Total correlations of the diagonal ensemble herald the many-body localization transition J. Goold1, C. Gogolin2,3, S. R. Clark4, J. Eisert5, A. Scardicchio1,6, and A. Silva1,7 1The Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy 5 2ICFO-The Institute of Photonic Sciences, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain Dahlem Center for Complex Quantum Systems, Freie Universit¨atBerlin, 14195 Berlin, Germany 6 3Max-Planck-Institut f¨urQuantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany INFN, Sezione di Trieste, I-34151, Trieste, Italy 7 4Department of Physics, Oxford University, Clarendon Laboratory, Parks Road, Oxford, UK SISSA-International School for Advanced Studies, via Bonomea, 265, 34136 Trieste, Italy The intriguing phenomenon of many-body localization (MBL) has attracted significant interest recently, but a complete characterization is still lacking. We introduce the total correlations, a concept from quantum information theory capturing multi-partite correlations, to the study of this phenomenon. We demonstrate that the total correlations of the diagonal ensemble provides a meaningful diagnostic tool to pin-down, probe, and better understand the MBL transition and ergodicity breaking in quantum systems. In particular we show that the total correlations has sub-linear dependence on the system size in delocalized, ergodic phases, whereas we find that it scales extensively in the localized phase developing a pronounced peak at the transition. Reference: arXiv:1504.06872.
The total correlations Heisenberg spin-1/2 chain . . .
! Definition: Consider an N-partite system, then . . . of N sites with coupling constants J, Jzz and disorder strenght h: N N X X h i i+1 i i+1 i i+1 i i T (ρ) := S(ρm) − S(ρ). H = J (σxσx + σyσy ) + Jzz σzσz + hi σz m=1 i=1
Where S(ρ) := −tr(ρ log2 ρ) is the von Neumann entropy. Where we set J = Jzz = 1 and take hi ∈ [−h, h] uniformly distributed. Intuition: Measures distinguishability from product state: Then it has a MBL transition at h = hc ∈ [2, 4] [1]. T (ρ) = min S(ρkπ) πproduct state Numerical results 2 Where S(ρkσ) := −tr(ρ log2 σ) − S(ρ) ≥ kρ − σk1/2 is the relative entropy. 1 Restrict to zero magnetization subspace Ergodicity 2 For product initial states compupte T (ω). 3 Average over all initial states and disorder → T (ω). Starting point: Ergodic Hypothesis: ergodic ⇔ systems explore phase space uniformly 5.00 Ergodic 5 h=4 ⇒ infinite time average = microcanonical average non-ergodic ∝ N h = 4 N=16 Essentially impossible in QM hence ask for less: 1.00 4 L N=14
0.2 0.50 h ≈h=00 ! = 3 Definition (in words): h T H N=12 T - A pair of Hamiltonian and initial state is ergodic (as oppose to MBL) if it T N=10 2 0.10 ergodic N=8 explores at least a constant fraction of the available Hilbert space. 0.05 1 ∝ log(N) N=6 0 Quantify: For a fixed initial state ρ and non-degenerate Hamiltonian H define 1 2 3 4 5 6 7 8 0 5 10 15 the dephased or time-averaged state (diagonal ensemble) h N Z τ X 1 −itH itH Figure 1: Initial increase for small h is Figure 3: For small h we see logarithmic ω := |EnihEn| ρ |EnihEn| = lim dt e ρ e , growth, as expected in an ergodic system, and τ→∞ τ power-law like with exponent 2.7(2). The n 0 deviation maks beginning of the MBL transition at large h linear growth. with |E i eigenvectors of H. region. n 0.2 A family of systems of increasing size N is ergodic if ∃λ > 0 such that for most 0.1 h=3.2 h=3.0 product initial states from some subspace (i.e., fixed magnetization) of N=6 h=2.8 0.0
Localized N=8 fit dimension d it holds (with high probability over the disorder average) h=2.6 N=10 0.20 -0.1 1 N N=12 linear 6 -
S(ω) ≥ log (λ d) T 2 N=14 5 h=2.4
- T N 0.00 0.05 0.10 0.15 0.2 0.10 N=16 4 T 4 3 -0.3 3 h=2.2
Scaling of T with N 0.05 peak 6 8 10 12 14 16 h 2 -0.4 6 8 N 10 12 14 16 1 PN N Non-ergodic: As T (ω) involves the sum S(ωm) of the N subsystem 1 2 5 10 20 50 100 m=1 Figure 4: Difference between T and best entropies: h possible linear fit for different values of h. T (ω) ∝ N Figure 2: For high h, T decays as a Crossover from a nearly linear scaling to a power-law with exponents of −0.9(1). Inset sub-linear scaling happens at h = 2.6(2) Ergodic: If a family of disordered systems is ergodic however, then ∃λ > 0 such shows position of peaks in Figure 1. (robust against omitting data points and holds that for most product initial states with high probability also for affine fits). The inset shows raw data. N X T (ω) ≤ S(ωm) − log2(λ d). Conclusions m=1 For a spin-1/2 chain the magentization zero subspace has Total correlations in the diagonal ensemble signal the MBL transition √ √ N N 2 −2 N They expose how this transition involves reorganization of correlations d = N/2 = N!/ 2 ! ≥ 8 π e 2 / N and S(ωm) ≤ log2 2 = 1, so that √ −2 Results suggests two step transition in line with [2, 3, 4] T (ω) ≤ log2(N)/2 − log2(λ 8 π e ). ergodic non-ergodic localized Intuition: Transport in ergodic systems makes parts so mixed that their distinguishability from the closest product state only grows logarithmically. 0 1 2 3 4 5 6 h
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