Table of contents

Finding the precise correspondence between lattice operators and the continuum fields that describe their long-distance properties is a largely open problem for strongly interacting critical points. Here, we solve this problem essentially completely in the case of the three-state Potts model, which exhibits a phase transition described by a strongly interacting 'parafermion' conformal field theory. Using symmetry arguments, insights from integrability, and extensive simulations, we construct lattice analogues of nearly all the relevant and marginal physical fields governing this transition. This construction includes chiral fields such as the parafermion. Along the way we also clarify the structure of operator product expansions between order and disorder fields, which we confirm numerically. Our results both suggest a systematic methodology for attacking non-free field theories on the lattice and find broader applications in the pursuit of exotic topologically ordered phases of matter.

We develop a field theory for spin glasses using replica Fourier transforms (RFT). We present the formalism for the case of replica symmetry and the case of replica symmetry breaking on an ultrametric tree, with the number of replicas n and the number of replica symmetry breaking steps R generic integers. We show how the RFT applied to the two-replica fields allows one to construct a new basis which block-diagonalizes the four-replica mass-matrix, into the replicon, anomalous and longitudinal modes. The eigenvalues are given in terms of the mass RFT and the propagators in the RFT space are obtained by inversion of the block-diagonal matrix. The formalism allows one to express any i-replica vertex in the new RFT basis and hence enables one to perform a standard perturbation expansion. We apply the formalism to calculate the contribution of the Gaussian fluctuations around the Parisi solution for the free-energy of an Ising spin glass.

The effects of a non-Markovian reservoir on the performance of an ideal quantum Otto heat engine are investigated. By coupling the cavity to a hot non-Markovian reservoir, the cavity can be enabled to reach a steady state corresponding to an effective temperature higher than the temperature of the hot reservoir. This observation shows that the quantum Otto heat engine can transport heat from a cold reservoir to a hot reservoir with positive net work output under certain conditions. Also, when the temperature of the hot reservoir lies in certain regions, the efficiency of the quantum Otto heat engine can exceed the efficiency of classical Carnot heat engine.

The fractional Langevin equation has been used to describe the evolution of the location of a stochastically growing interface in both planar and radial geometries. The interface dynamics can be driven by both local and non-local effects since the fractional derivative $-{{(-{{\nabla }^{2}})}^{\zeta /2}}h$ is non-local in terms of h unless ζ is an even, nonnegative integer. In this paper, we consider a general two-term Langevin equation and calculate the interface location in planar and radial geometries. We study the dynamics of the system for the stabilizing and destabilizing cases, and for different time limits. The analytical results are confirmed by numerically solving the fractional stochastic equations.

Real space condensation is known to occur in stochastic models of mass transport in the regime in which the globally conserved mass density is greater than a critical value. It has been shown within models with factorized stationary states that the condensation can be understood in terms of sums of independent and identically distributed random variables: these exhibit condensation when they are conditioned to a large deviation of their sum. It is well understood that the condensation, whereby one of the random variables contributes a finite fraction to the sum, occurs only if the underlying probability distribution (modulo exponential) is heavy-tailed, i.e. decaying slower than exponential. Here we study a similar phenomenon in which condensation is exhibited for non-heavy-tailed distributions, provided random variables are additionally conditioned on a large deviation of certain linear statistics. We provide a detailed theoretical analysis explaining the phenomenon, which is supported by Monte Carlo simulations (for the case where the additional constraint is the sample variance) and demonstrated in several physical systems. Our results suggest that the condensation is a generic phenomenon that pertains to both typical and rare events.

