Free-energy barriers in spin glasses mean-field vs short-range models

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Free-energy barriers in spin glasses mean-field vs short-range models Powered By Docstoc
					Free-energy barriers in spin glasses:
  mean-field vs short-range models



                       Elmar Bittner
                       Universität Leipzig



           Collaborators: A. Nußbaumer and W. Janke


 Spring School on Monte Carlo Simulations of Disordered Systems
                     Leipzig, April 4, 2008.
                 Outlook

●
    Intro
●
    Models
●
    Problems
●
    Algorithms
●
    Motivation
●
    Results
                        Spin Glass Systems
●
    There are real experimental spin glass systems.
    (dilute solutions of magnetic transition metal impurities in noble metal
    hosts, for instance Au-2.98% Mn or Cu-0.9% Mn)




                                        cos2 k F R
    RKKY interaction:   J eff R= J0         3
                                                       ,k F R≫1, k F Fermi wave number
                                            R
    Basic ingredients for spin-glass behaviour
●
    randomness in course of the dilution process the
    positions of the impurity moments are randomly
    distributed
●
    competing interactions due to the oscillations in the
    effective interaction as a function of the distance R




                                        cos2 k F R
    RKKY interaction:   J eff R= J0         3
                                                       ,k F R≫1, k F Fermi wave number
                                            R
                  Spherical Cow

Most theory uses the simplest model with these
ingredients:

the Edward-Anderson Model (EA)

              H=− ∑ J ij S i S j −∑ hi Si
                   〈i , j 〉       i


with Si =±1 lie on a regular lattice and the quenched
coupling constants Jij .
                                        Spherical Cow

Most theory uses the simplest model with these
ingredients:

the Edward-Anderson Model (EA)

                               H=− ∑ J ij S i S j −∑ hi Si
                                          〈i , j 〉                   i




bimodal distribution Jij =±1

3D: T c ~1.16, hi =0
[M. Palassini and S. Caracciolo, Phys. Rev. Lett. 82, 5128 (1999)]
no solution
             What is a spin glass?

A system with disorder (randomness) and frustration.


                        +

                +               +

                        -
             What is a spin glass?

A system with disorder (randomness) and frustration.


                        +

                +               +

                        -
                                 Another Cow
                                                             fully connected


the Sherrington-Kirkpatrick Model (SK)

                        H=−∑ Jij S i S j −∑ hi S i
                                 i j               i


Gaussian distribution with:
    〈 J ij 〉=0
        2          J
                   2
  〈 J 〉−〈 J ij 〉 =
        ij
                   N
N is the number of spins.
T c =1, hi =0
mean field, Parisi's replica solution [PRL 43 (1979) 1754]
                  Overlap parameter
                                     N
                                1      1 2
                           q=     ∑
                                N i=1
                                      Si S i
                                                   1      2



for to (real) replica S , S and given coupling constants J={ J ij }
                       1
                       i
                           2
                           i


  P J q               probability density of q

  x J q=∫ dq ' P J q ' cumulative distribution of P J q

average over the disorder
                                     1
               P q=[P J q]av =     ∑ P J q
                                     NJ J
                                     1
               x J q=[ x J q]av = ∑ x J q
                                     NJ J
                Slow Dynamics
                 (free) energy

                                           barrier

The dynamics is very
slow at low T.
System is not in
equilibrium due to                             DE
complicated energy
landscape: system
trapped     in   one
“valley”   for  long             valley
times.                                               valley


                                    configuration
                                  Algorithms

Parallel tempering (PT)

Exchange at regular intervals                           T1
system i and i+1 with                                        T2
                                                                  T3
                                                                       T4

 P i ,i1=min [1, exp   E ]

 expectation values for single system:

               〈 A〉 T =〈 Ai 〉
                        i
                                                                  System can
                                                                  decorrelate at high T

[K. Hukushima and K. Nemoto, J. Phys. Soc. Jpn. 65 (1996) 1604]
Talk of H. Katzgraber
                                  Algorithms

