Interaction-tuned compressible-to-incompressible phase transitions in quantum Hall systems

We analyze transitions between quantum Hall ground states at prominent filling factors $\nu$ in the spherical geometry by tuning the width parameter of the Zhang-Das Sarma interaction potential. We find that incompressible ground states evolve adiabatically under this tuning, whereas the compressible ones are driven through a first-order phase transition. Overlap calculations show that the resulting phase is increasingly well described by appropriate analytic model wave functions (Laughlin, Moore-Read, Read-Rezayi). This scenario is shared by both odd $\left(\nu =1/3,1/5,3/5,7/3,11/5,13/5\right)$ and even denominator states $\left(\nu =1/2,1/4,5/2,9/4\right)$. In particular, the Fermi-liquid-like state at $\nu =1/2$ gives way, at large enough value of the width parameter, to an incompressible state identified as the Moore-Read Pfaffian on the basis of its entanglement spectrum.

Figure 1

(Color online) Overlap $\left|⟨{\Psi }_{\text{L}}|{\Psi }_{\text{exact}}⟩\right|$ between the exact Coulomb state for finite width (ZDS model) at $\nu =1/3$ and the Laughlin wave-function $N=4–12$ particles. Inset: same quantity but in the first-excited Landau level, i.e., $\nu =7/3$.

Figure 2

(Color online) Overlap $\left|⟨{\Psi }_{\text{L}}|{\Psi }_{\text{exact}}⟩\right|$ between the exact Coulomb state for finite width (ZDS model) at $\nu =1/5$ and the Laughlin wave function for $N=4–9$ particles. Inset: same quantity but in the first-excited Landau level, i.e., $\nu =11/5$.

Figure 3

(Color online) Overlap $\left|⟨{\Psi }_{\text{Pf}}|{\Psi }_{\text{exact}}⟩\right|$ between the exact Coulomb state for finite width (ZDS model) at $\nu =1/2$ and the Pfaffian for $N=8–18$ particles. Inset: same quantity but in the first-excited Landau level, i.e., $\nu =5/2$. Only nonaliased states are shown. Note: the critical width of the QPT increases with system size; however, for the three available points and $N\to \infty$, it extrapolates to a value of $4l_B$.

Figure 4

(Color online) Overlap $\left|⟨{\Psi }_{\text{exact}}\left(w=0\right)|{\Psi }_{\text{exact}}\left(w\right)⟩\right|$ between the exact Coulomb state for finite width (ZDS model) at $\nu =1/2$ and the CF sea state defined to be exact Coulomb ground state for zero thickness. Red circles represent filled CF shells $\left(N=n^2,n=3,4\right)$, blue triangles are the lowest excited states $\Delta N=N-n^2=\pm 1$ and so on.

Figure 5

Entanglement spectrum of the exact ground state for $N=18$ particles at $\nu =1/2$ in the LLL, just before $\left(w/l_B=0.8\right)$ and after $\left(w/l_B=1.0\right)$ the QPT, and the spectrum of Moore-Read Pfaffian for comparison. Vertical axes show the quantity $\xi =-\text{log}{\lambda }_A$, where ${\lambda }_A$ are the eigenvalues of the reduced density matrix of the subsystem $A$ which comprises of 8 particles and 15 orbitals, given as a function of angular momentum $L_z^A$. Data shown is only for the partitioning denoted by $\left[0\mid 0\right]$ in Ref. 23; other sectors give a similar result.