Entanglement spectrum and entanglement thermodynamics of quantum Hall bilayers at $\nu =1$

We study the entanglement spectra of bilayer quantum Hall systems at total filling factor $\nu =1$. In the interlayer-coherent phase at layer separations smaller than a critical value, the entanglement spectra show a striking similarity to the energy spectra of the corresponding monolayer systems around half-filling. The transition to the incoherent phase can be followed in terms of low-lying entanglement levels. Finally, we describe the connection between those two types of spectra in terms of an effective temperature leading to relations for the entanglement entropy which are in full analogy to canonical thermodynamics.

Figure 1

Left: Entanglement spectra obtained from balanced bilayers at layer separation $d=0.1\ell$ for various system sizes $N$. Right: Energy spectra of half-filled monolayers with $N/2$ electrons.

Figure 2

Entanglement spectra of balanced bilayers at layer separation $d=0.7\ell$ (left) and $d=1.3\ell$ (right).

Figure 3

The four lowest entanglement levels as a function of the layer separation for different system sizes. At the phase boundary at $d\approx 1.3\ell$ the entanglement spectrum qualitatively changes.

Figure 4

Top panels: Entanglement spectra of $N=12$ electrons with $T^z\in \{1,2\}$ are compared with the corresponding monolayer spectra. Left bottom panel: Entanglement entropy $S$ along with its variance $\Delta S$ as a function of the layer separation $d$ for a balanced bilayer system of $N=12$ electrons. Right bottom panels: The “inverse temperature” $\beta$ and the “specific heat” $C$ as a function of layer separation.