Classical & quantum glasses - lpthe
109
Classical & quantum glasses Leticia F. Cugliandolo Univ. Pierre et Marie Curie – Paris VI [email protected] www.lpthe.jussieu.fr/ ˜ leticia Slides & useful notes in front page Cargèse, June 2011
Transcript of Classical & quantum glasses - lpthe
Classical & quantum glasses
Leticia F. Cugliandolo
Univ. Pierre et Marie Curie – Paris VI
www.lpthe.jussieu.fr/ leticia
Slides & useful notes in front page
Cargèse, June 2011
Plan
• First lesson .
Introduction. Overview of disordered systems and methods.
• Second lesson.
Statics of classical and quantum disordered systems.
• Third lesson .
Classical dynamics. Coarsening. Formalism.
• Fourth lesson .
Quantum dynamics. Formalism. Results for mean-field models.
Isolated systems
Dynamics of a classical isolated system .
Foundations of statistical physics .
Question : does the dynamics of a particular system reach a flat distri-
bution over the constant energy surface in phase space ?
Ergodic theory , ∈ mathematical physics at present.
Dynamics of a quantum isolated system :
a problem of current interest, recently boosted by cold atom experiments.
Question : after a quantum quench, i.e. a rapid variation of a parameter
in the system, are at least some observables described by thermal ones ?
When, how, which ? but we shall not discuss these issues here .
Dissipative systemsAim
Our interest is to describe the statics and dynamics of a classical or
quantum system coupled to a classical or quantum environment .
The Hamiltonian of the ensemble is
H = Hsyst + Henv + HintE∆
env
syst
The dynamics of all variables are given by Newton or Heisenberg rules, depen-
ding on the variables being classical or quantum.
The total energy is conserved, E = ct but each contribution is not, in particular,
Esyst 6= ct, and we’ll take Esyst ≪ Eenv .
Reduced systemModel the environment and the interaction
E.g., an esemble of harmonic oscillators and a bi-linear coupling :
Henv + Hint =N∑
α=1
[p2
α
2mα
+mαω2
α
2q2α
]
+N∑
α=1
cαqαx
Classically (coupled Newton equations) and quantum mechanically (easier in
a path-integral formalism) one can integrate out the oscillator variables.
Assuming the environment is coupled to the sample at the initial time, T , and
that its variables are characterized by a Gibbs-Boltzmann distribution or den-
sity function at inverse temperature β one finds a colored Langevin equation
(classical) or a reduced dynamic generating functional Zred (quantum me-
chanically). (see later explicit calculations)
What we know
Collective phenomena lead to phase transitions .
E.g., thermal PM - FM transitions in classical magnetic systems .
We understand the nature of the equilibrium phases and the phase transi-
tions . We can describe the phases with mean-field theory and the critical be-
havior with the renormalization group .
Quantum and thermal fluctuations conspire against the ordered phases.
We understand the equilibrium and out of equilibrium relaxation at the cri-
tical point or within the phases . We describe it with the dynamic RG at the
critical point or the dynamic scaling hypothesis in the ordered phase.
E.g., growth of critical structures or ordered domains .
What we do not know
In systems with competing interactions there is no consensus upon :
• whether there are phase transitions,
• which is the nature of the putative ordered phases,
• which is the dynamic mechanism.
Examples are :
• systems with quenched disorder ;
• systems with geometric frustration ;
• glasses of all kinds.
Static and dynamic mean-field theory has been developed – both classically
and quantum mechanically – and they yield new concepts and predictions.
Extensions of the RG have been proposed and are currently being explored.
Quenched disorder
Quenched variables are frozen during time-scales over which other va-
riables fluctuate.
Time scales τ0 ≪ τexp ≪ τ qdeq
τ qdeq could be the diffusion time-scale for magnetic impurities the magnetic
moments of which will be the variables of a magnetic system ;
or the flipping time of impurities that create random fields acting on
other magnetic variables.
Weak disorder (modifies the critical properties but not the phases) vs.
strong disorder (that modifies both).
e.g. random ferromagnets vs. spin-glasses .
The case of spin-glassesMagnetic impurities (spins) randomly placed in an inert host
Quenched random interactions
Magnetic impurities in a metal hostRKKY potential
V (rij) ∝cos 2kF rij
r3ij
sisj
very rapid oscillations about 0
and slow power law decay.
Standard lore : there is a 2nd order static phase transition at Ts
separating a paramagnetic from a spin-glass phase.
No dynamic precursor above Ts. Glassy dynamics below Ts with aging,
memory effects, etc.
Measurements
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J. Mydosh et al. 81. Hérisson & Ocio 02.Transition Dynamics
Pinning by impuritiesCompetition between elasticity and quenched randomness
d-dimensional elastic manifold in a transverse N -dimensional quenched
random potential.
OilWater
Interface between two phases ;vortex line in type-II supercond ;stretched polymer.
Distorted Abrikosov lattice
Goa et al. 01.
Frustration
HJ [{s}] = −∑
〈ij〉 Jijsisj Ising model
+
++ + +
+
+
+ +
+
Disordered Geometric
Efrustgs > EFM
gs and Sfrustgs > SFM
gs
Frustration enhances the ground-state enegy and entropy
Disordered example
Efrustgs = −2J > EFM
gs = −4J
Sfrustgs = ln 6 > SFM
gs = ln 2
Geometric case
Efrustgs = 3J > EFM
gs = −3J
Sfrustgs = ln 3 > SFM
gs = ln 2
Frustration
HJ [{s}] = −∑
〈ij〉 Jijsisj Ising model
One cannot satisfy all couplings simultaneously if∏
loop Jij < 0 .
One can expect to have metastable states too.
Trick : Avoid frustra-
tion by defining a spin
model on a tree, with
no loops, or in a di-
lute graph, with loops of
length lnN .
But it could be an over-simplication.
Heterogeneity
Each variable, spin or other, feels a different local field, hi =∑z
j=1 Jijsj ,
contrary to what happens in a ferromagnetic sample, for instance.
Homogeneous Heterogeneous
hi = −4J ∀ i. hj = 2J hk = 0 hl = −2J .
Each sample is a priori different but,
do they all have a different thermodynamic and dynamic behavior ?
Self-averagenessThe disorder-induced free-energy density distribution approaches a Gaus-
sian with vanishing dispersion in the thermodynamic limit :
limN→∞fN(β, J) = f∞(β) independently of disorder
– Experiments : all typical samples behave in the same way.
– Theory : one can perform a (hard) average of disorder,
−βf∞(β) = limN→∞[lnZN(β, J)]
Exercise : Prove it for the 1d Ising chain ; argument for finite d systems.
