Exact Results for the One-Dimensional Many-Body Problem With Contact Interaction: Including a Tunable Impurity

Abstract

The one-dimensional problem of N 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.

Exact energy spectrum for models with equally spaced point potentials

“…For M = 1, we recover the result given in [11]. To get the spectrum, one has to solve in k the following equations…”

“…Since our construction is based on W N , the Bethe ansatz is consistent if and only if y(k) satisfies the Yang-Baxter equation [5,14] and Z + (k) and y(k) satisfy the reflection equation [15,16] (see, for example [11], for details). These relations hold true by direct computation, implying the exact solvability of the model.…”

“…For instance, in [8,9], the Kronig-Penney model is extended to the case where the δ potential is replaced by a more general point potential. In [10], boundary Bethe ansatz equations were derived by putting bosons with δ interactions in a box while in [11], impurity Bethe ansatz equations appeared by including a general external contact potential in a system of particles with δ interactions.…”

Many-particle quantum graphs and Bose-Einstein condensation

“…In a graph language, the interval ends are vertices of degree one. Adding a vertex of degree two in the context of a Bethe ansatz was first done in [CC07], where is was found that this is incompatible with δ-pair interactions. Instead, the interactions were modified to include another contribution that formally looks like δ(x 1 + x 2 ).…”

“…An extension of this method to arbitrary metric graphs with generalisations of the interactions introduced in [CC07] has been done in [BG17a], and an extension to N particles can be found in [BG17b]. The first step is to define the singular pair interactions, and this is most clearly done on a star graph of d half-lines.…”