Structural glasses No obvious structural change but slowing down ! 6.0 g AA (r;t,t) 1.0 T f =0.1 g AA (r) 4.0 q=7.23 T f =0.3 T f =0.4 F s (q;t,t w ) 4.0 0.8 t w =63100 3.0 2.0 0.6 t w =10 2.0 T f =0.435 0.4 0.0 0.9 1.0 1.1 1.2 1.3 1.4 r t w =0 1.0 0.2 t=0 T f =0.4 T f =0.4 t=10 0.0 0.0 −1 10 0 1 2 3 4 5 1.0 2.0 3.0 4.0 10 10 10 10 10 10 r t−tw �� σ αβ � 6 � � 12 − � σ αβ V αβ ( r ) = 4 ǫ αβ LJ mixture r r τ 0 ≪ τ exp ≪ τ eq that changes by > 10 orders of magnitude ! Time-scale separation & slow non-equilibrium dynamics Molecular dynamics – J-L Barrat & Kob 99.
Time-scales Calorimetric measurement of entropy What is making the relaxation so slow ? Is there growth of static order ? Phase space picture ?
Fluctuations Energy scales Thermal fluctuations – irrelevant for granular matter since mgd ≫ k B T ; dynamics is induced by macroscopic external forces. – important for magnets, colloidal suspensions, etc. Quantum fluctuations > ∼ k B T quantum fluctuations are important. – when � ω – one can even set T → 0 and keep just quantum fluctuations. Examples : quantum magnets, Wigner crystals, etc.
Summary Structural glasses Crytallization at T m is avoided by cooling fast enough. Liquid Supercooled liquid Glass � �� � Exponential relax Non -exponential relax Non-equilibrium Equilibrium Metastable equilibrium � �� � Separation of time-scales & An exponential number � of metastable states ! � �� Stationary Aging correlations and reponses depend on t and t w Aging means that ac susceptibilities depend on ω and t w There might be an equilibrium transition to an ideal glass at T s .
Methods for classical and quantum disordered systems Statics TAP Thouless -Anderson-Palmer fully-connected (complete graph) Replica theory Gaussian approx. to field-theories � dilute (random graph) Cavity or Peierls approx. Bubbles & droplet arguments finite dimensions functional RG 1 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 transition P . 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 statistical mechanics (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 -glasses K. 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. Diep ed. (World Scientific, 2004). M. Talagrand, Spin glasses, a challenge for mathematicians (Springer-Verlag, 2003).
References – Disorder elastic systems T. 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 kinetics A. J. Bray, Theory of phase ordering kinetics , Adv. Phys. 43 , 357 (1994). S. Puri, Kinetics of Phase Transitions , (Vinod Wadhawan, 2009). – Glasses 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 statistical mechanics (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 Colloids (e.g. d ∼ 162 nm in water) Molecular (Sodium Silicate) Experiment Simulation Granular matter Polymer melt
Correlation functions Structure and dynamics The two -time dependent density-density correlation : g ( r ; t, t w ) ≡ � δρ ( � x, t ) δρ ( � y, t w ) � r = | � x − � y | with The average over different dynamical histories (simulation/experiment) � . . . � implies isotropy (all directions are equivalent) invariance under translations of the reference point � x . r j ( t w )) � . Its Fourier transform, F ( q ; t, t w ) = N − 1 � N i,j =1 � e i� q ( � r i ( t ) − � The incoherent intermediate or self correlation : r i ( t w )) � F s ( q ; t, t w ) = N − 1 � N r i ( t ) − � i =1 � e i� q ( � Hansen & McDonald 06.
