Bound States in the Continuum Realized in the One-Dimensional Two-Particle Hubbard Model with an Impurity (original) (raw)

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

Physical Review A

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 Uc1 of U (the on-site atom-atom interaction), below which the ground state is a bound state. Interestingly, close to the critical value (U≲Uc1), the ground state can be described by the Chandrasekhar-type variational wave function, which was initially proposed for H−. For U>Uc1, the ground state is no longer a bound state. However, there exists a second (larger) critical value Uc2 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 Z2 symmetry (parity) which allows classification of all eigenstates into even and odd states. 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 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.

Bound States in the One-dimensional Hubbard Model

1998

The Bethe Ansatz equations for the one-dimensional Hubbard model are reexamined. A new procedure is introduced to properly include bound states. The corrected equations lead to new elementary excitations away from half-filling.

Physical picture of the gapped excitation spectrum of the one-dimensional Hubbard model

Nuclear Physics B, 1999

A simple picture for the spectrum of the one-dimensional Hubbard model is presented using a classification of the eigenstates based on an intuitive bound-state Bethe-Ansatz approach. This approach allows us to prove a "string hypothesis" for complex momenta and derive an exact formulation of the Bethe-Ansatz equations including all states. Among other things we show that all gapped eigenstates have the Bethe Ansatz form, contrary to assertions in the literature 1. The simplest excitations in the upper Hubbard band are computed: we find an unusual dispersion close to half-filling.

Two-particle States in One-dimensional Coupled Bose-Hubbard Models

2022

We study dynamically coupled one-dimensional Bose-Hubbard models and solve for the wave functions and energies of two-particle eigenstates. Even though the wave functions do not directly follow the form of a Bethe Ansatz, we describe an intuitive construction to express them as combinations of Choy-Haldane states for models with intra- and inter-species interaction. We find that the two-particle spectrum of the system with generic interactions comprises in general four different continua and three doublon dispersions. The existence of doublons depends on the coupling strength ΩΩΩ between two species of bosons, and their energies vary with ΩΩΩ and interaction strengths. We give details on one specific limit, i.e., with infinite interaction, and derive the spectrum for all types of two-particle states and their spatial and entanglement properties. We demonstrate the difference in time evolution under different coupling strengths, and examine the relation between the long-time behavior...

Scattering resonances and two-particle bound states of the extended Hubbard model

Journal of Physics B: Atomic, Molecular and Optical Physics, 2009

We present a complete derivation of two-particle states of the one-dimensional extended Bose-Hubbard model involving attractive or repulsive on-site and nearest-neighbour interactions. We find that this system possesses scattering resonances and two families of energy-dependent interactionbound states which are not present in the Hubbard model with the on-site interaction alone.

Two-particle states in the Hubbard model

Journal of Physics B: Atomic, Molecular and Optical Physics, 2008

We consider a pair of bosonic particles in a one-dimensional tight-binding periodic potential described by the Hubbard model with attractive or repulsive on-site interaction. We derive explicit analytic expressions for the two-particle states, which can be classified as (i) scattering states of asymptotically free particles, and (ii) interaction-bound dimer states. Our results provide a very transparent framework to understand the properties of interacting pairs of particles in a lattice.

Comment on the paper ``Bound States in the One-dimensional Hubbard Model

1998

We comment on the preprint cond-mat/9805103 by D. Braak and N. Andrei . We point out that the "new" Bethe Ansatz equations presented in [1] are identical to the Bethe equations for strings introduced by M. Takahashi for the description of thermodynamics in 1972 . Some physics suggested in [1] is incorrect. In particular, all former conclusions made on the basis of the string Bethe equations remain valid.

Entanglement spectrum of one-dimensional extended Bose-Hubbard models

Physical Review B, 2011

The entanglement spectrum provides crucial information about correlated quantum systems. We show that the study of the block-like nature of the reduced density matrix in number sectors and the partition dependence of the spectrum in finite systems leads to interesting unexpected insights, which we illustrate for the case of a 1D extended Hubbard model. We show that block symmetry provides an intuitive understanding of the spectral double degeneracy of the Haldane-insulator, which is remarkably maintained at low on-site interaction, where triple or higher site occupation is significant and particle-hole symmetry is broken. Moreover, surprisingly, the partition dependence of the spectral degeneracy in the Haldane-isulator, and of a partial degeneracy in the Mott-insulator, are directly linked to the, in principle unrelated, density-density correlations, and presents an intriguing periodic behavior in superfluid and supersolid phases.

One-dimensional extended Bose–Hubbard model with a confining potential: a DMRG analysis

Journal of Physics B: Atomic, Molecular and Optical Physics, 2006

The extended Bose-Hubbard model in a quadratic trap potential is studied using a finitesize density-matrix renormalization group method (DMRG). We compute the boson density profiles, the local compressibility and the hopping correlation functions. We observed the phase separation induced by the trap in all the quantities studied and conclude that the local density approximation is valid in the extended Bose-Hubbard model. From the plateaus obtained in the local compressibility it was possible to obtain the phase diagram of the homogeneous system which is in agreement with previous results.

Solvable model of bound states in the continuum (BIC) in one dimension

Physica Scripta

Historically, most of the quantum mechanical results have originated in one dimensional model potentials. However, Von-Neumann's Bound states in the Continuum (BIC) originated in specially constructed, three dimensional, oscillatory, central potentials. One dimensional version of BIC has long been attempted, where only quasi-exactly-solvable models have succeeded but not without instigating degeneracy in one dimension. Here, we present an exactly solvable bottomless exponential potential barrier V (x) = −V 0 [exp(2|x|/a) − 1] which for E < V 0 has a continuum of non-square-integrable, definite-parity, degenerate states. In this continuum, we show a surprising presence of discrete energy, square-integrable, definite-parity, non-degenerate states. For E > V 0 , there is again a continuum of complex scattering solutions ψ(x) whose real and imaginary parts though solutions of Schrödinger equation yet their parities cannot be ascertained as Cψ(x) is also a solution where C is an arbitrary complex non-real number.