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arXiv:0804.0364 · Physical Review A

Second-order quantum phase transition of a homogeneous Bose gas with attractive interactions

Paweł Zin, Barbara Oleś, Marek Trippenbach, Krzysztof Sacha

Published Aug 15, 2008• 10 citations
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Abstract

We consider a homogeneous Bose gas of particles with an attractive interaction in an elongated three-dimensional box with periodic boundary conditions. Mean-field theory predicts for this system a spontaneous breaking of the translational symmetry at a certain value of the interaction strength. We show that at this point a second-order quantum phase transition occurs. We investigate the system in the vicinity of the critical point using Bogoliubov theory and a continuous description, that allows us to analyze quantum fluctuations in the system even when the Bogoliubov approach breaks down.

DOI
10.1103/physreva.78.023620

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We consider a homogeneous Bose gas of particles with an attractive interaction in an elongated three-dimensional box with periodic boundary conditions. Mean-field theory predicts for this system a spontaneous breaking of the translational symmetry at a certain value of the interaction strength. We show that at this point a second-order quantum phase transition occurs. We investigate the system in the vicinity of the critical point using Bogoliubov theory and a continuous description, that allows us to analyze quantum fluctuations in the system even when the Bogoliubov approach breaks down.
We consider a Bose gas with an attractive interaction in a symmetric double-well potential. In the mean-field approximation, the ground-state solution spontaneously breaks the symmetry of the trapping potential above a certain value of the interaction strength. We demonstrate how the Landau-Ginzburg scheme of the second-order phase transition emerges from the quantum model and show the link to the spontaneous symmetry breaking mentioned above. We identify the order parameter, the critical point and analyze quantum fluctuations around it.
We consider a Bose-Einstein condensate with an attractive particle interaction in a symmetric double well potential. It is known that in the mean field description a symmetric solution for the condensate wavefunction becomes unstable when the interactions are stronger than some critical value. We analyze the system around the critical point applying the number conserving Bogoliubov theory and exact diagonalization of the two mode hamiltonian. It allows for estimation of the density fluctuations in the system. Fluctuations of the order parameter, which is defined in the mean field description, turn out to be maximal at the critical point.
We generalize the scheme to characterize phase transitions of finite systems in a complex temperature plane and approach the classifications of phase transitions in ideal and weakly interacting Bose gases of a finite number of particles, confined in a cubic box of volume L3 with different boundary conditions. For this finite ideal Bose system, by extending the classification parameters to all regions, we predict that the phase transition for periodic boundary conditions is of second order, while the transition in Dirichlet boundary conditions is of first order. For a weakly interacting Bose gas with periodic boundary conditions, we discuss the effects of finite particle numbers and inter-particle interactions on the nature of the phase transitions. We show that this homogenous weakly interacting Bose gas undergoes a second-order phase transition, which is in accordance with universality arguments for infinite systems. We also discuss the dependence of transition temperature on interaction strengths and particle numbers.
Abstract We consider the homogeneous Bose gas in the three-dimensional unit torus, where N particles interact via a two-body potential of the form N−1v(x)N^{-1} v(x)N−1v(x) N - 1 v ( x ) . The system is studied at inverse temperatures of order N−2/3N^{-2/3}N−2/3 N - 2 / 3 , which corresponds to the temperature scale of the Bose–Einstein condensation phase transition. We show that spontaneous U (1) symmetry breaking occurs if and only if the system exhibits Bose–Einstein condensation in the sense that the one-particle density matrix of the Gibbs state has a macroscopic eigenvalue.
We generalize the concept of quantum phase transitions, which is conventionally defined for a ground state and usually applied in the thermodynamic limit, to one for metastable states in finite-size systems. In particular, we treat the one-dimensional Bose gas on a ring in the presence of both interactions and rotation. To support our study, we bring to bear mean-field theory, i.e., the nonlinear Schr\"odinger equation, and linear perturbation or Bogoliubov--de Gennes theory. Both methods give a consistent result in the weakly interacting regime: there exist two topologically distinct quantum phases. The first is the typical picture of superfluidity in a Bose-Einstein condensate on a ring: average angular momentum is quantized and the superflow is uniform. The second is where one or more dark solitons appear as stationary states, breaking the symmetry, the average angular momentum becomes a continuous quantity. The phase of the condensate can therefore be continuously wound and unwound.
We employ mean-field, Bogoliubov and many-body theories to study critical fluctuations in the position and momentum of a Bose-Einstein condensate whose translation symmetry is spontaneously broken due to attractive interactions. In a homogeneous system, the many-body ground state of the symmetry-preserving Hamiltonian is very fragile against superposition of low-lying states, while the mean-field theory predicts a stable bright soliton which spontaneously breaks translation symmetry. We show that weak symmetry-breaking perturbations cause the translation-symmetric many-body ground state to cross over to a many-body bright soliton. We argue that the center-of-mass fluctuations in the soliton state arise primarily from the depletion of the condensate to translation modes. We develop an extended mean-field theory to analytically reproduce these results obtained by the exact diagonalization method.