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Exact and variational investigations of Hubbard rings

William Hodge

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

We study the one-dimensional Hubbard model for interacting electrons in a crystal lattice, using two methods: solution of the Lieb-Wu equations in the atomic limit of no intersite hopping, and variational techniques to obtain a lower bound for the ground state energy. From the Lieb-Wu equations we derive an exact expression for the ground state energy in the atomic limit. We obtain our result by exploiting the spin symmetry of the Hubbard Hamiltonian. Our energy is lower than that obtained by previous published results, and we are able to explain why our results differ. Our method also gives information about the magnetic ordering of the ground state as a function of the number of electrons. Our results are valid for arbitrary electron density values. The variational techniques we use are applied to the two-particle reduced-density matrix (2-RDM), because this quantity suffices to obtain expectation values of all one- and two-particle operators, and because it is less computationally intensive than obtaining the complete N -particle wave function. Because the 2-RDM must be positive semidefinite, we use an efficient programming algorithm known as the Semidefinite Programming Algorithm. A naive search through the space of two-particle matrices encounters a difficult problem, known as the " N -representability problem", which is to assure that the trial density matrices actually come from a wave function that has the required exchange symmetry. This problem is ameliorated by imposing constraints on the variational search that eliminate trial matrices with the incorrect exchange symmetry. These constraints express known physical properties of the ground state (for example its total spin and its translational symmetry) and the so-called D, Q , and G two-positivity conditions that express the required exchange symmetry. To calculate matrix elements, we use the two-particle eigenstates of the reduced Hubbard Hamiltonian; this choice of basis significantly decreases the number of matrix elements needed for energy calculations. After an optimal two-particle density matrix has been obtained, we calculate additional properties of the ground state, including spin correlation functions, chemical potentials, and the momentum distribution. We compare all of these properties to values obtained by exact numerical solution of the Schrödinger equation.

