(De)-localization in mean-field quantum glasses

Transcription

1 QMATH13, Georgia Tech, Atlanta October 10, 2016 (De)-localization in mean-field quantum glasses Chris R. Laumann (Boston U) Chris Baldwin (UW/BU) Arijeet Pal (Oxford) Antonello Scardicchio (ICTP) CRL, Pal, Scardicchio, PRL 113 (2014) Baldwin, CRL, Pal, Scardicchio, PRB 93 (2016) Baldwin, CRL, Pal, Scardicchio, In Preparation

2 Plan Why? Localization and Thermalization Quantum Random Energy Model (QREM) Thermodynamics and eigenstates Quantum p-spin Model Thermodynamics and eigenstates Summary 2

3 Highly excited eigenstates Isolated Quantum Many-Body Systems 3

4 Highly excited eigenstates Isolated Quantum Many-Body Systems Eigenstate Thermalization Hypothesis (ETH) Many-Body Localization (MBL) 3

5 Highly excited eigenstates Isolated Quantum Many-Body Systems Eigenstate Thermalization Hypothesis (ETH) Excitations delocalized Many-Body Localization (MBL) Excitations localized 3

6 Highly excited eigenstates Isolated Quantum Many-Body Systems Eigenstate Thermalization Hypothesis (ETH) Excitations delocalized Eigenstates thermal Many-Body Localization (MBL) Excitations localized Eigenstates frozen 3

7 Highly excited eigenstates Isolated Quantum Many-Body Systems Eigenstate Thermalization Hypothesis (ETH) Excitations delocalized Eigenstates thermal Energies repel (GOE) Many-Body Localization (MBL) Excitations localized Eigenstates frozen No repulsion (Poisson) 3

8 Highly excited eigenstates Isolated Quantum Many-Body Systems Eigenstate Thermalization Hypothesis (ETH) Excitations delocalized Eigenstates thermal Energies repel (GOE) Many-Body Localization (MBL) Excitations localized Eigenstates frozen No repulsion (Poisson) No `Ψ-Chaos `Ψ-Chaos 3

9 Eigenstate Thermalization Hypothesis A System ETH systems behave as their own bath. Many-body eigenstates reproduce equilibrium on subsystems. H E i i = E i E i i A =Tr S\A E i ihe i =Tr S\A e H (At correct temperature for E) Deutsch (1991), Srednicki (1994), Rigol, Dunjko, Olshanii (2008), Pal and Huse (2010), 4

10 Frozen Eigenstates A System Localized systems can t exchange energy Many-body eigenstates locally frozen in particular patterns Eigenstate Edwards-Anderson order parameter q ES = 1 X z h i i 2 N i Note: possibly distinct from qea in Gibbs state Deutsch (1991), Srednicki (1994), Rigol, Dunjko, Olshanii (2008), Pal and Huse (2010), 5

11 Eigenstate Chaos ( Ψ-Chaos ) ETH MBL E E 2 N 2 N Local observables smooth M(n) =hn S z 0 ni = M( n ) M(n) n dm( ) d n M 0 ( )e Ns( ) Local observables frozen per eigenstate M n = hn +1 Sz 0 n +1i hn S z 0 ni = O(1)

12 Highly excited eigenstates Isolated Quantum Many-Body Systems Eigenstate Thermalization Hypothesis (ETH) Excitations delocalized Eigenstates thermal Energies repel (GOE) Many-Body Localization (MBL) Excitations localized Eigenstates frozen No repulsion (Poisson) No `Ψ-Chaos `Ψ-Chaos 7

13 Highly excited eigenstates Isolated Quantum Many-Body Systems Delocalized Eigenstate Thermalization Nonergodic Many-Body Hypothesis (ETH) Localization (MBL) Excitations delocalized Excitations delocalized Excitations localized Eigenstates thermal Eigenstates frozeneigenstates frozen Poisson level statistics Energies repel (GOE) Ψ-Chaos No repulsion (Poisson) No `Ψ-Chaos `Ψ-Chaos 7

14 Where we looked Analytically tractable localization transitions/phases? Localization is ultimate glass any connection?

