Dear Quanta,
I’ll be the surrogate Aram for tomorrow’s meeting. Donuts and coffee will be provided at
3:45, and the meeting will start promptly at 4:00.
After going around the room briefly, we will have our visitors Jacob Bridgeman and Chris
Chubb from the University of Sydney giving two short (20 min) talks about these two
papers:
Approximate symmetries of Hamiltonians
We explore the relationship between approximate symmetries of a gapped Hamiltonian and the
structure of its ground space. We start by showing that approximate symmetry
operators---unitary operators whose commutators with the Hamiltonian have norms that are
sufficiently small---which possess certain mutual commutation relations can be restricted
to the ground space with low distortion. We generalize the Stone-von Neumann theorem to
matrices that approximately satisfy the canonical (Heisenberg-Weyl-type) commutation
relations, and use this to show that approximate symmetry operators can certify the
degeneracy of the ground space even though they only approximately form a group.
Importantly, the notions of "approximate" and "small" are all
independent of the dimension of the ambient Hilbert space, and depend only on the
degeneracy in the ground space. Our analysis additionally holds for any gapped band of
sufficiently small width in the excited spectrum of the Hamiltonian, and we discuss
applications of these ideas to topological quantum phases of matter and topological
quantum error correcting codes. Finally, in our analysis we also provide an exponential
improvement upon bounds concerning the existence of shared approximate eigenvectors of
approximately commuting operators which may be of independent interest.
https://arxiv.org/abs/1608.02600 <https://arxiv.org/abs/1608.02600>
Detecting Topological Order with Ribbon Operators
We introduce a numerical method for identifying topological order in two-dimensional
models based on one-dimensional bulk operators. The idea is to identify approximate
symmetries supported on thin strips through the bulk that behave as string operators
associated to an anyon model. We can express these ribbon operators in matrix product form
and define a cost function that allows us to efficiently optimize over this ansatz class.
We test this method on spin models with abelian topological order by finding ribbon
operators for ℤd quantum double models with local fields and Ising-like terms. In
addition, we identify ribbons in the abelian phase of Kitaev's honeycomb model which
serve as the logical operators of the encoded qubit for the quantum error-correcting code.
We further identify the topologically encoded qubit in the quantum compass model, and show
that despite this qubit, the model does not support topological order. Finally, we discuss
how the method supports generalizations for detecting nonabelian topological order.
https://arxiv.org/abs/1603.02275 <https://arxiv.org/abs/1603.02275>
Best,
Steve
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