Intensive quantities are also self-averaging.
Replica theory
−βf∞(β) = limN→∞ limn→0Zn
N(β, J) − 1
Nn
Everyday-life glasses
• 3000 BC Glass discovered in the Middle East. LUXURIOUS OBJECTS.
• 1st century BC Blowpipe discovered on the Phoenician coast. Glass
manufacturing flourished in the Roman empire. EVERYDAY-LIFE USE.
• By the time of the Crusades glass manufacture had been revived in
Venice. CRISTALLO
• After 1890, the engineering of glass as a material developed very fast
everywhere.
What do glasses look like ?
Simulation Confocal microscopy
Molecular (Sodium Silicate) Colloids (e.g. d ∼ 162 nm in water)
Experiment Simulation
Granular matter Polymer melt
Structural glassesCharacteristics
• Selected variables (molecules, colloidal particles, vortices or polymers
in the pictures) are coupled to their surroundings (other kinds of
molecules, water, etc.) that act as thermal baths in equilibrium .
• There is no quenched disorder.
• The interactions each variable feels are still in competition, e.g. Lenard-
Jones potential, frustration .
• Each variables feels a different set of forces, time-dependent hetero-
geneity .
Sometimes one talks about self-generated disorder .
Correlation functionsStructure and dynamics
The two-time dependent density-density correlation :
g(r; t, tw) ≡ 〈 δρ(~x, t)δρ(~y, tw) 〉 with r = |~x − ~y|
The average over different dynamical histories (simulation/experiment)
〈. . .〉 implies isotropy (all directions are equivalent)
invariance under translations of the reference point ~x.
Its Fourier transform, F (q; t, tw) = N−1∑N
i,j=1〈 ei~q(~ri(t)−~rj(tw)) 〉.The incoherent intermediate or self correlation :
Fs(q; t, tw) = N−1∑N
i=1〈 ei~q(~ri(t)−~ri(tw)) 〉
Hansen & McDonald 06.
Structural glasses
No obvious structural change but slowing down !
1.0 2.0 3.0 4.00.0
1.0
2.0
3.0
4.0
r
g AA(r
;t,t)
g AA(r
)
r
t=0
t=10
Tf=0.1Tf=0.3Tf=0.4
Tf=0.435
Tf=0.4
0.9 1.0 1.1 1.2 1.3 1.40.0
2.0
4.0
6.0
10−1
100
101
102
103
104
1050.0
0.2
0.4
0.6
0.8
1.0
tw=63100
tw=10
t−twF
s(q;
t,tw
)
tw=0
q=7.23
Tf=0.4
LJ mixture Vαβ(r) = 4ǫαβ
[(σαβ
r
)12 −(σαβ
r
)6]
τ0 ≪ τexp ≪ τeq that changes by > 10 orders of magnitude !
Time-scale separation & slow non-equilibrium dynamics
Molecular dynamics – J-L Barrat & Kob 99.
Time-scalesCalorimetric measurement of entropy
What is making the relaxation so
slow ?
Is there growth of static order ?
Phase space picture ?
FluctuationsEnergy scales
Thermal fluctuations
– irrelevant for granular matter since mgd ≫ kBT ; dynamics is
induced by macroscopic external forces.
– important for magnets, colloidal suspensions, etc.
Quantum fluctuations
– when ~ω>∼ kBT quantum fluctuations are important.
– one can even set T → 0 and keep just quantum fluctuations.
Examples : quantum magnets, Wigner crystals, etc.
SummaryStructural glasses
Crytallization at Tm is avoided by cooling fast enough.
Liquid Supercooled liquid Glass︸ ︷︷ ︸
Exponential relax Non-exponential relax
Equilibrium Metastable equilibrium Non-equilibrium︸ ︷︷ ︸
Separation of time-scales &
An exponential number︸ ︷︷ ︸
of metastable states !
Stationary Aging
Aging means that correlations and reponses depend on t and tw
ac susceptibilities depend on ω and tw
There might be an equilibrium transition to an ideal glass at Ts.
Methodsfor classical and quantum disordered systems
Statics
TAP Thouless-Anderson-Palmer
Replica theory
fully-connected (complete graph)
Gaussian approx. to field-theories
Cavity or Peierls approx.
}
dilute (random graph)
Bubbles & droplet arguments
functional RG1
finite dimensions
Dynamics
Generating functional for classical field theories (MSRJD).
Schwinger-Keldysh closed-time path-integral for quantum dissipative models
(the previous is recovered in the ~ → 0 limit).
Perturbation theory, renormalization group techniques, self-consistent approx.1 See P. Le Doussal’s seminar.
References
– Liquids & glass transitionP. G. Debenedetti, Metastable liquids (Princeton Univ. Press, 1997).E. J. Donth, The glass transition : relaxation dynamics in liquids and disordered materials (Springer,2001).K. Binder and W. Kob, Glassy materials and disordered solids : an introduction to their statisticalmechanics (World Scientific, 2005).A. Cavagna, Supercooled liquids for pedestrians, Phys. Rep. 476, 51 (2009).L. Berthier & G. Biroli, A theoretical perspective on the glass transition and nonequilibrium pheno-mena in disordered materials, arXiv :1011.2578
– Spin-glassesK. H. Fischer and J. A. Hertz, Spin glasses (Cambridge Univ. Press, 1991).M. Mézard, G. Parisi, and M. A. Virasoro, Spin glass theory and beyond (World Scientific, 1986).T. Castellani & A. Cavagna, Spin-glass theory for pedestrians, J. Stat. Mech. (2005) P05012.F. Zamponi, Mean field theory of spin glasses, arXiv :1008.4844.N. Kawashima & H. Rieger, Recent progress in spin glasses in Frustrated spin systems, H. T. Dieped. (World Scientific, 2004).M. Talagrand, Spin glasses, a challenge for mathematicians (Springer-Verlag, 2003).
References
– Disorder elastic systemsT. Giamarchi & P. Le Doussal, Statics and dynamics of disordered elastic systems, arXiv :cond-mat/9705096.T. Giamarchi, A. B. Kolton, A. Rosso, Dynamics of disordered elastic systems, arXiv :cond-mat/0503437.
– Phase ordering kineticsA. J. Bray, Theory of phase ordering kinetics, Adv. Phys. 43, 357 (1994).S. Puri, Kinetics of Phase Transitions, (Vinod Wadhawan, 2009).
