Critical properties of disordered XY model on sparse random graphs

Transcription

1 Manuale di Identità Visiva Sapienza Università di Roma Sapienza Università di Roma Scuola di Dottorato Vito Volterra Dipartimento di Fisica Critical properties of disordered XY model on sparse random graphs Cosimo Lupo PhD Thesis Defense February 6, 27 Supervisor: Prof. Federico Ricci -Tersenghi PhD in Physics XXIX cycle In collaboration also with Prof. Giorgio Parisi Cosimo Lupo Critical properties of disordered XY model on sparse random graphs February 6, 27 / 36

2 Preliminaries Preliminaries 2 XY and clock model 3 Random field XY model 4 Spin glass XY model in a field 5 Conclusions Cosimo Lupo Critical properties of disordered XY model on sparse random graphs February 6, 27 2 / 36

3 Preliminaries What a spin glass is It is a magnet in which spins are not ordered according to a regular pattern, due to the presence of some impurities randomly positioned, e. g. as it happens for diluted magnetic alloys (MnAu, MnCu,... ). Two key elements of spin glasses: randomness in the exchange couplings? + frustration along loops σ k σ j + σ i The spin glass Ising model is the best known theoretical model of a spin glass H[{σ i }] = J ij σ i σ j H i σ i, σ i = ± (i,j) i with several results on both mean field topologies (SK model with Parisi solution) and finite dimensional lattices (RG, NPRG, MC,... ). Cosimo Lupo Critical properties of disordered XY model on sparse random graphs February 6, 27 3 / 36

4 Preliminaries Why vector spin glasses? The spin glass Ising model can be easily generalized to the m-component case m H[{ σ i }] = J ij σ i Û({ω ij }) σ j Hi σ i, σi,µ 2 = (i,j) i µ= There are several reasons for which it is interesting: different critical properties, due to continuous symmetries different types of phase transition, e. g. Kosterlitz-Thouless glassy features in experiments (rejuvenation, memory,... ) reduced by strong spin anisotropy continuous variables allow to study low-energy excitations through the Hessian mappable into continuous optimization problems Cosimo Lupo Critical properties of disordered XY model on sparse random graphs February 6, 27 4 / 36

5 Preliminaries From complete graphs... Vector spin glasses can be solved on fully connected graphs via the Parisi ansatz, just as for the SK model: Sherrington, Kirkpatrick 75 de Almeida, Jones, Kosterlitz, Thouless 78 Gabay, Toulouse 8 Sharma, Young In fully connected models only one - point correlations are taken into account: σ i σ j = σ i σ j Too far from finite dimensional lattices! no heterogeneity infinite neighbours diverging critical field at T = no distance no proper correlations Cosimo Lupo Critical properties of disordered XY model on sparse random graphs February 6, 27 5 / 36

6 Preliminaries... to diluted graphs A more realistic picture is provided by sparse random graphs, such as Erdős-Rényi Graphs (ERGs) and Random Regular Graphs (RRGs). Analytical solution is still possible, thanks to the locally treelike structure of the graphs. Loops length of order log N, nearest neighbours almost uncorrelated when cutting the edge between them. Bethe Peierls approximation completely neglects these correlations: just one - point and two - point marginals have to be considered. Advantages: heterogeneity, local fluctuations finite degree, notion of distance proper correlation functions finite critical field at T = Cosimo Lupo Critical properties of disordered XY model on sparse random graphs February 6, 27 6 / 36

7 Preliminaries Vector spin glasses on Bethe lattice Several motivations for studying vector spin glasses but very few results on diluted graphs! Why? Because it is more difficult to deal with these models: continuous symmetries (soft modes, spin waves,... ) spins described by continuous variables prob. density functions instead of discrete prob. distributions P(σ i ) = + m i σ i 2 involved analytic computations demanding numerical simulations P( σ i ) function of m variables! Cosimo Lupo Critical properties of disordered XY model on sparse random graphs February 6, 27 7 / 36

