Physical Review Letters

Random-Field Critical Behavior: Finite-Size Effects

Surjit Singh, Jayanth R. Banavar

Published Nov 11, 1985• 4 citations

Abstract

We present the results of the first analytical study of finite-size effects in a random-field model. In particular we consider the random-field spherical model in partially infinite geometries (infinite in ${d}^{\ensuremath{'}}$ dimensions and finite in $d\ensuremath{-}{d}^{\ensuremath{'}}$ dimensions) under periodic and antiperiodic boundary conditions in the finite directions. We find a simultaneous dimensional reduction by 2 in $d$ and ${d}^{\ensuremath{'}}$ for $4<d<6$ and $2<~{d}^{\ensuremath{'}}<~4$. For $0<~{d}^{\ensuremath{'}}<2$, we find new power laws describing the approach to bulk in different geometries.

DOI: 10.1103/physrevlett.55.2220

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Physical Review Letters

Random-Field Critical Behavior: Finite-Size Effects

Surjit Singh, Jayanth R. Banavar

Published Nov 11, 1985• 4 citations

Publisher

Abstract

{d}^{\ensuremath{'}}

dimensions and finite in

d\ensuremath{-}{d}^{\ensuremath{'}}

dimensions) under periodic and antiperiodic boundary conditions in the finite directions. We find a simultaneous dimensional reduction by 2 in

d

and

{d}^{\ensuremath{'}}

for

4<d<6

and

2<~{d}^{\ensuremath{'}}<~4

. For

0<~{d}^{\ensuremath{'}}<2

, we find new power laws describing the approach to bulk in different geometries.

DOI: 10.1103/physrevlett.55.2220

Related papers

Finite–size effects in the spherical model of finite thickness

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

Finite-size effects in the spherical model of finite thickness

H Chamati

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

Finite-size effects in the spherical model of ferromagnetism: Zero-field susceptibility under antiperiodic boundary conditions

Surjit Singh, Raj K. Pathria, Michael Ellis Fisher

The overall zero-field susceptibility \ensuremath{\chi}\ifmmode\bar\else\textasciimacron\fi{} of a finite-sized spherical model of spins under various antiperiodic boundary conditions is reexamined with a view to explaining the finite-size effects of an algebraic nature found recently by Singh and Pathria. The cause of this ``unexpected'' behavior at temperatures above the bulk critical temperature ${T}_{c}$ (\ensuremath{\infty}) is seen to lie in the spatial variation of the local susceptibility which, on averaging over the system, leads precisely to the effects found previously. Below ${T}_{c}$ (\ensuremath{\infty}), the influence of antiperiodic conditions is even more severe, in that not only are the finite-size amplitudes for \ensuremath{\chi}\ifmmode\bar\else\textasciimacron\fi{} modified but, for the local susceptibility, new exponents also appear.The overall zero-field susceptibility \ensuremath{\chi}\ifmmode\bar\else\textasciimacron\fi{} of a finite-sized spherical model of spins under various antiperiodic boundary conditions is reexamined with a view to explaining the finite-size effects of an algebraic nature found recently by Singh and Pathria. The cause of this ``unexpected'' behavior at temperatures above the bulk critical temperature ${T}_{c}$(\ensuremath{\infty}) is seen to lie in the spatial variation of the local susceptibility which, on averaging over the system, leads precisely to the effects found previously. Below ${T}_{c}$(\ensuremath{\infty}), the influence of antiperiodic conditions is even more severe, in that not only are the finite-size amplitudes for \ensuremath{\chi}\ifmmode\bar\else\textasciimacron\fi{} modified but, for the local susceptibility, new exponents also appear.

Physical review. B, Condensed matterMay 1, 198610 citations

Phase transitions in finite systems: Influence of geometry on approach towards bulk critical behavior

Surjit Singh, Raj K. Pathria

We predict the manner in which a physical system, of size ${\mathrm{L}}^{\mathrm{d}\mathrm{\ensuremath{-}}\mathrm{d}\mathcal{'}}$\ifmmode\times\else\texttimes\fi{}${\mathrm{\ensuremath{\infty}}}^{\mathrm{d}\mathcal{'}}$, subject to periodic boundary conditions, approaches bulk critical behavior as L\ensuremath{\rightarrow}\ensuremath{\infty}. While for T> ${\mathrm{T}}_{\mathrm{c}}$ (\ensuremath{\infty}) the approach is exponential, for T ${\mathrm{T}}_{\mathrm{c}}$ (\ensuremath{\infty}) it is generally governed by power laws whose indices are determined by the bulk exponents for the corresponding d- and d'-dimensional systems. Specific predictions on the spherical model of ferromagnetism and the relativistic Bose gas with pair production are verified by analytical results.We predict the manner in which a physical system, of size ${\mathrm{L}}^{\mathrm{d}\mathrm{\ensuremath{-}}\mathrm{d}\mathcal{'}}$\ifmmode\times\else\texttimes\fi{}${\mathrm{\ensuremath{\infty}}}^{\mathrm{d}\mathcal{'}}$, subject to periodic boundary conditions, approaches bulk critical behavior as L\ensuremath{\rightarrow}\ensuremath{\infty}. While for T>${\mathrm{T}}_{\mathrm{c}}$(\ensuremath{\infty}) the approach is exponential, for T${\mathrm{T}}_{\mathrm{c}}$(\ensuremath{\infty}) it is generally governed by power laws whose indices are determined by the bulk exponents for the corresponding d- and d'-dimensional systems. Specific predictions on the spherical model of ferromagnetism and the relativistic Bose gas with pair production are verified by analytical results.

