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Thin torus perturbative analysis of elementary excitations in the Gaffnian and Haldane-Rezayi quantum Hall states

We present a systematic perturbative approach to study excitations in the thin cylinder/torus limit of the quantum Hall states. The approach is applied to the Haldane-Rezayi and Gaffnian quantum Hall states, which are both expected to have gapless excitations in the usual two-dimensional thermodynamic limit. For the Haldane-Rezayi state, we confirm that gapless excitations are present also in the"one-dimensional"thermodynamic limit of an infinite thin cylinder, in agreement with earlier considerations based on the wave functions alone. In contrast, we identify the lowest excitations of the Gaffnian state in the thin cylinder limit, and conclude that they are gapped, using a combination of perturbative and numerical means. We discuss possible scenarios for the cross-over between the two-dimensional and the one-dimensional thermodynamic limit in this case.


I. INTRODUCTION
Quantum Hall states represent a prime example of phases of matter for which ideas of dimensional reduction are of central importance. This is rooted in the bulkedge correspondence for topological phases described by Chern-Simons quantum field theories. 1 This correspondence is also manifest in certain preferred or "special" microscopic trial wave functions used to described quantum Hall phases, and whose analytic structure is that of conformal blocks of the unitary rational conformal field theory (CFT) describing the edge of the same phase. 2 This situation extends to trial wave functions whose analytic structure is derived from conformal blocks in non-unitary CFTs. Examples of the latter kind are the Gaffnian state 3 and the Haldane-Rezayi 4 state. Here, the physical interpretation of this correspondence is more subtle, as the respective non-unitary CFT is not acceptable as the description of a physical edge. In these cases, it has been argued 3,5,6 that a local microscopic Hamiltonian stabilizing such a wave function as ground state must have gapless excitations. In other words, such wave functions are not expected to describe topological (gapped) phases. This conjecture has stimulated numerous theoretical and numerical investigations, 3,6-10 though direct evidence and/or microscopic characterization of the gapless excitations remains an interesting problem. In the case of the Haldane-Rezayi state, some insight has been obtained by analyzing a thin torus (TT) -or thin cylinder -limit. 11 The TT limit is yet another way to achieve a two-dimensional -one-dimensional (2D-1D) correspondence in the context of quantum Hall systems. [12][13][14][15] In Ref. 11, the very knowledge of the TT limit of the Haldane-Rezayi (HR) wave-functions was used to argue that charge-neutral gapless excitations must exist in the TT limit, and the latter have been characterized as certain extended equal-amplitude superpositions of defects (see below). In that argument, the detailed form of the HR parent-Hamiltonian was not used, merely the knowledge that it exists and that it has a zero energy ground state. In this paper, we will show how the features inferred in Ref. 11 can be straightforwardly derived in a perturbative framework, which, as a byproduct, also reveals the proper dependence of the quadratic dispersion on the (thin) cylinder radius. As we will review below, it has been cautioned in Ref. 11 that while the finding of gapless excitations in the thin torus limit is quite plausible evidence for their existence in the 2D thermodynamic limit, the converse is not necessarily true. Indeed, we apply the same perturbative scheme to the Gaffnian state, and find conclusive analytical and numerical evidence that gapless excitations are absent in the TT limit. We give an asymptotic formula describing the gap where first the thermodynamic limit is taken in one of two spatial directions and then the TT limit is taken in the other direction. As we discuss in detail in Sec. IV, this does not preclude the existence of gapless excitations in the usual 2D thermodynamic limit, though unfortunately, we cannot say more about this from a TT point of view. We hope that nonetheless our investigation will shine interesting light on the different possible relations between various types of quantum Hall states and their TT limits.

II. GAPLESS EXCITATIONS IN THE HALDANE-REZAYI STATE
It has been argued in previous studies 11 that in the TT limit, the gapless character of the Haldane-Rezayi state is manifest in the limiting forms of the associated wave functions. Below we develop a perturbative framework that makes these claims explicit. We focus on the top state in the HR sequence with fermionic filling fraction ν = 1/2.
The two-component HR state is tenfold degenerate 16 on the torus, with eight of ten ground states approaching one of two patterns in the TT limit, up to translations, given in Fig.1 eight ground states. Note that we refer to the two components of fermions as spin-up and spin-down here and in the following. There are two other special ground states whose TT limits are not fully described by a simple unit cell. These states are in fact closely related to the presence of gapless excitations in HR state. One of these special thin torus HR ground state patterns is given in Fig.2, which can be understood as a delocalized singlet immersed into and separating two ground states of the first kind in Fig.1. An explicit calculation using perturbation theory will be given explaining how these excitations acquire zero energy.
