Plasma picture of the fractional quantum Hall effect with internal $\text{SU}\left(K\right)$ symmetries

We consider trial wave functions exhibiting $\text{SU}\left(K\right)$ symmetry, which may be well suited to grasp the physics of the fractional quantum Hall effect with internal degrees of freedom. Systems of relevance may be either spin-unpolarized states $\left(K=2\right)$, semiconductor bilayers $\left(K=2,4\right)$, or graphene $\left(K=4\right)$. We find that some introduced states are unstable, undergoing phase separation or phase transition. This allows us to strongly reduce the set of candidate wave functions eligible for a particular filling factor. The stability criteria are obtained with the help of Laughlin’s plasma analogy, which we systematically generalize to the multicomponent $\text{SU}\left(K\right)$ case. The validity of these criteria is corroborated by exact-diagonalization studies for SU(2) and SU(4). Furthermore, we study the pair-correlation functions of the ground state and elementary charged excitations within the multicomponent plasma picture.

Figure 1

Stability of the $\left(73n\right)$ wave function. Both filling factors, ${\nu }_1$ (long dashed line) and ${\nu }_2$ (short dashed line), and the ${\lambda }_{-}$ eigenvalue (solid line) of the $\left(73n\right)$ wave function are plotted as a function of $n$. For $n\le 3$ (part I), all quantities are positive and the corresponding state is stable. In part II, one of the filling factors is negative. Part III exhibits positive filling factors, but the state is still unstable because of the negative eigenvalue; the system eventually undergoes a phase separation.

Figure 2

Sktech of the plasma corresponding to the $\left(73n\right)$ wave function for (a) $n\le 3$ and (b) $7\le n$. (a) Intracomponent repulsions are stronger than intercomponent repulsions $\left(n\le m_1,m_2\right)$. Type-(1) particles (black) are therefore, on the average, surrounded by type-(2) particles (gray). This yields a stable state of two homogeneous interpenetrating plasmas. (b) If the intercomponent repulsion is stronger than the intracomponent repulsion, the plasmas have a tendency to phase separate to minimize the number of neighbors from different types.

Figure 3

Stability of the $\left(33n\right)$ wave function. Equal filling factors, ${\nu }_1$ and ${\nu }_2$ (long dashed line), and the ${\lambda }_{-}$ eigenvalue (solid line) of the $\left(33n\right)$ wave function are plotted as a function of $n$. For $n\le 3$ (part I), the state is stable as in Fig. 1. There is no corresponding part II, since the densities never vanish. In part III, the plasmas tend to phase separate.

Figure 4

Stability of the $\left(3535,n22\right)$ wave functions. Filling factors ${\nu }_1$ and ${\nu }_3$ (long dashed line), ${\nu }_2$ and ${\nu }_4$ (short dashed line), and the third eigenvalue ${\lambda }_3$ (solid line) of the $\left(3535,n22\right)$ generalized Halperin wave function are plotted as a function of $n$. For $n\le 2$ (part I), all quantities are positive and the corresponding state is stable. Part II corresponds to unphysical negative densities with a negative eigenvalue $\left(-0.08\right)$. For $n\ge 4$, because of the eigenvalue ${\lambda }_3$ being still negative, the plasmas (1)–(3) phase separate from (2)–(4).

Figure 5

Phase diagram for (a) 331 and (b) 113 ground states. Exact diagonalization is performed with $N=6$ electrons and (a) $2S=9$ (b) $2S=11$ flux quanta. An unpolarized state is assumed. Pseudopotentials $V_{\uparrow \uparrow }=V_{\downarrow \downarrow }$ are denoted $V\left(\text{intra}\right)$, whereas $V_{\uparrow \downarrow }$ is $V\left(\text{inter}\right)$. The overlap between Halperin’s wave functions (exact ground state of the unperturbed model) and the exact ground state is plotted in gray scale. We also indicate the separation between incompressible and compressible states (i.e., whether the ground state is in the $L=0$ sector or not) by a black line.

Figure 6

Pair-correlation functions related to (a) (331) states for $N=10$ particles and (b) (113) states for $N=8$ particles, plotted as a function of distance (in units of magnetic length). Intracomponent pairs $\left(g_{11}=g_{22}\right)$ are plotted as a solid line and intercomponent pairs $\left(g_{12}=g_{21}\right)$ as dashed lines. In the (331) graph, all functions go to 1 at infinity, which is typical of uniform density. In contrast, the (113) intracomponent function vanishes at large distances, thus endowing a particle aggregate.

Figure 7

Pair-correlation functions related to (a) (3333,111) and (b) (3333,033) states for $N=8$ particles, plotted as a function of distance (in units of magnetic length). Intracomponent pairs $\left(g_{ii}\right)$ are plotted as a solid line. In the (3333,111) graph, the dashed line corresponds to the intercomponent pairs ($g_{i\ne j}$, all being equal). In the (3333,033) graph, the dotted line shows the correlation function for electrons with opposite spin within any of each layer or valley. The dashed line corresponds to intercomponent pairs between two different valleys or layers, which are trivially constant in this noncorrelated valley or layer example.