Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1304.2134 · Journal of Statistical Mechanics Theory and Experiment

Dynamical barriers of pure and random ferromagnetic Ising models on fractal lattices

Cécile Monthus, Thomas Garel

Published Jun 14, 2013• 6 citations
arXivHTMLPDF

Abstract

We consider the stochastic dynamics of the pure and random ferromagnetic Ising model on the hierarchical diamond lattice of branching ratio $K$ with fractal dimension $d_f=(\ln (2K))/\ln 2$. We adapt the Real Space Renormalization procedure introduced in our previous work [C. Monthus and T. Garel, J. Stat. Mech. P02037 (2013)] to study the equilibrium time $t_{eq}(L)$ as a function of the system size $L$ near zero-temperature. For the pure Ising model, we obtain the behavior $t_{eq}(L) \sim L^α e^{β2J L^{d_s}} $ where $d_s=d_f-1$ is the interface dimension, and we compute the prefactor exponent $α$. For the random ferromagnetic Ising model, we derive the renormalization rules for dynamical barriers $B_{eq}(L) \equiv (\ln t_{eq}/β)$ near zero temperature. For the fractal dimension $d_f=2$, we obtain that the dynamical barrier scales as $ B_{eq}(L)= c L+L^{1/2} u$ where $u$ is a Gaussian random variable of non-zero-mean. While the non-random term scaling as $L$ corresponds to the energy-cost of the creation of a system-size domain-wall, the fluctuation part scaling as $L^{1/2}$ characterizes the barriers for the motion of the system-size domain-wall after its creation. This scaling corresponds to the dynamical exponent $ψ=1/2$, in agreement with the conjecture $ψ=d_s/2$ proposed in [C. Monthus and T. Garel, J. Phys. A 41, 115002 (2008)]. In particular, it is clearly different from the droplet exponent $θ\simeq 0.299$ involved in the statics of the random ferromagnet on the same lattice.

DOI
10.1088/1742-5468/2013/06/p06007

Related papers

Dynamical barriers in the Dyson hierarchical model via real space renormalization

Cécile Monthus, Thomas Garel

The Dyson hierarchical one-dimensional Ising model of parameter σ > 0 contains long-ranged ferromagnetic couplings decaying as 1/ r 1+σ in terms of the distance r . We study the stochastic dynamics near zero temperature via the real space renormalization introduced in our previous work (Monthus and Garel, 2012 arXiv: 1212.0643 ) in order to compute explicitly the equilibrium time t eq ( L ) as a function of the system size L . For σ < 1 where the static critical temperature for the ferromagnetic transition is finite, T c > 0, we obtain that dynamical barriers grow as the power law ln t eq ( L ) ≃ β(4 J 0 /3(2 1−σ − 1)) L 1−σ . For σ = 1 where the static critical temperature vanishes, T c = 0, we obtain that dynamical barriers grow logarithmically as ln t eq ( L ) ≃ [β(4 J 0 /3ln2) − 1]ln L . We also compute finite contributions to the dynamical barriers that can depend on the choice of transition rates satisfying detailed balance.

arXiv:1212.4361•Journal of Statistical Mechanics Theory and Experiment•Feb 12, 2013•11 citations

Dynamics of Ising models near zero temperature: real-space renormalization approach

Cécile Monthus, Thomas Garel

We consider the stochastic dynamics of Ising ferromagnets (either pure or random) near zero temperature. The master equation satisfying detailed balance can be mapped onto a quantum Hamiltonian which has an exact zero-energy ground state representing the thermal equilibrium. The largest relaxation time $t_{eq}$ governing the convergence towards this Boltzmann equilibrium in finite-size systems is determined by the lowest non-vanishing eigenvalue $E_1=1/t_{eq}$ of the quantum Hamiltonian $H$. We introduce and study a real-space renormalization procedure for the quantum Hamiltonian associated to the single-spin-flip dynamics of Ising ferromagnets near zero temperature. We solve explicitly the renormalization flow for two cases. (i) For the one-dimensional random ferromagnetic chain with free boundary conditions, the largest relaxation time $t_{eq}$ can be expressed in terms of the set of random couplings for various choices of the dynamical transition rates. The validity of these RG results in $d=1$ is checked by comparison with another approach. (ii) For the pure Ising model on a Cayley tree of branching ratio $K$, we compute the exponential growth of $t_{eq}(N)$ with the number $N$ of generations.

arXiv:1212.0643•Journal of Statistical Mechanics Theory and Experiment•Feb 25, 2013•7 citations

