arXiv:cond-mat/9707280

Low-temperature regimes and finite-size scaling in a quantum spherical model

Hassan Chamati, Daniel M. Danchev, Ekaterina S. Pisanova, N. S. Tonchev

Published Jul 28, 1997• 3 citations

Abstract

A $d$--dimensional quantum model in the spherical approximation confined to a general geometry of the form $L^{d-d^{\prime}} \times\infty^{d^{\prime}}\times L_τ^{z}$ ($L$--linear space size and $L_τ$--temporal size) and subjected to periodic boundary conditions is considered. Because of its close relation with the quantum rotors model it can be regarded as an effective model for studying the low-temperature behavior of the quantum Heisenberg antiferromagnets. Due to the remarkable opportunity it offers for rigorous study of finite-size effects at arbitrary dimensionality this model may play the same role in quantum critical phenomena as the popular Berlin-Kac spherical model in classical critical phenomena. Close to the zero-temperature quantum critical point, the ideas of finite-size scaling are utilized to the fullest extent for studying the critical behavior of the model. For different dimensions $1

DOI: 10.48550/arxiv.cond-mat/9707280

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Theory of a spherical-quantum-rotors model: Low-temperature regime and finite-size scaling

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arXiv:cond-mat/9707280

Low-temperature regimes and finite-size scaling in a quantum spherical model

Hassan Chamati, Daniel M. Danchev, Ekaterina S. Pisanova, N. S. Tonchev

Published Jul 28, 1997• 3 citations

arXiv HTML PDF

Abstract

d

--dimensional quantum model in the spherical approximation confined to a general geometry of the form

L^{d-d^{\prime}} \times\infty^{d^{\prime}}\times L_τ^{z}

(

L

--linear space size and

L_τ

--temporal size) and subjected to periodic boundary conditions is considered. Because of its close relation with the quantum rotors model it can be regarded as an effective model for studying the low-temperature behavior of the quantum Heisenberg antiferromagnets. Due to the remarkable opportunity it offers for rigorous study of finite-size effects at arbitrary dimensionality this model may play the same role in quantum critical phenomena as the popular Berlin-Kac spherical model in classical critical phenomena. Close to the zero-temperature quantum critical point, the ideas of finite-size scaling are utilized to the fullest extent for studying the critical behavior of the model. For different dimensions $1

DOI: 10.48550/arxiv.cond-mat/9707280

Related papers

Theory of a spherical-quantum-rotors model: Low-temperature regime and finite-size scaling

Hassan Chamati, Ekaterina S. Pisanova, N. S. Tonchev

The quantum-rotors model can be regarded as an effective model for the low-temperature behavior of the quantum Heisenberg antiferromagnets. Here, we consider a $d$ -dimensional model in the spherical approximation confined to a general geometry of the form ${L}^{d\ensuremath{-}{d}^{\ensuremath{'}}}\ifmmode\times\else\texttimes\fi{}{\ensuremath{\infty}}^{{d}^{\ensuremath{'}}}\ifmmode\times\else\texttimes\fi{}{L}_{\ensuremath{\tau}}^{z}$ $(L$ -linear space size and ${L}_{\ensuremath{\tau}}$-temporal size) and subjected to periodic boundary conditions. Due to the remarkable opportunity it offers for rigorous study of finite-size effects at arbitrary dimensionality this model may play the same role in quantum critical phenomena as the popular Berlin-Kac spherical model in classical critical phenomena. Close to the zero-temperature quantum critical point, the ideas of finite-size scaling are utilized to the fullest extent for studying the critical behavior of the model. For different dimensions $1<d<3$ and $0<~{d}^{\ensuremath{'}}<~d$ a detailed analysis, in terms of the special functions of classical mathematics, for the susceptibility and the equation of state is given. Particular attention is paid to the two-dimensional case.The quantum-rotors model can be regarded as an effective model for the low-temperature behavior of the quantum Heisenberg antiferromagnets. Here, we consider a $d$-dimensional model in the spherical approximation confined to a general geometry of the form ${L}^{d\ensuremath{-}{d}^{\ensuremath{'}}}\ifmmode\times\else\texttimes\fi{}{\ensuremath{\infty}}^{{d}^{\ensuremath{'}}}\ifmmode\times\else\texttimes\fi{}{L}_{\ensuremath{\tau}}^{z}$ $(L$-linear space size and ${L}_{\ensuremath{\tau}}$-temporal size) and subjected to periodic boundary conditions. Due to the remarkable opportunity it offers for rigorous study of finite-size effects at arbitrary dimensionality this model may play the same role in quantum critical phenomena as the popular Berlin-Kac spherical model in classical critical phenomena. Close to the zero-temperature quantum critical point, the ideas of finite-size scaling are utilized to the fullest extent for studying the critical behavior of the model. For different dimensions $1<d<3$ and $0<~{d}^{\ensuremath{'}}<~d$ a detailed analysis, in terms of the special functions of classical mathematics, for the susceptibility and the equation of state is given. Particular attention is paid to the two-dimensional case.

