Coulomb Gas Research Articles

Mapping a dissipative quantum spin chain onto a generalized Coulomb gas

An XXZ spin chain at zero magnetization is subject to spatially correlated baths acting as dissipation. We show that the low-energy excitations of this model are described by a dissipative sine-Gordon field theory, i.e. a sine-Gordon action with an additional long-range interaction emerging from dissipation. The field theory is then exactly mapped onto a generalized Coulomb gas which, in addition to the usual integer charges, displays half-integer charges that originate from the dissipative baths. These new charges come in pairs linked by a charge independent logarithmic interaction. In the Coulomb gas picture, we identify a Berezinsky–Kosterlitz–Thouless-like phase transition corresponding to the binding of charges and derive the associated perturbative renormalization group equations. For superohmic baths, the transition is due to the binding of the integer charges, while for subohmic baths, it is due to the binding of the half-integer charges, thereby signaling a dissipation-induced transition.

Electrostatic-elastic coupling in colloidal crystals

Abstract Electrostatic-elastic coupling in colloidal crystals, composed of a mobile Coulomb gas permeating a fixed background crystalline lattice of charged colloids, is studied on the continuum level in order to analyze the lattice-mediated interactions between mobile charges. The linearized, Debye-H{" u}ckel-like mean-field equations incorporating a minimal coupling between electrostatic and displacement fields imply an additional effective attractive interaction between mobile charges. For small screening lengths, the interactions between like mobile charges exhibit colloid lattice-mediated effective interaction, ranging from weak to strong attraction, while for large screening lengths the lattice mediated interaction is purely attractive. Continuum theory incorporating the standard lattice elasticity and electrostatics of mobile charges, augmented by the minimal electrostatic-elastic coupling terms, can serve as a baseline for more detailed microscopic models.

Coulomb universality.

Motivated by a number of realizations of long-range interacting systems, including ultracold atomic and molecular gases, we study a neutral plasma with power-law interactions longer ranged than Coulombic. We find that beyond a crossover length, such interactions are universally screened down to a standard Coulomb form in all spatial dimensions. This implies, counterintuitively, that in two dimensions and below, such a "super-Coulombic" gas is asymptotically Coulombically confining at low temperatures. At higher temperatures, the plasma undergoes a deconfining transition that in two dimensions is the same Kosterlitz-Thouless transition that occurs in a conventional Coulomb gas, but at an elevated temperature that we calculate. We also predict that, in contrast, above two dimensions, even when naively the bare potential is confining, there is no confined phase of the plasma at any nonzero temperature. In addition, the super-Coulomb to Coulomb crossover is followed at longer length scales by an unconventional "Debye-Huckel" screening, which leads to faster-than-Coulombic, power-law decay of the screened potential, in contrast to the usual exponentially decaying Yukawa potential. Furthermore, we show that power-law potentials that fall off more rapidly than Coulomb are screened down to a shorter-ranged power law rather than an exponential Debye-Huckel Yukawa form. We expect these prediction to be testable in simulations and hope they will inspire experimental studies in various platforms.

Interactions between different birds of prey as a random point process

The two-dimensional (2D) Coulomb gas is a one-parameter family of random point processes, depending on the inverse temperature β. Based on previous work, it is proposed as a simple statistical measure to quantify the intra- and interspecies repulsion among three different highly territorial birds of prey. Using data from the area of the Teutoburger Wald over 20 years, we fit the nearest-neighbour and next-to-nearest neighbour spacing distributions between the respective nests of the goshawk, eagle owl and the previously examined common buzzard to β of the Coulomb gas. Within each species, the repulsion measured in this way deviates significantly from the Poisson process of independent points in the plane. In contrast, the repulsion amongst each of two species is found to be considerably lower and closer to Poisson. Methodologically, we investigate the influence of the terrain, of a shorter interaction range given by the 2D Yukawa interaction, and the statistical independence of the time moving average we use for the yearly ensembles of occupied nests. We also check that an artificial random displacement of the original nest positions of the order of the mean level spacing quickly destroys the repulsion measured by β > 0. A simple, approximate analytical expression for the nearest-neighbour spacing distribution derived from non-Hermitian random matrix theory proves to be very useful.

