Is there an Almeida-Thouless line in Spin Glasses?
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1 p.1 Is there an Almeida-Thouless line in Spin Glasses? Peter Young Work supported by the Collaborator: H. G. Katzgraber Talk at the APS March Meeting, New Orleans, March 11, 2008 This talk can be downloaded at peter/talks/aps2008.pdf
2 What is a spin glass? A system with disorder and frustration. or Most theory uses the simplest model with these ingredients: the Edwards-Anderson Model: H = i,j J ij S i S j i h i S i, [J ij ] av = 0, [J 2 ij ] av = J 2. S i = ±1, (the Ising spin glass) Take a Gaussian distribution for the J ij. The h i are fields (either uniform or random; see later). p.2
3 p.3 SK Model Motivated by pioneering work of Edwards and Anderson (1975), David Sherrington and Scott Kirkpatrick proposed the Sherrington-Kirkpatrick Model, an infinite-range spin glass model. They proposed that the exact solution be the mean field theory of spin glasses. It was hoped that the solution would be: simple a reasonable approximation to real spin glasses To what extent are these hopes realized?
4 Is the SK model simple? The SK solution (replica-symmetric solution) is simple. However SK realized that it cannot be correct at low temperature (negative entropy). Situation clarified by Almeida and Thouless (1978). In a magnetic field, the replica-symmetric solution is unstable below a line in the H-T plane (the AT line). H 0 AT line RSB solution (Parisi) (complicated) RS solution (simple) T c T Below the AT line the exact solution, which is complicated, was found in a tour-de-force by Parisi (replica symmetry breaking). The Parisi solution was subsequently found to be: stable (de Dominicis and Kondor) exact (Tallegrand) p.4
5 Parisi Solution The Parisi (RSB) solution has two important features: There is a transition in a field (AT line), as we have seen. The order parameter is not just a single number but a whole function. Defining the spin overlap by q = 1 N S (1) i S (2) i [(1)and (2) refer to 2 copies] N i=1 then q has a probability distribution P(q) (Parisi) which, for zero field, looks like this: P(q) Above the AT line P(q) is just a single delta function (the RS solution). 1 1 q Do these features occur in real spin glasses, or are they artefacts of the infinite-range SK model? p.5
6 p.6 Is the SK model relevant? Equilibrium state below T SG. Two main scenarios: Replica Symmetry Breaking (RSB), (Parisi). Droplet picture (DP) (Fisher and Huse, also Bray and Moore, and McMillan). Assume short-range is similar to the infinite-range SK model. P(q) is non-trivial. There is an AT line. Focus on the geometrical aspects of the low-energy excitations. P(q) is trivial (for L ). There is no AT line.
7 p.7 Models We perform Monte Carlo simulations on the following models: Nearest-neighbor model in d = 3. No. of spins is N = L 3, where L is not very big (even with parallel tempering to help speed things up.) Would also like to do higher-d. But there the range of L is even smaller. Hence we also study... Long-range model in d = 1. J ij ǫ ij, [ǫ rij σ ij ] av = 0, [ǫ 2 ij ] av = 1. Can have a finite T SG for a range of σ (next slide). When we apply a field we use a Gaussian distribution of random fields, [h 2 i ] = H2 r, rather than a uniform field. (Useful equilibration test.) For the SK model, there is an AT line for Gaussian random fields, as for a uniform field. (Note: the sign of the field can be gauged away.)
8 Long-Range model in d = 1 J ij ǫ ij, [ǫ rij σ ij ] av = 0, [ǫ 2 ij ] av = 1. For the short-range model: there is a lower critical dimension d l (equal to about 2.5, and an upper critical dimension d l (equal to 6). Make an analogy between varying d for short-range, and varying σ for long-range d = 1. (Works for ferromagnets, e.g. Fisher, Ma and Nickel.) In otherwords there is a σ l above which T SG = 0, and a σ u below which the zero-field critical behavior is mean-field like, where σ l = 1, σ u = 2/3, (Fisher and Huse) (Kotliar, Stein and Anderson) Note too that σ = 0 is SK model The range 0 σ 1/2 is infinite-range since j [J2 ij ] av diverges. σ is the nearest-neighbor model. p.8
9 p.9 Dependence on σ The following figure summarizes the behavior of the d = 1 long-range model as a function of σ. SK MF Non MF /2 2/3 1 σ Infinite range Finite T T = 0 SG SG The interesting range is 1/2 < σ < 1. For each σ we rescale the interactions so that T MF SG = 1 (for zero field).