The time fractional Fokker–Planck equation approach is an important tool for modeling subdiffusion. When the external field is time modulated, two types of time-dependent time fractional Fokker–Planck equations have been proposed, both reduced to the same time-dependent time fractional Fokker–Planck equation when the external field is time uncorrelated. The first type is strictly deduced as the continuous limit of the continuous time random walk with time modulated Boltzmann jumping weight, while the second type is derived by ideally assuming that the jump probabilities can be evaluated at the start of the waiting time prior to jumping. For the first time we obtain the linear response characteristic for the first type of the time fractional Fokker–Planck equation systems, and for a comparison we revisit the corresponding result for the second type of the time fractional Fokker–Planck equation systems, and the similarity and difference between them is discussed with an application example. The investigation not only helps in understanding the competition between subdiffusion and time-dependent modulation, but also has significance in accessing the spectral properties of spontaneous fluctuation and the linear dynamic susceptibility of external perturbation in subdiffusive processes.

During an epidemic, people may adapt or alter their social contacts to avoid infection. Various adaptation mechanisms have been studied previously. Recently, a new adaptation mechanism was presented in (Tunc et al 2013 J. Stat. Phys.151 355), where susceptible nodes temporarily deactivate their links to infected neighbors and reactivate when their neighbors recover. Considering the same adaptation mechanism on a scale-free network, we find that the topology of the subnetwork consisting of active links is fundamentally different from the original network topology. We predict the scaling exponent of the active degree distribution and derive mean field equations by using improved moment closure approximations based on the conditional distribution of active degree given the total degree. These mean field equations show better agreement with numerical simulation results than the standard mean field equations based on a homogeneity assumption.

Few analytical methods exist for quantitative studies of large fluctuations in stochastic systems. In this article, we develop a simple diagrammatic approach to the chemical master equation that allows us to calculate multi-time correlation functions which are accurate to any desired order in van Kampenʼs system size expansion. Specifically, we present a set of Feynman rules from which this diagrammatic perturbation expansion can be constructed algorithmically. We then apply the methodology to derive in closed form the leading order corrections to the linear noise approximation of the intrinsic noise power spectrum for general biochemical reaction networks. Finally, we illustrate our results by describing noise-induced oscillations in the Brusselator reaction scheme which are not captured by the common linear noise approximation.

In this work, we explore a prototypical example of a genuine continuum breather (i.e., not a standing wave) and the conditions under which it can persist in a $\mathcal{P}\mathcal{T}$-symmetric medium. As our model of interest, we will explore the sine-Gordon equation in the presence of a $\mathcal{P}\mathcal{T}$-symmetric perturbation. Our main finding is that the breather of the sine-Gordon model will only persist at the interface between gain and loss that $\mathcal{P}\mathcal{T}$-symmetry imposes but will not be preserved if centered at the lossy or at the gain side. The latter dynamics is found to be interesting in its own right giving rise to kink–antikink pairs on the gain side and complete decay of the breather on the lossy side. Lastly, the stability of the breathers centered at the interface is studied. As may be anticipated on the basis of their 'delicate' existence properties such breathers are found to be destabilized through a Hopf bifurcation in the corresponding Floquet analysis.

Karlin and McGregorʼs d-variable Hahn polynomials are shown to arise in the $(d+1)$-dimensional singular oscillator model as the overlap coefficients between bases associated with the separation of variables in Cartesian and hyperspherical coordinates. These polynomials in d discrete variables depend on $d+1$ real parameters and are orthogonal with respect to the multidimensional hypergeometric distribution. The focus is put on the d = 2 case for which the connection with the three-dimensional singular oscillator is used to derive the main properties of the polynomials: forward/backward shift operators, orthogonality relation, generating function, recurrence relations, bispectrality (difference equations) and explicit expression in terms of the univariate Hahn polynomials. The extension of these results to an arbitrary number of variables is presented at the end of the paper.

It was recently proven that the total multiplicity in the decomposition into irreducibles of the tensor product $\lambda \otimes \mu$ of two irreducible representations of a simple Lie algebra is invariant under conjugation of one of them; at a given level, this also applies to the fusion multiplicities of affine algebras. Here, we show that, in the case of SU(3), the lists of multiplicities, in the tensor products $\lambda \otimes \mu$ and $\lambda \otimes \bar{\mu }$, are identical up to permutations. This latter property does not hold in general for other Lie algebras. We conjecture that the same property should hold for the fusion product of the affine algebra of SU(3) at finite levels, but this is not investigated in the present paper.

