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Physical Review A

Bose—Einstein Condensation in a One-Dimensional Model with Random Impurities

J. M. Luttinger, H.K. Sy

Published Feb 1, 1973• 25 citations
Publisher

Abstract

We study the thermodynamics of a one-dimensional system of noninteracting bosons in the presence of random fixed finite-range repulsive potentials. It is shown that a Bose---Einstein condensation into the lowest state occurs. The critical temperature increases with the number of impurities. The specific heat exhibits a discontinuity and is infinitely differentiable from both sides of the critical point. For a suitably defined pressure the isothermals exhibit a flat portion below the "transition length."

DOI
10.1103/physreva.7.712

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Journal of the Physical Society of Japan•Aug 1, 2002•1 citation
We study the thermodynamics of a one-dimensional system of noninteracting bosons in the presence of random fixed finite-range repulsive potentials. It is shown that a Bose---Einstein condensation into the lowest state occurs. The critical temperature increases with the number of impurities. The specific heat exhibits a discontinuity and is infinitely differentiable from both sides of the critical point. For a suitably defined pressure the isothermals exhibit a flat portion below the "transition length."
The thermodynamic behavior of the one-dimensional Bose-gas–attractive-δ impurity system is studied in this paper. The system is shown to undergo the Bose–Einstein condensation and the cause of the phase transition is attributed to the bound state introduced by the impurity in the free-particle energy spectrum. The condensed phase is composed of particles captured by the impurity, forming a drop of particles well localized in space. This gives to the Bose–Einstein condensation in this system the appearance of the ordinary vapor–liquid phase transition. The expression for the pressure is obtained and is plotted as a function of the temperature and the system length. The shapes of the isothermal curves reinforce the interpretation of a vapor–liquid transition. The evaluation of the heat capacity at constant length shows the existence of a finite discontinuity at the transition temperature.
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 examine bosons hopping on a one-dimensional lattice in the presence of a random potential at zero temperature. Bogoliubov excitations of the Bose-Einstein condensate formed under such conditions are localized, with the localization length diverging at low frequency as l(omega) approximately 1/omega(alpha). We show that the well-known result alpha=2 applies only for sufficiently weak random potential. As the random potential is increased beyond a certain strength, alpha starts decreasing. At a critical strength of the potential, when the system of bosons is at the transition from a superfluid to an insulator, alpha=1. This result is relevant for understanding the behavior of the atomic Bose-Einstein condensates in the presence of random potential, and of the disordered Josephson junction arrays.
The exact density of states for one particle in the field of an attractive $\ensuremath{\delta}$ impurity center is obtained from the system Green's function. The impurity is shown to introduce a bound state in the free-particle energy spectrum without any further modification. This result is then used to study the effects of the impurity on the thermodynamical properties of a gas of noninteracting bosons. The "Bose-Einstein phase transition" is not destroyed by the impurity, but a true condensation in position space occurs due to the existence of the bound state. The captured bosons form a cloud of well-localized particles modifying the macroscopic behavior of the gas for temperatures below a transition temperature Tc{T}_{c}Tc​. The constant-volume specific heat, the entropy, and the pressure are evaluated as functions of the temperature and of the energy of the bound state, Eb{E}_{b}Eb​. The transition temperature and the discontinuity of the specific heat at Tc{T}_{c}Tc​ are also obtained as a function of Eb{E}_{b}Eb​.
The phase transition of the one-dimensional Bose gas, interacting weakly with a repulsive potential and trapped in an external parabolic potential, is studied in a mean field approximation. The Bose–Einstein condensed (BEC) state is identified by the non-vanishing thermal average of a bose field. Free energies between the BEC state and normal state are compared. A crossing of the free energies occurring at low temperature suggests the first order phase transition in an interacting one-dimensional bose system. From fitting of numerical results, the relation among the transition temperature T c , the total number of bosons N and the interaction V is also provided as a simple power law, T c ∝( N 2 V ) 2/3 .