arXiv subjects

A. I. Sokolov

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

At least 19 recordsLinked to original sources

arXiv2021

Critical behavior of weakly disordered Ising model: Six-loop $\sqrt \varepsilon$ expansion study

The critical behavior of three-dimensional weakly diluted quenched Ising model is examined on the base of six-loop renormalization group expansions obtained within the minimal subtraction scheme in $4-ε$ space dimensions. For this purpose the $ϕ^4$ field theory with cubic symmetry was analyzed in the replica limit $n\rightarrow 0$. Along with renormalization group expansions in terms of renormalized couplings the $\sqrt{\varepsilon}$ expansions of critical exponents are presented. Corresponding numerical estimates for the physical, three-dimensional system are obtained by means of different resummation procedures applied both to the $\sqrt{\varepsilon}$ series and to initial renormalization group expansions. The results given by the latter approach are in a good agreement with their counterparts obtained experimentally and within the Monte Carlo simulations, while resumming of $\sqrt{\varepsilon}$ series themselves turned out to be disappointing.

cond-mat.stat-mech↗

arXiv2021

Six-loop $\varepsilon$ expansion of three-dimensional $\text{U}(n)\times \text{U}(m)$ models

We analyze the Landau-Wilson field theory with $\text{U}(n)\times\text{U}(m)$ symmetry which describes the finite-temperature phase transition in QCD in the limit of vanishing quark masses with $n=m=N_f$ flavors and unbroken anomaly at the critical temperature. The six-loop expansions of the renormalization group functions are calculated within the Minimal Subtraction scheme in $4 - \varepsilon$ dimensions. The $\varepsilon$ series for the upper marginal dimensionality $n^{+}(m,4-\varepsilon)$ -- the key quantity of the theory -- are obtained and resummed by means of different approaches. The numbers found are compared with their counterparts obtained earlier within lower perturbative orders and the pseudo-$\varepsilon$ analysis of massive six-loop three-dimensional expansions. In particular, using an increase in the accuracy of numerical results for $n^{+}(m,3)$ by one order of magnitude, we strengthen the conclusions obtained within previous order in perturbation theory about fairness of the inequality $n^{+}(m,3)>m$. This, in turn, indicates the absence of a stable three-dimensional fixed point for $n=m$, and as a consequence a first-order kind of finite-temperature phase transition in light QCD.

hep-th↗

arXiv2020

Universal effective couplings of the three-dimensional $n$-vector model and field theory

We calculate the universal ratios $R_{2k}$ of renormalized coupling constants $g_{2k}$ entering the critical equation of state for the generalized Heisenberg (three-dimensional $n$-vector) model. Renormalization group (RG) expansions of $R_8$ and $R_{10}$ for arbitrary $n$ are found in the four-loop and three-loop approximations respectively. Universal octic coupling $R_8^*$ is estimated for physical values of spin dimensionality $n = 0, 1, 2, 3$ and for $n = 4,...64$ to get an idea about asymptotic behavior of $R_8^*$. Its numerical values are obtained by means of the resummation of the RG series and within the pseudo-$\varepsilon$ expansion approach. Regarding $R_{10}$ our calculations show that three-loop RG and pseudo-$\varepsilon$ expansions possess big and rapidly growing coefficients for physical values of $n$ what prevents getting fair numerical estimates.

cond-mat.stat-mech↗

arXiv2019

Six-loop $\varepsilon$ expansion study of three-dimensional $O(n)\times O(m)$ spin models

The Landau-Wilson field theory with $O(n)\times O(m)$ symmetry which describes the critical thermodynamics of frustrated spin systems with noncollinear and noncoplanar ordering is analyzed in $4 - \varepsilon$ dimensions within the minimal subtraction scheme in the six-loop approximation. The $\varepsilon$ expansions for marginal dimensionalities of the order parameter $n^H(m,4-\varepsilon)$, $n^-(m,4-\varepsilon)$, $n^+(m,4-\varepsilon)$ separating different regimes of critical behavior are extended up to $\varepsilon^5$ terms. Concrete series with coefficients in decimals are presented for $m=\{2, \dots, 6\}$. The \textit{diagram of stability} of nontrivial fixed points, including the chiral one, in $(m,n)$ plane is constructed by means of summing up of corresponding $\varepsilon$ expansions using various resummation techniques. Numerical estimates of the chiral critical exponents for several couples $\{m,n\}$ are also found. Comparative analysis of our results with their counterparts obtained earlier within the lower-order approximations and by means of alternative approaches is performed. It is confirmed, in particular, that in physically interesting cases $n=2, m=2$ and $n=2, m=3$ phase transitions into chiral phases should be first-order.

