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arXiv:1409.5484 · Journal of Physics A Mathematical and Theoretical

Ground-state Bethe root densities and quantum phase transitions

Jon Links, Ian Marquette

Published Jan 5, 2015• 8 citations
arXivHTMLPDF

Abstract

Exactly solvable models provide a unique method, via qualitative changes in the distribution of the ground-state roots of the Bethe ansatz equations, to identify quantum phase transitions. Here we expand on this approach, in a quantitative manner, for two models of Bose–Einstein condensates. The first model deals with the interconversion of bosonic atoms and molecules. The second is the two-site Bose–Hubbard model, widely used to describe tunneling phenomena in Bose–Einstein condensates. For these systems we calculate the ground-state root density. This facilitates the determination of analytic forms for the ground-state energy, and associated correlation functions through the Hellmann–Feynman theorem. These calculations provide a clear identification of the quantum phase transition in each model. For the first model we obtain an expression for the molecular fraction expectation value. For the two-site Bose–Hubbard model we find that there is a simple characterization of condensate fragmentation.

DOI
10.1088/1751-8113/48/4/045204

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Exactly solvable models provide a unique method, via qualitative changes in the distribution of the ground-state roots of the Bethe ansatz equations, to identify quantum phase transitions. Here we expand on this approach, in a quantitative manner, for two models of Bose–Einstein condensates. The first model deals with the interconversion of bosonic atoms and molecules. The second is the two-site Bose–Hubbard model, widely used to describe tunneling phenomena in Bose–Einstein condensates. For these systems we calculate the ground-state root density. This facilitates the determination of analytic forms for the ground-state energy, and associated correlation functions through the Hellmann–Feynman theorem. These calculations provide a clear identification of the quantum phase transition in each model. For the first model we obtain an expression for the molecular fraction expectation value. For the two-site Bose–Hubbard model we find that there is a simple characterization of condensate fragmentation.
We study the ground-state properties of an atomic-molecular Bose–Einstein condensate model through an exact Bethe Ansatz solution. For a certain range of parameter choices, we prove that the ground-state Bethe roots lie on the positive real-axis. We then use a continuum limit approach to obtain a singular integral equation characterising the distribution of these Bethe roots. Solving this equation leads to an analytic expression for the ground-state energy. The form of the expression is consistent with the existence of a line of quantum phase transitions, which has been identified in earlier studies. This line demarcates a molecular phase from a mixed phase. Certain correlation functions, which characterise these phases, are then obtained through the Hellmann–Feynman theorem.
The extended Bose-Hubbard model for a double-well potential with atom-pair tunneling is studied. Starting with a classical analysis we determine the existence of three quantum phases: self-trapping, phase-locking, and Josephson states. From this analysis we build the parameter space of quantum phase transitions between degenerate and nondegenerate ground states driven by the atom-pair tunneling. Considering only the repulsive case, we confirm the phase transition by the measure of the energy gap between the ground state and the first excited state. We study the structure of the solutions of the Bethe ansatz equations for a small number of atoms. An inspection of the roots for the ground state suggests a relationship to the physical properties of the system. By studying the energy gap we find that the profile of the roots of the Bethe ansatz equations is related to a quantum phase transition.
The attractive two-site Bose-Hubbard model is studied within the framework of the analytical solution obtained by the application of the quantum inverse scattering method. The structure of the ground and excited states is analyzed in terms of solutions of Bethe equations, and an approximate solution for the Bethe roots is given. This yields approximate formulas for the ground-state energy and for the first excited-state energy. The obtained formulas work with remarkable precision for a wide range of parameters of the model, and are confirmed numerically. An expansion of the Bethe state vectors into a Fock space is also provided for evaluation of expectation values, although this does not have accuracy similar to that of the energies.
The two-site version of the Bose-Hubbard model can be used as a simplified model of N bosons confined to two quantum wells. We study the behavior of low-temperature Bose-Einstein condensates as a function of the strength of the repulsive interactions between the bosons and of the size of the quantum-mechanical tunneling between the quantum wells. The ground-state of weakly interacting bosons tends to be one in which almost all particles are in the same quantum-mechanical state (‘Bose-Einstein Condensate’, or ‘BEC’). It is found that even very small amounts of interwell tunneling lead to a BEC, but that zero tunneling leads to two separate BECs, one in each well (‘fragmentation’). We study systematically the transition between a single BEC comprised of bosons in both wells, and a fragmented condensate consisting of two BECs, one for each well. Our analysis will use matrix computational methods to solve the Schrödinger equation of the two-well boson system, and obtain the energies and wavefunctions as eigenvalues and eigenvectors of the Hamiltonian matrix. We will compare the results obtained with previous analyses, and against analytical approximations developed for the limiting case of a large number of bosons (large N).
In this work we define a formal notion of a quantum phase crossover for certain Bethe ansatz solvable models. The approach we adopt exploits an exact mapping of the spectrum of a many-body integrable system, which admits an exact Bethe ansatz solution, into the quasi-exactly solvable spectrum of a one-body Schrödinger operator. Bifurcations of the minima for the potential of the Schrödinger operator determine the crossover couplings. By considering the behaviour of particular ground-state correlation functions, these may be identified as quantum phase crossovers in the many-body integrable system with finite particle number. In this approach the existence of the quantum phase crossover is not dependent on the existence of a thermodynamic limit, rendering applications to finite systems feasible. We study two examples of bosonic Hamiltonians which admit second-order crossovers. Keywords: quantum integrability (Bethe ansatz), Bose-Einstein condensation (theory) 1
We show that the SU(N) Fermi-Hubbard model (FHM) on two sites, where N is the number of flavors of each fermion, corresponds to an exactly solvable two-level many-boson model that Richardson [J. Math. Phys. 9, 1327 (1968)] analytically solved long ago. We express the Bethe ansatz solutions as a function of the physical parameters of the SU(N) FHM, and recast its eigenvalues and eigenstates in terms of the Richardson pair energies and creation operators. In this context, the connection with the well-studied Lipkin-Meshkov-Glick (LMG) model, known as equivalent to the Richardson model, is established and serves as a guideline to the prediction of some N-body physics phenomena in the two-site SU(N) FHM with N particles. In particular, the LMG second-order quantum phase transition (QPT) is shown to occur in the SU(N) FHM for an attractive density-density interaction U equal to U_c=-1/(2N), in units of the (absolute value of the) tunneling amplitude between the two sites. We show the finite-size energies, the gap, and the kinetic energy, which all reveal the transition, as a function of U for values of N from N = 3 to N = 36, suggesting that the QPT could be experimentally achieved with current technologies involving SU(N) ultracold atoms or molecules. Finally, we show the entanglement entropy of the first site with respect to the second, and it scales like N at the transition, in contrast with several two-mode models.