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理论物理通讯：英文版

Exact Numerical Solutions of Bose-Hubbard Model

ZHANGDan, PANFeng

Published Jan 1, 2004• 0 citations
OpenAlex

Abstract

Hamiltonian of a one-dimensional Bose Hubbard model is re-formulated by using differential realization of the boson algebra.Energy matrices can then be generated systematically by using a Mathematica package.The output can be taken as the input of other diagonalization codes.As examples exact energy cigenvalues and the corresponding wavefunctions for some cases are obtained with a Fortran diagonalization code.Phase transition of the model is analyzed.

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Hamiltonian of a one-dimensional Bose Hubbard model is re-formulated by using differential realization of the boson algebra.Energy matrices can then be generated systematically by using a Mathematica package.The output can be taken as the input of other diagonalization codes.As examples exact energy cigenvalues and the corresponding wavefunctions for some cases are obtained with a Fortran diagonalization code.Phase transition of the model is analyzed.
We take the Bose-Hubbard model to illustrate exact diagonalization techniques in a pedagogical way. We follow the road of first generating all the basis vectors, then setting up the Hamiltonian matrix with respect to this basis, and finally using the Lanczos algorithm to solve low lying eigenstates and eigenvalues. Emphasis is placed on how to enumerate all the basis vectors and how to use the hashing trick to set up the Hamiltonian matrix or matrices corresponding to other quantities. Although our route is not necessarily the most efficient one in practice, the techniques and ideas introduced are quite general and may find use in many other problems.
This paper explores the numerical study of quantum many-body systems with an emphasis on exact diagonalization techniques. The complexity of strongly correlated systems, often governed by large Hilbert spaces, presents significant computational challenges, making exact solutions difficult. In this work, we examine methods to simplify these systems by leveraging techniques such as the Schrieffer-Wolff transformation, which decouples high-energy and low-energy states, and the use of symmetry operators to block-diagonalize Hamiltonians and so on. These approaches are demonstrated with examples such as the hydrogen atom and a lambda system. The second part of the paper focuses on specific case studies, including a one-dimensional spin model and Bose-Hubbard model. The latter is crucial for understanding the behavior of interacting bosons in lattice systems and phenomena such as the superfluid-Mott insulator transition. We present a detailed investigation of the phase diagram for the one-dimensional Bose-Hubbard model using both exact diagonalization and mean field theory, providing insights into its quantum phase transitions. This study underscores the potential of exact diagonalization in quantum simulations and highlights its relevance for experimental setups involving trapped ions and superconducting qubits.
Abstract The Bose–Hubbard model is a paradigm for the study of strongly correlated bosonic systems. We study the two-site Bose–Hubbard model with generic integrable open boundaries specified by the most general non-diagonal reflecting matrices. Besides the inhomogeneous parameters, the model itself has three free boundary parameters, which break the U(1)-symmetry, in other words, break the particle number conservation. The Hamiltonian H under these circumstances is constructed. With the help of the off-diagonal Bethe Ansatz method, we successfully obtain the corresponding Bethe Ansatz equations as well as the eigenvalues.
The Bose-Hubbard dimer Hamiltonian is a simple yet effective model for describing tunneling phenomena of Bose-Einstein condensates.One of the significant mathematical properties of the model is that it can be exactly solved by Bethe ansatz methods.Here we review the known exact solutions, highlighting the contributions of V.B. Kuznetsov to this field.Two of the exact solutions arise in the context of the Quantum Inverse Scattering Method, while the third solution uses a differential operator realisation of the su(2) Lie algebra.
We present a deep neural network (DNN) -based model (HubbardNet) to variationally find the ground-state and excited-state wave functions of the one-dimensional and two-dimensional Bose-Hubbard model. Using this model for a square lattice with MMM sites, we obtain the energy spectrum as an analytical function of the on-site Coulomb repulsion, UUU, and the total number of particles, NNN, from a single training. This approach bypasses the need to solve a new Hamiltonian for each different set of values (U,N)(U,N)(U,N) and generalizes well even for out-of-distribution (U,N)(U,N)(U,N). Using HubbardNet, we identify the two ground-state phases of the Bose-Hubbard model (Mott insulator and superfluid). We show that the DNN-parametrized solutions are in excellent agreement with results from the exact diagonalization of the Hamiltonian, and it outperforms exact diagonalization in terms of computational scaling. These advantages suggest that our model is promising for efficient and accurate computation of exact phase diagrams of many-body lattice Hamiltonians.