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

Finite-size corrections and numerical calculations for long spin 1/2 Heisenberg chains in the critical region

Ferenc Woynarovich, Hans-Peter Eckle

Published Feb 1, 1987• 222 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/20/2/010

Related papers

Finite-size corrections for spin-S Heisenberg chains and conformal properties

B -D Dorfel

The author compares recent results on finite-size corrections for the isotropic (XXX) Heisenberg chain (and its integrable generalisations) with former numerical and analytic calculations. For the ground state and its lowest excitations, leading and non-leading terms are considered. The analysis confirms the predictions of conformal invariance and sheds some new light on the fine structure of finite-size corrections.

Journal of Physics A Mathematical and General•Jul 7, 1989•8 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-size corrections for the low lying states of a half-filled Hubbard chain

Ferenc Woynarovich, H-P Eckle

The finite-size corrections to the ground state and the energy of the low magnetisation (S O). It is found that the contribution of the charge degrees of freedom is negligible, while the contribution of the spin degrees is the same as that in the one-dimensional isotropic Heisenberg model. The analytical results are compared to numerical ones obtained for the chain lengths up to N=512.

Advanced series in mathematical physics•May 1, 1994•0 citations

Finite-size study of the one-dimensional spin-(1/2) dimerized Heisenberg chain

G. Spronken, B. Fourcade, Yves Lépine

We present an analysis based on the extrapolation of finite-size results to the infinite one-dimensional spin-(1/2) dimerized isotropic Heisenberg system for the whole range of the dimerization parameter \ensuremath{\delta} (\ensuremath{\Vert}\ensuremath{\delta}\ensuremath{\Vert}\ensuremath{\le}1) at zero temperature. This system undergoes a transition at \ensuremath{\delta}=0, and a gap opens in the spectrum of elementary excitations. The exponent \ensuremath{\nu}, which characterizes the opening of the gap, is estimated with use of the finite-size results (with size up to N=18). We investigate two finite-size-scaling hypotheses, assuming a pure power-law behavior [case (1)], or taking into account logarithmic corrections [case (2)]. To estimate \ensuremath{\nu}, we use the derivative of the reciprocal of the gap, the derivative of the gap, and the Callan-Symanzik function. We show that the first of these is less affected by finite-size corrections than are the other two. Using it, we have obtained, in case (1), \ensuremath{\nu}=0.71\ifmmode\pm\else\textpm\fi{}0.01, in agreement with previous estimates, and in case (2), \ensuremath{\nu}=0.668\ifmmode\pm\else\textpm\fi{}0.001, in very good agreement with the value \ensuremath{\nu}=(2/3) conjectured by den Nijs. We also show that, far from criticality, the ground-state energy per site may be described by the form \ensuremath{\Vert}\ensuremath{\delta}${\ensuremath{\Vert}}^{x}$ with x=1.34\ifmmode\pm\else\textpm\fi{}0.02. However, results for the derivative of this quantity show a different functional dependence upon \ensuremath{\delta}, at least for \ensuremath{\delta}\ensuremath{\gtrsim}0.4. In fact, both the values of the ground-state energy and its derivative agree with the third-order perturbation theory of Harris, with agreement improving as \ensuremath{\delta} approaches 1.

Physical review. B, Condensed matter•Feb 1, 1986•50 citations

Finite-size corrections for the low lying states of a half-filled Hubbard chain

Ferenc Woynarovich, Hans-Peter Eckle

The finite-size corrections to the ground state and the energy of the low magnetisation (S<<N) states as a function of the size N are calculated analytically for the one-dimensional half-filled Hubbard model with on-site repulsion (U>O). It is found that the contribution of the charge degrees of freedom is negligible, while the contribution of the spin degrees is the same as that in the one-dimensional isotropic Heisenberg model. The analytical results are compared to numerical ones obtained for the chain lengths up to N=512.

Journal of Physics A Mathematical and General•May 11, 1987•101 citations
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.
We present an analysis based on the extrapolation of finite-size results to the infinite one-dimensional spin-(1/2) dimerized isotropic Heisenberg system for the whole range of the dimerization parameter \ensuremath{\delta} (\ensuremath{\Vert}\ensuremath{\delta}\ensuremath{\Vert}\ensuremath{\le}1) at zero temperature. This system undergoes a transition at \ensuremath{\delta}=0, and a gap opens in the spectrum of elementary excitations. The exponent \ensuremath{\nu}, which characterizes the opening of the gap, is estimated with use of the finite-size results (with size up to N=18). We investigate two finite-size-scaling hypotheses, assuming a pure power-law behavior [case (1)], or taking into account logarithmic corrections [case (2)]. To estimate \ensuremath{\nu}, we use the derivative of the reciprocal of the gap, the derivative of the gap, and the Callan-Symanzik function. We show that the first of these is less affected by finite-size corrections than are the other two. Using it, we have obtained, in case (1), \ensuremath{\nu}=0.71\ifmmode\pm\else\textpm\fi{}0.01, in agreement with previous estimates, and in case (2), \ensuremath{\nu}=0.668\ifmmode\pm\else\textpm\fi{}0.001, in very good agreement with the value \ensuremath{\nu}=(2/3) conjectured by den Nijs. We also show that, far from criticality, the ground-state energy per site may be described by the form \ensuremath{\Vert}\ensuremath{\delta}${\ensuremath{\Vert}}^{x}$ with x=1.34\ifmmode\pm\else\textpm\fi{}0.02. However, results for the derivative of this quantity show a different functional dependence upon \ensuremath{\delta}, at least for \ensuremath{\delta}\ensuremath{\gtrsim}0.4. In fact, both the values of the ground-state energy and its derivative agree with the third-order perturbation theory of Harris, with agreement improving as \ensuremath{\delta} approaches 1.
The finite-size corrections to the ground state and the energy of the low magnetisation (S<<N) states as a function of the size N are calculated analytically for the one-dimensional half-filled Hubbard model with on-site repulsion (U>O). It is found that the contribution of the charge degrees of freedom is negligible, while the contribution of the spin degrees is the same as that in the one-dimensional isotropic Heisenberg model. The analytical results are compared to numerical ones obtained for the chain lengths up to N=512.