University of Cambridge > > CQIF Seminar > Gapped and gapless phases of frustration-free spin-1/2 chains

Gapped and gapless phases of frustration-free spin-1/2 chains

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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.

This talk is part of the CQIF Seminar series.

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