Quantum phase transitions and entanglement in (quasi)1d spin and electron models

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1 Quantum phase transitions and entanglement in (quasi)1d spin and electron models Elisa Ercolessi - Università di Bologna Group in Bologna: G.Morandi, F.Ortolani, E.E., C.Degli Esposti Boschi, A.Anfossi (M.Roncaglia, L.Campos-Venuti, S.Pasini) Problemi Attuali di Fisica Teorica - Vietri sul Mare Marzo 2008

2 Strongly correlated electron/spin models in 1d Paradigm: integrable models on a chain (s=1/2) XXZ model Hubbard model H = J (Si x Si+1 x + S y i Sy i+1 + λsz i Si+1) z H = t (c i,σ c i+1,σ + c i+1,σ c i,σ)+u,σ n i, n i, 2

3 No finite temperature phase transitions (Mermin-Wagner theorem) Quantum phase transitions (T=O) driven by coupling constants in the Hamiltonian Spin-1 λ-d model H = J [Si x Si+1 x + S y i Sy i+1 + +λs z i S z i+1 + D(S z i ) 2 ] 3

4 Analytical techniques: mapping onto a 2d field theory in the continuum Conformal Field Theory (classification of phase transitions) (Massive) Integrable models (sine-gordon or Nonlinear Sigma Models) Perturbative approach and Renormalization Group Numerical techniques: energy spectra and/or correlation functions on finite lattices of increasing size Exact Diagonalization (up to 20~30 sites) Density Matrix Renormalization Group (up to 1000 sites) 4

5 Spin 1/2 two-leg Hiesenberg ladder H = J S i,l S i+1,l +J L S i,1 S i+1,2 +J L ( S i,1 S i+1,2 + S i,2 S i+1,1 ) l=1,2 J J J QPT at J =2J (<< J) critical theory: E 1 L J =0.200 In the continum limit: two sine-gordon theories (spin & charge) + interactions 5

6 Entanglement and QPT Entanglement as a resource: ground state of strongly correlated systems shows entanglement between even distant parties does it play a role in QPT? is it a new kind of indicator? Local vs. nonlocal entanglement Local measures: that give correlations between nearest sites Nonlocal measures: that encode correlations of a large size of the system 6

7 Measures of (bipartite) entanglement B (L-n sites) A (n sites) B The system A+B: pure/mixed state Reduced density matrix: ρ AB ρ A = Tr B [ρ AB ] Von-Neumann entropy: S L = Tr A [ρ A log ρ A ] 7

8 Local measures on-site entropy : A=single site B= all the rest S i two-site entropy: A= two sites B= all the rest S ij concurrence (only for qubits): A= two sites B= all the rest ψ ij ψ ij =(σ y σ y ) ψ ij C ij = ψ ij ψ ij 8

9 Ising model in transverse field H = (λσi x σi+1 x + hσi z ) QPT at λ c =1 : c=1/2 conformal field theory C ij = 0 unless i n.n. j dc i,i+1 dλ = 8 λ λc 3π 2 ln λ λ c + const. A.Osterloh et al. Nature 416 (2002) 608 9

10 Hubbard model H = t (c i,σ c i+1,σ + c i+1,σ c i,σ)+u,σ n i, n i, QPT at U =0 : c=1 conformal field theory at SU(2) point - BKT transition Single site Von Neumann entropy S i has a maximum at the transition S. Gu et al. PRL 93 (2004)

11 Local entanglement can be used to locate with accuracy the transition Spin-1 λ-d model derivative of single site entropy S i (Si z ) 2 has a cusp LCampos-Venuti et al. PRA 73 (2006) (R) 11

12 Local entanglement can be used to locate NEW transitions Hirsch model (bond charge) H = t [1 X(n i,σ + n i+1,σ )(c i,σ c i+1,σ + c i+1,σ c i,σ)] + U,σ n i n i Exact results Gap with finite size scaling x Single site entropy S i n i n i A.Anfossi et al. PRB 73 (2006)

13 Nonlocal measures fidelity: ψ(λ) ψ(λ ) block entropy: critical massive S L c 6 log L + const. S L c 6 log ξ + const. more precisely, in critical theories: S n = c 6 log 2 [ L π sin ( π L n )] + A single block of n sites in a lattice with L sites with periodic boundary conditions P.Calabrese, J.Cardy J.Stat.Mech. PO6002 (2004) 13

14 Block entropy can be used to classify phase transitions SU(3) AFM Heisenberg chain (in fundamental representation) H = J S i S i+1 critical theory E 1 L Text!E Sz=0 Qz=0 Sz=1/2 Qz=3/ /L 14

15 S n L=90, M=300 L=90, M=400 L=90, M=500 L=90, M=600 L=90, M=700 L=90, M=800 L=90, M=900 A M log 2 (sin(n!/l)l/!) c =1.98 ± 0.09 (c=2) WZW model SU(3) 1 M.Aguado et al. arxiv:0801:3565 (2007) 15