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Register a new userPhase Transitions in the Two-Dimensional Ising Model from the Microcanonical Perspective
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The continuous ferromagnetic-paramagnetic phase transition in the two-dimensional Ising model has already been excessively studied by conventional canonical statistical analysis in the past. We use the recently developed generalized microcanonical inflection-point analysis method to investigate the least-sensitive inflection points of the microcanonical entropy and its derivatives to identify transition signals. Surprisingly, this method reveals that there are potentially two additional transitions for the Ising system besides the critical transition.
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In the present work, we discuss how the functional form of thermodynamic observables can be deduced from the geometric properties of subsets of phase space. The geometric quantities taken into account are mainly extrinsic curvatures of the energy level sets of the Hamiltonian of a system under investigation. In particular, it turns out that peculiar behaviours of thermodynamic observables at a phase transition point are rooted in more fundamental changes of the geometry of the energy level sets in phase space. More specifically, we discuss how microcanonical and geometrical descriptions of phase-transitions are shaped in the special case of $phi^4$ models with either nearest-neighbours and mean-field interactions.
Evaporation/condensation transition of the Potts model on square lattice is numerically investigated by the Wang-Landau sampling method. Intrinsically system size dependent discrete transition between supersaturation state and phase-separation state is observed in the microcanonical ensemble by changing constrained internal energy. We calculate the microcanonical temperature, as a derivative of microcanonical entropy, and condensation ratio, and perform a finite size scaling of them to indicate clear tendency of numerical data to converge to the infinite size limit predicted by phenomenological theory for the isotherm lattice gas model.
The surface and bulk properties of the two-dimensional Q > 4 state Potts model in the vicinity of the first order bulk transition point have been studied by exact calculations and by density matrix renormalization group techniques. For the surface transition the complete analytical solution of the problem is presented in the $Q to infty$ limit, including the critical and tricritical exponents, magnetization profiles and scaling functions. According to the accurate numerical results the universality class of the surface transition is independent of the value of Q > 4. For the bulk transition we have numerically calculated the latent heat and the magnetization discontinuity and we have shown that the correlation lengths in the ordered and in the disordered phases are identical at the transition point.
A framework is presented for carrying out simulations of equilibrium systems in the microcanonical ensemble using annealing in an energy ceiling. The framework encompasses an equilibrium version of simulated annealing, population annealing and hybrid algorithms that interpolate between these extremes. These equilibrium, microcanonical annealing algorithms are applied to the thermal first-order transition in the 20-state, two-dimensional Potts model. All of these algorithms are observed to perform well at the first-order transition though for the system sizes studied here, equilibrium simulated annealing is most efficient.
The Ising model, with short-range interactions between constituents, is a basic mathematical model in statistical mechanics. It has been widely used to describe collective phenomena such as order-disorder phase transitions in various physical, biological, economical, and social systems. However, it was proven that spontaneous phase transitions do not exist in the one-dimensional Ising models. Besides low dimensionality, frustration is the other well-known suppressor of phase transitions. Here I show that surprisingly, a strongly frustrated one-dimensional two-leg ladder Ising model can exhibit a marginal finite-temperature phase transition. It features a large latent heat, a sharp peak in specific heat, and unconventional order parameters, which classify the transition as involving an entropy-favored intermediate-temperature ordered state and further unveil a crossover to an exotic normal state in which frustration effectively decouples the two strongly interacted legs in a counterintuitive non-mean-field way. These exact results expose a mathematical structure that has not appeared before in phase-transition problems, and shed new light on our understanding of phase transitions and the dynamical actions of frustration. Applications of this model and its mechanisms to various systems with extensions to consider higher dimensions, quantum characters, or external fields, etc. are anticipated and briefly discussed---with insights into the puzzling phenomena of strange strong frustration and intermediate-temperature orders such as the Bozin-Billinge orbital-degeneracy-lifting recently discovered in real materials.
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