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 ~