Connected spatial correlation functions and the order parameter of the 3D Ising spin glass David Yllanes for the Janus Collaboration 1 Dep. Física Teórica, Universidad Complutense de Madrid http://teorica.fis.ucm.es/grupos/grupo-tec.html J. Stat. Mech. (2010) P06026 and arxiv:1003.2943 Statphys 24, Cairns, 22 July 2010 1 R. Alvarez Baños, A. Cruz, L.A. Fernandez, J.M. Gil-Narvion, A. Gordillo-Guerrero, M. Guidetti, A. Maiorano, F. Mantovani, E. Marinari, V. Martin-Mayor, J. Monforte-Garcia, A. Muñoz Sudupe, D. Navarro, G. Parisi, S. Perez-Gaviro, J.J. Ruiz-Lorenzo, S.F. Schifano, B. Seoane, A. Tarancon, R. Tripiccione and D. Yllanes D. Yllanes (Univ. Complutense Madrid) Connected correlations in spin glasses Statphys 24 1 / 10
Introduction Spin glasses: experiments vs. simulations Spin glasses are characterised by very sluggish dynamics. Off-equilibrium large gap between simulations and experiments. Equilibrium At low T, PCs can only thermalise very small lattices. Conventional computers Experimental times 10-6 10-5 10-4 10-3 10-2 10-1 10 0 10 1 10 2 10 3 Time (seconds) D. Yllanes (Univ. Complutense Madrid) Connected correlations in spin glasses Statphys 24 2 / 10
Introduction Spin glasses: experiments vs. simulations Spin glasses are characterised by very sluggish dynamics. Off-equilibrium large gap between simulations and experiments. Equilibrium At low T, PCs can only thermalise very small lattices. Conventional computers JANUS Experimental times 10-6 10-5 10-4 10-3 10-2 10-1 10 0 10 1 10 2 10 3 Time (seconds) Janus bridges the gap. The Janus computer Janus is a custom-built computing system based on FPGAs: Massively parallel Reconfigurable Modular D. Yllanes (Univ. Complutense Madrid) Connected correlations in spin glasses Statphys 24 2 / 10
Our simulations L T min NMC min 8 0.150 5 10 6 8.30 10 8 12 0.414 1 10 7 1.53 10 10 16 0.479 4 10 8 2.79 10 11 24 0.625 1 10 9 1.81 10 12 32 0.703 4 10 9 7.68 10 11 N max MC The model and our parameters H = x,y J xyσ x σ y, P(J xy ) = δ(jxy 2 1). We thermalise L = 32 down to T = 0.703 0.64T c. 4000 samples for L 24 and 1000 samples for L = 32. Parallel tempering with sample-dependent simulation times. A total of 1.1 10 20 spin updates. D. Yllanes (Univ. Complutense Madrid) Connected correlations in spin glasses Statphys 24 3 / 10
The probability distribution of the order parameter P(q), T = 0.64T c 2.5 2 1.5 1 0.5 0.22 0.19 0.16 0.1 0 0.1 L = 8 L = 12 L = 16 L = 24 L = 32 0 1 0.5 0 0.5 1 q Our order parameter is the overlap q q = 1 q x = 1 σ (1) x σ (2) x V V x Its pdf P(q) has peaks at ±q EA. x D. Yllanes (Univ. Complutense Madrid) Connected correlations in spin glasses Statphys 24 4 / 10
The probability distribution of the order parameter P(q), T = 0.64T c 2.5 2 1.5 1 0.5 0.22 0.19 0.16 0.1 0 0.1 L = 8 L = 12 L = 16 L = 24 L = 32 0 1 0.5 0 0.5 1 q Our order parameter is the overlap q q = 1 q x = 1 σ (1) x σ (2) x V V x Its pdf P(q) has peaks at ±q EA. There are several conflicting theoretical pictures for P(q) in the thermodynamical limit: Droplet P(q) = δ(q 2 EA 1). RSB Non-zero probability density in q < q EA. x D. Yllanes (Univ. Complutense Madrid) Connected correlations in spin glasses Statphys 24 4 / 10
The probability distribution of the order parameter P(q), T = 0.64T c 2.5 2 1.5 1 0.5 0.22 0.19 0.16 0.1 0 0.1 L = 8 L = 12 L = 16 L = 24 L = 32 0 1 0.5 0 0.5 1 q Our order parameter is the overlap q q = 1 q x = 1 σ (1) x σ (2) x V V x Its pdf P(q) has peaks at ±q EA. There are several conflicting theoretical pictures for P(q) in the thermodynamical limit: Droplet P(q) = δ(q 2 EA 1). RSB Non-zero probability density in q < q EA. q EA very difficult to compute. x D. Yllanes (Univ. Complutense Madrid) Connected correlations in spin glasses Statphys 24 4 / 10
Clustering states and fixed-q correlation functions In the RSB picture, the spin-glass phase is composed of a multiplicity of clustering states. D. Yllanes (Univ. Complutense Madrid) Connected correlations in spin glasses Statphys 24 5 / 10
