We demonstrate that the occurrence of symmetry breaking phase transitions together with the emergence of a local order parameter in classical statistical physics is a consequence of the geometrical structure of probability space. To this end we investigate convex sets generated by expectation values of certain observables with respect to all possible probability distributions of classical q-state spins on a two-dimensional lattice, for several values of q. The extreme points of these sets are then given by thermal Gibbs states of the classical q-state Potts model. As symmetry breaking phase transitions and the emergence of associated order parameters are signaled by the appearance ruled surfaces on these sets, this implies that symmetry breaking is ultimately a consequence of the geometrical structure of probability space. In particular we identify the different features arising for continuous and first order phase transitions and show how to obtain critical exponents and susceptibilities from the geometrical shape of the surface set. Such convex sets thus also constitute a novel and very intuitive way of constructing phase diagrams for many body systems, as all thermodynamically relevant quantities can be very naturally read off from these sets.
We demonstrate that the occurrence of symmetry breaking phase transitions together with the emergence of a local order parameter in classical statistical physics is a consequence of the geometrical structure of probability space. To this end we investigate convex sets generated by expectation values of certain observables with respect to all possible probability distributions of classical q-state spins on a two-dimensional lattice, for several values of q. The extreme points of these sets are then given by thermal Gibbs states of the classical q-state Potts model. As symmetry breaking phase transitions and the emergence of associated order parameters are signaled by the appearance ruled surfaces on these sets, this implies that symmetry breaking is ultimately a consequence of the geometrical structure of probability space. In particular we identify the different features arising for continuous and first order phase transitions and show how to obtain critical exponents and susceptibilities from the geometrical shape of the surface set. Such convex sets thus also constitute a novel and very intuitive way of constructing phase diagrams for many body systems, as all thermodynamically relevant quantities can be very naturally read off from these sets.
The concept of symmetry breaking and the emergence of corresponding local order parameters constitute the pillars of modern day many body physics. We demonstrate that the existence of symmetry breaking is a consequence of the geometric structure of the convex set of reduced density matrices of all possible many body wavefunctions. The surfaces of these convex bodies exhibit non-analyticities, which signal the emergence of symmetry breaking and of an associated order parameter and also show different characteristics for different types of phase transitions. We illustrate this with three paradigmatic examples of many body systems exhibiting symmetry breaking: the quantum Ising model, the classical q -state Potts model in two-dimensions at finite temperature and the ideal Bose gas in three-dimensions at finite temperature. This state based viewpoint on phase transitions provides a unique novel tool for studying exotic many body phenomena in quantum and classical systems.
In this work, we have proposed a new geometrical method for calculating the critical temperature and critical exponents by introducing a set of bond breaking probability values. The probability value Pc corresponding to the Coniglio–Klein probability for the transition temperature is obtained among this set of trial probabilities. Critical temperature, thermal and magnetic exponents are presented for d = 2 and d = 3, q = 2 Potts model and for the application of the method to the system with first order phase transition, q = 7 Potts model on different size lattices are employed. The advantage of this method can be that the bond breaking probability can be applied, where the clusters are defined on a set of dynamic variables, which are different from the dynamic quantities of the actual Hamiltonian or the action of the full system. An immediate application can be to use the method on finite temperature lattice gauge theories.
Scaling, universality, and renormalization are three pillars of modern critical phenomena. According to the hypothesis of universality, continuous phase transitions fall into classes mainly determined by spatial dimensionality and symmetry of the order parameter. The latter is usually reflected by the degeneracy of the ground state of the Hamiltonian. However, for certain systems at criticality, a higher symmetry emerges in the order parameter, and the associated critical behavior may become very rich. Such emergent symmetry has been found in spin ice systems, deconfined quantum critical points, high-Tc superconductors, and so forth, and are often accompanied by very interesting critical phenomena. New results presented here are based on Monte Carlo simulations and finite-size scaling of a series of q-state Potts models on the simple cubic lattice with ferromagnetic interactions in one lattice direction and antiferromagnetic interactions in the other two directions. The staggered magnetization appears to display an emergent continuous O(n) symmetry with n=q-1, as illustrated by two-dimensional intersections of the distribution functions. Also the estimated critical exponents are consistent with the O(n=q-1) universality classes.
It is discussed how phase transitions of first order (with phase separation and surface tension), continuous transitions and (multi)-critical points can be defined and classified for finite systems from the topology of the energy surface eS(E,N) of the mechanical N-body phase space or more precisely of the curvature determinant D(E,N)=∂2S/∂E2∗∂2S/∂N2−(∂2S/∂E∂N)2 without taking the thermodynamic limit. The first calculation of the entire entropy surface S(E,N) for a q=3-states Potts lattice gas on a 50*50 square lattice is shown. There are two lines, where S(E,N) has a maximum curvature ∼0. One is the border between the regions in \{E,N\} with D(E,N)>0 and with D(E,N)<0, the other line is critical starting as a valley in D(E,N) running from the continuous transition in the ordinary q=3-Potts model, converting at Pm into a flat ridge/plateau (maximum) deep inside the convex intruder of S(E,N) which characterizes the first order liquid-gas transition. The multi-critical point Pm is their crossing.
A cornerstone of the theory of phase transitions is the observation that many-body systems exhibiting a spontaneous symmetry breaking in the thermodynamic limit generally show extensive fluctuations of an order parameter in large but finite systems. In this work, we introduce the dynamical analog of such a theory. Specifically, we consider local dissipative dynamics preparing an equilibrium steady-state of quantum spins on a lattice exhibiting a discrete or continuous symmetry but with extensive fluctuations in a local order parameter. We show that for all such processes, there exist asymptotically stationary symmetry-breaking states, i.e., states that become stationary in the thermodynamic limit and give a finite value to the order parameter. We give results both for discrete and continuous symmetries and explicitly show how to construct the symmetry-breaking states. Our results show in a simple way that, in large systems, local dissipative dynamics satisfying detailed balance cannot uniquely and efficiently prepare states with extensive fluctuations with respect to local operators. We discuss the implications of our results for quantum simulators and dissipative state preparation.
The effects of linear and quadratic symmetry-breaking perturbations on the continuous version of the three-state Potts model are analysed, using both Landau's theory of phase transitions and renormalisation-group techniques. Variation of the strength of the perturbations produces many different types of phase diagrams, featuring lambda lines, first-order lines, critical, tricritical and triple points and other complexities. Universal amplitude ratios characterising the multicritical points are calculated to first order in epsilon =4-d.