Learning About Spin Glasses Enzo Marinari (Cagliari, Italy) I thank the organizers... I am thrilled since... (Realistic) Spin Glasses: a debated, inte

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1 Learning About Spin Glasses Enzo Marinari (Cagliari, Italy) I thank the organizers... I am thrilled since... (Realistic) Spin Glasses: a debated, interesting issue. Mainly work with: Giorgio Parisi, Federico Ricci-Tersenghi, Juan J. Ruiz-Lorenzo and Francesco Zuliani. Connected work, among other, by Berg and Janke, Kawashima and Young. SCRI, Tallahassee, March 1999 March 1999 SCRI, Tallahassee Page 0

2 A schematic summary of this talk: 3 parts. 1. Mean Field Theory of spin glasses, and its dramatically new features. 2. Standard evidence about the nature of the phase transition and of the broken phase. 3. Sensitive issues and numerical evidences about them (interfaces, Gibbs states, droplet or non-droplet). I will not try to enter in all details of the open debate (there is too much to say) but only try to signal the ones that look like the crucial questions. Spin Glasses? Why do we study spin glasses? Systems with frustration and quenched disorder. Maybe frustration more crucial than disorder (real glasses). Traps that do not change in time. March 1999 SCRI, Tallahassee Page 1

3 The study of Spin Glasses has led, in the last years, to a large amount of new physics, and created a new eld, where things are indeed very dicult to understand. Solution of Mean Field Theory Replica Symmetry Breaking Many States MF, innite range,! short range (i.e. realistic) Concept of state numerical results are a priori, but a clear picture is crucial for a systematic treatment Overlap among two replicas will play a crucial role (order parameter): q (a;b) 1 V X together with its distribution i (a) i (b) i P J (q) ; P (q) P J (q) March 1999 SCRI, Tallahassee Page 2

4 N spins H[] = H fjg [] fjg: quenched random couplings h i thermal average, disorder average, E ( ) h i Finite volume P fjg [] / e,h fjg [] Finite D: Edwards Anderson, = 1, J = 0, J 2 = 1. H fjg [] = X i;j 0 i J i;j j + X i h i i,! Sherrington Kirkpatrick mean eld, J 2 = 1 N. H fjg [] = X i;j i J i;j j + X i h i i March 1999 SCRI, Tallahassee Page 3

5 Quenched disorder. Study: Replicas log Z J = lim n!0 Z n J, 1 n H (n) J = nx a=1 H J [ n ] Integrate over the quenched couplings fj g =) replicas are coupled. Q a;b hq a;b i, F[Q], Q such that e.v. 2 c;d a;b = 0, High T, one ground state, OK (Q a;b = constant). Low T, many ground states, symmetry breaking (RSB). Here symmetry is permutation of dierent replicas. Q a;b now depends on a and b. March 1999 SCRI, Tallahassee Page 4

6 Parisi scheme for RSB P (q) P J (q) is the equilibrium probability distribution of the overlap, characterized by a function q(x) or by its inverse x(q) such that R dq P (q) q s = R dx q(x) s, i.e. P (q) = dx dq. is the solution of mean eld. It contains many remarkable features. Important questions related to nite D systems where new things have been learned recently are: 1. pattern of RSB, state structure. Potential problem: interfaces. P(q) This 2. de Almeida Thouless line: there is a transition in a nite magnetic eld (equilibrium + approach to equilibrium). 3. dynamical behavior, pattern of violation of FDT. RSB looks more and more OK for MF (Guerra, Aizenman-Contucci) Potential problems for nite D (Newman and Stein, heuristic + rigorous: we discuss here some of the ndings that seem to imply that the RSB picture does indeed describe realistic models. q March 1999 SCRI, Tallahassee Page 5

7 1. One nds a non-trivial P (q) (as opposed to usual magnetic systems). In Mean Field, for h 6= 0, one nds that P (q) = a(q, q m ) + b(q, q M ) + r(q) 2. If for large volumes P (q) does not become a simple function correlation functions do not cluster. 3. Building of an ultrametric structure (also here there are numerical results). Sum rules. P J (q 1;2 )P J (q 3;4 ) = 2 3 P (q 1;2)P (q 3;4 ) P (q 1;2)(q 1;2, q 3;4 ) P J (q 1;2 )P J (q 2;3 )P J (q 3;1 ) = 1 2 P (q 1;2)x(q 1;2 )(q 1;2, q 2;3 )(q 1;2, q 3;1 )+ 1 2 [P (q 1;2)P (q 2;3 )(q 1;2, q 2;3 )(q 2;3, q 3;1 )+ two permutations] March 1999 SCRI, Tallahassee Page 6

