Quantum glasses - Frustration and collective behavior at T = 0 - Markus Müller (ICTP Trieste)
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Quantum glasses – Frustration and
collective behavior at T = 0
Markus Müller (ICTP Trieste)
Collaborators:
Alexei Andreanov (ICTP)
Xiaoquan Yu (SISSA)
Lev Ioffe (Rutgers)
Universität zu Köln, 17. November 2010
Arrow of time and ergodicity
The arrow of time
How to know the direction of increasing time?
• Entropy always increases
• Example: diffusion (continues forever in infinite systems)
t ≠ t
Arrow of time and ergodicity
The arrow of time
How to know the direction of increasing time?
• Entropy always increases
• Example: diffusion (continues forever in infinite systems)
t ≠ t
Interesting exception: quantum localized systems → time reversal
symmetry even in infinite systems!
Arrow of time and ergodicity
The arrow of time
How to know the direction of increasing time?
• Entropy always increases
• Example: diffusion (continues forever in infinite systems)
t ≠ t
Interesting exception: quantum localized systems → time reversal
symmetry even in infinite systems!
Ergodicity
Arrow of time and ergodicity
The arrow of time
How to know the direction of increasing time?
• Entropy always increases
• Example: diffusion (continues forever in infinite systems)
t ≠ t
Interesting exception: quantum localized systems → time reversal
symmetry even in infinite systems!
Ergodicity
Fundamental postulate of thermodynamics:
State of maximal entropy (=equilibrium) is reached in finite time.
Eq
But: NO full equilibration when ergodicity is broken
t=∞ (Eq)
Occurs in particular in a large class of disordered systems: Glasses
→ configurational entropy, memory, history dependence, etc.
Interrelation between arrow of time and ergodicity? (both notions associated with time evolution and dynamics) Look at two types of ergodicity breakers = « glasses »!
Glasses: defying equilibration Two types of glasses (i) Quantum localized systems (Anderson glasses) (ii) Frustrated, disordered or amorphous systems
Glasses: defying equilibration
Two types of glasses
(i) Quantum localized systems (Anderson glasses)
(ii) Frustrated, disordered or amorphous systems
Cold atoms
No diffusion at
Single particle QM (Anderson)
i i, j
( )
H = ∑ ε i ni − t ∑ ci+ c j + h.c. large disorder!
No arrow of time!
Glasses: defying equilibration
Two types of glasses
(i) Quantum localized systems (Anderson glasses)
(ii) Frustrated, disordered or amorphous systems
Cold atoms
No diffusion at
Single particle QM (Anderson)
i i, j
(
H = ∑ ε i ni − t ∑ ci+ c j + h.c. ) large disorder!
No arrow of time!
Many particles (Anderson, Fleishman, Altshuler et al., Mirlin et al., etc.) No diffusion at large
disorder! &&
H = ∑ εα nα −
α
∑
α , β , γ ,δ
(
Vαβγδ cα+ cβ+ cγ cδ + h.c. ) Transition to ‘super-
insulator’ at finite T?!
Glasses: defying equilibration
Two types of glasses
(i) Quantum localized systems (Anderson glasses)
(ii) Frustrated, disordered or amorphous systems
High
barriers in
complex
energy
landscape
Examples:
• spin glasses H = ∑ J ij siz s zj
i, jGlasses: defying equilibration
Two types of glasses
(i) Quantum localized systems (Anderson glasses)
(ii) Frustrated, disordered or amorphous systems
High
barriers in
complex
energy
landscape
Examples:
• spin glasses H = ∑ J ij siz s zj
i, j
• electron glasses e2
H = ∑ ni n j + ∑ ε i ni
• dirty superconductors, i, j rij i
underdoped high Tc’s
• defectful supersolids (He)Glasses: defying equilibration
Two types of glasses
(i) Quantum localized systems (Anderson glasses)
(ii) Frustrated, disordered or amorphous systems
High
barriers in
complex
energy
landscape
Examples:
• spin glasses H = ∑ J ij siz s zj
i, j
• electron glasses e2
H = ∑ ni n j + ∑ ε i ni
• dirty superconductors, i, j rij i
underdoped high Tc’s
• defectful supersolids (He)
• many complex systems beyond physics (biology, economy, society)Glasses: defying equilibration
Two types of glasses
(i) Quantum localized systems (Anderson glasses)
(ii) Frustrated, disordered or amorphous systems
High
barriers in
complex
energy
landscape
Examples:
• spin glasses H = ∑ J ij siz s zj Ergodicity breaking!
i, j
• electron glasses e2
H = ∑ ni n j + ∑ ε i ni Transport (diffusion,
• dirty superconductors, i, j rij i arrow of time) ?!