Physical Review LettersJul 22, 198535 citations

Finite-size effects in the spherical model of ferromagnetism: Antiperiodic boundary conditions

Surjit Singh, Raj K. Pathria

Explicit expressions are derived for the free energy, the specific heat, and the magnetic susceptibility of a spherical model of spins on a d-dimensional hypercubical lattice, of size ${L}_{1}$ \ifmmode\times\else\texttimes\fi{} ${L}_{2}$ \ifmmode\times\else\texttimes\fi{}...\ifmmode\times\else\texttimes\fi{} ${L}_{d}$ , under antiperiodic boundary conditions. The relevant scaling functions that govern the critical behavior of the system are obtained and, with the use of the asymptotic properties of these functions, various predictions of the Privman-Fisher hypothesis [Phys. Rev. B 30, 322 (1984)] on the hyperuniversality of finite systems are verified. Approach towards standard critical behavior, both for T ${T}_{c}$ (\ensuremath{\infty}) and T> ${T}_{c}$ (\ensuremath{\infty}), is examined. In the former case, the approach generally takes place through a power law; only in some special situations does one obtain an exponential law instead. In the latter case, the approach is generally exponential, except for the susceptibility of the system which (somewhat surprisingly) displays a finite-size effect dominantly determined by the surface-to-volume ratio of the lattice.Explicit expressions are derived for the free energy, the specific heat, and the magnetic susceptibility of a spherical model of spins on a d-dimensional hypercubical lattice, of size ${L}_{1}$\ifmmode\times\else\texttimes\fi{}${L}_{2}$\ifmmode\times\else\texttimes\fi{}...\ifmmode\times\else\texttimes\fi{}${L}_{d}$, under antiperiodic boundary conditions. The relevant scaling functions that govern the critical behavior of the system are obtained and, with the use of the asymptotic properties of these functions, various predictions of the Privman-Fisher hypothesis [Phys. Rev. B 30, 322 (1984)] on the hyperuniversality of finite systems are verified. Approach towards standard critical behavior, both for T${T}_{c}$(\ensuremath{\infty}) and T>${T}_{c}$(\ensuremath{\infty}), is examined. In the former case, the approach generally takes place through a power law; only in some special situations does one obtain an exponential law instead. In the latter case, the approach is generally exponential, except for the susceptibility of the system which (somewhat surprisingly) displays a finite-size effect dominantly determined by the surface-to-volume ratio of the lattice.

Physical review. B, Condensed matterOct 1, 198521 citations

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

Hassan Chamati, Daniel M. Danchev, Ekaterina S. Pisanova +1 more

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 $1A $d$--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

arXiv:cond-mat/9707280Jul 28, 19973 citations

{d}^{\ensuremath{'}}

dimensions and finite in

d\ensuremath{-}{d}^{\ensuremath{'}}

dimensions) under periodic and antiperiodic boundary conditions in the finite directions. We find a simultaneous dimensional reduction by 2 in

d

and

{d}^{\ensuremath{'}}

for

4<d<6

and

2<~{d}^{\ensuremath{'}}<~4

. For

0<~{d}^{\ensuremath{'}}<2

, we find new power laws describing the approach to bulk in different geometries.

{T}_{c}

(\ensuremath{\infty}) is seen to lie in the spatial variation of the local susceptibility which, on averaging over the system, leads precisely to the effects found previously. Below

{T}_{c}

(\ensuremath{\infty}), the influence of antiperiodic conditions is even more severe, in that not only are the finite-size amplitudes for \ensuremath{\chi}\ifmmode\bar\else\textasciimacron\fi{} modified but, for the local susceptibility, new exponents also appear.

{\mathrm{T}}_{\mathrm{c}}

(\ensuremath{\infty}) the approach is exponential, for T

{\mathrm{T}}_{\mathrm{c}}

(\ensuremath{\infty}) it is generally governed by power laws whose indices are determined by the bulk exponents for the corresponding d- and d'-dimensional systems. Specific predictions on the spherical model of ferromagnetism and the relativistic Bose gas with pair production are verified by analytical results.

Explicit expressions are derived for the free energy, the specific heat, and the magnetic susceptibility of a spherical model of spins on a d-dimensional hypercubical lattice, of size

{L}_{1}

\ifmmode\times\else\texttimes\fi{}

{L}_{2}

\ifmmode\times\else\texttimes\fi{}...\ifmmode\times\else\texttimes\fi{}

{L}_{d}

, under antiperiodic boundary conditions. The relevant scaling functions that govern the critical behavior of the system are obtained and, with the use of the asymptotic properties of these functions, various predictions of the Privman-Fisher hypothesis [Phys. Rev. B 30, 322 (1984)] on the hyperuniversality of finite systems are verified. Approach towards standard critical behavior, both for T

{T}_{c}

(\ensuremath{\infty}) and T>

{T}_{c}

(\ensuremath{\infty}), is examined. In the former case, the approach generally takes place through a power law; only in some special situations does one obtain an exponential law instead. In the latter case, the approach is generally exponential, except for the susceptibility of the system which (somewhat surprisingly) displays a finite-size effect dominantly determined by the surface-to-volume ratio of the lattice.

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