Graph-theory treatment of one-dimensional strongly repulsive fermions

Abstract

One-dimensional atomic mixtures of fermions can effectively realize spin chains and thus constitute a clean and controllable platform to study quantum magnetism. Such strongly correlated quantum systems are also of sustained interest to quantum simulation and quantum computation due to their computational complexity. In this article, we exploit spectral graph theory to completely characterize the symmetry properties of one-dimensional fermionic mixtures in the strong interaction limit. We also develop a powerful method to obtain the so-called Tan contacts associated with certain symmetry classes. In particular, compared to brute-force diagonalization that is already virtually impossible for a moderate number of fermions, our analysis enables us to make unprecedented efficient predictions about the energy gap of complex spin mixtures. Our theoretical results are not only of direct experimental interest but also provide important guidance for the design of adiabatic control protocols in strongly correlated fermion mixtures.

Universal graph description for one-dimensional exchange models

“…Here, the ground-state energy is −ρ( ν ) for the antiferromagnetic case J k < 0, and we can reach a conclusion similar to Ref. [21]. Nevertheless, the situation is different for the ferromagnetic case J k > 0, where ρ( ν ) corresponds to the maximal energy of the mixture ν, which is of course less physically interesting.…”

“…The proof of this statement is exactly the same as the one we described in Ref. [21]: First we note that the graph X (S ν ⊂ S N ,S C ) is bipartite and can be separated into even and odd spin configurations |χ , then we use this fact to construct a vector that belongs to the symmetry class [ν] and show that it belongs to the eigenspace with eigenvalue ρ( ν ). Note that the fact that k νν = 1 in Eq.…”

“…We also remark that in Ref. [21] the fact that the Laplacian spectrum has a negative contribution to the energy implies that one can only use Eq. (17) in the case N = κ where Eq.…”

“…with b = 1 in the OBC case and b = 2 in the PBC case. Equation (21) has been checked for several examples, such as the ones provided in Fig. 2.…”

“…In Ref. [21], we have shown that strongly repulsive SU (κ ) mixtures of ultracold fermions confined in 1D continuous potentials can also be efficiently described using the X (S ν ⊂ S N ,S C ) graphs. This is due to a mapping between this model and a discrete spin chain [the OBC version of H, Eq.…”