[PPT] - Random first-order transition of a spin glass model in three PowerPoint Presentation

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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 3d 14 August, 2015 1 / 30

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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 3d 14 August, 2015 2 / 30

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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 3d 14 August, 2015 3 / 30

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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 nth order differential of free energy is called nth order phase transition.

1st order transition

Internal energy, entropy and volume have a jump

at Tc.

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 3d 14 August, 2015 4 / 30

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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 nth order differential of free energy is called nth 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 Tc

divergence of length scale ξ
universality class by critical exponents.

Koji Hukushima (Univ. of Tokyo@Komaba) RFOT of spin glasses in 3d 14 August, 2015 4 / 30

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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 3d 14 August, 2015 5 / 30

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Spin glasses and Random First-Order Transition

RFOT and spin glasses

RFOT has been found in some spin glass models with one-step replica symmetry breaking. Overlap distribution P(q)

1 step RSB

p-state Potts glass with p ≥ 4
p-spin glass with p > 2

P(q) for full RSB P(q) for 1RSB

Koji Hukushima (Univ. of Tokyo@Komaba) RFOT of spin glasses in 3d 14 August, 2015 6 / 30

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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 3d 14 August, 2015 7 / 30

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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 3d 14 August, 2015 8 / 30

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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 3d 14 August, 2015 8 / 30

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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 3d 14 August, 2015 9 / 30

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Potts glass model

RFOT occurs in a three dimensional Potts glass model?

Potts glass Hamiltonian in three dimensions

HJ(S) = − ∑

(ij)

Jijδ (Si, Sj) , Potts variable: Si = {0, 1, · · · , p−1}

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 Td is smeared
ut in finite dimensions. Does the singularity completely disappear?

Koji Hukushima (Univ. of Tokyo@Komaba) RFOT of spin glasses in 3d 14 August, 2015 10 / 30

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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
f the order parameter.
No features of RFOT/1RSB were found.

Koji Hukushima (Univ. of Tokyo@Komaba) RFOT of spin glasses in 3d 14 August, 2015 11 / 30

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Potts glass model

Comment from Cammarota et al (2013)

Koji Hukushima (Univ. of Tokyo@Komaba) RFOT of spin glasses in 3d 14 August, 2015 12 / 30

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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 3d 14 August, 2015 12 / 30

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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

1

1

1 2

1

1 2

Koji Hukushima (Univ. of Tokyo@Komaba) RFOT of spin glasses in 3d 14 August, 2015 13 / 30

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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 3d 14 August, 2015 14 / 30

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Our numerical results Thermodynamic properties

MC results for 7-state PG in three dimensions

non-linear susceptibility χSG 5 10 15 20 25 30 35 40 45 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

χSG T/J

Nearest 3rd Nearest

L=4 L=6 L=7 L=8 L=9 L=10 L=4 L=8 L=16

7-state PG model exhibits a

finite temperature SG transition at Tc/J ≃ 0.42.

Finite-size scaling for χSG

0.5 1 1.5 2 2.5 3

10
5

5 10 15 χSG/L1+η/(1+aL-ω) (T-Tc)L1/ν/J L=7 L=8 L=9 L=10

Tc/J = 0.42(1)
1/ν = 1.53(1)

⇐ ⇒ ν = 2/d

η = 0.43(2)

⇐ ⇒ γ ≃ 0.94

Koji Hukushima (Univ. of Tokyo@Komaba) RFOT of spin glasses in 3d 14 August, 2015 15 / 30

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Our numerical results Thermodynamic properties

MC results for 7-state PG in three dimensions: 2

Scaled correlation length ξSG/L

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.2 0.3 0.4 0.5 0.6 0.7

ξL/L T/J

L=4 L=5 L=6 L=7 L=8 L=9 L=10

Length scale also diverges at

Tc/J ≃ 0.42.

FSS for ξL/L

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

4
2

2 4 6 8 10 12 14

(ξL/L)/(1+aL-ω) (T-Tc)L1/ν/J

L = 7 L = 8 L = 9 L = 10

0.2 0.25 0.3 0.35 0.4 0.45

2
1

1 2

Tc/J = 0.421(3)
ν = 0.68(9) ⇐

⇒ ν = 2/d

Koji Hukushima (Univ. of Tokyo@Komaba) RFOT of spin glasses in 3d 14 August, 2015 16 / 30

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Our numerical results Thermodynamic properties

MC results for 7-state PG in three dimensions: 3

Order-parameter distribution

P(q) = ⟨ δ ( q − √ q(2) )⟩

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

P(T)(Q) Q (a)

T/J=0.5002 T/J=0.4688 T/J=0.4421 T/J=0.4131 T/J=0.3841 T/J=0.3550 T/J=0.3260 T/J=0.2970

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 3d 14 August, 2015 17 / 30

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Our numerical results Thermodynamic properties

MC results for 7-state PG in three dimensions: 3

Order-parameter distribution

P(q) = ⟨ δ ( q − √ q(2) )⟩

0.01 0.02 0.03 0.04 0.05 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

P(T)(Q) Q (b)

L=4 L=6 L=7 L=8 L=9

L dep at T ≃ 0.7Tc.

a bimodal distribution of P(q) is compatible to 1step RSB.

