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arXiv:cond-mat/0509581 · Journal of Mathematical Physics

Extensive ground state entropy in supersymmetric lattice models

Hendrik van Eerten

Published Dec 1, 2005• 42 citations
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Abstract

We present the result of calculations of the Witten index for a supersymmetric lattice model on lattices of various type and size. Because the model remains supersymmetric at finite lattice size, the Witten index can be calculated using row-to-row transfer matrices and the calculations are similar to calculations of the partition function at negative activity −1. The Witten index provides a lower bound on the number of ground states. We find strong numerical evidence that the Witten index grows exponentially with the number of sites of the lattice, implying that the model has extensive entropy in the ground state.

DOI
10.1063/1.2142836

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We present the result of calculations of the Witten index for a supersymmetric lattice model on lattices of various type and size. Because the model remains supersymmetric at finite lattice size, the Witten index can be calculated using row-to-row transfer matrices and the calculations are similar to calculations of the partition function at negative activity −1. The Witten index provides a lower bound on the number of ground states. We find strong numerical evidence that the Witten index grows exponentially with the number of sites of the lattice, implying that the model has extensive entropy in the ground state.
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Abstract We investigate the entanglement entropy in the ground state of the supersymmetric fermion lattice model. The Hamiltonian of this model is block diagonal in the basis of a fixed number of particles, so we compute the entanglement entropy for the blocks of the Hamiltonian separately. We explain the way of calculating the correlation matrix and the entanglement entropy for the ground state of supersymmetric Hamiltonian on diverse graphs. Then we show that the entanglement entropy exhibits the volume law scaling in various choices of the number of fermions in regular graphs. Also, we give the analytical formula for the entanglement entropy in one scalable set of strongly regular graphs in terms of their parameters, and we prove that the entanglement entropy satisfies a volume law.
We study a model for itinerant, strongly interacting fermions where a judicious tuning of the interactions leads to a supersymmetric Hamiltonian. On the triangular lattice this model is known to exhibit a property called superfrustration, which is characterized by an extensive ground state entropy. Using a combination of numerical and analytical methods we study various ladder geometries obtained by imposing doubly periodic boundary conditions on the triangular lattice. We compare our results to various bounds on the ground state degeneracy obtained in the literature. For all systems we find that the number of ground states grows exponentially with system size. For two of the models that we study we obtain the exact number of ground states by solving the cohomology problem. For one of these, we find that via a sequence of mappings the entire spectrum can be understood. It exhibits a gapped phase at 1/4 filling and a gapless phase at 1/6 filling and phase separation at intermediate fillings. The gapless phase separates into an exponential number of sectors, where the continuum limit of each sector is described by a superconformal field theory.
We analyze the quantum ground state structure of a specific model of itinerant, strongly interacting lattice fermions.The interactions are tuned to make the model supersymmetric.Due to this, quantum ground states are in one-to-one correspondence with cohomology classes of the so-called independence complex of the lattice.Our main result is a complete description of the cohomology, and thereby of the quantum ground states, for a two-dimensional square lattice with periodic boundary conditions.Our work builds on results by Jonsson, who determined the Euler characteristic (Witten index) via a correspondence with rhombus tilings of the plane.We prove a theorem, first conjectured by Fendley, which relates dimensions of the cohomology at grade n to the number of rhombus tilings with n rhombi.