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A d-dimensional finite quantum model system confined to a general hypercubical geometry with linear spatial size L and ``temporal size'' 1/T (T - temperature of the system) is considered in the spherical approximation under periodic boundary conditions. Because of its close relation with the system of quantum rotors it represents an effective model for studying the low-temperature behaviour of quantum Heisenberg antiferromagnets. Close to the zero-temperature quantum critical point the ideas of finite-size scaling are used for studying the critical behaviour of the model. For a film geometry in different space dimensions $\halfσA d-dimensional finite quantum model system confined to a general hypercubical geometry with linear spatial size L and ``temporal size'' 1/T (T - temperature of the system) is considered in the spherical approximation under periodic boundary conditions. Because of its close relation with the system of quantum rotors it represents an effective model for studying the low-temperature behaviour of quantum Heisenberg antiferromagnets. Close to the zero-temperature quantum critical point the ideas of finite-size scaling are used for studying the critical behaviour of the model. For a film geometry in different space dimensions $\halfσ

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Finite-size effects in the spherical model of finite thickness

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A detailed analysis of the finite-size effects on the bulk critical behaviour of the d -dimensional mean spherical model confined to a film geometry with finite thickness L is reported. Along the finite direction different kinds of boundary conditions are applied: periodic ( p ), antiperiodic ( a ) and free surfaces with Dirichlet ( D ), Neumann ( N ) and a combination of Neumann and Dirichlet ( ND ) on both surfaces. A systematic method for the evaluation of the finite-size corrections to the free energy for the different types of boundary conditions is proposed. The free energy density and the equation for the spherical field are computed for arbitrary d . It is found, for 2 < d < 4, that the singular part of the free energy has the required finite-size scaling form at the bulk critical temperature only for ( p ) and ( a ). For the remaining boundary conditions the standard finite-size scaling hypothesis is not valid. At d = 3, the critical amplitude of the singular part of the free energy (related to the so-called Casimir amplitude) is estimated. We obtain Δ ( p ) = −2ζ(3)/(5π) = −0.153 051..., Δ ( a ) = 0.274 543... and Δ ( ND ) = 0.019 22..., implying a fluctuation-induced attraction between the surfaces for ( p ) and repulsion in the other two cases. For ( D ) and ( N ) we find a logarithmic dependence on L .A detailed analysis of the finite-size effects on the bulk critical behaviour of the d -dimensional mean spherical model confined to a film geometry with finite thickness L is reported. Along the finite direction different kinds of boundary conditions are applied: periodic ( p ), antiperiodic ( a ) and free surfaces with Dirichlet ( D ), Neumann ( N ) and a combination of Neumann and Dirichlet ( ND ) on both surfaces. A systematic method for the evaluation of the finite-size corrections to the free energy for the different types of boundary conditions is proposed. The free energy density and the equation for the spherical field are computed for arbitrary d . It is found, for 2 < d < 4, that the singular part of the free energy has the required finite-size scaling form at the bulk critical temperature only for ( p ) and ( a ). For the remaining boundary conditions the standard finite-size scaling hypothesis is not valid. At d = 3, the critical amplitude of the singular part of the free energy (related to the so-called Casimir amplitude) is estimated. We obtain Δ ( p ) = −2ζ(3)/(5π) = −0.153 051..., Δ ( a ) = 0.274 543... and Δ ( ND ) = 0.019 22..., implying a fluctuation-induced attraction between the surfaces for ( p ) and repulsion in the other two cases. For ( D ) and ( N ) we find a logarithmic dependence on L .

arXiv:0805.0715Journal of Physics A Mathematical and TheoreticalAug 11, 200813 citations

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N P Nedev, Ekaterina S. Pisanova

Abstract The entropy, the internal energy and the specific heat of a d-dimensional quantum spherical model with a long-range interaction, the decreasing of which is controlled by a parameter σ (0 < σ ≤ 2), are studied close to its quantum critical point in the context of the finite-size scaling (FSS) theory for d = σ. The obtained specific heat critical exponent σ does not depend on σ and satisfies the quantum hyperscaling relation. Based on the derived FSS forms and asymptotes of the universal scaling functions of the considered quantities, the leading temperature dependencies of the entropy, the internal energy and the specific heat are obtained in both the renormalized classical and the quantum disordered regions of the phase diagram. It has been shown that when the temperature T → +0, the entropy and the specific heat in the renormalized classical region tend to zero as powers of T, while in the quantum disordered region they tend to zero exponentially.Abstract The entropy, the internal energy and the specific heat of a d-dimensional quantum spherical model with a long-range interaction, the decreasing of which is controlled by a parameter σ (0 < σ ≤ 2), are studied close to its quantum critical point in the context of the finite-size scaling (FSS) theory for d = σ. The obtained specific heat critical exponent σ does not depend on σ and satisfies the quantum hyperscaling relation. Based on the derived FSS forms and asymptotes of the universal scaling functions of the considered quantities, the leading temperature dependencies of the entropy, the internal energy and the specific heat are obtained in both the renormalized classical and the quantum disordered regions of the phase diagram. It has been shown that when the temperature T → +0, the entropy and the specific heat in the renormalized classical region tend to zero as powers of T, while in the quantum disordered region they tend to zero exponentially.