Incompressible Navier–Stokes equations for gas mixtures are derived from Boltzmann kinetic models in a suitable fluid dynamic limit. We consider polyatomic gases, each one endowed with a discrete set of internal energy levels. Specifically, we deal with a mixture of four polyatomic gases also undergoing chemical reactions. In the Maxwell molecule case, diffusion coefficients and contributions due to inelastic scattering and to chemical reactions may be explicitly computed.

The energy spectrum of a 3-level atomic system in the Ξ-configuration is studied. This configuration presents a triple point independently of the number of atoms, which remains in the thermodynamic limit. This means that in a vicinity of this point any quantum fluctuation will drastically change the composition of the ground state of the system. We study the expectation values of the atomic population of each level, the number of photons, and the probability distribution of photons at the triple point.

We investigate the action of the first three levels of the Clifford hierarchy on sets of mutually unbiased bases comprising the Ivanovic mutually unbiased base (MUB) and the Alltop MUBs. Vectors in the Alltop MUBs exhibit additional symmetries when the dimension is a prime number equal to 1 modulo 3 and thus the set of all Alltop vectors splits into three Clifford orbits. These vectors form configurations with so-called Zauner subspaces, eigenspaces of order 3 elements of the Clifford group highly relevant to the SIC problem. We identify Alltop vectors as the magic states that appear in the context of fault-tolerant universal quantum computing, wherein the appearance of distinct Clifford orbits implies a surprising inequivalence between some magic states.

Mutually unbiased bases that can be cyclically generated by a single unitary operator are of special interest, for they can be readily implemented in practice. We show that, for a system of qubits, finding such a generator can be cast as the problem of finding a symmetric matrix over the field $\mathbb{F}$₂ equipped with an irreducible characteristic polynomial of a given Fibonacci index. The entanglement structure of the resulting complete sets is determined by two additive matrices of the same size.

We define a simple rule that allows us to describe sequences of projective measurements for a broad class of generalized probabilistic models. This class embraces quantum mechanics and classical probability theory, but, for example, also the hypothetical Popescu–Rohrlich box. For quantum mechanics, the definition yields the established Lüders rule, which is the standard rule for updating the quantum state after a measurement. For the general case, it can be seen as being the least disturbing or most coherent way of performing sequential measurements. We show, as an example, that the Spekkens toy model (Spekkens 2007 Phys. Rev. A 75 032110) provides an instance of our definition. We also demonstrate the possibility of strong postquantum correlations as well as the existence of triple-slit correlations for certain nonquantum toy models.

We show that the γ_i-deformation, which was proposed as candidate gauge theory for a non-supersymmetric three-parameter deformation of the anti-de Sitter / conformal field theory correspondence, is not conformally invariant due to a running double-trace coupling—not even in the 't Hooft limit. Moreover, this non-conformality cannot be cured when we extend the theory by adding at tree-level arbitrary multi-trace couplings that obey certain minimal consistency requirements. Our findings suggest a possible connection between this breakdown of conformal invariance and a puzzling divergence recently encountered in the integrability-based descriptions of two-loop finite-size corrections for the single-trace operator of two identical chiral fields. We propose a test to clarify this.

We establish features of so-called Yangian secret symmetries for AdS₃ type IIB superstring backgrounds, thus verifying the persistence of such symmetries to this new instance of the AdS/CFT correspondence. Specifically, we find two a priori different classes of secret symmetry generators. One class of generators, anticipated from the previous literature, is more naturally embedded in the algebra governing the integrable scattering problem. The other class of generators is more elusive and somewhat closer in its form to its higher-dimensional AdS₅ counterpart. All of these symmetries respect left-right crossing. In addition, by considering the interplay between left and right representations, we gain a new perspective on the AdS₅ case. We also study the $R\mathcal{T}\mathcal{T}$-realisation of the Yangian in AdS₃ backgrounds, thus establishing a new incarnation of the Beisert–de Leeuw construction.