Classical & quantum glasses Leticia F. Cugliandolo Univ. Pierre - - PowerPoint PPT Presentation
Classical & quantum glasses Leticia F. Cugliandolo Univ. Pierre et Marie Curie Paris VI leticia@lpthe.jussieu.fr www.lpthe.jussieu.fr/ leticia Slides & useful notes in front page Cargse, June 2011 Plan First lesson .
Slides & useful notes in front page Cargèse, June 2011
Statics of classical and quantum disordered systems.
Classical dynamics. Coarsening. Formalism.
Quantum dynamics. Formalism. Results for mean-field models.
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.
Aim 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
E ∆ 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,
Model the environment and the interaction E.g., an esemble of harmonic oscillators and a bi-linear coupling :
N
α
α
α
N
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)
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-
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.
In systems with competing interactions there is no consensus upon :
Examples are :
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 variables are frozen during time-scales over which other va- riables fluctuate. Time scales
eq
eq could be the diffusion time-scale for magnetic impurities the magnetic
moments of which will be the variables of a magnetic system ;
Weak disorder (modifies the critical properties but not the phases) vs. strong disorder (that modifies both). e.g. random ferromagnets vs. spin-glasses.
Magnetic impurities (spins) randomly placed in an inert host Quenched random interactions
Magnetic impurities in a metal host
RKKY potential
ij
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.
(
(
Hérisson & Ocio 02.
Transition Dynamics
Competition between elasticity and quenched randomness
random potential.
Oil Water Interface between two phases ; vortex line in type-II supercond ; stretched polymer. Distorted Abrikosov lattice Goa et al. 01.
ij Jijsisj
Ising model
Disordered Geometric
gs
gs
and
gs
gs
Frustration enhances the ground-state enegy and entropy
Disordered example
Efrust
gs
= −2J > EF M
gs
= −4J Sfrust
gs
= ln 6 > SF M
gs
= ln 2
Geometric case
Efrust
gs
= 3J > EF M
gs
= −3J Sfrust
gs
= ln 3 > SF M
gs
= ln 2
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 ln N.
But it could be an over-simplication.
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
Each sample is a priori different but, do they all have a different thermodynamic and dynamic behavior ?
The disorder-induced free-energy density distribution approaches a Gaus- sian with vanishing dispersion in the thermodynamic limit :
independently of disorder – Experiments : all typical samples behave in the same way. – Theory : one can perform a (hard) average of disorder,
Exercise : Prove it for the 1d Ising chain ; argument for finite d systems.
Intensive quantities are also self-averaging. Replica theory
N(β, J) − 1
manufacturing flourished in the Roman empire. EVERYDAY-LIFE USE.
everywhere.
Simulation Confocal microscopy Molecular (Sodium Silicate) Colloids (e.g. d ∼ 162 nm in water) Experiment Simulation Granular matter Polymer melt
Characteristics
in the pictures) are coupled to their surroundings (other kinds of molecules, water, etc.) that act as thermal baths in equilibrium.
Jones potential, frustration.
geneity. Sometimes one talks about self-generated disorder.
Structure and dynamics The two-time dependent density-density correlation :
with
The average over different dynamical histories (simulation/experiment)
isotropy (all directions are equivalent) invariance under translations of the reference point
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 :
i=1 ei q( ri(t)− ri(tw))
Hansen & McDonald 06.
No obvious structural change but slowing down !
1.0 2.0 3.0 4.0 0.0 1.0 2.0 3.0 4.0
r gAA(r;t,t)
gAA(r) r t=0 t=10 Tf=0.1 Tf=0.3 Tf=0.4 Tf=0.435 Tf=0.4
0.9 1.0 1.1 1.2 1.3 1.4 0.0 2.0 4.0 6.0
10
−1 10
10
1
10
2
10
3
10
4
10
5
0.0 0.2 0.4 0.6 0.8 1.0
tw=63100 tw=10
t−tw
Fs(q;t,tw)
tw=0
q=7.23
Tf=0.4
LJ mixture
r
r
Time-scale separation & slow non-equilibrium dynamics
Molecular dynamics – J-L Barrat & Kob 99.
Calorimetric measurement of entropy
What is making the relaxation so slow ? Is there growth of static order ? Phase space picture ?
Energy 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
>
– one can even set T → 0 and keep just quantum fluctuations. Examples : quantum magnets, Wigner crystals, etc.
