arXiv subjects

A. I. Sokolov

Publications and source records attributed to A. I. Sokolov.

At least 37 records · Page 2Linked to original sources

arXiv2007

Two-dimensional phase transition models and λϕ^4 field theory

The overview is given of the results obtained recently in the course of renormalization-group (RG) study of two-dimensional (2D) models. RG functions of the two-dimensional n-vector λϕ^4 Euclidean field theory are written down up to the five-loop terms and perturbative series are resummed by the Pade-Borel-Leroy techniques. An account for the five-loop term is shown to shift the Wilson fixed point only briefly, leaving it outside the segment formed by the results of the lattice calculations. This is argued to reflect the influence of the non-analytical contribution to the β-function. The evaluation of the critical exponents for n = 1, n = 0 and n = -1 in the five-loop approximation and comparison of the results with known exact values confirm the conclusion that non-analytical contributions are visible in two dimensions. The estimates obtained on the base of pseudo-εexpansions originating from the 5-loop 2D RG series are also discussed.

cond-mat.stat-mech↗

arXiv2007

Landau thermodynamic potential for BaTiO_3

In the paper, the description of the dielectric and ferroelectric properties of BaTiO_3 single crystals using Landau thermodynamic potential is addressed. Our results suggest that when using the sixth-power free energy expansion of the thermodynamic potential, remarkably different values of the fourth-power coefficient, β(the coefficient of P^4_i terms), are required to adequately reproduce the nonlinear dielectric behavior of the paraelectric phase and the electric field induced ferroelectric phase, respectively. In contrast, the eighth-power expansion with a common set of coefficients enables a good description for both phases at the same time. These features, together with the data available in literature, strongly attest to the necessity of the eighth-power terms in Landau thermodynamic potential of BaTiO_3. In addition, the fourth-power coefficients, βand ξ(the coefficient of P^2_i P^2_j terms), were evaluated from the nonlinear dielectric responses along [001], [011], and [111] orientations in the paraelectric phase. Appreciable temperature dependence was evidenced for both coefficients above T_C. Further analysis on the linear dielectric response of the single domain crystal in the tetragonal phase demonstrated that temperature dependent anharmonic coefficients are also necessary for an adequate description of the dielectric behavior in the ferroelectric phase. As a consequence, an eighth-power thermodynamic potential, with some of the anharmonic coefficients being temperature dependent, was proposed and compared with the existing potentials. In general, the potential proposed in this work exhibits a higher quality in reproducing the dielectric and ferroelectric properties of this prototypic ferroelectric substance.

cond-mat.stat-mech↗

arXiv2006

Anharmonicity of BaTiO_3 single crystals

By analyzing the dielectric non-linearity with the Landau thermodynamic expansion, we find a simple and direct way to assess the importance of the eighth order term. Following this approach, it is demonstrated that the eighth order term is essential for the adequate description of the para/ferroelectric phase transition of BaTiO_3. The temperature dependence of the quartic coefficient βis accordingly reconsidered and is strongly evidenced by the change of its sign above 165 C. All these findings attest to the strong polarization anharmonicity of this material, which is unexpected for classical displacive ferroelectrics.

cond-mat.mtrl-sci↗

arXiv2005

Pseudo-epsilon expansion and the two-dimensional Ising model

Starting from the five-loop renormalization-group expansions for the two-dimensional Euclidean scalar ϕ^4 field theory (field-theoretical version of two-dimensional Ising model), pseudo-εexpansions for the Wilson fixed point coordinate g*, critical exponents, and the sextic effective coupling constant g_6 are obtained. Pseudo-εexpansions for g*, inverse susceptibility exponent γ, and g_6 are found to possess a remarkable property - higher-order terms in these expansions turn out to be so small that accurate enough numerical estimates can be obtained using simple Pade approximants, i. e. without addressing resummation procedures based upon the Borel transformation.

