Paolo Zanardi (USC)

1. Long time dynamics of a quantum quench Paolo Zanardi (USC) Lorenzo Campos Venuti (ISI) Obergugl, Austria June 2010

2. Unitary Equilibration?? Hey wait a sec: Equilibration of a finite closed quantum system?!? What are you talking about dude??? • Unitary Evolution==>no non-trivial fixed points for t=∞ • I.e., no strong (norm) convergence • Finite size==>Point spectrum==>A(t)=measurable quantity is a • quasi-periodic function ==>no t=∞ limit (quasi-returns/revivals) ==> • not even weak op convergence Unitary equilibrations will have to be a different kind of convergence….

3. Loschmidt Echo: Spectral resolution Probability distribution(s) Different Time-Scales & Characteristic quantities • Relaxation Time (to get to a small value by dephasing and oscillate around it) • Revivals Time (signal strikes back due to re-phasing) Q1: how all these depend on H, , and system size? Q2: how the global statistical features of L(t) depend on H, and system size N?

4. Typical Time Pattern ofL(t) Transverse Ising (N=100)

5. For a given initial state L-echo is a RV over the time line [0,∞) with Prob Meas Characteristic function of Probability distribution of L-echo Goal: study P(y) to extract global information about the Equilibration process 1 Moments of P(y) Each moment is a RV over the unit sphere (Haar measure) of initial states

6. Mean: Long time average of L(t) is the purity of the time-average density matrix (or 1 -Linear Entropy) Remark Is a projection on the algebra of the fixed points Of the (Heisenberg) time-evolution generated by H Remark dephased statemin purity given the constraints I.e., constant of motion Question: How about the other moments e.g., variance and initial state dependence? Are there “typical” values?

7. 2 Moments of P(y) Dephasing CP-map of the n-copies Hamiltonian, S is a swap in Projection on the totally symm ss of d=dim(Hilb)==> exp (in N) small =(positivity)=> exp (in N) state-space concentration of around Remark We assumedNO DEGENERACY, in generalbounded above by Remark All L(t) moments are Lipschitz functions on the unit sphere of Hilb ==> Levi’s Lemma implies exp (in d) concentration around

8. Remark: non-resonance assumed Chebyshev’s inequality goes to zero with N system size ==> Mcan diverge While ==>Flucts of L(t) are exponentially rare in time …..! In the overwhelming majority of time instants L(t) is exponentially close to the “equilibrium value” This what we (morally) got :

9. Q: Can we do better? E.g., exp in d concentration? A: yes we can! HP: energies rationally independent ==> motion on the d-torus ergodic ==>Temporal averages=phase-space averages L is Lipschitz on the d-torus with metric D==> known measure concentration phenomenon! Remark The rate of meas-conc is the inv of purity of the dephased States I.e., mean of L==> Typically order d=epx(N), as promised…

10. Far from typicality: Small Quenches =measures how initial state fails to be a quenched Hamiltonian Eigenstate. For H(quench) close to H(0) we expect it to be small…. Ground State of an initial Hamiltonian Quench-Ham=init-Ham+perturbation Distribution on the eigenbasis of GS Fidelity: leading term! The linear entropy of the dephased state for a small quench Is given by the fidelity susceptibility: a well-known object! Remark:

11. Small Quenches: The Role of Criticality Local operator (trans inv) Spectral gap (Hastings 06) Gapped system For gapped systems 1-LE mean scales at most extensively Superextensive scaling implies gaplessness

12. Small Quenches: FS Critical scaling Continuum limit Scaling transformations Proximity of the critical point At the critical points Scal dim of FS: the smaller the faster the orthogonalization rate Super-extensivity Criticality it is not sufficient, one needs enough relevance….

13. THE XY Model =anysotropy parameter, =external magnetic field XX line III-order QPT QCPs: Ferro/para-magnetic II order QPT Jordan-Wigner mappingHFree-Fermion system:EXACTLY SOLVABLE! Quasi-particle spectrum: zeroes in the TDL in all the QCPs Gaplessnessof the many-body spectrum

14. Ising in transverse field: H.T Quan et al, Phys. Rev. Lett. 96, 140604 (2006) Large size limit (TDL)==>spec(H) quasi-continuous ==> Large tlimits exist (R-L Lemma) =time averages (m=band min, M=band max Inverting limits I.e., 1st t-average, 2nd TDL • g and s are qualitatively the same but when we consider different phases

15. First Two Moments

16. P(L=y) Different Regimes • Large==>Lfor (moderately)large (quasi) exponential • Small and off critical a)Exponential • b) Otherwise Gaussian • Small and close to criticality a)Exponential • b) Quasi critical I.e., universal “Batman Hood”

17. Approaching exp for large sizes L=10,20,30,40 h(1)=0.9, h(2)=1.2 L=20,30,40,60,120 h(1)=0.2, h(2)=0.6

18. Thanks for the attention!

19. Summary & Conclusions • Unitary equilibrations: measure convergence/concentration • Universal content of the short-time behavior • of L(t) and criticality • Moments of Probability distribution of LE (large time) • Transverse Ising chain: mean, variance, regimes for P(L) Phys. Rev. A 81,022113 (2010) Phys. Rev. A 81, 032113 (2010)

20. Short-time behavior Square of a characteristic function --> cumulant expansion Hsum of N local operator in the TDLN-> ∞ one expects CLT to hold I.e., Relaxation time Off critical (or large quench) Critical (& small quench) Gaussian • 0for for Non Gaussian (universal) Non Gaussian & non universal)

21. L=18, h(1)=0.3, h(2)=1.4 L=20, h(1)=0.1, h(2)=0.11 + n-body spectrum contributions Different regimes depend on how many frequencies have a non-negligible weight L=40, h(1)=0.99, h(2)=1.1

22. L=40, h(1)=0.99, h(2)=1.1 L=20, h(1)=0.1, h(2)=0.11 L=20, h(1)=0.1, h(2)=0.11 With just two frequencies

23. More generally (Strong non resonance) Therefore implies For small Higher Moments: direct operational meaning ! is a dephasing super-operator of the n-copies Hamiltonian S is a “swap” between 1st and 2nd n-copies Protocol: 1) Prepare 2n copies of I.e., 2) Dephase 2nd n-copies I.e., 3) Measure S