# C. MalyshevRussian Academy of Sciences | RAS · St.Petersburg Department of Steklov Mathematical Institute

C. Malyshev

PhD 1993, DSc 2014

## About

75

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Introduction

**Skills and Expertise**

## Publications

Publications (75)

Relations between quantum integrable models solvable by the quantum inverse scattering method and some aspects of enumerative combinatorics and partition theory are discussed. The main example is the Heisenberg spin chain in the limit cases of zero or infinite anisotropy. Form factors and some thermal correlation functions are calculated, and it is...

Three-dimensional Lagrangian field theory is investigated as the T(3)-gauge model of static defects in continuous solids. The gauge Lagrangian is proposed in the Hilbert–Einstein form, and the non-conventional incompatibility law is given by an Einstein-like gauge equation with the elastic stress tensor as “matter” source. The stress function metho...

The behavior of the shear modulus caused by proliferation of dipoles of non-singular screw dislocations with finite-sized core is considered. The representation of two-dimensional Coulomb gas with smoothed-out coupling is used, and the stress–stress correlation function is calculated. A convolution integral expressed in terms of the modified Bessel...

We obtain and investigate mean values of the exponential of the centroid operator for the periodic Heisenberg XX0 chain on a ring. The generating function of directed lattice paths is obtained in terms of circulant matrices which leads to generalizations of the Ramus identity. The two-time correlation function of the exponential of the centroid ope...

Relations between the mean values of distributions of flipped spins on periodic Heisenberg $XX$ chain and some aspects of enumerative combinatorics are discussed. The Bethe vectors, which are the state-vectors of the model, are considered both as on- and off-shell. It is this approach that makes it possible to represent and to study the correlation...

We discuss connection between the XX0 Heisenberg spin chain and some aspects of enumerative combinatorics. The representation of the Bethe wave functions via the Schur functions allows to apply the theory of symmetric functions to the calculation of the correlation functions. We provide a combinatorial derivation of the dynamical auto-correlation f...

We discuss connection between the XX0 Heisenberg spin chain and some aspects of enumerative combinatorics. The representation of the Bethe wave functions via the Schur functions allows to apply the theory of symmetric functions to the calculation of the correlation functions. We provide a combinatorial derivation of the dynamical auto-correlation f...

Determinantal representation for the generating function of plane partitions with fixed volumes of diagonal parts is investigated in limiting cases.

The exactly solvable four-vertex model with the fixed boundary conditions in the presence of inhomogeneous linearly growing external field is considered. The partition function of the model is calculated and represented in the determinantal form. The established connection with the boxed plane partitions allows us to calculate the generating functi...

The mean values of non-homogeneously parameterized generating exponential are obtained and investigated for the periodic Heisenberg XX model. The norm-trace generating function of boxed plane partitions with fixed volume of their diagonal parts is obtained as N-particles average of the generating exponential. The generating function of self-avoidin...

The exactly solvable four-vertex model on a square grid with the fixed boundary conditions in a presence of a special external field is considered. Namely, we study a system in a linear field acting on the central column of the grid. The partition function of the model is calculated by the quantum inverse scattering method. The answer is written in...

We consider the XY Heisenberg spin 12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \frac{1}{2} $$\end{document} chain in the fermion representation. The construction...

We discuss a connection between the XXZ Heisenberg spin chain in the limiting case of zero anisotropy and some aspects of enumerative combinatorics. The representation of the Bethe wave functions in terms of Schur functions allows us to apply the theory of symmetric functions to calculating correlation functions. We provide a combinatorial derivati...

The XY Heisenberg spin 1/2 chain is considered in the fermion representation. The construction of the ground state-vector is based on the group-theoretical approach. The exact expression for the ground state-vector will allow to study the combinatorics of the correlation functions of the model.

A quantum phase model is considered and the temporal evolution of the exponential of the first moment of particles distribution is studied. The calculation of the temporal evolution is based on the properties of Schur functions. The form-factor of the considered distribution function is expressed in the determinant form and it is proved that in the...

We consider random multi-dimensional lattice walks bounded by a hyperplane, calling them walks over multi-dimensional simplicial lattices. We demonstrate that generating functions of these walks are dynamical correlation functions of a certain type of exactly solvable quantum phase models describing strongly correlated bosons on a chain. Walks over...

The special limit of the Totaly Asymmetric Zero Range Process of the low dimensional non-equilibrium statistical mechanics described by the non-Hermitian Hamiltonian is considered. The calculation of the conditional probabilities of the model are based on the algebraic Bethe Ansatz approach. We demonstrate that the conditional probabilities may be...

Einstein-like Lagrangian field theory is developed to describe elastic solid containing dislocations with finite-sized core. The framework of the Riemann-Cartan geometry in three dimensions is used, and the core self-energy is expressed by the translational part of the general Lagrangian quadratic in torsion and curvature. In the Hilbert-Einstein c...

