On Quantum Systems of Particles with Pair Long-Range Magnetic Interaction in One Dimension

Quantum one-dimensional systems of particles interacting via a (singular) “collective” (depending on all the position vectors of particles) vector electromagnetic potential is considered in the thermodynamic limit .The Gibbs(grand-canonical) reduced density matrices for the Maxwell-Boltzmann statistics are computed in the limit for a pair interaction, generated by a pair magnetic scalar potential ϕ, which is a sum of a short-range, increasing and long-range decreasing potentials. The considered n-particle systems are integrable and have a trivial thermodynamics

Infinite Particle Hamiltonian Dynamics of Chern-Simons Type

In this paper we give an example of the d-dimensional integrable infinite particle Hamiltonian system, originating from the Topological Quantum Field Theory

On Quantum Systems of Particles with Singular Magnetic Interactions in One Dimension. M-B Statistics

Quantum one-dimensional systems of particles interacting via singular “collective” (depending on all the position vectors of particles) vector electromagnetic potential is considered in the thermodynamic limit. The reduced density matrices in the limit are computed for the cases of short-range interaction and one-dimensional analog of Chern-Simons interaction (j-th “collective” vector electromagnetic potential of n particles equals the partial derivative in the position vector of the j-th particle of the Coulomb potential energy of a system of n charged particles)

On the Functional Integrals Associated to a Special Gibbs Systems with Three Body Potentials

The Lagrangian Euclidean Quantum Field Theory of two interacting vector fields is found, which is equivalent to a special Gibbs system with three body potential

Two Order Parameters in Quantum XZ Spin Models with Gibbsian Ground States

We describe a family of quantum spin models which are generators of a discrete Markovian process. We show that that there exists an explicit expression for the ground state of such models and give a simple argument for the existence of two types of long-range order in such systems. Two special examples of these systems are analysed in detail

Three Order Parameters in Quantum XZ Spin-Oscillator Models with Gibbsian Ground States

Quantum models on the hyper-cubic d-dimensional lattice of spin-1/2 particles inter-acting with linear oscillators are shown to have three ferromagnetic ground state order parameters. Two order parameters coincide with the magnetization in the first and third directions and the third one is a magnetization in a continuous oscillator variable. The proofs use a generalized Peierls argument and two Griffiths inequalities. The class of spin-oscillator Hamiltonians considered manifest maximal ordering in their ground states. The models have relevance for hydrogen-bond ferroelectrics. The simplest of these is proven to have a unique Gibbsian ground state

Order Parameters in XXZ-Type Spin 1/2 Quantum Models with Gibbsian Ground States

A class of general spin 1/2 lattice models on hyper-cubic lattice Zd, whose Hamiltonians are sums of two functions depending on the Pauli matrices S¹, S² and S³, respectively, are found, which have Gibbsian eigen (ground) states and two order parameters for two spin components x, z simultaneously for large values of the parameter α playing the role of the inverse temperature. It is shown that the ferromagnetic order in x direction exists for all dimensions d ≥ 1 for a wide class of considered models (a proof is remarkably simple)

Gibbs System of Interacting Scalar Fields and Particles as an Origin of the Sine-Gordon Transformation

The Gibbs System, defined by the formal measure on the Cartesian product of spaces of particle and field variables, is introduced. The Sine-Gordon transformation for the Gibbs systems of particles interacting via many-body potentials is derived with the help of the reduction of the introduced system