We investigate the spectral properties and the dynamics of doublons in the one-dimensional Hubbard model at infinite temperature. Using a Chebyshev expansion approach formulated in the superfermionic representation, we compute the momentum- and frequency-resolved doublon spectral function across a wide range of interaction strengths
U and in the presence of an external electric field. Increasing the interaction, the spectrum gradually splits into separate bands associated with two-hole and two-particle excitations. Despite the presence of strong correlations, we find that doublons retain their coherence and undergo long-lived Bloch oscillations, as well as rich quantum walk dynamics characterized by light-cone spreadings at strong coupling, which we analyze through the time-and space-resolved Green's function.
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${}^{\ensuremath{-}}$) 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 ($U\ensuremath{\lesssim}{U}_{c1}$), the ground state can be described by the Chandrasekhar-type variational wave function, which was initially proposed for H${}^{\ensuremath{-}}$. 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 $\ensuremath{-}2V<U<\ensuremath{-}V$ and $\ensuremath{-}V<U<0$, 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.
We study the quantum walks of two interacting spin-1 bosons. We derive an exact solution for the time-dependent wave function, which describes the two-particle dynamics governed by the one-dimensional spin-1 Bose-Hubbard model. We show that propagation dynamics in real space and mixing dynamics in spin space are correlated via the spin-dependent interaction in this system. The spin-mixing dynamics has two characteristic frequencies in the limit of large spin-dependent interactions. One of the characteristic frequencies is determined by the energy difference between two bound states, and the other frequency relates to the cotunneling process of a pair of spin-1 bosons. Furthermore, we numerically analyze the growth of the spin correlations in quantum walks. We find that long-range spin correlations emerge showing a clear dependence on the sign of the spin-dependent interaction and the initial state.
This thesis deals with the determination of the eigenstates of the Hamiltonian and quantum dynamics in many-body interacting systems. As an example, the Bose- Hubbard model is used. Methods of generating the basis of system state space and calculation of Hamiltonian matrix in selected representations are studied. After the complete specification of the system, the energy spectrum and the eigenstates are determined. By using the appropriate techniques, the mapping of the vector index and quantum numbers is established for efficient access and manipulation for physical quantities evaluations and simulating dynamics in quantum interacting systems. Furthermore, the expected distribution of particles and momenta are examined, studying the behaviour of characteristic quantities such as energy gaps, the excitation spectra of such a system as a function of the Hamiltonian parameters is determined for a couple of cases. An overview of the formalism of second quantization is given in the paper as well as the derivation of the Bose-Hubbard model in this formalism. Also, the limit of infinitely strong repulsive interactions between the bosons is investigated when the system shows certain similarities with the system of non-interacting fermions onto which it is mapped. In the end, the time evolution of the system of bosons in an optical lattice is investigated in the presence of an external potential in the case of a constant and a time-varying potential.
In this dissertation, I present my work on ground state entanglement in Bose-Hubbard-type models. Bose-Hubbard-type models form a wide class of bosonic lattice models and describe systems that can be realized by cold atoms or molecules trapped in optical lattices. By studying ground state entanglement, one can understand quantum phases and phase transitions among them from a microscopic point of view. I will first introduce the two-component Bose-Hubbard model describing Bose-Bose mixtures, and discuss the role played by the interspecies entanglement in the Mott-insulator-to-superfluid phase transition of one component in the presence of a second component. I will show that interspecies entanglement provides a new perspective to understand quantum phases in mixtures. Then I will discuss long-range spatial entanglement in single-species Bose-Hubbard-type models. Specifically, I will build a generic framework for understanding under which conditions these models harbor certain types of topological order. My results provide guidance for future numerical studies which can pave the way for searching experimentally realizable bosonic lattice models harboring nontrivial topological order.
We study the ground state of the attractive one-dimensional Bose-Hubbard model, and in particular the nature of the crossover between the weak interaction and strong interaction regimes for finite system sizes. Indicator properties such as the gap between the ground and first excited energy levels, and the incremental ground-state wave function overlaps are used to locate different regimes. Using mean-field theory we predict that there are two distinct crossovers connected to spontaneous symmetry breaking of the ground state. The first crossover arises in an analysis valid for large
L with finite
N, where
L is the number of lattice sites and
N is the total particle number. An alternative approach valid for large
N with finite
L yields a second crossover. For small system sizes we numerically investigate the model and observe that there are signatures of both crossovers. We compare with exact results from Bethe ansatz methods in several limiting cases to explore the validity for these numerical and mean-field schemes. The results indicate that for finite attractive systems there are generically three ground-state phases of the model.