underdoped high Tc’s
• defectful supersolids (He)
• many complex systems beyond physics (biology, economy, society)Glasses: defying equilibration
Two types of glasses
(i) Quantum localized systems (Anderson glasses)
(ii) Frustrated, disordered or amorphous systems
High
barriers in
complex
energy
landscape
Examples:
• spin glasses H = ∑ J ij siz s zj + Γ ∑ six [LiYHF] Ergodicity breaking!
i, j 2 i
• electron glasses e
H = ∑ ni n j + ∑ ε i ni Transport (diffusion,
• dirty superconductors, i, j rij i arrow of time) ?!
underdoped high Tc’s + ∑ tij ci+ c j Quantum? ⎡⎣ H cl , H q ⎤⎦ ≠ 0
• defectful supersolids (He) i, j
• many complex systems beyond physics (biology, economy, society)Two types of « glasses » Frustrated disordered systems Quantum localized systems e.g. Spin glass e.g. Anderson insulator (Fermi glass)
Two types of « glasses » Frustrated disordered systems Quantum localized systems e.g. Spin glass e.g. Anderson insulator (Fermi glass) Arbitrarily large energy barriers ΔE Too small matrix elements between between metastable states distant states in Hilbert space
Two types of « glasses » Frustrated disordered systems Quantum localized systems e.g. Spin glass e.g. Anderson insulator (Fermi glass) Arbitrarily large energy barriers ΔE Too small matrix elements between between metastable states distant states in Hilbert space Destroyed by large T Robust to T (if fully localized) Robust to quantum dephasing/ Destroyed by a dephasing bath coupling to environment Important to avoid in Q-computing!
Two types of « glasses »
Frustrated disordered systems Quantum localized systems
e.g. Spin glass e.g. Anderson insulator (Fermi glass)
Arbitrarily large energy barriers ΔE Too small matrix elements between
between metastable states distant states in Hilbert space
Destroyed by large T Robust to T (if fully localized)
Robust to quantum dephasing/ Destroyed by a dephasing bath
coupling to environment Important to avoid in Q-computing!
Non-ergodicity AND no diffusion
(no clear arrow of time)Two types of « glasses » Frustrated disordered systems Quantum localized systems e.g. Spin glass e.g. Anderson insulator (Fermi glass) Arbitrarily large energy barriers ΔE Too small matrix elements between between metastable states distant states in Hilbert space Destroyed by large T Robust to T (if fully localized) Robust to quantum dephasing/ Destroyed by a dephasing bath coupling to environment Important to avoid in Q-computing! Non-ergodicity, BUT diffusion Non-ergodicity AND no diffusion usually possible (arrow of time there) (no clear arrow of time)
Two types of « glasses »
Frustrated disordered systems Quantum localized systems
e.g. Spin glass e.g. Anderson insulator (Fermi glass)
Arbitrarily large energy barriers ΔE Too small matrix elements between
between metastable states distant states in Hilbert space
Destroyed by large T Robust to T (if fully localized)
Robust to quantum dephasing/ Destroyed by a dephasing bath
coupling to environment Important to avoid in Q-computing!
Non-ergodicity, BUT diffusion Non-ergodicity AND no diffusion
usually possible (arrow of time there) (no clear arrow of time)
Joining two ingredients of ergodicity breaking:
(i) Mutual enhancement or competition?
(ii) Is there many particle-localization in quantum glasses?Intriguing aspects of ‘spin glasses’
N J ij = 0
Classical glass: SK model H =−∑σ J σ z
i ij
z
j J2
i < j =1 J =
2
ij
N
• Thermodynamic transition at Tc to a glass phase
• Unusual order parameter: QEA = 1 ∑ i si 2
N
• Many metastable statesIntriguing aspects of ‘spin glasses’
N J ij = 0
Classical glass: SK model H =−∑σ J σ z
i ij
z
j J2
i < j =1 J =
2
ij
N
• Thermodynamic transition at Tc to a glass phase
• Unusual order parameter: QEA = 1 ∑ i si 2
N
• Many metastable states
Glass phase is always (self-organized) critical! (SK: Kondor-DeDominicis)
Power law correlations in whole glass phase! (Droplets: Fisher-Huse)Intriguing aspects of ‘spin glasses’
N J ij = 0
Classical glass: SK model H =−∑σ J σ z
i ij
z
j J2
i < j =1 J =
2
ij
N
• Thermodynamic transition at Tc to a glass phase
• Unusual order parameter: QEA = 1 ∑ i si 2
N
• Many metastable states
Glass phase is always (self-organized) critical! (SK: Kondor-DeDominicis)
Power law correlations in whole glass phase! (Droplets: Fisher-Huse)
Physical consequences of this criticality:
• Pseudgap in the distribution of local fields! (Palmer, Sommers)
(electron glasses: Coulomb gap!) (MM, Ioffe ’04, MM, Pankov ’07)Intriguing aspects of ‘spin glasses’
N J ij = 0
Classical glass: SK model H =−∑σ J σ z
i ij
z
j J2
i < j =1 J =
2
ij
N
• Thermodynamic transition at Tc to a glass phase
• Unusual order parameter: QEA = 1 ∑ i si 2
N
• Many metastable states
Glass phase is always (self-organized) critical! (SK: Kondor-DeDominicis)
Power law correlations in whole glass phase! (Droplets: Fisher-Huse)
Physical consequences of this criticality:
• Pseudgap in the distribution of local fields! (Palmer, Sommers)
(electron glasses: Coulomb gap!) (MM, Ioffe ’04, MM, Pankov ’07)
• Power law distributed avalanches + Barkhausen noise!
