Exact results from the Quench Action Method for a certain

Exact results from the Quench Action Method for a certain class of initial states Dr. Gabriele Martelloni SISSA Trieste Collaborators: Andrea De Luca,...

0 downloads 136 Views 352KB Size
Exact results from the Quench Action Method for a certain class of initial states Dr. Gabriele Martelloni SISSA Trieste

Collaborators: Andrea De Luca, Jacopo Viti arXiv:1404.1319 [cond-mat.stat-mech],10.1103/PhysRevA.91.021603 , Andrea De Luca, G.M., Jacopo Viti Exact results from the Quench Action Method for a certain class of initial states, Andrea De Luca, G.M., Jacopo Viti, to appear

Dr. Gabriele Martelloni (SISSA)

May 2016, Firenze (GGI)

1 / 20

Summary

1

Quantum Quench and Motivations

2

State of art of transport properties

3

XX chains: NESS, GGE and exact results

Dr. Gabriele Martelloni (SISSA)

May 2016, Firenze (GGI)

2 / 20

Summary

1

Quantum Quench and Motivations

2

State of art of transport properties

3

XX chains: NESS, GGE and exact results

Dr. Gabriele Martelloni (SISSA)

May 2016, Firenze (GGI)

3 / 20

Quantum Quench we select an initial state ρ0 = |ψ0 ihψ0 | H0 → H: change of a parameter (global quench), change of the geometry of the problem (local quench) Unitary evolution ρ(t) = e−iHt ρ0 eiHt pure state → pure state...stationary state or statistical ensemble only in the thermodynamic limit (TL) P |ψ(t)i = n e−iEn t |nihn|ψ0 i, the importance of the overlaps technical difficult: the double sum in EV of an observable O X X −i(E −E )t n m hψ(t)|O|ψ(t)i = e hm|O|nihn|ψ0 ihm|ψ0 i n

m

we don’t solve exactly the dynamics, but we can compute the expectation value of observables in the limit t → ∞ it’s possible to obtain many results for integrable system, question integrable systems equilibrates to a particular ensemble? Dr. Gabriele Martelloni (SISSA)

May 2016, Firenze (GGI)

4 / 20

GGE and NESS states GGE conjecture: integrable systems does not relax to a thermal state, but the equilibrium is described by a Generalized Gibbs Ensemble M. Rigol et all., Phys. Rev. Lett. 98, 050405 (2007) I need to described the equilibrium states with all the local conserved charges of the theory Attention!!! If I use all the conserved charges we have a tautology: GGE or diagonal ensemble is only a change of basis!!!!

ρGGE =

1 ZGGE

[In , Im ] = 0 X exp(− λn In ),

Tr In ρGGE = hIn i0

n

failure of the GGE (Wouters et al., Pozsgay et al. 2014 for XXZ model) in the sense of local charges Quench in XXZ from Néel state: success of the Quench Action Method (QAM) (Caux, Essler2013) GGE in the sense of local charges is always valid in interacting to free Quantum Quench (Sotiriadis, Calabrese 2014, Sotiriadis, G.M. 2016) unbalanced of energy: NESS, persistent current in the TL Dr. Gabriele Martelloni (SISSA)

May 2016, Firenze (GGI)

5 / 20

Summary

1

Quantum Quench and Motivations

2

State of art of transport properties

3

XX chains: NESS, GGE and exact results

Dr. Gabriele Martelloni (SISSA)

May 2016, Firenze (GGI)

6 / 20

Framework

Universal properties in 1+1dimensions (Bernard,Doyon, 2012) (Karrasch et al., 2012), experimentally (Brantut et al, 2013) (Schmidutz et al., 2013) „ « πc 1 1 T tx = − 2 2 12 βL βR no additivity in d>1 ansatz by (Bhaseen,Doyon,Lucas,Schalm,2013), (Chang,Karch,Yarom,2013) and (Amado,Yarom,2015) „ d+1 d+1 « TL − TR tx T =a uL + uR Dr. Gabriele Martelloni (SISSA)

May 2016, Firenze (GGI)

7 / 20

D=1

Dr. Gabriele Martelloni (SISSA)

T tt =

” πc “ 2 2 TL + TR 12

T tx =

” πc “ 2 2 TL − TR 12 May 2016, Firenze (GGI)

8 / 20

D>1(1)

Universal heat flow and energy density determined imposing only ∇µ T µν = 0

Dr. Gabriele Martelloni (SISSA)

May 2016, Firenze (GGI)

9 / 20

D>1(2) Assumption: same structure of L and R moving waves describes the system

this solution is correct if and only if TL ' TR , the left-moving shock violates the second law of thermodynamics (Lucas,Schalm, Doyon, Bhaseen, 2015) Dr. Gabriele Martelloni (SISSA)

May 2016, Firenze (GGI)

10 / 20

Summary

1

Quantum Quench and Motivations

2

State of art of transport properties

3

XX chains: NESS, GGE and exact results

Dr. Gabriele Martelloni (SISSA)

May 2016, Firenze (GGI)

11 / 20

The Quench Action Method(QAM)(1) GOAL: we want to use the QAM to obtain the NESS for the two temperatures case O(t) =

XX

e

∗ −S{µ} −S{λ}

ei(ω{λ} −ω{µ} )t h{λ}|O|{µ}i.

{λ} {µ}

we go in the continuum limit Z “ ” YY X h{λ}|O|ρi −S ∗ −S O(t) = D[ρ]eSρ e {λ} ρ ei(ω{λ} −ωρ )t +λ↔ρ . 2 {λ}

Using a saddle point approximation (stationary phase) we obtain a new free energy Fρ = 2ReSρ − SρYY ∂Fρ |ρ = 0. ∂ρ s R +∞ 1 this equation is coupled with ρ(λ) + ρh (λ) = 2π + −∞ dλ0 K (λ − λ0 )ρ(λ0 ) lim hO(t)i = hρs |O|ρs i.

t→∞

the question is : in the case of the problem of the two temperature |ρs i is the NESS or the GGE or into the overlaps are present both the two states? Dr. Gabriele Martelloni (SISSA)

May 2016, Firenze (GGI)

12 / 20

The Quench Action Method(QAM)(2) the QAM works very well for translationally invariant quench (De Nardis et al., 2014 for LL model); how to treat a non translationally invariant quench? we study a very well known problem in literature: the NESS in the XX chain (free fermions) starting from two chain at different temperatures RESULTS: a persistent energy current when ((De Luca, Viti, Bernard, Doyon, 2013) and (Collura, Karevski, 2014)) ρS (φ) =

1 [Θ(φ)fl (φ) + Θ(−φ)fr (φ)] π

absence of an energy current when (Collura, Karevski, 2014) ρS (φ) =

1 (fl (φ) + fr (φ)) 2π

The linear size of the system L is sent to infinity before the observation time T, phisically T