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

Operator renormalization group and spin systems

Calvin Stubbins

Published Jul 15, 1991• 2 citations
Publisher

Abstract

The results of a new hybrid method for calculating ground-state energies for lattice Hamiltonians are presented. This method combines the $t$ 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 $t$. 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 $t$ expansion or real-space renormalization-group methods used separately.

DOI
10.1103/physrevd.44.488

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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→\ensuremath∞t\ensuremath{\rightarrow}\ensuremath{\infty}t\ensuremath→\ensuremath∞ 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.
We introduce a novel operator renormalization-group method. This is a new and more powerful variant of the t expansion combining that method with the real-space renormalization-group approach. The aim is to extract infinite-volume physics at t\ensuremath{\rightarrow}\ensuremath{\infty} from calculations of only a few powers of t. Good results are obtained for the (1+1)-dimensional Ising model. The method readily generalizes to higher-dimensional spin theories and to a gauge-invariant treatment of gauge theories. These theories will require greater computational power.
It is shown that some recently proposed iterative approaches to the ground state of quantum systems can be obtained as zero-temperature limits of free-energy-preserving renormalization transformations, if the perturbative expansion is handled in the appropriate way. Besides giving new insight into the ground-state approaches and their mutual relationships, these results allow us to perform consistent approximate calculations of the free energy at all temperatures and to get global descriptions of the critical properties. Free-energy and specific-heat calculations are reported for XY\mathrm{XY}XY and Heisenberg spin-\textonehalf{} chains and for the triangular XY\mathrm{XY}XY model. It is also demonstrated how this extension into finite temperatures allows us to compute in a consistent way the zzz exponent, and to obtain substantial improvement in the numerical values for the ground-state energy density of the transverse Ising model.
The COntractor REnormalization group (CORE) approximation, a new method for solving Hamiltonian lattice systems, is introduced. The approach combines variational and contraction techniques with the real-space renormalization group approach and is systematically improvable. Since it applies to lattice systems of infinite extent, the method is suitable for studying critical phenomena and phase structure; systems with dynamical fermions can also be treated. The method is tested using the 1+1-dimensional Ising model.
We present a real-space renormalization-group (RG) study of the nnn-component classical Heisenberg model in the large-nnn limit. We obtain exact expressions for correlation functions of block spins, given those of the original spins. We show how to extract the critical behavior of the model from such information. The model provides a good testing ground for evaluating the effectiveness of Monte Carlo RG techniques. We study numerically the implementation of the real-space RG transformation on finite lattices. In two dimensions, we obtain the $\ensuremath{\beta}$ function as a function of temperature. In three dimensions, we calculate the critical temperature and the thermal exponent. We discuss the effects of the use of finite lattices and of the use of different forms for the initial action on the accuracy of the results.
The application of real space renormalization group methods to quantum lattice models has become a topic of great interest following the development of the density matrix renormalization group by White.This method has been used to find the ground and low-lying excited state energies and wave functions of quantum spin models in which the form of the ground state is not clear, for instance because the interactions are frustrated.It has also been applied to fermion problems where the tendency for localization due to the strong Coulomb repulsion is opposed by the lowering of the kinetic energy which occurs as a result of electron transfer.The approach is particularly suitable for one-, or quasi-one-dimensional problems.The method involves truncating the Hilbert space in a systematic and optimised manner.Results for the ground state energy are thus variational bounds.The results for low-lying energies and correlation functions for one-dimensional systems have unprecedented accuracy and the method has become the method of choice for solving one-dimensional quantum spin problems.We review the method and results obtained for the spin-1 chain with biquadratic exchange as well as the spin-1/2 model with competing nearest and next nearest neighbour exchange will be described.More recently, the density matrix renormalization group has been applied to reformulate the coupling constant renormalization group approach which is appropriate for the study of critical properties.This approach has been applied to the anisotropic spin-1/2 Heisenberg chain.Finally, we discuss recent work which has borne promising applications in two dimensions -the Ising model and the two-dimensional Hubbard model.
The contractor renormalization group (CORE) method, a new approach to solving Hamiltonian lattice systems, is introduced. The method combines contraction and variational techniques with the real-space renormalization group approach. It applies to lattice systems of infinite extent and is ideal for studying phase structure and critical phenomena. The CORE approximation is systematically improvable and can treat systems with dynamical fermions. The method is tested using the (1+1)-dimensional Ising model.