We consider integer quantum Hall states and calculate the bulk entanglement spectrum by formulating the correlation matrix in the guiding center representation. Our analytical approach is based on the strategy of redefining the inner product of states in the Hilbert space, via a projection operator, to take care of the restriction imposed by the (rectangle-pixeled) checkerboard partition. The resultant correlation matrix contains various couplings between states of different guiding centers parameterized by the magnetic length and the pixel size. Given a fixed magnetic field, we find various patterns of band crossings by tuning the pixel size (quantified by the flux $\mathrm{\ensuremath{\Phi}}$ threading each pixel) and by changing the filling factor $\ensuremath{\nu}\ensuremath{\in}\mathbb{N}$ (determined by the Fermi level). When $\ensuremath{\nu}=1$ and $\mathrm{\ensuremath{\Phi}}=2\ensuremath{\pi}$, or $\ensuremath{\nu}=2$ and $\mathrm{\ensuremath{\Phi}}=\ensuremath{\pi}$, one Dirac band crossing is found. For $\ensuremath{\nu}=1$ and $\mathrm{\ensuremath{\Phi}}=\ensuremath{\pi}$, the band crossings are in the form of a nodal line, enclosing the Brillouin zone. As for $\ensuremath{\nu}=2$ and $\mathrm{\ensuremath{\Phi}}=2\ensuremath{\pi}$, the doubled Dirac point, or the quadratic point, is seen. Additionally, we infer that the quadratic point is protected by the ${\mathrm{C}}_{4}$ symmetry of the pixel since it evolves into two separate Dirac points when the symmetry is lowered to ${\mathrm{C}}_{2}$ . We also identify the emerging symmetries responsible for the symmetric bulk entanglement spectra, which are absent in the underlying quantum Hall states.We consider integer quantum Hall states and calculate the bulk entanglement spectrum by formulating the correlation matrix in the guiding center representation. Our analytical approach is based on the strategy of redefining the inner product of states in the Hilbert space, via a projection operator, to take care of the restriction imposed by the (rectangle-pixeled) checkerboard partition. The resultant correlation matrix contains various couplings between states of different guiding centers parameterized by the magnetic length and the pixel size. Given a fixed magnetic field, we find various patterns of band crossings by tuning the pixel size (quantified by the flux $\mathrm{\ensuremath{\Phi}}$ threading each pixel) and by changing the filling factor $\ensuremath{\nu}\ensuremath{\in}\mathbb{N}$ (determined by the Fermi level). When $\ensuremath{\nu}=1$ and $\mathrm{\ensuremath{\Phi}}=2\ensuremath{\pi}$, or $\ensuremath{\nu}=2$ and $\mathrm{\ensuremath{\Phi}}=\ensuremath{\pi}$, one Dirac band crossing is found. For $\ensuremath{\nu}=1$ and $\mathrm{\ensuremath{\Phi}}=\ensuremath{\pi}$, the band crossings are in the form of a nodal line, enclosing the Brillouin zone. As for $\ensuremath{\nu}=2$ and $\mathrm{\ensuremath{\Phi}}=2\ensuremath{\pi}$, the doubled Dirac point, or the quadratic point, is seen. Additionally, we infer that the quadratic point is protected by the ${\mathrm{C}}_{4}$ symmetry of the pixel since it evolves into two separate Dirac points when the symmetry is lowered to ${\mathrm{C}}_{2}$. We also identify the emerging symmetries responsible for the symmetric bulk entanglement spectra, which are absent in the underlying quantum Hall states.

arXiv:1506.00775Physical Review BAug 20, 20153 citations

Spectrum of Quantum Entanglement in Fractional Quantum Hall States

Hui Li, F. D. M. Haldane

Bulletin of the American Physical SocietyMar 10, 20080 citations

{\mathrm{C}}_{4}

symmetry of the pixel since it evolves into two separate Dirac points when the symmetry is lowered to

{\mathrm{C}}_{2}

. We also identify the emerging symmetries responsible for the symmetric bulk entanglement spectra, which are absent in the underlying quantum Hall states.