Berry phase of the composite-fermion Fermi sea: Effect of Landau-level mixing

We construct explicit lowest-Landau-level wave functions for the composite-fermion Fermi sea and its low-energy excitations following a recently developed approach [Pu, Wu, and Jain, Phys. Rev. B 96, 195302 (2017)] and demonstrate them to be very accurate representations of the Coulomb eigenstates. We further ask how the Berry phase associated with a closed loop around the Fermi circle, predicted to be $\pi$ in a Dirac composite fermion theory satisfying particle-hole symmetry [D. T. Son, Phys. Rev. X 5, 031027 (2015)], is affected by Landau-level mixing. For this purpose, we consider a simple model wherein we determine the variational ground state as a function of Landau-level mixing within the space spanned by two basis functions: the lowest-Landau-level projected and the unprojected composite-fermion Fermi sea wave functions. We evaluate Berry phase for a path around the Fermi circle within this model following a recent prescription, and find that it rotates rapidly as a function of Landau-level mixing. We also consider the effect of a particle-hole symmetry-breaking three-body interaction on the Berry phase while confining the Hilbert space to the lowest Landau level. Our study deepens the connection between the $\pi$ Berry phase and the exact particle-hole symmetry in the lowest Landau level.

Figure 1

The Berry phase $\gamma$ as a function of the LL mixing parameter $\kappa$ for a Fermi sea with $N=12,N=24,N=36$ and $N=10$ composite fermions. The first three correspond to a closed loop of a CF hole, while the last one is for the closed loop of a CF particle.

Figure 2

Exact (blue squares) and CF (red dots) energy spectra in momentum sectors $K_1-K_2$ for $N=10$ particles exposed to a flux $N_{\varphi }=20$. The momentum $K_1$ is given on the x axis, and $K_2$ is displayed at the top; for each $K_1$, $K_2$ is chosen to match the momentum of the lowest energy state. For torus geometry, the spectra for $K_1$ and $K_1+N$ are identical.

Figure 3

This figure depicts the various paths we have considered for four Fermi sea configurations. The CF Fermi sea configurations in (a)–(g) consist of $N=12$ composite fermions, those in (h)–(l) have $N=24$ composite fermions, and (m) contains $N=36$ composite fermions; all of these configurations are obtained by creating a CF hole at the Fermi energy of a “symmetric” CF Fermi sea configuration. In all cases, the quoted value of $N$ excludes the CF hole. The panels (n)–(p) to a CF Fermi sea state with $N=10$ composite fermions, including a CF particle (red dot) outside the Fermi contour. The red crosses connected by dashed green lines indicate the successive positions of the CF hole (particle) along the closed path.

Figure 4

(a) The dependence of kinetic energy per particle ${\epsilon }_K$ as a function of $\beta$ for $N=12$, $N=24$, $N=36$, and $N=10$. At $\beta =1$ the CFFS fully projected into LLL, so the kinetic energy is zero, while at $\beta =0$ we have ${\epsilon }_K\sim 0.1\hslash {\omega }_c$. (b) The dependence of $\beta$ as a function of LL mixing parameter $\kappa$ for $N=12$, $N=24$, $N=36$, and $N=10$. (c) The percentage gain in energy, $\%\mathrm{\Delta }E$, as a function of LL mixing parameter $\kappa$ for $N=12,N=24$, $N=36$, and $N=10$. The energy gain by LL mixing is on the same order as or larger than that found by an earlier diffusion Monte Carlo study [48].

Figure 5

The dependence of Berry phase as a function of kinetic energy per electron for $N=12,N=24,N=36$, and $N=10$ around the paths in Figs. 3, 3 3, and 3. The plot for $N=10$ has been translated by $4\pi$ relative to Fig. 1 to be fitted in the figure. The Berry phase rotates by approximately $3\pi$ for $N=12,5\pi$ for $N=24,7\pi$ for $N=36$, and $-7\pi$ for $N=10$ as the wave function is continuously projected. The Berry phase increases with kinetic energy when moving a CF hole around, while decreases with kinetic energy when moving a CF particle around.

Figure 6

Variation in the Berry Phase $\gamma$ for the CF Fermi sea for the PH-perturbed Hamiltonian $H=(1-x)H_{\mathrm{Coulomb}}+x{H}_{3\mathrm{B}}$, where $x$ sets the strength of the three-body interaction $H_{3\mathrm{B}}$. Panel (a) shows the parameter $\sigma$ as a function of $x$, where $\sigma$ is so defined (see text) that its deviation from unity is a measure of the degree of PH-symmetry breaking. Panels (b) and (c) depict the change in the Bery phase $\gamma$ as a function of $\sigma$ and $x$.