Towards Excitations and Dynamical Quantities in Correlated Lattices with Density Matrix Embedding Theory

TL;DR

The paper develops a route to excitations and dynamical quantities within density matrix embedding theory (DMET) by constructing a local excitation basis on top of the DMET ground state and approximating the relevant matrix elements using democratic partitioning. The method yields an effective Hamiltonian in the excitation space, enabling calculation of excitation energies and dynamical functions for the 1D Hubbard model, with benchmarks against Bethe Ansatz and DMRG. Results show that a multi-site (patch) generalized excitation basis captures key features of spinon and spinon–holon physics and reproduces the energy–momentum dispersion and spectral structures, though continua are not fully resolved and artifacts remain due to the single-mode ansatz and overlap truncations. The work demonstrates a computationally efficient framework for dynamical quantities in correlated lattices and outlines clear paths for improvement, such as expanding the excitation space and refining overlap handling.

Abstract

Density matrix embedding theory (DMET) provides a framework to describe ground-state expectation values in strongly correlated systems, but its extension to dynamical quantities is still an open problem. We show one route to obtaining excitations and dynamical spectral functions by using the techniques of DMET to approximate the matrix elements that arise in a single-mode inspired excitation ansatz. We demonstrate this approach in the 1D Hubbard model, comparing the neutral excitations, single-particle density of states, charge, and spin dynamical structure factors to benchmarks from the Bethe ansatz and density matrix renormalization group. Our work highlights the potential of these ideas in building computationally efficient approaches for dynamical quantities.

Figures (15)

• Figure 1: Illustration of single-site and double-site patches (red regions) for a 1D system with periodic boundaries.
• Figure 2: Illustration of a large impurity with the bra patch, ket patch and democratic partitioning index.
• Figure 3: Sorted eigenvalues of the approximate overlap matrix of the excitation basis for a half-filled one-dimensional Hubbard model with $18$ sites, $U = 8$, from $k=\pi / 9$ to $k = \pi$ using two-site patches. (a) Excitation basis is the projector basis; (b) Excitation basis is the generalized excitation basis.
• Figure 4: Comparison of DMET with DMRG excitation energies for the 1-band Hubbard model on 18 sites with periodic boundary conditions. We show the lowest excited state for each crystal momentum $k$. A 4-site patch is used to build an effective Hamiltonian for the excitations. For the DMRG calculation, bond dimension $M=1200$ is used to calculate the first 50 excited states with a discarded weight of $\sim 10^{-8}$.
• Figure 5: The difference betweeen the lowest excitation energy as a function of $k$ from DMET and DMRG for $L=18, U=6$. The DMRG results can be assumed to be exact on the scale of this plot, and we show the DMET results as a function of patch size.