Phase Transitions in Disordered Systems: The Example of the 4D Random Field Ising Model

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1 Phase Transitions in Disordered Systems: The Example of the 4D Random Field Ising Model Nikos Fytas Applied Mathematics Research Centre, Coventry University, United Kingdom Work in collaboration with V. Martín-Mayor (Madrid), M. Picco (Paris), and N. Sourlas (Paris). November 24, 2016

2 Introduction

3 Introduction The Hamiltonian of the random-field Ising model (RFIM): H (RFIM) = J x,y S x S y x h x S x, ; S x = ±1 ; J > 0.

4 Introduction The Hamiltonian of the random-field Ising model (RFIM): H (RFIM) = J x,y S x S y x h x S x, ; S x = ±1 ; J > 0. {h x } are independent quenched random fields via P(h, σ).

5 Introduction The Hamiltonian of the random-field Ising model (RFIM): H (RFIM) = J x,y S x S y x h x S x, ; S x = ±1 ; J > 0. {h x } are independent quenched random fields via P(h, σ). At low T and for σ J we encounter the ferromagnetic phase, provided that D 3.

6 Introduction The Hamiltonian of the random-field Ising model (RFIM): H (RFIM) = J x,y S x S y x h x S x, ; S x = ±1 ; J > 0. {h x } are independent quenched random fields via P(h, σ). At low T and for σ J we encounter the ferromagnetic phase, provided that D 3. For D = 2, the tiniest σ > 0 suffices to destroy the ferromagnetic phase.

7 Introduction The Hamiltonian of the random-field Ising model (RFIM): H (RFIM) = J x,y S x S y x h x S x, ; S x = ±1 ; J > 0. {h x } are independent quenched random fields via P(h, σ). At low T and for σ J we encounter the ferromagnetic phase, provided that D 3. For D = 2, the tiniest σ > 0 suffices to destroy the ferromagnetic phase. Perturbative RG (PRG) computations suggest D u = 6 (for D D u : mean-field exponents).

8 Supersymmetry and dimensional reduction

9 Supersymmetry and dimensional reduction The RFIM and the branched polymers are unique among disordered systems: supersymmetry makes it possible to analyze the PRG to all orders of perturbation theory.

10 Supersymmetry and dimensional reduction The RFIM and the branched polymers are unique among disordered systems: supersymmetry makes it possible to analyze the PRG to all orders of perturbation theory. Supersymmetry predicts dimensional reduction: RFIM (D) Ising (D 2). Yet, the RFIM orders in D = 3 while the Ising ferromagnet in D = 1 does not.

11 Supersymmetry and dimensional reduction The RFIM and the branched polymers are unique among disordered systems: supersymmetry makes it possible to analyze the PRG to all orders of perturbation theory. Supersymmetry predicts dimensional reduction: RFIM (D) Ising (D 2). Yet, the RFIM orders in D = 3 while the Ising ferromagnet in D = 1 does not. The failure of the PRG begs the question: Is there an intermediate dimension D int < D u such that the PRG is accurate for D > D int?

12 RG fixed point

13 RG fixed point The relevant RG fixed-point lies at T = 0 and the flow is described by three independent critical exponents, ν, η, and η, and two correlation functions, C xy (con) (connected) and C xy (dis) (disconnected): C xy (con) S x 1 (dis) ; C h y r D 2+η xy S x S y 1 r D 4+η.

14 RG fixed point The relevant RG fixed-point lies at T = 0 and the flow is described by three independent critical exponents, ν, η, and η, and two correlation functions, C xy (con) (connected) and C xy (dis) (disconnected): C xy (con) S x 1 (dis) ; C h y r D 2+η xy S x S y 1 r D 4+η. The relationship between the anomalous dimensions η and η is hotly debated for many years now and is one of the main themes of the present work.