A Dyson Brownian Motion Model for Weak Measurements in Chaotic Quantum Systems

We consider a toy model for the study of monitored dynamics in many-body quantum systems. We study the stochastic Schrödinger equation resulting from continuous monitoring with a rate Γ of a random Hermitian operator, drawn from the Gaussian unitary ensemble (GUE) at every time t. Due to invariance by unitary transformations, the dynamics of the eigenvalues {λα}α=1n of the density matrix decouples from that of the eigenvectors, and is exactly described by stochastic equations that we derive. We consider two regimes: in the presence of an extra dephasing term, which can be generated by imperfect quantum measurements, the density matrix has a stationary distribution, and we show that in the limit of large size n→∞ it matches with the inverse-Marchenko–Pastur distribution. In the case of perfect measurements, instead, purification eventually occurs and we focus on finite-time dynamics. In this case, remarkably, we find an exact solution for the joint probability distribution of λ’s at each time t and for each size n. Two relevant regimes emerge: at short times tΓ=O(1), the spectrum is in a Coulomb gas regime, with a well-defined continuous spectral distribution in the n→∞ limit. In that case, all moments of the density matrix become self-averaging and it is possible to exactly characterize the entanglement spectrum. In the limit of large times tΓ=O(n), one enters instead a regime in which the eigenvalues are exponentially separated log(λα/λβ)=O(Γt/n), but fluctuations ∼O(Γt/n) play an essential role. We are still able to characterize the asymptotic behaviors of the entanglement entropy in this regime.

On currents in the $O(n)$ loop model

Using methods from the conformal bootstrap, we study the properties of Noether currents in the critical O(n)O(n) loop model. We confirm that they do not give rise to a Kac-Moody algebra (for n≠ 2n≠2), a result expected from the underlying lack of unitarity. By studying four-point functions in detail, we fully determine the current-current OPEs, and thus obtain several structure constants with physical meaning. We find in particular that the terms :\!J\bar{J}\!::JJ‾: in the identity and adjoint channels vanish exactly, invalidating the argument made in [Nucl. Phys. B 419, 411 (1994)] that adding orientation-dependent interactions to the model should lead to continuously varying exponents in self-avoiding walks. We also determine the residue of the identity channel in the JJJJ two-point function, finding that it coincides both with the result of a transfer-matrix computation for an orientation-dependent correlation function in the lattice model, and with an earlier Coulomb gas computation of Cardy [Nucl. Phys. B 265, 409 (1986)]. This is, to our knowledge, one of the first instances where the Coulomb gas formalism and the bootstrap can be successfully compared.

Integrability of rank-two web models

We continue our work on lattice models of webs, which generalise the well-known loop models to allow for various kinds of bifurcations [1,2]. Here we define new web models corresponding to each of the rank-two spiders considered by Kuperberg [3]. These models are based on the A2, G2 and B2 Lie algebras, and their local vertex configurations are intertwiners of the corresponding q-deformed quantum algebras. In all three cases we define a corresponding model on the hexagonal lattice, and in the case of B2 also on the square lattice. For specific root-of-unity choices of q, we show the equivalence to a number of three- and four-state spin models on the dual lattice.The main result of this paper is to exhibit integrable manifolds in the parameter spaces of each web model. For q on the unit circle, these models are critical and we characterise the corresponding conformal field theories via numerical diagonalisation of the transfer matrix.In the A2 case we find two integrable regimes. The first one contains a dense and a dilute phase, for which we have analytic control via a Coulomb gas construction, while the second one is more elusive and likely conceals non-compact physics. Three particular points correspond to a three-state spin model with plaquette interactions, of which the one in the second regime appears to present a new universality class. In the G2 case we identify four regimes numerically. The B2 case is too unwieldy to be studied numerically in the general case, but it found analytically to contain a simpler sub-model based on generators of the dilute Birman-Murakami-Wenzl algebra.

Linear statistics for Coulomb gases: higher order cumulants

We consider N classical particles interacting via the Coulomb potential in spatial dimension d and in the presence of an external trap, at equilibrium at inverse temperature β. In the large N limit, the particles are confined within a droplet of finite size. We study smooth linear statistics, i.e. the fluctuations of sums of the form LN=∑i=1Nf(xi) , where x i ’s are the positions of the particles and where f(xi) is a sufficiently regular function. There exists at present standard results for the first and second moments of LN in the large N limit, as well as associated Central Limit Theorems in general dimension and for a wide class of confining potentials. Here we obtain explicit expressions for the higher order cumulants of LN at large N, when the function f(x)=f(|x|) and the confining potential are both rotationnally invariant. A remarkable feature of our results is that these higher cumulants depend only on the value of f′(|x|) and its higher order derivatives evaluated exactly at the boundary of the droplet, which in this case is a d-dimensional sphere. In the particular two-dimensional case d = 2 at the special value β = 2, a connection to the Ginibre ensemble allows us to derive these results in an alternative way using the tools of determinantal point processes. Finally we also obtain the large deviation form of the full probability distribution function of LN .

