Ground states of fermionic lattice Hamiltonians with permutation symmetry

We study the ground states of lattice Hamiltonians that are invariant under permutations, in the limit where the number of lattice sites $N\to \infty$. For spin systems, these are product states, a fact that follows directly from the quantum de Finetti theorem. For fermionic systems, however, the problem is very different, since mode operators acting on different sites do not commute, but anticommute. We construct a family of fermionic states, $\mathcal{F}$, from which such ground states can be easily computed. They are characterized by few parameters whose number only depends on $M$, the number of modes per lattice site. We also give an explicit construction for $M=1,2$. In the first case, $\mathcal{F}$ is contained in the set of Gaussian states, whereas in the second it is not. Inspired by that construction, we build a set of fermionic variational wave functions, and apply it to the Fermi-Hubbard model in two spatial dimensions, obtaining results that go beyond the generalized Hartree-Fock theory.

Figure 1

Example of the operators ${\overline{A}}_{\vec{\mu }}^{\left(k\right)}$, Eq. (29), for $N=4$ and $M=3$. The gray balls represent the operators $a_{n,\mu }^{†}$ on site $n$ in mode $\mu$.

Figure 2

Numerical results of various correlation functions for $(U,\mu )=(6,-1.5)$. (a) Equal-time spin-spin correlation function $C\left(\vec{y}\right)=\langle (n_{(\vec{x}+\vec{y})\uparrow }-n_{(\vec{x}+\vec{y})\downarrow })(n_{\vec{x}\uparrow }-n_{\vec{x}\downarrow })\rangle$. (b) Magnetic structure factor $S\left(\vec{k}\right)={\sum }_{\vec{x}}{e}^{i\vec{k}·\vec{x}}C\left(\vec{x}\right)$. The sharp peak at $(\pi ,\pi )$ is an indicator for the antiferromagnetic order inherent in the system. (c) Order parameter for antiferromagnetic order, $A\left(\vec{y}\right)=\langle n_{\vec{x}\uparrow }{n}_{(\vec{x}+\vec{y})\downarrow }\rangle$.

Figure 3

Sketch illustrating the derivation behind the calculation of the minimal mean energy for $H_d$.