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CATEGORIES:CQIF Seminar
SUMMARY:Gapped and gapless phases of frustration-free spin
-1/2 chains - David Gosset (Caltech)
DTSTART;TZID=Europe/London:20150416T141500
DTEND;TZID=Europe/London:20150416T151500
UID:TALK58496AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/58496
DESCRIPTION:We consider a family of translation-invariant quan
tum spin chains with nearest-neighbor interactions
and derive necessary and sufficient conditions fo
r these systems to be gapped in the thermodynamic
limit. More precisely\, let psi be an arbitrary tw
o-qubit state. We consider a chain of n qubits wit
h open boundary conditions and Hamiltonian which i
s defined as the sum of rank-1 projectors onto psi
applied to consecutive pairs of qubits. We show t
hat the spectral gap of the Hamiltonian is upper b
ounded 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 indep
endent of n (depending only on psi). A key ingredi
ent in the proof is a new operator inequality for
the ground space projector which expresses a monot
onicity 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.
LOCATION:MR12\, Centre for Mathematical Sciences\, Wilberf
orce Road\, Cambridge
CONTACT:William Matthews
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