Connected spatial correlation functions and the order parameter of the 3D Ising spin glass

Transcription

1 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 J. Stat. Mech. (2010) P06026 and arxiv: Statphys 24, Cairns, 22 July 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

2 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 Time (seconds) D. Yllanes (Univ. Complutense Madrid) Connected correlations in spin glasses Statphys 24 2 / 10

3 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 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

4 Our simulations L T min NMC min 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 = T c samples for L 24 and 1000 samples for L = 32. Parallel tempering with sample-dependent simulation times. A total of spin updates. D. Yllanes (Univ. Complutense Madrid) Connected correlations in spin glasses Statphys 24 3 / 10

5 The probability distribution of the order parameter P(q), T = 0.64T c L = 8 L = 12 L = 16 L = 24 L = 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

6 The probability distribution of the order parameter P(q), T = 0.64T c L = 8 L = 12 L = 16 L = 24 L = 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

7 The probability distribution of the order parameter P(q), T = 0.64T c L = 8 L = 12 L = 16 L = 24 L = 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

8 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

9 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

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

11 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

12 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

13 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

14 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

15 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

16 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

17 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

18 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

19 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

20 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 F q / L q 2 F q / L L = L = L = 16 L = 12 L = q 2 D. Yllanes (Univ. Complutense Madrid) Connected correlations in spin glasses Statphys 24 7 / 10

21 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

22 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

23 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

24 The computation of q EA (II) q (2) L,y 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

25 The computation of q EA (II) q (2) L,y 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

26 The computation of q EA (II) q (2) L,y 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

27 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

28 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 q EA L = 32 L = 24 L = 16 L = 12 L = 8 R q q L 1/νˆ(q q EA ) D. Yllanes (Univ. Complutense Madrid) Connected correlations in spin glasses Statphys 24 9 / 10

29 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 / 10

30 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 / 10

31 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 / 10

32 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 / 10

33 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 / 10

34 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 / 10