Structural glasses No obvious structural change but slowing down ! 6.0 g AA (r;t,t) 1.0 T f =0.1 g AA (r) 4.0 q=7.23 T f =0.3 T f =0.4 F s (q;t,t w ) 4.0 0.8 t w =63100 3.0 2.0 0.6 t w =10 2.0 T f =0.435 0.4 0.0 0.9 1.0 1.1 1.2 1.3 1.4 r t w =0 1.0 0.2 t=0 T f =0.4 T f =0.4 t=10 0.0 0.0 −1 10 0 1 2 3 4 5 1.0 2.0 3.0 4.0 10 10 10 10 10 10 r t−tw �� σ αβ � 6 � � 12 − � σ αβ V αβ ( r ) = 4 ǫ αβ LJ mixture r r τ 0 ≪ τ exp ≪ τ eq 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 τ 0 ≪ τ exp ≪ τ eq Time-scale separation – Hard to make them flow under external forces. Pinning, creep, slow non-linear rheology.
PM-FM transition e.g., up & down spins in a 2 d Ising model (IM) � φ � = 0 � φ � = 0 � φ � � = 0 T → ∞ T = T c T < T c In a canonical setting the control parameter is T/J .
Mean-field theory for PM-FM Fully connected Ising model Normalize J by the size of the system N to have O (1) local fields � i � = j s i s j − h � H = − J i s i 2 N � 1 − 1 du e − βN f ( u ) with Nu = � The partition function reads Z N = i s i 2 u 2 − hu + T � 1+ u � + 1 − u 2 ln 1 − u f ( u ) = − J 2 ln 1+ u 2 2 Energy terms and entropic contribution stemming from N ( { s i } ) yielding the same u value. Use the saddle-point , lim N →∞ f N ( βJ, βh ) = f ( u sp ) , with u sp = tanh ( βJu sp + βh ) = � u � Exercise : do these calculations, see notes.
Ginzburg-Landau for PM-FM Continuous scalar statistical field theory Coarse -grain the spin � r ) = V − 1 φ ( � r s i . i ∈ V � � r Set h = 0 . L � D φ e − βV f ( φ ) with V the volume and The partition function is Z V = � 1 � � r )] 2 + T − J r ) + λ d d r 2 φ 2 ( � 4 φ 4 ( � 2 [ ∇ φ ( � f ( φ ) = 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 : φ sp ( � r ) = � φ ( � r ) � = φ 0 . The free-energy is lim V →∞ f V ( β, J ) = f ( φ sp ) .
2nd order phase-transition bi -valued equilibrium states related by symmetry, e.g. Ising magnets upper critical lower � φ � f φ T Ginzburg-Landau free-energy Scalar order parameter
Features • Spontaneous symmetry breaking below T c . • 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 � � = 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 configurations e.g. up & down spins in a 2 d Ising model (IM) � φ � = 0 � φ � = 0 � φ � � = 0 T → ∞ T = T c T < T c In a canonical setting the control parameter is T/J .
Spin-glasses e.g. up & down spins in a 3 d Edwards-Anderson model (EA) H J = � [ J 2 ij ] = J 2 � ij � J ij s i s j [ J ij ] = 0 � φ � = 0 � φ � = 0 � φ � = 0 T → ∞ T = T c T < T c Is there another order -parameter ?
MF theory for spin-glasses Fully connected SG : Sherrington-Kirkpatrick model � i � = j J ij s i s j − � 1 H = − i h i s i √ 2 N ij ] = J 2 = O (1) . with J ij i.i.d. Gaussian variables, [ J ij ] = 0 and [ J 2 One finds the naive free -energy landscape N � � 1+ m i ln 1+ m i + 1 − m i ln 1 − m i 1 N f ( { m i } ) = − J ij m i m j + T . √ 2 2 2 2 2 N i � = j i =1 and the naive TAP equations m isp = tanh( β � j ( � = i ) J ij m jsp + βh i ) that determine the restricted averages m i = � s i � = m isp . 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-glasses A hint on the proof The more traditional one assumes independence of the spins, P ( { s i } ) = � i p i ( s i ) with p i ( s i ) = 1+ m i δ s i , 1 + 1 − m i δ s i , − 1 2 2 and uses this form to express � H � − T � S � with S = ln P ( { s i } ) ; see notes . A more powerful proof expresses f as the Legendre transform of − βF ( h i ) with m i = N − 1 ∂ [ − βF ( h i )] /∂h i = � s i � 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 m i � = 0 below T c are heterogeneous . • The fact that an equilibrium state is determined by { m i } can be made rigorous in MF. • There are N order parameters , m i , i = 1 , . . . , N . • The TAP equations have an exponential in N number of solutions { m iα } that are extrema of the TAP free-energy landscape, i.e. saddles of all types, at low temperatures. • For each solution { m iα } one also has {− m iα sp } 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 averages � O � = � α w α � O � α The average of a generic observable is In the FM case , each state � φ � = ± φ 0 has weigth w ± = e − βN f ± Z N = 1 / 2 and the sum is � O � = 1 2 � 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. e − βN f J w J α α = Within the TAP approach and γ e − βN f J P γ � d f e − β [ N f − T ln N J ( f ,β )] O ( f ) � O � = Z − 1 ( β, J ) where N J is the (possibly exponential in N ) number of solutions to the TAP equations with free-energy density f .
