Universality in the equilibration of Universality in the - PowerPoint PPT Presentation

Universality in the equilibration of Universality in the equilibration of isolated systems after a small quench isolated systems after a small quench Lorenzo Campos Venuti University of Southern California, Los Angeles Paolo Zanardi (USC) Sunil Yeshwanth GGI, Firenze, May 25 2012 GGI, Firenze, May 25 2012

Preliminaries ρ( t ) ⇒ ρ eq Equilibration? 1. Isolated, finite system ∥ ρ( t )−ρ eq ∥ = cnst 1. No strong convergence ρ 0 2. Prepare initial state 2. For finite systems no weak convergence − iH t ρ 0 e iH t ρ( t )= e 3. Evolve with H: 3. Stochastic convergence 4. Monitor observable A: a ( t )= 〈 A ( t ) 〉 = Tr A ρ( t ) T f ( t ) dt f = ∫ ● Observation window [ 0, T ] ● Time average T 0 P A ( a ) da = Prob (〈 A ( t )〉∈[ a,a + da ] , t ∈[ 0, T ]) P A ( a )=δ( a − a ( t )) Equiliration = concentration of P A (a)

Equilibration ⇒ ρ eq = ρ a ( t ) = Tr A ρ( t ) = Tr A ρ 2 ≤ Ran A 2 Tr ρ 2 a = a ( t ) 2 − a 2 Δ + ”gap” non-degeneracy E i − E j = E l − E m Chebyshev's inequality ⇒ i = j ,l = m ∨ i = l , j = m 2 a Prob (∣ a ( t )− a ∣≥ϵ)≤Δ 2 ϵ Reimann, PRL (2008) 2 = L ( t ) Tr ρ n 〉 c 2 ) n (− t 2 = exp2 ∑ 〈 H L ( t ) = ∣ 〈 Ψ ∣ e − it H ∣ Ψ 〉 ∣ Vf ( t ) 2n ! = e n = 1 2 ∝ e Free systems? −α V ⇒ Tr ρ

Quench: dynamical detection of QPT's superfluid-Mott transition: experiment H → ∣ Ψ 0 〉 H ' = H +δλ V Theoretical description Greiner et al., Nature (2002) Sengupta, Powell, Sachdev, PRA (2004)

Small* quench: full statistics δλ = 0.04  Small quench, off-critical L = 12 L ≫ξ Bi-modal: phase transition  Small quench, quasi-critical L = 16 ξ≫ L 2 χ F ≪ 1 ⇒ δλ≪ 1 (*) Small: δλ 1 /ν L

Small quench: CLT c ( E n ) = 〈 E n , Ψ 0 〉 〈 A ( t ) 〉 = ∑ − it ( E m − E n ) A n, m c ( E n ) c ( E m ) e n, m ≈ A + ∑ F n cos ( t ( E n − E 0 )) F n ≈ 2 A n, 0 c ( E n ) n > 0 a(t) sum of rational independent variables independence 2 a = 1 2 ⩽ 2 [ 〈 0 ∣ A 2 ∑ 2 ] ∝ L 2 ∣ 0 〉 − 〈 0 ∣ A ∣ 0 〉 Δ F n n > 0 Look at F n , c ( E n ) ”Generally” CLT

Small quench: explanation Off-critical distribution Scaling prediction at criticality: − 1 /ν c ( E ) ∼ E Critical case distribution sum rule 2 = 1 ∑ E ∣ c ( E ) ∣ ∣ E 〉 Only few states contribute: Allowed values Bi-modal distribution

Small quench: critical case  Q: How to break CLT?  A: most F n → 0 (Quantum) critical points are more stable against perturbations

Relaxation time (Talk by Michael Pustilnik) T R 〈 A ( T R ) 〉 : = A Loschmidt echo L ( T R ) : = L

Relaxation time II Loschmidt echo: short time - cumulant expansion Generally (+small quench off-critical) d ⇒ T R = O ( L d 2 t 2 −α L 2 ∼ L 0 ) −σ L = e , σ L ( t ) = e small quench criticality 2 ( d +ζ−Δ) ⇒ T R = O ( L 2 ( d −Δ) , χ F ∼ L 2 χ F , σ − 2 δλ ζ ) 2 ∼ L L ≃ e −ζ ν T R ∼ ∣ λ−λ c ∣

Relaxation time: Random Systems ~ Inguscio, Modugno, LENS † c x + 1 + c x + 1 † c x H = ∑ † ( c x c x )−μ x c x x Random field Loschmidt echo E [ L ( t ) ]

Relaxation time: Random Systems E [ 〈 N l ( t ) 〉 ] Number operator α L T Relax ∼ e

Equilibration & Integrability a ( t ) = 〈 A ( t ) 〉 → P A ( a )=δ( a − a ( t )) Generally, for A extensive: a ( t ) ∝ V 2 a ≤ O ( e −α V ) Δ † M x , y c y H = ∑ c x Integrable systems (free Fermions) † A x , y c y A = ∑ c x † c y 〉 R y, x = 〈 c x − it M R e it M ) a ( t ) = Tr ( A e = ∑ − it (ϵ k −ϵ q ) A q, k R k , q e k , q F k ,q / 2

Stat-mech parallel Rational independence λ a ( t ) = ∑ λ E (θ ' s ) = e All cumulants f (λ ) V e e extensive: θ ' s CLT E (θ ' s ) = ∑ F k , q cos (θ k −θ q ) k ,q F k ,q = F (∣ k − q ∣) Z = ( a ( t )− a ) Gaussian √ V Classical XY model on lattice F i,j (infinite temperature)

Equilibration & Integrability Generally, for A extensive: 2 a ≤ O ( e −α V ) a ( t ) ∝ V Δ Integrable systems (free Fermions) 2 a = O ( V ) a ( t ) ∝ V Δ Gaussian(poor) equilibration

Loschmidt echo [ R ,M ] = 0 ● Applies to XY model ● Generalizes to thermal L ( t )= ∏ 2 ( t ϵ k / 2 )) ( 1 −α k sin quenches k ● Generalizes to Ulman Fidelity Z = log L ( t )− log L ( t ) Gaussian, => L(t) Log-Normal √ L For general models (RI spectrum), work in progress 2k )) μ n ( L ( t ))= f ( Tr (ρ

Curiosity: Riemman zeta H primon gas: − it H ρ σ ) , ρ σ = e −σ H ζ(σ+ it ) = Tr ( e free bosons 2 Z : = log ∣ ζ(σ 0 + i t ) ∣ Satisfies CLT Very similar to Loschmidt Echo)

Conclusions  Finite systems Ingredients  Look at full time statistics  Small quench: a tool to detect criticality, engineer ” new quantum states of matter”  Relaxation time can be defined  Integrability & equilibration: integrable systems concentrate less Thank you