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Supersymmetric lattice fermions on the triangular \nlattice: superfrustration and criticality

Liza Huijse, Dhagash B. Mehta, Niall Moran, Kareljan Schoutens, Jiří Vala

Published Jan 1, 2012• 25 citations
OpenAlex

Abstract

We study a model for itinerant, strongly interacting fermions where \na judicious tuning of the interactions leads to a supersymmetric Hamiltonian. \nOn the triangular lattice this model is known to exhibit a property called \nsuperfrustration, which is characterized by an extensive ground state entropy. \nUsing a combination of numerical and analytical methods we study various \nladder geometries obtained by imposing doubly periodic boundary conditions \non the triangular lattice. We compare our results to various bounds on the \nground state degeneracy obtained in the literature. For all systems we find that \nthe number of ground states grows exponentially with system size. For two \nof the models that we study we obtain the exact number of ground states by \nsolving the cohomology problem. For one of these, we find that via a sequence \nof mappings the entire spectrum can be understood. It exhibits a gapped phase at \n1/4 filling and a gapless phase at 1/6 filling and phase separation at intermediate \nfillings. The gapless phase separates into an exponential number of sectors, \nwhere the continuum limit of each sector is described by a superconformal field \ntheory.

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We study a model for itinerant, strongly interacting fermions where \na judicious tuning of the interactions leads to a supersymmetric Hamiltonian. \nOn the triangular lattice this model is known to exhibit a property called \nsuperfrustration, which is characterized by an extensive ground state entropy. \nUsing a combination of numerical and analytical methods we study various \nladder geometries obtained by imposing doubly periodic boundary conditions \non the triangular lattice. We compare our results to various bounds on the \nground state degeneracy obtained in the literature. For all systems we find that \nthe number of ground states grows exponentially with system size. For two \nof the models that we study we obtain the exact number of ground states by \nsolving the cohomology problem. For one of these, we find that via a sequence \nof mappings the entire spectrum can be understood. It exhibits a gapped phase at \n1/4 filling and a gapless phase at 1/6 filling and phase separation at intermediate \nfillings. The gapless phase separates into an exponential number of sectors, \nwhere the continuum limit of each sector is described by a superconformal field \ntheory.
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 review recent results on lattice models for spin-less fermions with strong repulsive interactions. A judicious tuning of kinetic and interaction terms leads to a model possessing supersymmetry. In the 1D case, this model displays critical behavior described by superconformal field theory. On 2D lattices we generically find superfrustration, characterized by an extensive ground state entropy. For certain 2D lattices analytical results on the ground state structure reveal yet another quantum phase, which we tentatively call 'supertopological'.
We study a model of strongly correlated electrons on the square lattice which exhibits charge frustration and quantum critical behavior. The potential is tuned to make the interactions supersymmetric. We establish a rigorous mathematical result which relates quantum ground states to certain tiling configurations on the square lattice. For periodic boundary conditions this relation implies that the number of ground states grows exponentially with the linear dimensions of the system. We present substantial analytic and numerical evidence that for open boundary conditions the system has gapless edge modes.
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.
We study a model of spinless fermions on the honeycomb lattice with nearest-neighbor exclusion and extended repulsive interactions that exhibits ``lattice supersymmetry'' [P. Fendley et al., Phys. Rev. Lett. 90, 120402 (2003)]. Using a combination of exact diagonalization of large ($N\ensuremath{\le}56$ site) systems, mean-field numerics, and symmetry analysis, we establish a rich phase structure as a function of fermion density, that includes non-Fermi-liquid behavior, resonating charge stripes, domain-wall and bubble physics, and identify a finite range of fillings with extensive ground-state degeneracy and both gapped and gapless spectra. We comment on the stability of our results to relaxing the stringent requirements for supersymmetry, and on their possible broader relevance to systems of strongly correlated electrons with extended repulsive interactions.