Statistical averages Consequences The equilibrium free-energy is given by the saddle -point evaluation of the partition sum that implies Nf = N f min − TS c ( f min , β ) S c ( f , β ) = ln N J ( f , β ) The rhs is the Landau free-energy of the problem, with f playing the role of the energy and N − 1 ln N J ( 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 very warm liquid 1/T cage warm liquid 1/T d viscous liquid 1/T g glass ? ? 1/T "configuration space" The ruggedness of the free -energy landscape increases upon decreasing tem- perature until a configurational entropy crisis arises (at Kauzmann T K ). Numerical simulations : one cannot access f but one can explore e : po- tential energy landscape . D. J. Wales 03.
Replica calculation A sketch [ Z n N ( β, J )] − 1 − βf = lim N →∞ ln Z N ( β, J ) = lim N →∞ lim Nn n → 0 Z n N partition function of n independent copies of the system : replicas . Average over disorder : coupling between replicas �� � 2 � � j ⇒ N − 1 � i � = j J ij s a i s a a s a i s a j a i � = j Decoupling with the Hubbard -Stratonovich trick Q ab N − 1 � i − 1 i s a i s b 2 Q 2 ab Q ab is a 0 × 0 matrix but it admits an interpretation in terms of overlaps . The elements of Q ab can be evaluated by saddle-point if one exchanges the limits N → ∞ n → 0 with n → 0 N → ∞ .
Replica calculation Overlaps Take one sample and run it until it reaches equilibrium , measure { s i } . Re -initialize the same sample (same J ij ), run it until it reaches equilibrium , measure { σ i } . Construct the overlap q sσ ≡ N − 1 � N i =1 s i σ i . In a FM system there are four possibilities f sσ sσ s σ σ s q sσ = m 2 m 2 − m 2 − m 2 P ( q sσ ) = 1 2 δ ( q sσ − m 2 ) + 1 2 δ ( q sσ + m 2 ) Many repetitions :
Replica calculation Overlaps in disordered systems Parisi 79-82 prescription for the replica symmetry breaking Ansatz yields PM FM p-spin SK p -m 2 m 2 -q EA q EA -1 -q EA q EA -1 0 1 -1 0 1 -1 0 1 0 1 q q q q 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 calculation Applications to other problems Random elastic manifolds , e.g. vortex systems, dirty interfaces, etc. Models with self-generated disorder , e.g. Lennard -Jones particles. Lennard-Jones mixture Abrikosov lattice Water Oil Interface, vortex Mézard & Parisi 91, Giamarchi & Le Doussal 97, Mézard & Parisi 99. Tricks are necessary to introduce quenched randomness in the calculation.
Droplet picture 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 ) ≃ Υ ℓ θ . Proposition for P ( E ℓ ) . P ( q ) with two peaks at ± q EA . 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.
Summary 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 T c . 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 T c . SK . Different scenario from droplet model.
