4 research outputs found
Lattice Sigma Models with Exact Supersymmetry
We show how to construct lattice sigma models in one, two and four dimensions
which exhibit an exact fermionic symmetry. These models are discretized and
{\it twisted} versions of conventional supersymmetric sigma models with N=2
supersymmetry. The fermionic symmetry corresponds to a scalar BRST charge built
from the original supercharges. The lattice theories possess local actions and
in many cases admit a Wilson term to suppress doubles. In the two and four
dimensional theorie s we show that these lattice theories are invariant under
additional discrete symmetries. We argue that the presence of these exact
symmetries ensures that no fine tuning is required to achieve N=2 supersymmetry
in the continuum limit. As a concrete example we show preliminary numerical
results from a simulation of the O(3) supersymmetric sigma model in two
dimensions.Comment: 23 pages, 3 figures, formalism generalized to allow for explicit
Wilson mass terms. New numerical results added. Version to be published in
JHE
A geometrical approach to N=2 super Yang-Mills theory on the two dimensional lattice
We propose a discretization of two dimensional Euclidean Yang-Mills theories
with N=2 supersymmetry which preserves exactly both gauge invariance and an
element of supersymmetry. The approach starts from the twisted form of the
continuum super Yang Mills action which we show may be written in terms of two
real Kahler-Dirac fields whose components transform into each other under the
twisted supersymmetry. Once the theory is written in this geometrical language
it is straightforward to discretize by mapping the component tensor fields to
appropriate geometrical structures in the lattice and by replacing the
continuum exterior derivative and its adjoint by appropriate lattice covariant
difference operators. The lattice action is local and possesses a unique vacuum
state while the use of Kahler-Dirac fermions ensures the model does not exhibit
spectrum doubling.Comment: Minor typos fixed. Version to be published in JHE
Tensor network states and geometry
Tensor network states are used to approximate ground states of local
Hamiltonians on a lattice in D spatial dimensions. Different types of tensor
network states can be seen to generate different geometries. Matrix product
states (MPS) in D=1 dimensions, as well as projected entangled pair states
(PEPS) in D>1 dimensions, reproduce the D-dimensional physical geometry of the
lattice model; in contrast, the multi-scale entanglement renormalization ansatz
(MERA) generates a (D+1)-dimensional holographic geometry. Here we focus on
homogeneous tensor networks, where all the tensors in the network are copies of
the same tensor, and argue that certain structural properties of the resulting
many-body states are preconditioned by the geometry of the tensor network and
are therefore largely independent of the choice of variational parameters.
Indeed, the asymptotic decay of correlations in homogeneous MPS and MERA for
D=1 systems is seen to be determined by the structure of geodesics in the
physical and holographic geometries, respectively; whereas the asymptotic
scaling of entanglement entropy is seen to always obey a simple boundary law --
that is, again in the relevant geometry. This geometrical interpretation offers
a simple and unifying framework to understand the structural properties of, and
helps clarify the relation between, different tensor network states. In
addition, it has recently motivated the branching MERA, a generalization of the
MERA capable of reproducing violations of the entropic boundary law in D>1
dimensions.Comment: 18 pages, 18 figure