1D Lattice Models with Exact Ground States Describing Fractional Quantum Hall Systems(Research)

Zheng-Yuan Wang, Masaaki Nakamura

We introduce one-dimensional lattice models with exact matrix-product ground states describing the fractional quantum Hall (FQH) states in Laughlin series (given by filling factors $\ensuremath{\nu}=1/q$) on torus geometry. Surprisingly, the exactly solvable Hamiltonian has the same mathematical structure as that of the pseudopotential for the Laughlin wave function, and naturally derives the general properties of the Laughlin wave function such as the ${Z}_{2}$ properties of the FQH states and the fermion-boson relation. The obtained exact ground states have high overlaps with the Laughlin states and well describe their properties. Using the matrix product method, density functions and correlation functions are calculated analytically. Especially, obtained entanglement spectra reflect gapless edge states as was discussed by Li and Haldane.

Anne E. B. Nielsen, Germán Sierra, J. I. Cirac

Masaaki Nakamura, Zheng-Yuan Wang, Emil J. Bergholtz

We introduce an exactly solvable fermion chain that describes a ν=1/3 fractional quantum Hall (FQH) state beyond the thin-torus limit. The ground state of our model is shown to be unique for each center-of-mass sector, and it has a matrix product representation that enables us to exactly calculate order parameters, correlation functions, and entanglement spectra. The ground state of our model shows striking similarities with the BCS wave functions and quantum spin-1 chains. Using the variational method with matrix product ansatz, we analytically calculate excitation gaps and vanishing of the compressibility expected in the FQH state. We also show that the above results can be related to a ν=1/2 bosonic FQH state.

Ying-Hai Wu

龍也 中島

Emma Wikberg

We study a 1D lattice Hamiltonian, relevant for a wide range of interesting physical systems like, e.g., the quantum-Hall system, cold atoms or molecules in optical lattices, and TCNQ salts. Through a tuning of the interaction parameters and a departure from a strictly convex interaction, nontrivial ground-state degeneracies and fractionally charged excitations emerge. The excitations, being a generalization of the fractional charges known from the fractional quantum-Hall effect, appear as domain walls between inequivalent ground states and carry the charge +e/mq or -e/mq, where m is an integer and associated with the specified interaction, and v = p/q is the filling fraction in the lattice. The description points at an interesting resemblance to states connected to non-Abelian statistics, which is central for the concept of topological quantum computing.