Bound states in the one-dimensional two-particle Hubbard model with an impurity

TL;DR

This paper studies bound states in a one-dimensional two-particle Bose-Hubbard model with an impurity, revealing multiple types of bound states, critical interaction strengths, and analytical solutions using Bethe ansatz, with implications for quantum impurity problems.

Contribution

It introduces a detailed analysis of bound states in the 1D two-particle Bose-Hubbard model with impurity, including variational and Bethe ansatz solutions, and identifies novel bound states and symmetry classifications.

Findings

Existence of multiple bound state types depending on interaction parameters.

Identification of critical interaction strengths for bound state formation.

Analytical Bethe ansatz solutions for odd-parity states.

Abstract

We investigate bound states in the one-dimensional two-particle Bose-Hubbard model with an attractive ($V> 0$) impurity potential. This is a one-dimensional, discrete analogy of the hydrogen negative ion H$^-$ problem. There are several different types of bound states in this system, each of which appears in a specific region. For given $V$, there exists a (positive) critical value $U_{c1}$ of $U$, below which the ground state is a bound state. Interestingly, close to the critical value ($U\lesssim U_{c1}$), the ground state can be described by the Chandrasekhar-type variational wave function, which was initially proposed for H$^-$. For $U>U_{c1}$, the ground state is no longer a bound state. However, there exists a second (larger) critical value $U_{c2}$ of $U$, above which a molecule-type bound state is established and stabilized by the repulsion. We have also tried to solve for the eigenstates of the model using the Bethe ansatz. The model possesses a global $\Zz_2$-symmetry (parity) which allows classification of all eigenstates into even and odd ones. It is found that all states with odd-parity have the Bethe form, but none of the states in the even-parity sector. This allows us to identify analytically two odd-parity bound states, which appear in the parameter regions $-2V<U<-V$ and $-V<U<0$, respectively. Remarkably, the latter one can be \textit{embedded} in the continuum spectrum with appropriate parameters. Moreover, in part of these regions, there exists an even-parity bound state accompanying the corresponding odd-parity bound state with almost the same energy.