Quantum spin-glasses H syst = − � j + Γ � i − � ˆ σ z σ z σ x σ z ij J ij ˆ i ˆ i ˆ i h i ˆ i . σ a σ a , ˆ σ b ] = 2 iǫ abc ˆ σ c . ˆ with a = 1 , 2 , 3 the Pauli matrices , [ˆ i J ij 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. h i longitudinal local fields ij nearest neighbours on the lattice – finite d or fully connected – mean -field
Quantum SG : finite d Phase transitions in d = 2 , 3 T =0 2d T T c Classical PM spin glass PM transition ! " c c " 3d SG Quantum critical region T >0 c " " c Quantum spin glass phase Quantum phase transition Γ c Quantum critical point. T c Classical critical point. Quantum MC. Overviews in Rieger & Young 95, Kawashima & Rieger 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 { m i ( τ ) } and C ( τ − τ ′ ) with m i ( τ ) = � s i ( τ ) � h and C ( τ − τ ′ ) = N − 1 � i � s i ( τ ) s i ( τ ′ ) � h . Biroli & LFC 01. Replica trick The partition function is a trace � { s i ( β � ) } D{ s i ( τ ) } e − 1 Z N = Tr e − β ˆ H = � S e syst [ { s i } ] { s i (0) } with the Euclidean action
Mean-field methods � β � � ∂s i ( τ ) � 2 M � S e syst [ { s i } ] = dτ + J ij s i ( τ ) s j ( τ ) 2 ∂τ 0 ij s i ( β � ) = s i (0) and the ‘mass’ given by M = ( � τ 0 ) / 2 ln[ � / (Γ τ 0 )] as a function of the transverse field Γ . Feynman-Matsubara construction of functional integral over imaginary time. Overlap matrix -function : Q ab ( τ, τ ′ ) N − 1 � i s a i ( τ ) s b i ( τ ′ ) − 1 2 Q 2 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 ?
1st order phase transitions Quantum p -spin model 3.0 β =12 (a) 3 2.5 enter text here PM 2.0 χ m < 1 β =4 2 1.5 Γ 1.0 m=1 (b) β =12 1 0.5 * q EA T SG 0 0.0 0 1 2 3 0.0 0.2 0.4 0.6 T Γ 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 optimization 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 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 systems 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 x, p Syst. H = H syst + H env + H int { q a , π a } 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.
Reduced system Imagine the ensemble H = H syst + H env + H int 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 : � p 2 N N � + m α ω 2 � � α α q 2 H env + H int = + c α q α x α 2 m α 2 α =1 α =1 Either classical variables or quantum operators.
Reduced description Statics of a classical system The partition function of the coupled system is � e − β ( H − ηx ) Z [ η ] = osc, syst Integrating out the oscillator variables : „ c 2 aω 2 « P N H syst + H count + ηx − 1 x 2 a � − β a =1 2 ma Z red [ η ] = e syst Choosing H count to cancel the quadratic term in x 2 one recovers Z red [ η ] = Z syst [ η ] i.e., the partition function of the system of interest.
Reduced description Statics of a quantum system The density matrix of the coupled system is ρ ( x ′′ , q ′′ a ; x ′ , q ′ a ) = � x ′′ , q ′′ ρ | x ′ , q ′ a | ˆ a � � x ( � β )= x ′′ � q a ( � β )= q ′′ = 1 a D q a ( τ ) e − 1 � S e [ η ] D x ( τ ) Z x (0)= x ′ q a (0)= q ′ a One integrates the oscillator’s degrees of freedom to get the reduced density matrix � x ′′ R � β R τ “ ” 0 dτ ′ x ( τ ) K ( τ − τ ′ ) x ( τ ′ ) D x ( τ ) e − 1 S e syst − dτ ρ red ( x ′′ , x ′ ) = Z − 1 0 � red x ′ The counter -term was chosen to cancel a quadratic term in x 2 ( τ ) , Z red = Z / Z osc and Z osc the partition function of isolated ensemble of oscillators, but a non-local interaction in the imaginary time with kernel � ∞ ν 2 � ∞ 0 dω I ( ω ) 2 K ( τ ) = n + ω 2 exp( iν n τ ) remains. n n = −∞ ν 2 π � β ω
1st order phase transition 20 0.25 Dissipative Ising p -spin model with p ≥ 3 at T ≈ 0 � = 1 : 0 16 0.2 12 0.15 Magnetic susceptibility Averaged entropy density � = 0 : 5 8 0.1 � = 0 : 0 4 0.05 0.8 0.9 1 1.1 1.2 1.3 1.4 0.8 0.9 1 1.1 1.2 1.3 1.4 � � s � α = 0 , 0 . 5 , 1 LFC, Grempel & da Silva Santos 00 ; 1 1 LFC, Grempel, Lozano, Lozza, da Silva Santos 04.
