Ashvin Vishwanath UC Berkeley

Transcription

1 TOPOLOGY + LOCALIZATION: QUANTUM COHERENCE IN HOT MATTER Ashvin Vishwanath UC Berkeley arxiv: (to appear in Nature Comm.) Thanks to David Huse for inspiring discussions Yasaman Bahri (Berkeley) Ehud Altman (Weizmann) Ronen Vosk (Weizmann) Drew Potter (Berkeley)

2 OVERVIEW Many body system of spins, strongly interacting Time evolution of a high energy state? Quantum coherence lost by interactions. Apply field +h for time T and -h for time T. A coherent spin will `echo. Lost by decoherence. (time evolution of environment). Here, quantum coherent echo without cooling or isolation. Topological qbit + localization.

3 OUTLINE Introduction ETH (thermalization) MBL (localization) SPT (topology) A fully localized topological phase (MBL+SPT) Signatures of Quantum coherence without cooling. Theory question -which topological phases can we fully localize?

4 THERMALIZATION IN MACROSCOPIC QUANTUM SYSTEMS Energy spectrum of typical quantum many body system (N sites, N => infinity) Usually - ground state (cooling required) E } E JN2 N highly excited states e= E/N (finite) N

5 EIGENSTATE THERMALIZATION HYPOTHESIS (ETH) System serves as its own thermal bath Signature of thermal equilibrium present even in a single eigenstates (extreme micro canonical ensemble). highly excited states - thermal. Deutsch 91, Srednicki 94 A A e H i E A = Tr Ā [ ih ] S A = Tr[ A log A ] S A sv A s=0

6 MANY BODY LOCALIZATION Known exceptions: (i) integrable systems with translation invariance (special) (ii) localization in disordered systems (MBL) Model Hamiltonian 1D: H = µ i n i µ i t H = X i X i t i c i c i+1 Anderson Localization - wave functions localized H 0 = X i V i n i n i+1 Many body localization Anderson; Gornyi, Polyakov and Mirlin; Basko, Aleiner, Altshuler; Oganesyan & Huse; Pal & Huse

7 FULL MANY BODY LOCALIZATION Full MBL - states at all energies localized. All states like ground states E Local conserved quantities: ñ i = Zn i + (2body) + (3body) +... Oganesyan and Huse (2013), Serbyn, Papic & Abanin (2013) Experimental signature - Bloch group, cold atoms in quasi-periodic optical lattice. Is it boring? Quantum but disconnected pieces? NO collective phenomena like broken symmetry and topological properties.

8 TOPOLOGICAL PHASES E x Short range entangled topological phases (~Integer Quantum Hall Effect). Unique ground state with periodic boundary conditions. Edge states with boundaries. Protected by bulk gap. Ground states topological. Excited states, bulk excitations mix edges unless localized.

9 QUANTUM ORDERS FROM MANY BODY LOCALIZATION MBL excited states are like ground states. Localization protected quantum orders (Huse, Nandkishore, Oganesyan, Pal, Sondhi) Eg. can stabilize broken symmetry phases in 1D excited states (not possible at finite T). More than just `isolated quantum regions - collective effects.

10 TOPOLOGY + LOCALIZATION Haldane chain: Spin-one antiferromagnet. spin%½%% spin%½%% Bulk gapped. But S=1/2 at the edge (4 fold degeneracy). Protected by SO(3) spin rotation - can be broken down to Z2xZ2. All states in spectrum topological. E Energy Energy Topological Trivial hlog h/ i

11 LOCALIZATION + TOPOLOGY H = X i 1/2 τ" τ" σ" ( i z i+1/2 x i+1 z + i z i x i+1/2 z ) Hamiltonian - sum of commuting projectors. Unique ground state with periodic BC. With randomness - all states localized. With open boundaries - S=1/2 at edge Σ=" n z0, o x 0 1/2 z, y 0 1/2 z

12 QUANTUM COHERENCE AT `INFINITE TEMPERATURE Edge spin echo. Protocol: Start with spins along `z. (High Energy) Apply edge field along `x. (Bσ x 0σ z 1 ) Reverse field at time T =5,000. Spin recovers. Time constant grows exponentially with size L. TIME CONSTANT T 0 TIME FIT T 2 TIME T * 2 TIME SYSTEM SIZE L

13 EDGE SPIN ECHO

14 EDGE SPIN ECHO Observe topological phenomena with an easily prepared initial state and NO cooling. Useful in cold atom systems where cooling is hard but systems isolated

15 WHICH TOPOLOGICAL PHASE CAN BE LOCALIZED? Require all eigenstates to be in the same topological phase. If we have a commuting projector H => Full MBL possible. If full MBL => local integrals of motion can be made into commuting projectors. NO commuting projectors representation => Not possible to full MBL the phase. Which topological phases have commuting projector representations? (with Drew Potter - to appear)

16 LOCALIZABLE TOPOLOGICAL PHASES All 1D Topological phases can be written as commuting projectors. r Chiral phases in 2D (eg. Integer Quantum hall effect) has NO commuting projector representation. Need a thermal Hall effect (assume energy is conserved) but J E =[ r, r 0] Non-chiral 2D SPTs? eg. topological insulator bosonic phases admit commuting projector representations free fermions do not (no Wannier states) Even with interactions some fermion SPTs have no commuting projector H.

17 SUMMARY Disorder and localization can help stabilize topological phases of matter. No cooling is required to observe characteristic features of these states. `Localizability may point to fundamental distinctions between different topological phases.