David Gosset: Gapped and Gapless Phases of Frustration-free Spin-1/2 chains
0David Gosset (Caltech)
Gapped and Gapless Phases of Frustration-free Spin-1/2 chains
QuICS Workshop on the Frontiers of Quantum Information and Computer Science (October 1, 2015)
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 |ψ⟩ 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 |ψ⟩ 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 |ψ⟩ have equal non-zero absolute value. Otherwise, the spectral gap is lower bounded by a positive constant independent of n (depending only on |ψ⟩).
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. As an extension of our main result, we obtain a complete classification of gapped and gapless phases of frustration-free translation-invariant spin-1/2 chains with nearest-neighbor interactions.
This is joint work with Sergey Bravyi of IBM.
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