We discuss the connection between quantum integrable models and some aspects of enumerative combinatorics and the theory of partitions. As a basic example, we consider the spin XXZ Heisenberg chain in the limiting cases of zero and infinite anisotropy. The representation of the Bethe wave functions via Schur functions allows us to apply the theory...

The Einstein-like field theory is developed to describe an elastic solid containing distribution of screw dislocations with finite-sized core. The core self-energy is given by a gauge-translational Lagrangian that is quadratic in torsion tensor and corresponding to the three-dimensional Riemann–Cartan geometry. The Hilbert–Einstein gauge equation p...

A gauge-translational Lagrangian approach is developed to describe elastic solid containing static dislocations with finite-sized core. The core self-energy includes the translational part of the general Lagrangian quadratic in torsion and curvature which corresponds to the Riemann–Cartan geometry in three dimensions. In the Hilbert–Einstein case,...

The representation of the Bethe wave functions of certain integrable models
via the Schur functions allows to apply the well-developed theory of the
symmetric functions to the calculation of the thermal correlation functions.
The algebraic relations arising in the calculation of the scalar products and
the correlation functions are based on the Bin...

The XX0 Heisenberg model on a cyclic chain is considered. The representation
of the Bethe wave functions via the Schur functions allows to apply the
well-developed theory of the symmetric functions to the calculation of the
thermal correlation functions. The determinantal expressions of the
form-factors and of the thermal correlation functions are...

The XXZ Heisenberg chain is considered for two specific limits of the
anisotropy parameter: $\Dl\to 0$ and $\Dl\to -\infty$. The corresponding wave
functions are expressed by means of the symmetric Schur functions. Certain
expectation values and thermal correlation functions of the ferromagnetic
string operators are calculated over the base of N-pa...

We consider the Heisenberg spin-1/2 XXZ magnet in the case where the anisotropy parameter tends to infinity (the so-called Ising limit). We find the temperature correlation function of a ferromagnetic string above the ground state. Our approach to calculating correlation functions is based on expressing the wave function in the considered limit in...

A field theory is developed for a thermodynamical description of an array of parallel non-singular screw dislocations in an elastic cylinder. The partition function of the system is considered in the functional integral form. Self-energy of the dislocation cores is chosen in the form suggested by the gauge-translational model of non-singular screw...

The spin-1/2 XXZ Heisenberg magnet is considered for the case of the
anisotropy parameter tending to infinity (so-called, Ising limit). A thermal
correlation function of the ferromagnetic string is calculated over the ground
state. The approach to the calculation of the correlation functions in the
limit of infinite anisotropy is based on the obser...

A relationship of the random walks on one-dimensional periodic lattice and the correlation functions of the XX Heisenberg spin chain is investigated. The operator averages taken over the ferromagnetic state play a role of generating functions of the number of paths made by the so-called "vicious" random walkers (the vicious walkers annihilate each...

The path integral approach is used for the calculation of the correlation functions of the $XY$ Heisenberg chain. The obtained answers for the two-point correlators of the $XX$ magnet are of the determinantal form and are interpreted in terms of the generating functions for the random turns vicious walkers.

We consider a quantum field-theoretical model which describes spatially-nonhomogeneous, one-dimensional, repulsive Bose gas
in an external harmonic potential. The two-point correlation function is calculated in the framework of functional integration.
The corresponding functional integrals are estimated by means of stationary phase approximation. A...

A translational gauge approach of the Einstein type is proposed for obtaining the stresses that are due to non-singular screw dislocation. The stress distribution of second order around the screw dislocation is classically known for the hollow circular cylinder with traction-free external and internal boundaries. The inner boundary surrounds the di...

A quantum field-theoretical model, which describes spatially non-homogeneous repulsive Bose gas in an external harmonic potential is considered. Two-point thermal correlation functions of the Bose gas are calculated in the framework of the functional integration approach. Successive integration over the ``high-energy'' functional variables first an...

We treat the functional integration approach for calculation of longitudinal correlation functions of the XY Heisenberg magnet
in a constant homogeneous magnetic field. Generating functionals of the correlators are defined in the form of functional
integrals over anti-commuting variables. The peculiarity of the functional integrals consists of the...

A quantum field-theoretical model which describes spatially non-homogeneous one-dimensional non-relativistic repulsive Bose gas in an external harmonic potential is considered. We calculate the two-point thermal correlation function of the Bose gas in the framework of the functional integration approach. The calculations are done in the coordinate...

The generating function of static correlators of z-components of local spins in the XY and XX Heisenberg magnets is calculated as a combination of functional integrals over anticommuting variables. The peculiarity of the Gaussian integrals in question consists in the fact that the integration variables are subjected to an automorphic boundary condi...