8 XY and clock model Preliminaries 2 XY and clock model 3 Random field XY model 4 Spin glass XY model in a field 5 Conclusions Cosimo Lupo Critical properties of disordered XY model on sparse random graphs February 6, 27 8 / 36

9 XY and clock model The XY model We restrict to the m = 2 case XY model, just an angle θ i per spin: H[{θ i }] = (i,j) G J ij cos (θ i θ j ω ij ) H i cos (θ i φ i ), θ i [, 2π) i It is the simplest vector spin model, prototype for studying: granular superconductors superfluid Helium synchronization problem random lasers low-energy excitations Cosimo Lupo Critical properties of disordered XY model on sparse random graphs February 6, 27 9 / 36

10 XY and clock model BP equations for the XY model We exploit Belief Propagation approach equivalent to cavity method to solve the XY model on C = 3 RRGs. At Replica Symmetric stage, each directed edge brings a message, given by marginal η i j (θ i ). k 2 We can write iterative equations along the graph for these messages: k η k2 i k 3 η i j (θ i ) = Z i j e βh i cos (θ i φ i ) k i\j 2π called BP equations. dθ k e βj ik cos (θ i θ k ω ik ) η k i (θ k ) η k i η i j i j η k3 i Cosimo Lupo Critical properties of disordered XY model on sparse random graphs February 6, 27 / 36

11 XY and clock model RS solution in absence of field When H = BP equations can be solved analytically in the high T region via a Fourier expansion: { η i j (θ i ) = 2π + l= [ a (k i) l cos (lθ i ) + b (k i) ] } l sin (lθ i ) Paramagnetic solution is the flat measure over [, 2π) interval: η i j (θ i ) = 2π, {a l } = {b l } = l > We expect a second-order transition, so we expand up to the first-order. We get BP equations for these coefficients: a (i j) I (βj l = ik ) I k i\j (βj ik ) a(k i) l a = : a 2 = : P F transition P SG transition Cosimo Lupo Critical properties of disordered XY model on sparse random graphs February 6, 27 / 36

12 XY and clock model The Q-state clock model Below T c Fourier expansion does not work Numerical tools! We move to the clock model with Q states: { θ i [, 2π) θ i, Q 2π, 2 Q 2π,..., Q Q Probability distributions η i j (θ i ) s again over discrete sets Integrals in BP equations are replaced by sums } 2π Extreme cases: Q = 2: Ising Q = : XY What happens for Q finite but larger than 2? Cosimo Lupo Critical properties of disordered XY model on sparse random graphs February 6, 27 2 / 36

13 XY and clock model Fast convergence in Q for physical observables Physical observables converge very fast in Q toward the XY values! T > f (Q) (β) exp ( Q/Q ) T = f (Q) (β) exp [ (Q/Q ) b ] log f (Q) log f (Q) 3 5 f (Q) Q f (Q) Q Q Q Slower convergence also excluded by analytic arguments! Cosimo Lupo Critical properties of disordered XY model on sparse random graphs February 6, 27 3 / 36

14 XY and clock model Fast convergence in Q for critical lines Critical lines are detected via suitable order parameters: m = N i m i, q = N i m i 2, λ BP = lim t t log δη i j 2 (i j) T/J P SG p M F Q = 2 Q = 4 Q = 8 Q = 6 Q = 32 Phase diagram with purely bimodal couplings: P(J ik ) = p δ(j ik J) + ( p) δ(j ik + J) Main features: exponential convergence of critical lines presence of RSB mixed phase RSB at T = as soon as p < Cosimo Lupo Critical properties of disordered XY model on sparse random graphs February 6, 27 4 / 36

15 Random field XY model Preliminaries 2 XY and clock model 3 Random field XY model 4 Spin glass XY model in a field 5 Conclusions Cosimo Lupo Critical properties of disordered XY model on sparse random graphs February 6, 27 5 / 36