Koji Hukushima (Univ. of Tokyo@Komaba) RFOT of spin glasses in 3d 14 August, 2015 17 / 30

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Our numerical results Thermodynamic properties

MC results for 7-state PG in three dimensions: 4

Energy

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

e T/J

(a)

L = 4 L = 6 L = 7 L = 8 L = 9 L = 10 L = 14

Specific heat

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

c T/J

(b)

L=4 L=6 L=7 L=8 L=9 L=10 L=14

No divergence of c is found at T = Tc, meaning no latent heat.
These indicate that it is Random First Order transition.

Koji Hukushima (Univ. of Tokyo@Komaba) RFOT of spin glasses in 3d 14 August, 2015 18 / 30

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Our numerical results Phase diagram of ϵ-coupled system

Phase diagram of ϵ-coupled system

Phase transition under a symmetry-broken field:

Predictions for an ϵ-coupled system

HJ(Sα, Sβ) = HJ(Sα) + HJ(Sβ) − ϵ ∑

i

δ(Si,α, Si,β)

0 < ϵ < ϵ∗

discontinuous jump in q
1st order phase transition

with latent heat

ϵ∗ < ϵ

crossover in q
no phase transition

Koji Hukushima (Univ. of Tokyo@Komaba) RFOT of spin glasses in 3d 14 August, 2015 19 / 30

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Our numerical results Phase diagram of ϵ-coupled system

Clausius-Clapeyron relation of ϵ-coupled system

Carnot cycle in the ϵ-coupled system Carnot theorem Q1 T1 = Q2 T2 As T1 − T2 = ∆T, the first law gives the work W of the cycle W = Q1 − Q2 = Q1 − T2 T1 Q1 = Q1 ∆T T1 The work W is also expressed by the area as ∆q∆ϵ = Q1 ∆T T1 , = ⇒ ∆T ∆ϵ = T∆q Q1 = ⇒ dT dϵ = T∆q Q1

Koji Hukushima (Univ. of Tokyo@Komaba) RFOT of spin glasses in 3d 14 August, 2015 20 / 30

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Our numerical results Phase diagram of ϵ-coupled system

From Clausius-Clapeyron relation, ...

1 First-order phase boundary is

monotonic upward curve in T − ϵ phase diagram

∆q ≥ 0 and Q1 ≥ 0

2 If RFOT occurs in the limit ϵ = 0

dT dϵ

ϵ=0

= ∞, because Q1 = 0 and ∆q > 0.

CC relation

dT dϵ = T∆q Q1

3 A mean-field prediction for annealed system gives

Tc(ϵ) = Tc(0) + O(√ϵ) with ∆q = const., yielding = ⇒ Q1 = √ϵ + · · · .

4 About critical end point...

Koji Hukushima (Univ. of Tokyo@Komaba) RFOT of spin glasses in 3d 14 August, 2015 21 / 30

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Our numerical results Phase diagram of ϵ-coupled system

Thermodynamics of the ϵ-coupled system

Specific heat

0 < ϵ < ϵ∗:

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 c T/J ε=0.0008 L=8 L=10 L=12 L=14

ϵ∗ < ϵ:

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95

c T/J

ε=0.012 L=8 L=10 L=12 L=14

Overlap

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 q T/J ε=0.0008 L=8 L=10 L=12 L=14 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 q T/J ε=0.0120 L=8 L=10 L=12 L=14 Koji Hukushima (Univ. of Tokyo@Komaba) RFOT of spin glasses in 3d 14 August, 2015 22 / 30

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Our numerical results Phase diagram of ϵ-coupled system

Phase diagram of 3d Potts glass model in ϵ − T

0.4 0.45 0.5 0.55 0.6 0.65

0.002 0.004 0.006 0.008 0.01 0.012

T/J ε Tc from g4 Tc from ξL/L

Our result is consistent with 1st order transition temperature with

infinite slope at ϵ = 0, suggesting RFOT at ϵ = 0.