– GlassesE. J. Donth, The glass transition : relaxation dynamics in liquids and disordered materials (Springer,2001).K. Binder and W. Kob, Glassy materials and disordered solids : an introduction to their statisticalmechanics (World Scientific, 2005).L. F. Cugliandolo, Dynamics of glassy systems, Les Houches Session 77, arXiv :cond-mat/0210312.L. Berthier & G. Biroli, A theoretical perspective on the glass transition and nonequilibrium pheno-mena in disordered materials, arXiv :1011.2578.
End of 1st talk
Plan
• First lesson .
Introduction. Overview of disordered systems and methods.
• Second lesson.
Statics of classical and quantum disordered systems.
• Third lesson .
Classical dynamics. Coarsening. Formalism.
• Fourth lesson .
Quantum dynamics. Formalism. Results for mean-field models.
What do glasses look like ?
Simulation Confocal microscopy
Molecular (Sodium Silicate) Colloids (e.g. d ∼ 162 nm in water)
Experiment Simulation
Granular matter Polymer melt
Correlation functionsStructure and dynamics
The two-time dependent density-density correlation :
g(r; t, tw) ≡ 〈 δρ(~x, t)δρ(~y, tw) 〉 with r = |~x − ~y|
The average over different dynamical histories (simulation/experiment)
〈. . .〉 implies isotropy (all directions are equivalent)
invariance under translations of the reference point ~x.
Its Fourier transform, F (q; t, tw) = N−1∑N
i,j=1〈 ei~q(~ri(t)−~rj(tw)) 〉.The incoherent intermediate or self correlation :
Fs(q; t, tw) = N−1∑N
i=1〈 ei~q(~ri(t)−~ri(tw)) 〉
Hansen & McDonald 06.
Structural glasses
No obvious structural change but slowing down !
1.0 2.0 3.0 4.00.0
1.0
2.0
3.0
4.0
r
g AA(r
;t,t)
g AA(r
)
r
t=0
t=10
Tf=0.1Tf=0.3Tf=0.4
Tf=0.435
Tf=0.4
0.9 1.0 1.1 1.2 1.3 1.40.0
2.0
4.0
6.0
10−1
100
101
102
103
104
1050.0
0.2
0.4
0.6
0.8
1.0
tw=63100
tw=10
t−twF
s(q;
t,tw
)
tw=0
q=7.23
Tf=0.4
LJ mixture Vαβ(r) = 4ǫαβ
[(σαβ
r
)12 −(σαβ
r
)6]
τ0 ≪ τexp ≪ τeq
Time-scale separation & slow non-equilibrium dynamics
Molecular dynamics – J-L Barrat & Kob 99.
Why are they all ‘glasses’ ?What do they have in common ?
– No obvious spatial order : disorder (differently from crystals).
– Many metastable states
Rugged landscape
– Slow non-equilibrium relaxation
τ0 ≪ τexp ≪ τeq
Time-scale separation
– Hard to make them flow under external forces.
Pinning, creep, slow non-linear rheology.
PM-FM transitione.g., up & down spins in a 2d Ising model (IM)
〈φ〉 = 0 〈φ〉 = 0 〈φ〉 6= 0
T → ∞ T = Tc T < Tc
In a canonical setting the control parameter is T/J .
Mean-field theory for PM-FMFully connected Ising model
Normalize J by the size of the system N to have O(1) local fields
H = − J2N
∑
i6=j sisj − h∑
i si
The partition function reads ZN =∫ 1
−1du e−βNf(u) with Nu =
∑
i si
f(u) = −J2u2 − hu + T
[1+u
2ln 1+u
2+ 1−u
2ln 1−u
2
]
Energy terms and entropic contribution stemming from N ({si}) yielding
the same u value.
Use the saddle-point , limN→∞ fN(βJ, βh) = f(usp), with
usp = tanh (βJusp + βh) = 〈u〉
Exercise : do these calculations, see notes.
Ginzburg-Landau for PM-FMContinuous scalar statistical field theory
Coarse-grain the spin
φ(~r) = V −1~r
∑
i∈V~rsi.
Set h = 0.
L
The partition function is ZV =∫Dφ e−βV f(φ) with V the volume and
f(φ) =∫
ddr{
12[∇φ(~r)]2 + T−J
2φ2(~r) + λ
4φ4(~r)
}
Elastic + potential energy with the latter inspired by the results for the fully-
connected model (entropy around φ ∼ 0 and symmetry arguments.
Uniform saddle point in the V → ∞ limit : φsp(~r) = 〈φ(~r)〉 = φ0.
The free-energy is limV →∞ fV (β, J) = f(φsp).
2nd order phase-transition
bi-valued equilibrium states related by symmetry, e.g. Ising magnets
lowercriticalupper
φ
f
T
〈φ〉
Ginzburg-Landau free-energy Scalar order parameter
Features
• Spontaneous symmetry breaking below Tc.
• Two equilibrium states related by symmetry φ → −φ.
• The state is chosen by a pinning field .
• If the partition sum is performed over the whole phase space 〈φ〉 = 0
(a consequence of the symmetry of the action).
• Restricted statistical averages (half phase space) yield 〈φ〉 6= 0.
• Under a magnetic field the free-energy landscape is tilted and one of
the minima becomes a metastable state .
• The barrier in the free-energy landscape between the two states
diverges with the size of the system implying ergodicity breaking .
• With p > 2-uplet interactions one finds first order phase transitions .
These results were not fully accepted as realistic at the time.
Equilibrium configurationse.g. up & down spins in a 2d Ising model (IM)
〈φ〉 = 0 〈φ〉 = 0 〈φ〉 6= 0
T → ∞ T = Tc T < Tc
In a canonical setting the control parameter is T/J .
Spin-glassese.g. up & down spins in a 3d Edwards-Anderson model (EA)
HJ =∑
〈ij〉 Jijsisj [Jij] = 0 [J2ij] = J2
〈φ〉 = 0 〈φ〉 = 0 〈φ〉 = 0
T → ∞ T = Tc T < Tc
Is there another order-parameter ?
MF theory for spin-glassesFully connected SG : Sherrington-Kirkpatrick model
H = − 12√
N
∑
i6=j Jijsisj −∑
i hisi
with Jij i.i.d. Gaussian variables, [Jij ] = 0 and [J2ij ] = J2 = O(1).
One finds the naive free-energy landscape
N f({mi}) = − 12√
N
∑
i6=j
Jijmimj + TN∑
i=1
1+mi
2 ln 1+mi
2 + 1−mi
2 ln 1−mi
2 .
and the naive TAP equations
misp = tanh(β∑
j( 6=i) Jijmjsp + βhi)
that determine the restricted averages mi = 〈si〉 = misp.
