Entanglement signatures of quantum Hall phase transitions

We study quantum phase transitions involving fractional quantum Hall states using numerical calculations of entanglements and related quantities. We tune finite-size wave functions on spherical geometries by varying the interaction potential away from the Coulomb interaction. We uncover signatures of quantum phase transitions contained in the scaling behavior of the entropy of entanglement between two parts of the sphere. In addition to the entanglement entropy, we show that signatures of quantum phase transitions also appear in other aspects of the reduced density matrix of one part of the sphere.

Figure 1

(Color online) Possible behaviors of $s_0$, defined in Eq. (1), within a topologically ordered phase $\left(X<X_c\right)$ and after a phase transition into a different phase $\left(X>X_c\right)$.

Figure 2

(Color online) Overlaps of ground-state wave functions at interactions relevant to fillings 1/3 and 5/2, respectively, with Laughlin and Moore-Read wave functions. A large-$N$ and a small-$N$ case are shown for each fraction.

Figure 3

(Color online) Top four: $S_{l_A}\left(N\right)$ versus $1/N$ for several $\delta V_1$ values, displaying how the smooth behavior of these curves disappear. Bottom left: quality of fit $\left({\chi }^2\right)$ to a simple function, $c_0+c_1/N^2$. Bottom right: attempt to extract $s_0$. The quantity is less and less meaningful for more negative $\delta V_1$ (discussion in text). The topological value $\frac{1}{2}\text{ln}3\approx 0.55$ is shown by dashed horizontal line.

Figure 4

(Color online) Top panels: entanglement entropy plotted against circumference of $A$ block; $\nu =1/3$, $N=12$. The linearity is progressively destroyed as one reduces the pseudopotential $V_1$. Lower panel: ${\chi }^2$ for linear fits to the $S_{l_A}$ versus circumference data.

Figure 5

(Color online) Top row: entanglement entropies for hemisphere partitioning, plotted against sphere radius; $\nu =1/3$. Center row shows quality of linear fit as function of pseudopotential $\delta V_1$ for $\nu =1/3$. Bottom row shows ${\chi }^2$ for $\nu =5/2$.

Figure 6

(Color online) Top panels: entanglement spectrum, $\nu =1/3$, $N=12$, block $A$ containing $l_A=17$ orbitals and $N_A=6$ fermions. Main plot: ground state for $\delta V_1=0.0$ (Coulomb interaction). Ellipse indicates the most prominent “conformal” part of the spectrum. Arrow indicates the entanglement gap ${\delta }_0$ between CFT and non-CFT parts of the spectrum. Inset shows exact Laughlin state, which has no higher-lying non-CFT part. Lower panels: empty dots show two lowest levels at $L_z^{\left(A\right)}=54$, plotted against $\delta V_1$. Filled squares show entanglement gap, the difference of the two lowest levels.

Figure 7

(Color online) Left panels: entanglement entropy and majorization plotted against $\delta V_1$ for $N=12$, $\nu =1/3$. Right panels: same for $N=18$, $\nu =5/2$. Majorization at some $\delta V_1$ is taken to be 1 if the corresponding ${\rho }_A$ spectrum majorizes a spectrum at an adjacent value of $\delta V_1$; otherwise it is set to 0.