Physical Review

Generalized Scaling and the Critical Eigenvector in Ideal Bose Condensation

Martin J. Cooper, Melville S. Green

Published Dec 5, 1968• 13 citations

Abstract

The condensation of an ideal Bose gas, in which the phase symmetry of the boson states has been removed by a linear coupling of the creation and annihilation operators ${\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$, $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ to a fictitious external field $C(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$, is considered as a model of a second-order phase transition. In the vicinity of the transition, the spontaneous-order parameter (thermal expectation value of $\ensuremath{\psi}$) is related to the conjugate field by a series of terms of which the first represents a scaled equation of state that exhibits a power-law behavior with nonclassical exponents. The expectation value for the ordered product of boson operators ${〈{\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(1) {\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(2)\ensuremath{\cdots}{\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(M) \ensuremath{\psi}({1}^{\ensuremath{'}})\ensuremath{\cdots}\ensuremath{\psi}({N}^{\ensuremath{'}})〉}_{\ensuremath{\beta}}$ is determined and its behavior across the coexistence curve discussed. The existence of a conjectured critical eigenvector is demonstrated and a related asymptotic property of the ordered product determined.

DOI: 10.1103/physrev.176.302

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Physical Review

Generalized Scaling and the Critical Eigenvector in Ideal Bose Condensation

Martin J. Cooper, Melville S. Green

Published Dec 5, 1968• 13 citations

Publisher

Abstract

The condensation of an ideal Bose gas, in which the phase symmetry of the boson states has been removed by a linear coupling of the creation and annihilation operators

{\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})

\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})

to a fictitious external field

C(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})

, is considered as a model of a second-order phase transition. In the vicinity of the transition, the spontaneous-order parameter (thermal expectation value of

\ensuremath{\psi}

) is related to the conjugate field by a series of terms of which the first represents a scaled equation of state that exhibits a power-law behavior with nonclassical exponents. The expectation value for the ordered product of boson operators

{〈{\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(1) {\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(2)\ensuremath{\cdots}{\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(M) \ensuremath{\psi}({1}^{\ensuremath{'}})\ensuremath{\cdots}\ensuremath{\psi}({N}^{\ensuremath{'}})〉}_{\ensuremath{\beta}}

is determined and its behavior across the coexistence curve discussed. The existence of a conjectured critical eigenvector is demonstrated and a related asymptotic property of the ordered product determined.

DOI: 10.1103/physrev.176.302

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Peter B. Weichman, Mark Rasolt, Michael Ellis Fisher +1 more

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Quantum criticality of the imperfect Bose gas inddimensions

Paweł Jakubczyk, M Napiórkowski

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In this work we present a large-deviations approach to the calculation of the canonical partition function for free bosons. Three-dimensional Bose-Einstein condensation is studied in the fixed-density ensemble as a function of the dimensionless density $\varrho = ρλ_T^3$ , with $ρ=N/L^3$ the standard particle density, $λ_T$ the thermal wavelength, $L$ the linear size of the box and $N$ the total number of particles. A large-deviations approach in terms of the dimensionless parameter $\ell=L/λ_T$ allows us to provide exact asymptotic estimates of the canonical partition function both above and below the critical density $\varrho_c$ for Bose-Einstein condensation. We show how this approach allows to explicitly account for finite-size effects and how it fully captures the first-order aspects of the transition, allowing us to explicitate its driving mechanism in terms of the competing probabilities of normal and condensed phases. The proposed large-deviations approach allows then to obtain in all regimes explicit and simple analytical expressions, at the leading order in the large parameter $\ell$ , for both the average fraction of particles in the ground state, the condensate fraction $\langle n_0(\varrho) \rangle = \langle N_0(\varrho) \rangle/N$ , and for its fluctuations, $σ_0(\varrho) = \sqrt{\langle N_0^2(\varrho)\rangle - \langle N_0(\varrho)\rangle^2}/N$ , retrieving for instance the anomalous scaling $σ_0(\varrho)\sim 1/V^{1/3}$ in the condensed regime, $\varrho > \varrho_c$ . Our large-deviations asymptotic estimate, by analytically clarifying the mixed-order nature of Bose-Einstein condensation, allows then to reveal the similarity between this transition and other mixed-order transitions, as for instance the localization transition in the Discrete Non-Linear Schrödinger Equation.In this work we present a large-deviations approach to the calculation of the canonical partition function for free bosons. Three-dimensional Bose-Einstein condensation is studied in the fixed-density ensemble as a function of the dimensionless density $\varrho = ρλ_T^3$, with $ρ=N/L^3$ the standard particle density, $λ_T$ the thermal wavelength, $L$ the linear size of the box and $N$ the total number of particles. A large-deviations approach in terms of the dimensionless parameter $\ell=L/λ_T$ allows us to provide exact asymptotic estimates of the canonical partition function both above and below the critical density $\varrho_c$ for Bose-Einstein condensation. We show how this approach allows to explicitly account for finite-size effects and how it fully captures the first-order aspects of the transition, allowing us to explicitate its driving mechanism in terms of the competing probabilities of normal and condensed phases. The proposed large-deviations approach allows then to obtain in all regimes explicit and simple analytical expressions, at the leading order in the large parameter $\ell$, for both the average fraction of particles in the ground state, the condensate fraction $\langle n_0(\varrho) \rangle = \langle N_0(\varrho) \rangle/N$, and for its fluctuations, $σ_0(\varrho) = \sqrt{\langle N_0^2(\varrho)\rangle - \langle N_0(\varrho)\rangle^2}/N$, retrieving for instance the anomalous scaling $σ_0(\varrho)\sim 1/V^{1/3}$ in the condensed regime, $\varrho > \varrho_c$. Our large-deviations asymptotic estimate, by analytically clarifying the mixed-order nature of Bose-Einstein condensation, allows then to reveal the similarity between this transition and other mixed-order transitions, as for instance the localization transition in the Discrete Non-Linear Schrödinger Equation.

