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

Physical review. B, Condensed matter

New algorithm for high-temperature series: The planar rotator model

P. Butera, M. Comi, G. Marchesini

Published Apr 1, 1986• 26 citations
Publisher

Abstract

We describe a new algorithm to compute high-temperature expansions for lattice-spin models. The method, based on a recently proposed iterative solution of the Schwinger-Dyson equations, may be efficiently implemented by a computer program. As an application we calculate the (zero-field) two-point correlation functions and their moments up to 15th order for the planar rotator model on a two-dimensional square lattice. The high-temperature series previously known up to tenth order are thus substantially improved and are analyzed within the framework of the Kosterlitz-Thouless theory.

DOI
10.1103/physrevb.33.4725

Related papers

High-temperature critical behavior of two-dimensional planar models: A series investigation

Martin V. Ferer, Michael J. Velgakis

We have analyzed new twelfth-order high-temperature series for the susceptibility and correlation length of classical planar models on the triangular lattice using an $n$-fit method of analysis tailored to the form of the singularity $A\mathrm{exp}(b{t}^{\ensuremath{-}\ensuremath{\nu}})$ predicted by Kosterlitz and Thouless. Test-function analysis shows that the $n$-fit method is significantly more reliable in treating a number of possible corrections to the leading singularity than is the $D$ log Pad\'e analysis of the logarithm and logarithmic derivative used in earlier series work on tenth-order series. Our $n$-fit analysis leads to the results $\ensuremath{\nu}=0.5\ifmmode\pm\else\textpm\fi{}0.1$ and $\ensuremath{\eta}=0.27\ifmmode\pm\else\textpm\fi{}0.03$ in good agreement with the Kosterlitz-Thouless predictions $\ensuremath{\nu}=\frac{1}{2}$ and $\ensuremath{\eta}=\frac{1}{4}$.

Physical review. B, Condensed matter•Jan 1, 1983•21 citations

High-temperature expansions through order 24 for the two-dimensional classical XY model on the square lattice

P. Butera, M. Pernici

The high-temperature expansion of the spin-spin correlation function of the two-dimensional classical XY (planar rotator) model on the square lattice is extended by three terms, from order 21 through order 24, and analyzed to improve the estimates of the critical parameters.

arXiv:0710.0338•Physical Review B•Sep 26, 2007•9 citations

Further extensions of the high-temperature expansions for the two-dimensional classical XY model on the triangular and the square lattices

P. Butera, M. Pernici

The high-temperature expansions for the spin-spin correlation function of the two-dimensional classical XY (planar rotator) model are extended by two terms, from order 24 through order 26, in the case of the square lattice, and by five terms, from order 15 through order 20, in the case of the triangular lattice. The data are analyzed to improve the current estimates of the critical parameters of the models.

arXiv:0806.1496•Jun 9, 2008•0 citations

Quantitative study of the Kosterlitz-Thouless phase transition in anXYmodel of two-dimensional plane rotators: High-temperature expansions to order β20

P. Butera, M. Comi

High-temperature series expansions of the spin-spin correlation function for the plane rotator (or XY) model on the square lattice are extended by three terms through order ${\mathrm{\ensuremath{\beta}}}^{20}$. Tables of the expansion coefficients are reported for the correlation function spherical moments of order l=0,1,2. The expansion coefficients through ${\mathrm{\ensuremath{\beta}}}^{15}$ for the vorticity are also tabulated. Our analysis of the series supports the Kosterlitz-Thouless predictions on the structure of the critical singularities and leads to fairly accurate estimates of the critical parameters.

arXiv:hep-lat/9209011•Physical review. B, Condensed matter•May 1, 1993•46 citations

High-temperature study of the Kosterlitz-Thouless phase transition in theXYmodel on the triangular lattice

P. Butera, M. Comi

High-temperature series expansions of the spin-spin correlation function for the XY (or plane rotator) model on the triangular lattice are extended by two terms up to order ${\mathrm{\ensuremath{\beta}}}^{14}$. Tables of the expansion coefficients are reported for the correlaton function spherical moments of order l=0 and 2. Our analysis of the series supports the Kosterlitz-Thouless predictions on the structure of the critical singularities and leads to fairly accurate estimates of the critical parameters.

arXiv:cond-mat/9902326•Physical review. B, Condensed matter•Aug 1, 1994•35 citations

High temperature series expansions of S = 1/2 Heisenberg spin models: Algorithm to include the magnetic field with optimized complexity

