Level Of Symmetry Breaking Research Articles

Crisanti–Sommers Formula and Simultaneous Symmetry Breaking in Multi-species Spherical Spin Glasses

There is a rich history of expressing the limiting free energy of mean-field spin glasses as a variational formula over probability measures on $[0,1]$, where the measure represents the similarity (or "overlap") of two independently sampled spin configurations. At high temperatures, the formula's minimum is achieved at a measure which is a point mass, meaning sample configurations are asymptotically orthogonal up to a magnetic field correction. At low temperatures, though, a very different behavior emerges known as replica symmetry breaking (RSB). The deep wells in the energy landscape create more rigid structure, and the optimal overlap measure is no longer a point mass. The exact size of its support remains in many cases an open problem. Here we consider these themes for multi-species spherical spin glasses. Following a companion work in which we establish the Parisi variational formula, here we present this formula's Crisanti-Sommers representation. In the process, we gain new access to a problem unique to the multi-species setting. Namely, if RSB occurs for one species, does it necessarily occur for other species as well? We provide sufficient conditions for the answer to be yes. For instance, we show that if two species share any quadratic interaction, then RSB for one implies RSB for the other. Moreover, the level of symmetry breaking must be identical, even in cases of full RSB. In the presence of an external field, any type of interaction suffices.

Symmetry breaking and inter-clonal behavioural variability in a slime mould.

Cells are dynamic systems capable of switching from isotropic growth to polarized growth even in the absence of any pre-existing external asymmetry. Here, we study this process of symmetry breaking in the acellular slime mould Physarum polycephalum. In these experiments, slime moulds could grow on two identical opposed sources of calcium. We highlighted a positive correlation between growth dynamic, level of symmetry breaking and calcium concentration. We identified three populations of slime moulds within our clonal lineage with similar symmetry breaking behaviours but different motility characteristics. These behavioural differences between slime moulds emerged in the absence of any environmental differences. Such behavioural plasticity could generate cellular diversity, which can be critical for survival.

Replica symmetry breaking in mean-field spin glasses through the Hamilton–Jacobitechnique

During the last few years, through the combined effort of the insight coming from physicalintuition and computer simulation, and the exploitation of rigorous mathematical methods,the main features of the mean-field Sherrington–Kirkpatrick spin glass model have beenfirmly established. In particular, it has been possible to prove the existence and uniquenessof the infinite-volume limit for the free energy, and its Parisi expression, in terms of avariational principle involving a functional order parameter. Even the expected property ofultrametricity, for the infinite-volume states, seems to be near to a complete proof.The main structural feature of this model, and related models, is the deep phenomenon ofspontaneous replica symmetry breaking (RSB), discovered by Parisi many years ago. Byexpanding on our previous work, the aim of this paper is to investigate a generalframework, where replica symmetry breaking is embedded in a kind of mechanical schemeof the Hamilton–Jacobi type. Here, the analog of the ‘time’ variable is a parametercharacterizing the strength of the interaction, while the ‘space’ variables rule outquantitatively the broken replica symmetry pattern. Starting from the simple cases, whereannealing is assumed, or replica symmetry, we build up a progression of dynamical systems,with an increasing number of space variables, which allow us to weaken the effect of thepotential in the Hamilton–Jacobi equation as the level of symmetry breaking is increased.This new machinery allows us to work out mechanically the generalK-step RSB solutions, in a different interpretation with respect to the replica trick, andeasily reveals their properties such as existence or uniqueness.

Multiple levels of symmetry breaking

In the previous paper in this volume we have studied the p-spin interaction model just below the critical temperature, and we have rigorously proved several aspects of the physicists prediction that this model exhibits “one level of symmetry breaking”. In the present paper we show how to construct systems that exhibit an arbitrarily large, but finite number of “levels of symmetry-breaking”. As the temperature decreases, such systems exhibit many phase transitions, as the structure of the overlaps gains complexity. This phenomenon does not seem to have been described previously, even in the physics literature.

Rigorous low-temperature results for the mean field p-spins interaction model

We prove that, just below the critical temperature, the mean field p-spins interaction model, for p suitably large, spontaneously decomposes into different states. The asymptotic overlaps between any two different states are zero. Under a mild (unproven) hypothesis on the weight distribution of these states, we prove that they are pure states. This situation is called in physics “one level of symmetry breaking”.

Gauge-coupling-constant magnitudes in the Pati-Salam model

A doubly tiered hierarchy of symmetry breaking is proposed in which a unified gauge group is broken first to the Pati-Salam group SU(2)/sub L/ x SU(2)/sub R/ x SU(4), subsequently to the SU(2)/sub L/ x U(1) x SU(3)/sub c/ group containing the known elementary-particle interactions. The second level of symmetry breaking corresponds to the differentiation of baryon number from lepton number. Such a hierarchy is consistent with the phenomenology of the Pati-Salam model for integrally charged quarks. The proposed hierarchy allows the weak angle to remain in the range 3/8 < or = sin/sup 2/theta/sub W/ < or = 0.42 as the quark-gluon coupling constant becomes appropriately large. It is suggested that the first level of symmetry breaking corresponds to the separation of gravity from elementary-particle interactions. The hierarchy is also shown to apply for fractionally charged quarks, increasing the quark-gluon coupling constant (..cap alpha../sub s/ approx. = 0.1) while leaving the weak angle sin/sup 2/theta/sub W/ approx. = 3/8.