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Journal of Physics Condensed Matter

A density matrix renormalization group approach to interacting quantum systems on Cayley trees

Barry Friedman

Published Oct 20, 1997• 34 citations
Publisher

Abstract

I have examined the nearest-neighbour quantum Heisenberg model on a Cayley tree using a simple density matrix renormalization group scheme. For spin 1/2 on a three-legged Bethe lattice I find indications of long-range antiferromagnetic order in agreement with the spin wave result. Preliminary calculations using a more complicated density matrix renormalization group scheme give promise that one can treat interacting quantum systems on Cayley trees very accurately.

DOI
10.1088/0953-8984/9/42/016

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I have examined the nearest-neighbour quantum Heisenberg model on a Cayley tree using a simple density matrix renormalization group scheme. For spin 1/2 on a three-legged Bethe lattice I find indications of long-range antiferromagnetic order in agreement with the spin wave result. Preliminary calculations using a more complicated density matrix renormalization group scheme give promise that one can treat interacting quantum systems on Cayley trees very accurately.
A density matrix renormalization group (DMRG) algorithm is presented for the Bethe lattice with connectivity Z=3Z=3Z=3 and antiferromagnetic exchange between nearest-neighbor spins s=1/2s=1/2s=1/2 or 1 sites in successive generations ggg. The algorithm is accurate for s=1s=1s=1 sites. The ground states are magnetic with spin S(g)=2gsS(g)={2}^{g}sS(g)=2gs, staggered magnetization that persists for large g>20g>20g>20, and short-range spin correlation functions that decrease exponentially. A finite energy gap to S>S(g)S>S(g)S>S(g) leads to a magnetization plateau in the extended lattice. Closely similar DMRG results for sss === 1/2 and 1 are interpreted in terms of an analytical three-site model.
Ground-state properties of a spin-\textonehalf{} XXZ\mathrm{XXZ}XXZ antiferromagnet on the Bethe lattice with three nearest neighbors are investigated; especially, the nature of the phase transition driven by the anisotropy is examined. In actual calculations, we employ the density-matrix renormalization-group method which is so far used in the studies on the one-dimensional quantum systems; its applicability to the higher-dimensional system is thus exhibited in our calculations. Numerical data on local spin correlations imply that the model undergoes the first-order phase transition at the Heisenberg point Jxy=Jz{J}_{\mathrm{xy}}={J}_{z}Jxy​=Jz​ and the ground state is conjectured to be in the Ising-like ordered phase (the ordered phase in XY\mathrm{XY}XY plain) for Jxy<Jz{J}_{\mathrm{xy}}<{J}_{z}Jxy​<Jz​ (Jxy>Jz{J}_{\mathrm{xy}}>{J}_{z}Jxy​>Jz​). We also compare our results with those for other systems, say, the XYZ\mathrm{XYZ}XYZ model on the square lattice to discuss a possible type of symmetry breaking.
A real-space, zero-temperature renormalization-group method is used to study a Heisenberg antiferromagnetic spin chain with long-range interactions varying as ${(\mathrm{distance})}^{\ensuremath{-}p}$. Simple renormalization-group equations are obtained and studied analytically, and the results are checked against a more accurate numerical calculation. The calculations provide evidence for a phase transition at $p\ensuremath{\approx}1.85$ characterized by a change in the behavior of zero-temperature correlation functions. For $p\ensuremath{\gtrsim}1.85$ the large-distance physics is that of the nearest-neighbor antiferromagnet, while there is a line of fixed points for $p\ensuremath{\lesssim}1.85$. I conjecture that $p\ensuremath{\approx}1.85$ also marks an order-disorder transition at finite temperature such as occurs in the Ising model with power-law interactions.
A variant of White's density-matrix renormalization-group scheme, which is designed to compute low-lying energies of one-dimensional quantum lattice models with a large number of degrees of freedom per site is described. The method is tested on two exactly solvable models---the spin-12\frac{1}{2}21​ antiferromagnetic Heisenberg chain and a dimerized XY\mathrm{XY}XY spin chain. To illustrate the potential of the method, it is applied to a model of spins interacting with quantum phonons. It is shown that the method accurately resolves a number of energy gaps on periodic rings that are sufficiently large to afford an accurate investigation of critical properties via the use of finite-size scaling theory.
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.
A density matrix renormalization group (DMRG) approach is presented for quantum systems of two spatial dimensions. In particular, it is shown that it is possible to create a multichain-type two-dimensional (2D) DMRG approach that utilizes previously determined system and environment blocks at all points. One first builds up an effective quasi-1D system and environment blocks of width L and these quasi-1D blocks are then used to as the initial building blocks of the 2D infinite-lattice algorithm introduced here. This algorithm is found to be competitive with those results of previous 2D DMRG algorithms and also of the best of other approximate methods. An illustration of this is given for the spin-half and spin-one Heisenberg models on the square lattice. The best results for the ground-state energies per bond of the spin-half and spin-one square-lattice Heisenberg antiferromagnets for the $N=20\ifmmode\times\else\texttimes\fi{}20$ lattice using this treatment are given by ${E}_{g}{/N}_{B}=\ensuremath{-}0.3321$ and ${E}_{g}{/N}_{B}=\ensuremath{-}1.1525,$ respectively.