Lattice Monte Carlo for quantum Hall states on a torus

The Monte Carlo method is very useful for studying various model states proposed for the fractional quantum Hall systems. In this paper, we introduce a lattice Monte Carlo method based on an exact lattice formalism for quantum Hall problems defined on a torus. This “lattice representation” is applied to study the many body Berry phase of the composite-Fermi liquid phase in a half-filled Landau level. The Monte Carlo result agrees with the exact numerical result found on small systems sizes and is consistent with the prediction from the Dirac fermion low-energy effective theory. Many other aspects of other model states, including the structure factor, Coulomb energy and particle-hole symmetry, are studied and discussed.

Figure 1

$N_e=50,\nu =1/q$. Laughlin-hole state Berry phase ($\mathrm{\Phi }$) vs area the loop enclosed ($A$). The dot and the square show Monte Carlo data for $q=3$ and $q=5$, respectively. The Monte Carlo error is bounded by the line width. This demonstrates that $\mathrm{\Phi }=A/q$. The inset shows the overlap $|\langle \psi (0\left)\right|\psi \left(\mathrm{\Delta }x\right)\rangle |$ vs $\mathrm{\Delta }x$ which allows us to take step step length to be $\mathrm{\Delta }x=0.02$, i.e., the quantum distance between consecutive steps is small.

Figure 2

$N_e=13$ CFL Berry phase. Cross mark “x” represents the “composite fermion” we are moving. This is a consistency check with the same calculation done in Ref. [25] (but using a different numerical approach). This data can be interpreted through Eq. (24), which shows that in addition to a ${\mathbb{Z}}_2$ piece there is a piece depending on the direction of motion around the Fermi surface. When this is accounted for we find a residual $“-1”$ from the ${\mathbb{Z}}_2$ part whenever the composite fermion encloses the Fermi sea.

Figure 3

$N_e=69$ CFL Berry Phase. Cross mark “x” represents the “composite fermions” we are moving. The results are again consistent with Eq. (24)

Figure 4

The guiding center structure factor for $N_e=50$ electrons in the Laughlin $\nu =1/3$ state. (a) shows its plot together with error bars. The Gaussian function $e^{-\frac{1}{4}{\mathit{q}}^2{l}_B^2}$ limits us to see $S\left(\mathit{q}\right)$ only within a window. (b) We check the long-wavelength expansion ($c_4$ and $c_6$) in (28) by comparing it to the Monte Carlo data, where $c_4,c_6$ is given by (29) and all other $c_i=0$. It can be seen that the long-wavelength behavior of $S\left(\mathit{q}\right)$ is correctly described by (29).

Figure 5

CFL $S\left(\mathit{q}\right)$ and dipole configuration. The peak of the structure factor and composite fermion Fermi surface have the same shape. The radius of the former is twice of the latter.

Figure 6

Overlaps between a wave function and its PH conjugate. The red points from come from doing an exact second quantization of the model wave function as in Ref. [25], while the blue comes from a Monte Carlo calculation. For each $N_e$, the configuration of $d$'s with the largest overlap was used.