Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1007.1458 · Physical Review A

Dimensional crossover of a boson gas in multilayers

P. Salas, Francisco J. Sevilla, Mauricio Fortes, MANUEL DE LLANO, A. Camacho, M. A. Solís

Published Sep 30, 2010• 8 citations
arXivHTMLPDF

Abstract

We obtain the thermodynamic properties for a noninteracting Bose gas constrained on multilayers modeled by a periodic Kronig-Penney delta potential in one direction and allowed to be free in the other two directions. We report Bose-Einstein condensation (BEC) critical temperatures, chemical potential, internal energy, specific heat, and entropy for different values of a dimensionless impenetrability $P\ensuremath{\geqslant}0$ between layers. The BEC critical temperature ${T}_{c}$ coincides with the ideal gas BEC critical temperature ${T}_{0}$ when $P=0$ and rapidly goes to zero as $P$ increases to infinity for any finite interlayer separation. The specific heat ${C}_{V}$ as a function of absolute temperature $T$ for finite $P$ and plane separation $a$ exhibits one minimum and one or two maxima in addition to the BEC, for temperatures larger than that of BEC ${T}_{c}$. This highlights the effects due to particle confinement. We then discuss a distinctive dimensional crossover of the system through the specific heat behavior driven by the magnitude of $P$. For $T<{T}_{c}$ the crossover is revealed by a change in slope of $\mathrm{log}{C}_{V}(T)$ and when $T>{T}_{c}$, it is exhibited by a broad minimum in ${C}_{V}(T)$.

DOI
10.1103/physreva.82.033632

Related papers

Dimensional Crossover in the Bose–Einstein Condensation Confined to Anisotropic Three-Dimensional Lattices

Kornel Witkowski, T. K. Kopeć

Abstract The Bose–Einstein condensation (BEC) in three-dimensional (3D) anisotropic lattices is studied. We present theoretical results for the critical temperature for BEC, chemical potential, condensate fraction and relevant thermodynamic quantities like: internal energy, entropy, specific heat and compressibility as a function of anisotropy parameter being the ratio of the nearest-neighbor in-plane ( $$t_\parallel$$ t ‖ ) and out-of-plane ( $$t_\perp$$ t ⊥ ) hopping amplitudes. In particular, considered scenarios include weakly coupled two-dimensional (2D) planes ( $$t_\perp /t_\parallel \ll 1$$ t ⊥ / t ‖ ≪ 1 , relevant for layered structures) as well as a rod-like geometry of interacting one-dimensional (1D) chains ( $$t_\parallel /t_\perp \ll 1$$ t ‖ / t ⊥ ≪ 1 ). The impact of the dimensional crossover as the system is tuned away from a set of disconnected 2D layers, or traverses from a set of separate 1D chains to a regime where a fully isotropic 3D structure emerges is elucidated. Both numerical and analytic approaches are employed, (the latter in a form of series expansions involving $$t_\parallel ,t_\perp$$ t ‖ , t ⊥ amplitudes) for internal energy, entropy, specific heat and isothermal compressibility. The theoretical outcome of the present study may be of interest to a number of scenarios in solid-state physics, where the relevant quasi-particles are bosonic-like, as well as might be applicable to the physics of cold bosons loaded in artificially engineered 3D optical lattices.

Journal of Low Temperature Physics•Jul 25, 2020•4 citations

Condensation energy of superconducting BEC of non-interacting Cooper pairs in multilayers

I. Chávez, P. Salas, J. J. Valencia +1 more

Boson-Fermion models of superconductivity are getting attention as they are able to explain some of the high temperature superconductor's properties. Here we report on the condensation energy of a 3D non-interacting mixture of paired fermions (electrons) as Cooper pairs assumed to be composite bosons, which are responsible for carrying superconductivity, plus unpaired fermions both trapped in a periodic multilayer structure like that of the cuasi-two dimensional High-Temperature superconductor planes, generated by applying an external Dirac's comb potential in the direction perpendicular to the planes where superconductivity preferably occurs, while in the other two directions parallel to the planes the mixture moves freely. For bosons we give the Bose-Einstein condensation critical temperature, which we assume is equal to the superconducting critical one, while for both bosons and fermions we give the chemical potential, the internal energy and the entropy, all of them as functions of temperature, in order to calculate the Helmholtz free energy, which we use to obtain the condensation energy of a mixture of $N_F$ ideal fermions (electrons), which after turning on the attractive pair interaction become $N_B = N_{F}/2$ ideal bosons. For several plane impenetrability magnitudes, we calculate the condensation energy as the difference between the free energies of the fermions which are in the normal state minus that of the bosons in the condensed state, where we observe that as the plane impenetrability increases: the condensation energy increases; the critical temperature decreases, as expected for example for cuprate superconductors, and the behavior of the entropy and the internal energy show a dimensional crossover from 3D to 2D.