Three-dimensional random-field Ising magnet: Interfaces, scaling, and the nature of states

A. Alan Middleton, Daniel S. Fisher

The nature of the zero-temperature ordering transition in the three-dimensional Gaussian random-field Ising magnet is studied numerically, aided by scaling analyses. Various numerical calculations are used to consistently infer the location of the transition to a high precision. A variety of boundary conditions are imposed on large samples to study the order of the transition and the number of states in the large volume limit. In the ferromagnetic phase, where the domain walls have fractal dimension ${d}_{s}=2,$ the scaling of the roughness of the domain walls, $w\ensuremath{\sim}{L}^{\ensuremath{\zeta}},$ is consistent with the theoretical prediction $\ensuremath{\zeta}=2/3.$ As the randomness is increased through the transition, the probability distribution of the interfacial tension of domain walls scales in a manner that is clearly consistent with a single second-order transition. At the critical point, the fractal dimensions of domain walls and the fractal dimension of the outer surface of spin clusters are investigated: there are at least two distinct physically important fractal dimensions that describe domain walls. These dimensions are argued to be related by scaling to combinations of the energy scaling exponent $\ensuremath{\theta},$ which determines the violation of hyperscaling, the correlation length exponent $\ensuremath{\nu},$ and the magnetization exponent $\ensuremath{\beta}.$ The value $\ensuremath{\beta}=0.017\ifmmode\pm\else\textpm\fi{}0.005$ computed from finite-size scaling of the magnetization is very nearly zero: this estimate is supported by the study of the spin cluster size distribution at criticality. The variation of configurations in the interior of a sample with boundary conditions is consistent with the hypothesis that there is a single transition separating the disordered phase with one ground state from the ordered phase with two ground states. The array of results, including values for several exponents, are shown to be consistent with a scaling picture and a geometric description of the influence of boundary conditions on the spins. The details of the algorithm used and its implementation are also described.

arXiv:cond-mat/0107489•Physical review. B, Condensed matter•Mar 19, 2002•172 citations

Nonequilibrium dynamics of random field Ising spin chains: Exact results via real space renormalization group

Daniel S. Fisher, Pierre Le Doussal, Cécile Monthus

The nonequilibrium dynamics of classical random Ising spin chains with nonconserved magnetization are studied using an asymptotically exact real space renormalization group (RSRG). We focus on random field Ising model (RFIM) spin chains with and without a uniform applied field, as well as on Ising spin glass chains in an applied field. For the RFIM we consider a universal regime where the random field and the temperature are both much smaller than the exchange coupling. In this regime, the Imry-Ma length that sets the scale of the equilibrium correlations is large and the coarsening of domains from random initial conditions (e.g., a quench from high temperature) occurs over a wide range of length scales. The two types of domain walls that occur diffuse in opposite random potentials, of the form studied by Sinai, and domain walls annihilate when they meet. Using the RSRG we compute many universal asymptotic properties of both the nonequilibrium dynamics and the equilibrium limit. We find that the configurations of the domain walls converge rapidly toward a set of system-specific time-dependent positions that are independent of the initial conditions. Thus the behavior of this nonequilibrium system is pseudodeterministic at long times because of the broad distributions of barriers that occur on the long length scales involved. Specifically, we obtain the time dependence of the energy, the magnetization, and the distribution of domain sizes (found to be statistically independent). The equilibrium limits agree with known exact results. We obtain the exact scaling form of the two-point equal time correlation function <S0(t)Sx(t)> and the two-time autocorrelations <S0(t('))S0(t)>. We also compute the persistence properties of a single spin, of local magnetization, and of domains. The analogous quantities for the +/-J Ising spin glass in an applied field are obtained from the RFIM via a gauge transformation. In addition to these we compute the two-point two-time correlation function <S0(t)Sx(t)><S0(t('))Sx(t('))> which can in principle be measured by experiments on spin-glass-like systems. The thermal fluctuations are studied and found to be dominated by rare events; in particular all moments of truncated equal time correlations are computed. Physical properties which are typically measured in aging experiments are also studied, focusing on the response to a small magnetic field which is applied after waiting for the system to equilibrate for a time t(w). The nonequilibrium fluctuation-dissipation ratio X(t,t(w)) is computed. We find that for (t-t(w)) approximately t(alpha)(w) with alpha;<1, the ratio equal to its equilibrium value X=1, although time translational invariance does not hold in this regime. For t-t(w) approximately t(w) the ratio exhibits an aging regime with a nontrivial X=X(t/t(w)) not equal to 1, but the behavior is markedly different from mean field theory. Finally the distribution of the total magnetization and of the number of domains is computed for large finite size systems. General issues about convergence toward equilibrium and the possibilities of weakly history-dependent evolution in other random systems are discussed.

