### Abstract

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.

Original language | English (US) |
---|---|

Pages (from-to) | 409-426 |

Number of pages | 18 |

Journal | Journal of Statistical Physics |

Volume | 140 |

Issue number | 3 |

DOIs | |

State | Published - 2010 |

### Fingerprint

### Keywords

- Ising model
- Lattice gas variables
- Majority rule
- Renormalization group

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

**Renormalization group maps for ising models in lattice-gas variables.** / Kennedy, Thomas G.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 140, no. 3, pp. 409-426. https://doi.org/10.1007/s10955-010-0002-0

}

TY - JOUR

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

AU - Kennedy, Thomas G

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

UR - http://www.scopus.com/inward/record.url?scp=77954244349&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77954244349&partnerID=8YFLogxK

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 -