cond-mat.stat-mech↗

arXiv2019

Six-loop $\varepsilon$ expansion study of three-dimensional $n$-vector model with cubic anisotropy

The six-loop expansions of the renormalization-group functions of $φ^4$ $n$-vector model with cubic anisotropy are calculated within the minimal subtraction (MS) scheme in $4 - \varepsilon$ dimensions. The $\varepsilon$ expansions for the cubic fixed point coordinates, critical exponents corresponding to the cubic universality class and marginal order parameter dimensionality $n_c$ separating different regimes of critical behavior are presented. Since the $\varepsilon$ expansions are divergent numerical estimates of the quantities of interest are obtained employing proper resummation techniques. The numbers found are compared with their counterparts obtained earlier within various field-theoretical approaches and by lattice calculations. In particular, our analysis of $n_c$ strengthens the existing arguments in favor of stability of the cubic fixed point in the physical case $n = 3$.

cond-mat.stat-mech↗

arXiv2017

Effective potential of the three-dimensional Ising model: the pseudo-$ε$ expansion study

The ratios $R_{2k}$ of renormalized coupling constants $g_{2k}$ that enter the effective potential and small-field equation of state acquire the universal values at criticality. They are calculated for the three-dimensional scalar $λϕ^4$ field theory (3D Ising model) within the pseudo-$ε$ expansion approach. Pseudo-$ε$ expansions for the critical values of $g_6$, $g_8$, $g_{10}$, $R_6 = g_6/g_4^2$, $R_8 = g_8/g_4^3$ and $R_{10} = g_{10}/g_4^4$ originating from the five-loop renormalization group (RG) series are derived. Pseudo-$ε$ expansions for the sextic coupling have rapidly diminishing coefficients, so addressing Padé approximants yields proper numerical results. Use of Padé--Borel--Leroy and conformal mapping resummation techniques further improves the accuracy leading to the values $R_6^* = 1.6488$ and $R_6^* = 1.6490$ which are in a brilliant agreement with the result of advanced lattice calculations. For the octic coupling the numerical structure of the pseudo-$ε$ expansions is less favorable. Nevertheless, the conform-Borel resummation gives $R_8^* = 0.868$, the number being close to the lattice estimate $R_8^* = 0.871$ and compatible with the result of 3D RG analysis $R_8^* = 0.857$. Pseudo-$ε$ expansions for $R_{10}^*$ and $g_{10}^*$ are also found to have much smaller coefficients than those of the original RG series. They remain, however, fast growing and big enough to prevent obtaining fair numerical estimates.

cond-mat.stat-mech↗

arXiv2016

Anisotropy of a Cubic Ferromagnet at Criticality

Critical fluctuations change the effective anisotropy of cubic ferromagnet near the Curie point. If the crystal undergoes phase transition into orthorhombic phase and the initial anisotropy is not too strong, reduced anisotropy of nonlinear susceptibility acquires at $T_c$ the universal value $δ_4^* = {{2v^*} \over {3(u^* + v^*)}}$ where $u^*$ and $v^*$ -- coordinates of the cubic fixed point on the flow diagram of renormalization group equations. In the paper, the critical value of the reduced anisotropy is estimated within the pseudo-$ε$ expansion approach. The six-loop pseudo-$ε$ expansions for $u^*$, $v^*$, and $δ_4^*$ are derived for the arbitrary spin dimensionality $n$. For cubic crystals ($n = 3$) higher-order coefficients of the pseudo-$ε$ expansions obtained turn out to be so small that use of simple Padé approximants yields reliable numerical results. Padé resummation of the pseudo-$ε$ series for $u^*$, $v^*$, and $δ_4^*$ leads to the estimate $δ_4^* = 0.079 \pm 0.006$ indicating that detection of the anisotropic critical behavior of cubic ferromagnets in physical and computer experiments is certainly possible.

