Truncated configuration interaction expansions as solvers for correlated quantum impurity models and dynamical mean-field theory

Abstract

The development of polynomial cost solvers for correlated quantum impurity models, with controllable errors, is a central challenge in quantum many-body physics, where these models find applications ranging from nano-science to the dynamical mean-field theory (DMFT). Here we describe how configuration interaction (CI) approximations to exact diagonalization (ED) may be used as solvers in DMFT. CI approximations retain the main advantages of ED, such as the ability to treat general interactions and off-diagonal hybridizations and to obtain real spectral information, but are of polynomial cost. Furthermore, their errors can be controlled by monitoring the convergence of physical quantities as a function of the CI hierarchy. Using benchmark DMFT applications, such as single-site DMFT of the 1D Hubbard model and 2 × 2 cluster DMFT of the 2D Hubbard model, we show that CI approximations allow us to obtain near-exact ED results for a tiny fraction of the cost. This is true over the entire range of interaction strengths including "difficult" regimes, such as in the pseudogap phase of the 2D Hubbard model. We use the ability of CI approximations to treat large numbers of orbitals to demonstrate convergence of the bath representation in the 2 × 2 cluster DMFT using a 24 bath orbital representation. CI approximations thus form a promising route to extend ED to problems that are currently difficult to study using other solvers such as continuous-time quantum Monte Carlo, including impurity models with large numbers of orbitals and general interactions.

Efficient Temperature-Dependent Green’s Function Methods for Realistic Systems: Using Cubic Spline Interpolation to Approximate Matsubara Green’s Functions

“…It was shown in ref 45 that the coefficients of high-frequency expansion in a non-orthogonal orbital basis for Hamiltonians with full Coulomb interaction are given by In a typical calculation, see Fig. 1, the self-energy Σ is evaluated either on the Matsubara frequency or imaginary time grid by a variety of solvers ranging from quantum Monte Carlo methods [7,[46][47][48] to perturbative [49][50][51][52][53][54] and configuration interaction type of methods [55,56]. In the next step, a Green's function is calculated by means of the Dyson equation…”

Multireference symmetry-projected variational approaches for ground and excited states of the one-dimensional Hubbard model

“…10 Among such approximations, we have the quantum Monte Carlo, 11,12 the variational Monte Carlo, 13 the density matrix renormalization group [14][15][16][17] as well as approximations based on matrix product and tensor network states. 18 Both the dynamical mean field theory and its cluster extensions [19][20][21][22][23][24][25][26] have made important contributions to our present knowledge of the Hubbard model. Other embedding approaches are also available.…”

Hybrid Quantum-Classical Approach to Correlated Materials

“…This and the need to obtain the full Green's function means that practical calculations are limited to N so þ N d ≈ 25, in other words, to five or fewer correlated orbitals, often corresponding to just a single correlated atom within a unit cell, and with a very small number of bath sites per correlated orbital. Recent developments [30][31][32] based on quantum chemical methods to define reduced basis sets for the exact diagonalization calculation permit inclusion of somewhat larger numbers of bath orbitals, but, at least as presently formulated, these methods work in a natural orbital basis that strongly mixes the bath and correlated orbitals, so that the sparsity structure mentioned above cannot be exploited. In a parallel development, ideas to solve the impurity problem using tensor networks [33] have recently started to show great promise [34][35][36].…”