We present an analytic study of finite-size effects on critical diffusion above and below ${T}_{c}$ of three-dimensional Ising-like systems whose order parameter is coupled to a conserved density. We also calculate the finite-size relaxation time that governs the critical order-parameter relaxation towards a metastable equilibrium state below ${T}_{c}.$ Two universal dynamic amplitude ratios at ${T}_{c}$ are predicted and quantitative predictions of dynamic finite-size scaling functions are given that can be tested by Monte Carlo simulations.We present an analytic study of finite-size effects on critical diffusion above and below ${T}_{c}$ of three-dimensional Ising-like systems whose order parameter is coupled to a conserved density. We also calculate the finite-size relaxation time that governs the critical order-parameter relaxation towards a metastable equilibrium state below ${T}_{c}.$ Two universal dynamic amplitude ratios at ${T}_{c}$ are predicted and quantitative predictions of dynamic finite-size scaling functions are given that can be tested by Monte Carlo simulations.

arXiv:cond-mat/9802206Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsAug 1, 199821 citations

Relaxation dynamics of the toric code in contact with a thermal reservoir: Finite-size scaling in a low-temperature regime

C. Daniel Freeman, C. M. Herdman, Dylan J Gorman +1 more

We present an analysis of the relaxation dynamics of finite-size topological qubits in contact with a thermal bath. Using a continuous-time Monte Carlo method, we explicitly compute the low-temperature nonequilibrium dynamics of the toric code on finite lattices. In contrast to the size-independent bound predicted for the toric code in the thermodynamic limit, we identify a low-temperature regime on finite lattices below a size-dependent crossover temperature with nontrivial finite-size and temperature scaling of the relaxation time. We demonstrate how this nontrivial finite-size scaling is governed by the scaling of topologically nontrivial two-dimensional classical random walks. The transition out of this low-temperature regime defines a dynamical finite-size crossover temperature that scales inversely with the log of the system size, in agreement with a crossover temperature defined from equilibrium properties. We find that both the finite-size and finite-temperature scaling are stronger in the low-temperature regime than above the crossover temperature. Since this finite-temperature scaling competes with the scaling of the robustness to unitary perturbations, this analysis may elucidate the scaling of memory lifetimes of possible physical realizations of topological qubits.We present an analysis of the relaxation dynamics of finite-size topological qubits in contact with a thermal bath. Using a continuous-time Monte Carlo method, we explicitly compute the low-temperature nonequilibrium dynamics of the toric code on finite lattices. In contrast to the size-independent bound predicted for the toric code in the thermodynamic limit, we identify a low-temperature regime on finite lattices below a size-dependent crossover temperature with nontrivial finite-size and temperature scaling of the relaxation time. We demonstrate how this nontrivial finite-size scaling is governed by the scaling of topologically nontrivial two-dimensional classical random walks. The transition out of this low-temperature regime defines a dynamical finite-size crossover temperature that scales inversely with the log of the system size, in agreement with a crossover temperature defined from equilibrium properties. We find that both the finite-size and finite-temperature scaling are stronger in the low-temperature regime than above the crossover temperature. Since this finite-temperature scaling competes with the scaling of the robustness to unitary perturbations, this analysis may elucidate the scaling of memory lifetimes of possible physical realizations of topological qubits.

arXiv:1405.2315Physical Review BOct 7, 201417 citations

Finite Size Scaling Effects in Dynamics

Yadin Y. Goldschmidt

Current physics sources and commentsJan 1, 19881 citation

Statics and dynamics of the ten-state mean-field Potts glass model: a Monte Carlo study

Claudio Brangian, Walter Kob, Kurt Binder

We investigate by means of Monte Carlo simulations the fully connected p -state Potts model for different system sizes in order to see how the static and dynamic properties of a finite model compare with the, exactly known, behaviour of the system in the thermodynamic limit. Using p = 10 we are able to study the equilibrium dynamics for system sizes as large as N = 2560. We find that the static quantities, such as the energy, the entropy, the spin glass susceptibility as well as the distribution of the order parameter P ( q ) show very strong finite-size effects. From P ( q ) we calculate the fourth-order cumulant g 4 ( N , T ) and the Guerra parameter G ( N , T ) and show that these quantities cannot be used to locate the static transition temperature for the system sizes investigated. Also the spin-autocorrelation function C ( t ) shows strong finite-size effects in that it does not show a plateau even for temperatures around the dynamical critical temperature T D . We show that the dependence on N and T of the α-relaxation time can be understood by means of a dynamical finite-size scaling ansatz. C ( t ) does not obey the time-temperature superposition principle for temperatures around T D , but does so for significantly lower T . Finally we study the relaxation dynamics of the individual spins and show that their dependence on time depends strongly on the chosen spin, i.e. that the system is dynamically very heterogeneous, which explains the non-exponentiality of C ( t ).We investigate by means of Monte Carlo simulations the fully connected p -state Potts model for different system sizes in order to see how the static and dynamic properties of a finite model compare with the, exactly known, behaviour of the system in the thermodynamic limit. Using p = 10 we are able to study the equilibrium dynamics for system sizes as large as N = 2560. We find that the static quantities, such as the energy, the entropy, the spin glass susceptibility as well as the distribution of the order parameter P ( q ) show very strong finite-size effects. From P ( q ) we calculate the fourth-order cumulant g 4 ( N , T ) and the Guerra parameter G ( N , T ) and show that these quantities cannot be used to locate the static transition temperature for the system sizes investigated. Also the spin-autocorrelation function C ( t ) shows strong finite-size effects in that it does not show a plateau even for temperatures around the dynamical critical temperature T D . We show that the dependence on N and T of the α-relaxation time can be understood by means of a dynamical finite-size scaling ansatz. C ( t ) does not obey the time-temperature superposition principle for temperatures around T D , but does so for significantly lower T . Finally we study the relaxation dynamics of the individual spins and show that their dependence on time depends strongly on the chosen spin, i.e. that the system is dynamically very heterogeneous, which explains the non-exponentiality of C ( t ).

arXiv:cond-mat/0106314Journal of Physics A Mathematical and GeneralJan 7, 200230 citations

We present an analytic study of finite-size effects on critical diffusion above and below

{T}_{c}

of three-dimensional Ising-like systems whose order parameter is coupled to a conserved density. We also calculate the finite-size relaxation time that governs the critical order-parameter relaxation towards a metastable equilibrium state below

{T}_{c}.

Two universal dynamic amplitude ratios at

{T}_{c}

are predicted and quantitative predictions of dynamic finite-size scaling functions are given that can be tested by Monte Carlo simulations.