10 p.10 Parallel Tempering Problem: Very slow Monte Carlo dynamics at low-t ; system trapped in a valley. Needs more energy to overcome barriers. This is achieved by parallel tempering: simulate copies at many different temperatures: T T T T T T n 2 n 1 n Lowest T : system would be trapped: Highest T : system has enough energy to fluctuate quickly over barriers. Perform global moves in which spin configurations at neighboring temperatures are swapped. Result: temperature of each copy performs a random walk between T 1 and T n. Advantages: satisfies detailed balance simple system can now visit many valleys at low-t (with correct relative weight) T
11 p.11 Equilibration: 1 For a Gaussian distribution of random bonds and fields, one can show (by integrating by parts over the disorder distribution) that (for the short-range case) [U] av = z 2 J 2 where U is the average energy, T ([q l] av 1) + H2 r T ([q] av 1), q l = 1 S i S j 2 is the link overlap N b q = 1 N i,j Si 2 is the spin overlap, i z (= 6) is the lattice coordination number, and N b = Nz/2. Note: This connection is for quantities that have been averaged over disorder.
12 p.12 Equilibration: 2 Equilibration test (Data for long range model.) As the number of sweeps increases, [U] av monotonically decreases, while U(q l, q) monotonically increases, where U(q l, q) = J2 T ([q l] av 1)+ H2 r T ([q] av 1) (J and q l generalized to the long-range case). Once U = U(q l, q) the results don t change if the number of sweeps is increased further. From Katzgraber and APY, Phys. Rev. B, 72, (2005).
13 Correlation Length We will see that the most useful quantity for finite-size scaling is the correlation length of the finite system, ξ L. To determine ξ L first compute the spin glass correlation function at finite wavevector. In a non-zero magnetic field this involves the connected correlation function χ SG (k) = 1 [( S i S j S i S j ) 2 ] av e ik (R i R j ) N i,j For zero field, where S i = 0, one has χ SG (0) χ nl, which can be measured experimentally. However, in a field χ SG (0) is not related to χ nl. Whereas χ SG (0) diverges on the AT line (for the SK model) (it is the replicon mode), χ nl does not. Hence there is no static, experimentally measurable, divergent quantity on the AT line. However, there is such a quantity, χ SG (0), in the simulations. We then determime ξ L from the Ornstein Zernicke equation χ SG (k) = χ SG (0) 1 + ξ 2 L k2 +..., by fitting to k = 0 and k = k min = (2π/L)(1, 0, 0). p.13
14 p.14 Finite size scaling Assumption: size dependence comes from the ratio L/ξ bulk where ξ bulk (T T SG ) ν is the bulk correlation length. (Note: (L/ξ bulk ) 1/ν L 1/ν (T T c ).) In particular, the finite-size correlation length varies as ξ ( ) L L = X L 1/ν (T T SG ), since ξ L /L is dimensionless (and so has no power of L multiplying the scaling function X). Hence data for ξ L /L for different sizes should intersect at T SG and splay out below T SG. Let s first see how this works for the Ising SG in zero field, both for the short-range d = 3 and long-range d = 1 models...
15 Short-Range: H r = 0 Crossings of the correlation length for the 3- d short-range Ising spin glass. Method first used for SG by Ballesteros et al., but for the ±J distribution. The clean intersections (corrections to FSS visible for L = 4) imply T SG Prevously Marinari et al. found T SG = 0.95 ±0.04 using a different analysis. Katzgraber and APY (unpublished) p.15
16 Long-Range: H r = 0 A transition is observed in zero field for ALL these values of σ. H. G. Katzgraber and APY, Phys. Rev. B, 72, (2005). p.16
17 p.17 Now do H 0 Now we see now this procedure works for the case of non-zero field, again for both the short-range d = 3 and long-range d = 1 models. According to RSB ξ L diverges for L on the AT line, so we look for intersections.
18 p.18 Short-Range: H r = 0.1 Crossings of the correlation length for the 3- d short-range Ising spin glass in a (Gaussian random) field of strength H r = 0.1. The lack of intersections implies that there is no AT line down to this value of H r. Katzgraber and APY (unpublished and Phys. Rev. Lett. 93, (2004)).
19 Long-Range: H r = 0.1 A transition seems to be observed in a field for σ 2/3. H. G. Katzgraber and APY, Phys. Rev. B, 72, (2005). p.19
20 p.20 Summary Does an AT line occur:
21 p.20 Summary Does an AT line occur: 1. only for the infinite-range (SK) model?
22 p.20 Summary Does an AT line occur: 1. only for the infinite-range (SK) model? 2. in real short-range three-dimensional spin glasses?
23 p.20 Summary Does an AT line occur: 1. only for the infinite-range (SK) model? 2. in real short-range three-dimensional spin glasses? 3. in short-range spin glasses for dimensions greater than a critical value which is above 3?
24 p.20 Summary Does an AT line occur: 1. only for the infinite-range (SK) model? 2. in real short-range three-dimensional spin glasses? 3. in short-range spin glasses for dimensions greater than a critical value which is above 3? Based on Monte Carlo simulations (and using the analogy between the short-range and long-range models) the answer seems to be 3.
25 p.20 Summary Does an AT line occur: 1. only for the infinite-range (SK) model? 2. in real short-range three-dimensional spin glasses? 3. in short-range spin glasses for dimensions greater than a critical value which is above 3? Based on Monte Carlo simulations (and using the analogy between the short-range and long-range models) the answer seems to be 3. There is an AT line above a critical dimension which seems to be d c = 6, the upper critical dimension for the zero field transition.
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