15 Previous conjectures on the η vs. η relationship

16 Previous conjectures on the η vs. η relationship Supersymmetry (Parisi and Sourlas, 1979): η = η.

17 Previous conjectures on the η vs. η relationship Supersymmetry (Parisi and Sourlas, 1979): η = η. Phenomenological scaling (Fisher, Schwartz and coworkers, 1986): η = 2η.

18 Previous conjectures on the η vs. η relationship Supersymmetry (Parisi and Sourlas, 1979): η = η. Phenomenological scaling (Fisher, Schwartz and coworkers, 1986): η = 2η. Functional RG (Tarjus and coworkers, 2011): rare events spontaneously break supersymmetry at the intermediate dimension D int :

19 Previous conjectures on the η vs. η relationship Supersymmetry (Parisi and Sourlas, 1979): η = η. Phenomenological scaling (Fisher, Schwartz and coworkers, 1986): η = 2η. Functional RG (Tarjus and coworkers, 2011): rare events spontaneously break supersymmetry at the intermediate dimension D int : For D > D int : η = η.

20 Previous conjectures on the η vs. η relationship Supersymmetry (Parisi and Sourlas, 1979): η = η. Phenomenological scaling (Fisher, Schwartz and coworkers, 1986): η = 2η. Functional RG (Tarjus and coworkers, 2011): rare events spontaneously break supersymmetry at the intermediate dimension D int : For D > D int : η = η. For D < D int : η η.

21 Previous conjectures on the η vs. η relationship Supersymmetry (Parisi and Sourlas, 1979): η = η. Phenomenological scaling (Fisher, Schwartz and coworkers, 1986): η = 2η. Functional RG (Tarjus and coworkers, 2011): rare events spontaneously break supersymmetry at the intermediate dimension D int : For D > D int : η = η. For D < D int : η η. D int 5.1.

22 Latest numerical results at D = 3 2η η = (9) ; χ 2 /DOF = 10.5/17 2η η = 0 (fixed) ; χ 2 /DOF = 18.3/ (e ff) (2 η - η) G d G d G P (σ = 1 ) (σ = 2 ) ω L 1 N.G. Fytas and V. Martín-Mayor, PRL 110, (2013)

23 Targets of the present work at D = 4

24 Targets of the present work at D = 4 1 Provide high-accuracy estimates for the critical exponents ν, η, and η, as well as for the corrections-to-scaling exponent ω and of other RG-invariants.

25 Targets of the present work at D = 4 1 Provide high-accuracy estimates for the critical exponents ν, η, and η, as well as for the corrections-to-scaling exponent ω and of other RG-invariants. 2 Clear out the puzzle with the number of independent critical exponents, compared to the inconclusive case of the D = 3 RFIM.

26 Targets of the present work at D = 4 1 Provide high-accuracy estimates for the critical exponents ν, η, and η, as well as for the corrections-to-scaling exponent ω and of other RG-invariants. 2 Clear out the puzzle with the number of independent critical exponents, compared to the inconclusive case of the D = 3 RFIM. 3 Examine previous claims of universality violations for the RFIM when comparing different distributions of random fields.

27 Targets of the present work at D = 4 1 Provide high-accuracy estimates for the critical exponents ν, η, and η, as well as for the corrections-to-scaling exponent ω and of other RG-invariants. 2 Clear out the puzzle with the number of independent critical exponents, compared to the inconclusive case of the D = 3 RFIM. 3 Examine previous claims of universality violations for the RFIM when comparing different distributions of random fields. 4 Check the validity of dimensional reduction.

28 Simulation details

29 Simulation details We consider the RFIM on a D = 4 hyper-cubic lattice with periodic boundary conditions and energy units J = 1. Our random fields {h x } follow either a Gaussian or a Poissonian distribution: P G (h, σ) = 1 h2 e 2σ 2, P P (h, σ) = 1 h 2πσ 2 2 σ e σ, where < h <. For both distributions σ is our single control parameter.

