Table of contents

Gradient descent dynamics in complex energy landscapes, i.e. featuring multiple minima, finds application in many different problems, from soft matter to machine learning. Here, we analyze one of the simplest examples, namely that of soft repulsive particles in the limit of infinite spatial dimension d. The gradient descent dynamics then displays a jamming transition: at low density, it reaches zero-energy states in which particles' overlaps are fully eliminated, while at high density the energy remains finite and overlaps persist. At the transition, the dynamics becomes critical. In the d → ∞ limit, a set of self-consistent dynamical equations can be derived via mean field theory. We analyze these equations and we present some partial progress towards their solution. We also study the random Lorentz gas in a range of d = 2...22, and obtain a robust estimate for the jamming transition in d → ∞. The jamming transition is analogous to the capacity transition in supervised learning, and in the appendix we discuss this analogy in the case of a simple one-layer fully-connected perceptron.

We present an algorithm to compute the exact probability R_n(p) for a site percolation cluster to span an n × n square lattice at occupancy p. The algorithm has time and space complexity O(λⁿ) with λ ≈ 2.6. It allows us to compute R_n(p) up to n = 24. We use the data to compute estimates for the percolation threshold p_c that are several orders of magnitude more precise than estimates based on Monte-Carlo simulations.

The kicked top is one of the paradigmatic models in the study of quantum chaos (Haake et al 2018 Quantum Signatures of Chaos (Springer Series in Synergetics vol 54)). Recently it has been shown that the onset of quantum chaos in the kicked top can be related to the proliferation of Trotter errors in digital quantum simulation (DQS) of collective spin systems. Specifically, the proliferation of Trotter errors becomes manifest in expectation values of few-body observables strongly deviating from the target dynamics above a critical Trotter step, where the spectral statistics of the Floquet operator of the kicked top can be predicted by random matrix theory. In this work, we study these phenomena in the framework of Hamiltonian learning (HL). We show how a recently developed HL protocol can be employed to reconstruct the generator of the stroboscopic dynamics, i.e., the Floquet Hamiltonian, of the kicked top. We further show how the proliferation of Trotter errors is revealed by HL as the transition to a regime in which the dynamics cannot be approximately described by a low-order truncation of the Floquet–Magnus expansion. This opens up new experimental possibilities for the analysis of Trotter errors on the level of the generator of the implemented dynamics, that can be generalized to the DQS of quantum many-body systems in a scalable way. This paper is in memory of our colleague and friend Fritz Haake.

The field of movement ecology has seen a rapid increase in high-resolution data in recent years, leading to the development of numerous statistical and numerical methods to analyse relocation trajectories. Data are often collected at the level of the individual and for long periods that may encompass a range of behaviours. Here, we use the power spectral density (PSD) to characterise the random movement patterns of a black-winged kite (Elanus caeruleus) and a white stork (Ciconia ciconia). The tracks are first segmented and clustered into different behaviours (movement modes), and for each mode we measure the PSD and the ageing properties of the process. For the foraging kite we find 1/f noise, previously reported in ecological systems mainly in the context of population dynamics, but not for movement data. We further suggest plausible models for each of the behavioural modes by comparing both the measured PSD exponents and the distribution of the single-trajectory PSD to known theoretical results and simulations.

Experiments investigating particles floating on a randomly stirred fluid show regions of very low density, which are not well understood. We introduce a simplified model for understanding sparsely occupied regions of the phase space of non-autonomous, chaotic dynamical systems, based upon an extension of the skinny bakers' map. We show how the distribution of the sizes of voids in the phase space can be mapped to the statistics of the running maximum of a Wiener process. We find that the model exhibits a lacunarity transition, which is characterised by regions of the phase space remaining empty as the number of trajectories is increased.

We introduce a finite dimensional anharmonic soft spin glass in a field and show how it allows the construction a field theory at zero temperature and the corresponding loop expansion. The mean field level of the model coincides with a recently introduced fully connected model, the KHGPS model, and it has a spin glass transition in a field at zero temperature driven by the appearance of pseudogapped non-linear excitations. We analyze the zero temperature limit of the theory and the behavior of the bare masses and couplings on approaching the mean field zero temperature critical point. Focusing on the so called replicon sector of the field theory, we show that the bare mass corresponding to fluctuations in this sector is strictly positive at the transition in a certain region of control parameter space. At the same time the two relevant cubic coupling constants g₁ and g₂ show a non-analytic behavior in their bare values: approaching the critical point at zero temperature, g₁ → ∞ while g₂ ∝ T with a prefactor diverging at the transition. Along the same lines we also develop the field theory to study the density of states of the model in finite dimension. We show that in the mean field limit the density of states converges to the one of the KHGPS model. However the construction allows a treatment of finite dimensional effects in perturbation theory.