These should be corrected by the Onsager reaction term, that subtracts the
self-response of a spin, see notes . Thouless, Anderson, Palmer 77 .
MF theory for spin-glassesA hint on the proof
The more traditional one assumes independence of the spins,
P ({si}) =∏
i pi(si)
with pi(si) = 1+mi
2δsi,1 + 1−mi
2δsi,−1
and uses this form to express 〈H〉 − T 〈S〉 with S = ln P ({si}) ; see
notes .
A more powerful proof expresses f as the Legendre transform of−βF (hi)
with mi = N−1∂[−βF (hi)]/∂hi = 〈si〉h.Georges & Yedidia 91.
This proof is easier to generalize to dynamics (Biroli 00 ) and quantum
systems (Biroli & LFC 01 ), see later .
Features
• Saddle-points mi 6= 0 below Tc are heterogeneous .
• The fact that an equilibrium state is determined by {mi} can be made
rigorous in MF.
• There are N order parameters , mi, i = 1, . . . , N .
• The TAP equations have an exponential in N number of solutions {miα}
that are extrema of the TAP free-energy landscape, i.e. saddles of
all types, at low temperatures.
• For each solution {miα} one also has {−mi
αsp} but apart from this
trivial doubling, the remaining ones are not related by symmetry.
• One can study the temperature-dependent free-energy landscape
and compute, e.g., how many saddles of each kind exist and how
many of these at each level of f .
• One finds a hierarchy of metastable states , with high degeneracy,
separated by diverging barriers (infinite life-time).
Statistical averagesThe average of a generic observable is 〈O〉 =
∑
α wα〈O〉α
In the FM case , each state 〈φ〉 = ±φ0 has weigth w± = e−βNf±ZN =
1/2 and the sum is 〈O〉 = 12〈O〉+ + 1
2〈O〉− . For instance, the avera-
ged magnetization vanishes if one sums over the± states or it is different
from zero if one restricts the sum to one of them.
Within the TAP approach wJα = e−βNf
Jα
P
γ e−βNfJγ
and
〈O〉 = Z−1(β, J)∫
df e−β[Nf−T lnNJ (f ,β)] O(f)
where NJ is the (possibly exponential in N ) number of solutions to the
TAP equations with free-energy density f .
Statistical averagesConsequences
The equilibrium free-energy is given by the saddle-point evaluation of
the partition sum that implies
Nf = N fmin − TSc(fmin, β) Sc(f , β) = lnNJ(f , β)
The rhs is the Landau free-energy of the problem, with f playing the role
of the energy and N−1 lnNJ(f , β) the one of the entropy.
In the sum we do not distinguish the stability of the TAP solutions.
In some cases higher lying extrema (metastable states) can be so nume-
rous to dominate the partition sum with respect to lower lying ones.
This feature is proposed to describe super-cooled liquids .
Structural glasses
"configuration space"
? ?
1/T
1/T
1/T
1/T
d
g
cage
very warmliquid
warmliquid
viscousliquid
glass
The ruggedness of the free-energy landscape increases upon decreasing tem-
perature until a configurational entropy crisis arises (at Kauzmann TK ).
Numerical simulations : one cannot access f but one can explore e : po-
tential energy landscape . D. J. Wales 03.
Replica calculationA sketch
−βf = limN→∞
lnZN(β, J) = limN→∞
limn→0
[ZnN(β, J)] − 1
Nn
ZnN partition function of n independent copies of the system : replicas .
Average over disorder : coupling between replicas
∑
a
∑
i6=j Jijsai s
aj ⇒ N−1
∑
i6=j
(∑
a sai s
aj
)2
Decoupling with the Hubbard-Stratonovich trick
QabN−1
∑
i sai s
bi − 1
2Q2
ab
Qab is a 0×0 matrix but it admits an interpretation in terms of overlaps .
The elements of Qab can be evaluated by saddle-point if one exchanges
the limits N → ∞ n → 0 with n → 0 N → ∞.
Replica calculationOverlaps
Take one sample and run it until it reaches equilibrium , measure {si}.
Re-initialize the same sample (same Jij ), run it until it reaches equilibrium ,
measure {σi}.
Construct the overlap qsσ ≡ N−1∑N
i=1 siσi.
In a FM system there are four possibilities
sσf
sσ σs sσ
qsσ = m2 m2 −m2 −m2
Many repetitions : P (qsσ) = 12δ(qsσ − m2) + 1
2δ(qsσ + m2)
Replica calculationOverlaps in disordered systems
Parisi 79-82 prescription for the replica symmetry breaking Ansatz yields
-1 0 1
p
q
PM
-1 -m2 0 m2 1
q
FM
-1 -qEA 0 qEA 1
q
p-spin
-1 -qEA 0 qEA 1
q
SK
High temperature FM Structural glasses Spin-glasses
Thermodynamic quantities, in particular the equilibrium free-energy density, are
expressed in terms of the functional order parameter P (q).
The equilibrium free-energy density predicted by the replica theory was confir-
med by Guerra & Talagrand 00-04 indepedent mathematical-physics methods.
Replica calculationApplications to other problems
Random elastic manifolds , e.g. vortex systems, dirty interfaces, etc.
Models with self-generated disorder , e.g. Lennard-Jones particles.
OilWater
Interface, vortex
Abrikosov latticeLennard-Jones mixture
Mézard & Parisi 91, Giamarchi & Le Doussal 97, Mézard & Parisi 99.
Tricks are necessary to introduce quenched randomness in the calculation.
Droplet pictureA much simpler viewpoint for finite- d systems
Just two equilibrium states as in a FM, only that they look spatially
disordered.
Compact excitations of linear size ℓ have energy E(ℓ) ≃ Υℓθ.
Proposition for P (Eℓ).
P (q) with two peaks at ±qEA.
A series of scaling laws lead to predictions for themodynamics, etc.
Also applicable to random manifold problems.
In a sense, a more conventional picture.
Fisher & Huse 87-89.
Long-standing debate, no consensus ; very hard to decide.
Plan
• First lesson .
Introduction. Overview of disordered systems. Short presentation of
methods.
• Second lesson .
Statics of classical and quantum disordered systems .
• Third lesson .
Classical dynamics. Coarsening. Formalism.
• Fourth lesson .
Quantum dynamics. Formalism and results for mean-field models.
SummaryClassical disordered systems
The statics and structure of metastable states is fully described in MF
Consistent results from TAP, replicas, cavity for dilute systems and
formal arguments .
Attn ! the statistics of barriers is not fully known.