30 Simulation details We consider the RFIM on a D = 4 hyper-cubic lattice with periodic boundary conditions and energy units J = 1. Our random fields {h x } follow either a Gaussian or a Poissonian distribution: P G (h, σ) = 1 h2 e 2σ 2, P P (h, σ) = 1 h 2πσ 2 2 σ e σ, where < h <. For both distributions σ is our single control parameter. We use a home-made version of the push-relabel algorithm of Tarjan and Goldberg to generate the ground states of the system.

31 Simulation details We consider the RFIM on a D = 4 hyper-cubic lattice with periodic boundary conditions and energy units J = 1. Our random fields {h x } follow either a Gaussian or a Poissonian distribution: P G (h, σ) = 1 h2 e 2σ 2, P P (h, σ) = 1 h 2πσ 2 2 σ e σ, where < h <. For both distributions σ is our single control parameter. We use a home-made version of the push-relabel algorithm of Tarjan and Goldberg to generate the ground states of the system. We simulated lattice sizes from L = 4 to L = 60. For each pair (L, σ) we computed ground states for 10 7 samples.

32 Observables 2 N.G. Fytas and V. Martín-Mayor, PRE 93, (2016)

33 Observables From simulations at a given σ, we obtained σ-derivatives and extrapolated to neighboring σ values by means of a reweighting method. 2 We computed the following observables: 2 N.G. Fytas and V. Martín-Mayor, PRE 93, (2016)

34 Observables From simulations at a given σ, we obtained σ-derivatives and extrapolated to neighboring σ values by means of a reweighting method. 2 We computed the following observables: χ (con) and χ (dis), 2 N.G. Fytas and V. Martín-Mayor, PRE 93, (2016)

35 Observables From simulations at a given σ, we obtained σ-derivatives and extrapolated to neighboring σ values by means of a reweighting method. 2 We computed the following observables: χ (con) and χ (dis), ξ (con) and ξ (dis), 2 N.G. Fytas and V. Martín-Mayor, PRE 93, (2016)

36 Observables From simulations at a given σ, we obtained σ-derivatives and extrapolated to neighboring σ values by means of a reweighting method. 2 We computed the following observables: χ (con) and χ (dis), ξ (con) and ξ (dis), U 4 = m 4 / m 2 2, and 2 N.G. Fytas and V. Martín-Mayor, PRE 93, (2016)

37 Observables From simulations at a given σ, we obtained σ-derivatives and extrapolated to neighboring σ values by means of a reweighting method. 2 We computed the following observables: χ (con) and χ (dis), ξ (con) and ξ (dis), U 4 = m 4 / m 2 2, and U 22 = χ (dis) /[χ (con) ] 2 = 2η η. 2 N.G. Fytas and V. Martín-Mayor, PRE 93, (2016)

38 Finite-size scaling scheme

39 Finite-size scaling scheme Quotients method: We compare observables computed in pairs (L, 2L). Scale-invariance is imposed by seeking the L-dependent critical point: the value of σ such that ξ 2L /ξ L = 2. Here, we consider both ξ (con) /L and ξ (dis) /L.

40 Finite-size scaling scheme Quotients method: We compare observables computed in pairs (L, 2L). Scale-invariance is imposed by seeking the L-dependent critical point: the value of σ such that ξ 2L /ξ L = 2. Here, we consider both ξ (con) /L and ξ (dis) /L. ξ (c o n ) / L L = 6 L = 1 2 L = 1 6 L = 3 2 L = 2 6 L = G a u s s ia n R F IM σ

41 For dimensionful quantities O, scaling in the thermodynamic limit as ξ x O/ν, we consider the quotient Q O = O 2L /O L at the crossing. For dimensionless magnitudes g, we focus on g L or g 2L, whichever show less finite-size corrections. In either case, one has: Q cross O = 2 x O/ν + O(L ω ), g cross (L);(2L) = g + O(L ω ), where x O /ν, g and the scaling-corrections exponent ω are universal.

