Exact ground states of a staggered supersymmetric model for lattice fermions

We study a supersymmetric model for strongly interacting lattice fermions in the presence of a staggering parameter. The staggering is introduced as a tunable parameter in the manifestly supersymmetric Hamiltonian. We obtain analytic expressions for the ground states in the limit of small and large staggering for the model on the class of doubly decorated lattices. On this type of lattice there are two ground states, each with a different density. In one limit we find these ground states to be a simple Wigner crystal and a valence bond solid state. In the other limit we find two types of quantum liquids. As a special case, we investigate the quantum liquid state on the one dimensional chain in detail. It is characterized by a massless kink that separates two types of order.

Figure 1

Decorated square lattice. The sublattice $S_1$ is the original square lattice; the sites are depicted as squares. The sublattice $S_2$ is composed of the added sites, indicated by circles, on the links of the original lattice. For doubly periodic boundary conditions the right (bottom) sites are identified with the left (top) sites.

Figure 2

Mapping of elements of the cohomology of $Q_1$ at $F=N_{\Lambda }$ to projected clock model states. On the left we show the four possible states of each vertex. On the right we show, as an example, a cohomology element of ${\mathcal{H}}_{Q_1}$ with $F=N_{\Lambda }$ for the doubly decorated square lattice with open boundary conditions.

Figure 3

Mapping of elements of the cohomology of $Q_1$ at $F=L_{\Lambda }$ to states on the original square lattice, where the degrees of freedom are arrows living on the links.

Figure 4

Plot of the energies of low-lying excitations of open staggered SUSY chains with staggering $y=1/100$ and lengths $3k$ with $4⩽k⩽8$. $x=\frac{n}{2k+2}$. The equation used for the fit is $f\left(n\right)=4y^2{\sin }^2\left(\frac{n\pi }{2k+2}\right)$.

Figure 5

Density on staggered sites $\langle n_i\rangle$ for $i\mathrm{mod}3=1$ for (a) $y=1/100$ and (b) $y=1/10$. Shaded plot markers correspond to the predictions from perturbation theory. In (a) this is to first order using expressions in Eq. (30) and in (b) this is to fourth order using expressions in Eq. (31). Here we follow the notation in Ref. 31 and use $k_{\pm }=(L\pm 1)/2$ and $\tilde{L}=L/3+1$.

Figure 6

Numerically calculated entanglement entropy data for the ground state of the open $L=39$ staggered chain with staggering parameter $y=\frac{1}{50}$ plotted alongside the analytical results for the small staggering limit indicated by the black crosses. The dotted line corresponds to the result for the system in the continuum limit where the ${\mathbb{Z}}_3$ substructure vanishes. As discussed in the text this line has a very simple interpretation in terms of the kink.