Quantum Hall system in Tao-Thouless limit

We consider spin-polarized electrons in a single Landau level on a torus. The quantum Hall problem is mapped onto a one-dimensional lattice model with lattice constant $2\pi /L_1$, where $L_1$ is a circumference of the torus (in units of the magnetic length). In the Tao-Thouless limit $L_1\to 0$, the interacting many-electron problem is exactly diagonalized at any rational filling factor $\nu =p/q\le 1$. For odd $q$, the ground state has the same qualitative properties as a bulk $\left(L_1\to \infty \right)$ quantum Hall hierarchy state and the lowest-energy quasiparticle excitations have the same fractional charges as in the bulk. These states are the $L_1\to 0$ limits of the Laughlin and Jain wave functions for filling fractions where these exist. We argue that the exact solutions generically, for odd $q$, are continuously connected to the two-dimensional bulk quantum Hall hierarchy states—i.e., that there is no phase transition as $L_1\to \infty$ for filling factors where such states can be observed. For even-denominator fractions, a phase transition occurs as $L_1$ increases. For $\nu =1/2$ this leads to the system being mapped onto a Luttinger liquid of neutral particles at small but finite $L_1$; this then develops continuously into the composite fermion wave function that is believed to describe the bulk $\nu =1/2$ system. The analysis generalizes to non-Abelian quantum Hall states.

Figure 1

Mapping of a single Landau level onto a one-dimensional lattice model. The two-dimensional space is a cylinder with circumference $L_1$ (or a torus if the ends are identified) (top panel). The single-particle wave functions are centered along circles and are Gaussians of width one along the cylinder (central panel); they are numbered by their momenta $k2\pi /L_1$ which also give the position along the cylinder. This gives a one-dimensional lattice model with lattice constant $2\pi /L_1$ where each site is either empty (0) or occupied by en electron (1). As an example, the Tao-Thouless state 001 at $\nu =1/3$ is shown (bottom panel).

Figure 2

A general translationally invariant two-electron interaction consists of the terms where two electrons preserve the position of their center of mass. There are electrostatic terms $V_{k0}$, where the electrons do not move, and hopping terms $V_{km},m\ne 0$, where the electrons hop symmetrically.

Figure 3

The interaction energy of one movable electron with two fixed electrons is minimized when the distances to the two fixed electrons differ by at most one lattice site.

Figure 4

Ground state unit cell at $\nu =5/13$ by relaxation on circle. (a) p = 5 equidistant electrons on a circle with $q=13$ lattice sites. (b) Each electron moved to its closest site; unit cell is ${\left(0_{2}101\right)}_2{0}_{2}1$.

Figure 5

(Color online) The Tao-Thouless states and their stability. Each point $\left(p/q,1/q\right)$, $q$ odd, corresponds to a TT state that we claim is adiabatically connected to a bulk QH state. The gap, and hence the stability, increases with decreasing $1/q$. At the points marked by pluses or crosses indications of QH-states are observed (Ref. 67). The line is at constant gap and shows the range of the experiment in $\nu$. It is an approximate lower boundary in $1/q$ for the observed states.

Figure 6

The hierarchy state at $\nu =p/q$ is the parent state for two sequences of daughter states formed by condensation of quasielectrons and quasiholes in the parent state. These sequences approach $p/q$, with decreasing gap $\sim 1/q$, from above and below, respectively. Repeating this construction ad infinitum gives the fractal structure in Fig. 5.

Figure 7

Quasiparticles and emergent Landau levels in the TT limit. (a) The TT ground state 001 at $\nu =1/3$. In (b) three quasielectrons with charge $-e/3$ are created by inserting 01 in three places. When one 01 is inserted for every third unit cell 001, the TT ground state at 4/11 is obtained (c). In (d) the 2/5 ground state is obtained by inserting one quasielectron for every unit cell 001. The ground states in (c) and (d) can be interpreted as filling emergent Landau levels.

Figure 8

Examples of Fermi seas of momenta $\left\{k_{i\alpha }\right\}$ for $N_e=5$. For each sea the conserved momenta $\left(K_1,K_2\right)$ are displayed. (a) and (b) are related by a translation that corresponds to $K$ invariance and have the same quantum numbers; they describe the same state. (c) is one of three new states obtained from (a) by reflections in the $x$ and $y$ axes. These four states have different quantum numbers and are degenerate ground states for $5.3<L_1<L_2$ (cf. Fig. 9); translating the corresponding Fermi seas one step in the $x$ direction gives the remaining four ground states. (d) is obtained from (c) by a rotation by $\pi /2$; it is one of the ground states when $5.3<L_2<L_1$.

Figure 9

Ground states and corresponding quantum numbers $\left(K_1,K_2\right)$ at $\nu =1/2$ as a function of $L_1$ for 4 to 9 electrons (from top to bottom) [$K_1$ is given $\text{mod}\left(N_e\right)$]. The results are obtained in exact diagonalization using an unscreened Coulomb interaction. Marked points on the $L_1$ axis denote transitions to new ground states. For $L_1<5.3$ the (virtually exact) ground state is the TT state, whose unit cell is 01 when $L_1\to 0$. In each region $L_1>5.3$, the state is identified with a Rezayi-Read state with the displayed Fermi sea of momenta. The overlap with the Rezayi-Read state is around or above 0.99 for all $N_e$ (in all regions). The phase diagram is symmetric about $L_1=L_2=\sqrt{4\pi N_e}$, which are the largest marked points on the $L_1$ axis in the figure.