69 research outputs found
Exact results for the one-dimensional many-body problem with contact interaction: Including a tunable impurity
The one-dimensional problem of particles with contact interaction in the
presence of a tunable transmitting and reflecting impurity is investigated
along the lines of the coordinate Bethe ansatz. As a result, the system is
shown to be exactly solvable by determining the eigenfunctions and the energy
spectrum. The latter is given by the solutions of the Bethe ansatz equations
which we establish for different boundary conditions in the presence of the
impurity. These impurity Bethe equations contain as special cases well-known
Bethe equations for systems on the half-line. We briefly study them on their
own through the toy-examples of one and two particles. It turns out that the
impurity can be tuned to lift degeneracies in the energies and can create bound
states when it is sufficiently attractive. The example of an impurity sitting
at the center of a box and breaking parity invariance shows that such an
impurity can be used to confine asymmetrically a stationary state. This could
have interesting applications in condensed matter physics.Comment: 20 pages, 5 figures, version accepted for publication: some typos
corrected, references and comments adde
New integrable boundary conditions for the Ablowitz–Ladik model: From Hamiltonian formalism to nonlinear mirror image method
Using Sklyanin's classical theory of integrable boundary conditions, we use the Hamiltonian approach to derive new integrable boundary conditions for the Ablowitz–Ladik model on the finite and half infinite lattice. In the case of half infinite lattice, the special and new emphasis of this paper is to connect directly the Hamiltonian approach, based on the classical r-matrix, with the zero curvature representation and Bäcklund transformation approach that allows one to implement a nonlinear mirror image method and construct explicit solutions. It is shown that for our boundary conditions, which generalise (discrete) Robin boundary conditions, a nontrivial extension of the known mirror image method to what we call time-dependent boundary conditions is needed. A careful discussion of this extension is given and is facilitated by introducing the notion of intrinsic and extrinsic picture for describing boundary conditions. This gives the specific link between Sklyanin's reflection matrices and Bäcklund transformations combined with folding, in the case of non-diagonal reflection matrices. All our results reproduce the known Robin boundary conditions setup as a special case: the diagonal case. Explicit formulas for constructing multisoliton solutions on the half-lattice with our time-dependent boundary conditions are given and some examples are plotted
Symmetries of Spin Calogero Models
We investigate the symmetry algebras of integrable spin Calogero systems constructed from Dunkl operators associated to finite Coxeter groups. Based on two explicit examples, we show that the common view of associating one symmetry algebra to a given Coxeter group W is wrong. More precisely, the symmetry algebra heavily depends on the representation of W on the spins. We prove this by identifying two different symmetry algebras for a BL spin Calogero model and three for G₂ spin Calogero model. They are all related to the half-loop algebra and its twisted versions. Some of the result are extended to any finite Coxeter group
R-matrix presentation for (super)-Yangians Y(g)
We give a unified RTT presentation of (super)-Yangians Y(g) for so(n), sp(2n)
and osp(m|2n).Comment: 9 page
Bethe Equations for a g_2 Model
We prove, using the coordinate Bethe ansatz, the exact solvability of a model
of three particles whose point-like interactions are determined by the root
system of g_2. The statistics of the wavefunction are left unspecified. Using
the properties of the Weyl group, we are also able to find Bethe equations. It
is notable that the method relies on a certain generalized version of the
well-known Yang-Baxter equation. A particular class of non-trivial solutions to
this equation emerges naturally.Comment: 10 pages, 3 figure
The thermal conductivity of alternating spin chains
We study a class of integrable alternating (S1,S2) quantum spin chains with
critical ground state properties. Our main result is the description of the
thermal Drude weight of the one-dimensional alternating spin chain as a
function of temperature. We have identified the thermal current of the model
with alternating spins as one of the conserved currents underlying the
integrability. This allows for the derivation of a finite set of non-linear
integral equations for the thermal conductivity. Numerical solutions to the
integral equations are presented for specific cases of the spins S1 and S2. In
the low-temperature limit a universal picture evolves where the thermal Drude
weight is proportional to temperature T and central charge c.Comment: 15 pages, 1 figur
Analytical Bethe Ansatz for open spin chains with soliton non preserving boundary conditions
We present an ``algebraic treatment'' of the analytical Bethe ansatz for open
spin chains with soliton non preserving (SNP) boundary conditions. For this
purpose, we introduce abstract monodromy and transfer matrices which provide an
algebraic framework for the analytical Bethe ansatz. It allows us to deal with
a generic gl(N) open SNP spin chain possessing on each site an arbitrary
representation. As a result, we obtain the Bethe equations in their full
generality. The classification of finite dimensional irreducible
representations for the twisted Yangians are directly linked to the calculation
of the transfer matrix eigenvalues.Comment: 1
Sutherland Models for Complex Reflection Groups
There are known to be integrable Sutherland models associated to every real
root system -- or, which is almost equivalent, to every real reflection group.
Real reflection groups are special cases of complex reflection groups. In this
paper we associate certain integrable Sutherland models to the classical family
of complex reflection groups. Internal degrees of freedom are introduced,
defining dynamical spin chains, and the freezing limit taken to obtain static
chains of Haldane-Shastry type. By considering the relation of these models to
the usual BC_N case, we are led to systems with both real and complex
reflection groups as symmetries. We demonstrate their integrability by means of
new Dunkl operators, associated to wreath products of dihedral groups.Comment: 26 pages, 8 figures, latex; v3, acknowledgement adde
Integrable Models From Twisted Half Loop Algebras
This paper is devoted to the construction of new integrable quantum
mechanical models based on certain subalgebras of the half loop algebra of
gl(N). Various results about these subalgebras are proven by presenting them in
the notation of the St Petersburg school. These results are then used to
demonstrate the integrability, and find the symmetries, of two types of
physical system: twisted Gaudin magnets, and Calogero-type models of particles
on several half-lines meeting at a point.Comment: 22 pages, 1 figure, Introduction improved, References adde
Thermodynamical limit of general gl(N) spin chains: vacuum state and densities
We study the vacuum state of spin chains where each site carry an arbitrary
representation. We prove that the string hypothesis, usually used to solve the
Bethe ansatz equations, is valid for representations characterized by
rectangular Young tableaux. In these cases, we obtain the density of the center
of the strings for the vacuum. We work out different examples and, in
particular, the spin chains with periodic array of impurities.Comment: Latex file, 27 pages, 5 figures (.eps) A more detailed study of the
representations allowing string hypothesis has added. A simpler formula for
the densities is given. References added and misprint correcte
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