15 B. Derrida (1980,1981), Y. Goldschmidt (1990), T. Jorg, et al (2008), CRL, Pal, Scardicchio (2014), Baldwin, CRL, Pal, Scardicchio (2016) Quantum Random Energy Model H = E({ z i }) X i x i Classical Random Energy Model The simplest spin glass

16 B. Derrida (1980,1981), Y. Goldschmidt (1990), T. Jorg, et al (2008), CRL, Pal, Scardicchio (2014), Baldwin, CRL, Pal, Scardicchio (2016) Quantum Random Energy Model H = E({ z i }) X i x i Classical Random Energy Model The simplest spin glass Random energy for each z-state P (E) = p 1 e E2 N N N-body generalization of SK model

17 B. Derrida (1980,1981), Y. Goldschmidt (1990), T. Jorg, et al (2008), CRL, Pal, Scardicchio (2014), Baldwin, CRL, Pal, Scardicchio (2016) Quantum Random Energy Model H = E({ z i }) X i x i Classical Random Energy Model The simplest spin glass Random energy for each z-state P (E) = p 1 e E2 N N Transverse field Provides dynamics N-body generalization of SK model

18 B. Derrida (1980) Classical Limit: Statistical Mechanics Microcanonical = E N s( ) 0 = p log 2 = 1 2T Canonical T Paramagnet f = T log 2 T c = 1 2 p log 2 Glass f = p log 2 1 4T s( ) = log 2 2 n(e) =2 N P (E) e Ns(E/N)

19 Y. Goldschmidt (1990) Canonical Phase Diagram f = T log 2 Quantum PM 1 4T f = T log (2 cosh /T ) Classical PM Classical Glass f = p log 2

20 CRL, Pal, Scardicchio PRL 2014; Baldwin, CRL, Pal, Scardicchio PRB 2016 Eigenstate Phase Diagram Localized Poisson q = 1 ETH GOE q = 0 Quantum PM Classical PM Classical Glass T = 1 2

21 Methods Exact Diagonalization (N <= 14) Perturbation theory for eigenstates Forward scattering approximation Rough estimates Single resonance approximation 1RSB replica treatment of wavefunction Numerical transfer matrix treatment (N <= 20) 13

22 Numerical Phase Diagram E/N 0.0 XXXX N =8, 10, 12, Contours of level statistics ratio [r] =0.48 X marks the replica formula estimate

23 Forward Scattering Leading perturbative wavefunction `forward scattering X Y b = paths p:a!b i2p E a E i Directed random polymer on hypercube Uncorrelated Gaussian random potential Transfer matrix treatment (numerical) Replica treatment of polymer problem identifies transitions as well

24 E a = E N s( ) Simple Estimate E i b n! E a n Gap typically O(N) so expand b = b X Y E paths p:a!b i2p a b E a b E i 0 n X p i2p Fluctuations small in N 1 p N E i E a + 1 A Demand typical amplitude decreasing uniformly to n = N c = a (T c = 1 2 ) Paths eventually resonate: can we do better?

25 Proliferation of Resonances SRA: Paths typical till resonance at x Expected number of resonances e Nf(x, ) Resonances proliferate beyond At distance x = p 2 + O( 2 ) x 17

26 A bit more structure? Random energy model is completely non-local Local fields in a given configuration O(N) Should be O(1) More local / less tractable model? 18

27 Quantum p-spin Glass Classical p-spin Glass The (next) simplest spin glass

28 Quantum p-spin Glass Classical p-spin Glass The (next) simplest spin glass Recovers REM as p!1 p-body generalization of SK model Typical fields O(p)

29 Quantum p-spin Glass Classical p-spin Glass The (next) simplest spin glass Transverse field Provides dynamics Recovers REM as p!1 p-body generalization of SK model Typical fields O(p)

