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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Variational approach to Hamiltonian lattice theories

D. Horn

Published Apr 15, 1981• 9 citations
Publisher

Abstract

A variational calculation of the vacuum energy of a Hamiltonian lattice theory is formulated in terms of a finite box Hamiltonian for a cluster of points. The box Hamiltonian contains surface terms which are proportional to order parameters of the system. It is tested on the Ising model and applied to $Z(N)$ spin models in 1 + 1 and 2 + 1 dimensions. $Z(3)$ is found to have an exceptional phase-transition structure. The application of the method to local gauge theories is discussed.

DOI
10.1103/physrevd.23.1824

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Physical Review•May 31, 1965•2 citations
A variational calculation of the vacuum energy of a Hamiltonian lattice theory is formulated in terms of a finite box Hamiltonian for a cluster of points. The box Hamiltonian contains surface terms which are proportional to order parameters of the system. It is tested on the Ising model and applied to Z(N)Z(N)Z(N) spin models in 1 + 1 and 2 + 1 dimensions. Z(3)Z(3)Z(3) is found to have an exceptional phase-transition structure. The application of the method to local gauge theories is discussed.
In the context of the Hamiltonian variational approach we propose a systematic method for the improvement of any given trial wave function. The approach has many similarities with the Lanczos scheme and it may be used for quantum-mechanical problems as well as field-theory and statistical-mechanics systems. We apply the method to the Mathieu equation and to the Ising one-dimensional model in a finite lattice. The agreement between our results and the exact ones is excellent in the whole range of parameters. We also briefly discuss the application of these ideas to lattice gauge theories and the similarities with other recently proposed methods.
The Lanczos scheme for finding low-lying eigenvalues of a sparse matrix of large dimension is applied to solving the Hamiltonian formulation for Z(2)Z(2)Z(2) and Z(3)Z(3)Z(3) Ising spin systems in 1+1 dimensions. The location of the fixed points and the values of the critical indices are consistent with known results.
We show that in contrast to simple block-spin methods, a previously discussed Hamiltonian variational approach indicates the existence of two phase transitions for the vector Potts spin model Z(N)Z(N)Z(N) in 1 + 1 dimensions and $N\ensuremath{\gtrsim}5$.
For hamiltonian lattice gauge theory, we introduce the matrix product anzats inspired from density matrix renormalization group. In this method, wavefunction of the target state is assumed to be a product of finite matrices. As a result, the energy becomes a simple function of the matrices, which can be evaluated using a computer. The minimum of the energy function corresponds to the vacuum state. We show that the S = 1/2 Heisenberg chain model are well described with the ansatz. The method is also applied to the two‐dimensional S = 1/2 Heisenberg and U(1) plaquette chain models.
The partition function of the Ising model with NNN spins, written as a functional integral, is approximated by replacing the field ${\ensuremath{\psi}}_{l}$ at the lattice site lll by the root-mean-square value of these fields, and evaluating the integral by a saddle-point method. A phase transition, whose mechanism is mathematically similar to that of the spherical model, is predicted at a temperature lower than the Weiss critical temperature. After parametrizing the mean-square field, one may apply a general variational principle to choose the parameters which minimize the difference between the approximate free energy and the exact Ising-model free energy. This optimum choice leads to the Horwitz-Callen theory.