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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) cos2 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 cos2 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 ,i1=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 ,i1=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=i1 iteration 20-28: canonical expectation values: H i q W i1 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 21 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 =1.95T c 2 T c= log 21 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 i1 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 i1 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 log1 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 1c 2 lnN (Berg, Billoire, and Janke, 2000) Results for the SK Model fit rage B ∝expcN Results for the SK Model fit rage B ∝expcN 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 ∝expcN but goodness of fit ... Results for the EA Model B ∝expcN but goodness of fit ... Results for the EA Model B ∝expcN but goodness of fit ... therefore, different fit: B ∝cN Results for the EA Model B ∝expcN 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!

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phys. rev, glass transition, spin glasses, phys. rev. lett, j. phys, free energy, spin glass, ground state, j. chem. phys, supercooled liquids, electronic structure, relaxation time, density of states, molecular dynamics, molecular dynamics simulations

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posted: | 12/1/2009 |

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