Localization 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 .
Summary Statics 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 04 (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 kinetics Dynamics 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 upper critical lower � φ � f φ T Ginzburg-Landau free-energy Scalar order parameter
Evolution 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.
Evolution A quench or an annealing across a phase transition T c � φ � 0 t T Non -conserved order parameter � φ � ( t, T ) � = ct Development of magnetization in a ferromagnet after a quench.
The problem e.g. up & down spins in a 2 d Ising model (IM) 200 200 200 ’data’ ’data’ ’data’ 150 150 150 100 100 100 50 50 50 0 0 0 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 200 200 200 ’data’ ’data’ ’data’ 150 150 150 100 100 100 50 50 50 0 0 0 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 Question : starting from equilibrium at T 0 → ∞ or T 0 = T c how is equilibrium at T f = T c or T f < T c attained ?
Growth kinetics • At T f = T c the system needs to grow structures of all sizes. Critical coarsening. At T f < T c : 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 � N C ( r, t ) = 1 R ( t, g ) i,j =1 � δs i ( t ) δs j ( t ) � | � from r i − � r j | = r N • The relaxation time t r needed to reach ±|� φ � eq ( T ) | diverges with the size of the system, t r ( T, L ) → ∞ when L → ∞ for T ≤ T c .
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. � � r x j | = r ∼ m 2 C ( r, t ) ≡ � s i ( t ) s j ( t ) �| | � eq ( T ) f , x i − � R ( T, t ) � R ( T, t ) � C ( t, t w ) ≡ � s i ( t ) s i ( t w ) � ∼ m 2 eq ( T ) f c , R ( T, t w ) etc. when r ≫ ξ ( T ) , t, t w ≫ t 0 and C < m 2 eq ( 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 Review Bray, 1994. spherical ferromagnet. • Distribution of hull -enclosed areas in 2 d curvature driven coarsening.
Space-time correlation Magnetic model r R ( t, T ) ≃ λ ( T ) t 1 /z d a ≪ r ≪ L , Scaling regime R ( t,T ) , � � r � C ( r, t ) = N − 1 s i ( t ) s j ( t ) ≃ m 2 eq ( T ) f c R ( t, T ) ij/r ij = r 1 10 3 0.8 10 2 R2 10 1 0.6 C(r,t) 10 0 10 0 10 1 10 2 10 3 0.4 t 0.2 0 1 2 3 r/R(t)
Phase separation Spinodal decomposition in binary mixtures A spieces ≡ spin up ; B species ≡ spin down 2 d Ising model with Kawasaki dynamics at T locally conserved order parameter 50 : 50 composition ; Rounder boundaries
Dynamics in the 2 d XY model Schrielen pattern : gray scale according to sin 2 2 θ i ( 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/t 0 ( T )] } 1 / 2 Yurke et al 93, Bray & Rutenberg 94.
