Ground state spin and excitation energies in half-filled Lieb lattices

TL;DR

This paper investigates the ground state spin configurations and excitation energies in small half-filled Lieb lattices, revealing how long-range interactions influence magnetic states and energy spectra through detailed analytical and numerical methods.

Contribution

It provides new insights into the effects of long-range interactions on ground state spins in Lieb lattices, extending understanding beyond the Lieb theorem.

Findings

Lieb state is always lower in energy than Hund state for small lattices.

Long-range interactions can induce a transition to minimal spin ground states.

Analytical results agree with numerical exact diagonalization and mean-field methods.

Abstract

We present detailed spectral calculations for small Lieb lattices having up to $N=4$ number of cells, in the regime of half-filling, an instance of particular relevance for the nano-magnetism of discrete systems such as quantum dot arrays, due to the degenerate levels at mid-spectrum. While for the Hubbard interaction model -and even number of sites- the ground state spin is given by the Lieb theorem, the inclusion of long range interaction -or odd number of sites- make the spin state not a priori known, which justifies our approach. We calculate also the excitation energies, which are of experimental importance, and find significant variation induced by the interaction potential. One obtains insights on the mechanisms involved that impose as ground state the Lieb state with lower spin rather than the Hund one with maximum spin for the degenerate levels, showing this in the first and second order of the interaction potential for the smaller lattices. The analytical results concorde with the numerical ones, which are performed by exact diagonalization calculations or by a combined mean-field and configuration interaction method. While the Lieb state is always lower in energy than the Hund state, for strong long-range interaction, when possible, another minimal spin state is imposed as ground state.