Parallel tempering (PT)

Exchange at regular intervals
system i and i+1 with

 P i ,i1=min [1, exp   E ]

 expectation values for single system:

               〈 A〉 T =〈 Ai 〉
                        i




[K. Hukushima and K. Nemoto, J. Phys. Soc. Jpn. 65 (1996) 1604]
Talk of H. Katzgraber
                                    Algorithms

Multioverlap Algorthim (MuQ)

non-Boltzmann sampling with
multioverlap weights W q:


   exp[− H]W q

canonical expectation values:

            〈W O 〉
          can
     〈O 〉 =
             〈W 〉                                System can reach highly
                                                 suppressed states

[B. Berg, W. Janke, PRL 80 (1998) 4771]
                                    Algorithms

Multioverlap Algorthim (MuQ)                                              iteration 1-9:




non-Boltzmann sampling with
mutlioverlap weights W q:
                                            W 0 q=1


   exp[− H ]W q                           simulation
                                                                 i=i1   iteration 20-28:
canonical expectation values:                          H i q
                                          W i1 q=
                                                       W i q
           〈W O〉
          can
     〈O〉 =
            〈W 〉                             sampling


[B. Berg, W. Janke, PRL 80 (1998) 4771]
                         The 2D Ising Model


H=− ∑ J Si S j
         〈i , j 〉




Si =±1


            2
T c=               =2.269 ...
       log  21
                         1
            2       −4
m0=1−sinh   8
            T
Onsager solution, Onsager-Yang solution
                         The 2D Ising Model


H=− ∑ J Si S j
         〈i , j 〉




Si =±1
                             T =1.95T c
            2
T c=
       log  21
                         1
            2       −4
m0=1−sinh  8
            T


                                           L=40
        The 2D Ising Model




multimagnetical simulation
        The 2D Ising Model




multimagnetical simulation
                The 2D Ising Model

But there are still problems:



                                     droplet/strip




                                     evaporation/
                                     condensation
                The 2D Ising Model

But there are still problems:
                       The 2D Ising Model

But there are still problems:




 T. Neuhaus and J. S. Hager, J. Stat. Phys. 116 (2003) 47,
 see also poster from A. Nußbaumer
                                    Algorithms
combination of both methods: PT-MuQ

   iterate to improve
         weights
                                                  simulate (e.g.) 1 sweep with MuQ
                                                     at N different temperatures
                  H i q
     W i1 q=
                  W i q                               {
                                                    min 1, exp[−m  H ]
                                                                         W 'm q
                                                                         W m q    }
               no!
          
      weights OK?                                           exchange replica

               yes!

   sample with fixed
                                              {
                                         min 1, exp [ n−m  E n−E m  ]
                                                                            W m qn W n qm 
                                                                            W m qm W n qn    }
       weights

[E. Bittner, A. Nußbaumer, W. Janke, in preperation]
                                    Algorithms
combination of both methods: PT-MuQ

   iterate to improve
         weights


                  H i q
     W i1 q=                          ●
                                             weights “belong” to a temperature m   Wm
                  W i q
                                         ●
                                             if every replica is simulated on a
               no!
                                             different computer, all nodes have
          
      weights OK?                            to have all weights

               yes!                      ●
                                             in reality weights are computed with
                                             a more “sophisticated” approach
   sample with fixed
       weights

[E. Bittner, A. Nußbaumer, W. Janke, in preperation]
                  Algorithms




                               EA model, V=8x8x8




SK model, N=512
                Slow Dynamics
                 (free) energy

                                           barrier

The dynamics is very
slow at low T.
System is not in
equilibrium due to                             DE
complicated energy
landscape: system
trapped     in   one
“valley”   for  long             valley
times.                                               valley


                                    configuration
              Main objective: barrier heights
       2D Ising Model

       H=− ∑ J Si S j                        P max  L
            〈i , j 〉                                    ~exp [ F B  L ] ≡ L
                                             P min  L


