Abstract:
We introduce a new class of quantum models with time-dependent Hamiltonians of a special scaling form. By using a couple of time-dependent unitary transformations, the time evolution of these models is expressed in terms of related systems with time-independent Hamiltonians. The mapping of dynamics can be performed in any dimension, for an arbitrary number of interacting particles and for any type of the scaling interaction potential. The exact solvability of a "dual" time-independent Hamiltonian automatically means the exact solvability of the original problem with model time-dependence.

Abstract:
The model under consideration is a classical 2D Coulomb gas of pointlike positive and negative unit charges, interacting via a logarithmic potential. In the whole stability range of temperatures, the equilibrium statistical mechanics of this fluid is exactly solvable via an equivalence with the integrable 2D sine-Gordon field theory. The exact solution includes the bulk thermodynamics, special cases of the surface thermodynamics, and the large-distance asymptotic behavior of the two-body correlation functions.

Abstract:
The model under consideration is an asymmetric two-dimensional Coulomb gas of positively (q_1=+1) and negatively (q_2=-1/2) charged pointlike particles, interacting via a logarithmic potential. This continuous system is stable against collapse of positive-negative pairs of charges for the dimensionless coupling constant (inverse temperature) \beta<4. The mapping of the Coulomb gas is made onto the complex Bullough-Dodd model, and recent results about that integrable 2D field theory are used. The mapping provides the full thermodynamics (the free energy, the internal energy, the specific heat) and the large-distance asymptotics of the particle correlation functions, in the whole stability regime of the plasma. The results are checked by a small-\beta expansion and close to the collapse \beta=4 point. The comparison is made with the exactly solvable symmetric version of the model (q_1=+1, q_2=-1), and some fundamental changes in statistics caused by the charge asymmetry are pointed out.

Abstract:
The model under consideration is the two-dimensional (2D) one-component plasma of pointlike charged particles in a uniform neutralizing background, interacting through the logarithmic Coulomb interaction. Classical equilibrium statistical mechanics is studied by non-traditional means. The question of the potential integrability (exact solvability) of the plasma is investigated, first at arbitrary coupling constant \Gamma via an equivalent 2D Euclidean-field theory, and then at the specific values of \Gamma=2*integer via an equivalent 1D fermionic model. The answer to the question in the title is that there is strong evidence for the model being not exactly solvable at arbitrary \Gamma but becoming exactly solvable at \Gamma=2*integer. As a by-product of the developed formalism, the gauge invariance of the plasma is proven at the free-fermion point \Gamma=2; the related mathematical peculiarity is the exact inversion of a class of infinite-dimensional matrices.

Abstract:
We test the concepts of renormalized charge and potential saturation, introduced within the framework of highly asymmetric Coulomb mixtures, on exactly solvable Coulomb models. The object of study is the average electrostatic potential induced by a unique "guest" charge immersed in a classical electrolyte, the whole system being in thermal equilibrium at some inverse temperature $\beta$. The guest charge is considered to be either an infinite hard wall carrying a uniform surface charge or a charged colloidal particle. The systems are treated as two-dimensional; the electrolyte is modelled by a symmetric two-component plasma of point-like $\pm e$ charges with logarithmic Coulomb interactions. Two cases are solved exactly: the Debye-H\"uckel limit $\beta e^2\to 0$ and the Thirring free-fermion point $\beta e^2 = 2$. The results at the free-fermion point can be summarized as follows: (i) The induced electrostatic potential exhibits the asymptotic behavior, at large distances from the guest charge, whose form is different from that obtained in the Debye-H\"uckel (linear Poisson-Boltzmann) theory. This means that the concept of renormalized charge, developed within the nonlinear Poisson-Boltzmann theory to describe the screening effect of the electrolyte cloud, fails at the free-fermion point. (ii) In the limit of an infinite bare charge, the induced electrostatic potential saturates at a finite value in every point of the electrolyte region. This fact confirms the previously proposed hypothesis of potential saturation.