(Numeric: Pazmandi et al ’99, Analytic: Le Doussal, MM, Wiese ‘10)Intriguing aspects of ‘spin glasses’
What are the consequences
? of this criticality in the
quantum versions?
?Problem: Very little is known
about quantum glasses!
= Strongly correlated, disordered quantum systems!
Our strategy:
1. Solve mean field models (infinite
connectivity) - highly non-trivial!
2. Obtain physical understanding
3. Extend to finite dimensions
(large but finite connectivity)Quantum glass models
Disorder: frustration vs. localization?
• Collective excitations in quantum glasses ?
Transverse field Ising spin glass
H = − ∑ σ iz J ijσ zj − Γ ∑ σ ix J ij = 0
(Sherrington-Kirkpatrick SK)
i< j i
J2
J =
2
ij
N
• Glassiness and superfluidity - (spin ½ = hard core bosons)
σ iz ↔ 2ni − 1
t
“Superglass” = glassy supersolid H = − ∑ σ J σ − ∑ σ ixσ xj
z
i ij
z
j
i< j N i< jQuantum SK: Known properties
Transverse field SK model (fully connected, random Ising)
H = − ∑ σ iz J ijσ zj − Γ ∑ σ ix
i< j i
Goldschmidt, Lai, PRL (‘90): There is a
Para-
Static approximation
magnet quantum glass
transition also
at Γ > 0
GlassQuantum SK: Known properties
Transverse field SK model (fully connected, random Ising)
H = − ∑ σ iz J ijσ zj − Γ ∑ σ ix
i< j i
Goldschmidt, Lai, PRL (‘90): There is a
Para-
Static approximation
magnet quantum glass
transition also
at Γ > 0
Glass
Spectral gap closes (Miller, Huse ‘93)
Seems to remains closed in the glass!!
Why?? (Read, Sachdev, Ye ’93)Quantum SK: Known properties
Transverse field SK model (fully connected, random Ising)
H = − ∑ σ iz J ijσ zj − Γ ∑ σ ix
i< j i
Goldschmidt, Lai, PRL (‘90): There is a
Para-
Static approximation
magnet quantum glass
transition also
at Γ > 0
Glass
Spectral gap closes (Miller, Huse ‘93)
??? Seems to remains closed in the glass!!
Why?? (Read, Sachdev, Ye ’93)Understanding the quantum glass
MM, Ioffe ‘07
1. Physical approach
→ Nature of excitations
→ Approach generalizable
to finite dimensionsEffective potential
Construct free energy functional G(m) imposing magnetization m
Homogeneous mEffective potential
Construct free energy functional G(m) imposing magnetization m
• Two states!
• Barrier
becomes
very large at
low T
Homogeneous mEffective potential
Construct free energy functional G(m) imposing magnetization m
• Two states!
• Barrier
becomes
very large at
low T
Homogeneous m Disordered pattern of miEffective potential
Construct free energy functional G(m) imposing magnetization m
• Two states! • Many states!
• Barrier • Hierarchical
becomes valleys
very large at • Barriers never
low T very large:
As if always at
criticality!
Homogeneous m Disordered pattern of miLow energy excitations
XY or Heisenberg ferromagnet:
Goldstone modes:
Soft collective excitations along
flat directions of the potential
What about Ising glasses?Low energy excitations
Two possibilities:
Isolated stable
minimum in the
potential landscape
Many valleys
with rather flat
interconnections!
(in configuration space)Low energy excitations
Two possibilities:
Isolated stable
minimum in the
potential landscape
Many valleys
with rather flat
interconnections!
