Bulk-edge correspondence in entanglement spectra

Li and Haldane conjectured and numerically substantiated that the entanglement spectrum of the reduced density matrix of ground states of time-reversal-breaking topological phases [fractional quantum Hall (FQH) states] contains information about the counting of their edge modes when the ground state is cut in two spatially distinct regions and one of the regions is traced out. We analytically substantiate this conjecture for a series of FQH states defined as unique zero modes of pseudopotential Hamiltonians by finding a one-to-one map between the thermodynamic limit counting of two different entanglement spectra: the particle entanglement spectrum (PES), whose counting of eigenvalues for each good quantum number is identical to the counting of bulk quasiholes (up to accidental zero eigenvalues of the reduced density matrix), and the orbital entanglement spectrum (OES), considered by Li and Haldane. By using a set of clustering operators that have their origin in conformal-field-theory (CFT) operator expansions, we show that the counting of the OES eigenvalues in the thermodynamic limit must be identical to the counting of quasiholes in the bulk. The latter equals the counting of edge modes at a hard-wall boundary placed on the sample. Our results can be interpreted as a bulk-edge correspondence in entanglement spectra. Moreover, we show that the counting of the PES and OES is identical even for CFT states that are likely bulk gapless, such as the Gaffnian wave function.

Figure 1

Left: Sketch of the partition in orbital space (part $B$ in gray). The tracing procedure creates a virtual edge and the orbital entanglement spectrum (OES) probes the chiral edge mode(s) of part $A$. Right: OES of the $\nu =1/2$ Laughlin state of $N=9$ bosons, with orbital cut $l_A=8$ and $N_A=4$. The minimal angular momentum $L_{z,\min }^A$ defined in the text is the $L_{z,\min }^A=20$ sector in the plot. The entanglement level counting at $\ell =|L_z^A-L_{z,\min }^A|=0,1,...,4$ is $(1,1,2,3,5)$, which is the counting of modes of a U$\left(1\right)$ boson in the thermodynamic limit. Finite-size effects appear at $L_z^A=15$.

Figure 2

Left: Sketch of the partition in particle space. The particles of part $B$ that are traced out are denoted in gray. Right: particle entanglement spectrum (PES) of the $\nu =1/2$ Laughlin state of $N=9$ bosons, with particle cut $N_A=4$. The minimum angular momentum $L_{z,\min }^A$ defined in the text is $L_{z,\min }^A=20$ in the plot. The entanglement level counting is identical to the counting of quasiholes in a Laughlin state of four particles with total flux $N_{\phi }=16$ at all angular momenta $L_z^A$. For $\ell =|L_z^A-L_{z,\min }^A|=0,1,...,4$, the counting is universal, i.e., independent of $N_A$: $(1,1,2,3,5)$.

Figure 3

The configurations in the PES can be related to those of the OES using the clustering constraints. These constraints reveal the vanishing properties of the FQH state as particles are brought closer together. They relate the long-wavelength properties of the FQH state when two particles are far away from each other to the short-wavelength properties of the state when particles are close together, and, hence, can be used to drag particles from the PES Hilbert space into the more restrictive OES Hilbert space.

Figure 4

A cartoon of the various submatrices in the PEM block labeled by the angular momentum $L_z^A$. The block of the OEM labeled by $(N_A,L_z^A)$ ($\mathbf{C}$ in the figure) is a submatrix of the PEM block $\mathbf{P}$ at $L_z^A$. $\tilde{\mathbf{P}}$ is a submatrix of $\mathbf{P}$ containing all rows and columns that are labeled by $(k,2)$-admissible partitions of $N_A$ and $N_B$ particles subject to total flux $N_{\phi }$.

Figure 5

The occupation configuration of a generic partition $\mu$ with the unit cells, the number of intact unit cells ${\Delta }_{\mu }$, and the distance from cuts after $l_A=9$ (top) and $l_A=10$ (bottom) shown. $N_{\phi }=12$ here.

Figure 6

(a) Orbital entanglement spectrum of the $\nu =1/2$ Laughlin state with $N=4$, $N_A=2$, and orbital cut after $l_A=3$ orbitals. The entanglement level counting is equal to the rank of the OEM at each angular momentum. (b) Particle entanglement spectrum of the $\nu =1/2$ Laughlin state with particle cut $N_A=2$. The entanglement level counting at all angular momenta $L_z^A$ is equal to the rank of the PEM. $L_{z,\min }^A$ defined in the text is $L_z^A=4$ in the plots.