Germán Sierra, Instituto de Física Teórica UAM-CSIC, Madrid 9th Bolonia Workshop in CFT and Integrable Systems Bolonia, Sept 2014

: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

If is the ground state of a local Hamiltonian

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

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

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

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

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

PART I : The Rainbow Model

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

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)

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, …)

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

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 It is exact in the limit (fixed point of the RG)

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

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

Hopping matrix Particle-hole symmetry Ground state at half-filling

Non uniform model scaling behaviour Uniform model

The Fermi velocity only depends on

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

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 ?

The proximity of the CFT: Half-chain entropy

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

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

Entanglement Hamiltonian For free fermions In the rainbow state ( )

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

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)

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

Entanglement spacing for constant

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

Entanglement spacing for z constant even odd The fit has

Fitting functions

Entropy/gap relation

Generalization to other models Local hamiltonian AF Heisenberg

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

Non uniform model wave functions near E=0

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

PART II : Infinite MPS

MPS Infinite MPS

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

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

Renyi entropy b

Experimental implementation

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.

Thank you Grazie mille