(in configuration space)Effective potential
(Thouless, Anderson, Palmer ‘77: Classical SK model; Biroli, Cugliandolo ’01, MM, Ioffe ‘07)
H = − ∑ σ iz J ijσ zj − Γ ∑ σ ix
i< j i
Effective potential (exact at N = ∞)
G ({ σ z
i = mi }) = −Γ∑
i
2
i
1
2 i≠ j
1
2 i≠ j
2
0
∞
1 − m − ∑ mi J ij m j − ∑ J ij ∫ dτχ i (τ ) χ j (τ ) + O ( 1N )
Static approximation for susceptibility: χ i (ω → 0 ) = dmi dhi ≈ χ i ( mi )Effective potential
(Thouless, Anderson, Palmer ‘77: Classical SK model; Biroli, Cugliandolo ’01, MM, Ioffe ‘07)
H = − ∑ σ iz J ijσ zj − Γ ∑ σ ix
i< j i
Effective potential (exact at N = ∞)
G ({ σ z
i = mi }) = −Γ∑ i
2
i
1
2 i≠ j
1
2 i≠ j
2
0
∞
1 − m − ∑ mi J ij m j − ∑ J ij ∫ dτχ i (τ ) χ j (τ ) + O ( 1N )
Static approximation for susceptibility: χ i (ω → 0 ) = dmi dhi ≈ χ i ( mi )
Local minima ( ∂G ∂mi = 0 ) (in static approximation)
Γmi
1− m 2
= ∑ J ij m j − mi
j ≠i
∑J
j ≠i
2
ij ( )
χ j mj
i N coupled random
Effective field on σi in z-direction equations for {mi}.
With ~ Exp[αN] solutions!Quantum TAP equations
Thouless, Anderson, Palmer ‘77 (Classical); Biroli, Cugliandolo ’01; MM, Ioffe ’07 (quantum)
Local minima ( !G !mi = 0 )
Γmi
1− m 2
= ∑ J ij m j − mi
j ≠i
∑J
j ≠i
2
ij ( )
χ j mj
i
Environment of a local minimum
Curvatures (Hessian): ! ij = " 2G "mi "m j = J ij + diagonal termsQuantum TAP equations
Thouless, Anderson, Palmer ‘77 (Classical); Biroli, Cugliandolo ’01; MM, Ioffe ’07 (quantum)
Local minima ( !G !mi = 0 )
Γmi
1− m 2
= ∑ J ij m j − mi
j ≠i
∑J
j ≠i
2
ij ( )
χ j mj
i
Environment of a local minimum
Curvatures (Hessian): ! ij = " 2G "mi "m j = J ij + diagonal terms
Standard resolvent technique
Spectrum of “spring constants” in a minimum
Gapless spectrum
')
Spec !" H ij #$ % & H ( ' ) = const ( 2 in the whole glass phase,
J
ensured by marginality of minima!
(at small λ: No Gap!)Soft collective modes
λΓ
Spectrum of “spring constants” Spec ⎡⎣ H ij ⎤⎦ ≡ ρ H ( λ ) = const ×
J2Soft collective modes
λΓ
Spectrum of “spring constants” Spec ⎡⎣ H ij ⎤⎦ ≡ ρ H ( λ ) = const ×
J2
Semiclassical picture:
→ N collective oscillators with mass M ~ 1/Γ: FrequencySoft collective modes
λΓ
Spectrum of “spring constants” Spec ⎡⎣ H ij ⎤⎦ ≡ ρ H ( λ ) = const ×
J2
Semiclassical picture:
→ N collective oscillators with mass M ~ 1/Γ: Frequency
ω2
Mode density ρ (ω ) = const × 2
ΓJSoft collective modes
λΓ
Spectrum of “spring constants” Spec ⎡⎣ H ij ⎤⎦ ≡ ρ H ( λ ) = const ×
J2
Semiclassical picture:
→ N collective oscillators with mass M ~ 1/Γ: Frequency
ω2
Mode density ρ (ω ) = const × 2 Mean square displacement
ΓJ 1
x2 =
ω
Mω
Susceptibility:
Collective modes form a bath
with Ohmic spectral functionSoft collective modes
λΓ
Spectrum of “spring constants” Spec ⎡⎣ H ij ⎤⎦ ≡ ρ H ( λ ) = const ×
J2
Semiclassical picture:
→ N collective oscillators with mass M ~ 1/Γ: Frequency
ω2
Mode density ρ (ω ) = const × 2 Mean square displacement
ΓJ 1
x2 =
ω
Mω
Susceptibility:
Collective modes form a bath
with Ohmic spectral function
Physical picture + generalization of a known result at the glass transition!
