arXiv:cond-mat/9809052 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Amplitude ratios and the approach to bulk criticality in parallel plate geometries

Marcelo M. Leite, M. Sardelich, Mauricio D. Coutinho-Filho

Published Mar 1, 1999• 7 citations

Abstract

We present analytical and numerical results for the specific heat and susceptibility amplitude ratios in parallel plate geometries. The results are derived using field-theoretic techniques suitable to describe the system in the bulk limit, i.e., $(L/{\ensuremath{\xi}}_{\ifmmode\pm\else\textpm\fi{}})\ensuremath{\gg}1,$ where L is the distance between the plates and ${\ensuremath{\xi}}_{\ifmmode\pm\else\textpm\fi{}}$ is the correlation length above (+) and below $(\ensuremath{-})$ the bulk critical temperature. Advantages and drawbacks of our method are discussed in the light of other approaches previously reported in the literature.

DOI: 10.1103/physreve.59.2683

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Specific heat critical amplitudes and the approach to bulk criticality in parallel plate geometries

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arXiv:cond-mat/9809052 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Amplitude ratios and the approach to bulk criticality in parallel plate geometries

Marcelo M. Leite, M. Sardelich, Mauricio D. Coutinho-Filho

Published Mar 1, 1999• 7 citations

arXiv HTML PDF

Abstract

(L/{\ensuremath{\xi}}_{\ifmmode\pm\else\textpm\fi{}})\ensuremath{\gg}1,

where L is the distance between the plates and

{\ensuremath{\xi}}_{\ifmmode\pm\else\textpm\fi{}}

is the correlation length above (+) and below

(\ensuremath{-})

the bulk critical temperature. Advantages and drawbacks of our method are discussed in the light of other approaches previously reported in the literature.

DOI: 10.1103/physreve.59.2683

Related papers

Specific heat critical amplitudes and the approach to bulk criticality in parallel plate geometries

Marcelo M. Leite, A. M. Nemirovsky, Mauricio D. Coutinho-Filho

Journal of Magnetism and Magnetic MaterialsFeb 1, 19926 citations

Critical confluent corrections: Universality and estimates of amplitude ratios from field theory at d=3

C. Bagnuls, Claude Bervillier

We estimate, at $d=3$ , from ${\ensuremath{\varphi}}^{4}$ field theory the universal ratios of confluent critical correction amplitudes relative to the correlation length, the susceptibility, and the specific heat above ${T}_{c}$ . Our central values are comparable to these available from analysis of high-temperature series and the $\ensuremath{\epsilon}$ expansion. Furthermore, experiments in progress on a binary mixture are in good agreement with one of our results, the ratio of the confluent corrections for the specific heat and susceptibility which have not been previously estimated. We also present, in detail, a derivation of the universality of these quantities in the framework of field theory.We estimate, at $d=3$, from ${\ensuremath{\varphi}}^{4}$ field theory the universal ratios of confluent critical correction amplitudes relative to the correlation length, the susceptibility, and the specific heat above ${T}_{c}$. Our central values are comparable to these available from analysis of high-temperature series and the $\ensuremath{\epsilon}$ expansion. Furthermore, experiments in progress on a binary mixture are in good agreement with one of our results, the ratio of the confluent corrections for the specific heat and susceptibility which have not been previously estimated. We also present, in detail, a derivation of the universality of these quantities in the framework of field theory.

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In a critical system contained between parallel plates, it is known that the magnetization and energy density profiles close to one wall are modified due to the presence of the other wall. In addition, there is a long-range force between the walls. It is shown that the ratios of the ampiltudes governing these effects are universal, and moreover, independent of the boundary conditions at either wall. Their values are proportional to the scaling dimension of the density under consideration. Results are also given for the effects of local curvature of the wall on the density profiles. These results are shown to be a consequence of the conformal invariance of the critical system.In a critical system contained between parallel plates, it is known that the magnetization and energy density profiles close to one wall are modified due to the presence of the other wall. In addition, there is a long-range force between the walls. It is shown that the ratios of the ampiltudes governing these effects are universal, and moreover, independent of the boundary conditions at either wall. Their values are proportional to the scaling dimension of the density under consideration. Results are also given for the effects of local curvature of the wall on the density profiles. These results are shown to be a consequence of the conformal invariance of the critical system.