arXiv:2503.00473•Mar 1, 2025•0 citations

Bose gas in disordered, finite-layered systems

V. E. Barragán, Mauricio Fortes, M. A. Solís +1 more

Disorder effects in the thermodynamic properties of an ideal Bose gas confined in a semi-infinite multi-layer structure within a box of thickness [Formula: see text] and infinite lateral extent, are analyzed. The layers are first modeled by a periodic array of [Formula: see text] Dirac-delta functions of equal intensity. Then, we introduce structural and compositional disorder, as well as a random set of layer vacancies in the system to calculate the internal energy, chemical potential and the specific heat for different configurations. Whereas structural and compositional disorder do not reveal a significant change, a dramatic increase in the maximum of the specific heat is observed when the system is depleted a fraction of the order of 0.1–0.2 of random layers compared to the original, fully periodic array. Furthermore, this maximum, which is reminiscent of a Bose–Einstein condensation for an infinite array, occurs at higher temperatures.

arXiv:1512.05756•International Journal of Modern Physics B•May 20, 2016•2 citations

Thermodynamics of the noninteracting Bose gas in a two-dimensional box

Heqiu Li, Qiujiang Guo, Ji Jiang +1 more

Bose-Einstein condensation (BEC) of a noninteracting Bose gas of N particles in a two-dimensional box with Dirichlet boundary conditions is studied. Confirming previous work, we find that BEC occurs at finite N at low temperatures T without the occurrence of a phase transition. The conventionally-defined transition temperature T(E) for an infinite three-dimensional (3D) system is shown to correspond in a 2D system with finite N to a crossover temperature between a slow and rapid increase in the fractional boson occupation N(0)/N of the ground state with decreasing T. We further show that T(E)∼1/logN at fixed area per boson, so in the thermodynamic limit there is no significant BEC in 2D at finite T. Thus, paradoxically, BEC only occurs in 2D at finite N with no phase transition associated with it. Calculations of thermodynamic properties versus T and area A are presented, including Helmholtz free energy, entropy S, pressure p, ratio of p to the energy density U/A, heat capacity at constant volume (area) C(V) and at constant pressure C(p), isothermal compressibility κ(T) and thermal expansion coefficient α(p), obtained using both the grand-canonical ensemble (GCE) and canonical ensemble (CE) formalisms. The GCE formalism gives acceptable predictions for S, p, p/(U/A), κ(T) and α(p) at large N, T and A but fails for smaller values of these three parameters for which BEC becomes significant, whereas the CE formalism gives accurate results for all thermodynamic properties of finite systems even at low T and/or A where BEC occurs.

arXiv:1508.07656•Physical Review E•Dec 4, 2015•6 citations

A finite temperature study of one dimensional quantum gases in a quasi-periodic potential

Nilanjan Roy, Subhasis Sinha

We study the thermodynamics of ideal Bose gas as well as the transport properties of non interacting bosons and fermions in a one dimensional quasi-periodic potential, namely Aubry-Andre (AA) model at finite temperature. For bosons in finite size systems, the effect of quasi-periodic potential on the crossover phenomena corresponding to Bose-Einstein condensation (BEC), superfluidity and localization phenomena at finite temperatures are investigated. From the ground state number fluctuation we calculate the crossover temperature of BEC which exhibits a non monotonic behavior with the strength of AA potential and vanishes at the self-dual critical point following power law. Appropriate rescaling of the crossover temperatures reveals universal behavior which is studied for different quasi-periodicity of the AA model. Finally, we study the temperature and flux dependence of the persistent current of fermions in presence of a quasi-periodic potential to identify the localization at the Fermi energy from the decay of the current.

•Aug 14, 2017•0 citations

Thermodynamic behavior of a one-dimensional Bose gas at low temperature

Giulia De Rosi, G. E. Astrakharchik, S. Stringari

We show that the chemical potential of a one-dimensional (1D) interacting Bose gas exhibits a nonmonotonic temperature dependence which is peculiar of superfluids. The effect is a direct consequence of the phononic nature of the excitation spectrum at large wavelengths exhibited by 1D Bose gases. For low temperatures $T$, we demonstrate that the coefficient in ${T}^{2}$ expansion of the chemical potential is entirely defined by the zero-temperature density dependence of the sound velocity. We calculate that coefficient along the crossover between the Bogoliubov weakly interacting gas and the Tonks-Girardeau gas of impenetrable bosons. Analytic expansions are provided in the asymptotic regimes. The theoretical predictions along the crossover are confirmed by comparison with the exactly solvable Yang-Yang model in which the finite-temperature equation of state is obtained numerically by solving Bethe-ansatz equations. A 1D ring geometry is equivalent to imposing periodic boundary conditions and arising finite-size effects are studied in detail. At $T=0$ we calculated various thermodynamic functions, including the inelastic structure factor, as a function of the number of atoms, pointing out the occurrence of important deviations from the thermodynamic limit.

