Excitation and tunneling spectra of a fractional quantum Hall system in the thin cylinder limit

TL;DR

The paper develops a controlled perturbative framework for the ν = 1/3 FQHE in the thin-cylinder limit by mapping to a 1D lattice with dipole-moment conservation and treating off-diagonal hopping as a small perturbation. It derives analytic expressions for the dispersions of neutral dipole-based excitations (magneto-roton-like) and predicts sharply peaked LDOS spectra for charged excitations due to dipole conservation, with quantitative agreement to exact diagonalization at small sizes. The work connects these insights to STM experiments and composite-fermion theory, and discusses the relevance to the 2D limit, other fillings, and potential MPS representations. Overall, it provides a tractable, analytically tractable route to understanding excitations and tunneling spectra in the FQHE from a thin-torus perspective, with implications for interpreting STM data and guiding future numerical approaches.

Abstract

The excitations of fractional quantum Hall effect (FQHE) states have been largely inaccessible to experimental probes until recently. New electron scanning tunneling microscopy (STM) results from Hu et.al. (arXiv:2308.05789) show promise in detecting and identifying these excited states via the local density of states (LDOS) spectrum. On a torus, there exists a mapping {from the lowest Landau level states} to a 1D lattice {with a Hamiltonian that features} dipole moment conservation. In this work, we apply perturbation theory starting from the thin cylinder limit ($L_x \rightarrow \infty, L_y <l_B$ for torus dimensions $L_x$ and $L_y$ {and magnetic length $l_B$}) to obtain an analytical approach to the low-lying neutral and charged excitations of the $ν=1/3$ FQHE state. Notably, in the thin cylinder we can systematically enumerate all the low-lying excitations by the patterns of 'dipoles' formed by the electron occupation pattern on the 1D lattice. We find that the thin-cylinder limit predicts a significant dispersion of the low-lying neutral excitations but sharpness of the LDOS spectra, which measure charged excitations. We also discuss connections between our work and several different approaches to the FQHE STM spectra, including those using the composite fermion theory. Numerical exact diagonalization beyond the thin-cylinder limit suggests that the energies of charged excitations remain largely confined to a narrow range of energies, which in experiments might appear as a single peak.

Figures (7)

• Figure 1: (A) A schematic of the cylinder/torus to lattice mapping. On the cylinder the single electron wave functions form a Gaussian of width $\sim 2l_B$ in the $x$ direction. With our choice of gauge, the Gaussians are centered at $2\pi l_B^2 k/L_y, k \in 0,1,...,N_{\phi}-1$, forming a 1D lattice. The darker stripes (and the corresponding filled circles in the lattice picture below it) signify the occupied states. In a many body system we can then write down the state in terms of the occupation of these Gaussians. When $L_y \rightarrow 0$, the separation between sites increases and the overlap between wave functions decreases exponentially, allowing us to ignore the hopping terms. (B) The action of the hopping term of the Hamiltonian on the many body state. (Top) Applying the hopping term to the ground state produces a state with a dipole pair (see explanation in Section \ref{['ssec:uncharged']}). (Bottom) The $P=1$ sector, which are unreachable from the ground state. Applying $\hat{V}_{31}$ to this state moves the dipole to an adjacent position, creating another state with the same energy.
• Figure 2: The interaction coefficients for our choice of potential with $L_y = 4l_B$, in units of $e^2/l_B$ (Gaussian units, setting $\epsilon=1$). The main plot shows the static pairwise interaction $V_{k0}$. The inset shows $V_{k0}$ and the hopping terms $V_{k1}$ and $V_{k2}$ on a log-log plot. From the plot we see that $V_{km}$ has a power law dependence on $k$. The coefficients for higher $m$ are order of magnitudes smaller, reflecting the exponential dependence of $V_{km}$ on $m$ at small $L_y$. For our choice of $L_y$, $V_{k1} \leq V_{21} \ll V_{k0}$ for most values of $k$.
• Figure 3: The dipole mapping used in our work. A dipole of moment $p$ is formed when an electron is moved by $p$ sites from its initial position. We define the position of a dipole $d$ in such a way that $d$ is leftmost orbital that does not follow the ultrathin ground state pattern. In other words, if an electron is moved to the right (positive $p$, red in the figure above) then $d$ is the initial position of the electron while if an electron is moved to the left (negative $p$, blue in the figure above) then $d$ is the final positon of the electron. By defining the position in this way we can write down a simple expression for the effective interaction potential between two dipoles (bottom).
• Figure 4: The energy spectrum for the charge-neutral, $P=0$ to $P=4$ sector, $L_y = 4l_B$. Color corresponds to the different dipole moment sectors. The ground state belongs to the black $P=0$ sector. Note that the lowest excited $P=1$ state would become the magnetoroton state as we move away from the thin torus limit. (Left) Perturbation theory calculation for the lowest energy states in $P=0$ to $P=4$ sector and the next-lowest energy states in the $P=0$ sectors - the dipole-dipole $\vert (+)_i (-)_{i+k} \rangle$ states. (Right) Exact diagonalization results with $N_{\phi} = 24$ lattice sites and $N=8$ electrons on the torus. The energies are shifted by a constant amount such that the ground state is at 0 energy. The dotted lines corresponds to the perturbation theory result, shifted in position to account for finite size effects.
• Figure 5: The dispersion calculated by perturbation theory (lines) and by exact diagonalization (dots), for the the $\vert (+)_i (-)_{i+k} \rangle$ states which are in the $P=0$ sector (left) and the $\vert (\pm)_i (\pm)_{i+3} ... (\pm)_{i+3(P-1)}\rangle$ states which are in the $P=1-3$ sector (right). The y axes show the energy in units of $e^2/\epsilon l_B$. We obtain very good agreement except for the $P=4$ case, where the dispersion becomes very small such that higher-order contributions cannot be neglected.