30 Canonical Phase Diagram T c = p p + T d = r p ln p + T T s = O(1) Paramagnet Classical Glass Y. Goldschmidt (1990); Th. M. Nieuwenhuizen, F. Ritort (1998) 20

31 Baldwin, CRL, Pal, Scardicchio (soon) Classical Clustering Entropy of states at distance x with energy matching origin 1 (1 2x) p s(x, ) = x ln x (1 x)ln(1 x) 1+(1 2x) p 2

32 Baldwin, CRL, Pal, Scardicchio (soon) Classical Clustering Entropy of states at distance x with energy matching origin 1 (1 2x) p s(x, ) = x ln x (1 x)ln(1 x) 1+(1 2x) p 2

33 Baldwin, CRL, Pal, Scardicchio (soon) Classical Clustering Entropy of states at distance x with energy matching origin 1 (1 2x) p s(x, ) = x ln x (1 x)ln(1 x) 1+(1 2x) p 2 T d

34 Baldwin, CRL, Pal, Scardicchio (soon) Classical Clustering Entropy of states at distance x with energy matching origin 1 (1 2x) p s(x, ) = x ln x (1 x)ln(1 x) 1+(1 2x) p 2 T d

35 Baldwin, CRL, Pal, Scardicchio (soon) Classical Clustering Entropy of states at distance x with energy matching origin 1 (1 2x) p s(x, ) = x ln x (1 x)ln(1 x) 1+(1 2x) p 2 T d T s

36 Dynamical Phase Diagram T c = p p + T d = r p ln p + T T s = O(1) ETH GOE q=0 Nonergodic Poisson q =1 4T 2 2 /p 2 Paramagnet Equilibrium Glass Baldwin, CRL, Pal, Scardicchio (soon) 26

37 Proliferation of Resonances SRA: Paths typical till resonance at x Neglect initial cluster; ok at low temperature Underestimate resonance proliferation Gives O(1/p) correction to phase boundary f(x) x 27

38 Summary The p-spin and QREM provides a mean-field model of eigenstate localization, and the localized-eth transition at finite energy density mobility edge. First order eigenstate transition. Perturbative treatment in forward approximation directed random polymer on hypercube. De-localization transition inside canonical paramagnetic phase, but below T d.

39 Open Questions Complete analytic solution of p-spin/ QREM? Do thermodynamics reflect dynamical transition? (Existing canonical phase diagrams perhaps not complete.) Time scales in the ETH regime? No infinite temperature localization feature of long-range interactions? Connection to short-range MBL?

40 30

41 P. W. Anderson, Phys. Rev. (1958) Single-particle Localization H = t X ij a i a j + X i µ i n i µ i 2 [ W/2,W/2] Energy Site

42 P. W. Anderson, Phys. Rev. (1958) Single-particle Localization H = t X ij a i a j + X i µ i n i µ i 2 [ W/2,W/2] Energy Site Off-resonant hopping fails to hybridize sites at long-distances: H = X e a a N (r) 2 Localized e r/

43 Basko, Aleiner, Altshuler, Annals of Physics (2006), Huse and Oganesyan (2007), Pal and Huse (2009) Localization in Fock space Start with single particle localized states and add in interactions: H = X e a a + X V a a a a Can weak V cause hybridization of localized many-particle states? 2 N

44 Canonical Thermodynamics from Replicas Imaginary Time Replica trick in imaginary time Replica Time and replica dependent order parameters Static RS and 1RSB ansatzes give three phases Y. Goldschmidt (1990) Q 0 kk 0 ( )= 1 N X i i (k) i (k 0 )

45 E a = E N s( ) Central Energies E i Gap typically, fluctuations large b = X p N Y E paths p:a!b i2p a Bound by greedy path at distance n g E i N (N 1) pn p N 1 p N ( p N) n Demand amplitudes small gives upper bound on delocalization c < p 1 N Counting resonances more carefully gives log correction