arXiv:1704.04642•Physical Review A•Jul 11, 2017•30 citations
We obtain the thermodynamic properties for a noninteracting Bose gas constrained on multilayers modeled by a periodic Kronig-Penney delta potential in one direction and allowed to be free in the other two directions. We report Bose-Einstein condensation (BEC) critical temperatures, chemical potential, internal energy, specific heat, and entropy for different values of a dimensionless impenetrability P\ensuremath⩾0P\ensuremath{\geqslant}0P\ensuremath⩾0 between layers. The BEC critical temperature Tc{T}_{c}Tc​ coincides with the ideal gas BEC critical temperature T0{T}_{0}T0​ when P=0P=0P=0 and rapidly goes to zero as PPP increases to infinity for any finite interlayer separation. The specific heat CV{C}_{V}CV​ as a function of absolute temperature TTT for finite PPP and plane separation aaa exhibits one minimum and one or two maxima in addition to the BEC, for temperatures larger than that of BEC Tc{T}_{c}Tc​. This highlights the effects due to particle confinement. We then discuss a distinctive dimensional crossover of the system through the specific heat behavior driven by the magnitude of PPP. For T<TcT<{T}_{c}T<Tc​ the crossover is revealed by a change in slope of logCV(T)\mathrm{log}{C}_{V}(T)logCV​(T) and when T>TcT>{T}_{c}T>Tc​, it is exhibited by a broad minimum in CV(T){C}_{V}(T)CV​(T).
Abstract The Bose–Einstein condensation (BEC) in three-dimensional (3D) anisotropic lattices is studied. We present theoretical results for the critical temperature for BEC, chemical potential, condensate fraction and relevant thermodynamic quantities like: internal energy, entropy, specific heat and compressibility as a function of anisotropy parameter being the ratio of the nearest-neighbor in-plane ( t∥t_\parallelt∥​ t ‖ ) and out-of-plane ( t⊥t_\perpt⊥​ t ⊥ ) hopping amplitudes. In particular, considered scenarios include weakly coupled two-dimensional (2D) planes ( t⊥/t∥≪1t_\perp /t_\parallel \ll 1t⊥​/t∥​≪1 t ⊥ / t ‖ ≪ 1 , relevant for layered structures) as well as a rod-like geometry of interacting one-dimensional (1D) chains ( t∥/t⊥≪1t_\parallel /t_\perp \ll 1t∥​/t⊥​≪1 t ‖ / t ⊥ ≪ 1 ). The impact of the dimensional crossover as the system is tuned away from a set of disconnected 2D layers, or traverses from a set of separate 1D chains to a regime where a fully isotropic 3D structure emerges is elucidated. Both numerical and analytic approaches are employed, (the latter in a form of series expansions involving t∥,t⊥t_\parallel ,t_\perpt∥​,t⊥​ t ‖ , t ⊥ amplitudes) for internal energy, entropy, specific heat and isothermal compressibility. The theoretical outcome of the present study may be of interest to a number of scenarios in solid-state physics, where the relevant quasi-particles are bosonic-like, as well as might be applicable to the physics of cold bosons loaded in artificially engineered 3D optical lattices.
Boson-Fermion models of superconductivity are getting attention as they are able to explain some of the high temperature superconductor's properties. Here we report on the condensation energy of a 3D non-interacting mixture of paired fermions (electrons) as Cooper pairs assumed to be composite bosons, which are responsible for carrying superconductivity, plus unpaired fermions both trapped in a periodic multilayer structure like that of the cuasi-two dimensional High-Temperature superconductor planes, generated by applying an external Dirac's comb potential in the direction perpendicular to the planes where superconductivity preferably occurs, while in the other two directions parallel to the planes the mixture moves freely. For bosons we give the Bose-Einstein condensation critical temperature, which we assume is equal to the superconducting critical one, while for both bosons and fermions we give the chemical potential, the internal energy and the entropy, all of them as functions of temperature, in order to calculate the Helmholtz free energy, which we use to obtain the condensation energy of a mixture of NFN_FNF​ ideal fermions (electrons), which after turning on the attractive pair interaction become NB=NF/2N_B = N_{F}/2NB​=NF​/2 ideal bosons. For several plane impenetrability magnitudes, we calculate the condensation energy as the difference between the free energies of the fermions which are in the normal state minus that of the bosons in the condensed state, where we observe that as the plane impenetrability increases: the condensation energy increases; the critical temperature decreases, as expected for example for cuprate superconductors, and the behavior of the entropy and the internal energy show a dimensional crossover from 3D to 2D.