Clustering states and fixed-q correlation functions In the RSB picture, the spin-glass phase is composed of a multiplicity of clustering states. We isolate clustering states by considering correlations at fixed q = c C 4 (r c) = x q xq x+r δ(q c) δ(q c) D. Yllanes (Univ. Complutense Madrid) Connected correlations in spin glasses Statphys 24 5 / 10
Clustering states and fixed-q correlation functions In the RSB picture, the spin-glass phase is composed of a multiplicity of clustering states. We isolate clustering states by considering correlations at fixed q = c x q xq x+r δ(q c) δ(q c) Gaussian C 4(r c) = x q xq x+r exp[ V (q c) 2 /2] convolution V exp[ V (q c) 2 /2] We smooth the comb-like P(q) with a Gaussian convolution. D. Yllanes (Univ. Complutense Madrid) Connected correlations in spin glasses Statphys 24 5 / 10
Clustering states and fixed-q correlation functions In the RSB picture, the spin-glass phase is composed of a multiplicity of clustering states. We isolate clustering states by considering correlations at fixed q = c x q xq x+r δ(q c) δ(q c) Gaussian C 4(r c) = x q xq x+r exp[ V (q c) 2 /2] convolution V exp[ V (q c) 2 /2] We smooth the comb-like P(q) with a Gaussian convolution. For T < T c and q q EA one expects C 4 (r q) q 2 + A q r θ(q) D. Yllanes (Univ. Complutense Madrid) Connected correlations in spin glasses Statphys 24 5 / 10
Clustering states and fixed-q correlation functions In the RSB picture, the spin-glass phase is composed of a multiplicity of clustering states. We isolate clustering states by considering correlations at fixed q = c x q xq x+r δ(q c) δ(q c) Gaussian C 4(r c) = x q xq x+r exp[ V (q c) 2 /2] convolution V exp[ V (q c) 2 /2] We smooth the comb-like P(q) with a Gaussian convolution. For T < T c and q q EA one expects C 4 (r q) q 2 + A q r θ(q) In real space, one has to perform a subtraction that complicates the analysis. D. Yllanes (Univ. Complutense Madrid) Connected correlations in spin glasses Statphys 24 5 / 10
Correlations in Fourier space We consider instead the Fourier transform of C 4 (r q) Ĉ 4 (k q 2 < q 2 EA) k θ(q) D +... Ĉ 4 (k q 2 > q 2 EA) 1 k 2 + ξ 2 q D. Yllanes (Univ. Complutense Madrid) Connected correlations in spin glasses Statphys 24 6 / 10
Correlations in Fourier space We consider instead the Fourier transform of C 4 (r q) Ĉ 4 (k q 2 < q 2 EA) k θ(q) D +... We use the shorthand F q = Ĉ4(k min q) Ĉ 4 (k q 2 > q 2 EA) F (n) q = Ĉ4(nk min q) 1 k 2 + ξ 2 q D. Yllanes (Univ. Complutense Madrid) Connected correlations in spin glasses Statphys 24 6 / 10
Correlations in Fourier space We consider instead the Fourier transform of C 4 (r q) Ĉ 4 (k q 2 < q 2 EA) k θ(q) D +... We use the shorthand F q = Ĉ4(k min q) Ĉ 4 (k q 2 > q 2 EA) F (n) q = Ĉ4(nk min q) 1 k 2 + ξ 2 q Droplet and RSB disagree in the precise form of θ(q) D. Yllanes (Univ. Complutense Madrid) Connected correlations in spin glasses Statphys 24 6 / 10
Correlations in Fourier space We consider instead the Fourier transform of C 4 (r q) Ĉ 4 (k q 2 < q 2 EA) k θ(q) D +... We use the shorthand F q = Ĉ4(k min q) Ĉ 4 (k q 2 > q 2 EA) F (n) q = Ĉ4(nk min q) 1 k 2 + ξ 2 q Droplet and RSB disagree in the precise form of θ(q) Yet, both theories agree that a crossover appears in F q for finite L F q L D θ(q) for q < q EA F q 1 for q > q EA D. Yllanes (Univ. Complutense Madrid) Connected correlations in spin glasses Statphys 24 6 / 10
Correlations in Fourier space We consider instead the Fourier transform of C 4 (r q) Ĉ 4 (k q 2 < q 2 EA) k θ(q) D +... We use the shorthand F q = Ĉ4(k min q) Ĉ 4 (k q 2 > q 2 EA) F (n) q = Ĉ4(nk min q) 1 k 2 + ξ 2 q Droplet and RSB disagree in the precise form of θ(q) Yet, both theories agree that a crossover appears in F q for finite L F q L D θ(q) for q < q EA F q 1 for q > q EA For large L the crossover becomes a phase transition where ξ L= q (q 2 q 2 EA) ˆν (analogous to the study of the equation of state in Heisenberg ferromagnets). D. Yllanes (Univ. Complutense Madrid) Connected correlations in spin glasses Statphys 24 6 / 10