Laurent Pierre, B. Bernu, Laura Messio

This work presents an algorithm for calculating high temperature series expansions (HTSE) of Heisenberg spin models with spin S=1/2 S = 1 / 2 in the thermodynamic limit. This algorithm accounts for the presence of a magnetic field. The paper begins with a comprehensive introduction to HTSE and then focuses on identifying the bottlenecks that limit the computation of higher order coefficients. HTSE calculations involve two key steps: graph enumeration on the lattice and trace calculations for each graph. The introduction of a non-zero magnetic field adds complexity to the expansion because previously irrelevant graphs must now be considered: bridged graphs. We present an efficient method to deduce the contribution of these graphs from the contribution of sub-graphs, that drastically reduces the time of calculation for the last order coefficient (in practice increasing by one the order of the series at almost no cost). Previous articles of the authors have utilized HTSE calculations based on this algorithm, but without providing detailed explanations. The complete algorithm is publicly available, as well as the series on many lattice and for various interactions.

arXiv:2404.02271•SciPost Physics•Oct 7, 2024•9 citations
We describe a new algorithm to compute high-temperature expansions for lattice-spin models. The method, based on a recently proposed iterative solution of the Schwinger-Dyson equations, may be efficiently implemented by a computer program. As an application we calculate the (zero-field) two-point correlation functions and their moments up to 15th order for the planar rotator model on a two-dimensional square lattice. The high-temperature series previously known up to tenth order are thus substantially improved and are analyzed within the framework of the Kosterlitz-Thouless theory.
We have analyzed new twelfth-order high-temperature series for the susceptibility and correlation length of classical planar models on the triangular lattice using an nnn-fit method of analysis tailored to the form of the singularity $A\mathrm{exp}(b{t}^{\ensuremath{-}\ensuremath{\nu}})$ predicted by Kosterlitz and Thouless. Test-function analysis shows that the nnn-fit method is significantly more reliable in treating a number of possible corrections to the leading singularity than is the DDD log Pad\'e analysis of the logarithm and logarithmic derivative used in earlier series work on tenth-order series. Our nnn-fit analysis leads to the results $\ensuremath{\nu}=0.5\ifmmode\pm\else\textpm\fi{}0.1$ and $\ensuremath{\eta}=0.27\ifmmode\pm\else\textpm\fi{}0.03$ in good agreement with the Kosterlitz-Thouless predictions $\ensuremath{\nu}=\frac{1}{2}$ and $\ensuremath{\eta}=\frac{1}{4}$.
High-temperature series expansions of the spin-spin correlation function for the plane rotator (or XY) model on the square lattice are extended by three terms through order ${\mathrm{\ensuremath{\beta}}}^{20}$. Tables of the expansion coefficients are reported for the correlation function spherical moments of order l=0,1,2. The expansion coefficients through ${\mathrm{\ensuremath{\beta}}}^{15}$ for the vorticity are also tabulated. Our analysis of the series supports the Kosterlitz-Thouless predictions on the structure of the critical singularities and leads to fairly accurate estimates of the critical parameters.
High-temperature series expansions of the spin-spin correlation function for the XY (or plane rotator) model on the triangular lattice are extended by two terms up to order ${\mathrm{\ensuremath{\beta}}}^{14}$. Tables of the expansion coefficients are reported for the correlaton function spherical moments of order l=0 and 2. Our analysis of the series supports the Kosterlitz-Thouless predictions on the structure of the critical singularities and leads to fairly accurate estimates of the critical parameters.
This work presents an algorithm for calculating high temperature series expansions (HTSE) of Heisenberg spin models with spin S=1/2 S = 1 / 2 in the thermodynamic limit. This algorithm accounts for the presence of a magnetic field. The paper begins with a comprehensive introduction to HTSE and then focuses on identifying the bottlenecks that limit the computation of higher order coefficients. HTSE calculations involve two key steps: graph enumeration on the lattice and trace calculations for each graph. The introduction of a non-zero magnetic field adds complexity to the expansion because previously irrelevant graphs must now be considered: bridged graphs. We present an efficient method to deduce the contribution of these graphs from the contribution of sub-graphs, that drastically reduces the time of calculation for the last order coefficient (in practice increasing by one the order of the series at almost no cost). Previous articles of the authors have utilized HTSE calculations based on this algorithm, but without providing detailed explanations. The complete algorithm is publicly available, as well as the series on many lattice and for various interactions.