TOPOLOGY + LOCALIZATION: !

QUANTUM COHERENCE IN “HOT” MATTER

Ashvin Vishwanath UC Berkeley

arXiv:1307.4092 (to appear in Nature Comm.)

Yasaman Bahri (Berkeley) Drew Potter (Berkeley)Ehud Altman (Weizmann) Ronen Vosk (Weizmann)

Thanks to David Huse for inspiring discussions

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.

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?

THERMALIZATION IN MACROSCOPIC QUANTUM SYSTEMS

• Energy spectrum of typical quantum many body system (N sites, N => infinity)

• Usually - ground state (cooling required)

• highly excited states e= E/N (finite)

E

N

} �E ⇠ JN2�N

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.

A

⇢A = TrA [| ih |]

E

SA = �Tr [⇢A log ⇢A]

⇢A ⇠ e��H

SA ⇠ sVA

| i

s=0

Deutsch 91, Srednicki 94

MANY BODY LOCALIZATION

• Known exceptions: (i) integrable systems with translation invariance (special) (ii) localization in disordered systems (MBL)

• Model Hamiltonian 1D:

!µi

H = �X

i

µini

�H = �X

i

tic†i ci+1

t

Anderson Localization - wave functions localized

�H 0 = �X

i

Vinini+1Many body localization

Anderson; Gornyi, Polyakov and Mirlin; Basko, Aleiner, Altshuler ; Oganesyan & Huse; Pal & Huse

⇠

FULL MANY BODY LOCALIZATION• Full MBL - states at all energies localized.

!

• Local conserved quantities:

• 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.

E

Oganesyan and Huse (2013), Serbyn, Papic & Abanin (2013)

⇠

ni = Zni + (2body) + (3body) + . . .

All states like ground states

TOPOLOGICAL PHASES

• 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.

E

x

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.

TOPOLOGY + LOCALIZATION

• Haldane chain: Spin-one antiferromagnet.

• 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.

spin%½%%spin%½%%

E

!!

Topological!

!!

Trivial!

Trivial phase (disorder variances resp. (0.1,0.1,1)):

-5 -4 -3 -2 -1 0 1 2 3 4 5

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

Energy

Topological phase for N=10 with disorder variances (1,0.1,0.05) in \lambda, V, and \mu. Upper left uses two kinds of symbols (alternating circles or stars) to distinguish multiplets. Upper right plots one line per quadruplet; lower left plots 4 lines (so some show thicker lines).

-5 -4 -3 -2 -1 0 1 2 3 4 5

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Energy

-5 -4 -3 -2 -1 0 1 2 3 4 5

0

0.05

0.1

0.15

0.2

0.25

0.3

Energy

-5 -4 -3 -2 -1 0 1 2 3 4 5

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Energy

hlog |h/�|i

LOCALIZATION + TOPOLOGY

• 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"

τ"

H = �X

i

(�z

i

⌧xi+1/2�

z

i+1 + ⌧zi�1/2�

x

i

⌧zi+1/2)

σ"

τ"

n

�z

0 , �x

0 ⌧z

1/2, �y

0⌧z

1/2

o

Σ="

QUANTUM COHERENCE AT `INFINITE TEMPERATURE’

Edge spin echo. Protocol: Start with spins along `z’. (High Energy!) Apply edge field along `x’. (Bσx0σz1 ) Reverse field at time T =5,000.

!!

• Spin recovers. Time constant grows exponentially with size L.

5

to generate the three-spin interaction. We stress, how-ever, that three-spin interactions are by no means a fun-damental requirement for establishing the strongly dis-ordered topological phase. While the simplest realiza-tion of a 1D topological phase - the S=1 antiferromag-net - transitions into a random singlet phase with strongdisorder33,34, we believe that other models with two-bodyinteractions that realize this phase can be found. Finally,the topological states of the spin chain (1) are related toa much wider classification of symmetry protected topo-logical ground states13–18. An interesting question forfuture investigation is whether the interplay of localiza-tion and topology in the higher dimensional members ofthis classification35–40 can help protect the edge statesfrom mixing with the bulk and thereby lead to novel dy-namical phenomena at high energy densities.

V. ACKNOWLEDGEMENTS

We acknowledge illuminating discussions with D. Huseand D. Budker. This work was supported by the ISF(EA), Minerva foundation (EA), the ERC under the

UQUAM project (EA), NSF GRFP under Grant No.DGE 1106400 (Y.B.), NSF DMR 0645691 (AV) and aSimons Fellowship (AV). E. A. acknowledges the hospi-tality of the Miller institute of Basic research in Scienceand the Aspen Center for Physics under NSF Grant #1066293 for hospitality during the writing of this paper.

7 8 9 10 1110

2

103

104

105

106

SYSTEM SIZE L

TIM

E C

ON

ST

AN

T

T

0 TIME

FIT

T2 TIME

T*

2 TIME

FIG. 4. Scaling of all the time constants with system size.The zero field decay time T0 and spin echo decay timeT2 are nearly identical and scale exponentially with systemsize, whereas the oscillations decay time T ⇤

2 is size indepen-dent. Parameters used had zero mean and variances were(�

�

,�V

, �h

) = (1.0, 0.1, 0.05).

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EDGE SPIN ECHO

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

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)

LOCALIZABLE TOPOLOGICAL PHASES

• All 1D Topological phases can be written as commuting projectors.

• Chiral phases in 2D (eg. Integer Quantum hall effect) has NO commuting projector representation. Need a thermal Hall effect (assume energy is conserved) but

• 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.

JE = [⇧r, ⇧r0 ]

⇧r

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.