arXiv subjects
Publications and source records attributed to Andrei Mudrov.
We introduce triples of associative algebras as a tool for building solutions to the Yang-Baxter equation. It turns out that the class of R-matrices thus obtained is related to a Hecke-like condition, which is formulated for associative algebras with symmetric cyclic inner product. R-matrices for a subclass of the $A_n$-type Belavin-Drinfel'd triples are derived in this way.
The critical behavior of a complex N-component order parameter Ginzburg-Landau model with isotropic and cubic interactions describing antiferromagnetic and structural phase transitions in certain crystals with complicated ordering is studied in the framework of the four-loop renormalization group (RG) approach in $(4-\ve)$ dimensions. By using dimensional regularization and the minimal subtraction scheme, the perturbative expansions for RG functions are deduced and resummed by the Borel-Leroy transformation combined with a conformal mapping. Investigation of the global structure of RG flows for the physically significant cases N=2 and N=3 shows that the model has an anisotropic stable fixed point governing the continuous phase transitions with new critical exponents. This is supported by the estimate of the critical dimensionality $N_c=1.445(20)$ obtained from six loops via the exact relation $N_c={1/2} n_c$ established for the complex and real hypercubic models.
Within the four-loop $\ve$ expansion, we study the critical behavior of certain antiferromagnets with complicated ordering. We show that an anisotropic stable fixed point governs the phase transitions with new critical exponents. This is supported by the estimate of critical dimensionality $N_c^C=1.445(20)$ obtained from six loops via the exact relation $N_c^C={1/2} N_c^R$ established for the real and complex hypercubic models.
The critical thermodynamics of an $MN$-component field model with cubic anisotropy relevant to the phase transitions in certain crystals with complicated ordering is studied within the four-loop $\ve$ expansion using the minimal subtraction scheme. Investigation of the global structure of RG flows for the physically significant cases M=2, N=2 and M=2, N=3 shows that the model has an anisotropic stable fixed point with new critical exponents. The critical dimensionality of the order parameter is proved to be equal to $N_c^C=1.445(20)$, that is exactly half its counterpart in the real hypercubic model.
A new approach to summation of divergent field-theoretical series is suggested. It is based on the Borel transformation combined with a conformal mapping and does not imply the exact asymptotic parameters to be known. The method is tested on functions expanded in their asymptotic power series. It is applied to estimating the critical exponent values for an N-vector field model, describing magnetic and structural phase transitions in cubic and tetragonal crystals, from five-loop εexpansions.
We find the general solution to the twisting equation in the tensor bialgebra $T({\bf R})$ of an associative unital ring ${\bf R}$ viewed as that of fundamental representation for a universal enveloping Lie algebra and its quantum deformations. We suggest a procedure of constructing twisting cocycles belonging to a given quasitriangular subbialgebra ${\cal H}\subset T({\bf R})$. This algorithm generalizes Reshetikhin's approach, which involves cocycles fulfilling the Yang-Baxter equation. Within this framework we study a class of quantized inhomogeneous Lie algebras related to associative rings in a certain way, for which we build twisting cocycles and universal $R$-matrices. Our approach is a generalization of the methods developed for the case of commutative rings in our recent work including such well-known examples as Jordanian quantization of the Borel subalgebra of $sl(2)$ and the null-plane quantized Poincaré algebra by Ballesteros at al. We reveal the role of special group cohomologies in this process and establish the bicrossproduct structure of the examples studied.
The structure of the renormalization-group flows in a model with three quartic coupling constants is studied within the $ε$-expansion method up to three-loop order. Twofold degeneracy of the eigenvalue exponents for the three-dimensionally stable fixed point is observed and the possibility for powers in $\sqrtε$ to appear in the series is investigated. Reliability and effectiveness of the $ε$-expansion method for the given model is discussed.
The universal ${\cal R}$--matrix for a quantized Poincar{é} algebra ${\cal P}(3+1)$ introduced by Ballesteros et al is evaluated. The solution is obtained as a specific case of a formulated multidimensional generalization to the non-standard (Jordanian) quantization of $sl(2)$.
The complete analysis of a model with three quartic coupling constants associated with an O(2N)--symmetric, a cubic, and a tetragonal interactions is carried out within the three-loop approximation of the renormalization-group (RG) approach in $D=4-2ε$ dimensions. Perturbation expansions for RG functions are calculated using dimensional regularization and the minimal subtraction (MS) scheme. It is shown that for $N\ge 2$ the model does possess a stable fixed point in three dimensional space of coupling constants, in accordance with predictions made earlier on the base of the lower-order approximations. Numerical estimate for critical (marginal) value of the order parameter dimensionality $N_c$ is given using Padé-Borel summation of the corresponding $ε$--expansion series obtained. It is observed that two-fold degeneracy of the eigenvalue exponents in the one-loop approximation for the unique stable fixed point leads to the substantial decrease of the accuracy expected within three loops and may cause powers of $\sqrtε$ to appear in the expansions. The critical exponents $γ$ and $η$ are calculated for all fixed points up to $ε^3$ and $ε^4$, respectively, and processed by the Borel summation method modified with a conformal mapping. For the unique stable fixed point the magnetic susceptibility exponent $γ$ for N=2 is found to differ in third order in $ε$ from that of an O(4)--symmetric point. Qualitative comparison of the results given by $ε$--expansion, three-dimensional RG analysis, non-perturbative RG arguments, and experimental data is performed.