cond-mat.stat-mech↗

arXiv2016

Field theory and anisotropy of cubic ferromagnet near Curie point

Critical fluctuations are known to change the effective anisotropy of cubic ferromagnet near the Curie point. If the crystal undergoes phase transition into orthorhombic phase and the initial anisotropy is not too strong, effective anisotropy acquires at T_c the universal value A* = v*/u* where u* and v* are coordinates of the cubic fixed point entering the scaling equation of state and expressions for nonlinear susceptibilities. In the paper, the numerical value of the anisotropy parameter A at the critical point is estimated using the ε-expansion and pseudo-ε-expansion techniques. Pade resummation of six-loop pseudo-$ε$-expansions for u*, v*, and A* leads to the estimate A* = 0.13 close to that extracted from the five-loop ε-expansion but differing considerably from the value A* = 0.089 given by the analysis of six-loop expansions of the β-functions themselves. This discrepancy is discussed and its roots are cleared up.

cond-mat.stat-mech↗

arXiv2016

Universal effective coupling constants for the generalized Heisenberg model

The aim of this study is to find universal critical values of the dimensionless effective coupling constant $g_6$ and refined universal values $g_4$ for Heisenberg ferromagnets with $n$-component order parameters. These constants appear in the equation of state and determine the nonlinear susceptibilities $χ_4$ and $χ_6$ in the critical region. The sextic coupling $g_6$ is calculated as a function of $g_4$ in the three-loop approximation, the series obtained is resummed by the Pade-Borel method and then by substituting the Wilson fixed point coordinate $g_4^*$ into resultant expression numerical values of $g_6^*$ are obtained for different $n$. The numbers $g_4^*$ were determined from the six-loop expansions for $β$ function resummed using Borel-transformation-based techniques. The analysis of the accuracy of these $g_6^*$ values showed that they differ from the true values by no more than 1.6 %. The values of $g_6^*$ thus found were compared also with those given by the $1/n$ method what allowed to estimate the level of accuracy of this method.

cond-mat.stat-mech↗

arXiv2016

Critical thermodynamics of the two-dimensional systems in five-loop renormalization-group approximation

The RG functions of the 2D $n$-vector $ϕ^4$ model are calculated in the five-loop approximation. Perturbative series for the $β$ function and critical exponents are resummed by the Pade-Borel and Pade-Borel-Leroy techniques, resummation procedures are optimized and an accuracy of the numerical results is estimated. In the Ising case $n = 1$ as well as in the others ($n = 0$, $n = -1$, $n = 2, 3,...32$) an account for the five-loop term is found to shift the Wilson fixed point location only briefly, leaving it outside the segment formed by the results of the corresponding lattice calculations; even error bars of the RG and lattice estimates do not overlap in the most cases studied. This is argued to reflect the influence of the singular (non-analytical) contribution to the $β$ function that can not be found perturbatively. The evaluation of the critical exponents for $n = 1$, $n = 0$ and $n = -1$ in the five-loop approximation and comparison of the numbers obtained with their known exact counterparts confirm the conclusion that non-analytical contributions are visible in two dimensions.

hep-th↗

arXiv2016

Critical exponents and the pseudo-$ε$ expansion

We present the pseudo-$ε$ expansions ($τ$-series) for the critical exponents of a $λϕ^4$ three-dimensional $O(n)$-symmetric model obtained on the basis of six-loop renormalization-group expansions. Concrete numerical results are presented for physically interesting cases $n = 1$, $n = 2$, $n = 3$ and $n = 0$, as well as for $4 \le n \le 32$ in order to clarify the general properties of the obtained series. The pseudo-$ε$-expansions for the exponents $γ$ and $α$ have small and rapidly decreasing coefficients. So, even the direct summation of the $τ$-series leads to fair estimates for critical exponents, while addressing Pade approximants enables one to get high-precision numerical results. In contrast, the coefficients of the pseudo-$ε$ expansion of the scaling correction exponent $ω$ do not exhibit any tendency to decrease at physical values of $n$. But the corresponding series are sign-alternating, and to obtain reliable numerical estimates, it also suffices to use simple Padé approximants in this case. The pseudo-$ε$ expansion technique can therefore be regarded as a specific resummation method converting divergent renormalization-group series into expansions that are computationally convenient.