arXiv:2608.24625Aug 25, 20260 citations

On generalized Bose-Einstein condensation in the almost-ideal boson gas

V. V. Papoyan, Valentin Anatol'evich Zagrebnov

Jun 15, 19902 citations

The condensation of an ideal Bose gas, in which the phase symmetry of the boson states has been removed by a linear coupling of the creation and annihilation operators

{\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})

\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})

to a fictitious external field

C(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})

, is considered as a model of a second-order phase transition. In the vicinity of the transition, the spontaneous-order parameter (thermal expectation value of

\ensuremath{\psi}

{〈{\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(1) {\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(2)\ensuremath{\cdots}{\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(M) \ensuremath{\psi}({1}^{\ensuremath{'}})\ensuremath{\cdots}\ensuremath{\psi}({N}^{\ensuremath{'}})〉}_{\ensuremath{\beta}}

{s}^{4}

spin model is achieved which is asymptotically exact in the critical region. In this way, previously known renormalization-group results are used to calculate the scaling functions for ${\ensuremath{\rho}}_{s}$, for the off-diagonal susceptibility, etc., to first order in \ensuremath{\epsilon}=4-d. Good agreement is found with the experimental measurements by Reppy and co-workers of ${\ensuremath{\rho}}_{s}$(T) for helium absorbed at low coverages in Vycor glass (of pore size 50--80 A\r{}). An effective-mass ratio

{m}^{\mathrm{*}}

/m\ensuremath{\simeq}1.5, and an effective-scattering-length ratio

{a}^{\mathrm{*}}

/a\ensuremath{\simeq}1.3 are indicated for mobile helium in the Vycor (although the irregular character of Vycor has not, at this point, been fully allowed for in the theory). Other approaches in the literature to computing the critical behavior of quantum-mechanical systems are reviewed and related to the present analysis.

\varrho = ρλ_T^3

, with

ρ=N/L^3

the standard particle density,

λ_T

the thermal wavelength,

L

the linear size of the box and

N

the total number of particles. A large-deviations approach in terms of the dimensionless parameter

\ell=L/λ_T

allows us to provide exact asymptotic estimates of the canonical partition function both above and below the critical density

\varrho_c

for Bose-Einstein condensation. We show how this approach allows to explicitly account for finite-size effects and how it fully captures the first-order aspects of the transition, allowing us to explicitate its driving mechanism in terms of the competing probabilities of normal and condensed phases. The proposed large-deviations approach allows then to obtain in all regimes explicit and simple analytical expressions, at the leading order in the large parameter

\ell

, for both the average fraction of particles in the ground state, the condensate fraction

\langle n_0(\varrho) \rangle = \langle N_0(\varrho) \rangle/N

, and for its fluctuations,

σ_0(\varrho) = \sqrt{\langle N_0^2(\varrho)\rangle - \langle N_0(\varrho)\rangle^2}/N

, retrieving for instance the anomalous scaling

σ_0(\varrho)\sim 1/V^{1/3}

in the condensed regime,

\varrho > \varrho_c

. Our large-deviations asymptotic estimate, by analytically clarifying the mixed-order nature of Bose-Einstein condensation, allows then to reveal the similarity between this transition and other mixed-order transitions, as for instance the localization transition in the Discrete Non-Linear Schrödinger Equation.