Structural glasses Crytallization at Tm is avoided by cooling fast enough. Liquid Supercooled liquid Glass
Non-exponential relax Equilibrium Metastable equilibrium Non-equilibrium
An exponential number
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.
for 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.
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.
– Liquids & glass transition
P . G. Debenedetti, Metastable liquids (Princeton Univ. Press, 1997).
2001).
mechanics (World Scientific, 2005).
mena in disordered materials, arXiv :1011.2578
– Spin-glasses
F . Zamponi, Mean field theory of spin glasses, arXiv :1008.4844.
– Disorder elastic systems
. Le Doussal, Statics and dynamics of disordered elastic systems, arXiv :cond- mat/9705096.
– Phase ordering kinetics
– Glasses
2001).
mechanics (World Scientific, 2005).
. Cugliandolo, Dynamics of glassy systems, Les Houches Session 77, arXiv :cond-mat/0210312.
mena in disordered materials, arXiv :1011.2578.
Statics of classical and quantum disordered systems.
Classical dynamics. Coarsening. Formalism.
Quantum dynamics. Formalism. Results for mean-field models.
Simulation Confocal microscopy Molecular (Sodium Silicate) Colloids (e.g. d ∼ 162 nm in water) Experiment Simulation Granular matter Polymer melt
Structure and dynamics The two-time dependent density-density correlation :
with
The average over different dynamical histories (simulation/experiment)
isotropy (all directions are equivalent) invariance under translations of the reference point
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 :
i=1 ei q( ri(t)− ri(tw))
Hansen & McDonald 06.
No obvious structural change but slowing down !
1.0 2.0 3.0 4.0 0.0 1.0 2.0 3.0 4.0
r gAA(r;t,t)
gAA(r) r t=0 t=10 Tf=0.1 Tf=0.3 Tf=0.4 Tf=0.435 Tf=0.4
0.9 1.0 1.1 1.2 1.3 1.4 0.0 2.0 4.0 6.0
10
−1 10
10
1
10
2
10
3
10
4
10
5
0.0 0.2 0.4 0.6 0.8 1.0
tw=63100 tw=10
t−tw
Fs(q;t,tw)
tw=0
q=7.23
Tf=0.4
LJ mixture
r
r
Time-scale separation & slow non-equilibrium dynamics
Molecular dynamics – J-L Barrat & Kob 99.
What do they have in common ? – No obvious spatial order : disorder (differently from crystals). – Many metastable states Rugged landscape – Slow non-equilibrium relaxation
Time-scale separation – Hard to make them flow under external forces. Pinning, creep, slow non-linear rheology.
e.g., up & down spins in a 2d Ising model (IM)
In a canonical setting the control parameter is T/J.
Fully connected Ising model Normalize J by the size of the system N to have O(1) local fields
2N
i si
The partition function reads ZN =
−1 du e−βNf(u) with Nu = i si
2u2 − hu + T
2 ln 1+u 2
2 ln 1−u 2
the same u value. Use the saddle-point, limN→∞ fN(βJ, βh) = f(usp), with
Exercise : do these calculations, see notes.
Continuous scalar statistical field theory Coarse-grain the spin
r si.
Set h = 0.
L
The partition function is ZV =
2[∇φ(
2 φ2(
4φ4(
connected model (entropy around φ ∼ 0 and symmetry arguments.
Uniform saddle point in the V → ∞ limit : φsp(
The free-energy is limV →∞ fV (β, J) = f(φsp).
bi-valued equilibrium states related by symmetry, e.g. Ising magnets lower critical upper
Ginzburg-Landau free-energy Scalar order parameter
(a consequence of the symmetry of the action).
the minima becomes a metastable state.
diverges with the size of the system implying ergodicity breaking.
These results were not fully accepted as realistic at the time.
e.g. up & down spins in a 2d Ising model (IM)
In a canonical setting the control parameter is T/J.
e.g. up & down spins in a 3d Edwards-Anderson model (EA)
ij Jijsisj
ij] = J2
Is there another order-parameter ?
Fully connected SG : Sherrington-Kirkpatrick model
1 2 √ N
i hisi
with Jij i.i.d. Gaussian variables, [Jij] = 0 and [J2
ij] = J2 = O(1).
One finds the naive free-energy landscape
1 2 √ N
N
1+mi 2
2
2
2
and the naive TAP equations
j(=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.
