arXiv subjects

Ladislav Šamaj

Publications and source records attributed to Ladislav Šamaj.

34 records · Page 2Linked to original sources

arXiv2020

Full nonuniversality of the symmetric 16-vertex model on the square lattice

We consider the symmetric two-state 16-vertex model on the square lattice whose vertex weights are invariant under any permutation of adjacent edge states. The vertex-weight parameters are restricted to a critical manifold which is self-dual under the gauge transformation. The critical properties of the model are studied numerically by using the Corner Transfer Matrix Renormalization Group method. Accuracy of the method is tested on two exactly solvable cases: the Ising model and a specific version of the Baxter 8-vertex model in a zero field that belong to different universality classes. Numerical results show that the two exactly solvable cases are connected by a line of critical points with the polarization as the order parameter. There are numerical indications that critical exponents vary continuously along this line in such a way that the weak universality hypothesis is violated.

cond-mat.stat-mech↗

arXiv2019

Short-distance symmetry of pair correlations in two-dimensional jellium

We consider the two-dimensional one-component plasma (jellium) of mobile pointlike particles with the same charge $e$, interacting pairwisely by the logarithmic Coulomb potential and immersed in a fixed neutralizing background charge density. Particles are in thermal equilibrium at the inverse temperature $β$, the only relevant dimensionless parameter is the coupling constant $Γ\equivβe^2$. In the bulk fluid regime and for any value of the coupling constant $Γ=2\times{\rm integer}$, Šamaj and Percus [J. Stat. Phys. {\bf 80}, 811--824 (1995)] have derived an infinite sequence of sum rules for the coefficients of the short-distance expansion of particle pair correlation function. In the context of the equivalent fractional quantum Hall effect, by using specific methods of quantum geometry Haldane [PRL {\bf 107}, 116801 (2011) and arXiv:1112.0990v2] derived a self-dual relation for the Landau-level guiding-center structure factor. In this paper, we establish the relation between the guiding-center structure factor and the pair correlation function of jellium particles. It is shown that the self-dual formula, which provides an exact relation between the pair correlation function and its Fourier component, comes directly from the short-distance symmetry of the bulk jellium. The short-distant symmetry of pair correlations is extended to the semi-infinite geometry of a rectilinear plain hard wall with a fixed surface charge density, constraining particles to a half-space. The symmetry is derived for the original jellium model as well as its simplified version with no background charge (charged wall surface with ``counter-ions only''). The obtained results are checked at the exactly solvable free-fermion coupling $Γ=2$.

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arXiv2019

Fourth Moment of the Charge Density Induced Around A Guest Charge in Two-Dimensional Jellium

The model under consideration is the classical two-dimensional one-component plasma (jellium) of pointlike particles with charge $e$, interacting pairwisely via the logarithmic Coulomb potential and immersed in a uniform neutralizing background charge density. The system is in thermal equilibrium at the inverse temperature $β$, its thermodynamics depends only on the coupling constant $Γ=βe^2$. We put into an infinite (homogeneous and translationally invariant) plasma a guest particle of charge $Ze$ with $Z>-2/Γ$ in order to prevent from the collapse of the jellium charges onto it. The guest particle induces a screening cloud (the excess charge density) in the plasma. The zeroth and second moments of this screening cloud were derived previously for any fluid value of $Γ$. In this paper, we propose a formula for the fourth moment of the screening cloud. The derivation is based on the assumption that the fourth moment is, similarly as the second moment, analytic in $Z$ around $Z=0$. An exact treatment of the limit $Z\to\infty$ shows that it is a finite (cube) polynomial in $Z$. The $Γ$-dependence of the polynomial coefficients is determined uniquely by considering the limits $Z\to 0$ and $Z\to\infty$, and the compressibility sum rule for $Z=1$. The formula for the fourth moment of screening cloud is checked in the leading and first correction orders of the Debye-Hückel limit $Γ\to 0$ and at the exactly solvable free-fermion point $Γ=2$. Sufficient conditions for sign oscillations of the induced charge density which follow from the second-moment and fourth-moment sum rules are discussed.