cond-mat.stat-mech↗

arXiv2005

Critical thermodynamics of two-dimensional N-vector cubic model in the five-loop approximation

The critical behavior of the two-dimensional N-vector cubic model is studied within the field-theoretical renormalization-group (RG) approach. The beta-functions and critical exponents are calculated in the five-loop approximation, RG series obtained are resummed using Pade-Borel-Leroy and conformal mapping techniques. It is found that for N = 2 the continuous line of fixed points is well reproduced by the resummed RG series and an account for the five-loop terms makes the lines of zeros of both beta-functions closer to each another. For N > 2 the five-loop contributions are shown to shift the cubic fixed point, given by the four-loop approximation, towards the Ising fixed point. This confirms the idea that the existence of the cubic fixed point in two dimensions under N > 2 is an artifact of the perturbative analysis. In the case N = 0 the results obtained are compatible with the conclusion that the impure critical behavior is controlled by the Ising fixed point.

cond-mat.stat-mech↗

arXiv2004

Critical behavior of two-dimensional cubic and MN models in the five-loop renormalization-group approximation

The critical thermodynamics of the two-dimensional N-vector cubic and MN models is studied within the field-theoretical renormalization-group (RG) approach. The beta functions and critical exponents are calculated in the five-loop approximation and the RG series obtained are resummed using the Borel-Leroy transformation combined with the generalized Padé approximant and conformal mapping techniques. For the cubic model, the RG flows for various N are investigated. For N=2 it is found that the continuous line of fixed points running from the XY fixed point to the Ising one is well reproduced by the resummed RG series and an account for the five-loop terms makes the lines of zeros of both beta functions closer to each another. For the cubic model with N\geq 3, the five-loop contributions are shown to shift the cubic fixed point, given by the four-loop approximation, towards the Ising fixed point. This confirms the idea that the existence of the cubic fixed point in two dimensions under N>2 is an artifact of the perturbative analysis. For the quenched dilute O(M) models ($MN$ models with N=0) the results are compatible with a stable pure fixed point for M\geq1. For the MN model with M,N\geq2 all the non-perturbative results are reproduced. In addition a new stable fixed point is found for moderate values of M and N.

cond-mat.stat-mech↗

arXiv2003

Critical thermodynamics of three-dimensional chiral model for N > 3

The critical behavior of the three-dimensional $N$-vector chiral model is studied for arbitrary $N$. The known six-loop renormalization-group (RG) expansions are resummed using the Borel transformation combined with the conformal mapping and Padé approximant techniques. Analyzing the fixed point location and the structure of RG flows, it is found that two marginal values of $N$ exist which separate domains of continuous chiral phase transitions $N > N_{c1}$ and $N < N_{c2}$ from the region $N_{c1} > N > N_{c2}$ where such transitions are first-order. Our calculations yield $N_{c1} = 6.4(4)$ and $N_{c2} = 5.7(3)$. For $N > N_{c1}$ the structure of RG flows is identical to that given by the $ε$ and 1/N expansions with the chiral fixed point being a stable node. For $N < N_{c2}$ the chiral fixed point turns out to be a focus having no generic relation to the stable fixed point seen at small $ε$ and large $N$. In this domain, containing the physical values $N = 2$ and $N = 3$, phase trajectories approach the fixed point in a spiral-like manner giving rise to unusual crossover regimes which may imitate varying (scattered) critical exponents seen in numerous physical and computer experiments.

cond-mat.stat-mech↗

arXiv2003

Chiral critical behavior in two dimensions from five-loop renormalization-group expansions

We analyse the critical behavior of two-dimensional N-vector spin systems with noncollinear order within the five-loop renormalization-group approximation. The structure of the RG flow is studied for different N leading to the conclusion that the chiral fixed point governing the critical behavior of physical systems with N = 2 and N = 3 does not coincide with that given by the 1/N expansion. We show that the stable chiral fixed point for $N \le N^*$, including N = 2 and N = 3, turns out to be a focus. We give a complete characterization of the critical behavior controlled by this fixed point, also evaluating the subleading crossover exponents. The spiral-like approach of the chiral fixed point is argued to give rise to unusual crossover and near-critical regimes that may imitate varying critical exponents seen in numerous physical and computer experiments.