MF theory predicts a functional ordered parameter and three univer-
sality classes :
FM : 2nd order transition, two states below Tc. Curie-Weiss, GL
Structural glasses : metastable states combine to make the
super-cooled liquid, random first order transition p-spin .
Spin-glasses : 2nd order transition, many states below Tc. SK.
Different scenario from droplet model.
Quantum spin-glasses
Hsyst = −∑
ij Jijσzi σ
zj + Γ
∑
i σxi − ∑
i hiσzi .
σai with a = 1, 2, 3 the Pauli matrices, [σa, σb] = 2iǫabcσ
c.
Jij quenched random., e.g. Gaussian pdf conveniently normalized.
Γ transverse field. It measures quantum fluctuations.
In the limit Γ → 0 the classical limit should be recovered.
hi longitudinal local fields
ij nearest neighbours on the lattice – finite d
or fully connected – mean-field
Quantum SG : finite dPhase transitions in d = 2, 3
!
Classical
spin glass
transition
SG
PM
c"
c
T
3d
T
"
Quantum phase transitionQuantum spin glass phase
T =0c
"c
PM
Quantum critical region
"T >0
c
2d
Γc Quantum critical point. Tc Classical critical point.
Quantum MC. Overviews in Rieger & Young 95, Kawashima & Riege r 03.
Many special features in d = 1 obtained with the Dasgupta-Ma RG decimation
procedure. Recall E. Altman’s talk. D. S. Fisher 92.
Mean-field methods
TAP method
Legendre transform of f(β, h) with respect to {mi(τ)} and C(τ − τ ′)
with mi(τ) = 〈si(τ)〉h and C(τ − τ ′) = N−1∑
i〈si(τ)si(τ′)〉h.
Biroli & LFC 01.
Replica trick
The partition function is a trace
ZN = Tr e−βH =
∫ {si(β~)}
{si(0)}D{si(τ)} e−
1~
Sesyst[{si}]
with the Euclidean action
Mean-field methods
Sesyst[{si}] =
∫ β~
0dτ
M
2
(∂si(τ)
∂τ
)2
+∑
ij
Jijsi(τ)sj(τ)
si(β~) = si(0) and the ‘mass’ given by M = (~τ0)/2 ln[~/(Γτ0)] as
a function of the transverse field Γ.
Feynman-Matsubara construction of functional integral ov er imaginary time.
Overlap matrix-function : Qab(τ, τ′)N−1
∑
i sai (τ)sb
i(τ′) − 1
2Q2ab(τ, τ
′)
Slightly intricate imaginary-time & replica index structure. Recipes to deal
with them
Bray & Moore 80.
Can an argument à la Guerra-Talagrand can be extended to quantu m
spin-glass models ?
1st order phase transitionsQuantum p-spin model
0.0 0.2 0.4 0.60
1
2
3
m < 1
m=1
T*
SG
PMenter text here
Γ
T0 1 2 3
0.0
0.5
1.0 (b) β=12
q EA
Γ
1.5
2.0
2.5
3.0(a)
β=4
β=12
χ
Jump in the susceptibility across the dashed part of the critical line.
LFC, Grempel & da Silva Santos 00 ; Biroli & LFC 01.
Many more examples, e.g. with cavity method in a model with a superfluid/
glass transition Foini, Semerjian & Zamponi 11 ; see Yu’s poster.
Combinatorial optimizationOptimization problems consist in finding the configuration that renders minimal
a cost function , e.g. the road traveled by a salesman to visit each of N cities
once and only once.
The most interesting of these problems can be mapped onto a classical spin
model on a random (hyper-)graph with the cost function its Hamiltonian.
For instance, K-satisfiability is written in terms of p(≤ K)- spin models on
a random (hyper-)graph .
Quantum annealing
Kadowaki & Nishimori 98
Dipolar spin-glass
G. Aeppli et al. 90s
1st order transitions : trouble for quantum annealing techniques.
Jorg, Krzakala, Kurchan, Maggs & Pujos 09.
Dissipative systemsAim
Our interest is to describe the statics and dynamics of a classical or
quantum system coupled to a classical or quantum environment .
The Hamiltonian of the ensemble is
H = Hsyst + Henv + Hint
Syst. x, p
Env. {qa, πa}
What is the static and dynamic behaviour of the reduced system ?
Discuss it step by step :
1) equilibrium classical,
2) equilibrium quantum,
3) classical dynamics, 4) quantum dynamics.
Reduced system
Imagine the ensemble H = Hsyst + Henv + Hint is in equilibrium
at inverse temperature β−1 :
Model the environment and the interaction
E.g., an ensemble of harmonic oscillators and a bi-linear coupling :
Henv + Hint =N∑
α=1
[p2
α
2mα
+mαω2
α
2q2α
]
+N∑
α=1
cαqαx
Either classical variables or quantum operators.
Reduced descriptionStatics of a classical system
The partition function of the coupled system is
Z[η] =∑
osc, syst
e−β(H−ηx)
Integrating out the oscillator variables :
Zred[η] =∑
syst
e−β
„
Hsyst+Hcount+ηx− 12
PNa=1
c2aω2a
max2
«
Choosing Hcount to cancel the quadratic term in x2 one recovers
Zred[η] = Zsyst[η]
i.e., the partition function of the system of interest.
Reduced descriptionStatics of a quantum system
The density matrix of the coupled system is
ρ(x′′, q′′a ;x′, q′a) = 〈x′′, q′′a | ρ |x′, q′a〉
=1
Z
∫ x(~β)=x′′
x(0)=x′
Dx(τ)
∫ qa(~β)=q′′a
qa(0)=q′a
Dqa(τ)e−1~
Se[η]
One integrates the oscillator’s degrees of freedom to get the reduced density
matrix
ρred(x′′, x′) = Z−1
red
∫ x′′
x′
Dx(τ) e− 1
~
“
Sesyst−
R
~β0 dτ
R τ0 dτ ′ x(τ)K(τ−τ ′)x(τ ′)
”
The counter-term was chosen to cancel a quadratic term in x2(τ), Zred =
Z/Zosc and Zosc the partition function of isolated ensemble of oscillators, but
a non-local interaction in the imaginary time with kernel
K(τ) = 2π~β
∑∞n=−∞
∫ ∞0 dω I(ω)
ων2
n
ν2n+ω2 exp(iνnτ) remains.