[Miller, Huse (SK model); Read, Ye, Sachdev (rotor models)]Soft collective modes
λΓ
Spectrum of “spring constants” Spec ⎡⎣ H ij ⎤⎦ ≡ ρ H ( λ ) = const ×
J2
Semiclassical picture:
→ N collective oscillators with mass M ~ 1/Γ: Frequency
ω2
Mode density ρ (ω ) = const × 2 Mean square displacement
ΓJ 1
x2 =
ω
Mω
Susceptibility: Prediction:
Collective modes form a bath Independent of
with Ohmic spectral function Γ for ωRigorous confirmation
A. Andreanov, MM, ‘10
H = − ∑ σ iz J ijσ zj − Γ ∑ σ ix
i< j iRigorous confirmation
Effective action and selfconsistency problem
1
Effective action and selfconsistency problem A. Andreanov, MM, ‘10
ffective action and selfconsistency problem
Mean field equations (exact for N = ∞)
Zeff = Tr T exp Seff (
Zeff = Tr T exp Seff H = − ∑ σ iz J ijσ zj − Γ ∑ σ ix (1)
(Replica trick: exp[−βFeff ] = Tr T exp Seff (
(1
i< j i (2)
n→0 system copies ) (2
!β! !β! "#" # $$ # !β!
β
# 11# #
Seff =SJeff2 = J 2dτ dτ !
"
dτ dτ ! Q σ z (τ z)σ z (τz! ) +
abQ σ (τ )σ (τ !
) + QQ
aa (τ
(τ−−ττ )σ
!! zz
)σ (τ
(τ )σ
)σ zz (τ! ! ) + Γ
(τ ) $ + Γ dτ dτ
σ xσ x (τ )
(τa) (3) (
β
!! #
a
ab a b b 2 2
aa aa aa !β a
aRigorous confirmation
Effective action and selfconsistency problem
1
Effective action and selfconsistency problem A. Andreanov, MM, ‘10
ffective action and selfconsistency problem
Mean field equations (exact for N = ∞)
Zeff = Tr T exp Seff (
Zeff = Tr T exp Seff H = − ∑ σ iz J ijσ zj − Γ ∑ σ ix (1)
(Replica trick: exp[−βFeff ] = Tr T exp Seff (
(1
i< j i (2)
n→0 system copies ) (2
!β! !β! "#" # $$ # !β!
β
# 11# #
Seff =SJeff2 = J 2dτ dτ !
"
dτ dτ ! Q σ z (τ z)σ z (τz! ) +
abQ σ (τ )σ (τ !
) + QQ
aa (τ
(τ−−ττ )σ
!! zz
)σ (τ
(τ )σ
)σ zz (τ! ! ) + Γ
(τ ) $ + Γ dτ dτ
σ xσ x (τ )
(τa) (3) (
β
!! #
a
ab a b b 2 2
aa aa aa !β a
aEffective action and selfconsistency problem
exp[−βFeff ] = Tr T exp Seff
Rigorous confirmation
Effective action and selfconsistency problem
1
Effective action and selfconsistency problem A. Andreanov, MM, ‘10
ffective action and selfconsistency problem
Mean field equations!β! " (exact for N = ∞)
# Z = Tr T exp S1 #
$
#!
β
(
Seff = J 2 dτ dτ ! eff z
Qab σ (τ )σ
a = Tr (τ
z !
) + eff
b T exp Seff Q (τ − τ )σ
! z
(τ )σ (τ
z !
) + Γ dτ σ x
(τ ) (1)
(Replica trick: 0
Zeff 2 a
exp[−βFeff ] = Tr T exp Seff
aEffective action and selfconsistency problem
exp[−βFeff ] = Tr T exp Seff
Rigorous confirmation
Effective action and selfconsistency problem
1
Effective action and selfconsistency problem A. Andreanov, MM, ‘10
ffective action and selfconsistency problem
Mean field equations!β! " (exact for N = ∞)
# Z = Tr T exp S1 #
$
#!
β
(
Seff = J 2 dτ dτ ! eff z
Qab σ (τ )σ
a = Tr (τ
z !
) + eff
b T exp Seff Q (τ − τ )σ
! z
(τ )σ (τ
z !
) + Γ dτ σ x
(τ ) (1)
(Replica trick: 0
Zeff 2 a
exp[−βFeff ] = Tr T exp Seff
aEffective action and selfconsistency problem
exp[−βFeff ] = Tr T exp Seff
Rigorous confirmation
Effective action and selfconsistency problem
1
Effective action and selfconsistency problem A. Andreanov, MM, ‘10
ffective action and selfconsistency problem
Mean field equations!β! " (exact for N = ∞)
# Z = Tr T exp S1 #
$
#!
β
(
Seff = J 2 dτ dτ ! eff z
Qab σ (τ )σ
a = Tr (τ
z !