cond-mat.stat-mech↗

arXiv2016

Renormalized coupling constants for 3D scalar λϕ^4 field theory and pseudo-ε-expansion

Renormalized coupling constants g_{2k} that enter the critical equation of state and determine nonlinear susceptibilities of the system possess universal values g*_{2k} at the Curie point. They are calculated, along with the ratios R_{2k} = g_{2k}/g_4^{k-1}, for the three-dimensional scalar λϕ^4 field theory within the pseudo-ε-expansion approach. Pseudo-ε-expansions for g*_6, g*_8, R*_6, and R*_8 are derived in the five-loop approximation, numerical estimates are presented for R*_6 and R*_8. The higher-order coefficients of the pseudo-ε-expansions for the sextic coupling are so small that simple Pade approximants turn out to be sufficient to yield very good numerical results. Their use gives R*_6 = 1.650 while the most recent lattice estimate is R*_6 = 1.649(2). For the octic coupling pseudo-ε-expansions are less favorable from the numerical point of view. Nevertheless, Pade-Borel resummation leads in this case to R*_8 = 0.890, the number differing only slightly from the values R*_8 = 0.871, R*_8 = 0.857 extracted from the lattice and field-theoretical calculations.

hep-th↗

arXiv2014

Critical Exponents in Two Dimensions and Pseudo-ε Expansion

The critical behavior of two-dimensional $n$-vector $λϕ^4$ field model is studied within the framework of pseudo-$ε$ expansion approach. Pseudo-$ε$ expansions for Wilson fixed point location $g^*$ and critical exponents originating from five-loop 2D renormalization group series are derived. Numerical estimates obtained within Padé and Padé-Borel resummation procedures as well as by direct summation are presented for $n = 1$, $n = 0$ and $n = -1$, i. e. for the models which are exactly solvable. The pseudo-$ε$ expansions for $g^*$, critical exponents $γ$ and $ν$ have small lower-order coefficients and slow increasing higher-order ones. As a result, direct summation of these series with optimal cut off provides numerical estimates that are no worse than those given by the resummation approaches mentioned. This enables one to consider the pseudo-$ε$ expansion technique itself as some specific resummation method.

cond-mat.stat-mech↗

arXiv2014

Critical Exponents of Superfluid Helium and Pseudo-$ε$ Expansion

Pseudo-$ε$ expansions ($τ$-series) for critical exponents of 3D XY model describing $λ$-transition in liquid helium are derived up to $τ^6$ terms. Numerical estimates extracted from the $τ$-series obtained using Padé-Borel resummation technique, scaling relations and seven-loop ($τ^7$) estimate for the Fisher exponent $η$ are presented including those for exponents $α$ and $ν$ measured in experiments with record accuracy. For the exponent $α$ the procedure argued to be most reliable gives $α= -0.0117$. This number is very close to the most accurate experimental values differing appreciably from the results of numerous lattice and field-theoretical calculations. It signals that the pseudo-$ε$ expansion is a powerful tool robust enough to evaluate critical exponents with very small absolute error. The arguments in favour of such a robustness are presented.

