Screening properties and phase transitions in unconventional plasmas for Ising-type quantum Hall states

Utilizing large-scale Monte Carlo simulations, we investigate an unconventional two-component classical plasma in two dimensions which controls the behavior of the norms and overlaps of the quantum-mechanical wave functions of Ising-type quantum Hall states. The plasma differs fundamentally from that which is associated with the two-dimensional $XY$ model and Abelian fractional quantum Hall states. We find that this unconventional plasma undergoes a Berezinskii-Kosterlitz-Thouless phase transition from an insulator to a metal. The parameter values corresponding to Ising-type quantum Hall states lie on the metallic side of this transition. This result verifies the required properties of the unconventional plasma used to demonstrate that Ising-type quantum Hall states possess quasiparticles with non-Abelian braiding statistics.

Figure 1

Illustration of interactions between the particles in the 2D system. The $w$ particles only interact by the second Coulomb-like interaction with charge $-Q_2$, whereas the $z$ particles carry charge $Q_1$ for the first Coulomb-like interaction and $Q_2$ for the second Coulomb-like interaction. Thus, the intraspecies interaction among the $w$ particles, shown in (a), and the interspecies interaction between $w$ and $z$ particles, shown in (b), are given by $Q_2$ only, whereas the intraspecies interaction among the $z$ particles, shown in (c), are determined by $Q_1$ in addition to $Q_2$. Interactions between the $z$ particles and the neutralizing background are omitted from the figure.

Figure 2

Plot of the inverse dielectric constant ${\epsilon }_{22}^{-1}$ for the model in Eq. (7) with $Q_1=0$ and $1\le Q_2^2\le 4.8$. Results are presented for three different values of packing fraction $\eta$ and three different values of system size $N$.

Figure 3

Plot of the inverse dielectric constant ${\epsilon }_{22}^{-1}$ for the model in Eq. (7) with $Q_1=2$ and $1\le Q_2^2\le 4.8$. Results are presented for three different values of packing fraction $\eta$ and three different values of system size $N$.

Figure 4

Snapshots of the charge configuration at $Q_2^2=1,3,5$ when $Q_1=2$, $\eta =5\times {10}^{-4}$, and $N=200$. Red markers (solid circles) are $w$ particles, while blue markers (open circles) are $z$ particles. The marker diameters are about 5 times larger than hard disk diameter $d$.

Figure 5

The critical value of $Q_2^2$ found by curve fitting to Eq. (33) with two free parameters. Results are presented for four values of the packing fraction $\eta$ and for $Q_1=0$ and $Q_1=2$. Fourteen system sizes in the range $20\le N\le 2000$ have been used.

Figure 6

The universal jump value determined by curve fitting to Eq. (33) with two free parameters. Results are presented for four values of the packing fraction $\eta$ and for $Q_1=0$ and $Q_1=2$. Fourteen system sizes in the range $20\le N\le 2000$ have been used.

Figure 7

The fourth-order modulus ${\gamma }_{2222}$ as a function of coupling $Q_2^2$ for five different system sizes $N$, when $Q_1=2$ and $\eta =5\times {10}^{-4}$.

Figure 8

The size of the dip in the fourth-order modulus $|{\gamma }_{2222,\min }|$ as a function of inverse system size $N^{-1}$. The packing fraction is $\eta =5\times {10}^{-4}$, and results for $Q_1=0$ and $Q_1=2$ are shown. The inset shows the results on a log-log scale. System sizes in the range $60\le N\le 10000$ are used.

Figure 9

The coupling value at the minimum of the dip in the fourth-order modulus as a function of inverse system size $N^{-1}$. The packing fraction is $\eta =5\times {10}^{-4}$, and results for $Q_1=0$ and $Q_1=2$ are shown. The inset shows the results on a log-log scale. System sizes in the size $60\le N\le 10000$ are used.

Figure 10

The critical value of $Q_2^2$ found by curve fitting to Eq. (33) with one free parameter. Results are presented for five values of the packing fraction $\eta$ and for two values of $Q_1$.

Figure 11

Snapshots of the charge configuration at $Q_1=0$ and $Q_1=2$ when $Q_2^2=7$, $\eta =2\times {10}^{-3}$, and $N=200$. Red markers (solid circles) are $w$ particles and blue markers (open circles) are $z$ particles. The marker diameters are about 2.5 times larger than hard disk diameter $d$.

Figure 12

Plot of the type 2 inverse dielectric constant for $Q_1=0$ and $Q_1=2$ with $N=100$, $\eta =5\times {10}^{-3}$ in the range $3\le Q_2^2\le 12$.

Figure 13

Plot of the size dependence in the inverse dielectric constant ${\epsilon }_{22}^{-1}\left(N\right)$ for 14 different system sizes in the range $20\le N\le 2000$ at three different values of the coupling $Q_2^2$. The best fit according to the fit function in Eq. (33) with two free parameters, is given as the corresponding solid line in all three cases. The packing fraction is $\eta =2\times {10}^{-3}$ and $Q_1=0$.

Figure 14

Plot of the goodness of fit parameter $A^2$ and the corresponding free parameter ${\epsilon }_2^{-1}\left(\infty \right)$ obtained when curve fitting to the critical finite-size relation given in Eq. (33). The results are given as a function of $Q_2^2$. System sizes $N$ and $\eta$ and $Q_1$ are the same as in Fig. 13. Error estimates are obtained by the jackknife method.