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arXiv:hep-lat/0002021 · Physical review. B, Condensed matter

Contractor renormalization group and the Haldane conjecture

Marvin Weinstein

Published Apr 9, 2001• 19 citations

Abstract

The contractor renormalization group formalism (CORE) is a real-space renormalization group method which is the Hamiltonian analogue of the Wilson exact renormalization group equations. In an earlier paper [Phys. Rev. D 61, 034505 (2000)] I showed that the CORE method could be used to map a theory of free quarks and quarks interacting with gluons into a generalized frustrated Heisenberg antiferromagnet (HAF) and proposed using CORE methods to study these theories. Since generalizations of HAF's exhibit all sorts of subtle behavior which, from a continuum point of view, are related to topological properties of the theory, it is important to know that CORE can be used to extract this physics. In this paper I show that despite the folklore which asserts that all real-space renormalization group schemes are necessarily inaccurate, simple CORE computations can give highly accurate results even if one only keeps a small number of states per block and a few terms in the cluster expansion. In addition I argue that even very simple CORE computations give a much better qualitative understanding of the physics than naive renormalization group methods. In particular I show that the simplest CORE computation yields a first-principles understanding of how the famous Haldane conjecture works for the case of the $\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}\ensuremath{-}1/2$ and $\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}\ensuremath{-}1$ HAF.

DOI: 10.1103/physrevb.63.174421

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The contractor renormalization group formalism (CORE) is a real-space renormalization group method which is the Hamiltonian analogue of the Wilson exact renormalization group equations. In an earlier paper [Phys. Rev. D 61, 034505 (2000)] I showed that the CORE method could be used to map a theory of free quarks and quarks interacting with gluons into a generalized frustrated Heisenberg antiferromagnet (HAF) and proposed using CORE methods to study these theories. Since generalizations of HAF's exhibit all sorts of subtle behavior which, from a continuum point of view, are related to topological properties of the theory, it is important to know that CORE can be used to extract this physics. In this paper I show that despite the folklore which asserts that all real-space renormalization group schemes are necessarily inaccurate, simple CORE computations can give highly accurate results even if one only keeps a small number of states per block and a few terms in the cluster expansion. In addition I argue that even very simple CORE computations give a much better qualitative understanding of the physics than naive renormalization group methods. In particular I show that the simplest CORE computation yields a first-principles understanding of how the famous Haldane conjecture works for the case of the

\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}\ensuremath{-}1/2

and

\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}\ensuremath{-}1

HAF.

Contractor renormalization (CORE) is a numerical renormalization method for Hamiltonian systems that has found applications in particle and condensed matter physics. There have been few studies, however, on further understanding of what exactly it does and its convergence properties. The current work has two main objectives. First, we wish to investigate the convergence of the cluster expansion for a two-dimensional Heisenberg antiferromagnet. This is important because the linked cluster expansion used to evaluate this formula nonperturbatively is not controlled by a small parameter. Here we present a study of three different blocking schemes which reveals some surprises and, in particular, leads us to suggest a scheme for defining successive terms in the cluster expansion. Our second goal is to present some new perspectives on CORE in light of recent developments to make it accessible to more researchers, including those in quantum information science. We make some comparison to entanglement-based approaches and discuss how it may be possible to improve or generalize the method.

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.

Contractor renormalization (CORE) is a real-space renormalization-group method to derive effective Hamiltionians for microscopic models. The original CORE method is based on a real-space decomposition of the lattice into small blocks and the effective degrees of freedom on the lattice are tensor products of those on the small blocks. We present an extension of the CORE method that overcomes this restriction. Our generalization allows the application of CORE to derive arbitrary effective models whose Hilbert space is not just a tensor product of local degrees of freedom. The method is especially well suited to search for microscopic models to emulate low-energy exotic models and can guide the design of quantum devices.

The COntractor REnormalization group method (CORE), originally developed for application to lattice gauge theories, is very well adapted the study of spin systems and systems with fermions. As an warmup exercise for studying Hubbard models this method is applied to spin-12 and spin-1 anti-ferromagnets in one space dimension in order to see if it is able to explain the physics of the Haldane conjecture. The method not only provides support for Haldane's conjecture but provides insight into the physics of a more general class of spin-1 systems with Hamiltonians of the form H = Σj→s(j) · →s(j + 1) − β(→s(j) · →s(j + 1))2 about which, until now, little was known.

The COntractor REnormalization group method was devised in 1994 by Morningstar and Weinstein. It was primarily aimed at extracting the physics of lattice quantum field theories (like lattice Quantum Chromodynamics). However, it is a general method of analyzing Hamiltonian lattice systems, e.g. Ising, Heisenberg or Hubbard models. The aim of this work is to show the application of CORE to one-dimensional quantum systems -- the Heisenberg zig-zag model and the Hubbard chain. As a test of the method, the ground state energy of these systems will be calculated.

A review of the Contractor Renormalization (CORE) method, as a systematic derivation of the low energy effective hamiltonian, is given, with emphasis on its differences and advantages over traditional perturbative (weak/strong links) real space RG. For the low energy physics of the 2D Hubbard model, we derive the plaquette bosons (projected SO(5)) model which connects the microscopic model to phases and phenomenology of high‐Tc cuprates. For the S = 1/2 Pyrochlore and Kagomé antiferromagnets, the effective hamiltonians predict spin‐disordered, lattice symmetry breaking, ground states with a large density of low energy singlets as found by exact diagonalization of small clusters.