000 000 00 0 00 000 000 000 + 000 000 00 00 00 00 000 000 + 000 00 00 00 00 000 000 000 + 000 00 00 000 00 00 000 000 . . . The HR state is known to be the exact zero energy ground state of the "hollow-core" Hamiltonian. 4 This is just the V 1 Haldane pseudo-potential, 17 acting between any two electrons regardless of their spin. The name "hollow-core" is alluding to the fact that a V 0 -term is allowed between electrons of opposite spin, but is absent in the Hamiltonian. Here we will work mostly on an infinite cylinder with finite circumference L y = 2π/κ, κ being the inverse radius. A basis for the lowest Landau level in this geometry is naturally given by a set of translationally related orbitals labeled φ r , labeled by an integer r, such that the x-component of the guiding center is welldefined and equal to κr, and we set the magnetic length equal to 1. In second quantized form, the Hamiltonian then takes the form of a spin-SU(2) invariant two-body operator, where C r destroys a particle in the state φ r , and the ma-trix element V m n nm does not depend on spin indices α, β because of SU (2) invariance. This matrix element is therefore just the same as for the V 1 pseudo-potantial in a spin polarized setting, on a cylinder of radius 1/κ, which is standard in the literature (see, e.g., Ref. 18). This gives (with arbitrary but κ-independent overall normalization): n ,β C n,β C m,α (2) Now that spin-1/2 degrees of freedom are present in the problem, it should be noted that the pair-interaction defined by Eq. (2) still only acts on triplet pairs. This is natural, since in the infinite plane, the V 1 -pseudo-potential is defined as a two-particle projection operator projecting on states with relative orbital angular momentum 1. No pair forming a spin singlet can have this relative angular momentum. On the cylinder/torus geometry, however, relative angular momentum is not well-defined. Hence it is worth noting that the fact remains that the interaction annihilates any singlet pair. This follows already from the fact that the matrix element V m n nm is antisymmetric in m and n (as well as their primed counterparts).
We now use the second-quantized Hamiltonian (2) to set up a perturbative scheme designed to calculate energies and states in powers of x = e − 1 2 κ 2 . To this end we write the Hamiltonian as where λ = 1 is a formal parameter. H 0 contains all terms in the Hamiltonian (2) that are diagonal in the orbital indices. That is, all terms for which the unordered pairs (m, n) and (m , n ) are equal, whereas spin indices may or may not be equal. H 1 contains all the remaining, off-diagonal terms. We will perform a double expansion. The first of these is the formal expansion in the parameter λ. It turns out that each order in λ receives multiple contributions (infinitely many, for infinite system size) in the different powers of the parameter x. At any fixed order in λ, we will therefore retain only those orders of x that we are interested in. We claim that in this way, to get all contributions of a certain order x exactly, one needs to go only to a certain finite order in λ, which will depend on . We will not attempt a formal proof of this statement, but it will become quite apparent that for higher and higher orders in λ, the leading order in x will grow systematically. In our case, we will be interested in terms up to 12th order in x, for which second order perturbation theory in λ will be sufficient. We will first focus on the odd particle number sector, for which one has two degenerate ground state doublets on the torus. 19 The relevant thin torus states are discussed in Ref. 11. They are obtained as a superposition of states of the form shown in Fig.3, where a single spin-1/2 defect becomes delocalized in a ground state pattern of the A-type (the first of the ground state patterns in Fig. 1). 000 000 00 00 000 000 000 000 000 00 00 000 000 00 00 000 000 000 00 00 000 000 000 000 Clearly, all states contributing to this superposition are degenerate for H 0 , and hence we must apply degenerate perturbation theory in λ. The leading non-trivial order in x turns out to be x 12 , and we shall be content with this order. For this, it turns out to go to second order in λ. Order-λ 0 diagonal matrix elements are dominated by the interaction of the spin-1/2 defect with two neighboring singlets at distance 3. It is easy to see from Eq. (2) that each "bond" between a spin-1/2 defect and one such neighboring singlet costs an energy of We shall now consider corrections up to order x 12 arising in second order degenerate perturbation theory in λ. For simplicity, we will first consider a three particle system. The two (H 0 -)degenerate thin cylinder ground states are We also truncate the Hilbert space to consist of five orbitals r = 0 . . . 