A hint on the proof The more traditional one assumes independence of the spins,
i pi(si)
with pi(si) = 1+mi
2
2
and uses this form to express H − TS 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 = sih.
Georges & Yedidia 91.
This proof is easier to generalize to dynamics (Biroli 00) and quantum systems (Biroli & LFC 01), see later.
rigorous in MF.
that are extrema of the TAP free-energy landscape, i.e. saddles of all types, at low temperatures.
sp} but apart from this
trivial doubling, the remaining ones are not related by symmetry.
and compute, e.g., how many saddles of each kind exist and how many of these at each level of f.
separated by diverging barriers (infinite life-time).
The average of a generic observable is
α wαOα
In the FM case, each state φ = ±φ0 has weigth w± = e−βNf±ZN =
2O+ + 1 2O− . 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
α = e−βNfJ
α
P
γ e−βNfJ γ
and
where NJ is the (possibly exponential in N) number of solutions to the TAP equations with free-energy density f.
Consequences The equilibrium free-energy is given by the saddle-point evaluation of the partition sum that implies
The rhs is the Landau free-energy of the problem, with f playing the role
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.
"configuration space"
1/T 1/T 1/T 1/T
d g cage
very warm liquid warm liquid viscous liquid 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.
A sketch
N→∞ ln ZN(β, J) = lim N→∞ lim n→0
N(β, J)] − 1
N partition function of n independent copies of the system : replicas.
Average over disorder : coupling between replicas
i sa j ⇒ N −1 i=j
i sa j
Decoupling with the Hubbard-Stratonovich trick
i sa i sb i − 1 2Q2 ab
The elements of Qab can be evaluated by saddle-point if one exchanges the limits N → ∞ n → 0 with n → 0 N → ∞.
Overlaps
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
Many repetitions :
2δ(qsσ − m2) + 1 2δ(qsσ + m2)
Overlaps in disordered systems Parisi 79-82 prescription for the replica symmetry breaking Ansatz yields
1 p q PM
m2 1 q FM
qEA 1 q p-spin
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.
Applications to other problems Random elastic manifolds, e.g. vortex systems, dirty interfaces, etc. Models with self-generated disorder, e.g. Lennard-Jones particles.
Oil Water
Interface, vortex Abrikosov lattice Lennard-Jones mixture Mézard & Parisi 91, Giamarchi & Le Doussal 97, Mézard & Parisi 99.
Tricks are necessary to introduce quenched randomness in the calculation.
A 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).
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.
methods.
Statics of classical and quantum disordered systems.
Classical dynamics. Coarsening. Formalism.
Quantum dynamics. Formalism and results for mean-field models.
Classical 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.
ij Jijˆ
i ˆ
j + Γ i ˆ
i − i hiˆ
i .
i
with a = 1, 2, 3 the Pauli matrices, [ˆ
quenched random., e.g. Gaussian pdf conveniently normalized.
transverse field. It measures quantum fluctuations. In the limit Γ → 0 the classical limit should be recovered.
longitudinal local fields
nearest neighbours on the lattice – finite d
Phase transitions in d = 2, 3
Classical spin glass transition SG PM
c
c
3d
T
Quantum phase transition Quantum spin glass phase
T =0
c
"c
PM Quantum critical region
T >0
c
2d
Quantum MC. Overviews in Rieger & Young 95, Kawashima & Rieger 03. Many special features in d = 1 obtained with the Dasgupta-Ma RG decimation
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
isi(τ)si(τ ′)h.
Biroli & LFC 01. Replica trick The partition function is a trace
H =
{si(0)}
Se syst[{si}]
with the Euclidean action
syst[{si}] =
a function of the transverse field Γ.
Feynman-Matsubara construction of functional integral over imaginary time.
Overlap matrix-function : Qab(τ, τ ′)N−1
i sa i (τ)sb i(τ ′) − 1 2Q2 ab(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 quantum spin-glass models ?
Quantum p-spin model
0.0 0.2 0.4 0.6 1 2 3
m < 1 m=1 T
*
SG PM
enter text hereΓ T
1 2 3 0.0 0.5 1.0
(b)
β=12
qEA 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.
Optimization 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
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
1st order transitions : trouble for quantum annealing techniques.
Jorg, Krzakala, Kurchan, Maggs & Pujos 09.
