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The problem of finding a microscopic theory of phase transitions across a critical point is a central unsolved problem in theoretical physics. We find a general solution to that problem and present it here for the cases of Bose-Einstein condensation in an interacting gas and ferromagnetism in a lattice of spins, interacting via a Heisenberg or Ising Hamiltonian. For Bose-Einstein condensation, we present the exact, valid for the entire critical region, equations for the Green functions and order parameter, that is a critical-region extension of the Beliaev-Popov and Gross-Pitaevskii equations. For the magnetic phase transition, we find an exact theory in terms of constrained bosons in a lattice and obtain similar equations for the Green functions and order parameter. In particular, we outline an exact solution for the three-dimensional Ising model.
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We present a microscopic theory of the second order phase transition in an interacting Bose gas that allows one to describe formation of an ordered condensate phase from a disordered phase across an entire critical region continuously. We derive the exact fundamental equations for a condensate wave function and the Green functions, which are valid both inside and outside the critical region. They are reduced to the usual Gross-Pitaevskii and Beliaev-Popov equations in a low-temperature limit outside the critical region. The theory is readily extendable to other phase transitions, in particular, in the physics of condensed matter and quantum fields.
(abridged) In this paper, we present the issues we consider as essential as far as the statistical mechanics of finite systems is concerned. In particular, we emphasis our present understanding of phase transitions in the framework of information theory. Information theory provides a thermodynamically-consistent treatment of finite, open, transient and expanding systems which are difficult problems in approaches using standard statistical ensembles. As an example, we analyze is the problem of boundary conditions, which in the framework of information theory must also be treated statistically. We recall that out of the thermodynamical limit the different ensembles are not equivalent and in particular they may lead to dramatically different equation of states, in the region of a first order phase transition. We recall the recent progresses achieved in the understanding of first-order phase transition in finite systems: the equivalence between the Yang-Lee theorem and the occurrence of bimodalities in the intensive ensemble and the presence of inverted curvatures of the thermodynamic potential of the associated extensive ensemble.
We analytically and numerically study the Loschmidt echo and the dynamical order parameters in a spin chain with a deconfined phase transition between a dimerized state and a ferromagnetic phase. For quenches from a dimerized state to a ferromagnetic phase, we find that the model can exhibit a dynamical quantum phase transition characterized by an associating dimerized order parameters. In particular, when quenching the system from the Majumdar-Ghosh state to the ferromagnetic Ising state, we find an exact mapping into the classical Ising chain for a quench from the paramagnetic phase to the classical Ising phase by analytically calculating the Loschmidt echo and the dynamical order parameters. By contrast, for quenches from a ferromagnetic state to a dimerized state, the system relaxes very fast so that the dynamical quantum transition may only exist in a short time scale. We reveal that the dynamical quantum phase transition can occur in systems with two broken symmetry phases and the quench dynamics may be independent on equilibrium phase transitions.
We study the quantum phase transition from a super solid phase to a solid phase of rho = 1/2 for the extended Bose-Hubbard model on the honeycomb lattice using first principles Monte Carlo calculations. The motivation of our study is to quantitatively understand the impact of theoretical input, in particular the dynamical critical exponent z, in calculating the critical exponent nu. Hence we have carried out four sets of simulations with beta = 2N^{1/2}, beta = 8N^{1/2}, beta = N/2, and beta = N/4, respectively. Here beta is the inverse temperature and N is the numbers of lattice sites used in the simulations. By applying data collapse to the observable superfluid density rho_{s2} in the second spatial direction, we confirm that the transition is indeed governed by the superfluid-insulator universality class. However we find it is subtle to determine the precise location of the critical point. For example, while the critical chemical potential (mu/V)_c occurs at (mu/V)_c = 2.3239(3) for the data obtained using beta = 2N^{1/2}, the (mu/V)_c determined from the data simulated with beta = N/2 is found to be (mu/V)_c = 2.3186(2). Further, while a good data collapse for rho_{s2}N can be obtained with the data determined using beta = N/4 in the simulations, a reasonable quality of data collapse for the same observable calculated from another set of simulations with beta = 8N^{1/2} can hardly be reached. Surprisingly, assuming z for this phase transition is determined to be 2 first in a Monte Carlo calculation, then a high quality data collapse for rho_{s2}N can be achieved for (mu/V)_c ~ 2.3184 and nu ~ 0.7 using the data obtained with beta = 8N^{1/2}. Our results imply that one might need to reconsider the established phase diagrams of some models if the accurate location of the critical point is crucial in obtaining a conclusion.
We investigate the dynamics of a conservative version of Conways Game of Life, in which a pair consisting of a dead and a living cell can switch their states following Conways rules but only by swapping their positions, irrespective of their mutual distance. Our study is based on square-lattice simulations as well as a mean-field calculation. As the density of dead cells is increased, we identify a discontinuous phase transition between an inactive phase, in which the dynamics freezes after a finite time, and an active phase, in which the dynamics persists indefinitely in the thermodynamic limit. Further increasing the density of dead cells leads the system back to an inactive phase via a second transition, which is continuous on the square lattice but discontinuous in the mean-field limit.
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