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Physical review. B, Condensed matter

Real-space renormalization-group method for the ground-state properties of quantum lattice models

Sanjoy Kumar Sarker

Published Sep 1, 1984• 8 citations
Publisher

Abstract

A real-space renormalization-group transformation is constructed between blocks of linear dimensions $\mathrm{bL}$ and $L$ 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.

DOI
10.1103/physrevb.30.2752

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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.
We propose a real-space renormalization group (RG) transformation for quantum systems on a D-dimensional lattice. The transformation partially disentangles a block of sites before coarse-graining it into an effective site. Numerical simulations with the ground state of a 1D lattice at criticality show that the resulting coarse-grained sites require a Hilbert space dimension that does not grow with successive RG transformations. As a result we can address, in a quasi-exact way, tens of thousands of quantum spins with a computational effort that scales logarithmically in the system's size. The calculations unveil that ground state entanglement in extended quantum systems is organized in layers corresponding to different length scales. At a quantum critical point, each relevant length scale makes an equivalent contribution to the entanglement of a block.
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.
A real-space renormalization scheme for calculating the thermodynamic properties of models with continuous variables on a lattice having phase transitions in the Ising universality class is described. The first renormalization transformation maps the initial model onto a spin-\textonehalf{} Ising model with generalized interactions, which is then analyzed by an Ising renormalization transformation or by other means. Two versions of this procedure are used to calculate the phase diagram of a double-well model for structural phase transitions, which is identical with the ${\ensuremath{\varphi}}^{4}$ field theory on a lattice, in two dimensions. The results are in excellent agreement with the molecular-dynamics studies of Schneider and Stoll.
We study the ground state of one- and two-dimensional (square-lattice) spin-1/2 quantum antiferromagnets using a numerical real-space renormalization-group (RG) approach. In our RG approach we consider blocks of various sizes but with an odd number of sites; we retain only the doublet ground state and we integrate out the higher-energy states by means of second-order quasidegenerate perturbation theory. That is, we assume that the role of the excited states of a block, in the RG iteration process, is to renormalize the effective coupling parameters between blocks. We compute the ground-state energy of a spin-1/2 linear chain for various block sizes and find close agreement with the Bethe-ansatz exact solution. In the case of the spin-1/2 square-lattice quantum antiferromagnet, the obtained ground-state energy is in reasonable agreement with the available numerical estimates.
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.
The results of a new hybrid method for calculating ground-state energies for lattice Hamiltonians are presented. This method combines the ttt expansion with the real-space renormalization-group approach, with the hope of extracting infinite-volume physics at $t\ensuremath{\rightarrow}\ensuremath{\infty}$ from calculations of only a few powers of ttt. We calculate the ground-state energy for the (1+1)-dimensional anisotropic Heisenberg model and the (1+1)-dimensional Ising model. Using a blocking that treats the sites and links symmetrically, we were able to determine the exact critical point of the Ising system. In both of these systems we see that the operator renormalization-group method substantially improves upon the results of either the ttt expansion or real-space renormalization-group methods used separately.