cond-mat.stat-mech↗

arXiv2003

Five-loop renormalization-group expansions for two-dimensional Euclidean λϕ^4 theory

The renormalization-group functions of the two-dimensional n-vector λϕ^4 model are calculated in the five-loop approximation. Perturbative series for the β-function and critical exponents are resummed by the Pade-Borel-Leroy techniques. An account for the five-loop term shifts the Wilson fixed point location only briefly, leaving it outside the segment formed by the results of the lattice calculations. This is argued to reflect the influence of the non-analytical contribution to the β-function. The evaluation of the critical exponents for n = 1, n = 0 and n = -1 in the five-loop approximation and comparison of the results with known exact values confirm the conclusion that non-analytical contributions are visible in two dimensions. For the 2D Ising model, the estimate ω= 1.31 for the correction-to-scaling exponent is found.

hep-th↗

arXiv2002

Chiral phase transitions: focus driven critical behavior in systems with planar and vector ordering

The fixed point that governs the critical behavior of magnets described by the $N$-vector chiral model under the physical values of $N$ ($N =2, 3$) is shown to be a stable focus both in two and three dimensions. Robust evidence in favor of this conclusion is obtained within the five-loop and six-loop renormalization-group analysis in fixed dimension. The spiral-like approach of the chiral fixed point results in unusual crossover and near-critical regimes that may imitate varying critical exponents seen in physical and computer experiments.

cond-mat.stat-mech↗

arXiv2002

Nonlinear susceptibilities of a weakly-disordered uniaxial ferromagnet in the critical region

For the three-dimensional random Ising model, the effective sextic coupling constant v_6 and the nonlinear susceptibilities of the fourth (chi_4) and sixth (chi_6) orders are calculated at criticality. These quantities are shown to differ markedly from their counterparts for pure uniaxial magnets. In particular, the ratio v_6/v_{4}^2 entering the equation of state of the random Ising model turns out to be equal to 0.87, while in pure magnets v_6/v_{4}^2 = 1.65. The universal susceptibility ratios m^3 chi_4/chi^2 and m^6 chi_6/chi^3 (m - the inverse correlation length) are found to differ by factors 1.6 and 2.7, respectively, for random and uniform Ising models. These big differences of the universal quantities can be measured both in physical and computer experiments, and such measurements may be considered as a tool for an identification of the random critical behavior.

cond-mat.stat-mech↗

arXiv2002

Fluctuations and Landau-Devonshire expansion for barium titanate

The experimentally observed temperature dependence of the quartic coefficients in the Landau-Devonshire expansion for BaTiO_3 is naturally accounted for within a proper fluctuation model. It is explained, in particular, why one of the quartic coefficients varies with temperature above T_c while the second is constant. The tetragonal phase in BaTiO_3 is argued to exist essentially due to the thermal fluctuations, while the true Landau-Devonshire expansion with temperature-independent coefficients favours the rhombohedral ferroelectric phase. Certain conclusions concerning the temperature dependence of the sextic Landau-Devonshire coefficients are also made.

cond-mat.stat-mech↗

arXiv2001

Renormalization group and nonlinear susceptibilities of cubic ferromagnets at criticality

For the three-dimensional cubic model, the nonlinear susceptibilities of the fourth, sixth, and eighth orders are analyzed and the parameters δ^(i) characterizing their reduced anisotropy are evaluated at the cubic fixed point. In the course of this study, the renormalized sextic coupling constants entering the small-field equation of state are calculated in the four-loop approximation and the universal values of these couplings are estimated by means of the Pade-Borel-Leroy resummation of the series obtained. The anisotropy parameters are found to be: δ^(4) = 0.054 +/- 0.012, δ^(6) = 0.102 +/- 0.02, and δ^(8) = 0.144 +/- 0.04, indicating that the anisotropic (cubic) critical behavior predicted by the advanced higher-order renormalization-group analysis should be, in principle, visible in physical and computer experiments.