We calculate the dynamic single-particle and many-particle correlation functions at non-zero temperature in one-dimensional trapped repulsive Bose gases. The decay for increasing distance between the points of these correlation functions is governed by a scaling exponent that has a universal expression in terms of observed quantities. This expressi...

Representations for the generating functionals of static correlators of $z$-components of spins in the XY and $XX$ Heisenberg spin chains are obtained in the form of sums of the fermionic functional integrals. The peculiarity of the functional integrals in question is because of the fact that the integration variables depend on the imaginary time `...

A continual model of non-singular screw dislocation lying along a straight infinitely long circular cylinder is investigated in the framework of translational gauge approach with the Hilbert--Einstein gauge Lagrangian. The stress--strain constitutive law implies the elastic energy of isotropic continuum which includes the terms of second and third...

For the generating function of static correlators of the third components of spins in the XX Heisenberg model, we derive a new representation given by a combination of Gaussian functional integrals over anticommuting variables. A peculiarity of the resulting functional integral is that a part of the integration variables depend on the imaginary tim...

The fundamental solution (Green's function) of a first order ordinary differential equation for 2 x 2 matrices related to a couple of Weber's equations is calculated by two methods. The matrix differential equation under consideration arises in a problem of condensed matter physics which is reduced to the famous Landau-type problem of quantization...

We evaluate finite-temperature equilibrium correlators
\(\langle T_\tau \hat \psi ({\text{r}}_{\text{1}} )\hat \psi ^\dag ({\text{r}}_{\text{2}} )\rangle \) for thermal time τ ordered Bose fields \(\hat \psi ,{\text{ }}\hat \psi ^\dag \) to good approximations by new methods of functional integration in d=1, 2, 3 dimensions and with the trap potent...

A new representation for the generating function of static correlators of the third components of spins in the XXO-Heisenberg model is suggested in the form of the Gaussian functional integral. The generating function, as well as the partition function of the model, are calculated by means of the zeta-regularization procedure. Formulas for some cor...

New representation for the generating function of correlators of third components of spins in the XX Heisenberg spin chain is considered in the form given by the fermionic Gaussian path integrals. A part of the discrete anti-commuting integration variables is subjected to ``automorphic'' boundary conditions in respect of imaginary time. The situati...

The fundamental solution (Green's function) of a first order matrix ordinary differential equation arising in a Landau-type problem is calculated by two methods. The coincidence of the two representations results in the integral formula for the product of two parabolic cylinder functions $D_\nu(x) D_\nu(-x)$ at $Re \nu <0$.

There is a large literature (cf. eg. [1, 2]) which, under conditions of translational invariance, has used functional integral methods to calculate, ab initio, the equilibrium finite temperature 2-point correlation functions (Green ’s functions) \[\left\langle {\hat \psi (r,\tau ){{\hat \psi }^\dag }(r',\tau ')} \right\rangle \]
\(G\left( {r,r'} \r...

The paper is concerned with the application of path integration (successive integration over “fast” and “slow” variables and
the Hubbard-Stratonovich transformation) to the antiferromagnetic state in the three-band, two-dimensional repulsive Hubbard
model. It is shown that, in the lowest approximation, the model under consideration is equivalent to...

Previous functional integral methods for translationally invariant systems have been extended to the case of a confining trap potential. Essentially all finite-temperature properties of the repulsive Bose gas in a paraboloidal trap can be determined this way. New analytical results reported here are for the finite-temperature two-point correlation...

The mass current $\vec j$ in the weakly inhomogeneous superfluid A phase of helium-3 is calculated near zero temperature by means of an exact solution of the Dyson-Gorkov equation with linearized order parameter. Two general representations for $\vec j$ are obtained in the form of a series and an integral. The standard representation $\vec j_0$ for...

T(3)-gauge model of defects based on the gauge Lagrangian quadratic in the
gauge field strength is considered. The equilibrium equation of the medium is
fulfilled by the double curl Kroner's ansatz for stresses. The problem of
replication of the static edge dislocation along third axis is analysed under a
special, though conventional, choice of thi...

The Bose-spectrum of the collective excitations in the antiferromagnetic state of 2d weakly repulsive Hubbard model is calculated by means of the path integration both in the Ginzburg-Landau region and near zero temperature.

New representations for the mass current in the A-phase of helium-3 are obtained from an exact solution of the Dyson-Gor'kov equation. Expansions of these representations in powers of a small parameter which characterizes the London limit is considered. A previously known formula for the current is obtained to the main order for zero temperature. T...

The spectrum of collective modes in the antiferromagnetic state of the two-dimensional weakly repulsive Hubbard model is calculated by means of path integration with the use of “fast” and “slow” variables and the Hubbard-Stratonovich transformation both in the Ginzburg-Landau region and near zero temperature.

