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

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

Standard model of Quantum criticality

xn

zn

zn hJH 1

• Quantum Ising model:

x

zzzzz

Jh

h

Standard model of Quantum criticality

xn

zn

zn hJH 1

• Quantum Ising model:

Ferro-magnet

Para-magnet

x

z

QCP

• Phase diagram

hJ

T

Quantum critical regime

zQC hJT ~

zzzz

Jh

h

Disordered Quantum Ising model

xnn

zn

znn hJH 1

• Quantum Ising model:

QCP

Para-magnet

Ferro-magnet

• Phase diagram:

hJ

T

Quantum critical regime

hJExpTQC ~

zQC hJT ~

xn

zn

zn hJH 1

zzzz z x

zzzzz

Surprise: Transition in all excited states

xnn

zn

znn hJH 1

• Quantum Ising model:

FMPM

• Phase diagram:

hJ

T Hilbert glasstransition

~typ......

...... Or:...... [All eigenstates

doubly degenerate]

QCP

zzzz z x

z

Surprise: Transition in all excited states

xnn

zn

znn hJH 1

• Quantum Ising model:

• Phase diagram:

hJ

T Hilbert glasstransition

~typ......

...... Or:...... [All eigenstates

doubly degenerate]

QCP

zzzz z

xn

xnn

xnn

zn

znn JhJH 11 '

x xxx x

FMPM

x

z

Surprise: Transition in all excited states

FMPM

• Phase diagram:

hJ

THilbert glass

transition

~typ ......

...... Or:......

QCP

• Dynamical quantum phase transition.

• Temperature tuned, but with no Thermodynamic signatures.

• Accessible example for an MBL like transition.

Hilbert glassphase

x-phase

maxJ

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-wallexcitations

• neighboring fields: quantum fluctuations.

zzJ 21max

rightleft hhJ ,max

xlefth 1

Cluster ground state:

2121 ,

E

maxJ

2121 ,

1 2

• Choose ground-state manifold.

xrighth 2

maxJ

hhh rightleft

eff

1 2

zzleftJ 21

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.

xh 2max

rightleft JJh ,max

zzrightJ 32

maxh

JJJ rightleft

eff

E

Field aligned:

2

maxh

3

maxh

2Anti-aligned:

21 3

• Choose ground-state manifold.

X

RG sketch•Ferromagnetic phase:

•Paramagnetic phase:

hJ

X

X XX XhJ

Universal coupling distributions and RG flow

• Initially, h and J have some coupling distributions:

)(h )(J

JmaxJhmaxh

Universal coupling distributions and RG flow

• These functions are attractors for all initial distributions.

• gh and gJ flow:

Ferro-magnet

Para-magnet

hJ lnln

RG

-flo

w

0

• As renormalization proceeds, universal distributions emerge:

)(JRG

J

JgJ 1

1)(hRG

h

hgh 1

1

},max{ hJ

Jh gg , flow with RG

hJ gg

QCP

maxJDomain-wallexcitations

• neighboring fields: quantum fluctuations.

zzJ 21max

rightleft hhJ ,max

xlefth 1

Cluster ground state:

2121 ,

E

maxJ

2121 ,

1 2

xrighth 2

maxJ

hhh rightleftExcited

eff

1 2

What about excited states?

• Put domain walls in strongest bonds:

maxJ

hhh rightleftGS

eff No effect on coupling magnitude!

Pekker, GR, VO, Altman, Demler; Huse, Nandkishore, VO, Sondhi, Pal (2013)

maxh

• Make spins antialigned with strong fields:

1 2

xh 2max

rightleft JJh ,max

zzrightJ 32

maxh

JJJ rightleftExcited

eff Field aligned:

2

E

maxh

3

2Anti-aligned:

zzleftJ 21

2X1 3

maxh

JJJ rightleftGS

eff • No effect on coupling magnitude!