5, limiting ourselves to the following two excited states: Note that these are H 0 -eigenstates, though not of the same energy, having energies 6x and 2x + 96x 4 respectively, owing to spin fluctuations that are kept in H 0 . In second order degenerate perturbation theory, one then has the following correction to the diagonal matrix element: where we have only kept terms up to order x 12 . This reduces these diagonal matrix elements to an energy of order x 12 , which we denote by V : Similarly, the off-diagonal matrix element is obtained as At order x 12 , we thus have t = V , as was correctly inferred from less direct arguments in Ref. 11. At order x 12 , we thus obtain the effective Hamiltonian leading to a single zero energy eingenstate, which is the equal amplitude superposition of |Ω 1 and |Ω 2 , as expected. The energy of the other member of the formerly degenerate pair is E ext = 2V . One may ask if our perturbative scheme is valid, since both zeroth order and second order matrix elements in λ were of the same order x 9 in x. At least for the 3-particle problem, this question can be settled exactly. The full Hamiltonian in this truncated Hilbert space corresponds to the 4x4 matrix It can be shown exactly that this matrix has one lowest eigenvalue at zero, with the next higher up eigenvalue being Expanding the above up to order x 12 , one finds E ext = 2V in agreement with our perturbative approach. Higher orders in λ will thus only contribute subdominant terms in x.
Turning to the N -particle problem defined by the Hamiltonian (3) and the H 0 -degenerate subspace described in figure 3, we have, first of all, contributions to the effective Hamiltonian H ef f that are exactly analogous to those in the 3-particle problem discussed first. We still find no other processes, at second or higher oder in λ, that contribute to order x 12 or less in x. Therefore, the picture is similar to the 3-particle problem. At order x 12 , each state in Fig. 3 has a diagonal energy of 2V (V for each neighboring singlet of the defect). On top of that, we have a hopping matrix element of the form shown in Fig. 4, with t = V . The defect thus acquires a gapless quadratic dispersion of the form as predicted in Ref. 11 with V = 1296 x 12 + O(x 13 ). The state corresponding to k = 0 is the zero energy ground state corresponding, in the TT limit, to the equal amplitude superposition of the states shown in Fig. 3.
Next we discuss the case of even particle number. In this case, the relevant H 0 -degenerate subspace is given by all states of the form indicated in Fig. 2. Diagonal energies are now of the form 4V (except for states such as the first shown in the figure, see below), and we still have the effective defect-hopping shown in Fig. 4, with t = V . It is found that the equal amplitude superposition of Fig. 2 still gives a zero energy state. The only additional subtlety arises from configurations where the two defects are in closest proximity, as the first shown in the figure. It was conjectured in Ref. 11 that there should be no energy associated with the two neighboring defects, as long as the latter are forming a singlet. We have already discussed above why this is indeed the case, as the Hamiltonian only acts on triplet pairs. The diagonal energy of such configurations is thus 2V , and this is exactly required to satisfy the "detailed balance" condition giving a zero energy state. Moreover, it is clear on variational grounds that boosting the momentum of the delocalized pair would give rise to orthogonal states of arbitrarily small energy, in the thermodynamic limit.
We have thus verified all of the conjectures made in Ref. 11, going up to orders x 12 in a perturbative framework. At higher order in x, we expect that while corrections will be non-trivial, the resulting effective Hamiltonian will continue to have a zero mode essentially of the form given in Fig. 2, 20 while at the same time, it will remain local up to exponentially small terms. Under these circumstances, the above arguments for gapless excitations will carry over. 11 We expect that the same perturbative scheme developed here could in principle be used to demonstrate this for any given order in x, although we will not attempt to go to higher order in this work.

A. General considerations
The Gaffnian wavefunction is a state of particles at filling factor ν = 2/3(2/5) for bosons(fermions). 3 Its parent Hamiltonian is a three-particle interaction, and has been extensively discussed. 3, 21 We will focus on the bosonic case here for simplicity. On the torus, the state is 6fold degenerate, with thin torus states approaching the patterns 200200200 . . . , 110110110 . . . , including translations. We we wish to investigate if a scenario similar to that of the HR state is realized, and gapless excitations can be identified in the TT limit.
