i ( ) s(i) φ i+ φ i L 2 ij 1 cos θ ag 2 1 +θ 2 θ 3 θ 4 p ( ) e Ht Ψ Trotterize, etc Fermionic mass term: 1-body gate

24 Lattice (2+1)d QED with PEPS H = i 2a + ag2 2 i, j i, j ( φ + i e iθ ij φ j h.c. ) + m 1 i ( ) s(i) φ i+ φ i L 2 ij 1 cos θ ag 2 1 +θ 2 θ 3 θ 4 p ( ) e Ht Ψ Trotterize, etc Electric field energy: 1-body gate

25 Lattice (2+1)d QED with PEPS H = i 2a + ag2 2 i, j i, j ( φ + i e iθ ij φ j h.c. ) + m 1 i ( ) s(i) φ i+ φ i L 2 ij 1 cos θ ag 2 1 +θ 2 θ 3 θ 4 p ( ) e Ht Ψ Trotterize, etc Magnetic field energy: plaquette gate

26 Discussion

27 Lattice (2+1)d QED with PEPS Compact or non-compact? Both should be possible, change in plaquette gates (pure gauge term) Chiral condensate and number of flavours? More fermions, larger bond dimensions. Should be possible but costly Monopole density and number of flavours? Same as above Finite-temperature BKT confinement-deconfinement transition? Should be possible with ipepos and related approaches Which approach is better? Simple update, full update, variational... Depends on the regime. One needs to try.

28 Tensor networks in Mainz The team Funding (Feb. 2016) LGT / TN chat Collaborators