David Pekker (U. Pitt) Gil Refael (Caltech) Vadim Oganesyan (CUNY) Ehud Altman (Weizmann)

1. The Hilbert-glass transition: Figuring out excited states in strongly disordered systems David Pekker (U. Pitt) Gil Refael (Caltech) Vadim Oganesyan (CUNY) Ehud Altman (Weizmann) Eugene Demler (Harvard)

2. Outline • Quantum criticality in the quantum Ising model + preview of punchline • Disordered Quantum Ising model and real-space RG • Extending to excited states – the RSRG-X method • The Hilbert-glass transition

3. Standard model of Quantum criticality • Quantum Ising model: z z z z z x

4. Standard model of Quantum criticality • Quantum Ising model: z z z z z x • Phase diagram Quantum critical regime Ferro- magnet Para- magnet QCP

5. Disordered Quantum Ising model • Quantum Ising model: z z z z z z z z z z x • Phase diagram: Quantum critical regime Ferro- magnet Para- magnet QCP

6. Surprise: Transition in all excited states • Quantum Ising model: z z z z z z x Hilbert glass transition • Phase diagram: [All eigenstates doubly degenerate] Or: PM FM QCP

7. Surprise: Transition in all excited states • Quantum Ising model: z z z z z z x x x x x x • Phase diagram: Hilbert glass transition [All eigenstates doubly degenerate] Or: PM FM QCP

8. Surprise: Transition in all excited states • Dynamical quantum phase transition. • Temperature tuned, but with no Thermodynamic signatures. • Accessible example for an MBL like transition. Hilbert glass transition • Phase diagram: Hilbert glass phase x-phase Or: PM FM QCP

9. Disarming disorder: Real space RG [Ma, Dasgupta, Hu (1979), Bhatt, Lee (1979), DS Fisher (1995)] • Isolate the strongest bond (or field) in the chain. • Choose ground-state manifold. • neighboring fields: quantum fluctuations. Domain-wall excitations 1 2 1 2 Cluster ground state:

10. Disarming disorder: Real space RG [Ma, Dasgupta, Hu (1979), Bhatt, Lee (1979), DS Fisher (1995)] • Isolate the strongest bond (or field) in the chain. • Choose ground-state manifold. • neighboring fields: quantum fluctuations. Anti-aligned: X 2 3 2 1 3 1 Field aligned:

11. RG sketch • Ferromagnetic phase: X • Paramagnetic phase: X X X X

12. Universal coupling distributions and RG flow • Initially, h and J have some coupling distributions:

13. Universal coupling distributions and RG flow • As renormalization proceeds, universal distributions emerge: flow with RG • These functions are attractors for all initial distributions. Ferro- magnet Para- magnet RG-flow • gh and gJflow: QCP

14. What about excited states? Pekker, GR, VO, Altman, Demler; Huse, Nandkishore, VO, Sondhi, Pal (2013) • Put domain walls in strongest bonds: • neighboring fields: quantum fluctuations. Domain-wall excitations 1 2 1 2 Cluster ground state: No effect on coupling magnitude!

15. What about excited states? Pekker, GR, VO, Altman, Demler; Huse, Nandkishore, VO, Sondhi, Pal (2013) • Make spins antialigned with strong fields: Anti-aligned: X 2 2 3 1 Field aligned: 3 1 • No effect on coupling magnitude!

16. RSRG-X Tree of states • At each RG step, choose ground state or excitation: [six sites with large disorder]

17. RG sketch • Hilbert-glass phase: X • Paramagnetic phase: X X X X

18. Excited state flow • Universal distribution functions independent of choice: flow with RG • Transition persists: Random-domain clusters Hilbert-glass X-phase RG-flow HGT

19. Order in the Hilbert glass vs. T=0 Ferromagnet • Symmetry-broken T=0 Ferromagnetic state: or Order parameter: • Typical Hilbert-Glass excited state: Order parameter: • Temporal correlations:

20. Order in the Hilbert glass vs. T=0 Ferromagnet Hilbert Glass transition HGT PM FM QCP QCP

21. T-tuned Hilbert glass transition: hJJ’ model • Quantum Ising model+J’: z z z z 1 2 3 4 5 x x x x • J’>0increases h for low-energy states. Hilbert glass X-states • T (or energy-density) tuned transition • But: No thermodynamic signatures

22. RSRG-X Tree of states Color code: inverse T Energy RG step • Sampling method: Branch changing Monte Carlo steps.

23. RSRG-X results for the Hilbert glass transition Flows for different temperatures: Complete phase diagram:

24. Thermal conductivity • No thermodynamic signatures – only dynamical signatures exist. • Only energy is conserved: Signatures in heat conductivity? Engineering Dimension: • assume scaling form:

25. Numerical tests

26. Summary + odds and ends • New universality: -T-tuned dynamical quantum transition. • - No thermodynamic signatures. • Developed the RSRG-X • - access to excitations and thermal averaging of L~5000 chains. • Excited states entanglement entropy: - ‘area law’ in both phases - log(L) at the Hilbert glass transition (Follows from GR, Moore, 2004) • Other Hilbert glass like transitions?

27. Edwards-Anderson order parameter

28. Lifshitz localization – a subtle example Pure chain: Random J: • Tight-binding electrons on an irregular lattice. • Density of states: Dyson singularity

29. Method of attack: Real space RG Ma, Dasgupta, Hu (1979), Bhatt, Lee (1979) • Reduced the largest bond • Eliminated two sites. • New Heisenberg chain resulting with new suppressed effective coupling. J 2 4 3 1

30. J Universality of emerging distribution functions D.S. Fisher (1994) 5 6 7 8 2 3 4 1 • Functional flow and universal coupling distributions:

31. Universality of emerging distribution functions D.S. Fisher (1994) 5 6 7 8 2 3 4 1 • Functional flow and universal coupling distributions: 0

32. Random singlet phase D.S. Fisher (1995) 5 6 7 8 2 3 4 1 • Low lying excitations: excited long-range singlets: • Susceptibility: Dyson singularity again!

33. Engtanglement entropy in the Heisenberg model number of singlets entering region A. A B B (QFT Central charge, c=1) L How many qubitsin A determined by B • Pure chain: Holzhey, Larsen, Wilczek (1994). Vidal, Latorre, Rico, Kitaev (2002). • Random singlet phase: Every singlet connecting A to B → entanglement entropy 1.

34. Engtanglement entropy in the Heisenberg model Effective central charge A B B (CFT Central charge, c=1) L How many qubitsin A determined by B • Pure chain: Holzhey, Larsen, Wilczek (1994). Vidal, Latorre, Rico, Kitaev (2002). • Random singlet phase: GR, Moore (2004). For the experts:Does the effective c obey a c-theorem? No… Examples of enropy increasingtransitions in random non-abelian anyon chains. Fidkowski, GR, Bonesteel, Moore (2008).

35. Universality at the transition? MG BG RSG Insulator superfluid 1 Altman, Kafri, Polkovnikov, GR (2009) g=1 Mechanical analogy Average effectivespring constant = 0 when g=1. Stiffness ~ (ave of inverse J)