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Tuesday, March 31, 2015
12:00 PM - 1:00 PM

Mathematical Physics Seminar

Gapped and gapless phases of frustration-free spin-1/2 chains
David Gosset, Sherman Fairchild Postdoctoral Scholar in Theoretical Physics, Theoretical Physics, Caltech,

We consider a family of translation-invariant quantum spin
chains with nearest-neighbor interactions and derive necessary and
sufficient conditions for these systems to be gapped in the thermodynamic
limit. More precisely, let psi be an arbitrary two-qubit state. We
consider a chain of n qubits with open boundary conditions and Hamiltonian
which is defined as the sum of rank-1 projectors onto psi applied to
consecutive pairs of qubits. We show that the spectral gap of the
Hamiltonian is upper bounded by 1/(n-1) if the eigenvalues of a certain
two-by-two matrix simply related to psi have equal non-zero absolute
value. Otherwise, the spectral gap is lower bounded by a positive constant
independent of n (depending only on psi). A key ingredient in the proof is
a new operator inequality for the ground space projector which expresses a
monotonicity under the partial trace. This monotonicity property appears
to be very general and might be interesting in its own right.  This is
joint work with Sergey Bravyi.
 

For more information, please contact Rupert Frank by email at [email protected].