Correlations in Fourier space We consider instead the Fourier transform of C 4 (r q) Ĉ 4 (k q 2 < q 2 EA) k θ(q) D +... We use the shorthand F q = Ĉ4(k min q) Ĉ 4 (k q 2 > q 2 EA) F (n) q = Ĉ4(nk min q) 1 k 2 + ξ 2 q Droplet and RSB disagree in the precise form of θ(q) Yet, both theories agree that a crossover appears in F q for finite L F q L D θ(q) for q < q EA F q 1 for q > q EA For large L the crossover becomes a phase transition where ξ L= q (q 2 q 2 EA) ˆν (analogous to the study of the equation of state in Heisenberg ferromagnets). From very general RG arguments we can derive a scaling law: θ(q EA ) = 2/ˆν. D. Yllanes (Univ. Complutense Madrid) Connected correlations in spin glasses Statphys 24 6 / 10
The computation of q EA (I) According to Finite-Size Scaling, F q L D θ(q EA) G ( L 1/ˆν (q q EA ) ) D. Yllanes (Univ. Complutense Madrid) Connected correlations in spin glasses Statphys 24 7 / 10
The computation of q EA (I) According to Finite-Size Scaling, F q L D θ(q EA) G ( L 1/ˆν (q q EA ) ) We consider F q /L y, with y < D θ(0). In the large-l limit, these quantities diverge for q < q EA but vanish for q > q EA. 0.13 0.20 0.10 0.60 0.11 0.15 0.08 F q / L 2 0.40 0.09 0.20 0.4 0.45 0.00 0 0.2 0.4 0.6 0.8 1 q 2 F q / L 2.35 0.10 0.06 L = 32 0.05 L = 24 0.25 0.3 L = 16 L = 12 L = 8 0.00 0 0.2 0.4 0.6 0.8 1 q 2 D. Yllanes (Univ. Complutense Madrid) Connected correlations in spin glasses Statphys 24 7 / 10
The computation of q EA (II) We consider pairs (L, 2L). The crossing points q L,y scale as q L,y = q EA + A y L 1/ˆν (**) D. Yllanes (Univ. Complutense Madrid) Connected correlations in spin glasses Statphys 24 8 / 10
The computation of q EA (II) We consider pairs (L, 2L). The crossing points q L,y scale as q L,y = q EA + A y L 1/ˆν (**) We can use this formula to compute the value of q EA. D. Yllanes (Univ. Complutense Madrid) Connected correlations in spin glasses Statphys 24 8 / 10
The computation of q EA (II) We consider pairs (L, 2L). The crossing points q L,y scale as q L,y = q EA + A y L 1/ˆν (**) We can use this formula to compute the value of q EA. For a given y, we have only three crossings: (8,16), (12,24) (16,32). D. Yllanes (Univ. Complutense Madrid) Connected correlations in spin glasses Statphys 24 8 / 10
The computation of q EA (II) q (2) L,y 0.8 0.7 0.6 0.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 0 0.1 0.2 0.3 0.4 L 1/νˆ We consider pairs (L, 2L). The crossing points q L,y scale as q L,y = q EA + A y L 1/ˆν (**) We can use this formula to compute the value of q EA. For a given y, we have only three crossings: (8,16), (12,24) (16,32). We perform a joint fit to (**) for several values of y. D. Yllanes (Univ. Complutense Madrid) Connected correlations in spin glasses Statphys 24 8 / 10
The computation of q EA (II) q (2) L,y 0.8 0.7 0.6 0.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 0 0.1 0.2 0.3 0.4 L 1/νˆ We consider pairs (L, 2L). The crossing points q L,y scale as q L,y = q EA + A y L 1/ˆν (**) We can use this formula to compute the value of q EA. For a given y, we have only three crossings: (8,16), (12,24) (16,32). We perform a joint fit to (**) for several values of y. Obviously, the intersections for different y are correlated, but we can control this by computing their covariance matrix. D. Yllanes (Univ. Complutense Madrid) Connected correlations in spin glasses Statphys 24 8 / 10
