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

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

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

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

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

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

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

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

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. Domain-wall excitations neighboring fields: quantum fluctuations. Cluster ground state: 12 Choose ground-state manifold. 12

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. 1 2 neighboring fields: quantum fluctuations. Field aligned: 3 Anti-aligned: 2 1 3 Choose ground-state manifold. X

11. RG sketch Ferromagnetic phase: Paramagnetic phase: X 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 These functions are attractors for all initial distributions. g h and g J flow: Ferro- magnet Para- magnet RG-flow As renormalization proceeds, universal distributions emerge: flow with RG QCP

14. Domain-wall excitations neighboring fields: quantum fluctuations. Cluster ground state: 12 12 What about excited states? Put domain walls in strongest bonds: No effect on coupling magnitude! Pekker, GR, VO, Altman, Demler; Huse, Nandkishore, VO, Sondhi, Pal (2013)

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

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: Paramagnetic phase: X X X X X

18. Excited state flow Transition persists: Random-domain clusters Hilbert-glass X-phase RG-flow Universal distribution functions independent of choice: flow with RG 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. HGT Order in the Hilbert glass vs. T=0 Ferromagnet QCP FM PM QCP Hilbert Glass transition

21. T-tuned Hilbert glass transition: hJJ’ model Quantum Ising model+J’: X-states Hilbert glass T (or energy-density) tuned transition But: No thermodynamic signatures z z zz xxx x 12345 J’>0 increases h for low-energy states.

22. RSRG-X Tree of states Energy RG step Color code: inverse T 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. Excited states entanglement entropy: - ‘area law’ in both phases - log(L) at the Hilbert glass transition (Follows from GR, Moore, 2004) Developed the RSRG-X - access to excitations and thermal averaging of L~5000 chains. Other Hilbert glass like transitions?

27. Edwards-Anderson order parameter

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

29. Method of attack: Real space RG Ma, Dasgupta, Hu (1979), Bhatt, Lee (1979) J 1 3 42 Eliminated two sites. Reduced the largest bond New Heisenberg chain resulting with new suppressed effective coupling.

30. D.S. Fisher (1994) 1 342 J 5678 Functional flow and universal coupling distributions: Universality of emerging distribution functions

31. D.S. Fisher (1994) 1 342 5678 Functional flow and universal coupling distributions: Universality of emerging distribution functions 0

32. Random singlet phase D.S. Fisher (1995) 1 342 5678 Low lying excitations: excited long-range singlets: Susceptibility: Dyson singularity again!

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

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

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