Quantum simulation of Hubbard model Experimental realization of a long-range antiferromagnet in the Hubbard model with ultracold atoms

The Hubbard model, a lattice model for spin 1/2 fermions with nearest neighbor “hopping”and an on-site repulsion, plays a special role in condensed matter physics. It is the simplest theoretical model describing correlated electronic systems. Originally introduced to describe the effects of screened Coulomb interaction in narrow band materials, the Hubbard model defined on square lattice gains a different level of interest when it was proposed by Anderson[1] that it is the model for the copper-oxide (cuprate) high temperature superconductors[2]. Regardless of the spatial dimension and the geometry of the lattice the properties of the Hubbard model are determined by two dimensionless parameters: (1) U/t: which is proportional to the ratio between the local repulsion energy and the kinetic energy bandwidth, and (2) n: the average number of fermion per lattice site. Despite its simplicity the Hubbard model has escaped solution except in one space dimension[3]. Very recently, in Ref.[4], a nice quantum simulation of a 15 site doped one dimensional Hubbard model (with U/t = 7.25, n . 1) using trapped Li atoms is achieved. Among other things it is demonstrated that the finite temperature spin correlation of the doped system is the same as that of the Heisenberg model on a squeezed lattice, i.e. the lattice obtained by omitting the holes[5]. However unlike the original theory, which addresses the U/t → ∞ and T = 0 limit, the experiment is done for finite U/t and T . In the following I focus on the recommended paper by Mazurenko et al. Here an approximately 80 site square lattice of Li atoms with Hubbard interaction (with U/t ≈ 7.2 and 0 ≤ 1 − n . 0.25) was simulated. (To model the cuprate superconductors n should be close to 1 and U/t should be significantly larger than 1, e.g. ∼ 8.) Although the 2D Hubbard model has not not been solved it is well understood when n = 1. In that case the ground state is an insulator with checkerboard, (π, π), antiferromagnetic long range order, and the low energy excitations are spin waves. The question is how about away from n = 1 (especially for large U/t), does the ground state exhibit superconducting order? It turns out that studying the large U/t 2D Hubbard model away from n = 1 is a very challenging theoretical problem. The only “small parameter” we can identify is |n−1|. However numerical evidence suggests that as n deviates from 1 the nature of ground state changes rather quickly. For example numerical density matrix renormalization group calculation done on a small (6×7) square