8 To study all that we will need numerical simulations. It is dicult, since we need to inspect a very complex phase space. Optimized Monte Carlo methods are mandatory. We use parallel tempering (in the family of density scaling, multi-canonical, tempering). We consider S copies of the system (in the same realization of the quenched disordered couplings fj g) 1 ;:::S (that together with the S A copies we need for computing the overlap gives us 2S congurations of the system). We enlarge the space of the degrees of freedom, by promoting to be a dynamical variable. The congurations are allowed to exchange their value of. Start with (f 1 g) = 1 ; :::; (fng) = n, and after one full MC sweep of the spin variables propose to swap (f 1 g) = 2 ; (f 2 g) = 1, and so on for all adjacent couples. Use Metropolis weighting to accept the swap. Very eective. It does not need any normalization constant. No tuning, no ne tuning. Be careful to thermalization! March 1999 SCRI, Tallahassee Page 7

9 We start by discussing usual numerical results, proving the existence of a phase transition. We will try in the following to qualify it. Here 4D, clearer than 3D, that is very close to the lower critical dimension (for example, very peculiar behavior of the Binder parameter). Binary coupling, J = 1 (allowing very eective multi-spin coding). Many samples (from at V = 4 4 down to = : at V = 10 4 ). Use the Binder cumulant to study the distribution probability of the order parameter! g L = 1 3, hq4 i L (T ), 2 hq 2(T 2i L ) Shape of P (q) at T c is universal: For ferromagnets: g c lim L!1 g L (T c ) lim L!1 g L (T > T c ) = 0, lim L!1 g L (T < T c ) = 1 March 1999 SCRI, Tallahassee Page 8

10 At T c : crossing of dierent L curves of g L (T ). In the RSB solution of MF spin glasses the innite volume limit of g is not 1 in the broken phase. P(q) non-trivial =) g(t < T c ) = 3 2, 1 2 R,R dp (q) q 4 dp (q) q 2 2 It is dicult to be sure that at xed T < T c the dierent curves tend to a limit dierent from 1. We will gather the same evidence with other methods (see later). g L3 L4 L5 L6 L8 L T March 1999 SCRI, Tallahassee Page 9

11 Dierent kind of analysis (linearization, derivatives,...). Error by jack-knife and binning. Close to T c : g(l; T ) ' g L 1 (T, Tc ) Scaling is good. T c = 2:03 0:03 = 1:00 0:10 g L3 L4 L5 L6 L8 L L 1/ν (T-T c ) March 1999 SCRI, Tallahassee Page 10

12 The overlap susceptibility. In the Replica Symmetry Broken phase of a spin glass q V hq 2 i At T = T c (determined by Binder cumulant) scaling is like q (L; T = T c ) ' L 2, L3 L4 L5 L6 L8 L10 χ / L 2-η L 1/ν (T-T c ) March 1999 SCRI, Tallahassee Page 11

13 There are at least two more ways to determine. One is based on using o-critical data, with L. Here q (T ) ' (T, T c ),(2,) χ L=8 L=10 (T-T c ) -ν(2-η) T-2.03 The second method is based on the scaling P (q ' 0;T = T c ) ' L D,2+ 2 (see later). They all give consistent results. We nd =,0:30 0:05 March 1999 SCRI, Tallahassee Page 12

14 P (q), T < T c (here T ' 0:6T c ). It is non trivial (for a ferromagnet in the broken phase, two -functions center in m 2 ). Peaks are moving (for L! 1 (q q EA )). We should check they do not go to zero. We should show that P (q ' 0) does not change with L L=3 L=4 L=5 L=6 L=8 L= March 1999 SCRI, Tallahassee Page 13

15 q MAX such that P (q = q MAX ) = max versus L,1:3. The exponent (,1:3) is from the best t. q MAX (L) = q MAX (L = 1) + c L ' 1:3 is consistent with MF analytic computations L -1.3 March 1999 SCRI, Tallahassee Page 14

16 P (q ' 0) versus L. No decrease. P(0) L March 1999 SCRI, Tallahassee Page 15

17 Smoothness SK! EA? H = (1, )H 3D + H SK 3D, 8 3 spin, T =,1 = 0:7. P (q) smooth, P J (q) no. PJ(q) PJ(q) No modied droplets (with one couple of function for each sample, but support depending on the sample). (With this transparency we are entering the third and last phase of this talk, concerning debated issues). March 1999 SCRI, Tallahassee Page 16

18 Interfaces and Local Overlaps V = L 3, 1 B L Region of size B 3 : q B 1 B 3 Pi2B q i. P B (q) = lim L!1 P L B (q) In the Replica Symmetry Breaking Ansatz one nds that lim B!1 P B (q) = P (q) The presence of interfaces can make P B=1 (q) peculiar. Let us discuss about interfaces. A non-trivial P (q) can be obtained without the need of a non-trivial Symmetry Breaking (Fisher-Huse, van Enter). 3D ferromagnet. yz: periodic bc, x: antiperiodic bc, T < T c : domain wall. -1 bc bc -1 bc bc March 1999 SCRI, Tallahassee Page 17