1st order phase transitionDissipative Ising p-spin model with p ≥ 3 at T ≈ 0
Magnetic susceptibility Averaged entropy density
�
�
1.41.31.21.110.90.820161284
1
� = 1:0� = 0:5� = 0:0�
s
1.41.31.21.110.90.80.250.20.150.10.05
1α = 0, 0.5, 1
LFC, Grempel & da Silva Santos 00 ;
LFC, Grempel, Lozano, Lozza, da Silva Santos 04.
LocalizationThe Caldeira-Leggett problem
A quantum particle in a double-well potential coupled to a bath of quan-
tum harmonic oscillators in equilibrium at T = 0.
Quantum tunneling for 0 < α < 1/2
‘Classical tunneling’ for 1/2 < α < 1
Localization in initial well for 1 < α
Bray & Moore 82, Leggett et al. 87.
More later .
SummaryStatics of quantum disordered systems
• We introduced quantum spin-glasses.
• We very briefly ‘explained’ that the TAP and replica approaches as well
as the cavity method can be applied to them.
• We showed that models in the random first order phase transition
class have first order phase transitions in the low temperature limit.
• We gave an argument as to why a quantum environment can have a
highly non-trivial effect quantum mechanically.
Similar results for quantum Ising chains with FM and disordered interactions
LFC, Lozano & Lozza ; Chakravarty, Troyer, Voelker, Werner 0 4 (MC)
Schehr & Rieger 06 (decimation RG)
It was the end of the 2nd talk
Plan
• First lesson .
Introduction. Overview of disordered systems. Short presentation of
methods.
• Second lesson .
Statics of classical and quantum disordered systems.
• Third lesson .
Classical dynamics. Coarsening. Formalism.
• Fourth lesson .
Quantum dynamics. Formalism and results for mean-field models.
Phase ordering kineticsDynamics across a phase transition
• Equilibrium phases are known on both sides of the transition.
• The dynamic mechanism can be understood.
• Interesting as a theoretical problem , beyond perturbation theory.
• To cfr. the observables to similar ones in problems for which we do not
know the equilibrium phases nor the dynamic mechanisms.
• To investigate whether growth phenomena exist in problems with unk-
nown dynamic mechanisms. e.g. glasses
• To unveil “generic” features of macroscopic systems out of equilibrium
(classical or quantum).
out of equilibrium statistical mechanics or thermodynamics
2nd order phase-transition
bi-valued equilibrium states related by symmetry, e.g. Ising magnets
lowercriticalupper
φ
f
T
〈φ〉
Ginzburg-Landau free-energy Scalar order parameter
EvolutionThe system is in contact with a thermal bath
Thermal agitation
Non-conserved order parameter 〈φ〉(t, T ) 6= ct
e.g. single spin flips with Glauber or Monte Carlo stochastic rules.
Development of magnetization in a ferromagnet.
Conserved order parameter 〈φ〉(t, T ) = 〈φ〉(0, T ) = ct
e.g. pair of antiparallel spin flips with stochastic rules.
Phase separation in binary fluids.
EvolutionA quench or an annealing across a phase transition
t0
Tc
T
〈φ〉
Non-conserved order parameter 〈φ〉(t, T ) 6= ct
Development of magnetization in a ferromagnet after a quench.
The probleme.g. up & down spins in a 2d Ising model (IM)
0
50
100
150
200
0 50 100 150 200
’data’
0
50
100
150
200
0 50 100 150 200
’data’
0
50
100
150
200
0 50 100 150 200
’data’
0
50
100
150
200
0 50 100 150 200
’data’
0
50
100
150
200
0 50 100 150 200
’data’
0
50
100
150
200
0 50 100 150 200
’data’
Question : starting from equilibrium at T0 → ∞ or T0 = Tc how is
equilibrium at Tf = Tc or Tf < Tc attained ?
Growth kinetics
• At Tf = Tc the system needs to grow structures of all sizes.
Critical coarsening.
At Tf < Tc : the system tries to order locally in one of the two com-
peting equilibrium states at the new conditions.
Sub-critical coarsening.
In both cases the linear size of the equilibrated patches increases
in time .
In both cases one extracts a growing linear size of equilibrated patches
R(t, g) from C(r, t) = 1N
∑Ni,j=1〈δsi(t)δsj(t)〉|~ri−~rj |=r
• The relaxation time tr needed to reach ±|〈φ〉eq(T )| diverges with the
size of the system, tr(T, L) → ∞ when L → ∞ for T ≤ Tc.
Dynamic scaling
At late times there is a single length-scale, the typical radius of the do-
mains R(T, t), such that the domain structure is (in statistical sense)
independent of time when lengths are scaled by R(T, t), e.g.
C(r, t) ≡ 〈 si(t)sj(t) 〉||~xi−~xj |=r ∼ m2eq(T ) f
(r
R(T, t)
)
,
C(t, tw) ≡ 〈 si(t)si(tw) 〉 ∼ m2eq(T ) fc
( R(T, t)
R(T, tw)
)
,
etc. when r ≫ ξ(T ), t, tw ≫ t0 and C < m2eq(T ).
Suggested by experiments and numerical simulations. Proved for
• Ising chain with Glauber dynamics.
• Langevin dynamics of the O(N) model with N → ∞, and the
spherical ferromagnet. Review Bray, 1994.
• Distribution of hull-enclosed areas in 2d curvature driven coarsening.
Space-time correlationMagnetic model
Scaling regime a ≪ r ≪ L, rR(t,T )
, R(t, T ) ≃ λ(T )t1/zd
C(r, t) = N−1∑
ij/rij=r
si(t)sj(t) ≃ m2eq(T ) fc
(r
R(t, T )
)
0
0.2
0.4
0.6
0.8
1
1 2 3
C(r,t)
r/R(t)
100
101
102
103
100 101 102 103
R2
t
Phase separationSpinodal decomposition in binary mixtures
A spieces ≡ spin up ; B species ≡ spin down
2d Ising model with Kawasaki dynamics at T
locally conserved order parameter
50 : 50 composition ; Rounder boundaries
Dynamics in the 2d XY modelSchrielen pattern : gray scale according to sin2 2θi(t)
Defects are vortices (planar spins turn around these points)
After a quench vortices annihilate and tend to bind in pairs
R(t, T ) ≃ λ(T ){t/ ln[t/t0(T )]}1/2
Yurke et al 93, Bray & Rutenberg 94.
Growing lengthsUniversality classes
R(t, T ) ≃
λ(T ) t1/2 scalar NCOP zd = 2
λ(T ) t1/3 scalar COP zd = 3
λ(T )
(t
ln t/t0
)1/2
2-comp. vector NCOP in d = 2
etc.
Defined by the time-dependent. Temperature and other parameters ap-
pear in the prefactor.