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arXiv2019

Two-dimensional Wigner crystals of classical Lennard-Jones particles

The ground-state of two-dimensional (2D) systems of classical particles interacting pairwisely by the generalized Lennard-Jones potential is studied. Taking the surface area per particle $A$ as a free parameter and restricting oneself to periodic Bravais lattices with one particle per unit cell, Bétermin L [2018 Nonlinearity {\bf 31} 3973] proved that the hexagonal, rhombic, square and rectangular structures minimize successively the interaction energy per particle as $A$ increases. We show here that the second-order transitions between the rhombic/square and square/rectangular phases are of mean-field type. The aim of this paper is to extend Bétermin's analysis to periodic 2D lattices with more than one particle per elementary cell. Being motivated by previous works dealing with other kinds of models, we propose as the ground-state the extensions of the 2D rectangular (1-chain) lattice, namely the "zig-zag" (2-chain), 3-chain, 4-chain, etc. structures possessing 2, 3, 4, etc. particles per unit cell, respectively. By using a recent technique of lattice summation we find for the standard Lennard-Jones potential that their ground-state energy per particle approaches systematically as the number of particles per unit cell increases to the one of a phase separated state (the optimal hexagonal lattice). We analyze analytically the low-density limit $A\to\infty$ and the limiting hard-core case of the generalized Lennard-Jones potential.

cond-mat.other↗

arXiv2018

Logarithmic Finite-Size Correction in Non-neutral Two-Component Plasma on Sphere

We consider a general two-component plasma of classical pointlike charges $+e$ ($e$ is say the elementary charge) and $-Z e$ (valency $Z=1,2,\ldots$), living on the surface of a sphere of radius $R$. The system is in thermal equilibrium at the inverse temperature $β$, in the stability region against collapse of oppositely charged particle pairs $βe^2 < 2/Z$. We study the effect of the system excess charge $Q e$ on the finite-size expansion of the (dimensionless) grand potential $βΩ$. By combining the stereographic projection of the sphere onto an infinite plane, the linear response theory and the planar results for the second moments of the species density correlation functions we show that for any $βe^2 < 2/Z$ the large-$R$ expansion of the grand potential is of the form $βΩ\sim A_V R^2 + [χ/6 - β(Qe)^2/2] \ln R$, where $A_V$ is the non-universal coefficient of the volume (bulk) part and the Euler number of the sphere $χ=2$. The same formula, containing also a non-universal surface term proportional to $R$, was obtained previously for the disc domain ($χ=1$), in the case of the symmetric $(Z=1)$ two-component plasma at the collapse point $βe^2=2$ and the jellium model $(Z\to 0)$ of identical $e$-charges in a fixed neutralizing background charge density at any coupling $βe^2$ being an even integer. Our result thus indicates that the prefactor to the logarithmic finite-size expansion does not depend on the composition of the Coulomb fluid and its non-universal part $-β(Qe)^2/2$ is independent of the geometry of the confining domain.

cond-mat.stat-mech↗

arXiv2018

Original electric-vertex formulation of the symmetric eight-vertex model on the square lattice is fully non-universal

The partition function of the symmetric (zero electric field) eight-vertex model on a square lattice can be formulated either in the original "electric" vertex format or in an equivalent "magnetic" Ising-spin format. In this paper, both electric and magnetic versions of the model are studied numerically by using the Corner Transfer Matrix Renormalization Group method which provides reliable data. The emphasis is put on the calculation of four specific critical exponents, related by two scaling relations, and of the central charge. The numerical method is first tested in the magnetic format, the obtained dependencies of critical exponents on model's parameters agree with Baxter's exact solution and weak universality is confirmed within the accuracy of the method due to the finite size of the system. In particular, the critical exponents $η$ and $δ$ are constant as required by weak universality. On the other hand, in the electric format, analytic formulas based on the scaling relations are derived for the critical exponents $η_{\rm e}$ and $δ_{\rm e}$ which agree with our numerical data. These exponents depend on model's parameters which is an evidence for the full non-universality of the symmetric eight-vertex model in the original electric formulation.