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We study the one-dimensional Hubbard model for interacting electrons in a crystal lattice, using two methods: solution of the Lieb-Wu equations in the atomic limit of no intersite hopping, and variational techniques to obtain a lower bound for the ground state energy. From the Lieb-Wu equations we derive an exact expression for the ground state energy in the atomic limit. We obtain our result by exploiting the spin symmetry of the Hubbard Hamiltonian. Our energy is lower than that obtained by previous published results, and we are able to explain why our results differ. Our method also gives information about the magnetic ordering of the ground state as a function of the number of electrons. Our results are valid for arbitrary electron density values. The variational techniques we use are applied to the two-particle reduced-density matrix (2-RDM), because this quantity suffices to obtain expectation values of all one- and two-particle operators, and because it is less computationally intensive than obtaining the complete N -particle wave function. Because the 2-RDM must be positive semidefinite, we use an efficient programming algorithm known as the Semidefinite Programming Algorithm. A naive search through the space of two-particle matrices encounters a difficult problem, known as the " N -representability problem", which is to assure that the trial density matrices actually come from a wave function that has the required exchange symmetry. This problem is ameliorated by imposing constraints on the variational search that eliminate trial matrices with the incorrect exchange symmetry. These constraints express known physical properties of the ground state (for example its total spin and its translational symmetry) and the so-called D, Q , and G two-positivity conditions that express the required exchange symmetry. To calculate matrix elements, we use the two-particle eigenstates of the reduced Hubbard Hamiltonian; this choice of basis significantly decreases the number of matrix elements needed for energy calculations. After an optimal two-particle density matrix has been obtained, we calculate additional properties of the ground state, including spin correlation functions, chemical potentials, and the momentum distribution. We compare all of these properties to values obtained by exact numerical solution of the Schrödinger equation.
Variational reduced-density-matrix theory is applied to calculating the ground-state energy and two-electron reduced density matrices (2-RDMs) of the one-dimensional Hubbard model for a range of interaction strengths. The 2-RDM is constrained to represent an NNN-particle wave function by two sets of NNN-representability conditions, known as the 2- and partial 3-positivity conditions. Variational optimization of the energy with the 2-RDM constrained by NNN-representability conditions is performed using a first-order semidefinite-programming algorithm that was developed for treating atoms and molecules [D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. Accurate energies for a broad range of interaction strengths indicate that the variational 2-RDM method is a valuable tool for studying strongly correlated electrons.
In this thesis we develop the variational two-electron reduced-density-matrix method for extended systems. Extended systems are represented in two ways: i) lattice models describing the dominant valence electronic structure with periodic boundaries to account for their extended nature and ii) a crystalline-orbital basis built from atomic orbitals using the generalization of molecular orbital theory to polymers. The first part of this thesis (Ch. 3-4) examines the performance of the variational 2-RDM method on lattice systems with tunable electron correlation. The first of these systems is the classic Hubbard model with linear and ladder lattice topologies. Because electron correlation functions, such as charge- and spin-ordering, are linear functions of the 2-RDM, the difference in electronic structure between one- and quasi-one-dimensional systems is accurately characterized. The second model contains only two-body interactions and is unique among typical spin models in that it does not have a mean-field reference wave function. The ground state wave functions from all Hamiltonians in the model have the same 1-electron reduced density matrix; consequently, one-electron theories are largely inapplicable. The superconducting eta-pairing ground states make the model a unique tool for demonstrating the necessary N-representability in highly correlated environments. The second part of this thesis (Ch. 5-6) develops a formalism for modeling materials by solving the full Schrodinger equation. Crystalline-orbital Hartree-Fock provides a set of orbitals and integral tensors for the variational 2-RDM method. We demonstrate that time-reversal symmetry, which is implicitly included in position space electronic structure calculations, must be explicitly included as an N-representability constraint on the 2-RDM when using a momentum space basis. The necessity of these equality constraints is demonstrated by the accurate recovery of the binding energy of two polymers and the symmetry of their natural orbital occupations.
The reduced density matrix (RDM) method, which is a variational calculation based on the second-order reduced density matrix, is applied to the ground state energies and the dipole moments for 57 different states of atoms, molecules, and to the ground state energies and the elements of 2-RDM for the Hubbard model. We explore the well-known N-representability conditions (P, Q, and G) together with the more recent and much stronger T1 and T2(') conditions. T2(') condition was recently rederived and it implies T2 condition. Using these N-representability conditions, we can usually calculate correlation energies in percentage ranging from 100% to 101%, whose accuracy is similar to CCSD(T) and even better for high spin states or anion systems where CCSD(T) fails. Highly accurate calculations are carried out by handling equality constraints and/or developing multiple precision arithmetic in the semidefinite programming (SDP) solver. Results show that handling equality constraints correctly improves the accuracy from 0.1 to 0.6 mhartree. Additionally, improvements by replacing T2 condition with T2(') condition are typically of 0.1-0.5 mhartree. The newly developed multiple precision arithmetic version of SDP solver calculates extraordinary accurate energies for the one dimensional Hubbard model and Be atom. It gives at least 16 significant digits for energies, where double precision calculations gives only two to eight digits. It also provides physically meaningful results for the Hubbard model in the high correlation limit.
In this Thesis, we develop and apply an accurate method to determine the ground-state properties of strongly-correlated electrons in the framework of lattice model Hamiltonians. In lattice density-functional theory (LDFT) the basic variable is the one-particle density matrix γ. From a generalized HK theorem, the ground state energy Egs[γgs] = minγ E[γ] is obtained by minimizing the energy-functional E[γ] over all the physical or representable γ. The energy functional can be divided into two contributions: the kinetic-energy functional T [γ], which linear dependence on γ is exactly known, and the correlation-energy functional W [γ], which dependence on γ is not explicitly known. Finding accurate approximations for W [γ] constitutes the actual challenge of this thesis. Part of this work is based on previous studies which derive an approximation of W [γ] for the Hubbard Hamiltonian based on scaling hypothesis and an exact analytical results for the Hubbard dimer. However, this approach is limited to spin independent and homogeneous systems. In order to extend the scope of LDFT, we have developed three different approaches to derive W [γ] in order to study broken symmetry problems. First we extend the previous scaling functional to system having charge transfer. By a systematic study of the dependence of W [γ] on the charge distribution we find similar scaling properties as for the homogeneous case. An extension to the Hubbard model on bipartite lattices is then derived and applied to finite and infinite systems with repulsive and attractive interactions. The high accuracy of this functional is demonstrated. Nevertheless, this approach is difficult to transfer to more complex systems, since the calculation of W [γ] applies to the system as a whole. To overcome this problem, we derive another approximation based on local scaling properties. This functional is formulated on-site so that it can be applied to any ordered or disordered Hamiltonian with on-site interaction. As applications we study the metal-insulator transition in the ionic Hubbard model in oneand two-dimensions, as well as the one-dimensional Hubbard chain with first and second nearest neighbor hoppings. Finally, we develop a numerical approach of W [γ], on the basis on exact diagonalizations of an effective many-body Hamiltonian corresponding to a cluster surrounded by an effective medium. This effective Hamiltonian depends on the density matrix γ and allows to derive approximation to W [γ] which improve systematically with increasing cluster size. The formulation is spin-dependent and allows a straightforward generalization to multi-orbital correlated systems such spd-Hamiltonians. In addition it take into account the effects of short-range charge and spin fluctuations in the functional. The accuracy of the method is demonstrated for the Hubbard model by comparison with the Bethe-Ansatz solution (1D) and quantum Monte Carlo simulations (2D). Finally, an outlook or relevant future developments of the theory is provided.
The energy and properties of a many-electron atom or molecule may be directly computed from a variational optimization of a two-electron reduced density matrix (2RDM) that is constrained to represent many-electron quantum systems. In this paper we implement a variational 2RDM method with a representability constraint, known as the T2{T}_{2}T2​ condition. The optimization of the 2RDM is performed with a first-order algorithm for semidefinite programming [D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)] which, because of its lower computational cost in comparison to second-order methods, allows the treatment of larger basis sets. We also derive and implement a spin- and symmetry-adapted formulation of the T2{T}_{2}T2​ condition that significantly decreases the size of the largest block in the T2{T}_{2}T2​ matrix. The T2{T}_{2}T2​ condition, originally derived by Erdahl [Int. J. Quantum Chem. 13, 697 (1978)], was recently applied via a second-order algorithm to atoms and molecules [Z. Zhao et al., J. Chem. Phys. 120, 2095 (2004)]. While these calculations were restricted to molecules at equilibrium geometries in minimal basis sets, we apply the 2RDM method with the T2{T}_{2}T2​ condition to compute the electronic energies of molecules in both minimal and nonminimal basis sets at equilibrium as well as nonequilibrium geometries. Accurate potential energies curves are produced for BH, HF, and N2{\mathrm{N}}_{2}N2​. Results are compared with the 2RDM method without the T2{T}_{2}T2​ condition as well as several wave-function methods.
Using a straightforward extension of the analysis of Lieb and Wu, we derive a simple analytic form for the ground state energy of a one-dimensional Hubbard ring in the atomic limit. This result is valid for an \textit{arbitrary} number of lattice sites LLL and electrons N≤LN \leq LN≤L. Furthermore, our analysis, including an application of the theory of stochastic matrices, provides insight into the degeneracy and spin properties of the ground states in the atomic limit. We give numerical results which illustrate how the atomic limit is approached.