Critical endpoint:0.008 < ϵ∗ < 0.012

Koji Hukushima (Univ. of Tokyo@Komaba) RFOT of spin glasses in 3d 14 August, 2015 23 / 30

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Our numerical results Dynamical properties

Dynamics –auto-correlation function and susceptibility–

Autocorrelation function：C(t; tw) = 1

N ∑

i

Si(tw) · Si(t + tw) 1 − 1/p

Dynamical susceptibility (four-point correlation function) χ4:

χ4(t; tw) = N [ ⟨C(t; tw)2⟩ − ⟨C(t; tw)⟩2]

lim

t→∞ lim tw→∞ χ4(∞; ∞) = χSG: Static spin-glass susceptibility

(Left)χ4 increases

monotonically for no-RFOT case

(right)χ4 has a peak at

finite time t and its height diverges at T = Td for RFOT case.

non-RFOT case RFOT case

Koji Hukushima (Univ. of Tokyo@Komaba) RFOT of spin glasses in 3d 14 August, 2015 24 / 30

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Our numerical results Dynamical properties

Dynamical properties in 3d Potts glass model –Auto-correlation function C(t)–

L = 6

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 100 102 104 106 108 1010 c(t) t(MCS)

L = 6

T/J = 0.852 T/J = 0.781 T/J = 0.711 T/J = 0.640 T/J = 0.605 T/J = 0.570 T/J = 0.548 T/J = 0.527 T/J = 0.506 T/J = 0.491 T/J = 0.477 T/J = 0.463 T/J = 0.449 T/J = 0.435 T/J = 0.429 T/J = 0.421 T/J = 0.414 T/J = 0.407

L = 8

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 100 102 104 106 108 1010 c(t) t(MCS)

L = 8

T/J = 0.852 T/J = 0.781 T/J = 0.711 T/J = 0.640 T/J = 0.605 T/J = 0.570 T/J = 0.548 T/J = 0.527 T/J = 0.506 T/J = 0.491 T/J = 0.477 T/J = 0.463 T/J = 0.449 T/J = 0.435 T/J = 0.429 T/J = 0.421 T/J = 0.414 T/J = 0.407

L = 10

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 100 102 104 106 108 1010 c(t) t(MCS)

L = 10

T/J = 0.852 T/J = 0.781 T/J = 0.711 T/J = 0.640 T/J = 0.605 T/J = 0.570 T/J = 0.548 T/J = 0.527 T/J = 0.506 T/J = 0.491 T/J = 0.477 T/J = 0.463 T/J = 0.449 T/J = 0.435 T/J = 0.429 T/J = 0.421 T/J = 0.414 T/J = 0.407

L = 12

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 100 102 104 106 108 1010 c(t) t(MCS)

L = 12

T/J = 0.852 T/J = 0.781 T/J = 0.711 T/J = 0.640 T/J = 0.605 T/J = 0.570 T/J = 0.548 T/J = 0.527 T/J = 0.506 T/J = 0.491 T/J = 0.477 T/J = 0.463 T/J = 0.449 T/J = 0.435 T/J = 0.429 T/J = 0.421 T/J = 0.414 T/J = 0.407

Static SG transition is located on boundary between blue and red.
No plateau is found, in contrast to the mean-field prediction
No dynamical singularity at Td is also found.

Koji Hukushima (Univ. of Tokyo@Komaba) RFOT of spin glasses in 3d 14 August, 2015 25 / 30

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Our numerical results Dynamical properties

Dynamical properties in 3d Potts glass model –dynamical susceptibility χ4(t)–

L = 6

2 4 6 8 10 12 14 16 18 20 100 101 102 103 104 105 106 107

χ4(t; tw) t - tw(MCS) L = 6

T/J = 0.847 T/J = 0.796 T/J = 0.745 T/J = 0.694 T/J = 0.643 T/J = 0.593 T/J = 0.542 T/J = 0.516 T/J = 0.491 T/J = 0.478 T/J = 0.465 T/J = 0.453

L = 8

2 4 6 8 10 12 14 16 18 20 100 101 102 103 104 105 106 107

χ4(t; tw) t - tw(MCS) L = 8

T/J = 0.847 T/J = 0.796 T/J = 0.745 T/J = 0.694 T/J = 0.643 T/J = 0.593 T/J = 0.542 T/J = 0.516 T/J = 0.491 T/J = 0.478 T/J = 0.465 T/J = 0.453

L = 10

2 4 6 8 10 12 14 16 18 20 100 101 102 103 104 105 106 107

χ4(t; tw) t - tw(MCS) L = 10

T/J = 0.847 T/J = 0.796 T/J = 0.745 T/J = 0.694 T/J = 0.643 T/J = 0.593 T/J = 0.542 T/J = 0.516 T/J = 0.491 T/J = 0.478 T/J = 0.465 T/J = 0.453