What about excited states?Pekker, GR, VO, Altman, Demler; Huse, Nandkishore, VO, Sondhi, Pal (2013)

RSRG-X Tree of states

• At each RG step, choose ground state or excitation:

[six sites with large disorder]

RG sketch•Hilbert-glass phase:

•Paramagnetic phase:

hJ

X

X XX XhJ

Excited state flow

• Transition persists:

Random-domain clustersHilbert-glass

X-phase

RG

-flo

w

0

• Universal distribution functions independent of choice:

|)(| JRG

J

JgJ 1||

1|)(| hRG

h

hgh 1||

1

},max{ hJ

Jh gg , flow with RG

hJhJ gggg /hJ lnln

HGT

Order in the Hilbert glass vs. T=0 Ferromagnet• Symmetry-broken T=0 Ferromagnetic state:

...... ...... or

GSGSm znOrder parameter:

• Typical Hilbert-Glass excited state:

......

znm Order parameter:

z

nm

2z

nEAm

• Temporal correlations:

)()0( tm zn

zn

......

~typ

......

hJ HGT

Order in the Hilbert glass vs. T=0 Ferromagnet

QCP

FM PM

hJ

T

GSGSm zn

hJ QCP

...... ......

Hil

bert

Gla

ss tr

ansi

tion

2z

nEAm

T-tuned Hilbert glass transition: hJJ’ model

xn

xnn

xnn

zn

znn JhJH 11 '

• Quantum Ising model+J’:

X-states

hJ

T

1

Hilbertglass

• T (or energy-density) tuned transition

• But: No thermodynamic signatures

zzzzx x x x

1 2 3 4 5

• J’>0 increases h for low-energy states.

hJJ ,'

RSRG-X Tree of statesE

nerg

y

RG step

Color code: inverse T

01

T

01

T

• Sampling method: Branch changing Monte Carlo steps.

RSRG-X results for the Hilbert glass transition

Jh ~

Flows for different temperatures: Complete phase diagram:

Thermal conductivity

• No thermodynamic signatures – only dynamical signatures exist.

• Only energy is conserved: Signatures in heat conductivity?

~)( 3EngineeringDimension:• assume scaling form: || c

Numerical tests

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?

Edwards-Anderson order parameter

Lifshitz localization – a subtle example

• Tight-binding electrons on an irregular lattice.

• Density of states:

111 nnnnn JJH nJ

)(E

E

kJEk cos2Pure chain:

224

1~)(

EJE

Random J:

EEE

3ln

1~)(

Dyson singularity

Method of attack: Real space RGMa, Dasgupta, Hu (1979), Bhatt, Lee (1979)

J maxJ

leftJrightJ

1 3 42

32322

1

• Eliminated two sites.

• Reduced the largest bond

• New Heisenberg chain resulting with new suppressed effective coupling.

max2J

JJ rightleft

D.S. Fisher (1994)

1 3 42

J

5 6 7 8

• Functional flow and universal coupling distributions:

)(J

JmaxJ

Universality of emerging distribution functions

D.S. Fisher (1994)

1 3 42 5 6 7 8

• Functional flow and universal coupling distributions:

)(JRG

JmaxJ

1gJ

Universality of emerging distribution functions

|ln|

1~

maxJg

0

Random singlet phaseD.S. Fisher (1995)

1 3 42 5 6 7 8

• Low lying excitations: excited long-range singlets:

EEE

3ln

1~)(

• Susceptibility:

02 /ln

1~)(

TTTT

T

1~

)(E

Dyson singularity again!

Engtanglement entropy in the Heisenberg model

L2log3

1

AAAAB TrE 2logHolzhey, Larsen, Wilczek (1994).

• Random singlet phase:

B BA

L

• Pure chain:

Every singlet connecting A to B → entanglement entropy 1.

ABEHow many qubits

in A determined by B

(QFT Central charge, c=1)

Vidal, Latorre, Rico, Kitaev (2002).

LE number of singlets entering region A.

Engtanglement entropy in the Heisenberg model

L2log3

1

Holzhey, Larsen, Wilczek (1994).

• Random singlet phase:

B BA

L

• Pure chain: ABEHow many qubits

in A determined by B

(CFT Central charge, c=1)

Vidal, Latorre, Rico, Kitaev (2002).

LEL ln3

1 L2log2ln

3

1

Effective central charge 12ln1 randomcGR, 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.

Universality at the transition?Altman, Kafri, Polkovnikov, GR (2009)

Insulator superfluid

g0 (~ J )

RSGBG

MG

1

g=1)(JRG

JmaxJ

1gJ

Mechanical analogy

1J 2JnJ

Average effectivespring constant

= 1/1 J

(ave of inverse J)

0 when g=1.Stiffness ~