The computation of q EA (II) q (2) L,y 0.8 0.7 0.6 0.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 0 0.1 0.2 0.3 0.4 L 1/νˆ q EA = 0.52(3), 1/ˆν = 0.39(5) We consider pairs (L, 2L). The crossing points q L,y scale as q L,y = q EA + A y L 1/ˆν (**) We can use this formula to compute the value of q EA. For a given y, we have only three crossings: (8,16), (12,24) (16,32). We perform a joint fit to (**) for several values of y. Obviously, the intersections for different y are correlated, but we can control this by computing their covariance matrix. D. Yllanes (Univ. Complutense Madrid) Connected correlations in spin glasses Statphys 24 8 / 10
The computation of q EA (III) From the graphs of P(q, L) we can extrapolate, using our ˆν, q EA = q EA (L) + AL 1/ˆν = q EA = 0.54(3) compatible with our result of q EA = 0.52(3). D. Yllanes (Univ. Complutense Madrid) Connected correlations in spin glasses Statphys 24 9 / 10
The computation of q EA (III) From the graphs of P(q, L) we can extrapolate, using our ˆν, q EA = q EA (L) + AL 1/ˆν = q EA = 0.54(3) compatible with our result of q EA = 0.52(3). From the dimensionless ratio F q /F (2) q we can obtain scaling plots: R q 2 1 2 q EA L = 32 L = 24 L = 16 L = 12 L = 8 R q 1.5 0.5 0 0.2 0.4 0.6 q 2 0.5 0 0.5 1.0 L 1/νˆ(q q EA ) D. Yllanes (Univ. Complutense Madrid) Connected correlations in spin glasses Statphys 24 9 / 10
Conclusions We have studied the connected spatial correlation functions in the low-temperature phase of the 3D Edwards-Anderson-Ising spin glass. D. Yllanes (Univ. Complutense Madrid) Connected correlations in spin glasses Statphys 24 10 / 10
Conclusions We have studied the connected spatial correlation functions in the low-temperature phase of the 3D Edwards-Anderson-Ising spin glass. At fixed temperature, we identify a phase transition at q = q EA. D. Yllanes (Univ. Complutense Madrid) Connected correlations in spin glasses Statphys 24 10 / 10
Conclusions We have studied the connected spatial correlation functions in the low-temperature phase of the 3D Edwards-Anderson-Ising spin glass. At fixed temperature, we identify a phase transition at q = q EA. We use FSS arguments to provide the first reliable determination of q EA and of the exponent 1/ˆν that rules finite-size effects. D. Yllanes (Univ. Complutense Madrid) Connected correlations in spin glasses Statphys 24 10 / 10
Conclusions We have studied the connected spatial correlation functions in the low-temperature phase of the 3D Edwards-Anderson-Ising spin glass. At fixed temperature, we identify a phase transition at q = q EA. We use FSS arguments to provide the first reliable determination of q EA and of the exponent 1/ˆν that rules finite-size effects. Our starting hypotheses are agreed upon by all (otherwise incompatible) theories for the spin-glass phase. D. Yllanes (Univ. Complutense Madrid) Connected correlations in spin glasses Statphys 24 10 / 10
Conclusions We have studied the connected spatial correlation functions in the low-temperature phase of the 3D Edwards-Anderson-Ising spin glass. At fixed temperature, we identify a phase transition at q = q EA. We use FSS arguments to provide the first reliable determination of q EA and of the exponent 1/ˆν that rules finite-size effects. Our starting hypotheses are agreed upon by all (otherwise incompatible) theories for the spin-glass phase. There is a scaling law θ(q EA ) = 2/ˆν. D. Yllanes (Univ. Complutense Madrid) Connected correlations in spin glasses Statphys 24 10 / 10
Conclusions We have studied the connected spatial correlation functions in the low-temperature phase of the 3D Edwards-Anderson-Ising spin glass. At fixed temperature, we identify a phase transition at q = q EA. We use FSS arguments to provide the first reliable determination of q EA and of the exponent 1/ˆν that rules finite-size effects. Our starting hypotheses are agreed upon by all (otherwise incompatible) theories for the spin-glass phase. There is a scaling law θ(q EA ) = 2/ˆν. In the RSB setting, we can formulate the conjecture (compatible with our numerical results) θ(0) = 1/ˆν D. Yllanes (Univ. Complutense Madrid) Connected correlations in spin glasses Statphys 24 10 / 10