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arXiv2017

Tricriticality for dimeric Coulomb molecular crystals in ground state

We study the ground-state properties of a system of dimers. Each dimer consists in a pair of equivalent charges at a fixed distance, immersed in a neutralizing homogeneous background. All charges interact pairwisely by Coulomb potential. The dimer centers form a two-dimensional rectangular lattice with the aspect ratio $α\in [0,1]$ and each dimer is allowed to rotate around its center. The previous numerical simulations, made for the more general Yukawa interaction, indicate that only two basic dimer configurations can appear: either all dimers are parallel or they have two different angle orientations within alternating (checkerboard) sublattices. As the dimer size increases, two second-order phase transitions, related to two kinds of the symmetry breaking in dimer's orientations, were reported. In this paper, we use a recent analytic method based on an expansion of the interaction energy in Misra functions which converges quickly and provides an analytic derivation of the critical behaviour. Our main result is that there exists a specific aspect ratio of the rectangular lattice $α^*=0.71410684000071\ldots$ which divides the space of model's phases onto two distinct regions. If the lattice aspect ratio $α>α^*$, we recover both types of the second-order phase transitions and find that they are of mean-field type with the critical exponent $β= 1/2$. If $α<α^*$, the phase transition associated with the discontinuity of dimer's angles on alternating sublattices becomes of first order. For $α=α^*$, the first- and second-order phase transitions meet at the tricritical point, characterized by the different critical index $β= 1/4$. Such phenomenon is known from literature about the Landau theory of one-component fields, but in our two-component version the scenario is more complicated.

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arXiv2017

Finite-Size Effects in Non-Neutral Two-Dimensional Coulomb Fluids

Thermodynamic potential of a neutral two-dimensional (2D) Cou\-lomb fluid, confined to a large domain with a smooth boundary, exhibits at any (inverse) temperature $β$ a logarithmic finite-size correction term whose universal prefactor depends only on the Euler number of the domain and the conformal anomaly number $c=-1$. A minimal free boson conformal field theory, which is equivalent to the 2D symmetric two-component plasma of elementary $\pm e$ charges at coupling constant $Γ=βe^2$, was studied in the past. It was shown that creating a non-neutrality by spreading out a charge $Q e$ at infinity modifies the anomaly number to $c(Q,Γ) = - 1 + 3ΓQ^2$. Here, we study the effect of non-neutrality on the finite-size expansion of the free energy for another Coulomb fluid, namely the 2D one-component plasma (jellium) composed of identical pointlike $e$-charges in a homogeneous background surface charge density. For the disk geometry of the confining domain we find that the non-neutrality induces the same change of the anomaly number in the finite-size expansion. We derive this result first at the free-fermion coupling $Γ\equivβe^2=2$ and then, by using a mapping of the 2D one-component plasma onto an anticommuting field theory formulated on a chain, for an arbitrary coupling constant.

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arXiv2016

Critical properties of the eight-vertex model in a field

The general eight-vertex model on a square lattice is studied numerically by using the Corner Transfer Matrix Renormalization Group method. The method is tested on the symmetric (zero-field) version of the model, the obtained dependence of critical exponents on model's parameters is in agreement with Baxter's exact solution and weak universality is verified with a high accuracy. It was suggested longtime ago that the symmetric eight-vertex model is a special exceptional case and in the presence of external fields the eight-vertex model falls into the Ising universality class. We confirm numerically this conjecture in a subspace of vertex weights, except for two specific combinations of vertical and horizontal fields for which the system still exhibits weak universality.