) + eff
b T exp Seff Q (τ − τ )σ
! z
(τ )σ (τ
z !
) + Γ dτ σ x
(τ ) (1)
(Replica trick: 0
Zeff 2 a
exp[−βFeff ] = Tr T exp Seff
aEffective action and selfconsistency problem
exp[−βFeff ] = Tr T exp Seff
Rigorous confirmation
Effective action and selfconsistency problem
1
Effective action and selfconsistency problem A. Andreanov, MM, ‘10
ffective action and selfconsistency problem
Mean field equations!β! " (exact for N = ∞)
# Z = Tr T exp S1 #
$
#!
β
(
Seff = J 2 dτ dτ ! eff z
Qab σ (τ )σ
a = Tr (τ
z !
) + eff
b T exp Seff Q (τ − τ )σ
! z
(τ )σ (τ
z !
) + Γ dτ σ x
(τ ) (1)
(Replica trick: 0
Zeff 2 a
exp[−βFeff ] = Tr T exp Seff
aRigorous confirmation
A. Andreanov, MM, ‘10
Take home message for mean field (SK):
• Effective potential approach → consistent physical
picture of the low frequency dynamics
• Quantum glass is “self-organized critical” (gapless)
• The dynamics of active spins
(for ω < Γ) remains almost
independent of Γ, despite mass Γ
and spring constants renormalizing!What survives beyond mean
field?
SK ↔What survives beyond mean
field?
SK ↔
• Gapless collective modes?
• Spatial properties (localization?)
Expected:
Large connectivity → Mean field describes well
the spectrum, except at the lowest energiesBeyond mean field
Quantum ‘spin glass’ with
• Exchange matrix Jij random, |Jij| ~ J
• Large connectivity z
Argued to be a relevant model for electron glasses
close to Mott-Anderson (M-I) transition MM, Ioffe ‘07Beyond mean field
Quantum ‘spin glass’ with
• Exchange matrix Jij random, |Jij| ~ J
• Large connectivity z
Argued to be a relevant model for electron glasses
close to Mott-Anderson (M-I) transition MM, Ioffe ‘07
Repeat effective potential + semi-classics analysis!Beyond mean field
Quantum ‘spin glass’ with
• Exchange matrix Jij random, |Jij| ~ J
• Large connectivity z
Spectrum of spring constants (Hessian Hij) (d>3)
∞ > z 1
Semicircle
+ Corrections in
regime ~1/z5/6:
Delocalized • localization
spectrum! • spectral deviationsBeyond mean field
Quantum ‘spin glass’ with
• Exchange matrix Jij random, |Jij| ~ J
• Large connectivity z
Spectrum of spring constants (Hessian Hij) (d>3)
∞ > z 1
Semicircle
+ Corrections in
regime ~1/z5/6:
Delocalized • localization
spectrum! • spectral deviations
+ semiclassical treatment
Delocalized low-energy excitations, at least down to ω min ~ Jz J hloc
−1 6 typBeyond mean field?
Conclusions from our reasoning:
Criticality of quantum glass
→ large density of soft collective modes
delocalized to very low energies
→ Spin glass-type ergodicity breaking counteracts quantum localization!
→ Instead it enhances transport of energy and charge
(and decoherence in qubits)!Beyond mean field?
Conclusions from our reasoning:
Criticality of quantum glass
→ large density of soft collective modes
delocalized to very low energies
→ Spin glass-type ergodicity breaking counteracts quantum localization!
→ Instead it enhances transport of energy and charge
(and decoherence in qubits)!
→ NO many body localization! (only with weak interactions!)
Especially, not to be expected close to the MIT!Beyond mean field?
Conclusions from our reasoning:
Criticality of quantum glass
→ large density of soft collective modes
delocalized to very low energies
→ Spin glass-type ergodicity breaking counteracts quantum localization!
→ Instead it enhances transport of energy and charge
(and decoherence in qubits)!
→ NO many body localization! (only with weak interactions)
Especially, not to be expected close to the MIT!
Interesting open questions:
• Mobility edge for many body excitations?
• What happens at the quantum phase transition? (cf. MM ’09 regarding Bose glass)S=1/2 ↔ hard core bosons Bose condensation in disorder?
S=1/2 ↔ hard core bosons
How do glassy order
and
superfluidity compete?
Possibility of a “superglass” ?
