Abstract
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.
Highlights
In a series of ground breaking papers in the late 19th century, Gibbs [1,2,3] elegantly derived the thermodynamic stable state of a given substance through the minimization of some thermodynamic potential, by means of a geometric construction
Gibbs considered a surface given by the possible values of the thermodynamic extensive quantities of a system of interest and realized that points on this surface with tangent planes of equal orientation correspond to possible stable states of the substance at a temperature and pressure given by the orientation of the tangent plane
We investigated the convex structure of reduced density matrices (RDMs) and marginal probability distributions of many body systems and illustrated how the concept of symmetry breaking emerges very naturally through the appearance of ruled surfaces at the boundaries of these sets. As these sets exist without any prior notion of an underlying Hamiltonian, this shows that the reason for the occurrence of symmetry breaking lies in the geometrical structures of the convex sets of all possible RDMs or marginal probability distributions
Summary
Any further distribution of this work must maintain symmetry breaking is a consequence of the geometric structure of the convex set of reduced density attribution to the matrices of all possible many body wavefunctions The surfaces of these convex bodies exhibit nonauthor(s) and the title of the work, journal citation analyticities, which signal the emergence of symmetry breaking and of an associated order parameter and DOI. 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.
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