Trial wave functions for a composite Fermi liquid on a torus

We study the two-dimensional electron gas in a magnetic field at filling fraction $\nu =\frac{1}{2}$. At this filling the system is in a gapless state which can be interpreted as a Fermi liquid of composite fermions. We construct trial wave functions for the system on a torus, based on this idea, and numerically compare these to exact wave functions for small systems found by exact diagonalization. We find that the trial wave functions give an excellent description of the ground state of the system, as well as its charged excitations, in all momentum sectors. We analyze the dispersion of the composite fermions and the Berry phase associated with dragging a single fermion around the Fermi surface and comment on the implications of our results for the current debate on whether composite fermions are Dirac fermions.

Figure 1

Comparison between exact energy landscape and composite fermion energy landscape, given by (4) for $N_e=15$. The lowest (highest) energy is at the darkest (lightest) red. The ground states at $(K_1,K_2)=(6,6)$ and $(K_1,K_2)=(5,6)$, respectively, are accurately identified by the CF model, as well as the low energy features. The arrows show the directions of the two reciprocal unit vectors $\mathbf{G}$. Energy is in arbitrary units.

Figure 2

The FS configurations with the lowest CF energy for the global ground state at $N_e=15$ for a square and hexagonal torus. These are located at $(K_1,K_2)=(6,6)$ and $(K_1,K_2)=(5,6)$, respectively. The center of mass is at the blue small dot.

Figure 3

Plot of the overlap between all of the 30 FS states with smallest CF energy and their PH conjugated/orbital inverted duals for $N_e=10$ in the momentum sectors $(5,5)$ and $(2,3)$. Left = square torus, right = hexagonal torus. Upper = particle hole conjugation, Lower = orbital inversion. The momentum sectors $(2,3)$ and $(8,7)$ are related under PH conjugation/orbital inversion as $(K_1,K_2)\to (N_e-K_1,N_e-K_2)$. At $N_e=10$ the $(5,5)$ sector is self dual. As can be seen, in square geometry the overlaps are extremely high with both the inverted state and the PH conjugated state. In the hexagonal geometry, only the FS configurations with the lowest CF energy have high overlap with their inverted and PH conjugated duals. The FS conjurations are in general more inversion symmetric than PH symmetric.

Figure 4

Comparing variational energy of FS states to the exactly diagonalized energy for $N_e=10$ (upper) and $N_e=11$ (lower) on a square torus in all momentum sectors (up to $C_4$ rotations). The blue horizontal bars show the exact energy of the Coulomb interaction from diagonalization. The red discs are the variational energy of (one of) the FS configuration(s) that minimize the CF energy within a given $(K_1,K_2)$ sector. The green discs are the eigenenergies of the Coulomb interaction, projected onto the space spanned by all FS configurations (related by symmetry) that minimize the CF energy. The purple/cyan discs are eigenenergies of the Coulomb interaction, projected onto the space spanned by all FS configurations (related by symmetry) that minimize the CF energy as well as their PH conjugated/orbital inverted reversed flux duals. The colored numbers in each momentum sector show the squared overlap between the lowest energy FS superposition (of the corresponding color) and the Coulomb ground state in that momentum sector. As can be seen, the variational energy of the chosen FS configurations (green) is close to the Coulomb energy, and the squared overlap is almost always very close to one. The larger overlaps are found in momentum sectors with lower energies and where the energy gap is large. Adding in PH conjugation (purple) or orbital inversion (cyan) has most often negligible impact on the variational energy and overlap with the Coulomb ground state. This is related to the fact that most FS configurations are highly PH/inversion symmetric, see Fig. 3.

Figure 5

Comparing variational energy of FS states to the exactly diagonalized energy for $N_e=12$ (left) and $N_e=13$ (right) on a square torus for the lowest energy momentum sectors (up to $C_4$ rotations). The symbols are as in Fig. 4. Just as in Fig. 4, if there are more than one FS configurations that minimize the CF energy all of the FS states need to be taken into account to obtain a good approximation of the Coulomb ground states. Also here, PH conjugation/inversion has a minute effect on the wave function.

