2,412 research outputs found
Ansatz of Hans Bethe for a two-dimensional Bose gas
The method of q-oscillator lattices, proposed recently in [hep-th/0509181],
provides the tool for a construction of various integrable models of quantum
mechanics in 2+1 dimensional space-time. In contrast to any one dimensional
quantum chain, its two dimensional generalizations -- quantum lattices -- admit
different geometrical structures. In this paper we consider the q-oscillator
model on a special lattice. The model may be interpreted as a two-dimensional
Bose gas. The most remarkable feature of the model is that it allows the
coordinate Bethe Ansatz: the p-particles' wave function is the sum of plane
waves. Consistency conditions is the set of 2p equations for p one-particle
wave vectors. These "Bethe Ansatz" equations are the main result of this paper.Comment: LaTex2e, 12 page
Explicit Free Parameterization of the Modified Tetrahedron Equation
The Modified Tetrahedron Equation (MTE) with affine Weyl quantum variables at
N-th root of unity is solved by a rational mapping operator which is obtained
from the solution of a linear problem. We show that the solutions can be
parameterized in terms of eight free parameters and sixteen discrete phase
choices, thus providing a broad starting point for the construction of
3-dimensional integrable lattice models. The Fermat curve points parameterizing
the representation of the mapping operator in terms of cyclic functions are
expressed in terms of the independent parameters. An explicit formula for the
density factor of the MTE is derived. For the example N=2 we write the MTE in
full detail. We also discuss a solution of the MTE in terms of bosonic
continuum functions.Comment: 28 pages, 3 figure
Ground states of Heisenberg evolution operator in discrete three-dimensional space-time and quantum discrete BKP equations
In this paper we consider three-dimensional quantum q-oscillator field theory
without spectral parameters. We construct an essentially big set of eigenstates
of evolution with unity eigenvalue of discrete time evolution operator. All
these eigenstates belong to a subspace of total Hilbert space where an action
of evolution operator can be identified with quantized discrete BKP equations
(synonym Miwa equations). The key ingredients of our construction are specific
eigenstates of a single three-dimensional R-matrix. These eigenstates are
boundary states for hidden three-dimensional structures of U_q(B_n^1) and
U_q(D_n^1)$.Comment: 13 page
Quantum 2+1 evolution model
A quantum evolution model in 2+1 discrete space - time, connected with 3D
fundamental map R, is investigated. Map R is derived as a map providing a zero
curvature of a two dimensional lattice system called "the current system". In a
special case of the local Weyl algebra for dynamical variables the map appears
to be canonical one and it corresponds to known operator-valued R-matrix. The
current system is a kind of the linear problem for 2+1 evolution model. A
generating function for the integrals of motion for the evolution is derived
with a help of the current system. The subject of the paper is rather new, and
so the perspectives of further investigations are widely discussed.Comment: LaTeX, 37page
Zamolodchikov's Tetrahedron Equation and Hidden Structure of Quantum Groups
The tetrahedron equation is a three-dimensional generalization of the
Yang-Baxter equation. Its solutions define integrable three-dimensional lattice
models of statistical mechanics and quantum field theory. Their integrability
is not related to the size of the lattice, therefore the same solution of the
tetrahedron equation defines different integrable models for different finite
periodic cubic lattices. Obviously, any such three-dimensional model can be
viewed as a two-dimensional integrable model on a square lattice, where the
additional third dimension is treated as an internal degree of freedom.
Therefore every solution of the tetrahedron equation provides an infinite
sequence of integrable 2d models differing by the size of this "hidden third
dimension". In this paper we construct a new solution of the tetrahedron
equation, which provides in this way the two-dimensional solvable models
related to finite-dimensional highest weight representations for all quantum
affine algebra , where the rank coincides with the size
of the hidden dimension. These models are related with an anisotropic
deformation of the -invariant Heisenberg magnets. They were extensively
studied for a long time, but the hidden 3d structure was hitherto unknown. Our
results lead to a remarkable exact "rank-size" duality relation for the nested
Bethe Ansatz solution for these models. Note also, that the above solution of
the tetrahedron equation arises in the quantization of the "resonant three-wave
scattering" model, which is a well-known integrable classical system in 2+1
dimensions.Comment: v2: references adde
Quantum Geometry of 3-Dimensional Lattices and Tetrahedron Equation
We study geometric consistency relations between angles of 3-dimensional (3D)
circular quadrilateral lattices -- lattices whose faces are planar
quadrilaterals inscribable into a circle. We show that these relations generate
canonical transformations of a remarkable "ultra-local" Poisson bracket algebra
defined on discrete 2D surfaces consisting of circular quadrilaterals.
Quantization of this structure allowed us to obtain new solutions of the
tetrahedron equation (the 3D analog of the Yang-Baxter equation) as well as
reproduce all those that were previously known. These solutions generate an
infinite number of non-trivial solutions of the Yang-Baxter equation and also
define integrable 3D models of statistical mechanics and quantum field theory.
The latter can be thought of as describing quantum fluctuations of lattice
geometry.Comment: Plenary talk at the XVI International Congress on Mathematical
Physics, 3-8 August 2009, Prague, Czech Republi
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