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

Exact real-space renormalization group and new truncation algorithms for lattice theories

Helen R. Quinn, Marvin Weinstein

Published Mar 15, 1982• 10 citations
Publisher

Abstract

This paper discusses the theoretical basis of variational algorithms for thinning degrees of freedom in a lattice Hamiltonian theory. We show that such a thinning can, in principle, be an exact real-space renormalization-group transformation to a lattice with fewer sites. This exact transformation can only be constructed given the exact ground state of the theory. However, this insight teaches us how to significantly improve previous variational algorithms. Further improvements in ability to calculate quantities of interest are achieved by introducing "look-ahead" algorithms which allow us to maintain a renormalization-group interpretation while introducing long-distance physics into the determination of variational parameters. We find that algorithms incorporating both these features are much more powerful than previous truncation algorithms. For pedagogical reasons we present our algorithms using as examples two well-understood theories, namely, free scalar field theory and the Ising model in a transverse field in 1 + 1 dimensions. Discussion of higher dimensions and less trivial theories is included in the final section. An appendix of generalized mean-field theory is also included.

DOI
10.1103/physrevd.25.1661

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This paper discusses the theoretical basis of variational algorithms for thinning degrees of freedom in a lattice Hamiltonian theory. We show that such a thinning can, in principle, be an exact real-space renormalization-group transformation to a lattice with fewer sites. This exact transformation can only be constructed given the exact ground state of the theory. However, this insight teaches us how to significantly improve previous variational algorithms. Further improvements in ability to calculate quantities of interest are achieved by introducing "look-ahead" algorithms which allow us to maintain a renormalization-group interpretation while introducing long-distance physics into the determination of variational parameters. We find that algorithms incorporating both these features are much more powerful than previous truncation algorithms. For pedagogical reasons we present our algorithms using as examples two well-understood theories, namely, free scalar field theory and the Ising model in a transverse field in 1 + 1 dimensions. Discussion of higher dimensions and less trivial theories is included in the final section. An appendix of generalized mean-field theory is also included.
The contractor renormalization group method, a new approach to solving Hamiltonian lattice systems, is presented. The method defines a systematic and nonperturbative means of implementing Kadanoff-Wilson real-space renormalization group transformations using cluster expansion and contraction techniques. We illustrate the approach and demonstrate its effectiveness using scalar field theory, the Heisenberg antiferromagnetic chain, and the anisotropic Ising chain. Future applications to the Hubbard and $t\ensuremath{-}J$ models and lattice gauge theory are discussed.
In a previous paper we showed that a class of real-space renormalization-group calculations could be reinterpreted as an orthogonal transformation of coordinates followed by an approximate (variational) evaluation of the resulting Hamiltonian. Here we shall discuss the orthogonal-transform method and concentrate on analytic rather than numerical results. The transform method may ultimately be more accurate because even though similar approximations are made in coupling the lattice oscillators, at no time is the system split into noninteracting finite-sized blocks. Approximate variational solutions to a ${\ensuremath{\varphi}}^{4}$ lattice field theory in 2, 3, and 4 spacetime dimensions are constructed using an anharmonic-oscillator basis. The solutions exhibit an ordered and a disordered phase. Analytic expressions are obtained for certain critical surfaces of these theories in regions of parameter space where a spin approximation would be invalid. The continuum limit of our solutions is discussed. The resultant perturbation expansion is also discussed and a method for its evaluation is described.
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A real-space renormalization-group transformation is constructed between blocks of linear dimensions bL\mathrm{bL}bL and LLL to study the ground-state properties of quantum lattice models. In this scheme it is possible (i) to estimate the errors due to the irrelevant variables, including boundary effects, and (ii) to test scaling explicitly, so that an accurate calculation can be performed over the entire range of the parameter space. Application to the one-dimensional Ising model in a tranverse field yields excellent results.
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Ab-initio calculations of real-time evolution for lattice gauge theory have very interesting potential applications but present challenging computational aspects. We show that tensor renormalization group methods developed in the context of Euclidean-time lattice field theory can be applied to calculation of Trotterized evolution operators at real-time. We discuss the optimization of truncation procedures for the real-time equivalent of the partition function. We apply the numerical methods to the 1D Quantum Ising Model with an external transverse field in the ordered phase and compare with universal quantum computing for Ns=4N_{s}=4Ns​=4 and 8 sites.