Certain Homology Cycles of the Independence Complex of Grids

Jönsson, Jakob

doi:10.1007/s00454-009-9224-9

Cited by 33 publications

(44 citation statements)

References 15 publications

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“…In Section IV, we focus on the states found near either endpoint of the zero-energy ground state interval. The minimum filling with a zero energy state is exactly f = 1/7 as conjectured 11 ; however, from that filling up to f ≈ 0.156, the only zero-energy states are crystal-like and gapped states, and have no extensive entropy. At fillings slightly higher than 1/5, we find that certain zero-energy states exist, contradicting the conjecture 11 .…”

Section: Introductionmentioning

confidence: 96%

“…The minimum filling with a zero energy state is exactly f = 1/7 as conjectured 11 ; however, from that filling up to f ≈ 0.156, the only zero-energy states are crystal-like and gapped states, and have no extensive entropy. At fillings slightly higher than 1/5, we find that certain zero-energy states exist, contradicting the conjecture 11 . These states show a tendency to anisotropic forms of spatial order, and their wavefunctions exhibit surprising regularities, in that many inequivalent fermion configurations have the same, maximal amplitude in the wavefunction (Sec.…”

Section: Introductionmentioning

confidence: 96%

“…In contrast, on the triangular lattice the number of zero energy states appears exponential in the area, as recent exact diagonalization (ED) studies have indicated 10 . Jonsson, by studying finite triangular clusters, recently conjectured 11 , that on the triangular lattice, zero-energy states only appear in the interval 1/7 ≤ f ≤ 1/5, where f is the filling (number of fermions/site). 28 Huijse et al 10 recently investigated, using ED, the nature and number of zero-energy states on the triangular lattice.…”

Section: Introductionmentioning

confidence: 99%

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

2012

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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 non-relativistic 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 ∼ 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 ∼ 0.178. At the higher end of the interval, we find zero energy ground states at fillings above 1/5, in contradiction to previous numerical studies and analytical suggestions; these display non-trivial amplitude degeneracies.

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Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

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

2012

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“…The application motivating this method is in the study of the independence and dominance complexes of a graph, which have been extensively studied in the recent years [6,11,12,[18][19][20][21]27] because of their applications in graph theory, network analysis, and statistical mechanics. Our method allows us to obtain several results on the topology of the independence and dominance complexes.…”

Section: Introductionmentioning

confidence: 99%

Cores of Simplicial Complexes

Marietti

Testa

2008

Discrete Comput Geom

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We introduce a method to reduce the study of the topology of a simplicial complex to that of a simpler one. Applying this method to complexes arising from graphs, we give topological meaning to classical graph invariants. As a consequence, we answer some questions raised in (Ehrenborg and Hetyei in Eur. J. Comb. 27(6):906-923, 2006) on the independence complex and the dominance complex of a forest and obtain improved algorithms to compute their homotopy types.

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“…Cross-cycles have been used to construct homology classes in a number of contexts [1,3,13,20]. They appear as the main contribution to the homology of the clique complexes of random geometric graphs [15].…”

Section: Introductionmentioning

confidence: 99%

Algorithmic complexity of finding cross-cycles in flag complexes

Adamaszek

Stacho

2012

Proceedings of the Twenty-Eighth Annual Symposium on Computational Geometry

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A cross-cycle in a flag simplicial complex K is an induced subcomplex that is isomorphic to the boundary of a cross-polytope and that contains a maximal face of K. A cross-cycle is the most efficient way to define a non-zero class in the homology of K. For an independence complex of a graph G, a cross-cycle is equivalent to an induced matching containing a maximal independent set of G.We study the complexity of finding cross-cycles in independence complexes. We show that in general this problem is NP-complete when input is a graph whose independence complex we consider. This allows us to study special cases. Unfortunately, not a lot has been done in this direction besides the recent polynomial time algorithm for forests [16].In this contribution, we focus on the more general class of chordal graphs. This is a natural choice for the problem as the independence complexes of chordal graphs are quite well understood, namely they are wedges of spheres up to homotopy, and any wedge of spheres can be realized as the independence complex of a chordal graph, up to homotopy.As our main result, we present a polynomial time algorithm for detecting a cross-cycle in the independence complex of a chordal graph. Our algorithm is based on the geometric intersection representation of chordal graphs and has an efficient implementation.We further prove that for chordal graphs cross-cycles detect all of homology of the independence complex. As a corollary, we obtain polynomial time algorithms for such topological properties as contractibility or simple-connectedness of independence complexes of chordal graphs. These problems are undecidable for general independence complexes.We conclude with a discussion of some related cases and open problems.

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Certain Homology Cycles of the Independence Complex of Grids

Cited by 33 publications

References 15 publications

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

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

Cores of Simplicial Complexes

Algorithmic complexity of finding cross-cycles in flag complexes

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