How Universal Is the Entanglement Spectrum?

It is now commonly believed that the ground state entanglement spectrum (ES) exhibits universal features characteristic of a given phase. In this Letter, we show that this belief is false in general. Most significantly, we show that the entanglement Hamiltonian can undergo quantum phase transitions in which its ground state and low-energy spectrum exhibit singular changes, even when the physical system remains in the same phase. For broken symmetry problems, this implies that the low-energy ES and the Rényi entropies can mislead entirely, while for quantum Hall systems, the ES has much less universal content than assumed to date.

Figure 1

Phase diagram of the entanglement Hamiltonian $H_E(\delta )$ for an Ising model. The blue dot at ${\delta }_c^P$ represents a physical transition. The black star at ${\delta }_c^E$ represents a pseudo-QPT. ${\mathrm{FM}}_2$ is a spurious ordered phase. The dashed line separating ${\mathrm{FM}}_2$ and PM represents a pseudotransition in $d\ge 3$. If generic, $T_{Ec}^{+}=\infty$.

Figure 2

Left: The RK Ising model on the square lattice. Right: The three layer Chern insulator model with magnetic flux between the layers. Region $B$ is traced out in the ground state to obtain the entanglement Hamiltonian in $A$.

Figure 3

Entanglement spectra [Eq. (6)] of the ground states of the three layer model with $M_1=-2.5$, $M_2=-1$, $M_3=-3.5$, and $L_x=L_y=200$. (a) $\lambda =0$, ${\overrightarrow{A}}_l=0$. (b) $\lambda =1/2$, ${\overrightarrow{A}}_1=-{\overrightarrow{A}}_3=(-\pi /2,-\pi /2)$, ${\overrightarrow{A}}_2=(0,0)$. (c) ${\overrightarrow{A}}_1=-{\overrightarrow{A}}_3=(-\pi ,-\pi )$, ${\overrightarrow{A}}_2=(0,0)$.