Connection probabilities of multiple FK-Ising interfaces

We find the scaling limits of a general class of boundary-to-boundary connection probabilities and multiple interfaces in the critical planar FK-Ising model, thus verifying predictions from the physics literature. We also discuss conjectural formulas using Coulomb gas integrals for the corresponding quantities in general critical planar random-cluster models with cluster-weight q∈[1,4)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${q \\in [1,4)}$$\\end{document}. Thus far, proofs for convergence, including ours, rely on discrete complex analysis techniques and are beyond reach for other values of q than the FK-Ising model (q=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q=2$$\\end{document}). Given the convergence of interfaces, the conjectural formulas for other values of q could be verified similarly with relatively minor technical work. The limit interfaces are variants of SLEκ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {SLE}_\\kappa $$\\end{document} curves (with κ=16/3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa = 16/3$$\\end{document} for q=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q=2$$\\end{document}). Their partition functions, that give the connection probabilities, also satisfy properties predicted for correlation functions in conformal field theory (CFT), expected to describe scaling limits of critical random-cluster models. We verify these properties for all q∈[1,4)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q \\in [1,4)$$\\end{document}, thus providing further evidence of the expected CFT description of these models.

Large deviations and phase transitions in spectral linear statistics of Gaussian random matrices

We evaluate, in the large-N limit, the complete probability distribution P(A,m) of the values A of the sum ∑i=1N|λi|m , where λ i ( i=1,2,…,N ) are the eigenvalues of a Gaussian random matrix, and m is a positive real number. Combining the Coulomb gas method with numerical simulations using a matrix variant of the Wang–Landau algorithm, we found that, in the limit of N→∞ , the rate function of P(A,m) exhibits phase transitions of different characters. The phase diagram of the system on the (A, m) plane is surprisingly rich, as it includes three regions: (i) a region with a single-interval support of the optimal spectrum of eigenvalues, (ii) a region emerging for m < 2 where the optimal spectrum splits into two separate intervals, and (iii) a region emerging for m > 2 where the maximum or minimum eigenvalue ‘evaporates’ from the rest of eigenvalues and dominates the statistics of A. The phase transition between regions (i) and (iii) is of second order. Analytical arguments and numerical simulations strongly suggest that the phase transition between regions (i) and (ii) is of (in general) fractional order p=1+1/|m−1| , where 0<m<2 . The transition becomes of infinite order in the special case of m = 1, where we provide a more complete analytical and numerical description. Remarkably, the transition between regions (i) and (ii) for m⩽1 and the transition between regions (i) and (iii) for m > 2 occur at the ground state of the Coulomb gas which corresponds to the Wigner’s semicircular distribution.

The Singularities of Selberg- and Dotsenko–Fateev-Like Integrals

We discuss the meromorphic continuation of certain hypergeometric integrals modeled on the Selberg integral, including the 3-point and 4-point functions of BPZ’s minimal models of 2D CFT as described by Felder & Silvotti and Dotsenko & Fateev (the “Coulomb gas formalism”). This is accomplished via a geometric analysis of the singularities of the integrands. In the case that the integrand is symmetric (as in the Selberg integral itself) or, more generally, what we call “DF-symmetric,” we show that a number of apparent singularities are removable, as required for the construction of the minimal models via these methods.

Law of large numbers for the maximum of the two-dimensional Coulomb gas potential

Law of large numbers for the maximum of the two-dimensional Coulomb gas potential

Self-diffusion and structure of a quasi two-dimensional, classical Coulomb gas under increasing magnetic field and temperature

Self-diffusion and structure of a quasi two-dimensional, classical Coulomb gas under increasing magnetic field and temperature

