Chiral Central Charge from a Single Bulk Wave Function

A ($2+1$)-dimensional gapped quantum many-body system can have a topologically protected energy current at its edge. The magnitude of this current is determined entirely by the temperature and the chiral central charge, a quantity associated with the effective field theory of the edge. We derive a formula for the chiral central charge that, akin to the topological entanglement entropy, is completely determined by the many-body ground state wave function in the bulk. According to our formula, nonzero chiral central charge gives rise to a topological obstruction that prevents the ground state wave function from being real valued in any local product basis.

Figure 1

Partition of a disk-shaped region $ABC$ in the bulk ($\mathrm{\Lambda }$). Each subsystem is assumed to be sufficiently large compared with the correlation length.

Figure 2

First row: deformation of $A$ to $Aa$, where $a$ is a small disk separated from $B$. Second row: deformations of $B$ to $Bb$; (for the rightmost figure, $b\subset A$). All cases in the second row can be reduced to the cases in the first row by a change of subsystem.

Figure 3

(a) A disk $\mathfrak{D}$ and its partition into ${\mathfrak{D}}_{\mathrm{int}}$ and ${\mathfrak{D}}_{\partial }$. (b) A disk $\mathfrak{D}$ (blue) and the one-site terms $K_v$ (yellow), two-site terms $K_e$ (orange), and three-site terms $K_f$ (green) in the local decomposition [Eq. (7)] of modular Hamiltonian $K_{\mathfrak{D}}$.

Figure 4

The calculation of the edge modular current $I_{\sigma }$ [Eq. (12)]. The computation result of all the nonvanishing contribution from modular currents passing through a specific cut is summarized (see the Supplemental Material [36] for details).