Search Talks

I will summarize some recent results on systems of dipole-conserving point particles, 'fractons'. These exhibit non-equilibrium dynamics characterized by attractors, that cannot be characterized by Gibbsean statistical mechanics. Fracton dynamics generically possess a 'Janus pont' of low complexity around which a bidirectional arrow of time naturally obtains. Its Boltzmann entropy is unbounded and thus the dynamics evades 'heat death' at late times, suggesting a surprisingly clean resolution of the arrow-of-time paradox in non-equilibrium dynamics.

The hippocampus supports two important functions: spatial navigation and the storage of episodic memories. Yet, how these two seemingly distinct roles converge in a single circuit remains an open question. In this talk, I will present a neural model that leverages the low-dimensional attractor dynamics of spatial "grid cells" to implement associative, spatial, and episodic memory, thus unifying these distinct functions. Our model, Vector-HaSH (Vector Hippocampal Scaffolded Heteroassociative Memory) operates through a factorization of the content of memories from dynamics that generate error-correcting stable states. This leads to a graceful trade-off between number of stored items and recall detail, unlike the abrupt capacity limits found in classical Hopfield-like memory models. We find that the usage of pre-structured low-dimensional representations also enables high-capacity sequence memorization by recasting the chaining problem of high-dimensional states into one of learning low-dimensional transitions. Further, our presented approach reproduces several hippocampal experiments on spatial mapping and context-dependent representations, and provides a circuit model of the 'memory palaces' used by memory athletes. Thus, this work provides a unified framework for understanding how the hippocampus simultaneously supports spatial mapping, associative memory, and episodic memory.

Exact solutions for excited states in non-integrable quantum Hamiltonians have revealed novel dynamical phenomena that can occur in quantum many-body systems. This work proposes a method to analytically construct a specific set of volume-law-entangled exact excited eigenstates in a large class of spin Hamiltonians. In particular, we show that all spin chains that satisfy a simple set of conditions host exact volume-law eigenstates in the middle of their spectra. Examples of physically relevant spin chains of this type include the transverse-field Ising model, PXP model, spin-S XY model, and spin-S Kitaev chain. Although these eigenstates are highly atypical in their structure, they are thermal with respect to local observables. Our framework also unifies many recent constructions of volume-law entangled eigenstates in the literature. Finally, we show that a similar construction also generalizes to spin models on graphs in arbitrary dimensions.

We introduce a rejection-free, flat histogram, cluster algorithm to determine the density of states of hardcore lattice gases. We show that the algorithm is able to efficiently sample low entropy states that are usually difficult to access, even when the excluded volume per particle is large. The algorithm is based on simultaneously evaporating all the particles in a strip and reoccupying these sites with a new appropriately chosen configuration. We implement the algorithm for the particular case of the hard-core lattice gas in which the first k next-nearest neighbors of a particle are excluded from being occupied. It is shown that the algorithm is able to reproduce the known results for k = 1, 2, 3 both on the square and cubic lattices. We also show that, in comparison, the corresponding flat histogram algorithms with either local moves or unbiased cluster moves are less accurate and do not converge as the system size increases.

Nonlinear dynamics aims to elucidate the basic mechanisms necessary to reflect the temporal behavior of a natural system. The data analysis and modeling techniques proposed by artificial intelligence (deep networks, computational reservoirs, recurrent networks, as examples), on the other hand, ostensibly resign the mechanistic vision for a data-oriented modeling paradigm.  In these lectures, these apparently antagonistic approaches will be analyzed in parallel, using as examples my work on the physics of birdsong production and vocal learning.

Small systems consisting of a few particles are increasingly technologically relevant. In such systems, an intense debate in microcanonical statistical mechanics has been about the correctness of Boltzmann’s surface entropy versus Gibbs’ volume entropy. While the former considers states within a fixed energy window centered around the energy of the system, the latter considers all states with energy lesser than or equal to the energy of the system. Both entropies have shortcomings─while Boltzmann entropy predicts unphysical negative/infinite absolute temperatures for small systems with an unbounded energy spectrum, Gibbs entropy entirely disallows negative absolute temperatures, in disagreement with experiments. We consider a relative energy window, motivated by the Heisenberg energy-time uncertainty principle and an eigenstate thermalization time inversely proportional to the system energy. The resulting entropy ensures positive, finite temperatures for systems without a maximum limit on their energy and allows negative absolute temperatures in bounded energy spectrum systems, e.g., with population inversion. It also closely matches canonical ensemble predictions for prototypical systems, for instance, correctly describing the zero-point energy of an isolated quantum harmonic oscillator. Overall, we enable accurate thermodynamic models for isolated systems with few degrees of freedom.

We study the dynamics of an athermal inertial active particle moving in a shear-thinning medium in $d=1$. The viscosity of the medium is modeled using a Coulomb-tanh function, while the activity is represented by an asymmetric dichotomous noise with strengths $-\Delta$ and $\mu\Delta$, transitioning between these states at a rate $\lambda$. Starting from the Fokker-Planck~(FP) equation for the time-dependent probability distributions $P(v,-\Delta,t)$ and $P(v,\mu\Delta,t)$ of the particle's velocity $v$ at time $t$, moving under the influence of active forces $-\Delta$ and $\mu\Delta$ respectively, we analytically derive the steady-state velocity distribution function $P_s(v)$, explicitly dependent on $\mu$. Also, we obtain a quadrature expression for the effective diffusion coefficient $D_e$ for the symmetric active force case~($\mu=1$). For a given $\Delta$ and $\mu$, we show that $P_s(v)$ exhibits multiple transitions as $\lambda$ is varied. Subsequently, we numerically compute $P_s(v)$, the mean-squared velocity $\langle v^2\rangle(t)$, and the diffusion coefficient $D_e$ by solving the particle's equation of motion, all of which show excellent agreement with the analytical results in the steady-state. Finally, we examine the universal nature of the transitions in $P_s(v)$ by considering an alternative functional form of medium's viscosity that also capture the shear-thinning behavior.

Hard-core lattice-gas models serve as minimal yet powerful models to explore entropy-driven phase transitions. In these models, particles are restricted from occupying neighboring sites up to a specified kth next-nearest neighbor, effectively bridging the behavior from simple nearest-neighbor exclusion to the continuum hard-sphere gas. While most prior studies have examined cases up to k = 3, this talk presents a detailed investigation of the lattice-gas model on a triangular lattice up to k = 7, using a rejection free flat histogram algorithm enhanced with cluster moves. Our findings reveal that for k = 3 to k = 7, the system exhibits a single, discontinuous phase transition from a low-density disordered fluid to a high-density sublattice-ordered phase. This conclusion is supported by the analysis of partition function zeros and the nonconvexity of entropy.