Growing lengths Universality classes λ ( T ) t 1 / 2 z d = 2 scalar NCOP λ ( T ) t 1 / 3 z d = 3 scalar COP R ( t, T ) ≃ � � 1 / 2 t λ ( T ) 2 -comp. vector NCOP in d = 2 ln t/t 0 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 disorder 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 R ( t, T ) ≃ λ ( T ) t 1 /z d At longer time scales domain -wall pinning by disorder becomes impor- tant. Assume that a length-dependent barrier B ( R ) ≃ Υ R ψ B ( R ) The Arrhenius time needed to go over such a barrier is t ≃ t 0 e kBT This implies � k B T � 1 /ψ R ( t, T ) ≃ ln t/t 0 Υ
Weak disorder Still two ferromagnetic states related by symmetry [ λ ( T ) t ] 1 /z d R ≪ L c ( T ) curvature -driven R ( t, T ) ≃ L c ( T )(ln t/t 0 ) 1 /ψ R ≫ L c ( T ) activated with L c ( T ) a growing function of T . t ≃ [ R /λ ( T )] z d e R /L c ( 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 : t ≃ R z d ( T ) z d ( T ) ≃ z d [1 + ct /L c ( 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 dynamics Two-time dependence t = 0 initial time t w waiting -time t measuring time. Correlation C ( t, t w ) = � [ ˆ O ( t ) , ˆ O ( t w )] + � Symmetrized correlator Linear response � R ( t, t w ) = δ � ˆ O ( t ) � � = � [ ˆ O ( t ) , ˆ � O ( t w )] − � � δh ( t w ) � h =0 Antisymmetrized correlator
Linear response To a kick and to a step r( ) t h � � r(tw) � � � � � � �� �� �� �� r( ) t h δ − δ 0 t t t w w + � � 2 2 � � r(0) The perturbation couples linearly to the observable H → H − hB ( { � r i } ) The linear instantaneous response of another observable A ( { � r i } ) is � δA ( { � � � r i } )( t ) � R AB ( t, t w ) ≡ � δh ( t w ) � h =0 The linear integrated response or dc susceptibility is � t dt ′ R AB ( t, t ′ ) χ AB ( t , t w ) ≡ t w
Formalism • Path-integral Schwinger-Keldysh formalism. T > t Closed-time path to allow for � ˆ O ( t w ) ˆ O ( t ) � with t > t w . • Connect system to reservoirs at time t = 0 , factorize density matrix : ρ (0) = ˆ ˆ ρ syst (0) ⊗ ˆ ρ env (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 = S syst + S bath − syst
Formalism Some 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 ) . For � → 0 the quantum x + ( t ) and x − ( t ) combine into x ( t ) and i ˆ bath kernels become the ones of a colored classical Langevin process.
Formalism The rôle played by the initial condition & disorder • Typical initial conditions : ˆ ρ syst (0) = I ‘random’ (quench from PM) no need of replica trick to average over disorder ! � In the classical limit Z [ η = 0] = D ξP [ ξ ] = 1 , independently of disorder de Dominicis 78. In the quantum model ˆ ρ (0) is independent of disorder. LFC & Lozano 98. We simply average Z or ˆ ρ . • Derive Schwinger -Dyson equations for correlations and responses with saddle-point in the large N limit or a variational approximation . • Solve these equations.
Formalism An example : rotors with pair intereactions � � 2 � � � n ia ( t )) 2 + . S syst = a d t ( ˙ J ij n ia ( t ) n ja ( t ) 2Γ a = ± i i<j � − 1 � � d t d t ′ Σ B ab ( t, t ′ ) n ia ( t ) n ib ( t ′ ) , S bath − syst = 2 ab = ± i � a � � λ ia ( t )( n 2 ia ( t ) − M ) S λ = d t 2 a = ± i with the bath induced kernels ab ( t, t ′ ) Σ B that take different forms for different baths, e.g. oscillators, leads, etc.
1 1 � = 0 : 5 � = 0 : 5 Real-time dynamics � = 1 : 0 � = 1 : 0 � = 2 : 0 � = 2 : 0 � = 3 : 0 � = 3 : 0 0.5 0.5 Paramagnetic phase 0 0 -0.5 -0.5 ) ) w w -1 -1 t t 0 2 4 6 8 10 0 2 4 6 8 10 ; ; w w t t t t + + ( t ( t R C Dependence on the quantum parameter Γ ( T = 0 , α fixed.) LFC & G. Lozano 98-99.
Real-time dynamics Dependence on the coupling to the bath 1.5 0.8 Symmetric correlation Linear response 1 0.4 0.5 0 0 ) ) w w -0.4 -0.5 t t 0 2.5 5 7.5 10 12.5 15 0 2.5 5 7.5 10 12.5 15 ; ; w w t t t t + + ( t ( t R C Comparison between α = 0 . 2 (PM) and α = 1 (SG) LFC, Grempel, Lozano, Lozza & da Silva Santos 02.
