Momentum space entanglement in quantum spin chains

I will discuss work performed in collaboration with R. Thomale and A. Bernevig ({\sl Phys. Rev. Lett.} {\bf 105}, 116805 (2010)) on entanglement spectra in spin chains. Typically, bipartite entanglement entropy and spectra have been studied in the case of spatial partitions, {\it i.e.\/} A denotes the left half of a spin chain, B the right half, $\rho^{\vphantom{\dagger}}_{\rm A}={\rm Tr}^{\vphantom{\dagger}}_{\rm B}|\Psi_0\rangle\,\langle\Psi_0|$ is the reduced density matrix, and ${\rm spec}(\rho_{\rm A})$ is the entanglement spectrum (ES). We find for the $S={1\over 2}$ Heisenberg model that a remarkable structure in the ES is revealed if the partition is performed in momentum space, {\it i.e.\/} A = left-movers and B = right-movers. Further classifying the entanglement eigenstates by total crystal momentum, we observe a universal low-lying portion of the ES with specific multiplicities separated from a higher-lying nonuniversal set of levels by an {\it entanglement gap\/}, similar to what was observed by Li and Haldane ({\sl Phys. Rev. Lett.} {\bf 101}, 010504 (2008)) for the fractional quantum Hall effect. Indeed, the momentum space ES for the Heisenberg chain is understood in terms of the proximity of the Haldane-Shastry model, which corresponds to a fixed point with no nonuniversal corrections, and whose ground state wavefunction is related to that for the $\nu={1\over 2}$ Laughlin state. We further explore the behavior of the ES as one tunes through the spin-Peierls transition in a model with next-nearest- neighbor exchange. We also discuss entanglement gap scaling and applications to other systems.