Posts by Collection

Niche overlap and Hopfield-like interactions in generalized random Lotka-Volterra systems

Published in Physical Review E, 2023

We study communities emerging from generalized random Lotka-Volterra dynamics with a large number of species with interactions determined by the degree of niche overlap. Each species is endowed with a number of traits, and competition between pairs of species increases with their similarity in trait space. This leads to a model with random Hopfield-like interactions. We use tools from the theory of disordered systems, notably dynamic mean-field theory, to characterize the statistics of the resulting communities at stable fixed points and determine analytically when stability breaks down. Two distinct types of transition are identified in this way, both marked by diverging abundances but differing in the behavior of the integrated response function. At fixed points only a fraction of the initial pool of species survives. We numerically study the eigenvalue spectra of the interaction matrix between extant species. We find evidence that the two types of dynamical transition are, respectively, associated with the bulk spectrum or an outlier eigenvalue crossing into the right half of the complex plane.

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Fluctuation corrections to Lifshitz tails in disordered systems

Published in Physical Review E, 2024

Quenched disorder in semiconductors induces localized electronic states at the band edge, which manifest as an exponential tail in the density of states. For large impurity densities, this tail takes a universal Lifshitz form that is characterized by short-ranged potential fluctuations. We provide both analytical expressions and numerical values for the Lifshitz tail of a parabolic conduction band including its exact fluctuation prefactor. Our analysis is based on a replica field integral approach, where the leading exponential scaling of the tail is determined by an instanton profile and fluctuations around the instanton determine the subleading preexponential factor. This factor contains the determinant of a fluctuation operator, and we avoid a full computation of its spectrum by using a Gel’fand-Yaglom formalism, which provides a concise general derivation of fluctuation corrections in disorder problems. We provide a revised result for the disorder band tail in two dimensions.

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Universal fragmentation in annihilation reactions with constrained kinetics

Published in Physical Review Research, 2024

In reaction-diffusion models of annihilation reactions in low dimensions, single-particle dynamics provides a bottleneck for reactions, leading to an anomalously slow approach to the empty state. Here, we construct a reaction model with a reciprocal bottleneck of reactions on particle dynamics in which single-particle motion conserves the center of mass. We show that such a constrained reaction-diffusion dynamics does not approach an empty state but freezes at late times in a state with fragmented particle clusters. The late-time dynamics and final density are universal, and we provide exact results for the final density in the large-reaction rate limit. Thus, our setup constitutes a minimal model for the fragmentation of a one-dimensional lattice into independent particle clusters. We suggest that the universal reaction dynamics could be observable in experiments with cold atoms or in the Auger recombination of exciton gases.

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Instanton theory and fluctuation corrections to the thermal nucleation rate of a ferromagnetic superfluid

Published in arXiv, 2025

We provide a field-theoretical description of thermal nucleation in a one-dimensional ferromagnetic superfluid, a quantum-gas analogue of false-vacuum decay. The rate at which ground-state domains nucleate from fluctuations in the metastable phase follows an Arrhenius law, with an exponential factor determined by a saddle-point configuration of the energy functional – the critical droplet – and a magnitude fixed by small fluctuations of this configuration. We evaluate both contributions over the full parameter space, using a Gelfand-Yaglom approach to reduce the calculation of the fluctuation spectrum to an initial value problem. In addition, we obtain a closed-form expression for the critical droplet in the limit of small potential tilts, and use it to formulate an effective theory of domain nucleation and growth as a Kramers escape problem for the droplet size. Our results determine the parametric dependence of the nucleation rate and its signature on experimental images of a nucleating gas, and should allow for a rigorous comparison between nucleation theory and experiment.