Using a Hamiltonian formulation of the master equation, we carry out finite-size calculations for the dynamical critical exponent $z$ of the kinetic Ising model in one and two dimensions. In one dimension, different appropriately normalized transition probabilities give different values for $z$ ; this result is based on both exact and finite-size calculations. We show that this "nonuniversal" behavior is due to the fact that the critical temperature for the one-dimensional Ising model is zero. In two dimensions we combine our finite-size calculations with finite-size scaling theory to calculate $z$ . Even in this case we find that $z$ depends on the form of the transition probability---a result that contradicts the universality hypothesis for dynamical critical behavior. We discuss the possible reasons for this contradiction.Using a Hamiltonian formulation of the master equation, we carry out finite-size calculations for the dynamical critical exponent $z$ of the kinetic Ising model in one and two dimensions. In one dimension, different appropriately normalized transition probabilities give different values for $z$; this result is based on both exact and finite-size calculations. We show that this "nonuniversal" behavior is due to the fact that the critical temperature for the one-dimensional Ising model is zero. In two dimensions we combine our finite-size calculations with finite-size scaling theory to calculate $z$. Even in this case we find that $z$ depends on the form of the transition probability---a result that contradicts the universality hypothesis for dynamical critical behavior. We discuss the possible reasons for this contradiction.

Physical review. B, Condensed matterAug 1, 19819 citations

Dynamics of the Ising model

Huey Wen Huang

Physics Letters AJul 1, 19744 citations

Critical behaviour of driven bilayer systems: a field-theoretic renormalization group study

Uwe Claus Täuber, B. Schmittmann, Royce K. P. Zia

We investigate the static and dynamic critical behaviour of a uniformly driven bilayer Ising lattice gas at half-filling. Depending on the strength of the inter-layer coupling J , phase separation occurs across or within the two layers. The former transitions are controlled by the universality class of model A (corresponding to an Ising model with Glauber dynamics), with upper critical dimension d c = 4. The latter transitions are dominated by the universality class of the standard (single-layer) driven Ising lattice gas, with d c = 5 and a nonclassical anisotropy exponent. These two distinct critical lines meet at a nonequilibrium bicritical point which also falls into the driven Ising class. At all transitions, novel couplings and dangerous irrelevant operators determine corrections to scaling.We investigate the static and dynamic critical behaviour of a uniformly driven bilayer Ising lattice gas at half-filling. Depending on the strength of the inter-layer coupling J , phase separation occurs across or within the two layers. The former transitions are controlled by the universality class of model A (corresponding to an Ising model with Glauber dynamics), with upper critical dimension d c = 4. The latter transitions are dominated by the universality class of the standard (single-layer) driven Ising lattice gas, with d c = 5 and a nonclassical anisotropy exponent. These two distinct critical lines meet at a nonequilibrium bicritical point which also falls into the driven Ising class. At all transitions, novel couplings and dangerous irrelevant operators determine corrections to scaling.

arXiv:cond-mat/0108094Journal of Physics A Mathematical and GeneralOct 15, 20014 citations

Embedded dynamics for φ4 theory

Richard C. Brower, Pablo Tamayo

A new Monte Carlo dynamics is proposed for ${\mathrm{\ensuremath{\varphi}}}^{4}$ field theory that reduces critical slowing down. Discrete Ising variables are embedded into the field theory, so that large-scale tunneling events can be induced via a modified Swendsen-Wang algorithm with fluctuating site-bond percolation probabilities. The measured dynamical critical exponents ${\mathrm{z}}_{\mathrm{EL}=0.07\ifmmode\pm\else\textpm\fi{}}$0.07, 0.29\ifmmode\pm\else\textpm\fi{}0.09, and 0.87\ifmmode\pm\else\textpm\fi{}0.20 in one, two, and three dimensions, respectively, are consistent with our conjecture that this embedded dynamics lies in the same universality class as the Swendsen-Wang Ising dynamics.A new Monte Carlo dynamics is proposed for ${\mathrm{\ensuremath{\varphi}}}^{4}$ field theory that reduces critical slowing down. Discrete Ising variables are embedded into the field theory, so that large-scale tunneling events can be induced via a modified Swendsen-Wang algorithm with fluctuating site-bond percolation probabilities. The measured dynamical critical exponents ${\mathrm{z}}_{\mathrm{EL}=0.07\ifmmode\pm\else\textpm\fi{}}$0.07, 0.29\ifmmode\pm\else\textpm\fi{}0.09, and 0.87\ifmmode\pm\else\textpm\fi{}0.20 in one, two, and three dimensions, respectively, are consistent with our conjecture that this embedded dynamics lies in the same universality class as the Swendsen-Wang Ising dynamics.

Physical Review LettersMar 6, 1989129 citations

Dynamics of Quantum Ising Systems

Sei Suzuki, Jun‐ichi Inoue, Bikas K. Chakrabarti

Lecture notes in physicsDec 12, 20120 citations

Using a Hamiltonian formulation of the master equation, we carry out finite-size calculations for the dynamical critical exponent

z

of the kinetic Ising model in one and two dimensions. In one dimension, different appropriately normalized transition probabilities give different values for

z

; this result is based on both exact and finite-size calculations. We show that this "nonuniversal" behavior is due to the fact that the critical temperature for the one-dimensional Ising model is zero. In two dimensions we combine our finite-size calculations with finite-size scaling theory to calculate

z

. Even in this case we find that

z

depends on the form of the transition probability---a result that contradicts the universality hypothesis for dynamical critical behavior. We discuss the possible reasons for this contradiction.