Quantum computation and quantum information
Quantum computation and quantum information
Geometric entanglement of one-dimensional systems: bounds and scalings in the thermodynamic limit
Quantum Information & Computation
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The entanglement in quantum XY spin chains of arbitrary length is investigated viathe geometric measure of entanglement. The emergence of entanglement is explainedintuitively from the perspective of perturbations. The model is solved exactly and theenergy spectrum is determined and analyzed in particular for the lowest two levels forboth finite and infinite systems. The overlaps for these two levels are calculated analyti-cally for arbitrary number of spins. The entanglement is hence obtained by maximizingover a single parameter. The corresponding ground-state entanglement surface is thendetermined over the entire phase diagram, and its behavior can be used to delineate theboundaries in the phase diagram. For example, the field-derivative of the entanglementbecomes singular along the critical line. The form of the divergence is derived analyti-cally and it turns out to be dictated by the universality class controlling the quantumphase transition. The behavior of the entanglement near criticality can be understoodvia a scaling hypothesis, analogous to that for free energies. The entanglement den-sity vanishes along the so-called disorder line in the phase diagram, the ground space isdoubly degenerate and spanned by two product states. The entanglement for the super-position of the lowest two states is also calculated. The exact value of the entanglementdepends on the specific form of superposition. However, in the thermodynamic limit theentanglement density turns out to be independent of the superposition. This proves thatthe entanglement density is insensitive to whether the ground state is chosen to be thespontaneously Z2 symmetry broken one or not. The finite-size scaling of entanglement atcritical points is also investigated from two different view points. First, the maximum inthe field-derivative of the entanglement density is computed and fitted to a logarithmicdependence of the system size, thereby deducing the correlation length exponent for theIsing class using only the behavior of entanglement. Second, the entanglement densityitself is shown to possess a correction term inversely proportional to the system size,with the coefficient being universal (but with different values for the ground state andthe first excited state, respectively).