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Massimo Campostrini B, Andrea Pelissetto C, Paolo Rossi +1 more

We address the problem of determining the critical equation of state of three-dimensional $\mathrm{XY}$ systems. For this purpose we first consider the small-field expansion of the effective potential (Helmholtz free energy) in the high-temperature phase. We compute the first few nontrivial zero-momentum n-point renormalized couplings, which parametrize such expansion, by analyzing the high-temperature expansion of an improved lattice Hamiltonian with suppressed leading scaling corrections. These results are then used to construct parametric representations of the critical equation of state which are valid in the whole critical regime, satisfy the correct analytic properties (Griffith's analyticity), and take into account the Goldstone singularities at the coexistence curve. A systematic approximation scheme is introduced, which is limited essentially by the number of known terms in the small-field expansion of the effective potential. From our approximate representations of the equation of state, we derive estimates of universal ratios of amplitudes. For the specific-heat amplitude ratio we obtain ${A}^{+}{/A}^{\ensuremath{-}}=1.055(3),$ to be compared with the best experimental estimate ${A}^{+}{/A}^{\ensuremath{-}}=1.054(1).$We address the problem of determining the critical equation of state of three-dimensional $\mathrm{XY}$ systems. For this purpose we first consider the small-field expansion of the effective potential (Helmholtz free energy) in the high-temperature phase. We compute the first few nontrivial zero-momentum n-point renormalized couplings, which parametrize such expansion, by analyzing the high-temperature expansion of an improved lattice Hamiltonian with suppressed leading scaling corrections. These results are then used to construct parametric representations of the critical equation of state which are valid in the whole critical regime, satisfy the correct analytic properties (Griffith's analyticity), and take into account the Goldstone singularities at the coexistence curve. A systematic approximation scheme is introduced, which is limited essentially by the number of known terms in the small-field expansion of the effective potential. From our approximate representations of the equation of state, we derive estimates of universal ratios of amplitudes. For the specific-heat amplitude ratio we obtain ${A}^{+}{/A}^{\ensuremath{-}}=1.055(3),$ to be compared with the best experimental estimate ${A}^{+}{/A}^{\ensuremath{-}}=1.054(1).$

arXiv:cond-mat/0001440Physical review. B, Condensed matterSep 1, 200032 citations

(L/{\ensuremath{\xi}}_{\ifmmode\pm\else\textpm\fi{}})\ensuremath{\gg}1,

where L is the distance between the plates and

{\ensuremath{\xi}}_{\ifmmode\pm\else\textpm\fi{}}

is the correlation length above (+) and below

(\ensuremath{-})

the bulk critical temperature. Advantages and drawbacks of our method are discussed in the light of other approaches previously reported in the literature.

We estimate, at

d=3

, from ${\ensuremath{\varphi}}^{4}$ field theory the universal ratios of confluent critical correction amplitudes relative to the correlation length, the susceptibility, and the specific heat above

{T}_{c}

. Our central values are comparable to these available from analysis of high-temperature series and the $\ensuremath{\epsilon}$ expansion. Furthermore, experiments in progress on a binary mixture are in good agreement with one of our results, the ratio of the confluent corrections for the specific heat and susceptibility which have not been previously estimated. We also present, in detail, a derivation of the universality of these quantities in the framework of field theory.

{\mathit{T}}_{\mathit{c},}

b in spatial dimension d=4-\ensuremath{\varepsilon}. At distances z\ensuremath{\ll}L from one wall a the profile contains a perturbation

{\mathit{B}}_{\mathit{a},}

b(z/L

{)}^{\mathit{d}}

induced by the distant wall b. In contrast to previous expectations the ratio of the universal amplitude

{\mathit{B}}_{\mathit{a},}

b and the corresponding Casimir amplitude is not hyperuniversal, i.e., not independent of both surface universality classes characterizing the walls a and b.

We address the problem of determining the critical equation of state of three-dimensional

\mathrm{XY}

systems. For this purpose we first consider the small-field expansion of the effective potential (Helmholtz free energy) in the high-temperature phase. We compute the first few nontrivial zero-momentum n-point renormalized couplings, which parametrize such expansion, by analyzing the high-temperature expansion of an improved lattice Hamiltonian with suppressed leading scaling corrections. These results are then used to construct parametric representations of the critical equation of state which are valid in the whole critical regime, satisfy the correct analytic properties (Griffith's analyticity), and take into account the Goldstone singularities at the coexistence curve. A systematic approximation scheme is introduced, which is limited essentially by the number of known terms in the small-field expansion of the effective potential. From our approximate representations of the equation of state, we derive estimates of universal ratios of amplitudes. For the specific-heat amplitude ratio we obtain ${A}^{+}{/A}^{\ensuremath{-}}=1.055(3),$ to be compared with the best experimental estimate ${A}^{+}{/A}^{\ensuremath{-}}=1.054(1).$