The effect of the bath Summary • 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.
1 Real-time dynamics Lo alized Interactions against localization Glassy 0.6 rit � < � 0.2 � = 0 : 02 J = 0 � = 4 : 0 J = 0 � = 4 : 0 J = 0 : 5 � = 4 : 0 J = 1 ) w -0.2 t 0 5 10 15 20 ; w t t + ( t C LFC, Grempel, Lozano, Lozza & da Silva Santos 02
Dynamic vs static phase diagram Quantum p -spin model 3 enter text here PM m < 1 2 Γ m=1 1 * T SG 0 0.0 0.2 0.4 0.6 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-dissipation Equilibrium spontaneous ( C ) and induced ( R ) fluctuations p ( { � r } ) , t w ) = p eq ( { � r } ) If • The dynamics is stationary, C → C ( t − t w ) and R → R ( t − t w ) . ∂C ( t − t w ) 1 • The fluctuation -dissipation th. R ( t − t w ) = − θ ( t − t w ) k B T ∂t holds and implies � t t w dt ′ R ( t, t ′ ) = 1 χ ( t − t w ) ≡ k B T [ C (0) − C ( t − t w )] . In glassy systems below T g : breakdown of stationarity & FDT . � t t w dt ′ R ( t, t ′ ) χ ( t, t w ) ≡ C ( t, t w ) not obviously related. and
Fluctuation-dissipation Solvable cases : p spin-models 1 . 0 1 k B T 0 . 8 Parametric construction 1 χ ( t, t w ) k B T ∗ 0 . 6 t w fixed t : t w → ∞ or t w 1 0 . 4 t w 2 dt : 0 → ∞ . t w 3 0 . 2 T T ∗ 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 C ( t, t w ) LFC & Kurchan 93.
Fluctuation-dissipation 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 ∞ ] lim χ ( t, t w ) = f χ ( C ) tw →∞ C ( t,t w )= C LFC & Kurchan 94. For weakly forced non-equilibrium systems in the limit of small work lim χ ( t, t w ) = f χ ( C ) ǫ → 0 C ( t,t w )= C
FDT in relaxing glasses Experiments and simulations 0.8 ��� ��� −k B TM 0.6 ��� ��� � ���� * ��� ��� 0.4 � � �� � �� � � � � �� ��� ��� 0.2 ��� ��� 0.0 ��� ��� 0.0 0.2 0.4 0.6 0.8 1.0 ��� ��� ��� ��� ��� ��� ��� C ���������� 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.
Phenomenology Times scales In all these systems the dynamics occur 1 • In quasi -equilibrium [ χ = k B T (1 − C ) ] when > t ∼ t w q ea ≤ C ≤ 1 . or • Clearly out of equilibrium when C ≤ q ea . t > t w or In structural glassy systems one finds 1 1 k B T ∗ ( q ea − C ) + k B T (1 − q ea ) χ = Interpretation • In particle systems, rattling within cages vs. structural relaxation. • In magnetic coarsening, thermal fluctuations within domains vs. domain wall motion.
FDT & effective temperatures Can one interpret the slope as a temperature ? 1 . 0 Thermometer 1 k B T (coordinate x) ’ x 0 . 8 1 χ ( t, t w ) k B T ∗ 0 . 6 Coupling constant k Observable A M copies of the system t w 1 0 . 4 t w 2 A A A A t w 3 0 . 2 . . . T T ∗ 0 . 0 α=1 α=2 α=3 α=Μ 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 C ( t, t w ) Thermal bath (temperature T) ’ (1) Measurement with a thermometer with • Short internal time scale τ 0 , fast dynamics is tested and T is recorded. • Long internal time scale τ 0 , slow dynamics is tested and T ∗ is recorded. (2) Partial equilibration (3) Direction of heat-flow LFC, Kurchan & Peliti 97.
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