Pfaffian-like ground state for three-body hard-core bosons in one-dimensional lattices

We propose a Pfaffian-like ansatz for the ground state of bosons subject to three-body infinite repulsive interactions in a one-dimensional (1D) lattice. Our ansatz consists of symmetrization over all possible ways of distributing the particles in two identical Tonks-Girardeau gases. We support the quality of our ansatz with numerical calculations and propose an experimental scheme based on mixtures of bosonic atoms and molecules in 1D optical lattices in which this Pfaffian-like state could be realized. Our findings may open the way for the creation of non-Abelian anyons in 1D systems.

Figure 1

(Color online) Schematic representation of the operator ${\mathcal{P}}_{\ell }$ mapping the single-site four-dimensional Hilbert space of two species of hard-core bosons to the three-dimensional one of three-hard-core bosons.

Figure 2

(Color online) Overlap $⟨{\Psi }_{\mathrm{ex}}\mid {\Psi }_3⟩$. The main plot shows the dependence of the overlap on the filling factor $\nu =N∕M$ for a system of $M=10$ (squares) and 20 (circles) lattice sites. The inset shows the decrease of the overlap with increasing system size $M$, at $\nu =1$.

Figure 3

(Color online) Average occupation of sites with one (left figure) and two particles (right figure) for the exact (solid lines), ansatz (dotted lines), and Poissonian distributions (dot-dashed lines), as a function of the filling factor $\nu$. The system size is $M=20$.

Figure 4

(Color online) Quasimomentum distribution of particles $n_k\propto {\sum }_{\ell ,\Delta }{e}^{-ik\Delta }⟨a_{\ell +\Delta }^{†}{a}_{\ell }⟩$ (left axis) and of on-site pairs $n_k^{\left(2\right)}\propto {\sum }_{\ell ,\Delta }{e}^{-ik\Delta }⟨a_{\ell +\Delta }^{†}{a}_{\ell +\Delta }^{†}{a}_{\ell }{a}_{\ell }⟩$ (right axis), with $k=2\pi ∕Mn$, $n=0,\dots ,M-1$. Results are shown for both the exact ground state (solid lines) and the ansatz (dashed lines). The inset shows the long-distance scaling of the correlation functions $⟨a_{\ell +\Delta }^{†}{a}_{\ell }⟩\sim {\Delta }^{-{\alpha }_1}$ and $⟨a_{\ell +\Delta }^{†}{a}_{\ell +\Delta }^{†}{a}_{\ell }{a}_{\ell }⟩\sim {\Delta }^{-{\alpha }_2}$, for the exact ground state (circles, ${\alpha }_1=0.22$, ${\alpha }_2=0.83$) and the ansatz (diamonds, ${\alpha }_1=0.24$, ${\alpha }_2=0.99$). Parameters are $M=20$ and $\nu =1$.