Projection operator formalism applied to a harmonic chain

Johannes Kofler, Vlatko Vedral, Myungshik S. Kim +1 more

We investigate entanglement between collective operators of two blocks of oscillators in an infinite linear harmonic chain. These operators are defined as averages over local operators (individual oscillators) in the blocks. On the one hand, this approach of ``physical blocks'' meets realistic experimental conditions, where measurement apparatuses do not interact with single oscillators but rather with a whole bunch of them, i.e., where in contrast to usually studied ``mathematical blocks'' not every possible measurement is allowed. On the other, this formalism naturally allows the generalization to blocks which may consist of several noncontiguous regions. We quantify entanglement between the collective operators by a measure based on the Peres-Horodecki criterion and show how it can be extracted and transferred to two qubits. Entanglement between two blocks is found even in the case where none of the oscillators from one block is entangled with an oscillator from the other, showing genuine bipartite entanglement between collective operators. Allowing the blocks to consist of a periodic sequence of subblocks, we verify that entanglement scales at most with the total boundary region. We also apply the approach of collective operators to scalar quantum field theory.

David T. Ng

George D. J. Phillies

Randall W. Hall

A class of approximate projection operators is used to reduce the variance in path integral Monte Carlo calculations in a formally exact manner. Paths are classified according to the projection operators, allowing the identification of paths whose contribution to the variance is negligible. The approach is applied to two canonical systems: Two like-spin electrons in a three dimensional harmonic well and the two dimensional Hubbard model, representing localized and extended electronic states, respectively. Time savings of 15–900 over straightforward Monte Carlo calculations are observed.

A. Collins, Chris J. Hamer, Zheng Weihong

An alternative triplet-wave expansion formalism for dimerized spin systems is presented, a modification of the ``bond operator'' formalism of Sachdev and Bhatt. Projection operators are used to confine the system to the physical subspace, rather than constraint equations. The method is illustrated for the case of the alternating Heisenberg chain, and comparisons are made with the results of dimer series expansions and exact diagonalization. Some discussion is included of the phenomenon of ``quasiparticle breakdown'' as it applies to the two-triplon bound states in this model.

Masayoshi Nakamura, Nobuhiko Mishima