arXiv:cond-mat/0012290•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Nov 14, 2001•77 citations

Strong disorder RG principles within a fixed-cell-size real space renormalization: application to the random transverse field Ising model on various fractal lattices

Cécile Monthus, Thomas Garel

Strong disorder renormalization is an energy-based renormalization that leads to a complicated renormalized topology for the surviving clusters as soon as d > 1. In this paper, we propose to include strong disorder renormalization ideas within the more traditional fixed-cell-size real space RG framework. We first consider the one-dimensional chain as a test for this fixed-cell-size procedure: we find that all exactly known critical exponents are reproduced correctly, except for the magnetic exponent β (because it is related to more subtle persistence properties of the full RG flow). We then apply numerically this fixed-cell-size procedure to two types of renormalizable fractal lattice: (i) the Sierpinski gasket of fractal dimension D = ln3/ln2, where there is no underlying classical ferromagnetic transition, so that the RG flow in the ordered phase is similar to what happens in d = 1; (ii) a hierarchical diamond lattice of fractal dimension D = 4/3, where there is an underlying classical ferromagnetic transition, so that the RG flow in the ordered phase is similar to what happens on hypercubic lattices of dimension d > 1. In both cases, we find that the transition is governed by an infinite disorder fixed point: besides the measure of the activated exponent ψ, we analyze the RG flow of various observables in the disordered and ordered phases, in order to extract the 'typical' correlation length exponents of these two phases which are different from the finite-size correlation length exponent.

arXiv:1201.6136•Journal of Statistical Mechanics Theory and Experiment•May 3, 2012•11 citations

Critical dynamics of the kinetic Glauber-Ising model on hierarchical lattices

Xiang-Mu Kong, Zhao‐Di Yang

The critical dynamics of the kinetic Glauber-Ising model is studied on a family of the diamond-type hierarchical lattices with various branches. By carrying out the time-dependent real-space renormalization-group transformation to the master equation of the systems considered, the dynamic exponent is calculated. We find that the dynamic exponent depends on fractal dimension d(f) or the branch number m in a generator, and that it increases with the increase of d(f) or m. We notice that for the case of m=1 (one-dimensional spin chain, d(f)=1) our result z=2 is the same as the exact result obtained by Glauber, and for the case of m=2 (the simplest one in the diamond-type hierarchical lattices, d(f)=2) the exponent z=2.626 is higher than those of the two-dimensional regular lattice and the triangular lattice.