= amorphous+glassy supersolid
(e.g. dirty bosons [preformed pairs] with Coulomb frustration)Motivation
Supersolidity observed in defectful
(glassy) quantum solids
(Kim&Chan, Reppy, Dalibard)
Torsional oscillator filled
with Helium:
Superfluid fraction
observed in the solid
phase via non-classical
rotational inertia,
+ anomalous shear
properties etc.Superglasses ?! Xiaoquan Yu, MM ‘10
H = −Γ ∑ σ ix − ∑ σ iz J ijσ zj
i i< j
t
H = − ∑ σ ixσ xj − ∑ σ iz J ijσ zj
N i< j i< j
“Self-generated transverse field”Superglasses ?! Xiaoquan Yu, MM ‘10
H = −Γ ∑ σ ix − ∑ σ iz J ijσ zj
i i< j
t
H = − ∑ σ ixσ xj − ∑ σ iz J ijσ zj
N i< j i< j
“Self-generated transverse field”
New aspect:
Glass 1st order transition?
Or superglass?1
(Π̃0 (y) − Π̃ω (y))/ω ≡ A(y) ∼ 3 â(y/Γ),
2
â(x)
Superglasses ?! Xiaoquan Yu, MM ‘10 Γ
With the above assumptions one gets a set of self-consistency equations w
particular the above assumptions 1z
zimmediately lead to the coefficient 5B bei
(Π̃0 (y)
H =
− −Γ
Π̃
If they have a solution,
ω ∑σ
(y))/ω x2
−≡ ∑A(y) σ
i this proves J∼ σ
i ij that
â(y/Γ), â(x) ∼ min(1, 1/x ).
Γj3 the limit Γ $ Γc is indeed given
i< j
Order parameters superglass:
i
With the above assumptions one gets a set of self-consistency equations which are complete
particular the above assumptions immediately lead to the coefficient B being independent of
t that the
If they have a solution, thisHproves
Order parameters superglass:
=−
N i< j
∑ σ ixσ limit
x
j − ∑ Γσ$z JΓcσ is
i ij j
z indeed given by
M =
N
1 the
%σix & above scali
i< j
1 x 1 z 2
Competing order parameters: M= %σ & qEA = %σi &
N i N
M signals superfluidity of hard core bosons
1 z 2 σ z
i ↔ 2ni − 1
qEA = %σi &
N1
(Π̃0 (y) − Π̃ω (y))/ω ≡ A(y) ∼ 3 â(y/Γ),
2
â(x)
Superglasses ?! Xiaoquan Yu, MM ‘10 Γ
With the above assumptions one gets a set of self-consistency equations w
particular the above assumptions 1z
zimmediately lead to the coefficient 5B bei
(Π̃0 (y)
H =
− −Γ
Π̃
If they have a solution,
ω ∑σ
(y))/ω x2
−≡ ∑A(y) σ
i this proves J∼ σ
i ij that
â(y/Γ), â(x) ∼ min(1, 1/x ).
Γj3 the limit Γ $ Γc is indeed given
i< j
Order parameters superglass:
i
With the above assumptions one gets a set of self-consistency equations which are complete
particular the above assumptions immediately lead to the coefficient B being independent of
t that the
If they have a solution, thisHproves
Order parameters superglass:
=−
N i< j
∑ σ ixσ limit
x
j − ∑ Γσ$z JΓcσ is
i ij j
z indeed given by
M =
N
1 the
%σix & above scali
i< j
1 x 1 z 2
Competing order parameters: M= %σ & qEA = %σi &
N i N
M signals superfluidity of hard core bosons
1 z 2 σ z
i ↔ 2ni − 1
qEA = %σi &
N
If M and qEA exist at same time → Meanfield model of a superglass!« Superglass »
Tam, Geraedts, Inglis, Gingras, Melko, PRL (10) Carleo, Tarzia, Zamponi PRL (09)
QMC (3d) “Mean field”
QMC and cavity
Quenched randomness
Random, frustrated “Bethe” lattice« Superglass »
Tam, Geraedts, Inglis, Gingras, Melko, PRL (10) Carleo, Tarzia, Zamponi PRL (09)
QMC (3d) “Mean field”
QMC and cavity
Quenched randomness
Random, frustrated “Bethe” lattice« Superglass »
Tam, Geraedts, Inglis, Gingras, Melko, PRL (10) Carleo, Tarzia, Zamponi PRL (09)
QMC (3d) “Mean field”
QMC and cavity
Quenched randomness
Random, frustrated “Bethe” lattice
?? Low T behavior ? – QPT ? - Local structure of the superglass ??With the above assumptions one gets a set of self-consistency equations whic
1
particular the above assumptions immediately lead to the coefficient B being
Π̃ω (y))/ω 2 ≡ A(y) ∼ 3 â(y/Γ), â(x) ∼ min(1, 1/x5 ).
