Berry Phase and Model Wave Function in the Half-Filled Landau Level

We construct model wave functions for the half-filled Landau level parametrized by “composite fermion occupation-number configurations” in a two-dimensional momentum space, which correspond to a Fermi sea with particle-hole excitations. When these correspond to a weakly excited Fermi sea, they have a large overlap with wave functions obtained by the exact diagonalization of lowest-Landau-level electrons interacting with a Coulomb interaction, allowing exact states to be identified with quasiparticle configurations. We then formulate a many-body version of the single-particle Berry phase for adiabatic transport of a single quasiparticle around a path in momentum space, and evaluate it using a sequence of exact eigenstates in which a single quasiparticle moves incrementally. In this formulation the standard free-particle construction in terms of the overlap between “periodic parts of successive Bloch wave functions” is reinterpreted as the matrix element of a “momentum boost” operator between the full Bloch states, which becomes the matrix elements of a Girvin-MacDonald-Platzman density operator in the many-body context. This allows the computation of the Berry phase for the transport of a single composite fermion around the Fermi surface. In addition to a phase contributed by the density operator, we find a phase of exactly $\pi$ for this process.

Figure 1

The energy, energy variance, overlap with exact eigenstates, and overlap with $\mathcal{P}\mathcal{H}$ conjugates for a number of model states, with the $\{d\}$ configurations shown. We see that clustered $\{d\}$ configurations have low energies, good overlap with exact eigenstates, and are $\mathcal{P}\mathcal{H}$ symmetric, while less clustered configurations lose these properties.

Figure 2

The density operator in Eq. (1) adds an additional phase to our Berry phase calculation, with is imaginary due to particle-hole symmetry. An antiunitary reflection symmetry (present in the thermodynamic limit, as well as in the square lattice considered here) introduces a relative $-1$ between clockwise and counterclockwise hopping, while also forbidding hopping in a direction normal to the Fermi surface.

Figure 3

Berry phases observed for a variety of paths around the Fermi surface. The solid symbols represent the locations of composite fermions, and for each step on the path we remove a composite fermion at the location of the empty symbols (i.e., we are moving a composite hole around the Fermi surface). There are a number of effects which arise from the insertion of a density operator in Eq. (1). These differences lead to additional phases summarized in Eq. (7). We compute these Berry phase ${\tilde{\mathrm{\Phi }}}_{\text{exact}}$ for the exact ED states, and ${\tilde{\mathrm{\Phi }}}_{\text{model}}$ for the model wave function of Eq. (4). In both cases, the results, combined with Eq. (7), indicate a Berry phase of $\mathrm{\Phi }=\pi$ when the center of the Fermi surface is encircled.

Figure 4

Same as Fig. 3, but for $N=13$. The stars indicate an additional composite fermion which we are moving around the Fermi surface. At this size we are able to construct paths which do not enclose the origin, and we find ${\mathrm{\Phi }}_{\text{exact}}=0$ for these paths (c), and ${\mathrm{\Phi }}_{\text{exact}}=\pi$ for others.