cond-mat.stat-mech↗

arXiv2016

Amplitude Function of Asymptotic Correlations Along Charged Wall in Coulomb Fluids

In classical semi-infinite Coulomb fluids, two-point correlation functions exhibit a slow inverse-power law decay along a uniformly charged wall. In this work, we concentrate on the corresponding amplitude function which depends on the distances of the two points from the wall. Recently [L. Šamaj, J. Stat. Phys. {\bf 161}, 227 (2015)], applying a technique of anticommuting variables to a 2D system of charged rectilinear wall with "counter-ions only", we derived a relation between the amplitude function and the density profile which holds for any temperature. In this paper, using the Möbius conformal transformation of particle coordinates in a disc, a new relation between the amplitude function and the density profile is found for that model. This enables us to prove, at any temperature, the factorization property of the amplitude function in point distances from the wall and to express it in terms of the density profile. Presupposing the factorization property of the amplitude function and using specific sum rules for semi-infinite geometries, a relation between the amplitude function of the charge-charge structure function and the charge profile is derived for many-component Coulomb fluids in any dimension.

cond-mat.stat-mech↗

arXiv2016

Separation of Wigner structures for 2D equimolar binary mixtures of Coulomb particles

We study the lowest energy configurations of an equimolar binary mixture of classical pointlike particles with charges $Q_1$ and $Q_2$, such that $q=Q_2/Q_1\in [0,1]$. The particles interact pairwisely via 3D Coulomb potential and are confined to a 2D plane with a homogeneous neutralizing background charge density. In a recent paper by M. Antlanger and G. Kahl [Cond. Mat. Phys. {\bf 16}, 43501 (2013)], using numerical computations based on evolutionary algorithm, six fully mixed structures were identified for $0\le q\lesssim 0.59$, while the separation of $Q_1$ and $Q_2$ pure hexagonal phases minimizes the energy for $0.59\lesssim q<1$. Here, we introduce a novel structure which consists in the separation of two phases, the pure hexagonal one formed by a fraction of particles with the larger charge $Q_1$ and the other fixed one containing different numbers of $Q_1$ and $Q_2$ charges. Using an analytic method based on an expansion of the interaction energy in Misra functions we show that this novel structure provides the lowest energy in two intervals of $q$ values, $0<q\lesssim 0.04707$ and $0.58895\lesssim q\lesssim 0.61367$. This fact might inspire numerical methods, for both Coulomb and Yukawa interactions, to test more general separations which go beyond the separation of two pure phases.

cond-mat.other↗

arXiv2016

Reentrant disorder-disorder transitions in generalized multicomponent Widom-Rowlinson models

In the lattice version of the multicomponent Widom-Rowlinson (WR) model, each site can be either empty or singly occupied by one of $M$ different particles, all species having the same fugacity $z$. The only nonzero interaction potential is a nearest-neighbor hard-core exclusion between unlike particles. For $M z_d(M)$. If $M\ge M_0$, there is an intermediate ordered "crystal phase" (composed of two nonequivalent even and odd sublattices) for $z$ lying between $z_c(M)$ and $z_d(M)$ which is driven by entropy. We generalize the multicomponent WR model by replacing the hard-core exclusion between unlike particles by more realistic large (but finite) repulsion. The model is solved exactly on the Bethe lattice with an arbitrary coordination number. The numerical calculations, based on the corner transfer matrix renormalization group, are performed for the two-dimensional square lattice. The results for $M=4$ indicate that the second-order phase transitions from the disordered gas to the demixed phase become of first order, for an arbitrarily large finite repulsion. The results for $M\ge M_0$ show that, as the repulsion weakens, the region of crystal phase diminishes itself. For weak enough repulsions, the direct transition between the crystal and demixed phases changes into a separate pair of crystal-gas and gas-demixed transitions; this is an example of a disorder-disorder reentrant transition via an ordered crystal phase. If the repulsion between unlike species is too weak, the crystal phase disappears from the phase diagram. It is shown that the generalized WR model belongs to the Ising universality class.

