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Physical Review Letters

Real-space quantum renormalization groups

Steven R. White, R. M. Noack

Published Jun 15, 1992• 289 citations
Publisher

Abstract

Although originally thought to show great promise in solving quantum many-body problems on a lattice, numerical real-space renormalization-group techniques have had little success for such problems. We explore the nature of the difficulties involved by studying the applicaton of the method to the simple tight-binding model in one dimension. The standard approach fails dramatically for this model. We show that the key to successfully applying the renormalization-group technique lies in applying a variety of boundary conditions to a block in order to simulate the effect of neighboring blocks.

DOI
10.1103/physrevlett.68.3487

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Although originally thought to show great promise in solving quantum many-body problems on a lattice, numerical real-space renormalization-group techniques have had little success for such problems. We explore the nature of the difficulties involved by studying the applicaton of the method to the simple tight-binding model in one dimension. The standard approach fails dramatically for this model. We show that the key to successfully applying the renormalization-group technique lies in applying a variety of boundary conditions to a block in order to simulate the effect of neighboring blocks.
A formulation of numerical real-space renormalization groups for quantum many-body problems is presented and several algorithms utilizing this formulation are outlined. The methods are presented and demonstrated using S=1/2 and S=1 Heisenberg chains as test cases. The key idea of the formulation is that rather than keep the lowest-lying eigenstates of the Hamiltonian in forming a new effective Hamiltonian of a block of sites, one should keep the most significant eigenstates of the block density matrix, obtained from diagonalizing the Hamiltonian of a larger section of the lattice which includes the block. This approach is much more accurate than the standard approach; for example, energies for the S=1 Heisenberg chain can be obtained to an accuracy of at least ${10}^{\mathrm{\ensuremath{-}}9}$. The method can be applied to almost any one-dimensional quantum lattice system, and can provide a wide variety of static properties.
We show that the failure of the real-space RG method in the 1D tight-binding model is not intrinsic to the method as considered so far but depends on the choice of boundary conditions. For fixed BC’s the failure does happen. For free BC’s we present a new analytical block RG-method which gives the exact ground state of the model and the correct 1/N 2-law for the energy of the first excited state in the large N(size)-limit. We also give a reconstruction method for the wave-functions of the excited states.
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.
We discuss the application of a quantum real-space renormalization-group approach to the Anderson-localization problem. States obtained by applying combinations of fixed and free boundary conditions to the system are used as a truncated basis at each iteration. We discuss the problem of intruder states, which occurs when the renormalization-group procedure is used to find states at the center of the band; we also discuss some possible solutions. We present numerical results for the conductance, the localization length, and the scaling function for the Anderson-localization problem in one dimension.
The renormalization-group (RG) transformation for the one-dimensional quantum Ising model has been performed exactly for two different block transformations with arbitrary size of the block (L). The recursion equations have been generalized in the large-L limit for other models using finite-size scaling ideas. We show that the success of the real-space RG method for the quantum Ising model is accidental. For other models the RG transformation generally mixes up the surface and bulk properties of the model such that strong block-size corrections appear.