Dynamical Correlation Functions in the Two-Dimensional Kinetic Ising Model: A Real-Space Renormalization-Group Approach

Abstract

A real-space dynamic renormalization-group scheme is used to evaluate static and dynamic correlation functions for a kinetic Ising model on a two-dimensional square lattice. The critical exponents obtained from the correlation functions calculated using this method satisfy the proper static and dynamic scaling relations and are in excellent agreement with known values.PACS numbers: 64.60.-i The calculation of static and dynamic correlation functions in systems that undergo a phase transition is a problem of great interest. These functions can be measured in scattering experiments and exhibit a wide variety of phenomena in the region near the phase transition. Recently, 1 ' 2 a real-space dynamic renormalization-group (RSDRG) method was introduced for the investigation of lattice dynamical models. In this Letter we apply this method to the kinetic Ising (Kl) model on a square lattice and show how, within the formalism, one can calculate space-and timedependent correlation fucntions.The KI model 3 describes the stochastic time evolution of a set of Ising spins {cr}. The equilibrium properties are governed by the nearestneighbor Ising-model Hamiltonian, H a , characterized by a coupling K. The time evolution of this model is assumed to be generated by a single-spin-flip operator (SFO):o> ln W i [v]a i <j i (Dl where a is a relaxation rate, i is a site index, A 0t a / c<] sets all o k =o k ' except for site i, and the simplest W { [o], consistent with detailed balance and which reproduces itself under RG iterations, is of the "minimal-coupling" form: Wi[a]=l+A 1 a i a i r +A 2 a/ n) ,where A 1 =-itanh(2ir), A 2 =A t 2 9of is the sum of the four nearest neighbors about the spin at site i, and a/ n) is the symmetric sum of the products of two nearest-neighbor spins about the spin at site i.As a first step in our RSDRG analysis we divide the system into cells of four spins and the dynamical operator into intracell and inter cell parts: D a = n o. [i] where i now labels cells, A 0t 0 # L,J sets all o k =o k ' at all cells except i, axidSf depends only on the spins in the ith cell: 5 l°[ a|a'] = -ia 0 S.A 0bO^-^ali^t /^0P--3[a], W Q V* a] [a] = 1 +J x o it a (o it fl+1 +a < § «"!> +J 2 cr <t a+1 a lB ai-1 , The intracell part is of the formwhere the index a labels the four spins in the ith cell. The detailed balance condition for the uncoupled cells gives the condition J L = -i(tanh2tf 0 ) x (1 +J 2 ), where K 0 is the nearest-neighbor coupling for the uncoupled cells. As discussed in Ref. 2, the RSDRG method cor-I responds to choosing a mapping function T[\x\ O] , relating our original problem to a similar one on a lattice with a larger lattice constant, as the solution to the eigenvaluelike equation: S35[or|a]T[/i|a]=Sji^[Ml/i]T[/i|a],

Real-Space Renormalization Group Approach to the Kinetic Ising Model

“…14). 17) at Therefore, the transformation function R(,ua) should satisfy the orthogonality condition (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19) in addition to the normalization condition (2)(3)(4)(5)(6)(7). This orthogonality condition also ensures the positivity of P;t(fl) through P;t(fl) = 8(fllfl)P;t(fl) =(R(fliJ»2pstUn because (R(fl(J»2 and Pst«J) are positive.…”

“…17) In order to find r' (fllfl'), we use here the memory function formalism. 22)~24) The Laplace transform Gs(O"I(J') can be written as (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20) where (…”