Roman Krcmar

Spherical Deformation for One-Dimensional Quantum Systems

System-size dependence of the ground-state energy E^N is investigated for N-site one-dimensional (1D) quantum systems with open boundary condition, where the interaction strength decreases towards the both ends of the system. For the spinless Fermions on the 1D lattice we have considered, it is shown that the finite-size correction to the energy per site, which is defined as E^N / N - \lim_{N \to \infty} E^N / N, is of the order of 1 / N^2 when the reduction factor of the interaction is expressed by a sinusoidal function. We discuss the origin of this fast convergence from the view point of the spherical geometry.

Corner Transfer Matrix Renormalization Group Method Applied to the Ising Model on the Hyperbolic Plane

Critical behavior of the Ising model is investigated at the center of large scale finite size systems, where the lattice is represented as the tiling of pentagons. The system is on the hyperbolic p...

Ising model on a hyperbolic lattice studied by the corner transfer matrix renormalization group method

We study a two-dimensional ferromagnetic Ising model on a series of regular lattices, which are represented as a tessellation of polygons with p ≥ 5 sides, such as pentagons (p = 5), hexagons (p = 6), etc. Such lattices are on hyperbolic planes, which have constant negative scalar curvatures. We calculate critical temperatures and scaling exponents by the use of the corner transfer matrix renormalization group method. As a result, the mean-field-like phase transition is observed for all the cases p ≥ 5. Convergence of the calculated transition temperatures with respect to p is investigated toward the limit p → ∞, where the system coincides with the Ising model on the Bethe lattice.

Tricritical point of the J1-J2 Ising model on a hyperbolic lattice.

The ferromagnetic-paramagnetic phase transition of the two-dimensional frustrated Ising model on a hyperbolic lattice is investigated by use of the corner transfer matrix renormalization group method. The model contains a ferromagnetic nearest-neighbor interaction J1 and a competing antiferromagnetic interaction J2 . A mean-field-like second-order phase transition is observed when the ratio kappa=J_{2}J_{1} is less than 0.203. In the region 0.203<kappa<14 , the spontaneous magnetization is discontinuous at the transition temperature. Such tricritical behavior suggests that the phase transitions on hyperbolic lattices need not always be mean-field-like.

Weak correlation effects in the Ising model on triangular-tiled hyperbolic lattices.

The Ising model is studied on a series of hyperbolic two-dimensional lattices which are formed by tessellation of triangles on negatively curved surfaces. In order to treat the hyperbolic lattices, we propose a generalization of the corner transfer matrix renormalization group method using a recursive construction of asymmetric transfer matrices. Studying the phase transition, the mean-field universality is captured by means of a precise analysis of thermodynamic functions. The correlation functions and the density-matrix spectra always decay exponentially even at the transition point, whereas power-law behavior characterizes criticality on the Euclidean flat geometry. We confirm the absence of a finite correlation length in the limit of infinite negative Gaussian curvature.

Phase transition of clock models on a hyperbolic lattice studied by corner transfer matrix renormalization group method

Two-dimensional ferromagnetic N -state clock models are studied on a hyperbolic lattice represented by tessellation of pentagons. The lattice lies on the hyperbolic plane with a constant negative scalar curvature. We observe the spontaneous magnetization, the internal energy, and the specific heat at the center of sufficiently large systems, where fixed boundary conditions are imposed, for the cases N>or=3 up to N=30 . The model with N=3 , which is equivalent to the three-state Potts model on the hyperbolic lattice, exhibits a first-order phase transition. A mean-field-like phase transition of second order is observed for the cases N>or=4 . When N>or=5 we observe a Schottky-type specific heat below the transition temperature, where its peak height at low temperatures scales as N(-2) . From these facts we conclude that the phase transition of the classical XY model deep inside hyperbolic lattices is not of the Berezinskii-Kosterlitz-Thouless type.

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. Introduction. – The universality hypothesis states that for a statistical system with a given symmetry of microscopic state variables, critical exponents do not depend on model’s Hamiltonian parameters [1]. Historically, the first violation of the universality was observed in the symmetric (zero-field) eight-vertex model on the square lattice, whose critical exponents depend continuously on model’s parameters. Baxter solved the symmetric eight-vertex model by using the concept of commuting transfer matrices and the Yang-Baxter equation for the scattering matrix as the consistency condition [2–5]. This became a basis for generating and solving systematically integrable models within the “Quantum Inverse-Scattering method” [6, 7], see e.g. monographs [8, 9]. The next nonuniversal model, the Ashkin-Teller model [10–13], is in fact related to the eight-vertex model [14]. All these systems exhibit a “weak universality” as was proposed by Suzuki [15]: defining the singularities of statistical quantities near the critical point in terms of the inverse correlation length, rather than the temperature difference, the rescaled critical exponents are universal. The phenomenon of weak universality appears in many other physical systems, like interacting dimers [16], frustrated spins [17], quantum phase transitions [18] and so on. There are indications that both universality and weak universality are violated in the symmetric 16-vertex model on the 2D square and 3D diamond lattices [19, 20], Ising spin glasses [21], frustrated spin models [22], experimental measurements on composite materials [23, 24], etc. The general eight-vertex model on a square lattice can be formulated as an Ising model on the dual square lattice with (nearest-neighbour and diagonal) two-spin and (plaquette) four-spin interactions [25, 26]. The symmetric version of the eight-vertex model corresponds to two Ising models on two alternating sublattices, coupled with one another via plaquette couplings. Kadanoff and Wegner [26] suggested that the variation of critical indices is due to the special hidden symmetries of the zero-field eight-vertex model. If an external field is applied, they argued that the magnetic exponents should be constant and equivalent to those of the standard Ising model, see also monograph [5]. This conjecture was supported by renormalization group calculations [14, 27, 28]. Since the eight-vertex model in a field is non-integrable, the above conjecture about the Ising-type universality must be checked numerically. To our knowledge, no numerical test was done in the past, probably because of high demands on numerical precision. In this letter, in order to achieve a very high accuracy, we apply the Corner Transfer Matrix Renormalization Group (CTMRG) method, having its origin in the renormalization of the density matrix [29–32]. A subspace of vertex weights is chosen to ensure the symmetricity of the corner transfer matrix [5]. The CTMRG method is first tested on the zero-field version of the eight-vertex model, the obtained dependence of critical exponents on model’s parameters is in good agreement with Baxter’sThe 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 models parameters is in agreement with Baxters 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.

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 < M0 with some minimum M0 dependent on the lattice structure, as z increases from 0 to ∞ there is a direct transition from the disordered (gas) phase to a demixed (liquid) phase with one majority component at z > zd(M). If M ≥ M0, there is an intermediate ordered “crystal phase” (composed of two nonequivalent even and odd sublattices) for z lying between zc(M) and zd(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 twodimensional 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 ≥ M0 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.

Persistent current of correlated electrons in mesoscopic ring with impurity

The persistent current of correlated electrons in a continuous one-dimensional ring with a single scatterer is calculated by solving the many-body Schrodinger equation for several tens of electrons interacting via the electron-electron (e-e) interaction of finite range. The problem is solved by the configuration-interaction (CI) and diffusion Monte Carlo (DMC) methods. The CI and DMC results are in good agreement. In both cases, the persistent current I as a function of the ring length L exhibits the asymptotic dependence I α L -1-α typical of the Luttinger liquid, where the power a depends only on the e-e interaction. The numerical values of a agree with the known formula of the renormalization-group theory.

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

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 dependences 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 ηe and δe 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.