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Hassan Chamati, Universität Duisburg–essen Fachbereich Physik

A detailed analysis of the finite–size effects on the bulk critical behaviour of the d–dimensional mean spherical model confined to a film geometry with finite thickness L is reported. Along the finite direction different boundary conditions are applied: periodic (p), antiperiodic (a) and free surfaces with Dirichlet (D), Neumann (N) and a combination of Neumann and Dirichlet (ND) on both surfaces. A systematic method for the evaluation of the finite–size corrections to the free energy for the different types of boundary conditions is proposed. The free energy density and the equation for the spherical field are computed for arbitrary d. It is found, for 2 < d < 4, that the singular part of the free energy has the required finite–size scaling form at the bulk critical temperature only for (p) and (a). For the remaining boundary conditions the standard finite–size scaling hypothesis is not valid. At d = 3, the critical amplitude (related to the so called Casimir amplitude) is estimated. We obtainA detailed analysis of the finite–size effects on the bulk critical behaviour of the d–dimensional mean spherical model confined to a film geometry with finite thickness L is reported. Along the finite direction different boundary conditions are applied: periodic (p), antiperiodic (a) and free surfaces with Dirichlet (D), Neumann (N) and a combination of Neumann and Dirichlet (ND) on both surfaces. A systematic method for the evaluation of the finite–size corrections to the free energy for the different types of boundary conditions is proposed. The free energy density and the equation for the spherical field are computed for arbitrary d. It is found, for 2 < d < 4, that the singular part of the free energy has the required finite–size scaling form at the bulk critical temperature only for (p) and (a). For the remaining boundary conditions the standard finite–size scaling hypothesis is not valid. At d = 3, the critical amplitude (related to the so called Casimir amplitude) is estimated. We obtain

Aug 4, 20134 citations

Universal critical amplitudes in a quantum spherical model: Entropy, internal energy and specific heat

Ekaterina S. Pisanova, Ivan K. Ivanov

A d-dimensional quantum spherical model with a long-range interaction (decreasing at long distances r as r−d−σ, 0 <σ ≤ 2) is considered close to its quantum critical point. The low-temperature behaviors of the entropy, the internal energy and the specific heat capacity are studied for σ/2 < d < 3σ/2 in the context of the finite-size scaling (FSS) theory. For a σ-dimensional system and a d-dimensional system with short-range interaction (σ = 2) the results for the corresponding critical amplitudes as functions of the dimensionality of the system are presented.A d-dimensional quantum spherical model with a long-range interaction (decreasing at long distances r as r−d−σ, 0 <σ ≤ 2) is considered close to its quantum critical point. The low-temperature behaviors of the entropy, the internal energy and the specific heat capacity are studied for σ/2 < d < 3σ/2 in the context of the finite-size scaling (FSS) theory. For a σ-dimensional system and a d-dimensional system with short-range interaction (σ = 2) the results for the corresponding critical amplitudes as functions of the dimensionality of the system are presented.

AIP conference proceedingsJan 1, 20190 citations

d

--dimensional quantum model in the spherical approximation confined to a general geometry of the form

L^{d-d^{\prime}} \times\infty^{d^{\prime}}\times L_τ^{z}

(

L

--linear space size and

L_τ

The quantum-rotors model can be regarded as an effective model for the low-temperature behavior of the quantum Heisenberg antiferromagnets. Here, we consider a

d

-dimensional model in the spherical approximation confined to a general geometry of the form ${L}^{d\ensuremath{-}{d}^{\ensuremath{'}}}\ifmmode\times\else\texttimes\fi{}{\ensuremath{\infty}}^{{d}^{\ensuremath{'}}}\ifmmode\times\else\texttimes\fi{}{L}_{\ensuremath{\tau}}^{z}$

(L

-linear space size and ${L}_{\ensuremath{\tau}}$-temporal size) and subjected to periodic boundary conditions. Due to the remarkable opportunity it offers for rigorous study of finite-size effects at arbitrary dimensionality this model may play the same role in quantum critical phenomena as the popular Berlin-Kac spherical model in classical critical phenomena. Close to the zero-temperature quantum critical point, the ideas of finite-size scaling are utilized to the fullest extent for studying the critical behavior of the model. For different dimensions

1<d<3

and $0<~{d}^{\ensuremath{'}}<~d$ a detailed analysis, in terms of the special functions of classical mathematics, for the susceptibility and the equation of state is given. Particular attention is paid to the two-dimensional case.