Table of contents

This paper studies the conductance on the universal homology covering space Z of 2D orbifolds in a strong magnetic field, thereby removing the rationality constraint on the magnetic field in earlier works (Avron et al 1994 Phys. Rev. Lett.73 3255–3257; Mathai and Wilkin 2019 Lett. Math. Phys.109 2473–2484; Prieto 2006 Commun. Math. Phys.265 373–396) in the literature. We consider a natural Landau Hamiltonian on Z and study its spectrum which we prove consists of a finite number of low-lying isolated points and calculate the von Neumann degree of the associated holomorphic spectral orbibundles when the magnetic field B is large, and obtain fractional quantum numbers as the conductance.

In classical Markov jump processes, current fluctuations can only be reduced at the cost of increased dissipation. To explore how quantum effects influence this trade-off, we analyze the uncertainty of steady-state currents in Markovian open quantum systems. We first consider three instructive examples and then systematically minimize the product of uncertainty and entropy production for small open quantum systems. As our main result, we find that the thermodynamic cost of reducing fluctuations can be lowered below the classical bound by coherence. We conjecture that this cost can be made arbitrarily small in quantum systems with sufficiently many degrees of freedom. Our results thereby provide a general guideline for the design of thermal machines in the quantum regime that operate with high thermodynamic precision, meaning low dissipation and small fluctuations around average values.

Diffusion processes are important in several physical, chemical, biological and human phenomena. Examples include molecular encounters in reactions, cellular signalling, the foraging of animals, the spread of diseases, as well as trends in financial markets and climate records. Deviations from Brownian diffusion, known as anomalous diffusion (AnDi), can often be observed in these processes, when the growth of the mean square displacement in time is not linear. An ever-increasing number of methods has thus appeared to characterize anomalous diffusion trajectories based on classical statistics or machine learning approaches. Yet, characterization of anomalous diffusion remains challenging to date as testified by the launch of the AnDi challenge in March 2020 to assess and compare new and pre-existing methods on three different aspects of the problem: the inference of the anomalous diffusion exponent, the classification of the diffusion model, and the segmentation of trajectories. Here, we introduce a novel method (CONDOR) which combines feature engineering based on classical statistics with supervised deep learning to efficiently identify the underlying anomalous diffusion model with high accuracy and infer its exponent with a small mean absolute error in single 1D, 2D and 3D trajectories corrupted by localization noise. Finally, we extend our method to the segmentation of trajectories where the diffusion model and/or its anomalous exponent vary in time.

The Hohenberg–Mermin–Wagner (HMW) theorem states that infrared (IR) fluctuations prevent long-range order which breaks continuous symmetries in two dimensions (2D), at finite temperatures. We note that the theorem becomes physically effective for superconductivity (SC) only for astronomical sample sizes, so it does not prevent 2D SC in practice. We systematically explore the sensitivity of the magnetic and SC versions of the theorem to finite-size and disorder effects. For magnetism, finite-size effects, disorder, and perpendicular coupling can all restore the order parameter at a non-negligible value of T_c equally well, making the physical reason for finite T_c sample-dependent. For SC, an alternative version of the HMW theorem is presented, in which the temperature cutoff is set by Cooper pairing, in place of the Fermi energy in the standard version. It still allows 2D SC at 2–3 times the room temperature when the interaction scale is large and Cooper pairs are small, the case with high-T_c SC in the cuprates. Thus IR fluctuations do not prevent 2D SC at room temperatures in samples of any reasonable size, by any known version of the HMW argument. A possible approach to derive mechanism-dependent upper bounds for SC T_c is pointed out.

We compute exactly the statistics of the number of records in a discrete-time random walk model on a line where the walker stays at a given position with a nonzero probability 0 ⩽ p ⩽ 1, while with the complementary probability 1 − p, it jumps to a new position with a jump length drawn from a continuous and symmetric distribution f₀(η). We have shown that, for arbitrary p, the statistics of records up to step N is completely universal, i.e. independent of f₀(η) for any N. We also compute the connected two-time correlation function C_p(m₁, m₂) of the record-breaking events at times m₁ and m₂ and show it is also universal for all p. Moreover, we demonstrate that C_p(m₁, m₂) < C₀(m₁, m₂) for all p > 0, indicating that a nonzero p induces additional anti-correlations between record events. We further show that these anti-correlations lead to a drastic reduction in the fluctuations of the record numbers with increasing p. This is manifest in the Fano factor, i.e. the ratio of the variance and the mean of the record number, which we compute explicitly. We also show that an interesting scaling limit emerges when p → 1, N → ∞ with the product t = (1 − p)N fixed. We compute exactly the associated universal scaling functions for the mean, variance and the Fano factor of the number of records in this scaling limit.

Here we show that the Gross–Pitaevskii equation (GPE) for Bose–Einstein condensates (BECs) admits hydrodynamic interpretation in a general Riemannian metric, and show that in this metric the momentum equation has a new term that is associated with local curvature and density distribution profile. In particular conditions of steady state a new Einstein's field equation is determined in presence of negative curvature. Since GPE governs BECs defects that are useful, analogue models in cosmology, a relativistic form of GPE is also considered to show connection with models of analogue gravity, thus providing further grounds for future investigations of black hole dynamics in cosmology.

We obtain the exact solution of an anisotropic quantum spin chain including nearest neighbor (NN), next nearest neighbor (NNN), chiral three-spin interactions and non-diagonal boundary terms. Because the NNN couplings are involved, we find that the off-diagonal boundary reflections can not only induce the anisotropic Dzyloshinsky–Moriya interactions at boundaries but also enhance the anisotropy of first and last bonds. These properties are absent if only the NN spin-exchanging interactions are considered. Due to that the U(1) symmetry is broken, we analytically solve the system by using the off-diagonal Bethe ansatz. The inhomogeneous T − Q relation, energy spectrum and associated Bethe ansatz equations are given explicitly. This paper provides an universal scheme to construct new integrable models with certain interesting interactions.

