Exact time evolution of the asymmetric Hubbard dimer

TL;DR

This paper provides an exact analytical solution for the time evolution of an asymmetric Hubbard dimer with time-dependent hopping, revealing how symmetries simplify the dynamics under different physical scenarios.

Contribution

It demonstrates how symmetries reduce the complexity of the time evolution in an asymmetric Hubbard dimer, enabling exact solutions for specific cases.

Findings

Time evolution can be exactly solved for certain cases using symmetry analysis.

The Hamiltonian reduces to small blocks (3x3 or 2x2) depending on the physical scenario.

Analytical solutions involve solving a cubic equation for the dynamics.

Abstract

We examine the time evolution of an asymmetric Hubbard dimer, which has a different on-site interaction on the two sites. The Hamiltonian has a time-dependent hopping term, which can be employed to describe an electric field (which creates a Hamiltonian with complex matrix elements), or it can describe a modulation of the lattice (which has real matrix elements). By examining the symmetries under spin and pseudospin, we show that the former case involves at most a 3 x 3 block---it can be mapped onto the time evolution of a time-independent Hamiltonian, so the dynamics can be evaluated analytically and exactly (by solving a nontrivial cubic equation). We also show that the latter case reduces to at most 2 x 2 blocks, and hence the time evolution for a single Trotter step can be determined exactly, but the time evolution generically requires a Trotter product.