Random first-order transition of a spin glass model in three dimensions – Toward a mean-field theory for glass transitions – Koji Hukushima University of Tokyo, Dep. of Basic Sciences 14 August 2015 In collaboration with Takashi Takahasi. Japan-France Joint Seminar Koji Hukushima (Univ. of Tokyo@Komaba) RFOT of spin glasses in 3 d 14 August, 2015 1 / 30
Outline 1 Spin glasses and Random First-Order Transition 2 Potts glass model 3 Our numerical results Thermodynamic properties Phase diagram of ϵ -coupled system Dynamical properties 4 Summary Koji Hukushima (Univ. of Tokyo@Komaba) RFOT of spin glasses in 3 d 14 August, 2015 2 / 30
Spin glasses and Random First-Order Transition Outline 1 Spin glasses and Random First-Order Transition 2 Potts glass model 3 Our numerical results Thermodynamic properties Phase diagram of ϵ -coupled system Dynamical properties 4 Summary Koji Hukushima (Univ. of Tokyo@Komaba) RFOT of spin glasses in 3 d 14 August, 2015 3 / 30
Spin glasses and Random First-Order Transition Order of phase transition Ehrenhest’s criterion Phase transition is described by a singularity of free energy. Phase transition with a singularity in n th order differential of free energy is called n th order phase transition. 1st order transition • Internal energy, entropy and volume have a jump at T c . • Order parameter also has a jump. • latent heat and delta-function-type divergence in specific heat. Koji Hukushima (Univ. of Tokyo@Komaba) RFOT of spin glasses in 3 d 14 August, 2015 4 / 30
Spin glasses and Random First-Order Transition Order of phase transition Ehrenhest’s criterion Phase transition is described by a singularity of free energy. Phase transition with a singularity in n th order differential of free energy is called n th order phase transition. 2nd order transition • Internal energy, entropy S , order parameter O change continuously at critical temperature. • Specific heat, susceptibility χ follow power-law divergence at T c • divergence of length scale ξ • universality class by critical exponents. Koji Hukushima (Univ. of Tokyo@Komaba) RFOT of spin glasses in 3 d 14 August, 2015 4 / 30
Spin glasses and Random First-Order Transition 1.5 order phase transition? • There exists infinite order phase transition like KT transitions. • Meanwhile, a kind of phase transitions, not 1st order and not 2nd order, has attracted much attention... Random first-order transition (RFOT) Koji Hukushima (Univ. of Tokyo@Komaba) RFOT of spin glasses in 3 d 14 August, 2015 5 / 30
Spin glasses and Random First-Order Transition RFOT and spin glasses 1 step RSB RFOT has been found in some spin glass models with one-step replica • p -state Potts glass with p ≥ 4 symmetry breaking. • p -spin glass with p > 2 Overlap distribution P ( q ) P ( q ) for full RSB P ( q ) for 1RSB Koji Hukushima (Univ. of Tokyo@Komaba) RFOT of spin glasses in 3 d 14 August, 2015 6 / 30
Spin glasses and Random First-Order Transition Some features of a 1RSB transition Random First-Order Transition (RFOT) = ⇒ the phenomenology of glass transitions Koji Hukushima (Univ. of Tokyo@Komaba) RFOT of spin glasses in 3 d 14 August, 2015 7 / 30
Spin glasses and Random First-Order Transition Does RFOT provides a good theory of glass transitions? Some statistical-mechanical models of RFOT • mean-field Potts glass model • mean-field p -spin glass model • Biroli-M´ ezard model (lattice glass model) on Bethe lattice • K-SAT, . . . They are all mean-field models with 1RSB transition. Koji Hukushima (Univ. of Tokyo@Komaba) RFOT of spin glasses in 3 d 14 August, 2015 8 / 30
Spin glasses and Random First-Order Transition Does RFOT provides a good theory of glass transitions? Two difficulties –mean-field theory and quench disorder– 1 One of the next issues to be addressed is to clarify whether predictions from the mean-field theory survive in finite dimensional models. Today’s issue 2 The correct theory of glass transitions must be free from quench disorder. Universality class?? Koji Hukushima (Univ. of Tokyo@Komaba) RFOT of spin glasses in 3 d 14 August, 2015 8 / 30
Potts glass model Outline 1 Spin glasses and Random First-Order Transition 2 Potts glass model 3 Our numerical results Thermodynamic properties Phase diagram of ϵ -coupled system Dynamical properties 4 Summary Koji Hukushima (Univ. of Tokyo@Komaba) RFOT of spin glasses in 3 d 14 August, 2015 9 / 30
