Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics A Mathematical and General

Calculation of finite-size corrections for the antiferromagnetic Heisenberg chain

B -D Dorfel, St Meißner

Published Jun 21, 1995• 0 citations
Publisher

Abstract

The finite-size corrections for the anisotropic Heisenberg model in the gap region are calculated using an exact integral representation based on the Bethe ansatz equations (BAE). The energy corrections are determined by saddle-point approximations for states without complex roots. The case of symmetric states is discussed in greater detail.

DOI
10.1088/0305-4470/28/12/009

Related papers

An analytic treatment of finite-size corrections in the spin-1 antiferromagnetic XXZ chain

Andreas Klümper, Murray T. Batchelor

It is known that the standard method for calculating finite-size corrections in Bethe ansatz solvable systems is not applicable to the Takhtajan-Babujian model and its anisotropic XXZ generalisation. The authors develop a new analytic method explicitly avoiding root densities and associated problems. Nonlinear integral equations are derived whose solutions yield the correct central charges c=1 and c=3/2 for the spin-1/2 and spin-1 XXZ chains, respectively. In the spin-1 case the authors obtain as a by-product the finite-size deviation of the Bethe ansatz roots from the 2-string formation.

Journal of Physics A Mathematical and General•Mar 7, 1990•91 citations

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

Djalma Medeiros, G. G. Cabrera

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.

Physical review. B, Condensed matter•Jul 1, 1991•4 citations

Finite-size corrections for the XXX antiferromagnet

L.V. Avdeev, B -D Dorfel

The authors present a method of calculating finite-size corrections for the isotropic (XXX) Heisenberg chain in the antiferromagnetic case. The leading corrections to the energy and the root density of the ground state are compared with the numerical data up to N=256 sites in the chain.

Journal of Physics A Mathematical and General•Jan 11, 1986•56 citations

Finite Heisenberg Quantum Spin Chain

Hans-Peter Eckle

Abstract The Bethe ansatz genuinely considers a finite system. The extraction of finite-size results from the Bethe ansatz equations is of genuine interest, especially against the background of the results of finite-size scaling and conformal symmetry in finite geometries. The mathematical techniques introduced in chapter 19 permit a systematic treatment in this chapter of finite-size corrections as corrections to the thermodynamic limit of the system. The application of the Euler-Maclaurin formula transforming finite sums into integrals and finite-size corrections transforms the Bethe ansatz equations into Wiener–Hopf integral equations with inhomogeneities representing the finite-size corrections solvable using the Wiener–Hopf technique. The results can be compared to results for finite systems obtained from other approaches that are independent of the Bethe ansatz method. It briefly discusses higher-order corrections and offers a general assessment of the finite-size method.

•Jul 25, 2019•0 citations

Anisotropic spin chain: Heisenberg’s xxz model for 4 inverted spins

А.М. Калиев

we study the numerical solutions of the Bethe ansatz equations for the inhomogeneous anisotropic XXZ Heisenberg model with a finite number of spins, with four inverted spins.Having determined the phases of the wave functions and using N-periodicity, we obtained numerical solutions for the system of Bethe equations of the anisotropic antiferromagnetic XXZ Heisenberg model.Using algebraic Bethe ansatz, various pulse and energy configurations for these states were given.

International scientific review•Jan 1, 2019•0 citations

Solution of the Bethe ansatz equations with complex roots for finite size: the spin S1 isotropic and anisotropic chains

Hector J. de Vega, Ferenc Woynarovich

The Bethe ansatz equations for spin-S (S>or=1) integrable vertex models (and magnetic chains) where the ground state is formed by complex roots are investigated for finite-size N. It is shown that the finite-size corrections to the imaginary parts of the roots (Bethe strings) for N>>1 are given by alpha m /(N sigma ( eta )) where eta is the real part of the roots, sigma ( eta ) is the density of the real parts, and m is the index of a root within a string. The constants alpha m are determined by a set of algebraic equations, and are given explicitly by alpha m =1/ pi ln cos(1/2 pi (S-m-1)/(S+1))/cos(1/2 pi (S-m)/(S+1)). For the best known, S=1, case alpha 0 =ln 2/(2 pi ). These results are found through a generalisation of the Euler-Maclaurin formula including nonanalytic contributions in N -1 which turn out to be essential in the solution of the present problem.

Journal of Physics A Mathematical and General•May 7, 1990•21 citations
The finite-size corrections for the anisotropic Heisenberg model in the gap region are calculated using an exact integral representation based on the Bethe ansatz equations (BAE). The energy corrections are determined by saddle-point approximations for states without complex roots. The case of symmetric states is discussed in greater detail.
It is known that the standard method for calculating finite-size corrections in Bethe ansatz solvable systems is not applicable to the Takhtajan-Babujian model and its anisotropic XXZ generalisation. The authors develop a new analytic method explicitly avoiding root densities and associated problems. Nonlinear integral equations are derived whose solutions yield the correct central charges c=1 and c=3/2 for the spin-1/2 and spin-1 XXZ chains, respectively. In the spin-1 case the authors obtain as a by-product the finite-size deviation of the Bethe ansatz roots from the 2-string formation.
Abstract The Bethe ansatz genuinely considers a finite system. The extraction of finite-size results from the Bethe ansatz equations is of genuine interest, especially against the background of the results of finite-size scaling and conformal symmetry in finite geometries. The mathematical techniques introduced in chapter 19 permit a systematic treatment in this chapter of finite-size corrections as corrections to the thermodynamic limit of the system. The application of the Euler-Maclaurin formula transforming finite sums into integrals and finite-size corrections transforms the Bethe ansatz equations into Wiener–Hopf integral equations with inhomogeneities representing the finite-size corrections solvable using the Wiener–Hopf technique. The results can be compared to results for finite systems obtained from other approaches that are independent of the Bethe ansatz method. It briefly discusses higher-order corrections and offers a general assessment of the finite-size method.
The Bethe ansatz equations for spin-S (S>or=1) integrable vertex models (and magnetic chains) where the ground state is formed by complex roots are investigated for finite-size N. It is shown that the finite-size corrections to the imaginary parts of the roots (Bethe strings) for N>>1 are given by alpha m /(N sigma ( eta )) where eta is the real part of the roots, sigma ( eta ) is the density of the real parts, and m is the index of a root within a string. The constants alpha m are determined by a set of algebraic equations, and are given explicitly by alpha m =1/ pi ln cos(1/2 pi (S-m-1)/(S+1))/cos(1/2 pi (S-m)/(S+1)). For the best known, S=1, case alpha 0 =ln 2/(2 pi ). These results are found through a generalisation of the Euler-Maclaurin formula including nonanalytic contributions in N -1 which turn out to be essential in the solution of the present problem.