16 Random field XY model Inserting a random field We change our point of view: all J ij s equal to +J Hi randomly oriented on each site We get the Random Field XY Model (RFXYM): H[{θ i }] = J (i,j) G cos (θ i θ j ) H cos (θ i φ i ), φ i Unif(, 2π) i We saw the spin glass XY model is more glassy than spin glass Ising model. Is it true also for the RFXYM? Cosimo Lupo Critical properties of disordered XY model on sparse random graphs February 6, 27 6 / 36

17 Random field XY model Negative correlations? Random Field Ising Model (RFIM) does never show a RSB phase! (Krzakala, Ricci-Tersenghi, Zdeborovà ) Connected correlations are always non negative if J ij > : σ i σ j c For continuous spins it is no longer true! Continuous nature of variables can produce negative correlations: Hi σ j δ σ i δ σ j < Above argument does not apply, possible SG phase in the RFXYM. σ i J ij Hj Cosimo Lupo Critical properties of disordered XY model on sparse random graphs February 6, 27 7 / 36

18 Random field XY model Computing critical lines At H = stability has to be computed numerically. We perturb the fixed point {ηi j } of BP: η i j (θ i ) = ηi j (θ i) + δη i j (θ i ), dθ i δη i j (θ i ) = Perturbations evolve according to linearized BP equations, and we measure the growth rate of their global norm: λ (t) BP log δη(t) δη (t ) In the t limit we get the stability parameter: λ BP < : stability λ BP > : instability Cosimo Lupo Critical properties of disordered XY model on sparse random graphs February 6, 27 8 / 36

19 Random field XY model RFXYM breaks Replica Symmetry! λbp T= T=.4 T=.8 T=.2 By using Population Dynamics we always reach a fixed point of cavity message distribution P[η i j ]. RFXYM shows unexpected features: H/J H..5 Hc(T = ) P SG.95.9 M.85 F.8 mc point Hc(T = ) Instability toward RSB solution for very low temperatures close to the P-F critical line Both spin glass and mixed phases F phase marginally stable (weak coupling between m and H) T/J Cosimo Lupo Critical properties of disordered XY model on sparse random graphs February 6, 27 9 / 36

20 Spin glass XY model in a field Preliminaries 2 XY and clock model 3 Random field XY model 4 Spin glass XY model in a field 5 Conclusions Cosimo Lupo Critical properties of disordered XY model on sparse random graphs February 6, 27 2 / 36

21 Spin glass XY model in a field RS instability in a field Spin glass Ising model has a unique RS instability line in a field: dat line. What happens for the XY model? More degrees of freedom, more instability directions more critical lines! Hi = H i e iφ i 5 Randomness in direction is essential: HGT 4.82 H i = H, φ i P(φ i ) 4 Homogeneous field: GT line 3 GT line H P(φ i ) = δ(φ i ) H (δt c ) /2 2 HdAT.6 Randomly oriented field: dat line P(φ i ) = 2π H (δt c ) 3/2 dat line Tc T Cosimo Lupo Critical properties of disordered XY model on sparse random graphs February 6, 27 2 / 36

22 Spin glass XY model in a field Nature of GT and dat instabilities dat line GT line T =.3, H =.9 cos ϑi T =.2, H = T =., H = mi Critical properties of disordered XY model on sparse random graphs.8 T =., H = cos ϑi.8 T =.2, H = cos ϑi.4 T =.3, H = Cosimo Lupo.9 T =.4, H = Breaking of transverse spin symmetry Transverse perturbations, cos ϑi '.4.5 dat line Longitudinal perturbations, cos ϑi ' {, +}.45 T =.4, H =.7 - No breaking of spin symmetries GT line cos ϑi What is the physical interpretation of these instabilities? BP tells us it! δ~mi ~ Hi cos ϑi δmi H mi February 6, / 36

