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arXiv:cond-mat/0412358 · Physical Review B

Contractor renormalization group theory of SU(N) chains and ladders

Peng Li, Shun-Qing Shen

Published Jun 1, 2005• 5 citations
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Abstract

The contractor renormalization group (CORE) method is applied to the $\mathrm{SU}(N)$ chain and ladders in this paper. In our designed schemes, we show that these two classes of systems can return to their original form of Hamiltonian after CORE transformation. Successive iteration of the transformation leads to a fixed point so that the ground state energy and the energy gap to the ground state can be deduced. The result of $\mathrm{SU}(N)$ chain is compared with the one by Bethe ansatz method. The transformation on spin-$1∕2$ ladders gives a finite gap in the excited energy spectra to the ground state in an intuitive way. The application to SU(3) ladders is also discussed.

DOI
10.1103/physrevb.71.212401

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The contractor renormalization group (CORE) method is applied to the SU(N)\mathrm{SU}(N)SU(N) chain and ladders in this paper. In our designed schemes, we show that these two classes of systems can return to their original form of Hamiltonian after CORE transformation. Successive iteration of the transformation leads to a fixed point so that the ground state energy and the energy gap to the ground state can be deduced. The result of SU(N)\mathrm{SU}(N)SU(N) chain is compared with the one by Bethe ansatz method. The transformation on spin-1∕21∕21∕2 ladders gives a finite gap in the excited energy spectra to the ground state in an intuitive way. The application to SU(3) ladders is also discussed.
The contractor renormalization (CORE) group method is applied to the even-leg spin tori to reveal their ground-state energies and the lowest excitation gaps. The spin tori are referred to the spin ladders that are closed along the rung direction. We designed an improved iterative CORE algorithm and applied it to the even-leg tori. The improvement is made by introducing an extrapolation based on a sequence of hierarchical data that is obtained by varying the size of the elementary block. Quite good results are obtained and compared with the ones by other methods.
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.
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 contractor renormalization group method 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.
We develop an efficient method to perform density matrix renormalization group simulations of the SU(NNN) Heisenberg chain with open boundary conditions taking full advantage of the SU(NNN) symmetry of the problem. This method is an extension of the method previously developed for exact diagonalizations and relies on a systematic use of the basis of standard Young tableaux. Concentrating on the model with the fundamental representation at each site (i.e., one particle per site in the fermionic formulation), we have benchmarked our results for the ground-state energy up to N=8N=8N=8 and up to 420 sites by comparing them with Bethe ansatz results on open chains, for which we have derived and solved the Bethe ansatz equations. The agreement for the ground-state energy is excellent for SU(3) (12 digits). It decreases with NNN, but it is still satisfactory for N=8N=8N=8 (six digits). Central charges ccc are also extracted from the entanglement entropy using the Calabrese-Cardy formula and agree with the theoretical values expected from the SU(N)1\mathrm{SU}{(N)}_{1}SU(N)1​ Wess-Zumino-Witten conformal field theories.
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.
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.