       Si =±1                   P max  L           F B~  L~2 L


               L=30
                                  T T c


P min L
                        L=100
      Main objective: barrier heights
Spin glasses:




                                     T T c




                How do we measure
                   the size of the
                  largest barrier?
         1d Markov chain/transition matrix
 Definition:




                                                             
           1−w1,2    w1,2         0                           ⋯
            w2,1  1−w2,1−w2,3    w2,3                         ⋯
        T=   0       w3,2     1−w3,2−w3,4                     ⋯
             0        0          w4,3                         ⋯
             ⋮        ⋮           ⋮                           ⋱

                 1
         wi , j = min 1,
                 2           [
                         P q j 
                         P qi             ]
[B.A. Berg, A. Billoire and W. Janke, PRB 61 (2000), 12143]
    1d Markov chain/transition matrix
Definition:




                                            
      1−w1,2    w1,2          0            ⋯
       w2,1  1−w2,1−w 2,3    w 2,3         ⋯
   T=   0       w3,2      1−w3,2−w 3,4     ⋯
        0        0           w 4,3         ⋯
        ⋮        ⋮            ⋮            ⋱


T fulfills detailed balance   only real eigenvalues
                             q      1
autocorrelation time for q: B=
                                N log1 
1 second largest eigenvalue 0=1
                         Motivation
theoretical predictions for mean-field model (SK):
barrier between time-reversed states scales with
system size as
                       N1/3
(Rodgers and Moore, 1988)
(Kinzelbach and Horner, 1991)



results for short-ranged models (EA) are far away
from the mean-field theory limit

                             c 1c 2 lnN

(Berg, Billoire, and Janke, 2000)
           Results for the SK Model

                      fit rage
B ∝expcN  
             Results for the SK Model

                                       fit rage
B ∝expcN 




      non-self-averaging (SK)
      [L.A. Pastur and M.V. Shcherbina, J. Stat. Phys. 62 (1992) 1]
                  Results for the SK Model

                                                       extreme-value distribution




                                                                  fat tails




Fit integrated probability density:


               [                 ]
                                 −1/
                           x−          T T c   0
F  ;  ;   x=exp − 1
                                                fat tailed (algebraic)
                                                 Fréchet distribution
Results for the SK Model
           Results for the SK Model
Peaked probability distribution D  F B / F B 
                                            med
           Results for the SK Model
Peaked probability distribution D  F B / F B 
                                            med




                                         no scaling,
                                         i.e. non-self-averaging
           Results for the SK Model
Peaked probability distribution D  F B / F B 
                                            med




                                         scaling,
                                         i.e. self-averaging

                                                  (EA Model)
          Results for the EA Model
          
B ∝expcN 


                           but goodness of fit ...
          Results for the EA Model
          
B ∝expcN 


                           but goodness of fit ...
          Results for the EA Model
          
B ∝expcN 


                                                but goodness of fit ...




                                            
         therefore, different fit: B ∝cN
          Results for the EA Model
          
B ∝expcN 


                                                but goodness of fit ...




                                            
         therefore, different fit: B ∝cN
                  Results for the EA Model




            SK Model




Fit integrated probability density:                    EA Model

               [                 ]
                                 −1/
                           x−          T T c   0
F  ;  ;   x=exp − 1
                                                fat tailed (algebraic)
                                                 Fréchet distribution
            Results for the EA Model
Peaked probability distribution D  F B / F B 
                                            med




          SK Model


            no scaling,
            i.e. non-self-averaging
                                         EA Model
                               Conclusion
●
    Algorithmic
    - PT is good to decrease the autocorrelation time
    - MuQ gives the full P q information
    - the combination of PT+MuQ makes it possible
        to get P q down to T ≈0.5 T c


●
    Physical
    - the free energy barriers of the SK and EA model are
          a) non-self-averaging
          b) follow the Fréchet extreme-value distribution
    -   the free energy barriers of the SK model diverge with
        an exponent of =1/3
    -   the last is not true for the EA model
Thank you!