Abstract
We introduce a notion of homogeneous topological order, which is
obeyed by most, if not all, known examples of topological order
including fracton phases on quantum spins (qudits). The notion is a
condition on the ground state subspace, rather than on the Hamiltonian,
and demands that given a collection of ball-like regions, any linear
transformation on the ground space be realized by an operator that
avoids the ball-like regions. We derive a bound on the ground state
degeneracy \mathcal D𝒟
for systems with homogeneous topological order on an arbitrary closed
Riemannian manifold of dimension dd,
which reads [ D c (L/a)^{d-2}.] Here, LL
is the diameter of the system, aa
is the lattice spacing, and cc
is a constant that only depends on the isometry class of the manifold,
and \muμ
is a constant that only depends on the density of degrees of freedom. If
d=2d=2,
the constant cc
is the (demi)genus of the space manifold. This bound is saturated up to
constants by known examples.examples.
Subject
General Physics and Astronomy
Cited by
10 articles.
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