Germ á n Sierra, Instituto de Física Teórica UAM-CSIC, Madrid

1. Breaking the area law of entanglement Germán Sierra, Instituto de Física Teórica UAM-CSIC, Madrid 9th Bolonia Workshop in CFT and Integrable Systems Bolonia, 15-18 Sept 2014

2. Entanglement entropy :a pure state in H choosen at random EE is almost maximal (Page) :number of sites of A Volumen law like for the thermodynamic entropy

3. Area law of entanglement entropy If is the ground state of a local Hamiltonian

4. Physics happens at a corner of the Hilbert space Experiments occur in the Lab not in a Hilbert space (A. Peres)

5. Basis of Tensor Networks (MPS, PEPS, MERA,..) (c) MERA

6. Hastings theorem (2007): • Conditions: • Finite range interactions • Finite interaction strengths • Existence of a gap in the spectrum In 1D In these cases the GS can be well approximated by a MPS

7. Violations of the area law in 1D require one of the following • non local interactions • divergent interactions • gapless systems Best well known examples are CFT and quenched disordered systems -> Log violations of entropy Here we shall investigate a stronger violation Entanglement entropy satisfies a volumen law

8. Part I: The rainbow model: arXiv:1402.5015 G. Ramírez, J. Rodríguez-Laguna, GS Part II: Infinite Matrix Product States: arXiv:1103.2205 A.E.B. Nielsen, GS, J.I.Cirac

9. PART I : The Rainbow Model

10. The Model Inhomogenous free fermion model in an open chain with 2L sites Introduced by Vitigliano, Riera and Latorre (2010)

11. Other inhomogenous Hamiltonians • Smooth boundary conditions (Vekic and White 93) • Quenched disordered: J’s random (Fisher, Refael-Moore 04) • - Scale free Hamiltonian and Kondo (Okunishi, Nishino 10) -Hyperbolic deformations (Nishino, Ueda, Nakano, Kusabe 09)

12. Renormalization group Dasgupta-Ma method (1980) At the i-th bond there is a bonding state In second order perturbation This method is exact for systems with quenched disorder (Fisher, …)

13. Choosing the J’s at random -> infinite randomness fixed point Average entanglement entropy and Renyi entropies Refael, Moore 04 Laflorencie 05 Fagotti,Calabrese,Moore 11 Ramirez,Laguna,GS 14 CFT Renyi

14. If the strongest bond is between sites i=1,-1 RG gives the effective coupling: This new bond is again the strongest one because Repeating the process one finds the GS: valence bond state (fixed point of the RG) It is exact in the limit

15. Density matrix of the rainbow state B: a block number of bonds joining B with the rest of the chain has an eigenvalue with multiplicity von Neumann entropy Moreover all Renyi entropies are equal to von Neumann

16. Take B to be the half-chain then Maximal entanglement entropy for a system of L qubits The energy gap is proportional to the effective coupling of the last effective bond Hasting’s theorem is satisfied Define Uniform case

17. Exact diagonalization Hopping matrix Particle-hole symmetry Ground state at half-filling

18. Uniform model Non uniform model scaling behaviour

19. The Fermi velocity only depends on

20. Entanglement entropy Correlation method (Peschel,…) Two point correlator in the block B of size Diagonalize finding its eigenvalues Reduced density matrix of the block von Neumann entropy

22. For small and L large there is a violation of the area law that becomes a volumen law. This agrees with the analysis based on the Dasgupta-Ma RG What about the limit ?

23. The proximity of the CFT: Half-chain entropy

24. CFT formula for open chain Boundary entropy Luttinger parameter Fitting curve The fits have

25. c(z) decreases with z: similar to the c-theorem (z similar to mR) d(z) increases with z: the g-theorem does not apply because the bulk is not critical Origin of the volumen law

26. Entanglement spectrum Entanglement Hamiltonian For free fermions In the rainbow state ( )

27. Entanglement energies L=40 L=41 L: even L: odd

28. Make the approximation one can estimate the EE • - Critical model : Peschel, Truong (87), Cardy, Peschel (88), …Corner Transfer Matrix ES: energy spectrum of a boundary CFT (Lauchli, 14)

29. - Massive models in the scaling limit Cardy, Calabrese (04) using CTM Ercolessi, Evangelisti, Francini, Ravanini 09,…14 Castro-Alvaredo, Doyon, Levi, Cardy, 07,…14 - Rainbow model for for L sufficiently large

30. Entanglement spacing for constant

31. Based on equations And one is lead to the ansatz for the entanglement spacing depend on the parity of L

32. Entanglement spacing for z constant even odd The fit has

33. Fitting functions

34. Entropy/gap relation

35. Generalization to other models Local hamiltonian AF Heisenberg

36. Continuum limit of the rainbow model (work in progress) Uniform model Fast-low factorization CFT with c=1

37. Non uniform model wave functions near E=0

38. theory numerical It is expected to predict some of the scaling functions c(z)

39. PART II : Infinite MPS

40. MPS Infinite MPS

41. Vertex operators in CFT (Cirac, GS 10) Renyi 2 entropy Good variational ansatz for the XXZ model

42. Truncate the vertex operator to the first M modes (Nielsen,Cirac,GS) The wave function is

43. Renyi entropy b

45. Experimental implementation

46. Conclusions and suggestions We have shown that rather simple local Hamiltonians can give rise to ground states that violate the area law. They can be thought of as conformal transformation on a critical model that preserves some of the entanglement properties. In the strong coupling limit they become valence bond states: provide a way to interpolate continuously between the CFT and the VBS. The infinite MPS based on CFT lie in the boundary of the states that satisfy the area law.

47. Thank you Grazie mille