Approximate solution of variational wavefunctions for strongly correlated systems: description of a correlated insulator

Balázs Hetényi

An approximate solution scheme, similar to the Gutzwiller approximation, is presented for the Baeriswyl and the Baeriswyl-Gutzwiller variational wave functions. The phase diagram of the one-dimensional Hubbard model as a function of interaction strength and particle density is determined. For the Baeriswyl wave function a metal-insulator transition is found at half filling, where the metallic phase $(U<{U}_{c})$ corresponds to the Hartree-Fock solution, the insulating phase is one with finite double occupations arising from bound excitons. This transition can be viewed as the ``inverse'' of the Brinkman-Rice transition. Close to but away from half filling, the $U>{U}_{c}$ phase displays a finite Fermi step, as well as double occupations originating from bound excitons. As the filling is changed away from half-filling bound excitons are suppressed. For the Baeriswyl-Gutzwiller wave function at half filling a metal-insulator transition between the correlated metallic and excitonic insulating state is found. Away from half-filling bound excitons are suppressed quicker than for the Baeriswyl wave function.

Balázs Hetényi, Hans Gerd Evertz, Xiancong Lu +1 more

Chao Xu, Yixin Ma, Shenghan Jiang

Abstract The study of topological band insulators has revealed fascinating phases characterized by band topology indices and anomalous boundary modes protected by global symmetries. In strongly correlated systems, where the traditional notion of electronic bands becomes obsolete, it has been established that topological insulator phases persist as stable phases, separate from the trivial insulators. However, due to the inability to express the ground states of such systems as Slater determinants, the formulation of generic variational wave functions for numerical simulations is highly desirable. In this paper, we tackle this challenge for two-dimensional topological insulators by developing a comprehensive framework for fermionic tensor network states. Starting from simple assumptions, we obtain possible sets of tensor equations for any given symmetry group, capturing consistent relations governing symmetry transformation rules on tensor legs. We then examine the connection between these tensor equations and non-chiral topological insulators by constructing edge theories and extracting quantum anomaly data from each set of tensor equations. By exhaustively exploring all possible sets of equations, we achieve a systematic classification of non-chiral topological insulator phases. Imposing the solutions of a given set of equations onto local tensors, we obtain generic variational wavefunctions for the corresponding topological insulator phases. Our methodology provides an important step toward simulating topological insulators in strongly correlated systems. We discuss the limitations and potential generalizations of our results, paving the way for further advancements in this field.

Franjo Franjic

P. Fazekas

Paul A. Johnson

Standard approaches to approximating solutions to the Schroedinger equation rely upon the assumption that there is a single dominant electronic configuration with a small number of low-lying excited states. As such, they are referred to as single reference approaches and are accurate only when the band gap is small in comparison with the inter-electronic repulsion of the valence electrons. Systems for which this is the case are weakly-correlated. If the inter-electronic repulsion larger than the band gap, then a single electronic reference is not an appropriate ansatz for the physical wavefunction. As the strength of the interaction is increased many Slater determinants become near-degenerate, each providing non-negligible contributions. Such systems are strongly-correlated. The goal of the thesis was to employ wavefunctions which are: non-trivial, tractable, size-consistent and extensive, and not given in an active space. We have outlined approaches based on wavefunction forms more suited to describe strong correlation. Built on the Lie algebra su(2), our first trial wavefunction was the Bethe ansatz solution to the reduced BCS Hamiltonian derived by Richardson. This wavefunction is expressed as a product of quasi-particles each parametrized in terms of quasi-momenta. Consistency among the quasi-momenta is ensured by a set of non-linear equations to be solved numerically. This approach is feasible in both variational and projected flavours. We next considered a wavefunction of the same form without requiring it to be an eigenvector of the reduced BCS Hamiltonian. A variational approach in this case is only feasible provided there are two equivalent descriptions in terms of particles and holes. In any case, a bivariational principle and a projected approach are both feasible. Also based on su(2), there is AP1roG (coupled cluster pair doubles). Open-shell singlets and triplets were constructed with the Lie algebras sp(N) and so(4N). The resulting formulae appear to be infeasible unless it is possible to efficiently compute generalizations of determinants to three-dimensional objects. Treating each spatial orbital individually, all four possible occupations close the Lie superalgebra gl(2|2).