L = 12

2 4 6 8 10 12 14 16 18 20 100 101 102 103 104 105 106 107

χ4(t; tw) t - tw(MCS) L = 12

T/J = 0.847 T/J = 0.796 T/J = 0.745 T/J = 0.694 T/J = 0.643 T/J = 0.593 T/J = 0.542 T/J = 0.516 T/J = 0.491 T/J = 0.478 T/J = 0.465 T/J = 0.453

Koji Hukushima (Univ. of Tokyo@Komaba) RFOT of spin glasses in 3d 14 August, 2015 26 / 30

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Our numerical results Dynamical properties

Dynamical properties in 3d Potts glass model –dynamical susceptibility χ4(t)–

L = 6

2 4 6 8 10 12 14 16 18 20 100 101 102 103 104 105 106 107

χ4(t; tw) t - tw(MCS) L = 6

T/J = 0.847 T/J = 0.796 T/J = 0.745 T/J = 0.694 T/J = 0.643 T/J = 0.593 T/J = 0.542 T/J = 0.516 T/J = 0.491 T/J = 0.478 T/J = 0.465 T/J = 0.453

L = 12

2 4 6 8 10 12 14 16 18 20 100 101 102 103 104 105 106 107

χ4(t; tw) t - tw(MCS) L = 12

T/J = 0.847 T/J = 0.796 T/J = 0.745 T/J = 0.694 T/J = 0.643 T/J = 0.593 T/J = 0.542 T/J = 0.516 T/J = 0.491 T/J = 0.478 T/J = 0.465 T/J = 0.453

Before equilibrium limit t → ∞, χ4(t) has a peak at certain time scale
The time scale and the peak height have a divergent tendency
χ4 in equilibrium limit diverges at Tc.

Koji Hukushima (Univ. of Tokyo@Komaba) RFOT of spin glasses in 3d 14 August, 2015 26 / 30

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Our numerical results Dynamical properties

Dynamical properties in 3d Potts glass model

Temperature dependence of the peak in χ4

1 10 0.1 1 10

χ4(t*;tw) Tc/(T-Tc)

L = 12 L = 10 L = 8 L = 6

Peak height χ4(t∗) in χ4 diverges at Tc not Td: χ4(t∗) ∼ |T − Tc|−γ∗
χ4(t∗) has singularity stronger than static χ4(∞): χ4(∞) ∼ |T − Tc|−γ

γ∗ > γ = ⇒Separation between static and dynamics singularities

Koji Hukushima (Univ. of Tokyo@Komaba) RFOT of spin glasses in 3d 14 August, 2015 27 / 30

SLIDE 33

Summary

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 3d 14 August, 2015 28 / 30

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Summary

Random first-order transition (RFOT) of spin glasses
7-state Potts glass model in three dimensions with 1st, 2nd, and 3rd

neighbor couplings.

Our findings

Phys. Rev. E 91, 020102(R)(2015)
SG transition at finite temperature Tc.
A critical exponent ν = 0.68(9) ∼ 2/d.
Overlap has a jump at Tc without latent heat.
P(q) has double-peak structure at and below Tc.
These features are compatible to 1RSB.
This model is a strong candidate which displays RFOT in three

dimensions.

A dynamical singularity in flucutuation is found, indicating a possibility
f Td ≃ Tc.

Koji Hukushima (Univ. of Tokyo@Komaba) RFOT of spin glasses in 3d 14 August, 2015 29 / 30

SLIDE 35

Summary

This research is one step toward the understanding of thermodynamic glass transitions as a transition from liquid to disordered solid.

Koji Hukushima (Univ. of Tokyo@Komaba) RFOT of spin glasses in 3d 14 August, 2015 29 / 30

SLIDE 36

Summary

This research is one step toward the understanding of thermodynamic glass transitions as a transition from liquid to disordered solid. Thank you for your attention.

Koji Hukushima (Univ. of Tokyo@Komaba) RFOT of spin glasses in 3d 14 August, 2015 29 / 30

SLIDE 37

Summary

Financial support

2013-2018 Grant-in-Aid for Scientific Research on Innovative Areas,

MEXT, Japan “Initiative for High-Dimensional Data-Driven Science through Deepening of Sparse Modeling”

JPS Core-to-Core program 2013-2015, “Non-equilibrium dynamics of

soft matter and information”

Koji Hukushima (Univ. of Tokyo@Komaba) RFOT of spin glasses in 3d 14 August, 2015 30 / 30