Cornering Gapless Quantum States via Their Torus Entanglement

The entanglement entropy (EE) has emerged as an important window into the structure of complex quantum states of matter. We analyze the universal part of the EE for gapless systems on tori in 2D and 3D, denoted by $\chi$. Focusing on scale-invariant systems, we derive general nonperturbative properties for the shape dependence of $\chi$ and reveal surprising relations to the EE associated with corners in the entangling surface. We obtain closed-form expressions for $\chi$ in 2D and 3D within a model that arises in the study of conformal field theories (CFTs), and we use them to obtain Ansätze without fitting parameters for the 2D and 3D free boson CFTs. Our numerical lattice calculations show that the Ansätze are highly accurate. Finally, we discuss how the torus EE can act as a fingerprint of exotic states such as gapless quantum spin liquids, e.g., Kitaev’s honeycomb model.

Figure 1

(a),(b) 2D space with a torus topology. We study the EE of a cylindrical region $A$. (c) Constraints on the torus EE function $\chi (\theta )$ result from dividing $A$ into three parts and applying strong subadditivity. (d) Region with a sharp corner.

Figure 2

Comparing the universal torus and corner EE of various CFTs in 2D.

Figure 3

Torus function $\chi$ for the massless scalar in 2D for various aspect ratios where $b=L_x/L_y$, with a vertical offset for clarity. The points represent numerical data, and the lines are the predictions obtained using ${\chi }_{\mathrm{EMI}}$ without any fitting parameters. (Inset) Dirac fermion data [21] at $b=1$, the corresponding ${\chi }_{\mathrm{EMI}}$, and the complex scalar data for comparison. The axes represent the same quantities as in the main plot.

Figure 4

Torus function ${\chi }^{3\mathrm{D}}$ for the massless free boson in 3D for various aspect ratios where $b_y=b_z=b$, with a vertical offset. The points represent numerical data and the line is the prediction from the Ansatz ${\chi }_{\mathrm{EMI}}^{3\mathrm{D}}$ without any fitting. (Inset) Opposing faces of the box are identified to give a 3-torus. Region $A$ is a hypercylinder extending along $x$.