42 For dimensionful quantities O, scaling in the thermodynamic limit as ξ x O/ν, we consider the quotient Q O = O 2L /O L at the crossing. For dimensionless magnitudes g, we focus on g L or g 2L, whichever show less finite-size corrections. In either case, one has: Q cross O = 2 x O/ν + O(L ω ), g cross (L);(2L) = g + O(L ω ), where x O /ν, g and the scaling-corrections exponent ω are universal. Dimensionless quantities: ξ (con) /L, ξ (dis) /L and U 4.

43 For dimensionful quantities O, scaling in the thermodynamic limit as ξ x O/ν, we consider the quotient Q O = O 2L /O L at the crossing. For dimensionless magnitudes g, we focus on g L or g 2L, whichever show less finite-size corrections. In either case, one has: Q cross O = 2 x O/ν + O(L ω ), g cross (L);(2L) = g + O(L ω ), where x O /ν, g and the scaling-corrections exponent ω are universal. Dimensionless quantities: ξ (con) /L, ξ (dis) /L and U 4. Dimensionful quantities: Derivatives of ξ (con), ξ (dis) [x ξ = 1 + ν], Derivatives of χ (con) and χ (dis) [x χ (con) = ν(2 η), x χ (dis) = ν(4 η)], U 22 [x U22 = ν(2η η)].

44 Fitting details

45 Fitting details 1 We restrict ourselves to data with L L min. To determine an acceptable L min we employ the standard χ 2 -test for goodness of fit, where χ 2 is computed using the complete covariance matrix.

46 Fitting details 1 We restrict ourselves to data with L L min. To determine an acceptable L min we employ the standard χ 2 -test for goodness of fit, where χ 2 is computed using the complete covariance matrix. 2 We fit 4 data sets:

47 Fitting details 1 We restrict ourselves to data with L L min. To determine an acceptable L min we employ the standard χ 2 -test for goodness of fit, where χ 2 is computed using the complete covariance matrix. 2 We fit 4 data sets: 2 random-field distributions: Gaussian and Poissonian,

48 Fitting details 1 We restrict ourselves to data with L L min. To determine an acceptable L min we employ the standard χ 2 -test for goodness of fit, where χ 2 is computed using the complete covariance matrix. 2 We fit 4 data sets: 2 random-field distributions: Gaussian and Poissonian, 2 crossing points: ξ (con) /L and ξ (dis) /L,

49 Fitting details 1 We restrict ourselves to data with L L min. To determine an acceptable L min we employ the standard χ 2 -test for goodness of fit, where χ 2 is computed using the complete covariance matrix. 2 We fit 4 data sets: 2 random-field distributions: Gaussian and Poissonian, 2 crossing points: ξ (con) /L and ξ (dis) /L, We denote these as: G (con), G (dis), P (con), and P (dis).

50 A spectacular example of non-monotonic behavior Possible explanation of previously reported universality violations G (c o n ) P (c o n ) L m a x = U / L

51 A spectacular example of non-monotonic behavior Possible explanation of previously reported universality violations G (c o n ) P (c o n ) L m a x = U / L Higher-order corrections are necessary: X L = X + a 1 L ω + a 2 L 2ω

52 Universality in the 4D RFIM Joint fit of ξ (con) /L and η ω = 1.30(9) ; ξ (con) /L = (8) ; η = (13) χ 2 /DOF = 27.85/29 ξ (c o n ) / L (e ff) η (a ) (b ) G (c o n ) (d is ) G P (c o n ) (d is ) P L -ω

53 Universal ratio ξ (dis) /L ξ (dis) /L = (36)(34) χ 2 /DOF = 16/ ξ (d is ) / L G (c o n ) (d is ) G P (c o n ) (d is ) P L -ω

54 Binder cumulant U 4 U 4 = (32)(14) χ 2 /DOF = 10/11 U G (c o n ) (d is ) G P (c o n ) (d is ) P L -ω

55 Extrapolation of ν ν = (58)(19) χ 2 /DOF = 62.9/ ξ (c o n ) / L G (c o n ) (d is ) G P (c o n ) (d is ) P (e ff) ν ξ (d is ) / L 0.7 G (c o n ) (d is ) G P (c o n ) (d is ) G L -ω

