21 research outputs found
Upper bounds on the Witten index for supersymmetric lattice models by discrete Morse theory
The Witten index for certain supersymmetric lattice models treated by de
Boer, van Eerten, Fendley, and Schoutens, can be formulated as a topological
invariant of simplicial complexes arising as independence complexes of graphs.
We prove a general theorem on independence complexes using discrete Morse
theory: If G is a graph and D a subset of its vertex set such that G\D is a
forest, then sum_i \dim H_i(Ind(G);Q) \leq |Ind}(G[D])|. We use the theorem
to calculate upper bounds on the Witten index for several classes of lattices.
These bounds confirm some of the computer calculations by van Eerten on small
lattices. The cohomological method and the 3-rule of Fendley et al. is a
special case of when G\D lacks edges. We prove a generalized 3-rule and
introduce lattices in arbitrary dimensions satisfying it.Comment: European Journal of Combinatorics, accepted 200
A staggered fermion chain with supersymmetry on open intervals
A strongly-interacting fermion chain with supersymmetry on the lattice and
open boundary conditions is analysed. The local coupling constants of the model
are staggered, and the properties of the ground states as a function of the
staggering parameter are examined. In particular, a connection between certain
ground-state components and solutions of non-linear recursion relations
associated with the Painlev\'e VI equation is conjectured. Moreover, various
local occupation probabilities in the ground state have the so-called
scale-free property, and allow for an exact resummation in the limit of
infinite system size.Comment: 21 pages, no figures; v2: typos correcte
Detailed analysis of the continuum limit of a supersymmetric lattice model in 1D
We present a full identification of lattice model properties with their field
theoretical counter parts in the continuum limit for a supersymmetric model for
itinerant spinless fermions on a one dimensional chain. The continuum limit of
this model is described by an superconformal field theory
(SCFT) with central charge c=1. We identify states and operators in the lattice
model with fields in the SCFT and we relate boundary conditions on the lattice
to sectors in the field theory. We use the dictionary we develop in this paper,
to give a pedagogical explanation of a powerful tool to study supersymmetric
models based on spectral flow. Finally, we employ the developed machinery to
explain numerically observed properties of the particle density on the open
chain presented in Beccaria et al. PRL 94:100401 (2005).Comment: 28 pages, 7 figures, 3 tables, 1 appendix, this work is based on
chapter 4 of the authors PhD Thesis: L. Huijse, A supersymmetric model for
lattice fermions, University of Amsterdam (2010