TY - JOUR

T1 - Renormalization group maps for ising models in lattice-gas variables

AU - Kennedy, Tom

N1 - Funding Information:
This research was supported in part by the National Science Foundation under grant

PY - 2010

Y1 - 2010

N2 - Real-space renormalization group maps, e.g., the majority rule transformation, map Ising-type models to Ising-type models on a coarser lattice. We show that each coefficient in the renormalized Hamiltonian in the lattice-gas variables depends on only a finite number of values of the renormalized Hamiltonian. We introduce a method which computes the values of the renormalized Hamiltonian with high accuracy and so computes the coefficients in the lattice-gas variables with high accuracy. For the critical nearest neighbor Ising model on the square lattice with the majority rule transformation, we compute over 1,000 different coefficients in the lattice-gas variable representation of the renormalized Hamiltonian and study the decay of these coefficients. We find that they decay exponentially in some sense but with a slow decay rate. We also show that the coefficients in the spin variables are sensitive to the truncation method used to compute them.

AB - Real-space renormalization group maps, e.g., the majority rule transformation, map Ising-type models to Ising-type models on a coarser lattice. We show that each coefficient in the renormalized Hamiltonian in the lattice-gas variables depends on only a finite number of values of the renormalized Hamiltonian. We introduce a method which computes the values of the renormalized Hamiltonian with high accuracy and so computes the coefficients in the lattice-gas variables with high accuracy. For the critical nearest neighbor Ising model on the square lattice with the majority rule transformation, we compute over 1,000 different coefficients in the lattice-gas variable representation of the renormalized Hamiltonian and study the decay of these coefficients. We find that they decay exponentially in some sense but with a slow decay rate. We also show that the coefficients in the spin variables are sensitive to the truncation method used to compute them.

KW - Ising model

KW - Lattice gas variables

KW - Majority rule

KW - Renormalization group

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U2 - 10.1007/s10955-010-0002-0

DO - 10.1007/s10955-010-0002-0

M3 - Article

AN - SCOPUS:77954244349

VL - 140

SP - 409

EP - 426

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 3

ER -