1. P. Zanardi USC Ground State and out of equilibrium fidelities: from quantum phase transitions to equilibration dynamics KITPC April 2011

2. (Quantum) Phase Transition: dramatic change of the (ground) state properties of a quantum system with respect A smooth change of some control parameter e.g., temperature, external field, coupling constant,… Characterization of PTs: “Traditional”: Local Order parameter (OP), symmetry breaking (SB) (correlation functions/length) Landau-Ginzburg framework Smoothness of the (GS) Free-energy (I, II,.. order QPTs) Quantum Information views: Quantum Entanglement (concurrence, block entanglement,..) Geometrical Phases Question: How to map out the phase diagram of a system with no a priori knowledge of its symmetries and ops? Question: How to map out the phase diagram of a system with no a priori knowledge of its symmetries and ops?

3. Ground State of F is a universal (Hilbert space) geometrical quantity: No a priori understanding of the SB pattern or OP is needed PZ, N. Paunkovic, PRE 74 031123 (2006) At the quantum critical points (QCPs) F should have a sharp drop i.e., the induced metric d should have a sharp increase. Weak topology vs norm topology: a METRIC APPROACH At the quantum critical points (QCPs) F should have a sharp drop i.e., the induced metric d should have a sharp increase. Weak topology vs norm topology: a METRIC APPROACH THE KEY (naive) IDEA Answer(?): distinguishability =distance in the information space

4. THE XY Model (I) =anysotropy parameter, =external magnetic field QCPs: XX line II-order QPT Ferro/para-magnetic II order QPT Jordan-Wigner mapping H Free-Fermion system: EXACTLY SOLVABLE! Quasi-particle spectrum: zeroes in the TDL in all the QCPs Gaplessness of the many-body spectrum

5. THE XY Model (II) Ground State Fidelity Strong L-dependent peak at universality features in the TDL Orthogonalization in the TDL: enhanced @ QCPs! PZ, N. Paunkovic, Phys. Rev E (2006)

6. XY Model (III): Overlap Functions

7. Differential Geometry of QPTs:QGT Hamiltonians Ground states Control parameters Hermitean metric over the projective space Riemannian metric 2-form (=Berry curvature!)

8. Metric scaling behaviour: the XY model Ricci Scalar

9. QGT: Spectral representations Local operator (trans inv) Spectral gap (Hastings 06) Gapped system For gapped systems the QGT entries scales at most extensively Superextensive scaling implies gaplessness

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

11. Dynamical Fidelity: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?

12. Typical Time Pattern of L(t) Transverse Ising (N=100)

13. 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 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

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

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

16. First Two Moments

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

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

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

20. Joint work with Nikola Paunkovic Marco Cozzini Paolo Giorda Radu Ionicioiu L. Campos-Venuti and now @USC Damian Abasto Toby Jacoboson Silvano Garnerone Stephan Haas

21. THANKS!

22. Summary Part I We analyze the phase diagram of a systems in terms of the induced geometric tensor Q on the parameter space Boundaries bewteen different phases can be detected by singularities of Q (real & immaginary parts) Q and related functions show critical scaling behaviour and Universality (response function representation) Quantum Tensor(s) approach to (Q)PTs: Geometrical, QI-theoretic, Universal Quantum Tensor(s) approach to (Q)PTs: Geometrical, QI-theoretic, Universal Extensions to finite temperature and to classical systems

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

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

25. Quantum Fidelity and Thermodynamics Uhlmann fidelity between Gibbs states of the same Hamiltonian at different temperatures = partition function Singularities of the specific heat shows up at the fidelity level: Divergencies e.g., lambda points, results in a singular drop of fidelity Temperature driven phase transitions can be detected by mixed state fidelity in both classical and quantum systems! Temperature driven phase transitions can be detected by mixed state fidelity in both classical and quantum systems!

26. The “Hey dude! If you have the GS you have EVERYTHING …” Slide The “Hey dude! If you have the GS you have EVERYTHING …” Slide Ok cool. Let’s suppose you’re somehow given the GS e.g.,a MPS Please, now tell me where the QCPs are…. “Great! Now I compute the OP and its correlations! ….” Well… you have infinitely many candidates: Which one?!? and perhaps none of them is gonna work out (e.g., TOPorder) … “Ok, fair enough, then I compute the energy!” I see, so you want also the H? You didn’t ask for that so far… Anyways, here we go: identically vanishing E(g)=0…. You may answer:

27. Qualitative: the difference between the means should be (much) bigger than the finite size variances…. Quantitative: Fisher Information metric Problem: Quantumly each Q-preparation defines infinitely many prob distributions, one for each observable: one has to maximize over all possible experiments! d(P,Q)= number of (asymptotically) distinguishable preparations between P and Q: ds=dX/Variance(X) d(P,Q)= number of (asymptotically) distinguishable preparations between P and Q: ds=dX/Variance(X) Distinguishability of ExpsDistance in Prob space

28. Surprise! Projective Hilbert space distance! (Wootters 1981) Statistical distance and geometrical one collapse: Hilbert space geometry is (quantum) information geometry….. Question: What about non pure preparations ?!? Answer: Bures metric & Uhlmann fidelity! (Braunstein Caves 1994) Remark: In the classical case one has commuting objects, simplification …. Q-fidelity!