23 Spin glass XY model in a field Toward the T = axis We choose a random field only dat line survives! H[{θ i }] = (i,j) G J ij cos (θ i θ j ) H cos (θ i φ i ), φ i Unif(, 2π) i T = limit of BP equations: η i j (θ i ) exp { βh i j (θ i ) } h i j (θ i ) [ ] = H cos (θ i φ i ) + max J ik cos (θ i θ k ) + h k i (θ k ) θ k i\j k Linearization of BP equations gives the RS stability: δh i j (θ i ) = k i\j [ ] δh k i (θk (θ i)), θk (θ i) = argmax J ik cos (θ i θ k ) + h k i (θ k ) θ k At T = it is crucial the interpolation over the real-valued angles! Cosimo Lupo Critical properties of disordered XY model on sparse random graphs February 6, / 36

24 Spin glass XY model in a field How to reach ground state? BP always reaches the actual ground state {θi } in the RS region: BP on the Q-state clock model gets very close to the minimum Greedy descent (T = MC) on real-valued angles reaches the bottom Below the critical line (RSB region) BP relaxes to metastable minima. MC would always stop on metastable minima: BP is better than MC! The T = limit allows to reach the inherent structures: characterization of the free energy landscape Hessian matrix, low-energy excitations H ij 2 F = 2 H [N N matrix] θ i θ j θ i θ j connection with RSB? T=,{θ i } {θ i } Cosimo Lupo Critical properties of disordered XY model on sparse random graphs February 6, / 36

25 Spin glass XY model in a field Spectral density of Hessian Eigenvalues of H: low-energy excitations. Corresponding eigenvectors: flat directions. ρ(λ) = N N δ(λ λ i ) i= How to compute ρ(λ) for sparse matrices? Direct diagonalization ( N 3 ) High H expansion ( N 3 ) Arnoldi method ( N 2 ) Resolvent approach ( N) Gap closes well above H dat : H gap 4.72 H dat.6 ρ(λ) ρ(λ) ρ(λ) ρ(λ) H = 25 H = H = 5. H = H = 2. H = H =.8 H = λ λ Cosimo Lupo Critical properties of disordered XY model on sparse random graphs February 6, / 36

26 Spin glass XY model in a field Density of soft modes In the gapless region ρ(λ) goes as: ρ(λ) λ α Numerical computation gives: α 3 ( α FC = ) 2 2 χ SG does not diverge, null modes of Hessian do not imply RSB! log l i log li H = N = 3 N = N = 5 N = H = 4. H = 3. H = 2. H =. N = log (i/n ) There is an argument (Gurarie, Chalker 3) for soft excitations in random media: ρ(ω) ω 4 ρ(λ) λ 3/2 computed on local minima of a random potential. Cosimo Lupo Critical properties of disordered XY model on sparse random graphs February 6, / 36

27 Spin glass XY model in a field What causes RSB? Is χ SG then linked to the delocalization of soft modes? Eigenvector components have a power-law decay even for large H Standard IPR analysis is inconclusive (strong heterogeneity, Υ ) It would not seem so. Υ min N = 3 N = 4 N = 5 N = Another scenario is the competition between a large number of minima, inside which there are rather localized flat modes. Υ.3.2. N = 3 N = 4 N = 5 N = H Connection between Hessian and RS instability has to be investigated deeper! Cosimo Lupo Critical properties of disordered XY model on sparse random graphs February 6, / 36

28 Conclusions Preliminaries 2 XY and clock model 3 Random field XY model 4 Spin glass XY model in a field 5 Conclusions Cosimo Lupo Critical properties of disordered XY model on sparse random graphs February 6, / 36

29 Conclusions Conclusions and perspectives Q-states clock model is a reliable and efficient proxy for XY model XY is more glassy than Ising: closer to structural glasses and to experimental results? RSB behaviour of RFXYM Heterogeneous long-range correlations, different from the Ising case Low-energy excitations even far from critical point Hessian seems not to be easily connected with RS instability Cosimo Lupo Critical properties of disordered XY model on sparse random graphs February 6, / 36