Potts glass model RFOT occurs in a three dimensional Potts glass model? Potts glass Hamiltonian in three dimensions ∑ H J ( S ) = − J ij δ ( S i , S j ) , Potts variable: S i = { 0 , 1 , · · · , p − 1 } ( ij ) Questions • Does the model have a SG phase transition beyond a mean-field theory? Is it RFOT? • Low-temperature properties are described by replica symmetry breaking? • Most of researchers believe that dynamical singularity at T d is smeared out in finite dimensions. Does the singularity completely disappear? Koji Hukushima (Univ. of Tokyo@Komaba) RFOT of spin glasses in 3 d 14 August, 2015 10 / 30
Potts glass model Summary of the previous numerical studies 3 dimensional Potts glass models • No phase transition for p = 3 , Scheucher et al(1990). • No phase transition for p = 10 , Brangian et al(2002). • Finite T transition for p = 3 , Lee-Katzgraber-Young(2006). • Finite T transition for p = 4 JANUS project, Cruz et. al(2009). • Finite T transition for p = 5 and 6 JANUS project, Ba˜ nos et. al(2010). • It is found that p -state PG model with p = 3 , 4 , 5 and 6 shows a finite temperature SG transition by extended-ensemble MC simulations. • The transitions are continuous and there is no finite discontinuous jump of the order parameter. • No features of RFOT/1RSB were found. Koji Hukushima (Univ. of Tokyo@Komaba) RFOT of spin glasses in 3 d 14 August, 2015 11 / 30
Potts glass model Comment from Cammarota et al (2013) Koji Hukushima (Univ. of Tokyo@Komaba) RFOT of spin glasses in 3 d 14 August, 2015 12 / 30
Potts glass model Comment from Cammarota et al (2013) Detect 1RSB transition/RFOT in finite dimensions, 1 The number of Potts states, p , have to be large enough. 2 A rate of antiferromagnetic coupling have to be increased for preventing a ferromagnetic ordering. 3 The connectivity must be increased in order to keep the frustration. (Large p suppresses the frustration in general) Thus, in a naive sense, it is very hard to meet these conditions simultaneously on a three dimensional lattice. It might be possible in higher dimensions like d = 9 , 10 , ... . Our strategy: • we don’t want to go to high dimensions. • Instead, interaction range is enlarged up to 1st, 2nd and 3rd neighbors. Koji Hukushima (Univ. of Tokyo@Komaba) RFOT of spin glasses in 3 d 14 August, 2015 12 / 30
Potts glass model Our Model 7-state Potts glass model • 3rd neighbor random interactions with ± J type. (# of neighbors = 26 ). • System sizes: L = 4 , . . . , 10 , Number of samples: 256 ∼ 1024 . • Exchange MC(parallel tempering). 1 0 -1 -1 -1 0 0 1 1 2 2 Koji Hukushima (Univ. of Tokyo@Komaba) RFOT of spin glasses in 3 d 14 August, 2015 13 / 30
Our numerical results Outline 1 Spin glasses and Random First-Order Transition 2 Potts glass model 3 Our numerical results Thermodynamic properties Phase diagram of ϵ -coupled system Dynamical properties 4 Summary Koji Hukushima (Univ. of Tokyo@Komaba) RFOT of spin glasses in 3 d 14 August, 2015 14 / 30
Our numerical results Thermodynamic properties MC results for 7-state PG in three dimensions non-linear susceptibility χ SG Finite-size scaling for χ SG 45 3 L =7 40 L =8 3rd Nearest L=4 2.5 L =9 35 L=6 L =10 L=7 χ SG / L 1+ η /(1+a L - ω ) 2 30 L=8 L=9 25 L=10 χ SG 1.5 Nearest 20 L=4 L=8 1 15 L=16 10 0.5 5 0 0 -10 -5 0 5 10 15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ( T - T c ) L 1/ ν / J T/J • T c /J = 0 . 42(1) • 7-state PG model exhibits a • 1 /ν = 1 . 53(1) ⇐ ⇒ ν = 2 /d finite temperature SG transition • η = 0 . 43(2) ⇐ ⇒ γ ≃ 0 . 94 at T c /J ≃ 0 . 42 . Koji Hukushima (Univ. of Tokyo@Komaba) RFOT of spin glasses in 3 d 14 August, 2015 15 / 30
Our numerical results Thermodynamic properties MC results for 7-state PG in three dimensions: 2 Scaled correlation length ξ SG /L FSS for ξ L /L 0.6 0.5 0.45 L=4 0.55 L=5 0.45 0.5 0.4 L=6 0.45 0.4 0.35 L=7 0.4 L=8 ( ξ L /L)/(1+aL - ω ) 0.3 0.35 0.35 ξ L /L L=9 0.3 0.25 L=10 0.3 0.25 0.2 0.2 0.25 0.15 -2 -1 0 1 2 0.1 0.2 0.05 L = 7 0.2 0.3 0.4 0.5 0.6 0.7 0.15 L = 8 T/J L = 9 0.1 L = 10 • Length scale also diverges at -4 -2 0 2 4 6 8 10 12 14 (T-T c )L 1/ ν /J T c /J ≃ 0 . 42 . • T c /J = 0 . 421(3) • ν = 0 . 68(9) ⇐ ⇒ ν = 2 /d Koji Hukushima (Univ. of Tokyo@Komaba) RFOT of spin glasses in 3 d 14 August, 2015 16 / 30
Our numerical results Thermodynamic properties MC results for 7-state PG in three dimensions: 3 Order-parameter distribution ⟨ ( )⟩ √ q (2) P ( q ) = δ q − 0.14 T/J=0.5002 (a) 0.12 T/J=0.4688 T/J=0.4421 0.1 T/J=0.4131 T/J=0.3841 P (T) (Q) 0.08 T/J=0.3550 T/J=0.3260 0.06 T/J=0.2970 0.04 0.02 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Q Temp. dep for L = 9 . • a bimodal distribution of P ( q ) is compatible to 1step RSB. Koji Hukushima (Univ. of Tokyo@Komaba) RFOT of spin glasses in 3 d 14 August, 2015 17 / 30
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