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arXiv:1206.2044 · Physical Review B

Order and supersymmetry at high filling zero-energy states on the triangular lattice

D. Galanakis, Christopher L. Henley, Stefanos Papanikolaou

Published Nov 5, 2012• 9 citations
arXivHTMLPDF

Abstract

We perform exact-diagonalization studies in $d=2$ dimensions for the Fendley and Schoutens model of hard-core and nearest-neighbor-excluding fermions that displays an exact nonrelativistic supersymmetry. Using clusters of all possible shapes up to 46 sites, we systematically study the behavior of the ground-state phase diagram as a function of filling. We focus on the highly degenerate zero-energy states found at fillings between $1/7$ and $\ensuremath{\sim}$1/5. At the lower end of that interval, at filling $1/7$, we explicitly show that the ground states are gapped crystals. Consistent with previous suggestions, we find that the extensive entropy of zero states peaks at a filling of $\ensuremath{\sim}$0.178. At the higher end of the interval, we find zero-energy ground states at fillings above $1/5$, contrary to previous expectations, which display nontrivial amplitude degeneracies.

DOI
10.1103/physrevb.86.195105

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We perform exact-diagonalization studies in d=2d=2d=2 dimensions for the Fendley and Schoutens model of hard-core and nearest-neighbor-excluding fermions that displays an exact nonrelativistic supersymmetry. Using clusters of all possible shapes up to 46 sites, we systematically study the behavior of the ground-state phase diagram as a function of filling. We focus on the highly degenerate zero-energy states found at fillings between 1/71/71/7 and \ensuremath∼\ensuremath{\sim}\ensuremath∼1/5. At the lower end of that interval, at filling 1/71/71/7, we explicitly show that the ground states are gapped crystals. Consistent with previous suggestions, we find that the extensive entropy of zero states peaks at a filling of \ensuremath∼\ensuremath{\sim}\ensuremath∼0.178. At the higher end of the interval, we find zero-energy ground states at fillings above 1/51/51/5, contrary to previous expectations, which display nontrivial amplitude degeneracies.
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.
We investigate the phase diagram of spinless fermions with nearest- and next-nearest-neighbor density-density interactions on the honeycomb lattice at half-filling. Using exact diagonalization techniques of the full Hamiltonian and constrained subspaces, combined with a careful choice of finite-size clusters, we determine the different charge orderings that occur for large interactions. In this regime, we find a two-sublattice N\'eel-like state, a charge modulated state with a tripling of the unit cell, a zigzag phase, and a charge ordered state with a 12-site unit cell we call N\'eel domain wall crystal, as well as a region of phase separation for attractive interactions. A sizable region of the phase diagram is classically degenerate, but it remains unclear whether an order-by-disorder mechanism will lift the degeneracy. For intermediate repulsion, we find evidence for a Kekul\'e or plaquette bond-order wave phase. We also investigate the possibility of a spontaneous Chern insulator phase (dubbed topological Mott insulator), as previously put forward by several mean-field studies. Although we are unable to detect convincing evidence for this phase based on energy spectra and order parameters, we find an enhancement of current-current correlations with the expected spatial structure compared to the noninteracting situation. While for the studied $t\ensuremath{-}{V}_{1}\ensuremath{-}{V}_{2}$ model, the phase transition to the putative topological Mott insulator is preempted by the phase transitions to the various ordered states, our findings might hint at the possibility for a topological Mott insulator in an enlarged Hamiltonian parameter space, where the competing phases are suppressed.
We study the quantum phase diagram of a system of hard-core bosons on the kagome lattice with nearest-neighbor repulsive interactions, for arbitrary densities, by means of the hierarchical mean-field theory and exact diagonalization techniques. This system is isomorphic to the spin S=1/2S=1/2S=1/2 XXZ model in presence of an external magnetic field, a paradigmatic example of frustrated quantum magnetism. In the nonfrustrated regime, we find two crystal phases at densities 1/3 and 2/3 that melt into a superfluid phase when increasing the hopping amplitude, in semiquantitative agreement with quantum Monte Carlo computations. In the frustrated regime and away from half-filling, we find a series of plateaux with densities commensurate with powers of 1/3. The broader density plateaux (at densities 1/3 and 2/3) are remnants of the classical degeneracy in the Ising limit. For densities near half-filling, this staircase of crystal phases melts into a superfluid, which displays finite chiral currents when computed with clusters having an odd number of sites. Both the staircase of crystal phases and the superfluid phase prevail in the noninteracting limit, suggesting that the lowest dispersionless single-particle band may be at the root of this phenomenon.
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.
The ground state of hard-core bosons on the square lattice with nearest and next-nearest-neighbor repulsion is studied by quantum Monte Carlo simulations. A supersolid phase with vacancy condensation and ``star'' diagonal ordering is found for filling $\ensuremath{\rho}<0.25$. At filling $\ensuremath{\rho}>0.25$, a supersolid phase exists between the star and the stripe crystal at $\ensuremath{\rho}=0.5$. No supersolid phase occurs for $\ensuremath{\rho}>0.25$ if the ground state at half-filling is either a checkerboard crystal or a superfluid. No commensurate supersolid phase is observed.
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.