Abstract:
This paper is a continuation of a previous one [L. {\v{S}}amaj, {\it J. Stat. Phys.} {\bf 120}:125 (2005)] dealing with the renormalization of a guest charge immersed in a two-dimensional logarithmic Coulomb gas of pointlike $\pm$ unit charges, the latter system being in the stability-against-collapse regime of reduced inverse temperatures $0\le \beta <2$. In the previous work, using a sine-Gordon representation of the Coulomb gas, an exact renormalized-charge formula was derived for the special case of the {\em pointlike} guest charge $Q$, in its stability regime $\beta | Q| < 2$. In the present paper, we extend the renormalized-charge treatment to the guest charge with a hard core of radius $\sigma$, which allows us to go beyond the stability border $\beta| Q| = 2$. In the limit of the hard-core radius much smaller than the correlation length of the Coulomb-gas species and at a strictly finite temperature, due to the counterion condensation in the extended region $\beta| Q| >2$, the renormalized charge $Q_{\rm ren}$ turns out to be a periodic function of the bare charge $Q$ with period 1. The renormalized charge therefore does not saturate at a specific finite value as $| Q| \to\infty$, but oscillates between two extreme values. In the high-temperature Poisson-Boltzmann scaling regime of limits $\beta\to 0$ and $Q\to\infty$ with the product $\beta Q$ being finite, one reproduces correctly the monotonic dependence of $\beta Q_{\rm ren}$ on $\beta Q$ in the guest-charge stability region $\beta| Q| <2$ and the Manning-Oosawa type of counterion condensation with the uniform saturation of $\beta Q_{\rm ren}$ at the value $4/\pi$ in the region $\beta| Q|\ge 2$.

Abstract:
In the equilibrium statistical mechanics of classical Coulomb fluids, the long-range tail of the Coulomb potential gives rise to the Stillinger-Lovett sum rules for the charge correlation functions. For the jellium model of mobile particles of charge $q$ immersed in a neutralizing background, the fixing of one of the $q$-charges induces a screening cloud of the charge density whose zeroth and second moments are determined just by the Stillinger-Lovett sum rules. In this paper, we generalize these sum rules to the screening cloud induced around a pointlike guest charge $Z q$ immersed in the bulk interior of the 2D jellium with the coupling constant $\Gamma=\beta q^2$ ($\beta$ is the inverse temperature), in the whole region of the thermodynamic stability of the guest charge $Z>-2/\Gamma$. The derivation is based on a mapping technique of the 2D jellium at the coupling $\Gamma$ = (even positive integer) onto a discrete 1D anticommuting-field theory; we assume that the final results remain valid for all real values of $\Gamma$ corresponding to the fluid regime. The generalized sum rules reproduce for arbitrary coupling $\Gamma$ the standard Z=1 and the trivial Z=0 results. They are also checked in the Debye-H\"uckel limit $\Gamma\to 0$ and at the free-fermion point $\Gamma=2$. The generalized second-moment sum rule provides some exact information about possible sign oscillations of the induced charge density in space.

Abstract:
The Debye-H\"uckel theory describes rigorously the thermal equilibrium of classical Coulomb fluids in the high-temperature $\beta\to 0$ regime ($\beta$ denotes the inverse temperature). It is generally believed that the Debye-H\"uckel theory and the systematic high-temperature expansion provide an adequate description also in the region of small {\em strictly positive} values of $\beta>0$. This hypothesis is tested in the present paper on a two-dimensional Coulomb gas of pointlike $+/-$ unit charges interacting via a logarithmic potential which is equivalent to an integrable sine-Gordon field model. In particular, we apply a form factor method to obtain the exact asymptotic large-distance behavior of particle correlation functions, considered in the charge and number density combinations. We first determine the general forms of the leading and subleading asymptotic terms at strictly positive $\beta>0$ and then evaluate their high-temperature $\beta\to 0$ forms. In the case of the {\em charge} correlation function, the leading asymptotic term at a strictly positive $\beta>0$ is also the leading one in the high-temperature $\beta\to 0$ regime. On the contrary, the $\beta\to 0$ behavior of the {\em number density} correlation function is accompanied by an interference between the first two asymptotic terms. Consequently, the large-distance behavior of this function exhibits a discontinuity when going from strictly positive values of $\beta>0$ to the Debye-H\"uckel limit $\beta\to 0$. This is the crucial conclusion of the paper: the large-distance asymptotics and the high-temperature limit do not commute for the density correlation function of the two-dimensional Coulomb gas.

Abstract:
We study physical situations when one or two "guest" arbitrarily-charged particles are immersed in the bulk of a classical electrolyte modelled by a Coulomb gas of positive/negative unit point-like charges, the whole system being in thermal equilibrium. The models are treated as two-dimensional with logarithmic pairwise interactions among charged constituents; the (dimensionless) inverse temperature $\beta$ is considered to be smaller than 2 in order to ensure the stability of the electrolyte against the collapse of positive-negative pairs of charges. Based on recent progress in the integrable (1+1)-dimensional sine-Gordon theory, exact formulas are derived for the chemical potential of one guest charge and for the asymptotic large-distance behavior of the effective interaction between two guest charges. The exact results imply, under certain circumstances, anomalous effects such as an effective attraction (repulsion) between like-charged (oppositely-charged) guest particles and the charge inversion in the electrolyte vicinity of a highly-charged guest particle. The adequacy of the concept of renormalized charge is confirmed in the whole stability region of inverse temperatures and the related saturation phenomenon is revised.