Disorder effects in the thermodynamic properties of an ideal Bose gas confined in a semi-infinite multi-layer structure within a box of thickness [Formula: see text] and infinite lateral extent, are analyzed. The layers are first modeled by a periodic array of [Formula: see text] Dirac-delta functions of equal intensity. Then, we introduce structural and compositional disorder, as well as a random set of layer vacancies in the system to calculate the internal energy, chemical potential and the specific heat for different configurations. Whereas structural and compositional disorder do not reveal a significant change, a dramatic increase in the maximum of the specific heat is observed when the system is depleted a fraction of the order of 0.1–0.2 of random layers compared to the original, fully periodic array. Furthermore, this maximum, which is reminiscent of a Bose–Einstein condensation for an infinite array, occurs at higher temperatures.
Bose-Einstein condensation (BEC) of a noninteracting Bose gas of N particles in a two-dimensional box with Dirichlet boundary conditions is studied. Confirming previous work, we find that BEC occurs at finite N at low temperatures T without the occurrence of a phase transition. The conventionally-defined transition temperature T(E) for an infinite three-dimensional (3D) system is shown to correspond in a 2D system with finite N to a crossover temperature between a slow and rapid increase in the fractional boson occupation N(0)/N of the ground state with decreasing T. We further show that T(E)∼1/logN at fixed area per boson, so in the thermodynamic limit there is no significant BEC in 2D at finite T. Thus, paradoxically, BEC only occurs in 2D at finite N with no phase transition associated with it. Calculations of thermodynamic properties versus T and area A are presented, including Helmholtz free energy, entropy S, pressure p, ratio of p to the energy density U/A, heat capacity at constant volume (area) C(V) and at constant pressure C(p), isothermal compressibility κ(T) and thermal expansion coefficient α(p), obtained using both the grand-canonical ensemble (GCE) and canonical ensemble (CE) formalisms. The GCE formalism gives acceptable predictions for S, p, p/(U/A), κ(T) and α(p) at large N, T and A but fails for smaller values of these three parameters for which BEC becomes significant, whereas the CE formalism gives accurate results for all thermodynamic properties of finite systems even at low T and/or A where BEC occurs.
We study the thermodynamics of ideal Bose gas as well as the transport properties of non interacting bosons and fermions in a one dimensional quasi-periodic potential, namely Aubry-Andre (AA) model at finite temperature. For bosons in finite size systems, the effect of quasi-periodic potential on the crossover phenomena corresponding to Bose-Einstein condensation (BEC), superfluidity and localization phenomena at finite temperatures are investigated. From the ground state number fluctuation we calculate the crossover temperature of BEC which exhibits a non monotonic behavior with the strength of AA potential and vanishes at the self-dual critical point following power law. Appropriate rescaling of the crossover temperatures reveals universal behavior which is studied for different quasi-periodicity of the AA model. Finally, we study the temperature and flux dependence of the persistent current of fermions in presence of a quasi-periodic potential to identify the localization at the Fermi energy from the decay of the current.
We show that the chemical potential of a one-dimensional (1D) interacting Bose gas exhibits a nonmonotonic temperature dependence which is peculiar of superfluids. The effect is a direct consequence of the phononic nature of the excitation spectrum at large wavelengths exhibited by 1D Bose gases. For low temperatures TTT, we demonstrate that the coefficient in T2{T}^{2}T2 expansion of the chemical potential is entirely defined by the zero-temperature density dependence of the sound velocity. We calculate that coefficient along the crossover between the Bogoliubov weakly interacting gas and the Tonks-Girardeau gas of impenetrable bosons. Analytic expansions are provided in the asymptotic regimes. The theoretical predictions along the crossover are confirmed by comparison with the exactly solvable Yang-Yang model in which the finite-temperature equation of state is obtained numerically by solving Bethe-ansatz equations. A 1D ring geometry is equivalent to imposing periodic boundary conditions and arising finite-size effects are studied in detail. At T=0T=0T=0 we calculated various thermodynamic functions, including the inelastic structure factor, as a function of the number of atoms, pointing out the occurrence of important deviations from the thermodynamic limit.