Aim 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
Syst.
Env.
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.
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 :
N
α
α
α
N
Either classical variables or quantum operators.
Statics of a classical system The partition function of the coupled system is
Integrating out the oscillator variables :
−β „ Hsyst+Hcount+ηx− 1
2
PN
a=1 c2 aω2 a ma
x2 «
Choosing Hcount to cancel the quadratic term in x2 one recovers
i.e., the partition function of the system of interest.
Statics of a quantum system
The density matrix of the coupled system is
a; x′, q′ a) = x′′, q′′ a| ˆ
a
x(0)=x′
a
qa(0)=q′
a
Se[η]
One integrates the oscillator’s degrees of freedom to get the reduced density matrix
red
x′
Se
syst−
R β dτ R τ
0 dτ ′ x(τ)K(τ−τ ′)x(τ ′)
”
The counter-term was chosen to cancel a quadratic term in x2(τ), Zred =
a non-local interaction in the imaginary time with kernel
2 πβ
n=−∞
0 dω I(ω) ω ν2
n
ν2
n+ω2 exp(iνnτ) remains.
Dissipative Ising p-spin model with p ≥ 3 at T ≈ 0 Magnetic susceptibility Averaged entropy density
LFC, Grempel & da Silva Santos 00 ; LFC, Grempel, Lozano, Lozza, da Silva Santos 04.
The 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.
Statics of quantum disordered systems
as the cavity method can be applied to them.
class have first order phase transitions in the low temperature limit.
highly non-trivial effect quantum mechanically. Similar results for quantum Ising chains with FM and disordered interactions LFC, Lozano & Lozza ; Chakravarty, Troyer, Voelker, Werner 04 (MC) Schehr & Rieger 06 (decimation RG)
methods.
Statics of classical and quantum disordered systems.
Classical dynamics. Coarsening. Formalism.
Quantum dynamics. Formalism and results for mean-field models.
Dynamics across a phase transition
know the equilibrium phases nor the dynamic mechanisms.
nown dynamic mechanisms. e.g. glasses
(classical or quantum).
bi-valued equilibrium states related by symmetry, e.g. Ising magnets lower critical upper
Ginzburg-Landau free-energy Scalar order parameter
The system is in contact with a thermal bath Thermal agitation Non-conserved order parameter φ(t, T) = 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.
A quench or an annealing across a phase transition
Non-conserved order parameter φ(t, T) = ct Development of magnetization in a ferromagnet after a quench.
e.g. up & down spins in a 2d Ising model (IM)
50 100 150 200 50 100 150 200 ’data’ 50 100 150 200 50 100 150 200 ’data’ 50 100 150 200 50 100 150 200 ’data’ 50 100 150 200 50 100 150 200 ’data’ 50 100 150 200 50 100 150 200 ’data’ 50 100 150 200 50 100 150 200 ’data’
Question : starting from equilibrium at T0 → ∞ or T0 = Tc how is equilibrium at Tf = Tc or Tf < Tc attained ?
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
from
N
i,j=1δsi(t)δsj(t)| ri− rj|=r
size of the system, tr(T, L) → ∞ when L → ∞ for T ≤ Tc.
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.
xi− xj|=r ∼ m2 eq(T) f
eq(T) fc
eq(T).
Suggested by experiments and numerical simulations. Proved for
spherical ferromagnet. Review Bray, 1994.
Magnetic model Scaling regime
r R(t,T),
eq(T) fc
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
Spinodal decomposition in binary mixtures
locally conserved order parameter 50 : 50 composition ; Rounder boundaries
Schrielen 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
Yurke et al 93, Bray & Rutenberg 94.
Universality classes
scalar NCOP
scalar COP
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
e.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
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 e
B(R) kBT
This implies
Still two ferromagnetic states related by symmetry
curvature-driven
activated with Lc(T) a growing function of T . Inverting times as a function of length
At short times this equation can be approximated by an effective power law with a T -dependent exponent :
Bustingorry et al. 09.
methods.
Classical dynamics. Coarsening. Formalism.
Quantum dynamics. Formalism. Mean-field models.
Two-time dependence
Correlation
Symmetrized correlator Linear response
Antisymmetrized correlator
To a kick and to a step
w w
r(tw) t r( ) r( ) t
h
The perturbation couples linearly to the observable H
The linear instantaneous response of another observable A({
tw
Closed-time path to allow for ˆ
and take the reservoir in equilibrium at its own β (and µ).