19 m(i x ) = 1 N y N z Pi y ;i z (i x ;i y ;i z ) = f (i x, I) where I is the position of the interface in the x direction. I can be everywhere, and f can tend to,m for large negative argument and to +m for large positive argument, or it can tend to +m for large negative argument and to,m for large positive argument. In these conditions P (q) = 1 2m 2 for,m 2 q m 2. The RSB approach can tell the dierence between true RSB and such a trivially non-trivial P(q). Both local overlaps and correlation functions do so. In the situation that we have described, where RS is not broken: lim B!1 P B (q) = 1 2,, q, m 2 +, q + m 2 since the probability of hitting an interface is negligible when B L. Local measurements can show when RSB predictions are valid and when they are not. March 1999 SCRI, Tallahassee Page 18

20 Numerical simulations show, as we have seen, a clear, non-trivial P (q). P (q ' 0) 6= 0, and the maximum at q MAX becomes sharper with increasing L. Maybe this is an eect faked by interfaces? From the work by Newman and Stein: Essentially all the simulations of which we are aware compute the overlap distribution in the entire box. [...] We suspect that the overlaps computed over the entire box are observing domain wall eects arising solely from the imposed boundary conditions rather than revealing spin glass ordering [...] In other words if overlap computations were measured in small windows far from any boundary, one should nd only a pair of {functions [...] This is a very interesting issue, worth to check. March 1999 SCRI, Tallahassee Page 19

21 Block size B = 4. Triangles L = 8. Squares L = 12. T ' 0:7 T c, 3D. Parallel Tempering =) equilibrium ok. 1. Small nite size eects: better than full P (q) 2. non-trivial 3. no substantial dependence on L 4. no approach to -function March 1999 SCRI, Tallahassee Page 20

22 T ' 0:7 T c, 3D. Fixed lattice size L = 12. From B = 2 (lower curve, triangles). To B = 12 (sharper peaks, arrows. The major changes in P B (q) are for increasing B. They are related to the usual (strong) L-dependence of the full P (q). These data are from cond-mat/ , published in J. Phys. A. March 1999 SCRI, Tallahassee Page 21

23 Overlap Sum Rules In Mean Field (Mezard, Parisi, Sourlas, Toulouse and Virasoro) we have seen already some of the relations that relate expectations of P J (q). Consider 4 or 3 real replicas. One nds that () E(q12 2 q2 34 ) = 2 3 E(q2 12 ) E(q4 12 ), E( ) h i () E(q12 2 q2 23 ) = 1 2 E(q2 12 ) E(q4 12 ) with one replica in common (Building blocks of ultrametricity). q 12 and q 34 are not independent. Parisi, Ritort, Ruiz, EM (*) holds with good accuracy for 3D spin glasses. Guerra, Aizenman Contucci and Parisi: rigorous proof and more relations. Proof is also valid for nite D EA model if some continuity hypothesis holds (Guerra). This is the main point of the discussion that emerges again: are typical equilibrium congurations of the, say, 3D model, similar to the ones of the SK model? March 1999 SCRI, Tallahassee Page 22

24 Two data points (LHS and RHS) on each plot point. The dierence cannot be appreciated on this graph. But shows the non-trivial q 2 in the broken phase D +/- J rule: E(q 12 2 q34 2 ) = 2/3 E(q 2 ) 2 + 1/3 E(q 4 ) Temperature L=6 L=8 L=10 L=12 Satised with very good accuracy. Stronger eects close to T c. Converges to zero with a power law E(q 12 2 q34 2 ) - 2/3 E(q 2 ) 2-1/3 E(q 4 ) L=6 L= L=10 L= Temperature March 1999 SCRI, Tallahassee Page 23

25 It works perfectly also for the 3 replica case. This is a strong step toward ultrametricity D +/- J rule: E(q 12 2 q23 2 ) = 1/2 E(q 2 ) 2 + 1/2 E(q 4 ) Temperature L=6 L=8 L=10 L= E(q 12 2 q23 2 ) - 1/2 E(q 2 ) 2-1/2 E(q 4 ) L= L=8 L=10 L= Temperature March 1999 SCRI, Tallahassee Page 24

26 Coupling Replicas Couple two copies of the system. You can use or for example Hq[; ] = H 0 []+H 0 []+ V Hq E [; ] = H 0 []+H 0 []+ ~ V P i ii P i; ii+ii+ Compute lim!0 + q() and lim!0, q() or do it for ~ (avoid possible troubles due to interfaces) RSB: lim!0 + q() = q + + A + lim!0, q() = q, + A,jj 0.7 EA L=8 q (L) (ε) EA L=12 MKA L=4,8, EA L= sign(ε) ε 1/2 March 1999 SCRI, Tallahassee Page 25