30 Conclusions Publications C. Lupo, F. Ricci-Tersenghi: Approximating the XY model on a random graph with a q-state clock model, arxiv 6.24 (26), to appear in Phys. Rev. B C. Lupo, F. Ricci-Tersenghi: Comparison of Gabay-Toulouse and de Almeida-Thouless instabilities for the spin glass XY model in a field on sparse random graphs, in preparation C. Lupo, F. Ricci-Tersenghi: The random field XY model shows a replica symmetry broken phase, in preparation C. Lupo, G. Parisi, F. Ricci-Tersenghi: Exact computation of soft modes in inherent structures for the disordered XY model, in preparation Cosimo Lupo Critical properties of disordered XY model on sparse random graphs February 6, 27 3 / 36

31 Backup slides Backup slides Cosimo Lupo Critical properties of disordered XY model on sparse random graphs February 6, 27 3 / 36

32 Backup slides The gauge glass XY model Good model for granular superconductors: H[{θ}] = J (i,j) G cos (θ i θ j ω ij ).4.2 P Angular shifts randomly distributed: P ω (ω ij ) = ( ) δ(ω ij ) + Unif(, 2π) T/J.8.6 Q = 2 Q = 3.4 Q = 4 F Q = 8.2 Q = 6 SG Q = 32 M Main features: odd values of Q allowed again RSB at T = with small disorder different critical line F M Cosimo Lupo Critical properties of disordered XY model on sparse random graphs February 6, / 36

33 Backup slides RSB BP algorithm We try to go beyond RS RSB cavity method. RSB BP equations basis on a two-level hierarchy of populations: where: ( P i j [η i j ] = k i\j ) [ ) x Dη k i P k i [η k i ] δ η i j F[{η k i }] ](Z i j ( ) [ ] P[P i j ] = DP k i P[P k i ] δ P i j G[{P k i }] k i\j N distr marginals η, described by functional distribution P[η] N pop populations of marginals, described by functional distribution P[P] Algorithmic complexity drastically increases! Cosimo Lupo Critical properties of disordered XY model on sparse random graphs February 6, / 36

34 Backup slides Universality class of the Q-state clock model RSB solution is a proxy for the exact solution (frsb) of the Q-state clock model and the XY model. Universality class is approximated by RSB parameters at T T c : Σ(x ) = x, q (x ), q (x ) Complexity Σ(x) very small for systems with continuous RSB transition. We use T = T c /2: Q = 2, 4: same class of Ising Q 5: same class of XY Q = 3: same class of 3-state Potts Cosimo Lupo Critical properties of disordered XY model on sparse random graphs February 6, / 36

35 Backup slides Growth of perturbations at T = t=2 t=4 t=6 t=8 t= t=4 t=8 t=22 t=3 t=4 t=5 H= t=4 t=8 t=2 t=6 t=2 t=3 t=5 t=7 t=9 t= t=3 H= log (var) t=4 t=8 t=2 t=6 t=2 t=3 t=5 t=7 t=9 t= t=3 H= log (var) log (var) Perturbations are very heterogeneous! The variances of δη i j s span several orders of magnitude. λ BP just measures how these distributions move. Cosimo Lupo Critical properties of disordered XY model on sparse random graphs February 6, / 36

36 Backup slides Correlation functions Connected correlation functions computed along a chain of length r: C(r) µ(θ, θ r ) µ(θ )µ(θ r ) 5 dat instability at T =. L 2 norm of correlation functions in the RFXY model at T=. [Q=64, N=.e+5] H c = H =.8-5 H =. H =.2-2 H =.4 H =.6-25 H =.8 H = 2. H = H = 3. H = r d Cosimo Lupo Critical properties of disordered XY model on sparse random graphs February 6, / 36