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Physical review. B, Condensed matter

Ground-state properties of Heisenberg spin chains

Adriana Moreo

Published Jun 1, 1987• 65 citations
Publisher

Abstract

The ground-state properties of some finite antiferromagnetic Heisenberg rings with N spins at the isotropic point have been investigated using a modified Lanczos method. Rings with spin 1 and (3/2 have been studied and new results are presented for the correlation functions and the structure factor up to N=16 (12) for S=1 ((3/2). Evidence in support of Haldane's conjecture is found. The correlation functions for S=(3/2 behave as predicted by Affleck.

DOI
10.1103/physrevb.35.8562

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The ground-state properties of some finite antiferromagnetic Heisenberg rings with N spins at the isotropic point have been investigated using a modified Lanczos method. Rings with spin 1 and (3/2 have been studied and new results are presented for the correlation functions and the structure factor up to N=16 (12) for S=1 ((3/2). Evidence in support of Haldane's conjecture is found. The correlation functions for S=(3/2 behave as predicted by Affleck.
Ground state properties of finite anisotropic antiferromagnetic Heisenberg chains are studied for odd half-integer spins S= 1 / 2 , 3 / 2 ... as well as integer spins S=1, 2 .... Finite size scaling analysis of the results clearly distinguishes the half-integer (S= 1 / 2 , 3 / 2 ) from the integer (S=1) spin situation. It gives strong support to a recent conjecture which postulates that the T=0 phase structure is very different in the two cases. According to this idea there exists a new phase between the planar and the antiferromagnetic region, for integer spins only. This phase, which includes the isotropic point, has a finite energy gap and no long range order.
A comprehensive investigation has been made of the spectral excitations and static properties of Heisenberg antiferromagnetic chains of spin 1/2, 1, 3/2, and 2, using Lanczös, Bethe Ansatz, and Monte Carlo techniques. An unusual and unanticipated crossover mechanism for spin chains with 1/2≤S≤∞ has been discovered. The validity of the Haldane conjecture concerning the presence of a spectral excitation gap for integer-spin chains has been investigated by exact finite chains calculations of (a) the primary singlet-triplet excitation gap, (b) higher excitation gaps, and (c) the Fourier transform of the ground state correlation functions. A new Monte Carlo method has extended the spin-1 gap calculations to N=32.
We present results of exact diagonalization of the isotropic antiferromagnetic S=1 Heisenberg chain by the Lanczos method, for finite rings of up to N=22 sites. The Haldane gap G(N) and the ground-state energy per site e(N) converge, with increasing N, faster than a power law. By VBS and Shanks transformations, the extrapolated values are G(\ensuremath{\infty})=0.41049(2) and e(\ensuremath{\infty})=-1.401485(2). The spin-spin correlation function is well fit by exp(-r/\ensuremath{\xi})/ \ensuremath{\surd}r with \ensuremath{\xi}=6.2.
We recall a simple class of translation invariant states for an infinite quantum spin chain, which was introduced by Accardi. Those states have exponential decay of correlation functions, and a subclass contains the groun states of a certain class of finite-range interactions. We consider, in particular, a family of states for integer spin chains, containing as its simplest member the ground state of a spin-1 Heisenberg antiferromagnet recently studied by Affleck, Kennedy, Lieb and Tasaki. For this family we compute explicitly the correlation functions and other properties.