Energy projection and modified Laughlin states

We develop a method to efficiently calculate trial wave functions for quantum Hall systems which involve projection onto the lowest Landau level. The method essentially replaces the lowest Landau level projection by projection onto the $M$ lowest eigenstates of a suitably chosen Hamiltonian acting within the lowest Landau level. The resulting “energy projection” is a controlled approximation to the exact lowest Landau level projection which improves with increasing $M$. It allows us to study the projected trial wave functions for system sizes close to the maximal sizes that can be reached by exact diagonalization and can be straightforwardly applied in any geometry. As a first application and test case, we study a class of trial wave functions first proposed by Girvin and Jach [Girvin and Jach, Phys. Rev. B 29, 5617 (1984)], which are modifications of the Laughlin states involving a single real parameter. While these modified Laughlin states probably represent the same universality class exemplified by the Laughlin wave functions, we show by extensive numerical work for systems on the sphere and torus that they provide a significant improvement of the variational energy, overlap with the exact wave function and properties of the entanglement spectrum.

Figure 1

[Left (sphere)] Overlaps and variational energies for $N=10$ electrons on a sphere (full Hilbert space dimension is of order ${10}^7$, after use of symmetries 319), using $1.2\times {10}^8$ MC samples. Eigenstates calculated using iterative diagonalisation methods discussed in Sec. 6. (Top) Exact and approximate (indicated by $d=0)$ overlaps of the $\nu =1/3$ Laughlin wave function with consecutive LLL-Coulomb eigenstates (with the Coulomb ground state on the left). (Middle) Cumulative square overlap $f$ and energy estimate. The expected limit values for $f$ and $E$ are indicated. (Bottom) Absolute differences between $E_n$ and $f_n$ (note the logarithmic scale). [Right (torus)] Overlaps and variational energies for $N=10$ electrons on a square torus (full Hilbert space dimension is of order ${10}^7$, after use of symmetries of order $5\times {10}^5)$, using $2\times {10}^7$ MC samples. Panel content is the same as for panels on the left-hand side. The appearance of many near zero overlaps in the upper panel is due to the $C_4$ symmetry of the system on the square torus.

Figure 2

Overlaps (upper panels in each figure) and cumulative squared overlaps (lower panels) with eigenstates of the lowest Landau level (LLL) and second Landau level (SLL) Coulomb potentials. Eigenstates calculated using iterative diagonalization methods discussed in Sec. 6. [Left (Sphere)] Results for the modified Laughlin states on a sphere at filling $\nu =\frac{1}{3}$ for $d=0$ (top), $d=1$ (middle), $d=7$ (lower), for $N=10$ electrons. [Right (torus)] Results for the modified Laughlin states on a square torus at filling $\nu =\frac{1}{3}$ for $d=0$ (top), $d=1$ (middle), $d=6$ (lower), for $N=6$ electrons.

Figure 3

LLL content $f$ of the modified Laughlin state as a function of $d$ and $N$ on the sphere (left) and torus (right).

Figure 4

Overlaps (upper panel in each figure) and cumulative squared overlaps (lower panels) with eigenstates of the lowest Landau level (LLL) Coulomb potentials. Eigenstates calculated using iterative diagonalization methods discussed in Sec. 6. [Left (sphere)] Results for modified Laughlin states at filling $\nu =\frac{1}{3}$ for $N=10$ particles on the sphere with $d=1$ (top) and $d=2$ (bottom). [Right (torus)] Results for modified Laughlin states on a square torus at filling $\nu =\frac{1}{3}$ for $d=1$ (top) and $d=2$ (bottom) for $N=10$ electrons.

Figure 5

Squared overlap of the modified Laughlin states with the Coulomb ground state, as a function of $d$ and $N$ on the sphere (left) and torus (right). The insets zoom in around the optimal values of $d$. Note that error bars are included but so small they are not visible.

Figure 6

Variational energy per particle with varying system size on the sphere (left) and torus (right), after background subtraction (see main text) for various values of $d$ (indicated in the figure). Energies plotted against $1/N$ to indicate scaling behavior. Errors are indicated where they are larger than the symbols used. The sphere plot shows limit values of the energy as $N\to \infty$ obtained from linear extrapolations in $1/N$. The torus plot shows linear extrapolations for the Coulomb and Laughlin states.

Figure 7

(Top) Variational LLL Coulomb energy per particle of the modified Laughlin states as a function of $d$ for $N=10$ particles on a sphere (left) and on a torus (right). The exact Coulomb energy is subtracted. The plots show clearly that a very significant reduction of the variational energy is obtained upon changing from the Laughlin $\left(d=0\right)$ state to the modification with the optimal value of $d$. (Bottom) Scaling with $N$ of the value of $d$ where the minimum energy is obtained. The minimum $d$ values at various $N$ are obtained from parabolic fits such as those shown in the upper panels.

Figure 8

Two-particle correlation function of the modified Laughlin wave functions on the sphere (left panels) and torus (right panels) for $N=10$ particles. (Top) Dashed lines show the correlation function of the unprojected alternative Laughlin wave functions. Solid lines show the correlations after projection to the LLL. On the sphere, the distance measure is the chord length $r_{12}=2R|u_1{v}_2-u_2{v}_1|.$ On the torus, the curve represents a diagonal cut of the square torus. The inset is a plot for the full torus, indicating the cut shown in the main panel. (Bottom) The same functions as in the upper panels, but with the correlation function for the $\left(d=0\right)$ Laughlin wave function subtracted, to highlight the differences between Coulomb, Laughlin and modified Laughlin correlations.

Figure 9

Entanglement spectra for $N=10$ particles on a sphere as function of $d=0$ (upper left), $d=1$ (upper right), $d=1.3$ (lower left), and $d=5$ (lower right). All figures also show the Coulomb entanglement spectrum (dashes) for comparison.

Figure 10

Entanglement spectra for $N=10$ particles on a torus. (Left) ES of the exact Laughlin state (crosses) superimposed on the ES for the energy projected $d=0$ state (dashes). The difference between these spectra is due to Monte Carlo error in determining the $d=0$ state. We see that the projected data cannot be trusted for entanglement energies above $\xi =12$. (Middle) Comparison of exact Coulomb (dashes) and Laughlin spectra (crosses). We see that, as in the case of the sphere ES, many states are missing from the Laughlin ES at entanglement energies above $\xi \approx 8$. (Right) ES of the $d=1.5$ modified Laughlin state (crosses) superimposed on the exact Coulomb ES (dashes). The missing states in the Laughlin ES are accounted for, and the fit on the other states is also improved over the middle panel.