Phase diagram for bilayer quantum Hall effect at total filling ${\nu }_T=5$

We evaluate the phase diagram of the bilayer quantum Hall effect at total filling ${\nu }_T=5$, which is a bilayer phase coherent state at small separations and two uncoupled $5∕2$ states at large separations. Based on a combination of variational and exact calculations, we estimate that the transition between these states occurs at a layer separation of approximately one magnetic length. The composite fermion Fermi sea is not found to be relevant for any parameters.

Figure 1

(Color online) Phase diagram in the $w\text{−}d$ plane for a bilayer system, where $w$ is the width of each layer and $d$ is the interlayer separation (measured from center to center), both given in units of the magnetic length. The transition from the 111 wave function to the ${\mathrm{Pf}}^2$ wave function (see text for definition) occurs at $d\sim 0.9$ magnetic length, almost independent of the layer width. The region with $d<w$ is unphysical.

Figure 2

(Color online) Thermodynamic values of the interaction energies of the Pfaffian and the CF Fermi sea wave functions are shown as functions of $w$, i.e., the width of the layer. The former has lower energy in the entire parameter range considered. We have used the interaction in Eq. (A18) with three adjustable parameters.

Figure 3

(Color online) Thermodynamic limit of the energy per particle of the 111 wave function (curve) as a function of the layer separation $d$ for two values of width $w=0$ and 0.9 magnetic length. The energies have been computed for two approximations (with three and four adjustable parameters, shown by black dots and boxes, respectively), which are indistinguishable for $d\sim 1$. The energy of the Pfaffian wave function ${\mathrm{Pf}}^2$ (dashed horizontal line) is independent of layer separation. ${\mathrm{Pf}}^2$ has lower energy for $d$ greater than approximately one magnetic length.

Figure 4

(Color online) Overlaps of variational bilayer wave functions with the exact ground state wave functions as a function of the layer separation for 16 and 20 electrons. ${\mathrm{Pf}}^2$ is compared to the exact ground state at $2Q=2N-3$, and the 111 wave function to the ground state at $2Q=2N-1$. The layer thickness is set to $w=0$ in these calculations.

Figure 5

(Color online) Overlaps of CF fermi sea wave function with the exact ground state wave function as a function of the layer separation for 16 and 18 electrons. The layer thickness is set to $w=0$ in these calculations. For $2N=18$ particles, the CF Fermi sea in each layer has $L=0$ because we have a filled shells state (with $1+3+5$ particles in the lowest three $\Lambda$ levels). For $2N=16$, the state has one CF-hole in the third $\Lambda$ level, giving $L=2$. The nonmonotonic behavior of the overlap for $2N=18$ indicates nearly degenerate ground states in the exact solution, which was also confirmed by the long convergence times for the Lanczos procedure. Please note that the $y$-axis scales are different in Figs. 5, 4.

Figure 6

(Color online) Expectation values of the pseudospin operator ${\mathit{S}}^2$ of the true ground states as a function of the layer separation $d$ for $2Q=2N-3$ and $2Q=2N-1$.

Figure 7

(Color online) Open squares show the percent difference between $n=0$ LL and $n=1$ LL pseudopotentials of Coulomb interaction. The open circles show the percent differences between $n=0$ LL pseudopotentials of $V_{\text{long}}\left(r\right)$ and $n=1$ LL pseudopotentials of Coulomb interaction. Only pseudopotentials with odd $m$ are considered here.

Figure 8

(Color online) Percent difference between $n=0$ LL pseudopotentials of $V_{\mathrm{eff}}$ and $n=1$ LL pseudopotentials of the Coulomb interaction with three and four adjustable parameters.

Figure 9

(Color online) Thermodynamic extrapolation ($1∕N\to 0$ of three individual terms in $V_{\mathrm{eff}}$. $I_3$ is defined in text, and $P_0=e^{-r^2}$. The black squares (red dots) are obtained with the interparticle distance defined as the chord (great circle arc) distance between their coordinates on the sphere. The curves represent best quadratic fits. Similar fits are obtained for other terms in the interaction.