cond-mat.stat-mech↗

arXiv2000

Five-loop renormalization-group expansions for the three-dimensional n-vector cubic model and critical exponents for impure Ising systems

The renormalization-group (RG) functions for the three-dimensional n-vector cubic model are calculated in the five-loop approximation. High-precision numerical estimates for the asymptotic critical exponents of the three-dimensional impure Ising systems are extracted from the five-loop RG series by means of the Pade-Borel-Leroy resummation under n = 0. These exponents are found to be: γ= 1.325 +/- 0.003, η= 0.025 +/- 0.01, ν= 0.671 +/- 0.005, α= - 0.0125 +/- 0.008, β= 0.344 +/- 0.006. For the correction-to-scaling exponent, the less accurate estimate ω= 0.32 +/- 0.06 is obtained.

cond-mat.stat-mech↗

arXiv2000

Critical behavior of frustrated systems: Monte Carlo simulations versus Renormalization Group

We study the critical behavior of frustrated systems by means of Pade-Borel resummed three-loop renormalization-group expansions and numerical Monte Carlo simulations. Amazingly, for six-component spins where the transition is second order, both approaches disagree. This unusual situation is analyzed both from the point of view of the convergence of the resummed series and from the possible relevance of non perturbative effects.

cond-mat.stat-mech↗

arXiv1999

Renormalized sextic coupling constant for the two-dimensional Ising model from field theory

The field-theoretical renormalization group approach is used to estimate the universal critical value g_6^* of renormalized sextic coupling constant for the two-dimensional Ising model. Four-loop perturbative expansion for g_6 is calculated and resummed by means of the Pade-Borel-Leroy technique. Under the optimal value of the shift parameter b providing the fastest convergence of the iteration procedure the estimates g_6^* = 1.10, g_6^*/{g_4^*}^2 = 2.94 are obtained which agree quite well with those deduced recently by S.-Y. Zinn, S.-N. Lai, and M. E. Fisher (Phys. Rev. E 54 (1996) 1176) from the high-temperature expansions.

cond-mat.stat-mech↗

arXiv1999

Universal critical coupling constants for the three-dimensional n-vector model from field theory

The field-theoretical renormalization group approach in three dimensions is used to estimate the universal critical values of renormalized coupling constants g_6 and g_8 for the O(n)-symmetric model. The RG series for g_6 and g_8 are calculated in the four-loop and three-loop approximations respectively and then resummed by means of the Pade-Borel-Leroy technique. Under the optimal value of the shift parameter b providing the fastest convergence of the iteration procedure numerical estimates for the universal critical values g_6^*(n) are obtained for n = 1, 2, 3,...40 with the accuracy no worse than 0.3%. The RG expansion for g_8 demonstrates stronger divergence and results in considerably cruder numerical estimates. They are found to be consistent with the values of g_8^* deduced from the exact RG equations and, for n > 8, with those given by a constrained analysis of corresponding ε-expansion.

hep-th↗

arXiv1999

Critical behavior of three-dimensional magnets with complicated ordering from three-loop renormalization-group expansions

The critical behavior of a model describing phase transitions in 3D antiferromagnets with 2N-component real order parameters is studied within the renormalization-group (RG) approach. The RG functions are calculated in the three-loop order and resummed by the generalized Pade-Borel procedure preserving the specific symmetry properties of the model. An anisotropic stable fixed point is found to exist in the RG flow diagram for N > 1 and lies near the Bose fixed point; corresponding critical exponents are close to those of the XY model. The accuracy of the results obtained is discussed and estimated.

cond-mat.stat-mech↗