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Nonstandard critical behavior of the imperfect Bose gas on a lattice

Paweł Jakubczyk, Jacek Wojtkiewicz

Published Sep 27, 2016• 0 citations
OpenAlex

Abstract

We study an anisotropic variant of the $d$-dimensional imperfect Bose gas on a lattice, where the asymptotic behavior of the dispersion $\epsilon_{\bf k}$ at vanishing momentum $\bf{k}$ may differ from the standard quadratic form. We derive a general expression for the lower critical dimension $d_L$ of the model and show that it may be pushed to arbitrarily high values by engineering the hopping parameters. The same applies to the upper critical dimension $d_U$. For $d>d_L$ we find a general, exact expression for the Bose-Einstein condensation temperature $T_c(\mu)$ as a function of the chemical potential $\mu$. The asymptotic behavior of $T_c(\mu)$ at $T_c\to 0$ is governed by the shift exponent $\psi$, whose value turns out to fully determine whether the system is below or above the upper critical dimension. The critical exponents at condensation as well as the exponent $\psi$ may attain non-standard values for specific choices of the lattice kinetic couplings. Finally, we study the mechanism leading to the affinity between the imperfect Bose gas and the classical spherical model of spins in the critical regime.

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We study an anisotropic variant of the ddd-dimensional imperfect Bose gas on a lattice, where the asymptotic behavior of the dispersion ϵk\epsilon_{\bf k}ϵk​ at vanishing momentum k\bf{k}k may differ from the standard quadratic form. We derive a general expression for the lower critical dimension dLd_LdL​ of the model and show that it may be pushed to arbitrarily high values by engineering the hopping parameters. The same applies to the upper critical dimension dUd_UdU​. For d>dLd>d_Ld>dL​ we find a general, exact expression for the Bose-Einstein condensation temperature Tc(μ)T_c(\mu)Tc​(μ) as a function of the chemical potential μ\muμ. The asymptotic behavior of Tc(μ)T_c(\mu)Tc​(μ) at Tc→0T_c\to 0Tc​→0 is governed by the shift exponent ψ\psiψ, whose value turns out to fully determine whether the system is below or above the upper critical dimension. The critical exponents at condensation as well as the exponent ψ\psiψ may attain non-standard values for specific choices of the lattice kinetic couplings. Finally, we study the mechanism leading to the affinity between the imperfect Bose gas and the classical spherical model of spins in the critical regime.
Phase diagram and correlation functions of the anisotropic imperfect Bose gas in ddd dimensions
We study an anisotropic variant of the ddd-dimensional imperfect Bose gas, where the asymptotic behaviour of the dispersion εkε_{\bf k}εk​ at vanishing momentum k\bf{k}k may differ from the standard quadratic form. The analysis reveals the key role of the shift exponent ψψψ governing the asymptotic behaviour of the critical temperature Tc(μ)T_c(μ)Tc​(μ) as a function of the chemical potential μμμ at Tc→0T_c\to 0Tc​→0. We argue that the universality classes of Bose-Einstein condensation admitted by the model may be classified according to the allowed values of ψψψ so that spatial dimensionality has only an indirect impact on the transition properties. We analyse the correlation function of the model and discuss its asymptotics depending on the direction. Both for the perfect and imperfect anisotropic Bose gases, the correlation function χ(x)χ({\bf x})χ(x) at T>TcT>T_cT>Tc​ turns out to show either exponential decay or exponentially damped oscillatory behaviour depending on the orientation of x{\bf x}x with respect to the dispersion anisotropies.
Abstract We study an anisotropic variant of the d -dimensional imperfect Bose gas, where the asymptotic behaviour of the dispersion at vanishing momentum may differ from the standard quadratic form. The analysis reveals the key role of the shift exponent ψ governing the asymptotic behaviour of the critical temperature as a function of the chemical potential μ at . We argue that the universality classes of Bose–Einstein condensation admitted by the model may be classified according to the allowed values of ψ so that spatial dimensionality has only an indirect impact on the transition properties. We analyse the correlation function of the model and discuss its asymptotics depending on the direction. Both for the perfect and imperfect anisotropic Bose gases, the correlation function at turns out to show either exponential decay or exponentially damped oscillatory behaviour depending on the orientation of with respect to the dispersion anisotropies.