Overcrowding and separation estimates for the Coulomb gas

AbstractWe prove several results for the Coulomb gas in any dimension that follow from isotropic averaging, a transport method based on Newton's theorem. First, we prove a high‐density Jancovici–Lebowitz–Manificat law, extending the microscopic density bounds of Armstrong and Serfaty and establishing strictly sub‐Gaussian tails for charge excess in dimension 2. The existence of microscopic limiting point processes is proved at the edge of the droplet. Next, we prove optimal upper bounds on the ‐point correlation function for merging points, including a Wegner estimate for the Coulomb gas for . We prove the tightness of the properly rescaled th minimal particle gap, identifying the correct order in and a three term expansion in , as well as upper and lower tail estimates. In particular, we extend the two‐dimensional “perfect‐freezing regime” identified by Ameur and Romero to higher dimensions. Finally, we give positive charge discrepancy bounds which are state of the art near the droplet boundary and prove incompressibility of Laughlin states in the fractional quantum Hall effect, starting at large microscopic scales. Using rigidity for fluctuations of smooth linear statistics, we show how to upgrade positive discrepancy bounds to estimates on the absolute discrepancy in certain regions.

Classical conformal blocks, Coulomb gas integrals, and quantum integrable models

In this paper, we recall Richardson’s solution of the reduced BCS model, its relationship with the Gaudin model, and the known implementation of these models in conformal field theory. The CFT techniques applied here are based on the use of the free field realization, or more precisely, on the calculation of saddle-point values of Coulomb gas integrals representing certain (perturbed) WZW conformal blocks. We identify the saddle-point limit as the classical limit of conformal blocks. We show that this observation implies a new method for calculating classical conformal blocks and can be further used in the study of quantum integrable models.

Bounding Free Energy Difference with Flow Matching

We introduce a method for computing the Helmholtz free energy using the flow matching technique. Unlike previous work that utilized flow-based models for variational free energy calculations, this method provides bounds for free energy estimation based on targeted free energy perturbation by performing calculations on samples from both ends of the mapping. We demonstrate applications of the present method by estimating the free energy of a classical Coulomb gas in a harmonic trap.

Dynamical signatures of Chaos to integrability crossover in generalized random matrix ensembles

We introduce a two-parameter ensemble of generalized real symmetric random matrices called the β-Rosenzweig–Porter ensemble (β-RPE), parameterized by β, a fictitious inverse temperature of the analogous Coulomb gas model, and γ, controlling the relative strength of disorder. β-RPE encompasses RPE from all of the Dyson’s threefold symmetry classes: orthogonal, unitary and symplectic for . Firstly, we study the energy correlations by calculating the density and 2nd moment of the nearest neighbor spacing (NNS) and robustly quantify the crossover among various degrees of level repulsions. Secondly, the dynamical properties are determined from an exact calculation of the temporal evolution of the fidelity enabling an identification of the characteristic Thouless and the equilibration timescales. The relative depth of the correlation hole in the average fidelity serves as a dynamical signature of the crossover from chaos to integrability and enables us to construct the phase diagram of β-RPE in the γ-β plane. Our results are in qualitative agreement with numerically computed fidelity for matrix ensembles. Furthermore, we observe that for large N the 2nd moment of NNS and the relative depth of the correlation hole exhibit a second order phase transition at γ = 2.

Determinantal Coulomb Gas Ensembles with a Class of Discrete Rotational Symmetric Potentials

Determinantal Coulomb Gas Ensembles with a Class of Discrete Rotational Symmetric Potentials

Hidden Critical Points in the Two-Dimensional O(n>2) Model: Exact Numerical Study of a Complex Conformal Field Theory.

The presence of nearby conformal field theories (CFTs) hidden in the complex plane of the tuning parameter was recently proposed as an elegant explanation for the ubiquity of "weakly first-order" transitions in condensed matter and high-energy systems. In this work, we perform an exact microscopic study of such a complex CFT (CCFT) in the two-dimensional O(n) loop model. The well-known absence of symmetry-breaking of the O(n>2) model is understood as arising from the displacement of the nontrivial fixed points into the complex temperature plane. Thanks to a numerical finite-size study of the transfer matrix, we confirm the presence of a CCFT in the complex plane and extract the real and imaginary parts of the central charge and scaling dimensions. By comparing those with the analytic continuation of predictions from Coulomb gas techniques, we determine the range of validity of the analytic continuation to extend up to n_{g}≈12.34, beyond which the CCFT gives way to a gapped state. Finally, we propose a beta function which reproduces the main features of the phase diagram and which suggests an interpretation of the CCFT as a liquid-gas critical point at the end of a first-order transition line.

Jastrow wave function for the spin-1 Heisenberg chain: The string order revealed by the mapping to the classical Coulomb gas

Jastrow wave function for the spin-1 Heisenberg chain: The string order revealed by the mapping to the classical Coulomb gas