46 Replicated Polymers Typical amplitudes given by quenched average ln =lim n!0 n 1 n n interacting paths on hypercube n = r i = X p 1 p n Y i nx 1 [i 2 p a ] a=1 w r i(p 1 p n ) i

47 Replicated Polymers 1RSB Ansatz: polymers clump in n/x groups of x paths Z n X p Y w x i 1 A n/x =exp[nf(x)] i2p f(x) = L x (log L 1 + log wx i ) w x = Z de p N e E2 /N E a E x

48 Finite-size form at zero energy Replica recipe: f 1RSB = min x2[0,1] f(x) f(x) = L x (log L 1 + log wx i ) w x = Z de p N e E2 /N E a E 1RSB holds at finite L but is swamped slowly by L! paths x Demanding amplitude decays at size L = N c = p 2 p N log p 2/ N +

49 Perturbative Rigidity!!i = E N In forward scattering: E 0 ( )=E 0 2 NX i=1 E p 2 log(2) 1 E i E 0 + GS( = 0)i 0 + O 2 O(N) O(1) All orders give O(1) corrections to extensive energies States localized strongly to corners of hypercube Cross with paramagnetic state at boundary T. Jorg et al (2008); Baldwin, CRL, Pal, Scardicchio (2016) N Inverse Participation Ratio: IPR = 1 O(1/N )

50 Quantum Annealing Annealing transverse field ground state search Final finite energy density approximate search Scaling of final energy density with time how hard is approximation? = E N!!i GS( = 0)i 0 + O 2 N

51 Annealing the QREM Unstructured cost function exponential lower bounds E f (T ) N 1 2 ln 2 ln 2 = ln T N Brick Wall Classical Bound p p 2 0 Quantum Bound 41

52 Annealing the QREM Linear ramping iterated Landau-Zener problem E f (T ) N 1 2 ln 2 ln 2 = ln T N Brick Wall Classical Bound p p 2 0 Quantum Bound 42

53 Many-body Level Statistics Level statistics diagnose dynamical phase transition Ratio diagnostic cancels DOS fluctuations MBL: Poisson level statistics E [r] 0.39 ETH: GOE level statistics 2 N [r] 0.53

54 Local observables In ergodic phase local observables are smooth M(n) =hn S z 0 ni = M( n ) M(n) n dm( ) d n M 0 ( )e Ns( ) Local observables frozen in MBL phase M n = hn +1 Sz 0 n +1i hn S z 0 ni = O(1) In QREM, z-magnetization is local H = E({ z i }) X i x i

55 Spider diagrams Histogram of delta Z-Magnetization across eigenstates N=8 N=10 hs z 0i N=12 N=14 hs z 0i = E/N = E/N

56 Spider cuts (flow)

57 [r] flow (fixed field cuts)

58 QREM IPRs

59 ` lim 1/T T!1 Z dt < S z (t)s z (0) > c

60 Y. Goldschmidt (1990) Replica Solution of QREM Replica trick in Suzuki-Trotter representation log Z =lim n!0 Z n 1 n Introduce time-replica dependent order parameter Z n = Q 0 kk 0 ( )= 1 N Z Y ( k)6=( 0 k 0 ) X i i (k) i (k 0 ) dq 0 kk 0 Z Y d 0 kk 0 e NG

61 Y. Goldschmidt (1990) Replica Solution of QREM Exact solutions in static approximation kk = 0 Q 0, 0 kk = 0 0 6= 0 kk =, 0 kk 0 = Q 0 Two replica-symmetric paramagnetic phases Q 0 kk = Q, 0 0 kk0 = 6= 0 kk =, 0 kk 0 = One 1RSB phase, n/m groups of m replicas Q L,L 0 = q 2 Q L,L0 0 = q 11 L 6= L 0 L,L 0 = 2 L,L 0 0 = 11