IfΓ they have a solution, this proves that the limit Γ $ Γc is indeed given by
Order parameters superglass:
gets a set of self-consistency equations which are completely independent of Γ! In
Mean field superglass
mmediately lead to the coefficient B being independent of Γ.
es that the limit Γ $ Γc is indeed given by the above scaling form.X. Yu, 1 x
= ‘10
M MM %σi &
N
t
H = − ∑ σ ixσ xj − ∑ σ iz J ijσ zj
1 x N i< j i< j 1 z 2
M= %σi & qEA = (29)
%σ i&
N N
1 z 2
qEA = %σi & (30)
NWith the above assumptions one gets a set of self-consistency equations whic
1
particular the above assumptions immediately lead to the coefficient B being
Π̃ω (y))/ω 2 ≡ A(y) ∼ 3 â(y/Γ), â(x) ∼ min(1, 1/x5 ).
IfΓ they have a solution, this proves that the limit Γ $ Γc is indeed given by
Order parameters superglass:
gets a set of self-consistency equations which are completely independent of Γ! In
Mean field superglass
mmediately lead to the coefficient B being independent of Γ.
es that the limit Γ $ Γc is indeed given by the above scaling form.X. Yu, 1 x
= ‘10
M MM %σi &
N
t
H = − ∑ σ ixσ xj − ∑ σ iz J ijσ zj
1 x N i< j i< j 1 z 2
M= %σi & qEA = (29)
%σ i&
N N
1• Obtain
qEA = %σiz &2 T = 0 phase transition glass-to-superglass (30)
Nexactly! (BCS instability of glass!)
• For superfluid-to-superglass transition: Use static
approximation (but exact upper bound)
• Analytical proof of existence of superglass phase!Phase diagram
PM PM Classical
SK transition1
(Π̃0 (y) − Π̃ω (y))/ω 2 ≡ A(y) ∼ â(y/Γ), â(x) ∼ min(1, 1/x5 ).
Phase diagram Γ3
h the above assumptions one gets a set of self-consistency equations which are completely independent of Γ! In
ular the above assumptions immediately lead to the coefficient B being independent of Γ.
ey have a solution, this proves that the limit Γ $ Γc is indeed given by the above scaling form.
er parameters superglass:
PM PM Classical
1
M = %σix & SK transition (29)
N
1 z 2
qEA = %σ & (30)
N i
∂%sx &y
χxx
y = . (31)
∂hx
!
Suppression of superfluidity
BCS-like condition: t dyP (y)χxx
y = 1. (32)
by the pseudogap!
Transition at finite J/t ≅ 1.001
(Π̃0 (y) − Π̃ω (y))/ω 2 ≡ A(y) ∼ â(y/Γ), â(x) ∼ min(1, 1/x5 ).
Phase diagram Γ3
h the above assumptions one gets a set of self-consistency equations which are completely independent of Γ! In
ular the above assumptions immediately lead to the coefficient B being independent of Γ.
ey have a solution, this proves that the limit Γ $ Γc is indeed given by the above scaling form.
er parameters superglass:
PM PM Classical
1
M = %σix & SK transition (29)
N
1 z 2
qEA = %σ & (30)
N i
∂%sx &y
χxx
y = . (31)
∂hx
!
Suppression of superfluidity
BCS-like condition: t dyP (y)χxx
y = 1. (32)
by the pseudogap!
Transition at finite J/t ≅ 1.00
Analogue for 1/r (Coulomb gap) in d∂%sx &y T →0 1
χxx
y = →
∂hx y
Local structuret ofdyPthe
(y)χ superglass
!
= 1. xx
y
• Superfluid and glass try to avoid each other:
%sxi & and %szi &2 are anticorrelated∂%sx &y T →0 1
χxx
y = →
∂hx y
Local structuret ofdyPthe
(y)χ superglass
!
= 1. xx
y
• Superfluid and glass try to avoid each other:
%sxi & and %szi &2 are anticorrelated
• Superfluid order non-monotonous with T!
M = sx
Superfluid enhanced
due to weakened glassy disorderConclusions • Understanding of collective low energy excitations in quantum glasses • Glassy order counteracts many particle quantum localization Ergodicity is broken – but arrow of time is intact! • Frustrated bosons: superfluid and glassy order can coexist
Conclusions
• Understanding of collective low energy
excitations in quantum glasses
• Glassy order counteracts many particle quantum localization
Ergodicity is broken – but arrow of time is intact!
• Frustrated bosons: superfluid and glassy order can coexist
Perspectives
Quantum phase transitions in disordered systems
(MIT, SIT, Bose glass, quantum glass transition):
• Nature of excitations and many particle localization?
Where/when does it occur? Implications on quantum information?
• New effects due to competition of different orders
• Quantum annealing, adiabatic quantum computation?You can also read
