Physical Review B

Power laws in Ising strips: Density-matrix renormalization-group calculations

A. Drzewiński, J. Szṅajd, Katarzyna Szota

Published Jul 18, 2005• 2 citations

Abstract

The results of high-accuracy density-matrix renormalization-group calculations for infinite Ising strips of finite widths $40\ensuremath{\leqslant}L\ensuremath{\leqslant}400$ with open boundary conditions are presented. It is shown that in the presence of a small external magnetic field the infinite-system critical power laws can be observed for $L$ of the order of hundreds of nanometers. The single power law describes the field dependence of the magnetization or the longitudinal correlation length only on the infinite-system critical isotherm independently of the value of $L$. The approximate power law which describes how the magnetization varies with distance from the infinite-system critical point for several directions in the plane (temperature, external field) is also studied.

DOI: 10.1103/physrevb.72.014441

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Physical Review B

Power laws in Ising strips: Density-matrix renormalization-group calculations

A. Drzewiński, J. Szṅajd, Katarzyna Szota

Published Jul 18, 2005• 2 citations

Publisher

Abstract

The results of high-accuracy density-matrix renormalization-group calculations for infinite Ising strips of finite widths

40\ensuremath{\leqslant}L\ensuremath{\leqslant}400

with open boundary conditions are presented. It is shown that in the presence of a small external magnetic field the infinite-system critical power laws can be observed for

L

of the order of hundreds of nanometers. The single power law describes the field dependence of the magnetization or the longitudinal correlation length only on the infinite-system critical isotherm independently of the value of

L

. The approximate power law which describes how the magnetization varies with distance from the infinite-system critical point for several directions in the plane (temperature, external field) is also studied.

DOI: 10.1103/physrevb.72.014441

Related papers

Real-space renormalization-group analysis for finite ferromagnetic systems

Lawrence G. Dunfield, Jaan Noolandi

The real-space renormalization-group transformation based on the cumulant expansion is modified for systems with free-surface boundary conditions. A discussion is given of the accuracy and limitations of the method and of the extension to higher orders in the cumulant expansion. Free energies and heat capacities of ferromagnetic strips ($n\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$) and slabs ($n\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$) are determined over a wide temperature range. Exact results are obtained by a transfer matrix technique for strips with $n<~8$ , and compared to the renormalization results. The critical or pseudocritical behavior is found to approach the bulk limit as $n\ensuremath{\rightarrow}\ensuremath{\infty}$ according to a simple power law $|{T}_{c}(n)\ensuremath{-}{T}_{c}(\ensuremath{\infty})|\ensuremath{\propto}{n}^{\ensuremath{-}\ensuremath{\lambda}}$, where the shift exponent $\ensuremath{\lambda}$ is equal to the reciprocal of the critical length exponent. Renormalization equations are used for the first time to calculate the heat capacity of the three-dimensional Ising model over a wide temperature range. In addition, the effect of surface anisotropy is considered for ferromagnetic strips, and procedures are outlined for obtaining analytic derivatives of the free energy with respect to the renormalized parameters.The real-space renormalization-group transformation based on the cumulant expansion is modified for systems with free-surface boundary conditions. A discussion is given of the accuracy and limitations of the method and of the extension to higher orders in the cumulant expansion. Free energies and heat capacities of ferromagnetic strips ($n\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$) and slabs ($n\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$) are determined over a wide temperature range. Exact results are obtained by a transfer matrix technique for strips with $n<~8$, and compared to the renormalization results. The critical or pseudocritical behavior is found to approach the bulk limit as $n\ensuremath{\rightarrow}\ensuremath{\infty}$ according to a simple power law $|{T}_{c}(n)\ensuremath{-}{T}_{c}(\ensuremath{\infty})|\ensuremath{\propto}{n}^{\ensuremath{-}\ensuremath{\lambda}}$, where the shift exponent $\ensuremath{\lambda}$ is equal to the reciprocal of the critical length exponent. Renormalization equations are used for the first time to calculate the heat capacity of the three-dimensional Ising model over a wide temperature range. In addition, the effect of surface anisotropy is considered for ferromagnetic strips, and procedures are outlined for obtaining analytic derivatives of the free energy with respect to the renormalized parameters.

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The density-matrix renormalization-group (DMRG) method with periodic boundary conditions is introduced for two-dimensional (2D) classical spin models. It is shown that this method is more suitable for derivation of the properties of infinite 2D systems than the DMRG with open boundary conditions, despite the fact that the latter describes much better strips of finite width. For calculation at criticality, phenomenological renormalization at finite strips is used together with a criterion for optimum strip width for a given order of approximation. For this width the critical temperature of the 2D Ising model is estimated with seven-digit accuracy for a not too large order of approximation. Similar precision is reached for critical exponents. These results exceed the accuracy of similar calculations for the DMRG with open boundary conditions by several orders of magnitude. The method is applied to the calculation of critical exponents of the $q=3,$ 4 Potts model, as well.The density-matrix renormalization-group (DMRG) method with periodic boundary conditions is introduced for two-dimensional (2D) classical spin models. It is shown that this method is more suitable for derivation of the properties of infinite 2D systems than the DMRG with open boundary conditions, despite the fact that the latter describes much better strips of finite width. For calculation at criticality, phenomenological renormalization at finite strips is used together with a criterion for optimum strip width for a given order of approximation. For this width the critical temperature of the 2D Ising model is estimated with seven-digit accuracy for a not too large order of approximation. Similar precision is reached for critical exponents. These results exceed the accuracy of similar calculations for the DMRG with open boundary conditions by several orders of magnitude. The method is applied to the calculation of critical exponents of the $q=3,$ 4 Potts model, as well.

arXiv:cond-mat/0004024Physical review. B, Condensed matterDec 11, 20004 citations

The results of high-accuracy density-matrix renormalization-group calculations for infinite Ising strips of finite widths

40\ensuremath{\leqslant}L\ensuremath{\leqslant}400

with open boundary conditions are presented. It is shown that in the presence of a small external magnetic field the infinite-system critical power laws can be observed for

L

L

n<~8

, and compared to the renormalization results. The critical or pseudocritical behavior is found to approach the bulk limit as $n\ensuremath{\rightarrow}\ensuremath{\infty}$ according to a simple power law $|{T}_{c}(n)\ensuremath{-}{T}_{c}(\ensuremath{\infty})|\ensuremath{\propto}{n}^{\ensuremath{-}\ensuremath{\lambda}}$, where the shift exponent $\ensuremath{\lambda}$ is equal to the reciprocal of the critical length exponent. Renormalization equations are used for the first time to calculate the heat capacity of the three-dimensional Ising model over a wide temperature range. In addition, the effect of surface anisotropy is considered for ferromagnetic strips, and procedures are outlined for obtaining analytic derivatives of the free energy with respect to the renormalized parameters.

{10}^{6}

lattice spacings. A phenomenological renormalization-group analysis indicates that there are strong corrections to simple power-law scaling near the zero-temperature critical point, as is to be expected near a lower critical dimension. We examine models with Gaussian, exp(-

{J}^{2}

/2), and exponential, exp(-\ensuremath{\Vert}J\ensuremath{\Vert}), distributions of couplings; the Gaussian distribution shows stronger finite-size corrections. The correlation-length exponent is estimated to be \ensuremath{\nu}=4.2\ifmmode\pm\else\textpm\fi{}0.5, although we do not want to rule out the possibility that \ensuremath{\nu} is significantly larger than this.

q=3,

4 Potts model, as well.