Quantum homogeneous spaces are noncommutative spaces with quantum group covariance. Their semiclassical counterparts are Poisson homogeneous spaces, which are quotient manifolds of Lie groups M = G/H equipped with an additional Poisson structure π which is compatible with a Poisson–Lie structure Π on G. Since the infinitesimal version of Π defines a unique Lie bialgebra structure δ on the Lie algebra $\mathfrak{g}=\text{Lie}(G)$, we exploit the idea of Lie bialgebra duality in order to study the notion of complementary dual homogeneous space M^⊥ = G*/H^⊥ of a given homogeneous space M with respect to a coisotropic Lie bialgebra. Then, by considering the natural notions of reductive and symmetric homogeneous spaces, we extend these concepts to M^⊥ thus showing that an even richer duality framework between M and M^⊥ arises from them. In order to analyze physical implications of these notions, the case of M being a Minkowski or (anti-) de Sitter Poisson homogeneous spacetime is fully studied, and the corresponding complementary dual reductive and symmetric spaces M^⊥ are explicitly constructed in the case of the well-known κ-deformation, where the cosmological constant Λ is introduced as an explicit parameter in order to describe all Lorentzian spaces simultaneously. In particular, the fact that M^⊥ is a reductive space is shown to provide a natural condition for the representation theory of the quantum analogue of M that ensures the existence of physically meaningful uncertainty relations between the noncommutative spacetime coordinates. Finally, despite these dual spaces M^⊥ are not endowed in general with a G*-invariant metric, we show that their geometry can be described by making use of K-structures.

The Helmholtz equation in one dimension, which describes the propagation of electromagnetic waves in effectively one-dimensional systems, is equivalent to the time-independent Schrödinger equation. The fact that the potential term entering the latter is energy-dependent obstructs the application of the results on low-energy quantum scattering in the study of the low-frequency waves satisfying the Helmholtz equation. We use a recently developed dynamical formulation of stationary scattering to offer a comprehensive treatment of the low-frequency scattering of these waves for a general finite-range scatterer. In particular, we give explicit formulas for the coefficients of the low-frequency series expansion of the transfer matrix of the system which in turn allow for determining the low-frequency expansions of its reflection, transmission, and absorption coefficients. Our general results reveal a number of interesting physical aspects of low-frequency scattering particularly in relation to permittivity profiles having balanced gain and loss.

We consider the extreme value statistics of N independent and identically distributed random variables, which is a classic problem in probability theory. When N → ∞, fluctuations around the maximum of the variables are described by the Fisher–Tippett–Gnedenko theorem, which states that the distribution of maxima converges to one out of three limiting forms. Among these is the Gumbel distribution, for which the convergence rate with N is of a logarithmic nature. Here, we present a theory that allows one to use the Gumbel limit to accurately approximate the exact extreme value distribution. We do so by representing the scale and width parameters as power series, and by a transformation of the underlying distribution. We consider functional corrections to the Gumbel limit as well, showing they are obtainable via Taylor expansion. Our method also improves the description of large deviations from the mean extreme value. Additionally, it helps to characterize the extreme value statistics when the underlying distribution is unknown, for example when fitting experimental data.

We prove an optimal version of Uhlhorn's generalization of Wigner's unitary–antiunitary theorem. The main tool in our proof is Gleason's theorem.

Entropic uncertainty relations play an important role in both fundamentals and applications of quantum theory. Although they have been well-investigated in quantum theory, little is known about entropic uncertainty in generalized probabilistic theories (GPTs). The current study explores two types of entropic uncertainty relations, preparation and measurement uncertainty relations, in a class of GPTs which can be considered generalizations of quantum theory. Not only a method for obtaining entropic preparation uncertainty relations but also an entropic measurement uncertainty relation similar to the quantum one by Buscemi et al (2014 Phys. Rev. Lett.112 050401) are proved in those theories. These results manifest that the entropic structure in the uncertainty relations is not restricted to quantum theory and therefore is universal one. Concrete calculations of our relations in GPTs called the regular polygon theories are also demonstrated.

An observation space$\mathcal{S}$ is a family of probability distributions $\left\langle {\mathcal{P}}_{\mathit{i}}:\mathit{i}\in I\right\rangle$ sharing a common sample space Ω in a consistent way. A grounding for $\mathcal{S}$ is a signed probability distribution $\mathcal{P}$ on Ω yielding the correct marginal distribution ${\mathcal{P}}_{\mathit{i}}$ for every i. A wide variety of quantum scenarios can be formalized as observation spaces. We describe all groundings for a number of quantum observation spaces. Our main technical result is a rigorous proof that Wigner's distribution is the unique signed probability distribution yielding the correct marginal distributions for position and momentum and all their linear combinations.

The (1 + 1)-dimensional classical φ⁴ theory contains stable, topological excitations in the form of solitary waves or kinks, as well as a non-topological one, such as the oscillon. Both are used in effective descriptions of excitations throughout myriad fields of physics. The oscillon is well-known to be a coherent, particle-like structure when introduced as an ansatz in the φ⁴ theory. Here, we show that oscillons also arise naturally in the dynamics of the theory, in particular as the result of kink–antikink collisions in the presence of an impurity. We show that in addition to the scattering of kinks and the formation of a breather, both bound oscillon pairs and propagating oscillons may emerge from the collision. We discuss their resonances and critical velocity as a function of impurity strength and highlight the role played by the impurity in the scattering process.