We propose a systematic method for constructing integrable delay-difference and delay-differential analogues of known soliton equations such as the Lotka–Volterra, Toda lattice (TL), and sine-Gordon equations and their multi-soliton solutions. It is carried out by applying a reduction and delay-differential limit to the discrete KP or discrete two-dimensional TL equations. Each of the delay-difference and delay-differential equations has the N-soliton solution, which depends on the delay parameter and converges to an N-soliton solution of a known soliton equation as the delay parameter approaches 0.

We quantise orbits of the adjoint group action on elements of the sl$(2,\mathbb{R})$ Lie algebra. The path integration along elliptic slices is akin to the coadjoint orbit quantization of compact Lie groups, and the calculation of the characters of elliptic group elements proceeds along the same lines as in compact groups. The computation of the trace of hyperbolic group elements in a diagonal basis as well as the calculation of the full group action on a hyperbolic basis requires considerably more technique. We determine the action of hyperbolic one-parameter subgroups of PSL$(2,\mathbb{R})$ on the adjoint orbits and discuss global subtleties in choices of adapted coordinate systems. Using the hyperbolic slicing of orbits, we describe the quantum mechanics of an irreducible sl$(2,\mathbb{R})$ representation in a hyperbolic basis and relate the basis to the mathematics of the Mellin integral transform. We moreover discuss the representation theory of the double cover SL$(2,\mathbb{R})$ of PSL$(2,\mathbb{R})$ as well as that of its universal cover. Traces in the representations of these groups for both elliptic and hyperbolic elements are computed. Finally, we motivate our treatment of this elementary quantization problem by indicating applications.

We propose a systematic procedure to construct polynomial algebras from intermediate Casimir invariants arising from (semisimple or non-semisimple) Lie algebras $\mathfrak{g}$. In this approach, we deal with explicit polynomials in the enveloping algebra of $\mathfrak{g}\oplus \mathfrak{g}\oplus \mathfrak{g}$. We present explicit examples of low-dimensional Lie algebras (up to dimension six) to show how they can display different behaviours and can lead to abelian algebras, quadratic algebras or more complex structures involving higher order nested commutators. Within this framework, we also demonstrate how virtual copies of the Levi factor of a Levi decomposable Lie algebra can be used as a tool to construct 'copies' of polynomial algebras. Different schemes to obtain polynomial algebras associated to algebraic Hamiltonians have been proposed in the literature, among them the use of commutants of various type. The present approach is different and relies on the construction of intermediate Casimir invariants in the enveloping algebra $\mathcal{U}(\mathfrak{g}\oplus \mathfrak{g}\oplus \mathfrak{g})$.

We apply the bootstrap technique to find the moments of certain multi-trace and multi-matrix random matrix models suggested by noncommutative geometry. Using bootstrapping we are able to find the relationships between the coupling constant of these models and their second moments. Using the Schwinger–Dyson equations, all other moments can be expressed in terms of the coupling constant and the second moment. Explicit relations for higher mixed moments are also obtained.

For a given diagrammatic approximation in many-body perturbation theory it is not guaranteed that positive observables, such as the density or the spectral function, retain their positivity. For zero-temperature systems we developed a method [2014 Phys. Rev. B90 115134] based on so-called cutting rules for Feynman diagrams that enforces these properties diagrammatically, thus solving the problem of negative spectral densities observed for various vertex approximations. In this work we extend this method to systems at finite temperature by formulating the cutting rules in terms of retarded N-point functions, thereby simplifying earlier approaches and simultaneously solving the issue of non-vanishing vacuum diagrams that has plagued finite temperature expansions. Our approach is moreover valid for nonequilibrium systems in initial equilibrium and allows us to show that important commonly used approximations, namely the GW, second Born and T-matrix approximation, retain positive spectral functions at finite temperature. Finally we derive an analytic continuation relation between the spectral forms of retarded N-point functions and their Matsubara counterparts and a set of Feynman rules to evaluate them.