Figure 6

Comparing variational energy of FS states to the exactly diagonalized energy for $N_e=10$ on the $\tau =2ı$ rectangular torus (upper) and the hexagonal torus (lower). The symbols are as in Fig. 4. Here we see that the composite fermion wave function is generally less good at approximating the Coulomb ground state than was the case on the square torus. On the rectangular torus, adding in PH/inversion symmetry does little to improve the wave functions, but on the hexagonal tours a definite improvement can be seen in some momentum sectors, e.g., $(2,4)$ and $(5,5)$. This is not surprising given that the FS configurations have less PH/inversion symmetric on the hexagonal torus than on the square torus. See Fig. 3.

Figure 7

Extrapolation of the exact average ground state (GS) energy and average (momentum preserving) first excited (FE) state for a square torus to the thermodynamic limit. Data for hexagon looks qualitatively the same. Inset: The gap (cyan) in all momentum sectors, and the gap averaged over all momentum sectors (black bars) extrapolated to the thermodynamic limit (blue line). We conclude that the average gap is closing in the thermodynamic limit. Main plot: The ground state (purple) and first exited state (cyan) energy density for all momentum sectors. Average (red) and global (green) ground state energy extrapolated to the thermodynamic limit. The blue line is the gap size extracted from the inset superimposed on the average ground state energy (red). Note: The horizontal position of the energies of the GS and FE states are slightly shifted with respect to each other for better readability.

Figure 8

Main plot: The overlap between the Coulomb ground state at $N_e$ electrons and $N_s=2N_e\pm 1$ flux quantum and the CF wave function (7) for that flux. The overlap is higher than 0.99 for all system sizes considered, except for two data points at $N_e=9,10$. There seems to be an even odd effect as to whether $\mathrm{CF}(\pm )$ has higher overlap at a given system size. Inset: Overlap reduction factor per particle $\epsilon$, related to the overlap as $|〈{\psi }_{\mathrm{CF}}|{\psi }_{\mathrm{ED}}〉{|=(1-\epsilon )}^{N_e}$. This is stable at $\epsilon \sim 8\times {10}^{-3}$ almost independently of system size.

Figure 9

Inset: The exact gap at $N_s=2N_e+1$ (red) and $N_s=2N_e-1$ (cyan) extrapolated to the thermodynamic limit for a square torus. The gap appears to be closing in the thermodynamic limit, both from higher and lower number of fluxes, but we are not yet in a scaling regime. Main plot: The energy density of the exact ground state (GS), first excited (FE) state, and composite fermion state (CF) for a square torus with $N_s=2N_e\pm 1$, and their extrapolations to the thermodynamic limit. The red and cyan lines are the gap size extracted from the inset, superimposed on the ground state energies (green and yellow). The composite fermions model the ground states very well, and variational energy of the composite fermion states are almost indistinguishable from the energies of the exact ground states on the scale of the plot. The black dashed line is the extrapolation of the global ground state energy for a half filled system (same as green line in Fig. 7).

Figure 10

Main plot: The charge gap at constant torus area $A$ (as opposed to constant magnetic length ${\ell }_B)$ for adding (removing) a magnetic flux at $\nu =1/2$. Although the finite size effects are large, it is clearly seen that the energy needed to add (remove) a magnetic flux is negative (positive). See comparison with $\nu =1/3$ in inset. The dashed lines show the extrapolation of the gap to the thermodynamic limit. Inset: The charge gap at constant torus area $A$ for adding (removing) a magnetic flux at $\nu =1/3$, as comparison. Here there is a (positive) energy gap both for adding or removing a magnetic flux.

Figure 11

The estimate of the effective mass for nonrelativistic CFs or the speed off light for Dirac CFs for a square and hexagonal torus. We find that the masses and velocities are stable in a wide range of $N_e$ and for both square and hexagonal tori.

Figure 12

Path of dragging two CFs (red discs) around the Fermi disk such that momentum is conserved. The two colors (blue,green) show the paths for the two CFs on either side of the Fermi disk. After the move, the Fermi disk is returned to its original configuration. There is a phase of $\pi$ from the exchange of the electrons and in addition a Berry phase.