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🗀 Archive 2010-2015

Department of Mathematical Methods in Theoretical Physics

- Recently it has been shown that the proposed by us two- and three-parameter deformed analogs of the Heisenberg algebra of position and momentum operators can be mapped onto respective algebras of non-standard deformed quantum oscillators. We have established that such relation dictates an unusual rule (instead of Hermitian) of mutual conjugation for the creation and annihilation operators a+and a-, namely the η(N)-pseudo-Hermitian conjugation, η(N) being certain operator function of the excitation number operator N. Respectively, that leads to η(N)-pseudo-Hermiticity of the position (or momentum) operator, whereas the Hamiltonian is Hermitian. Diverse possible cases have been described and specific properties related with η(N)-pseudo-Hermitian conjugation have been revealed.
*A.M. Gavrilik* - Integrable generalizations of the Jaynes-Cummings-Dicke models and the Bose-Hubbard-type dimers model, which are associated with general non-skew-symmetric classical r-matrices and Lie algebras of higher ranks, have been constructed. The spectrum of the Hamiltonians of integrable generalizations of Jaynes-Cummings-Dicke models and Bose-Hubbard-type dimers, associated with the twisted rational r-matrices and Lie algebras gl(n), have been found.
*T. Skrypnyk* - Within the problem of realization of composite (or quasi-) fermions, each of these consisting of a fermion and a deformed boson, the physical interpretation of the solutions and relevant parameters is analyzed in terms of deformed Fermi-oscillators. For the case of 2 modes of composite fermions and 2 or 3 modes of their constituents, i.e., ordinary bosons and fermions, the solutions are presented in the form symmetric under the interchange of quasi-fermion modes. General solution has been obtained in the case of non-deformed constituent boson and arbitrary number of modes of both composite fermions and their constituents. Particular solutions have been given, including those for maximally entangled states of composite fermions. The (inter-constituent) entanglement entropy has been obtained for composite quasi-fermions admitting the mentioned realization.
*A.M. Gavrilik, Yu.A. Mishchenko*

- We establish the relation of the second virial coefficient of certain (μ, q)-deformed Bose gas model, recently proposed by us, to the interaction and compositeness parameters when either of these factors is taken into account separately. The deformation parameter relative to interaction, is now linked directly with the scattering length for some potential, and the effective radius of interaction (in general, with scattering phases). A novel feature is found: the appearance of temperature dependence in the deformation parameters μ and q, absent in the previous approaches to deformed Bose gas models. The problem of temperature dependence is analyzed in detail and its possible solution is proposed.
*A.M. Gavrilik, Yu.A. Mishchenko* - Formerly, for an effective description of the gas of composite bosons with interaction, we have designed two-parameter ??, q-deformation of the Bose gas model. Now, similar μ,q-Bose gas model is developed through deforming the distributions and correlation functions. In this model, the explicit expressions for one- and two-particle distribution functions and for the intercept of two-particle correlation function are obtained: all depending on μ and q. The results are presented on graphics, and confronting them with data for μ-correlations extracted in the STAR/RHIC experiments on heavy ion collisions show qualitative agreement.
*A.M. Gavrilik, Yu.A. Mishchenko* - It is proven that the Fourier transform of the conformal blocks (which are special functions related to general n-point correlation functions of two-dimensional conformal field theory) gives the explicit solution of the Riemann-Hilbert problem of constructing multi-valued analytic function with prescribed SL(2,C)-monodromy on Riemann sphere with n punctures. In the case of 4 punctures, the obtained result yields asymptotic behavior of tau-functions of Painleve equations.
*N. Iorgov*

- A new deformed Bose gas model is proposed based on the (deformation) structure function combining q-deformation and quadratically nonlinear deformation. The model enables an effective description of the interacting gas of (2-fermion or 2-boson) composite bosons or quasi-bosons. Using certain extension of Jackson derivative, we derive the expression for the total mean number of particles, deformed virial expansion of the equation of state along with five first virial coefficients. That corresponds to virial expansion of the equation of state for non-ideal gas of composite bosons with some nontrivial interaction between them.
*A.M. Gavrilik, Yu.A. Mishchenko* - A multiparametric deformation of two-dimensional conformal field theory is constructed. The pole structure is found for the product of holomorphic component of the energy-momentum tensor and a primary conformal field. A realization of the deformed Virasoro algebra on conformal fields is given; the two-point correlation function of this theory is calculated.
*I. Burban* - General four-point conformal blocks of quantum conformal field theory with central charge c=1 on the Riemann sphere are expressed as the coefficients of Fourier transform of the tau-function of the Painleve VI equation with respect to one of the integration constants. On this base it is shown that for c=1 the fusion matrix (matrix of the transition between the s- and t-channel conformal blocks) coincides with the coefficient of connection which relates the asymptotic expansionsof the tau-function in the vicinity of distinct critical points. Final result for these quantities is given explicitly in terms of the ratio of two products of the Barnes G-functions with the arguments, expressed through the conformal dimensions or the monodromy data.
*N.Z. Iorgov, O. Lisovyy, Yu.V. Tykhyy*

- Using earlier proven result about realization of the creation, annihilation, and particle number operators of composite bosons (quasibosons) through the algebra of respective operators of deformed oscillator, what implies non-Bose statistics of quasibosons, a direct connection is esta-blished between such important concept in quantum information theory as the entanglement of constituents of quasiboson, and the deformation of quantum oscillator. The characteristics of entanglement such as entropy and Schmidt number are calculated for two-component quasiboson and expressed through the parameter of deformation. These bipartite entanglement characteristics are extended to arbitrary multi-quasibosonic states (Fock type, coherent, etc.) and given in terms of the deformation parameter too.
*A.M. Gavrilik, Yu.A. Mishchenko* - Matrix of induced transformation of spin operator on the algebra of fermions is found and, on its base, the form-factors of spin operator (matrix elements between eigenstates of transfer matrix) of two-dimensional Ising model on finite lattice are calculated. It is proven that such multi-particle form-factors, parameterized by elliptic functions, are expressed through Pfaffians of the matrix built from two-particle form-factors.
*N.Z. Iorgov, O. Lisovyy* - Integrable quantum models which generalize the two-level models of the Jaines-Cummings-Dicke type as well as Bose-Hubbard type integrable boson models related to general non-skewsymmetric classical r-matrices are constructed.
*T.W. Skrypnik*

- The quasi-bosons were considered which are composite bosons (like mesons, excitons etc.). They are composed by two fermions with creation and annihilation operators satisfying non-standard commutation relations. The conditions are clarified which allow to realize the quasi-boson operators by the operators of deformed (nonlinear) oscillator. It is proved that such deformed oscillator exists and it is unique in the family of deformations under consideration.
*A.M. Gavrilik, Yu.A. Mishchenko* - For the superintegrable ZN-symmetric chiral Potts quantum chain of finite length, on the base of the notion of Onsager sectors (the spaces of irreducible representations of Onsager algebra), the spin matrix elements between eigenstates of Hamiltonian of quantum chain were found in a factorized form up to a common scalar factor, known for the Onsager sectors with the lowest energies. For the quantum Ising chain in a transverse field (the case of N=2) the uknown scalar factors were found for all the Onsager sectors by the method of fermion operators. For general N, the matrix elements between vacuum states in the thermodynamic limit were obtained and the formula for the order parameters was derived.
*N.Z. Iorgov, V.N. Shadura, Yu.V. Tykhyy*