Gapless excitations in the Haldane-Rezayi state: The thin-torus limit

We study the thin-torus limit of the Haldane-Rezayi state. Eight of the ten ground states are found to assume a simple product form in this limit, as is known to be the case for many other quantum Hall trial wave functions. The two remaining states have a somewhat unusual thin-torus limit, where a broken pair of defects forming a singlet is completely delocalized. We derive these limits from the wave functions on the cylinder, and deduce the dominant matrix elements of the thin-torus hollow-core Hamiltonian. We find that there are gapless excitations in the thin-torus limit. This is in agreement with the expectation that local Hamiltonians stabilizing wave functions associated with nonunitary conformal field theories are gapless. We also use the thin-torus analysis to obtain explicit counting formulas for the zero modes of the hollow-core Hamiltonian on the torus, as well as for the parent Hamiltonians of several other paired and related quantum Hall states.

Figure 1

Transitions between different ground-state sectors of the HR state by quasihole insertion. We start with the polynomial Eq. (2), which corresponds to the unique ground state on the sphere, and consider the associated state on the cylinder (a). The thin-cylinder limit of this state is given by the $A$ pattern, Eq. (1a). Insertion of two quasiholes (b) leads to a state whose thin cylinder limit is shown in Fig. 4a, with a $B$ pattern appearing inside the $A$ pattern, and separated from it by two domain walls. When the quasihole positions are taken to $\pm \infty$, the state approaches Eq. (11), whose thin-cylinder limit is described by the $B$ pattern (2). We may iterate the process by inserting a second quasihole pair (c). For this one has the choice of breaking a pair in the pairing wave function of the state or not (Ref.25). In the latter case, another $A$ pattern will appear in the thin-cylinder limit, surrounded by $B$ patterns. If, on the other hand, the pair is broken, one obtains an “$A^{\prime }$ string” instead. With the quasiholes taken to $\pm \infty$, the state approaches the wave function Eq. (16). Its thin-cylinder limit is the superposition of states represented in Fig. 2, which features a delocalized broken pair, and which we will loosely refer to as an $A^{\prime }$ string. [See also Fig. 4e for the appearance of an $A^{\prime }$ string terminating in domain walls before quasiholes are taken to $\pm \infty$.]

Figure 2

The thin-cylinder limit of the $A^{\prime }$ ground state, Eq. (16). (a) One of many squeezed lattice configurations, corresponding to a choice of the permutation $\rho$ that maximizes the quantity $S$, Eq. (4). A particular choice of $\rho$ corresponds to a squeezed-lattice configuration where each spin carries a particle number index. This is not shown, since permutation of like-spin indices does not change the value of $S$, as must be the case by the antisymmetry of the wave function. The short underscores represent squeezed lattice sites, and the long underscores indicate the pairing $\mathcal{P}$, Eq. (23). Pairs must be nearest neighbors on the squeezed lattice, whereas the positions of the members of the broken pair are arbitrary. Together, $\rho$ and $\mathcal{P}$ completely define the $p$ matrix, Eq. (22). (b) The thin-cylinder limit of the $A^{\prime }$ state, obtained as an equal-amplitude superposition of all product states derived by unsqueezing all squeezed-lattice configurations with given pairings of the form described under (a). This is done by using the associated $p$ matrix described in (a) to form monomials according to Eqs. (18) and (5). The unsqueezed version of (a) is shown in the last line. As a result, the state is one where a completely delocalized pair of charge-neutral defects forming a singlet separates two mutually out-of-phase $A$ patterns.

Figure 3

The off-diagonal matrix element present in the thin-torus hollow-core Hamiltonian. This, together with the detailed balance condition Eq. (24), ensures that states containing an $A^{\prime }$-type string with one or two delocalized defects are zero-energy states in the thin-torus limit.

Figure 4

Charge-1/4 domain walls between $A$ or $A^{\prime }$ and $B$ strings. Underscores label the domain wall positions as defined in the text. The hopping process indicated in (a) moves the right domain wall in the $ABA$ sequence to the left by four orbitals. (b) shows the situation after three such moves, which shrink the size of the $B$ string to zero and lead to an additional 0 between mutually shifted $A$ strings. (c),(d) show the same for a $BAB$ sequence. Note that for the conventions defined in the text, the merged domain walls now coincide at the same orbital. In any case, each allowed zero-energy sequence of patterns can be assigned a sequence of domain wall positions satisfying Eq. (28) in a unique manner. (e) and (f) show patterns involving $A^{\prime }$ strings with two (e) or one (f) delocalized defect. Only a “snapshot” of the state is shown with defects in fixed positions, where it is understood that these defects are delocalized as indicated and stated in the text. The right defect in (e) is shown at the rightmost possible position.