One main difference between the Gaffnian and HR case is the fact that none of the Gaffnian ground states look "suspicious" in the TT limit, whereas the HR state has ground states (among others) whose TT limit is the equal amplitude superposition shown in Fig. 2. From the latter, all features derived in the preceding section have been correctly inferred previously. 11 Here we investigate a scenario that could nonetheless explain the existence of Gaffnian gapless excitations of a similar flavor to those discussed for the HR. Unfortunately, we find that details of this scenario do not work out, and the excitations we discuss are gapped in the TT limit. However, from comparison with numerics, we do find that these excitations are indeed the lowest energy excitations in the TT limit, and hence the TT limit of the Gaffnian state is gapped.
A class of parent Hamiltonians for the Gaffnian state can be written as 3 where V 0 and V 2 are positive constants, and P 0 3 and P 2 3 are 3-particle projection operators that project onto the subspace of relative angular momentum 0 and 2, respectively. Using the results of Ref. 22, this interaction is readily presented in second quantized form: Where, where C > 0 is a constant that controls the relative strength between the two terms in Eq. (17), and we have chosen an overall normalization. 23 In Eq. (18), the summation over R is over all values such that 3R is an integer. We now wish to investigate a possible scenario for gapless neutral excitations similar to those of the HR state in the TT limit. Charge neutrality is a key aspect of the domain-wall type defects studied in the preceding section. Only a neutral defect is necessarily delocalized in the manner seen there, allowing for the gapless character. Charged defects would be subject to greater constraints from "center-of-mass conservation" 24 (momentum conservation around the cylinder axis). A natural neutral defect between two different Gaffnian TT ground state patterns is given by the following configuration: . . . 200200200201011011011011011 . . . (19) As written, the defect should cost a finite energy, as it violates the Gaffnian "generalized Pauli principle" 25,26 of having no more than 2 particles in any three adjacent sites. The question is if this energy cost can be fully compensated by delocalization, as was the case for the HR state. Moreover, on the torus, defects such as the above could only occur in pairs. Assuming, then, that there is some contact energy when two such defects are in proximity, unlike it is the case for a singlet pair of defects in the HR state, this could explain why a true zero energy state featuring such delocalized defects is only possible in the thermodynamic limit. This would explain why no exact zero mode wave functions are known featuring these delocalized defects, unlike in the HR case. Alas, the above scenario does not come to pass. We will find the asymptotic energy of defects as shown in Eq. (19) in the TT limit using the same perturbative approach used in the preceding section. We find that, unlike in the HR case, diagonal and off-diagonal energies for this defect are of different orders of magnitude in x in the TT limit, with the positive diagonal part dominating. We thus find the energy of such defect, and numerical calculations will show that it is indeed the energy of the lowest excited state in the TT limit. Our analytic result will show that this energy does not vanish in the thermodynamic limit.

B. TT perturbation theory
Equation (18) describes a center of mass conserving three particle hopping process. It is useful to explicitly spell out the first few dominant processes in the TT limit: The above four terms penalize states having three particles in three adjacent sites. It is apparent how the Hamiltonian assigns an energy to configurations (030), (210), (111) and (201) that is large compared to (most) off-diagonal processes. A detailed analysis similar to the one carried out in Ref. 27 could show that any zero mode of this Hamiltonian is necessarily dominated, in the usual sense, 25,26 by occupation number eigenstates free of such configurations. This is of course known to be the case for the Gaffnian wave function. 25,26 This last observation is quite generally equivalent to saying that the TT limit must be free of such configurations. In Eq. (19), we see that the excited state we consider has one (201) configuration. As in the preceding section, we write where H 0 contains all diagonal terms, and H 1 contains all off-diagonal terms, and subtleties concerning spin fluctuations are now absent. We see from Eq. (20) that H 0 assigns an energy of order e −8κ 2 /3 to the (201) defect. For comparison, the ground state patterns (200) and (110) have an H 0 -energy of O(e −14κ 2 /3 ) and O(e −18κ 2 /3 ) per unit cell, respectively. We know, however, that the energy associated with the (200) and (110) unit cells will cancel order by order in x = exp(−κ 2 /3) in perturbation theory, since we know that the ground states corresponding to these respective TT limits have zero energy. Hence, we will for now be interested in terms of order x 8 and lower order in x, and need to worry about higher order in x only if cancellation