Super-universality ?
Are scaling functions independent of temperature and other parameters ?
Review Bray 94
Weak disordere.g., random ferromagnets
At short time scales the dynamics is relatively fast and independent of
the quenched disorder ; domain walls accomodate in places where the
disorder is the weakest, thus
R(t, T ) ≃ λ(T )t1/zd
At longer time scales domain-wall pinning by disorder becomes impor-
tant.
Assume that a length-dependent barrier B(R) ≃ ΥRψ
The Arrhenius time needed to go over such a barrier is t ≃ t0 eB(R)kBT
This implies
R(t, T ) ≃(
kBT
Υln t/t0
)1/ψ
Weak disorderStill two ferromagnetic states related by symmetry
R(t, T ) ≃
[λ(T )t]1/zd R ≪ Lc(T ) curvature-driven
Lc(T )(ln t/t0)1/ψ R ≫ Lc(T ) activated
with Lc(T ) a growing function of T .
Inverting times as a function of length t ≃ [R/λ(T )]zd eR/Lc(T )
At short times this equation can be approximated by an effective power
law with a T -dependent exponent :
t ≃ Rzd(T ) zd(T ) ≃ zd [1 + ct/Lc(T )]
Bustingorry et al. 09.
End of 3rd lecture
Plan
• First lesson .
Introduction. Overview of disordered systems. Short presentation of
methods.
• Second lesson . Static methods.
• Third lesson .
Classical dynamics. Coarsening. Formalism.
• Fourth lesson .
Quantum dynamics. Formalism. Mean-field models.
Real-time dynamicsTwo-time dependence
t = 0 initial time tw waiting-time t measuring time.
Correlation
C(t, tw) = 〈[O(t), O(tw)]+〉
Symmetrized correlator
Linear response
R(t, tw) =δ〈O(t)〉δh(tw)
∣∣∣∣∣h=0
= 〈[O(t), O(tw)]−〉
Antisymmetrized correlator
Linear responseTo a kick and to a step
− δ δ+
h
t t2 2
w w0 t����
��������
����
����
r(0)
r(tw)
tr( )
r( )th
The perturbation couples linearly to the observable H → H−hB({~ri})
The linear instantaneous response of another observable A({~ri}) is
RAB(t, tw) ≡⟨
δA({~ri})(t)δh(tw)
∣∣∣∣h=0
⟩
The linear integrated response or dc susceptibility is
χAB(t, tw) ≡∫ t
tw
dt′ RAB(t, t′)
Formalism• Path-integral Schwinger-Keldysh formalism.
T > t
Closed-time path to allow for 〈O(tw)O(t)〉 with t > tw.
• Connect system to reservoirs at time t = 0, factorize density matrix :
ρ(0) = ρsyst(0) ⊗ ρenv(0)
and take the reservoir in equilibrium at its own β (and µ).
• Integrate out the bath – assumed to be in equilibrium.
• Obtain an effective action S = Ssyst + Sbath−syst
FormalismSome important technical remarks
• Integration over bath variables ⇒Two real-time long-range interactions (recall static cases ).
• The vanishing quantum fluctuations limit (~ → 0) of the Schwinger-
Keldysh generating functional is :
if no environment, Newton dynamics .
if environment on, Langevin dynamics with coloured noise .
How to prove it : linear combinations of the forward and backward variables
x+(t) and x−(t) combine into x(t) and ix(t). For ~ → 0 the quantum
bath kernels become the ones of a colored classical Langevin process.
FormalismThe rôle played by the initial condition & disorder
• Typical initial conditions : ρsyst(0) = I ‘random’ (quench from PM)
no need of replica trick to average over disorder !
In the classical limit Z[η = 0] =∫DξP [ξ] = 1, independently of
disorder de Dominicis 78.
In the quantum model ρ(0) is independent of disorder.
LFC & Lozano 98.We simply average Z or ρ.
• Derive Schwinger-Dyson equations for correlations and responses with
saddle-point in the large N limit or a variational approximation .
• Solve these equations.
FormalismAn example : rotors with pair intereactions
Ssyst =∑
a=±a
∫
dt
~
2
2Γ
∑
i
(nia(t))2 +
∑
i<j
Jijnia(t)nja(t)
.
Sbath−syst = −1
2
∑
ab=±
∫
dtdt′ ΣBab(t, t
′)∑
i
nia(t)nib(t′) ,
Sλ =∑
a=±
a
2
∫
dt∑
i
λia(t)(n2ia(t) −M)
with the bath induced kernels
ΣBab(t, t
′)
that take different forms for different baths, e.g. oscillators, leads, etc.
Real-time dynamicsParamagnetic phase
� = 3:0� = 2:0� = 1:0� = 0:5t
C(t+t w;tw)
108642010.50-0.5-1
� = 3:0� = 2:0� = 1:0� = 0:5t
R(t+t w;tw)
108642010.50-0.5-1
Dependence on the quantum parameter Γ (T = 0, α fixed.)
LFC & G. Lozano 98-99.
Real-time dynamicsDependence on the coupling to the bath
Symmetric correlation Linear responset
C(t+t w;tw)
1512.5107.552.500.80.40-0.4 t
R(t+t w;tw)
1512.5107.552.501.510.50-0.5
Comparison between α = 0.2 (PM) and α = 1 (SG)
LFC, Grempel, Lozano, Lozza & da Silva Santos 02.
The effect of the bathSummary
• Classical statics is not altered by the bath.
• Quantum statics is altered by the bath.
• Classical dynamics becomes a coloured noise Langevin equation and,
although the evolution can depend on the bath, the target equilibrium
does not.
• Quantum dynamics is very importantly altered by the bath, e.g. locali-
zation and modification of the phase transition line.
Quantum aging
Other examples : SK (Chamon, Kennett & Yu), SU(N) in large N (Biroli & Par-
collet), rotors (Rokni & Chandra ; Aron et al), Wigner crystals (LFC, Giamarchi
& Le Doussal), etc.
Real-time dynamicsInteractions against localization
� = 4:0 J = 1� = 4:0 J = 0:5� = 4:0 J = 0� = 0:02 J = 0 � < � ritGlassyLo alized
tC(t+t w;t
w)
2015105010.60.2-0.2
LFC, Grempel, Lozano, Lozza & da Silva Santos 02
Dynamic vs static phase diagramQuantum p-spin model
0.0 0.2 0.4 0.60
1
2
3
m < 1
m=1
T*
SG
PMenter text here
Γ
T
Dynamic evidence of high-lying metastable states !
The relaxational dynamics gets trapped in a region of phase space na-
med threshold .