56 Extrapolation of 2η η 2η η = (23)(1) χ 2 /DOF = 16.0/ (e ff) (2 η - η) G (c o n ) (d is ) G P (c o n ) ω 1.6 (d is ) P (2 η - η) L -ω

57 Critical fields σ c,l = σ c + b 1 L (ω+ 1 ν ) + b 2 L (2ω+ 1 ν ) σ c (G) = (4)(2) ; χ 2 /DOF = 5.6/7 σ c (P) = (3)(8) ; χ 2 /DOF = 8.85/11 (c ro s s ) σ P (c o n ) G (d is ) P G (c o n ) (d is ) /L

58 Summary of universal ratios and exponents QF χ 2 =DOF ω 1.30(9) ξ ðconþ =L (8) 27.85=29 η (13) σ c ðgþ (4)(2) 5.6=7 σ c ðpþ (3)(8) 8.85=11 U (32)(14) 10=11 ξ ðdisþ =L (36)(34) 16=15 ν (58)(19) 62.9=55 2η η (23)(1) 16.0=19 Hartmann, PRB 65, (2002): σ c (G) = 4.18(1); ν = 0.78(10) Middleton, arxiv:cond-mat/ : σ c (G) = 4.179(2); ν = 0.82(6)

59 Conclusions

60 Conclusions We have been able to show universality by comparing different field distributions. To reach this conclusion, we had to identify and control the role of scaling corrections. In doing so, we provided:

61 Conclusions We have been able to show universality by comparing different field distributions. To reach this conclusion, we had to identify and control the role of scaling corrections. In doing so, we provided: An original estimate of the exponent ω.

62 Conclusions We have been able to show universality by comparing different field distributions. To reach this conclusion, we had to identify and control the role of scaling corrections. In doing so, we provided: An original estimate of the exponent ω. Original estimates of RG invariants: ξ (con) /L, ξ (dis) /L, and U 4.

63 Conclusions We have been able to show universality by comparing different field distributions. To reach this conclusion, we had to identify and control the role of scaling corrections. In doing so, we provided: An original estimate of the exponent ω. Original estimates of RG invariants: ξ (con) /L, ξ (dis) /L, and U 4. We determined with high accuracy the three independent critical exponents ν, η, and η, that are needed to describe the transition.

64 Conclusions We have been able to show universality by comparing different field distributions. To reach this conclusion, we had to identify and control the role of scaling corrections. In doing so, we provided: An original estimate of the exponent ω. Original estimates of RG invariants: ξ (con) /L, ξ (dis) /L, and U 4. We determined with high accuracy the three independent critical exponents ν, η, and η, that are needed to describe the transition. We stress the non-trivial difference 2η η = (24) which is 10 times larger than its corresponding 3D value (9).

65 Conclusions We have been able to show universality by comparing different field distributions. To reach this conclusion, we had to identify and control the role of scaling corrections. In doing so, we provided: An original estimate of the exponent ω. Original estimates of RG invariants: ξ (con) /L, ξ (dis) /L, and U 4. We determined with high accuracy the three independent critical exponents ν, η, and η, that are needed to describe the transition. We stress the non-trivial difference 2η η = (24) which is 10 times larger than its corresponding 3D value (9). We provided decisive evidence in favor of the three-exponent scaling scenario and the spontaneous supersymmetry breaking at some D int > 4.

66 Acknowledgements Our L = 52, 60 lattices were simulated in the MareNostrum and Picasso supercomputers. We thankfully acknowledge the computer resources and assistance provided by the staff at the Red Española de Supercomputación. Computational time in the cluster Memento (BIFI Institute, Zaragoza). Coventry University for providing a Research Sabbatical Fellowship during which this work has been completed.

67 Work in progress: RFIM at D = 5 ν (5D RFIM) = 0.626(15) (4) = ν (3D IM) = D int (e ff) ν G (c o n ) (d is ) G G (U ) 4 P (c o n ) (d is ) P P (U ) L -ω

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