Physical Review E•Jan 13, 2004•4 citations
We consider the stochastic dynamics of the pure and random ferromagnetic Ising model on the hierarchical diamond lattice of branching ratio KKK with fractal dimension df=(ln⁡(2K))/ln⁡2d_f=(\ln (2K))/\ln 2df​=(ln(2K))/ln2. We adapt the Real Space Renormalization procedure introduced in our previous work [C. Monthus and T. Garel, J. Stat. Mech. P02037 (2013)] to study the equilibrium time teq(L)t_{eq}(L)teq​(L) as a function of the system size LLL near zero-temperature. For the pure Ising model, we obtain the behavior teq(L)∼Lαeβ2JLdst_{eq}(L) \sim L^α e^{β2J L^{d_s}} teq​(L)∼Lαeβ2JLds​ where ds=df−1d_s=d_f-1ds​=df​−1 is the interface dimension, and we compute the prefactor exponent ααα. For the random ferromagnetic Ising model, we derive the renormalization rules for dynamical barriers Beq(L)≡(ln⁡teq/β)B_{eq}(L) \equiv (\ln t_{eq}/β)Beq​(L)≡(lnteq​/β) near zero temperature. For the fractal dimension df=2d_f=2df​=2, we obtain that the dynamical barrier scales as Beq(L)=cL+L1/2u B_{eq}(L)= c L+L^{1/2} uBeq​(L)=cL+L1/2u where uuu is a Gaussian random variable of non-zero-mean. While the non-random term scaling as LLL corresponds to the energy-cost of the creation of a system-size domain-wall, the fluctuation part scaling as L1/2L^{1/2}L1/2 characterizes the barriers for the motion of the system-size domain-wall after its creation. This scaling corresponds to the dynamical exponent ψ=1/2ψ=1/2ψ=1/2, in agreement with the conjecture ψ=ds/2ψ=d_s/2ψ=ds​/2 proposed in [C. Monthus and T. Garel, J. Phys. A 41, 115002 (2008)]. In particular, it is clearly different from the droplet exponent θ≃0.299θ\simeq 0.299θ≃0.299 involved in the statics of the random ferromagnet on the same lattice.
The Dyson hierarchical one-dimensional Ising model of parameter σ > 0 contains long-ranged ferromagnetic couplings decaying as 1/ r 1+σ in terms of the distance r . We study the stochastic dynamics near zero temperature via the real space renormalization introduced in our previous work (Monthus and Garel, 2012 arXiv: 1212.0643 ) in order to compute explicitly the equilibrium time t eq ( L ) as a function of the system size L . For σ < 1 where the static critical temperature for the ferromagnetic transition is finite, T c > 0, we obtain that dynamical barriers grow as the power law ln t eq ( L ) ≃ β(4 J 0 /3(2 1−σ − 1)) L 1−σ . For σ = 1 where the static critical temperature vanishes, T c = 0, we obtain that dynamical barriers grow logarithmically as ln t eq ( L ) ≃ [β(4 J 0 /3ln2) − 1]ln L . We also compute finite contributions to the dynamical barriers that can depend on the choice of transition rates satisfying detailed balance.
We consider the stochastic dynamics of Ising ferromagnets (either pure or random) near zero temperature. The master equation satisfying detailed balance can be mapped onto a quantum Hamiltonian which has an exact zero-energy ground state representing the thermal equilibrium. The largest relaxation time teqt_{eq}teq​ governing the convergence towards this Boltzmann equilibrium in finite-size systems is determined by the lowest non-vanishing eigenvalue E1=1/teqE_1=1/t_{eq}E1​=1/teq​ of the quantum Hamiltonian HHH. We introduce and study a real-space renormalization procedure for the quantum Hamiltonian associated to the single-spin-flip dynamics of Ising ferromagnets near zero temperature. We solve explicitly the renormalization flow for two cases. (i) For the one-dimensional random ferromagnetic chain with free boundary conditions, the largest relaxation time teqt_{eq}teq​ can be expressed in terms of the set of random couplings for various choices of the dynamical transition rates. The validity of these RG results in d=1d=1d=1 is checked by comparison with another approach. (ii) For the pure Ising model on a Cayley tree of branching ratio KKK, we compute the exponential growth of teq(N)t_{eq}(N)teq​(N) with the number NNN of generations.
The nature of the zero-temperature ordering transition in the three-dimensional Gaussian random-field Ising magnet is studied numerically, aided by scaling analyses. Various numerical calculations are used to consistently infer the location of the transition to a high precision. A variety of boundary conditions are imposed on large samples to study the order of the transition and the number of states in the large volume limit. In the ferromagnetic phase, where the domain walls have fractal dimension ds=2,{d}_{s}=2,ds​=2, the scaling of the roughness of the domain walls, $w\ensuremath{\sim}{L}^{\ensuremath{\zeta}},$ is consistent with the theoretical prediction $\ensuremath{\zeta}=2/3.$ As the randomness is increased through the transition, the probability distribution of the interfacial tension of domain walls scales in a manner that is clearly consistent with a single second-order transition. At the critical point, the fractal dimensions of domain walls and the fractal dimension of the outer surface of spin clusters are investigated: there are at least two distinct physically important fractal dimensions that describe domain walls. These dimensions are argued to be related by scaling to combinations of the energy scaling exponent $\ensuremath{\theta},$ which determines the violation of hyperscaling, the correlation length exponent $\ensuremath{\nu},$ and the magnetization exponent $\ensuremath{\beta}.$ The value $\ensuremath{\beta}=0.017\ifmmode\pm\else\textpm\fi{}0.005$ computed from finite-size scaling of the magnetization is very nearly zero: this estimate is supported by the study of the spin cluster size distribution at criticality. The variation of configurations in the interior of a sample with boundary conditions is consistent with the hypothesis that there is a single transition separating the disordered phase with one ground state from the ordered phase with two ground states. The array of results, including values for several exponents, are shown to be consistent with a scaling picture and a geometric description of the influence of boundary conditions on the spins. The details of the algorithm used and its implementation are also described.