cond-mat.stat-mech↗

arXiv2015

Counter-Ions Near a Charged Wall: Exact Results for Disc and Planar Geometries

Macromolecules, when immersed in a polar solvent like water, become charged by a fixed surface charge density which is compensated by ``counter-ions'' moving out of the surface. Such classical particle systems exhibit poor screening properties at any temperature and the trivial bulk regime (far away from the charged surface) with no particles, so the validity of standard Coulomb sum rules is questionable. In the present paper, we concentrate on the two-dimensional version of the model with the logarithmic interaction potential. We go from the finite disc to the semi-infinite planar geometry. The system is exactly solvable for two values of the coupling constant $Γ$: in the Poisson-Boltzmann mean-field limit $Γ\to 0$ and at the free-fermion point $Γ=2$. We show that the finite-size expansion of the free energy does not contain universal term as is usual for Coulomb fluids. For the coupling constant being an arbitrary positive even integer, using an anticommuting representation of the partition function and many-body densities we derive a sequence of sum rules. As a result, the contact density of counter-ions at the wall is available for the disc. The amplitude function, which characterizes the asymptotic inverse-power law behavior of the two-body density along the wall, is found to be related to the particle density profile. The dielectric susceptibility tensor, calculated exactly for an arbitrary coupling and the particle number, exhibits the anticipated disc value in the thermodynamic limit, in spite of zero contribution from the bulk region. Some of the results obtained in the Poisson-Boltzmann limit are generalized to an arbitrary Euclidean dimension.

cond-mat.stat-mech↗

arXiv2014

Taking one charge off a two-dimensional Wigner crystal

A planar array of identical charges at vanishing temperature forms a Wigner crystal with hexagonal symmetry. We take off one (reference) charge in a perpendicular direction, hold it fixed, and search for the ground state of the whole system. The planar projection of the reference charge should then evolve from a six-fold coordination (center of a hexagon) for small distances to a three-fold arrangement (center of a triangle), at large distances $d$ from the plane. The aim of this paper is to describe the corresponding non-trivial lattice transformation. For that purpose, two numerical methods (direct energy minimization and Monte Carlo simulations), together with an analytical treatment, are presented. Our results indicate that the $d=0$ and $d\to\infty$ limiting cases extend for finite values of $d$ from the respective starting points into two sequences of stable states, with intersecting energies at some value $d_t$; beyond this value the branches continue as metastable states.

cond-mat.soft↗

arXiv2008

Guest Charge and Potential Fluctuations in Two-Dimensional Classical Coulomb Systems

A known generalization of the Stillinger-Lovett sum rule for a guest charge immersed in a two-dimensional one-component plasma (the second moment of the screening cloud around this guest charge) is more simply retrieved, just by using the BGY hierarchy for a mixture of several species; the zeroth moment of the excess density around a guest charge immersed in a two-component plasma is also obtained. The moments of the electric potential are related to the excess chemical potential of a guest charge; explicit results are obtained in several special cases.

cond-mat.stat-mech↗

arXiv2000

Microscopic Calculation of the Dielectric Susceptibility Tensor for Coulomb Fluids

In a Coulomb fluid confined to a domain $V$, the dielectric susceptibility tensor $χ_V$ depends on the shape of $V$, even in the thermodynamic $V\to\infty$ limit. This paper deals with the classical two-dimensional one-component plasma formulated in an elliptic $V$-domain, equilibrium statistical mechanics is used. For the dimensionless coupling constant $Γ=$ even positive integer, the mapping of the plasma onto a discrete one-dimensional anticommuting-field theory provides a new sum rule. This sum rule confirms the limiting value of $χ_V$ predicted by macroscopic electrostatics and gives the finite-size correction term to $χ_V$.

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