The three-band two-dimensional repulsive Hubbard model in the formalism of the temperature Green function is investigated.
The applicability of the weak coupling approach at the values of the dimensionless coupling constant U/t≈10–11 is justified.
The normal state, the antiferromagnetic state with Neel magnetic ordering on the “copper” sublattice,...

This paper is devoted to the calculation of superfluid current in the A-phase of helium-3.This calculation is connected with the solution of a non-homogeneous ordinary differential equation arising in the low-temperature limit from the Gor'kov equation for normal and anomalous Green functions. The new solution is obtained in the form of the expansi...

Two new general representations (the series and the integral) for the mass current ~j in weakly inhomogeneous superfluid A-phase of Helium-3 are obtained near zero of temperature by solving the Dyson-Gorkov equation. These representations result in additional correcting contribution to the standard leading expression for ~j which is of first order...

Both the formulae for j (14) and (20), and therefore the Eq. (15) have to be multiplied by two. The estimation (17) has to be corrected as ¢,o~ (y) 1 Iql x(yZ + q2) (I) by equivalence f(s, y) ,,, s exp(-y2s 2) in the integral (16). Multiplying by two and using (I) one obtains that J (the integration over u runs from a to ~) is not zero but (3a2) d...

Two new general representations (the series and the integral) for the mass current $\vj$ in weakly inhomogeneous superfluid A-phase of Helium--3 are obtained near zero of temperature by solving the Dyson-Gorkov equation. These representations result in additional correcting contribution to the standard leading expression for $\vj$ which is of first...

A method is proposed to reduce the Cartan structure equations and the Bianchi identities of the non-Abelian ISO(3)-gauge model of defects in solids to the appropriate relations of the theory of disclinations considered as the Abelian iso(3)-gauge model. As the result, the possibility arises to identify the ISO(3)-gauge potentials in terms of the de...

The investigation of the two-dimensional repulsive Hubbard model is continued for the case in which the Fermi level is close
to one of the saddle Van Hove points of the quasiparticle energy function. The Bethe-Salpeter equation for the two-particle
scattering amplitude and the system of Dyson-Gor'kov equations for normal and anomalous Green functio...

A new representation for the thermal Green's function of 3He-A and its zero the corresponding supercurrent at an arbitrary temperature. Expansion of this general expression in the orbital vector gradients is considered at zero temperature; the widely accepted first-order contribution with the anomalous part and the second-order corrections to it ar...

The three-band Hubbard model, known also as the Emery model, is investigated in the framework of thermal Green functions. It is shown that antiferromagnetic and superconductive states can exist in this model at appropriate doping and sufficiently low temperatures. In the two-dimensional repulsive case under consideration, superconductivity turns ou...

The first part of the paper deals with the model of minimal coupling of an Abelian Chern-Simons gauge field with some matter current. The elements of exterior calculus are used to solve the gauge field equations of motion under transversal, Weyl, and Coulomb gauges. The second part reviews the model proposed by G. Semenoff (a scalar matter field co...

A representation of the algebra g(3)=t(3) ⊕ so(3, ℝ) by differential Schaefer's operators is proposed, and an external algebra of g(3)-valued differential forms is constructed. The requirement of local gauge invariance is formulated in the model of the g(3)-valued field, which enables a group of gauge transformations of the continual theory of defe...

The earlier suggested superconductive state in the two-dimensional repulsive Hubbard model is investigated. Cooper pairing in this state arises due to an effective attraction of fermions in channels with odd angular momenta. It is shown that superconductivity can exist only if one of the Van Hove saddle points of the quasiparticle energy lies near...

The main purpose of the paper is to derive the group of gauge transformations of the theory of disclinations with the help of the Schaefer’s motor calculus [H. Schaefer, Z. Angew. Math. Mech. 47, No. 8, 319-328 (1967; Zbl 0171.233), and ibid. 47, 485-498 (1967; Zbl 0189.271)]. The Lie algebraic aspect of the problem under consideration is specially...

The rod-like defects arise in the process of the oxygen precipitation at the low
temperature annealing of the Czochralski grown silicon crystals. The elasticity theory enables to obtain two continual models of the rod-like defect. The electron-microscopical patterns of the rod-like defects are compared with the computer simulated images correspond...

The electron-microscopic contrast of the rod-like defects is studied. The rod-like defects arise in the oxygen-containing silicon after the low-temperature annealing. The dislocation solutions of the elasticity theory are used to build up the model of the defect, the diffraction patterns are calculated for the case of two-beam dynamical electron di...

A Cooper instability for a weakly interacting 2D repulsive Hubbard model on a square lattice is found at low fermion occupancy. The point is that the previously known results concerning superconductivity under the conditions presented claim the absence of both s-and p-pairings when only nearest neighbors are accounted for. Taking into account next-...

## Projects

Project (1)