Some important technical remarks
Two real-time long-range interactions (recall static cases).
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
bath kernels become the ones of a colored classical Langevin process.
The rôle played by the initial condition & disorder
no need of replica trick to average over disorder ! In the classical limit Z[η = 0] =
disorder
de Dominicis 78.
In the quantum model ˆ
LFC & Lozano 98.
We simply average Z or ˆ
saddle-point in the large N limit or a variational approximation.
An example : rotors with pair intereactions
ab(t, t′)
ia(t) − M)
with the bath induced kernels
ab(t, t′)
that take different forms for different baths, e.g. oscillators, leads, etc.
Paramagnetic phase
Dependence on the quantum parameter Γ (T = 0, α fixed.)
LFC & G. Lozano 98-99.
Dependence on the coupling to the bath
Comparison between α = 0.2 (PM) and α = 1 (SG)
LFC, Grempel, Lozano, Lozza & da Silva Santos 02.
Summary
although the evolution can depend on the bath, the target equilibrium does not.
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.
Interactions against localization
LFC, Grempel, Lozano, Lozza & da Silva Santos 02
Quantum p-spin model
0.0 0.2 0.4 0.6 1 2 3
m < 1 m=1 T
*
SG PM
enter text hereΓ 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
Equilibrium spontaneous (C) and induced (R) fluctuations If
1 kBT ∂C(t−tw) ∂t
holds and implies
tw dt′ R(t, t′) = 1 kBT [C(0) − C(t − tw)] .
In glassy systems below Tg : breakdown of stationarity & FDT.
tw dt′ R(t, t′)
and
Solvable cases : p spin-models
0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
Parametric construction
LFC & Kurchan 93.
Proposal 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∞]
tw→∞
C(t,tw)=C
LFC & Kurchan 94.
For weakly forced non-equilibrium systems in the limit of small work
ǫ→0
C(t,tw)=C
Experiments and simulations
0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8
C −kBTM
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.
Times scales
In all these systems the dynamics occur
1 kBT (1 − C) ] when
>
In structural glassy systems one finds
1 kBT ∗ (qea − C) + 1 kBT (1 − qea)
Interpretation
wall motion.
Can one interpret the slope as a temperature ?
M copies of the system 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 ∗ T tw3 tw2 tw1 1 kBT ∗ 1 kBT
χ(t, tw) C(t, tw)
0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
(1) Measurement with a thermometer with
(2) Partial equilibration (3) Direction of heat-flow
LFC, Kurchan & Peliti 97.
Sheared binary Lennard-Jones mixture
T T b m tr m tr hv 2 z i 10 8 10 7 10 6 10 5 10 4 10 3 10 2 10 1 10 10 1 1 0.8 0.6 0.4 0.2 FDT TLeft : The kinetic energy of a tracer particle (the thermometer) as a func- tion of its mass (τ0 ∝ √mtr)
1 2mtr v2 z = 1 2kBTeff.
Right : χk(Ck) plot for different wave-vectors k : partial equilibrations.
J-L Barrat & Berthier 00.
Dissipative quantum glassy models
The quantum equilibrium FDT
−∞
becomes
1 TeffC(t, tw)
if the integral is dominated by ω(t − tw) ≪ 1 and T → Teff > 0 such that βeffω → 0.
LFC & G. Lozano 98-99.
Glassy phase
LFC & G. Lozano 98-99.
with quenched random interactions.
states.
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.
Suggest the existence of three classes of systems :
glasses but the response decays very fast.
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.
equilibrium states, all kinds of overlaps between them, very slow dynamics. A many morepeculiar features that I have not discussed here !
What is special of quantum models ?
tion.
scales ; e.g. for coarsening systems the time-dependence of the growing length,
in all these cases the FDT becomes classical.
The main one
A quantum quench Γ0 → Γ of the isolated Ising chain Here : to its critical point Γ = 1
0.2 0.4 0.6 0.8 1 2 4 6 8 10 t C(t) G=1 G0=0.2 R(t) G=1 G0=0.2 1 2 3 4 5 6 7 8 9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Gamma_0 b_eff=tau*Ar/Ac b_eff lim omega->0 qFDT
Dissipative dynamics of the quantum Ising chain
Part of Laura Foini’s PhD project.