Rue de Fossés St. Jacques et rue St. Jacques
Paris 5ème Arrondissement.
LFC
Fluctuation-dissipationEquilibrium spontaneous ( C) and induced ( R) fluctuations
If p({~r}), tw) = peq({~r})• The dynamics is stationary, C → C(t − tw) and R → R(t − tw).
• The fluctuation-dissipation th. R(t− tw) = − 1kBT
∂C(t−tw)∂t
θ(t− tw)
holds and implies
χ(t − tw) ≡∫ ttw
dt′ R(t, t′) = 1kBT [C(0) − C(t − tw)] .
In glassy systems below Tg : breakdown of stationarity & FDT.
χ(t, tw) ≡∫ ttw
dt′ R(t, t′) and C(t, tw) not obviously related.
Fluctuation-dissipationSolvable cases : p spin-models
T ∗Ttw3
tw2
tw1
1
kBT ∗
1
kBT
χ(t
,tw)
C(t, tw)
0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
Parametric construction
tw fixed
t : tw → ∞ or
dt : 0 → ∞.
LFC & Kurchan 93.
Fluctuation-dissipationProposal
For non-equilibrium systems, relaxing slowly towards an asymptotic limit
(cfr. threshold in p spin models) such that one-time quantities [e.g. the
energy-density E(t)] approach a finite value [e.g. E∞]
limtw→∞
C(t,tw)=C
χ(t, tw) = fχ (C)
LFC & Kurchan 94.
For weakly forced non-equilibrium systems in the limit of small work
limǫ→0
C(t,tw)=C
χ(t, tw) = fχ (C)
FDT in relaxing glassesExperiments and simulations
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0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
C−
k BT
M
Spin-glass (thiospinel) Lennard-Jones binary mixture
Hérisson & Ocio 02. J-L Barrat & Kob 99.
also in glycerol (Grigera & Israeloff 99 ) and after this paper many others in colloidal
suspensions & polymer glasses (exps.), silica, vortex glasses, dipolar glasses, etc.
PhenomenologyTimes scales
In all these systems the dynamics occur
• In quasi-equilibrium [ χ = 1kBT
(1 − C) ] when
t>∼ tw or qea ≤ C ≤ 1.
• Clearly out of equilibrium when
t > tw or C ≤ qea.
In structural glassy systems one finds
χ = 1kBT ∗ (qea − C) + 1
kBT(1 − qea)
Interpretation
• In particle systems, rattling within cages vs. structural relaxation.
• In magnetic coarsening, thermal fluctuations within domains vs. domain
wall motion.
FDT & effective temperaturesCan one interpret the slope as a temperature ?
M c
opie
s of
the
sys
tem
Observable A
’
’
Thermometer
(coordinate x)
Coupling constant k
Thermal bath (temperature T)
A A A A. . .
α=1 α=3 α=Μ
x
α=2
T ∗Ttw3
tw2
tw1
1
kBT ∗
1
kBT
χ(t
,tw)
C(t, tw)
0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
(1) Measurement with a thermometer with
• Short internal time scale τ0, fast dynamics is tested and T is recorded.
• Long internal time scale τ0, slow dynamics is tested and T ∗ is recorded.
(2) Partial equilibration (3) Direction of heat-flow
LFC, Kurchan & Peliti 97.
FDT & effective temperaturesSheared binary Lennard-Jones mixture
TT bmtr
m trhv2 zi
10810710610510410310210110010�110.80.60.40.20
FDT T �FDT TSelf BSelf A11:22; (A+B)7:47; (A+B)3:74; (A+B)1:87
Correlation
Sus eptibility
10.80.60.40.2043210
Left : The kinetic energy of a tracer particle (the thermometer) as a func-
tion of its mass (τ0 ∝√
mtr) 12mtr〈 v2
z〉 = 12kBTeff .
Right : χk(Ck) plot for different wave-vectors k : partial equilibrations.
J-L Barrat & Berthier 00.
FDT & effective temperaturesDissipative quantum glassy models
The quantum equilibrium FDT
R(t, tw) =i
~
∫ ∞
−∞
dω
πe−iω(t−tw) tanh
(β~ω
2
)
C(ω, tw)
becomes
χ(t, tw) ≈ − 1Teff
C(t, tw) t ≫ tw
if the integral is dominated by ω(t − tw) ≪ 1 and T → Teff > 0
such that βeff~ω → 0.
LFC & G. Lozano 98-99.
Real-time dynamicsGlassy phase
Symmetric correlation Linear response
LFC & G. Lozano 98-99.
Summary I
• We analyzed fully-connected classical and quantum spin models
with quenched random interactions .
• The TAP method yields the free-energy landscape in phase space.
• The replica trick yields complementary information on equilibrium
states.
• The out of equilibrium relaxation from a random initial condition cap-
tures a slow aging decay of the correlation and linear response. Gen-
eration of an effective temperature .
These methods can be adapted to deal with particles in interac-
tion moving in a continuous space.
Summary IISuggest the existence of three classes of systems :
• FM-like two-equilibrium states, coarsening. Droplet picture of spin-
glasses but the response decays very fast.
• p-spin-like random first order phase transition. Have many metastable
states that block the relaxation (super-cooled liquids, structural glasses)
Different dynamic and static transitions. The effective temperature is
finite and takes the ‘microcanonic’ value where the configuration-
al entropy is the relevant one.
• SK-like . 2nd order phase transition, exponentially large number of
equilibrium states, all kinds of overlaps between them, very slow
dynamics.
A many morepeculiar features that I have not discussed here !
Summary IIIWhat is special of quantum models ?
• First order phase transitions in models of the p-psin/random first
order kind.
• The phase transition depends on the bath .
• For the out of equilibrium relaxation, interactions go against localiza-
tion .
• Effective temperature ‘driven’ decoherence at long time or spatial
scales ; e.g.
for coarsening systems the time-dependence of the growing length,
R(t; T, Γ), should be the same as for the classical counterpart ;
in all these cases the FDT becomes classical.
ChallengesThe main one
Get a convincing real-space translation of the com-plex phase space and real-time information.
Two problems in progressTeff from FDT
A quantum quench Γ0 → Γ of the isolated Ising chain
Here : to its critical point Γ = 1
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10
t
C(t) G=1 G0=0.2R(t) G=1 G0=0.2
0
1
2
3
4
5
6
7
8
9
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Gamma_0
b_eff=tau*Ar/Acb_eff lim omega->0 qFDT
Dissipative dynamics of the quantum Ising chain
Part of Laura Foini’s PhD project.
The end
