We study analytically the superfluid flow of a Bose-Einstein condensate in a ring geometry in the presence of a rotating barrier. We show that a phase transition breaking a parity symmetry among two topological phases occurs at a critical value of the height of the barrier. Furthermore, a discontinuous (accompanied by hysteresis) phase transition is observed in the ordered phase when changing the angular velocity of the barrier. At the critical point where the hysteresis area vanishes, the chemical potential of the ground state develops a cusp (a discontinuity in the first derivative). Along this path, the jump between the two corresponding states having a different winding number shows analogies with a topological phase transition. We finally study the current-phase relation of the system and compare some of our calculations with published experimental results.
Quantum phase transition in the spin-boson model was claimed on the basis of various numerical studies, but not strictly proven. Here by using a unitary transformation to decompose the Hamiltonian into two branches of odd and even parity we obtained the necessary and sufficient condition for degeneracy to occur between states of opposite parity in the spin-boson model, and the analytical expression for such degenerate energies. It can be strictly proven that the ground state of spin-boson model with non-vanishing tunneling amplitude must have an energy lower than the lowest possible such degenerate energy, and have definite parity. Starting from the invariancy of the parity operator we show that finite expansion by numerical calculation induces the breaking of parity symmetry responsible for the phase transition. The critical dissipation parameter we obtained for parity-symmetry breaking, as a logarithmic function of summed diagonal matrix elements in the finite expansion for the bosonic part of the parity operator, can reproduce the phase diagram derived with quantum Monte Carlo method and logarithmically discretized numberical renormalization group approach. It reveals that the quantum phase transition in spin-boson model claimed by numerical procedures arises from symmetry breaking caused by finite expansion in practical calculation. The method we developed here may also be applicable to the discussion of quantum chaos and other similar problems.
Degeneracy patterns in quantum mechanics stem from the system symmetries. In particular, the broken-symmetry phase in the well-known Lipkin-Meshkov-Glick (LMG) model is composed of doubly-degenerate states of different parity. In this work, we show that such doublets can exist even if parity is not conserved. For this purpose, our starting point is an anharmonic LMG Hamiltonian with a second-order ground-state quantum phase transition (GSQPT) and a rich spectrum, with two different excited-state quantum phase transitions. The inclusion in the Hamiltonian of a term inducing a first-order GSQPT breaks the parity symmetry but conserving the exponential degeneracy in the energy doublets. We demonstrate that this phenomenon can be traced back to the existence of a
Z2 symmetry (reflection symmetry) in the system's classical limit phase space that leads to an anti-unitary
Z2 symmetry in the quantum system.