Gapped and gapless phases of frustrationfree spin /1 2 chains
Abstract
We consider a family of translationinvariant quantum spin chains with nearestneighbor interactions and derive necessary and sufficient conditions for these systems to be gapped in the thermodynamic limit. More precisely, let ψ be an arbitrary twoqubit state. We consider a chain of n qubits with open boundary conditions and Hamiltonian Hn(ψ) which is defined as the sum of rank1 projectors onto ψ applied to consecutive pairs of qubits. We show that the spectral gap of Hn(ψ) is upper bounded by 1/(n − 1) if the eigenvalues of a certain 2 × 2 matrix simply related to ψ have equal nonzero 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 frustrationfree translationinvariant spin1/2 chains with nearestneighbor interactions.
 Publication:

Journal of Mathematical Physics
 Pub Date:
 June 2015
 DOI:
 10.1063/1.4922508
 arXiv:
 arXiv:1503.04035
 Bibcode:
 2015JMP....56f1902B
 Keywords:

 Quantum Physics;
 Mathematical Physics
 EPrint:
 v3: published version