The nonequilibrium dynamics of classical random Ising spin chains with nonconserved magnetization are studied using an asymptotically exact real space renormalization group (RSRG). We focus on random field Ising model (RFIM) spin chains with and without a uniform applied field, as well as on Ising spin glass chains in an applied field. For the RFIM we consider a universal regime where the random field and the temperature are both much smaller than the exchange coupling. In this regime, the Imry-Ma length that sets the scale of the equilibrium correlations is large and the coarsening of domains from random initial conditions (e.g., a quench from high temperature) occurs over a wide range of length scales. The two types of domain walls that occur diffuse in opposite random potentials, of the form studied by Sinai, and domain walls annihilate when they meet. Using the RSRG we compute many universal asymptotic properties of both the nonequilibrium dynamics and the equilibrium limit. We find that the configurations of the domain walls converge rapidly toward a set of system-specific time-dependent positions that are independent of the initial conditions. Thus the behavior of this nonequilibrium system is pseudodeterministic at long times because of the broad distributions of barriers that occur on the long length scales involved. Specifically, we obtain the time dependence of the energy, the magnetization, and the distribution of domain sizes (found to be statistically independent). The equilibrium limits agree with known exact results. We obtain the exact scaling form of the two-point equal time correlation function <S0(t)Sx(t)> and the two-time autocorrelations <S0(t('))S0(t)>. We also compute the persistence properties of a single spin, of local magnetization, and of domains. The analogous quantities for the +/-J Ising spin glass in an applied field are obtained from the RFIM via a gauge transformation. In addition to these we compute the two-point two-time correlation function <S0(t)Sx(t)><S0(t('))Sx(t('))> which can in principle be measured by experiments on spin-glass-like systems. The thermal fluctuations are studied and found to be dominated by rare events; in particular all moments of truncated equal time correlations are computed. Physical properties which are typically measured in aging experiments are also studied, focusing on the response to a small magnetic field which is applied after waiting for the system to equilibrate for a time t(w). The nonequilibrium fluctuation-dissipation ratio X(t,t(w)) is computed. We find that for (t-t(w)) approximately t(alpha)(w) with alpha;<1, the ratio equal to its equilibrium value X=1, although time translational invariance does not hold in this regime. For t-t(w) approximately t(w) the ratio exhibits an aging regime with a nontrivial X=X(t/t(w)) not equal to 1, but the behavior is markedly different from mean field theory. Finally the distribution of the total magnetization and of the number of domains is computed for large finite size systems. General issues about convergence toward equilibrium and the possibilities of weakly history-dependent evolution in other random systems are discussed.
Strong disorder renormalization is an energy-based renormalization that leads to a complicated renormalized topology for the surviving clusters as soon as d > 1. In this paper, we propose to include strong disorder renormalization ideas within the more traditional fixed-cell-size real space RG framework. We first consider the one-dimensional chain as a test for this fixed-cell-size procedure: we find that all exactly known critical exponents are reproduced correctly, except for the magnetic exponent β (because it is related to more subtle persistence properties of the full RG flow). We then apply numerically this fixed-cell-size procedure to two types of renormalizable fractal lattice: (i) the Sierpinski gasket of fractal dimension D = ln3/ln2, where there is no underlying classical ferromagnetic transition, so that the RG flow in the ordered phase is similar to what happens in d = 1; (ii) a hierarchical diamond lattice of fractal dimension D = 4/3, where there is an underlying classical ferromagnetic transition, so that the RG flow in the ordered phase is similar to what happens on hypercubic lattices of dimension d > 1. In both cases, we find that the transition is governed by an infinite disorder fixed point: besides the measure of the activated exponent ψ, we analyze the RG flow of various observables in the disordered and ordered phases, in order to extract the 'typical' correlation length exponents of these two phases which are different from the finite-size correlation length exponent.
The critical dynamics of the kinetic Glauber-Ising model is studied on a family of the diamond-type hierarchical lattices with various branches. By carrying out the time-dependent real-space renormalization-group transformation to the master equation of the systems considered, the dynamic exponent is calculated. We find that the dynamic exponent depends on fractal dimension d(f) or the branch number m in a generator, and that it increases with the increase of d(f) or m. We notice that for the case of m=1 (one-dimensional spin chain, d(f)=1) our result z=2 is the same as the exact result obtained by Glauber, and for the case of m=2 (the simplest one in the diamond-type hierarchical lattices, d(f)=2) the exponent z=2.626 is higher than those of the two-dimensional regular lattice and the triangular lattice.