Physical review. B, Condensed matter

Universality of correlation-length amplitudes and their relation with critical exponents for quantum systems

K. A. Penson, M. Kolb

Published Mar 1, 1984• 24 citations

Abstract

Recently it has been found that for several two-dimensional classical models the amplitude of the correlation length of a finite system at a critical point is universal and related to the correlation-function exponent $\ensuremath{\eta}$. Here we propose that for quantum systems the ratios of two such amplitudes are universal and are related to the ratios of the corresponding exponents. This proposition is confirmed numerically for the transverse Ising model for spin $S=\frac{1}{2},1,\frac{3}{2}, \mathrm{and} 2$. It is used to calculate $\ensuremath{\eta}$ in a simplified way.

DOI: 10.1103/physrevb.29.2854

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Physical review. B, Condensed matter

Universality of correlation-length amplitudes and their relation with critical exponents for quantum systems

K. A. Penson, M. Kolb

Published Mar 1, 1984• 24 citations

Publisher

Abstract

\ensuremath{\eta}

. Here we propose that for quantum systems the ratios of two such amplitudes are universal and are related to the ratios of the corresponding exponents. This proposition is confirmed numerically for the transverse Ising model for spin

S=\frac{1}{2},1,\frac{3}{2}, \mathrm{and} 2

. It is used to calculate

\ensuremath{\eta}

in a simplified way.

DOI: 10.1103/physrevb.29.2854

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K. A. Penson, M. Kolb

The phase transitions in the ground state of (1+1)-dimensional Ising models in a transverse field are investigated for spin $S=\frac{1}{2}, 1, \frac{3}{2}, \mathrm{and} 2$ . The critical fields and "thermal" and correlation exponents are calculated as a function of $S$ by the finite-size scaling method. The universality of the transition with respect to $S$ is fully confirmed. Furthermore, we elaborate on a recent conjecture by Luck and others relating the correlation exponent $\ensuremath{\eta}$ to the amplitude of a correlation length for a two-dimensional (2D) classical isotropic system. The extension of this relation to quantum chains as extremely anisotropic 2D systems has been considered. We find that while no simple prescription can be given to calculate various correlation exponents from the amplitudes, the ratios of the amplitudes of different quantities still have universal character.The phase transitions in the ground state of (1+1)-dimensional Ising models in a transverse field are investigated for spin $S=\frac{1}{2}, 1, \frac{3}{2}, \mathrm{and} 2$. The critical fields and "thermal" and correlation exponents are calculated as a function of $S$ by the finite-size scaling method. The universality of the transition with respect to $S$ is fully confirmed. Furthermore, we elaborate on a recent conjecture by Luck and others relating the correlation exponent $\ensuremath{\eta}$ to the amplitude of a correlation length for a two-dimensional (2D) classical isotropic system. The extension of this relation to quantum chains as extremely anisotropic 2D systems has been considered. We find that while no simple prescription can be given to calculate various correlation exponents from the amplitudes, the ratios of the amplitudes of different quantities still have universal character.

Physical review. B, Condensed matterAug 1, 198410 citations

Amplitude-exponent relation for the correlation length in the spherical model

Malte Henkel

Journal of Physics A Mathematical and GeneralFeb 21, 198815 citations

\ensuremath{\eta}

S=\frac{1}{2},1,\frac{3}{2}, \mathrm{and} 2

. It is used to calculate

\ensuremath{\eta}

in a simplified way.

1/2

XY model. From the ground state wave functions, we calculate classical and quantum correlations and mutual information. All of the correlations are found to exhibit a power-law decay with the increments of the lattice distance for both the transition lines of the Ising universality class and the Gaussian universality class. Such power-law scaling behaviors of the correlations manifest the existence of diversing correlation lengths, which means scale invariance. The critical features of the correlations can be characterized by introducing a critical exponent of the power-law decaying correlations. Similar to the critical exponent

η

of the spin-spin correlation for the universality classes in the transverse-field XY model, we calculate the critical exponents of the two-spin classical and quantum correlations as well as that of the corresponding mutual information. All of the correlations have the same critical exponents, i.e.,

η^{I}=η^{C}=η^{D}

at a critical point, where the superscripts

I

C

, and

D

stand for mutual information, classical correlation, and quantum correlation, respectively. Furthermore, the critical exponent

η

of the spin-spin correlation is shown to relate to

η= η^α/2

with

α\in \{ I, C, D\}

The phase transitions in the ground state of (1+1)-dimensional Ising models in a transverse field are investigated for spin

S=\frac{1}{2}, 1, \frac{3}{2}, \mathrm{and} 2

. The critical fields and "thermal" and correlation exponents are calculated as a function of

S

by the finite-size scaling method. The universality of the transition with respect to

S

is fully confirmed. Furthermore, we elaborate on a recent conjecture by Luck and others relating the correlation exponent $\ensuremath{\eta}$ to the amplitude of a correlation length for a two-dimensional (2D) classical isotropic system. The extension of this relation to quantum chains as extremely anisotropic 2D systems has been considered. We find that while no simple prescription can be given to calculate various correlation exponents from the amplitudes, the ratios of the amplitudes of different quantities still have universal character.