Spontaneous Symmetry Breaking and Mean-Field Theory

Spartak T. Belyaev

Masuo Suzuki, Hidenori Suzuki

The basic problem how spontaneous symmetry breaks is studied rigorously for local fields in a finite domain. Another basic problem is to investigate whether or not there exist lower dimensions (of the domain to which a field H is applied) below which no spontaneous symmetry breaking (SSB) occurs. This is studied using scaling, renormalization group and mean-field theory for partial fields in an infinite domain. Our main results are the following: (i) The total magnetization is proportional to the square of the spontaneous magnetization when a field is applied only to a finite region (an extension of Yang's result [Phys. Rev. 85 (1952) 808], M = N m 2 s for a fixed single spin). A general formulation is given to calculate the proportionality coefficient which is shown to vanish as the field goes to zero. This indicates no SSB in this situation. (ii) On the other hand, if we apply partial fields to a d '-dimensional region and make them vanish, we find that SSB occurs, at least, for d '> d ' c =β/ν. In the region 0< d '≤ d ' c , the situation is subtle, as is exemplified by an exactly soluble model with such a field on one-side boundary as depends on the system-size.

A. Beraudo, A. De Pace, Marco Martini +1 more

Qizhong Zhu, Qi Zhang, Biao Wu

Abstract The extended Bose–Hubbard model for a double-well potential with pair tunneling is studied through both exact diagonalization and mean field theory (MFT). When pair tunneling is strong enough, the ground state wavefunction predicted by the MFT is complex and doubly degenerate while the quantum ground state wavefunction is always real and unique. The time reversal symmetry is spontaneously broken when the system transfers from the quantum ground state into one of the mean field ground states upon a small perturbation. As the gap between the lowest two levels decreases exponentially with particle number, the required perturbation inducing the spontaneous symmetry breaking (SSB) is infinitesimal for particle number of typical cold atom systems. The quantum ground state is further analyzed with the Penrose–Onsager criterion, and is found to be a fragmented condensate. The state also develops the pair correlation and has non-vanishing pair order parameter instead of the conventional single particle order parameter. When this model is generalized to optical lattice, a pair superfluid can be generated. The mean field ground state can be regarded as effective ground state in this simple model. The detailed computation for this model enables us to offer an in-depth discussion of the relation between SSB and effective ground state, giving a glimpse on how nonlinearity arises in the SSB of a quantum system.

Haijun Zhou

Spontaneous symmetry breaking (SSB) in statistical physics is a macroscopic collective phenomenon. For the paradigmatic Q-state Potts model it means a transition from the disordered color-symmetric phase to an ordered phase in which one color dominates. Existing mean field theories imply that SSB in the microcanonical statistical ensemble (with energy being the control parameter) should be a continuous process. Here we study microcanonical SSB on the random-graph Potts model, and discover that the entropy is a kinked function of energy. This kink leads to a discontinuous phase transition at certain energy density value, characterized by a jump in the density of the dominant color and a jump in the microcanonical temperature. This discontinuous SSB in random graphs is confirmed by microcanonical Monte Carlo simulations, and it is also observed in bond-diluted finite-size lattice systems.

Dieter Vollhardt, Krzysztof Byczuk, Marcus Kollar