cond-mat.stat-mech↗

arXiv2014

Fisher Exponent from Pseudo-$ε$ Expansion

Critical exponent $η$ for three-dimensional systems with $n$-vector order parameter is evaluated in the frame of pseudo-$ε$ expansion approach. Pseudo-$ε$ expansion ($τ$-series) for $η$ found up to $τ^7$ term for $n$ = 0, 1, 2, 3 and within $τ^6$ order for general $n$ is shown to have a structure rather favorable for getting numerical estimates. Use of Padé approximants and direct summation of $τ$-series result in iteration procedures rapidly converging to the asymptotic values that are very close to most reliable numerical estimates of $η$ known today. The origin of this fortune is discussed and shown to lie in general properties of the pseudo-$ε$ expansion machinery interfering with some peculiarities of the renormalization group expansion of $η$.

cond-mat.stat-mech↗

arXiv2014

Pseudo-$ε$ Expansion and Renormalized Coupling Constants at Criticality

Universal values of dimensional effective coupling constants $g_{2k}$ that determine nonlinear susceptibilities $χ_{2k}$ and enter the scaling equation of state are calculated for $n$-vector field theory within the pseudo-$ε$ expansion approach. Pseudo-$ε$ expansions for $g_6$ and $g_8$ at criticality are derived for arbitrary $n$. Analogous series for ratios $R_6 = g_6/g_4^2$ and $R_8 = g_8/g_4^3$ figuring in the equation of state are also found and the pseudo-$ε$ expansion for Wilson fixed point location $g_{4}^*$ descending from the six-loop RG expansion for $β$-function is reported. Numerical results are presented for $0 \le n \le 64$ with main attention paid to physically important cases $n = 0, 1, 2, 3$. Pseudo-$ε$ expansions for quartic and sextic couplings have rapidly diminishing coefficients, so Padé resummation turns out to be sufficient to yield high-precision numerical estimates. Moreover, direct summation of these series with optimal truncation gives the values of $g_4^*$ and $R_6^*$ almost as accurate as those provided by Padé technique. Pseudo-$ε$ expansion estimates for $g_8^*$ and $R_8^*$ are found to be much worse than that for the lower-order couplings independently on the resummation method employed. Numerical effectiveness of the pseudo-$ε$ expansion approach in two dimensions is also studied. Pseudo-$ε$ expansion for $g_4^*$ originating from the five-loop RG series for $β$-function of 2D $λϕ^4$ field theory is used to get numerical estimates for $n$ ranging from 0 to 64. The approach discussed gives accurate enough values of $g_{4}^*$ down to $n = 2$ and leads to fair estimates for Ising and polymer ($n = 0$) models.

cond-mat.stat-mech↗

arXiv2009

Fluctuations, Higher Order Anharmonicities, and Landau Expansion for Barium Titanate

Correct phenomenological description of ferroelectric phase transitions in barium titanate requires accounting for eighth-order terms in the free energy expansion, in addition to the conventional sixth-order contributions. Another unusual feature of BaTiO_3 crystal is that the coefficients B_1 and B_2 of the terms P_x^4 and P_x^2*P_y^2 in the Landau expansion depend on the temperature. It is shown that the temperature dependence of B_1 and B_2 may be caused by thermal fluctuations of the polarization, provided the fourth-order anharmonicity is anomalously small, i. e. the nonlinearity of P^4 type and higher-order ones play comparable roles. Non-singular (non-critical) fluctuation contributions to B_1 and B_2 are calculated in the first approximation in sixth-order and eighth-order anharmonic constants. Both contributions increase with the temperature, which is in agreement with available experimental data. Moreover, the theory makes it possible to estimate, without any additional assumptions, the ratio of fluctuation (temperature dependent) contributions to coefficients B_1 and B_2. Theoretical value of B_1/B_2 appears to be close to that given by experiments.

cond-mat.mtrl-sci↗

arXiv2009

Paraelectric in a Strong High-Frequency Field

A change in the effective permittivity of a ferroelectric film in the paraelectric phase under the action of a strong high-frequency field (nonequilibrium soft mode heating) is considered. It is shown that this effect must be most clearly pronounced far from the resonance (ω_0 << ω_sm), rather than for the external field frequency ω_0 close to the soft mode frequency ω_sm. The effective permittivity as a function of the high-frequency field amplitude is calculated using the phenomenological approach and within the microscopic theory based on the simple model of a displacement-type ferroelectric.

cond-mat.mtrl-sci↗