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Physical review. B, Condensed matter

Numerical test of finite-size corrections for the anisotropic Heisenberg antiferromagnet chain

Djalma Medeiros, G. G. Cabrera

Published Jul 1, 1991• 4 citations
Publisher

Abstract

A leading exponential dependence of the mass gap with size has been numerically tested in the whole axial-anisotropy region for the spin-1/2 XXZ antiferromagnetic chain. Exact results for finite chains are obtained via a modified Lanczos method due to Paige, allowing calculations up to N=28 spins. Our results show that the ground state is doubly degenerate and displays long-range order throughout the whole anisotropic regime.

DOI
10.1103/physrevb.44.848

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A leading exponential dependence of the mass gap with size has been numerically tested in the whole axial-anisotropy region for the spin-1/2 XXZ antiferromagnetic chain. Exact results for finite chains are obtained via a modified Lanczos method due to Paige, allowing calculations up to N=28 spins. Our results show that the ground state is doubly degenerate and displays long-range order throughout the whole anisotropic regime.
The Heisenberg-Ising Hamiltonian has been exactly solved us- ing the Lanczos method, for chain of up to N = 28 spins. In the whole anisotropic region, the mass gap displays a dominant exponential behavior with size. The extrapolated ground state is doubly degenerate, and presents long-range order with dominant components of the Neel type. Quantum fluctuations become more important as long as the anisotropy is reduced, leading to a disordered singlet ground state for the infinite system at the isotropic point.
Highly precise numerical calculations using a variation of the Lanczos method devised by Paige were performed for s=1/2 antiferromagnetic Heisenberg finite chains up to sizes N=28. Several interesting physical quantities, including the ground-state energy, the mass gap, and the spin-wave velocity, were computed and fitted for logarithmic finite-size corrections, as suggested by conformal invariance and Bethe ansatz calculations. A crossover size for a maximum of the spin-wave velocity is predicted for N\ensuremath{\sim}50, well above the current available sizes in a typical Lanczos simulation. A list of our numerical results is given along with extrapolated values using standard algorithms.
The antiferromagnetic spin-1/2 Heisenberg model, with axial anisotropy and applied magnetic field parallel to the axis, is solved numerically by Lanczos diagonalization of finite systems. We devise an extremely accurate method that allows for a reliable extrapolation to the infinite size limit. In the above limit, our results are compared with exact analytical solutions obtained through the Bethe ansatz. As an important example, we probe the behavior of the Critical Magnetic Field in the whole anisotropic region. We find indications that suggest the presence of two different regimes for the scaling of size effects as a function of the anisotropy.
Leading and next-to-leading-order finite-size corrections to the ground and first excited states are calculated for the spin-1/2 anisotropic Heisenberg model in the critical region. The analytic results are compared to numerical data obtained for chains up to a length of N=1024. It is found that, near the isotropic point, the asymptotic region where the results obtained for N to infinity are applicable sets in at very large N values, and for obtaining good accuracy in fitting the numerical data one has to take into account several correction terms, even at large (N>100) chain lengths.
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.
Ground-state properties of ferrimagnetic XXZ chains with alternating spins S A and S B with nearest-neighbor exchange interaction are discussed based upon the results for finite-size systems obtained mainly by the exact diagonalization method and partly by the density matrix renormalization group method. For -1<Δ<1, where Δ is the parameter of anisotropy of the interactions, the ground state is unique or doubly degenerate and gapless. When Δ=1, the ground state of 2 N sites is the state with total spin N | S A - S B |, and has nonvanishing Néel long-range order parameter. For Δ>1, the ground state is doubly degenerate Néel long-range ordered state with a finite energy gap.