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Journal of Low Temperature Physics

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

Kornel Witkowski, T. K. Kopeć

Published Jul 25, 2020• 4 citations
Publisher

Abstract

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.

DOI
10.1007/s10909-020-02499-y

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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.
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Abstract Dimensionality serves as an indispensable ingredient in any attempt to formulate low-dimensional physics, and studying the dimensional crossover at a fundamental level is challenging. The purpose of this work is to study the hierarchical dimensional crossovers, namely the crossover from three dimensions (3D) to quasi-2D and then to 1D. Our system consists of a 3D Bose–Einstein condensate trapped in an anisotropic 2D optical lattice characterized by the lattice depths V 1 along the x direction and V 2 along the y direction, respectively, where the hierarchical dimensional crossover is controlled via V 1 and V 2. We analytically derive the ground-state energy, quantum depletion and the superfluid density of the system. Our results demonstrate the 3D-quasi-2D-1D dimensional crossovers in the behavior of quantum fluctuations. Conditions for possible experimental realization of our scenario are also discussed.
We investigate systems of interacting bosonic particles confined within slablike boxes of size ${L}^{2}\ifmmode\times\else\texttimes\fi{}Z$ with $Z\ensuremath{\ll}L$, at their three-dimensional (3D) Bose-Einstein-condensation (BEC) transition temperature Tc{T}_{c}Tc​, and below Tc{T}_{c}Tc​ where they experience a quasi-two-dimensional (quasi-2D) Berezinskii-Kosterlitz-Thouless (BKT) transition (at TBKT<Tc{T}_{\mathrm{BKT}}<{T}_{c}TBKT​<Tc​ depending on the thickness ZZZ). The low-temperature phase below TBKT{T}_{\mathrm{BKT}}TBKT​ shows quasi-long-range order: the planar correlations decay algebraically as predicted by the 2D spin-wave theory. This dimensional crossover, from a 3D behavior for $T\ensuremath{\gtrsim}{T}_{c}$ to a quasi-2D critical behavior for $T\ensuremath{\lesssim}{T}_{\mathrm{BKT}}$, can be described by a transverse finite-size scaling limit in a slab geometry. Numerical evidence of the $3\mathrm{D}\ensuremath{\rightarrow}2\mathrm{D}$ dimensional crossover is presented for the Bose-Hubbard model defined in anisotropic ${L}^{2}\ifmmode\times\else\texttimes\fi{}Z$ lattices with $Z\ensuremath{\ll}L$. We extend this scaling analysis to the case the slab geometry of the gas is effectively realized by a transverse (inhomogeneous) harmonic trap. Finally, we discuss off-equilibrium behaviors arising from slow time variations of the temperature across the BEC transition of gases confined within a slab geometry. We argue that the system develops an off-equilibrium transverse finite-size scaling under these time-dependent protocols.
Quasi-one-dimensional (Q1D) systems, i.e., three- and two-dimensional (3D/2D) arrays composed of weakly coupled one-dimensional lattices of interacting quantum particles, exhibit rich and fascinating physics. They are studied across various areas of condensed matter and ultracold atomic lattice-gas physics, and are often marked by dimensional crossover as the coupling between one-dimensional systems is increased or temperature decreased, i.e., the Q1D system goes from appearing largely 1D to largely 3D. Phase transitions occurring along the crossover can strongly enhance this effect. Understanding these crossovers and associated phase transitions can be challenging due to the very different elementary excitations of 1D systems compared to higher-dimensional ones. In the present work, we combine numerical matrix product state (MPS) methods with mean-field (MF) theory to study paradigmatic cases of dimensional crossovers and the associated phase transitions in systems of both hard-core and soft-core lattice bosons, with relevance to both condensed matter physics and ultracold atomic gases. We show that the superfluid-to-insulator transition is a first order one, as opposed to the isotropic cases, and calculate transition temperatures for the superfluid states, finding excellent agreement with analytical theory. At the same time, our MPS+MF\mathrm{MPS}+\mathrm{MF}MPS+MF approach keeps functioning well where the current analytical framework cannot be applied. We further confirm the qualitative and quantitative reliability of our approach by comparison to exact quantum Monte Carlo calculations for the full 3D arrays.
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.