We consider the Bose gas on a ddd-dimensional anisotropic lattice employing the imperfect (mean-field) gas as a prototype example. We study the dimensional crossover arising as a result of varying the dispersion relation at finite temperature TTT. We analyze in particular situations where one of the relevant effective dimensionalities is located at or below the lower critical dimension, so that the Bose-Einstein condensate becomes expelled from the system by anisotropically modifying the lattice parameters controlling the kinetic term in the Hamiltonian. We clarify the mechanism governing this phenomenon. Subsequently we study the thermodynamic Casimir effect occurring in this system. We compute the exact profile of the scaling function for the Casimir energy. As an effect of strongly anisotropic scale invariance, the Casimir force below or at the critical temperature Tc{T}_{c}Tc​ may be repulsive even for periodic boundary conditions. The corresponding Casimir amplitude is universal only in a restricted sense, and the power law governing the decay of the Casimir interaction becomes modified. We also demonstrate that, under certain circumstances, the scaling function is constant for sufficiently large values of the scaling variable, and in consequence is not an analytical function. At T>TcT>{T}_{c}T>Tc​ the Casimir-like interactions reflect the structure of the correlation function, and, for certain orientations of the confining walls, show exponentially damped oscillatory behavior so that the corresponding force is attractive or repulsive depending on the distance.
We study the low-temperature limit of the d -dimensional imperfect Bose gas. Relying on an exact analysis of the microscopic model, we establish the existence of a second-order quantum phase transition to a phase involving the Bose–Einstein condensate. The transition is triggered by varying the chemical potential and persists at non-zero temperatures T for d > 2. We extract the exact phase diagram and identify the scaling regimes in the vicinity of the quantum-critical point focusing on the behavior of the correlation length ξ. The length ξ develops an essential singularity exclusively for d = 2. We follow the evolution of the phase diagram varying d . For d > 2 our results agree with renormalization-group-based analysis of the effective bosonic order-parameter models with the dynamical exponent z = 2.
Abstract Some properties of an ideal gas of massive bosons in a mean field potential and, confined between two infinite parallel slabs in a d-dimensional configuration space are investigated systematically. Here, one particle density of states approach is employed to study the critical temperature, shift of density, Casimir effects and critical exponents, starting from the evaluation of the grand canonical free energy in d-dimension. We have found that, the shift of density, Casimir force and the critical temperature depend on the space dimensionality. But the Casimir force decays as an inverse power law of the distance between two slabs in the condensate and, decays exponentially in the non-condensed state situated very close to the point of phase transition. Specifically, this study enabled us to predict the shift of the density of boson in the excited states due to mean field potential, and also the dimensional dependence of the critical exponent. The form of the critical exponent is found to be 1 d − 2 for the imperfect Bose gas. This leads to a value of critical exponent 1 for d = 3. Most interestingly, in the limit when the mean field potential goes to zero the resulting expressions for the above properties coincide with those of the ideal Bose gas.
Recent experimental progress on the realization of quantum systems with highly controllable long-range interactions has impelled the study of quantum phase transitions in low-dimensional systems with power-law couplings. Long-range couplings mimic higher-dimensional effects in several physical contexts. Here, we provide the exact relation between the spectral dimension d at the band bottom and the exponent α that tunes the range of power-law hoppings of a one-dimensional ideal lattice Bose gas. We also develop a finite-size scaling analysis to obtain some relevant critical exponents and the critical temperature of the BEC transition. In particular, an irrelevant dangerous scaling field has to be taken into account when the hopping range is sufficiently large to make the effective dimensionality d>4.