is found at order x 8 , as it did similarly happen in the HR case. The zeroth order (in λ ≡ 1) energy of the state (19) can be inferred from Eq. (20) as We look for corrections at second order in λ that are also proportional to x 8 = e −8κ 2 /3 . We first consider diagonal processes only. The relevant virtual transition is From this we obtain the following energy correction: At the order we are interested in, it is safe to neglect E 0 in the denominator. We see that this correction is of order x 8 , thus of the same order as E 0 and of opposite sign. So far, this is similar to the HR case. Unlike in the latter, however, there is no complete cancellation between the leading orders in x in E 0 and E 2 . A positive order x 8 energy therefore remains. It turns out that this energy dominates contributions from any other processes at second or higher order in perturbation theory. We have checked explicitly up to forth order perturbation theory that all other such processes contribute only higher powers in x. This is true for both diagonal processes and off-diagonal processes that effectively translate the defect. While the latter processes will certainly delocalize the defect in exact eigenstates, they do not affect the energy to the leading order in x. Taking into account the fact that defects of the kind considered here only occur in pairs on the torus, we have the following relation for the gap in the TT limit: Eq. (25) assumes that the defect (19) does indeed correspond to the lowest (thin torus) excitation of the Gaffnian parent Hamiltonian (18). In order to avoid having to consider many alternatives in the same manner, we compare Eq. (25) to numerics carried out for C = 1, Fig.  5. The figure shows both N = 8 and N = 10 particle data. It is evident that there is small discrepancy both between the N = 8 and the N = 10 particle energy gap, as well as between the latter and the prediction Eq. (25). Relative deviations between numerical gaps and Eq. (25) at κ = 3 are 0.05%. As particle number was not particularly relevant to the considerations leading to Eq. (25), this equation is expected to correspond to the thermodynamic (infinite cylinder) limit at fixed but large κ (fixed cylinder radius, small compared to a magnetic length). The N = 8 and N = 10 particle data conform to this expectation. We thus conclude that in the 1D thermodynamic limit of a thin, infinite cylinder, the Gaffnian parent Hamiltonian does not have gapless excitations.

IV. DISCUSSION AND CONCLUSION
The perturbative scheme used here explicitly confirms the existence of gapless excitations in the TT limit of the Haldane-Rezayi state. All the results obtained here regarding this state had been anticipated earlier, 11 based on the somewhat anomalous TT limit of some of the HR ground states on the torus, featuring delocalized defects. In contrast, all Gaffnian ground states have inconspicuous and simple thin torus limits. This alone could cast doubt on the existence of gapless excitations in the Gaffnian TT limit, though we have argued in Sec. III A that such reasoning would be naive. Instead we have applied the same perturbative scheme employed in Sec. II for the HR state to the problem of thin torus Gaffnian excitation. We have identified certain charge-neutral defects as natural suspects for gapless excitations. Alas, detailed calculation has shown that these excitations are gapped, and numerics strongly suggest that they are indeed the lowest excitations in the TT limit. This implies that the 1D, thin cylinder thermodynamic limit of the Gaffnian parent Hamiltonian is gapped, unlike the similar limit for the HR parent Hamiltonian. Powerful arguments, however, suggest that both states are gapless in the ordinary, 2D thermodynamic limit. On the torus, this opens up the interesting question of what happens if this 2D limit is approached asymmetrically, by first taking the 1D infinite cylinder limit at small cylinder radius, and subsequently taking the cylinder radius to infinity. During the latter step, gapless excitations are expected to appear, under the assumption that the 2D limit is indeed gapless. This could happen either at a critical point at some finite cylinder radius (finite κ), or only in the limit where the radius approaches infinity (κ → 0). The latter is completely consistent with the idea of adiabatic continuity as a function of radius, at least for any finite radius. For this very reason, it was cautioned in Ref. 11 that finding gapless excitations in the TT limit is actually a more significant indication for their existence in the 2D limit compared to converse situation, where finding their absence in the TT limit does not necessarily imply the existence of a gap in the 2D limit, even if adiabatic continuity is assumed. The latter part of this cautionary remark seems to apply to the Gaffnian state. Barring any level crossings, it is possible that the delocalized defects identified in Sec. III A are adiabatically connected to gapless excitations in the 2D limit. This and other interesting open questions, such as the identification of the underlying cause why gapless excitations are sometimes detectable in the TT limit and sometimes not, are left for future investigation.