Landau-level-mixing and the ground state of the 5/2 quantum Hall effect

TL;DR

This paper investigates the impact of Landau-level mixing on the ground state of the 5/2 fractional quantum Hall effect, finding that it favors the anti-Pfaffian state through a perturbative approach.

Contribution

It introduces a perturbation theory method that unbiasedly determines the ground state, highlighting the importance of a previously omitted term in the effective Hamiltonian.

Findings

Landau-level mixing favors the anti-Pfaffian state.

A critical term in the effective Hamiltonian reverts the ground state to anti-Pfaffian.

The method provides an unbiased approach to determine the ground state.

Abstract

Inter-Landau-level transitions break particle hole symmetry and will choose either the Pfaffian or the anti-Pfaffian state as the absolute ground state at 5/2 filling of the fractional quantum Hall effect. An approach based on truncating the Hilbert space has favored the anti-Pfaffian. A second approach based on an effective Hamiltonian produced the Pfaffian. In this letter perturbation theory is applied to finite sizes without bias to any specific pseudo-potential component. This method also singles out the anti-Pfaffian. A critical piece of the effective Hamiltonian, which was absent in previous studies, reverts the ground state at 5/2 to the anti-Pfaffian.

Figures

Tables

Table 1. Table 1: Comparison of the differences between 3-body pseudo-potentials V M subscript 𝑉 𝑀 V_{M} for polarized electrons and finite flux sizes and their infinite size counterparts. The percent differences are given in the parenthesis.

$N_{\varphi }$	$V_6-V_3$	$V_9-V_3$	$V_{12}-V_3$	$V_5-V_3$
30	0.0044 (8.3%)	0.0051 (12%)	0.0088	0.0091 (2.2%)
32	0.0044 (8.3%)	0.0051 (12%)	0.0090	0.0092 (1.1%)
34	0.0045 (7.0%)	0.0052 (10%)	0.0090	0.0092 (1.1%)
$\infty$	0.0048	0.0058	NA	0.0093

Table 2. Table 2: The LL-mixing corrections δ ​ v m 𝛿 subscript 𝑣 𝑚 \delta v_{m} for 2-body pseudo-potentials v m subscript 𝑣 𝑚 v_{m} (in units of ε ​ κ 𝜀 𝜅 \varepsilon\kappa ) relative to their respective δ ​ v 1 𝛿 subscript 𝑣 1 \delta v_{1} values and for 3 different flux sizes on the torus. For polarized electrons only odd values of relative angular momentum m 𝑚 m are relevant. The percent differences from infinite PT results (last row) are given in parenthesis.

$N_{\varphi }$	$\delta v_3-\delta v_1$	$\delta v_5-\delta v_1$	$\delta v_7-\delta v_1$	$\delta v_9-\delta v_1$
30	0.1094 (0.45%)	0.1767 (0.62%)	0.1994 (0.75%)	0.2067 (0.86%)
32	0.1094 (0.45%)	0.1769 (0.51%)	0.1996 (0.65%)	0.2069 (0.77%)
34	0.1095 (0.36%)	0.1770 (0.45%)	0.1998 (0.55%)	0.2071 (0.67%)
$\infty$	0.1099	0.1778	0.2009	0.2085

Equations

$$\mathcal{H}=\frac{1}{2}\sum _{\{m_i\}}⟨m_1{m}_2|V|m_3{m}_4⟩C_{m_1}^{†}{C}_{m_2}^{†}{C}_{m_3}{C}_{m_4},$$

$$H_{\text{eff}}=H_1+\kappa \sum _p^{\prime }\frac{\mathcal{H}|p⟩⟨p|\mathcal{H}}{E_0-E_p},$$

$$H(M=9)=$$

Full text

Topological phases of matter have been the focus of much recent theoretical interest. A prime example of such a phaseWen (1992), which is realized in experiment, is the fractional quantum Hall effectTsui and Gossard (1982); Laughlin (1983a) (FQHE). The most intriguing of these quantum Hall states occurs in the half-filled first excited Landau level (1LL)Willett et al. (1987). Breaking the odd-denominator trend of the lowest Landau level (LLL), the state at $\nu=5/2$ is one of the strongest FQHE states in the 1LLPan et al. (1999); Xia et al. (2004). It has been the subject of numerous studies and is widely believed to be in the universality class of the Moore-Read (MR) stateMoore and Read (1991), which is a fully polarized $p_{x}+ip_{y}$ paired stateRead and Green (2000) of composite fermionsJain (1989). The MR state possesses richer excitations than states dominating the LLL. In particular, the presence of the neutral Majorana fermion mode results in quasi-particle excitations exhibiting non-Abelian statisticsMoore and Read (1991); Nayak and Wilczek (1994); Read and Rezayi (1996); Read (2009). This property has greatly increased interest in the 5/2 effect, in part due to its potential for quantum information processingKitaev (2006); Nayak et al. (2008). In this letter we will assume the MR state to be the underlying phase of the 5/2 effect. Morf (1998); Rezayi and Haldane (2000); Wan et al. (2008); Zhao et al. (2011)

Particle-hole symmetry and the anti-Pfaffian- Earlier studies of the 5/2 state disallowed virtual inter-LL transitions, which are only exact at infinite cyclotron energy $\hbar\omega$. In that case, because of particle-hole (P-H) symmetry, both the MR or (Pfaffian) state and its P-H conjugateLevin et al. (2007); Lee et al. (2007) (the anti-Pfaffian) are equally valid candidates for the 5/2 state. However, while both states display non-Abelian statistics, they are distinct topological phasesLevin et al. (2007); Lee et al. (2007) of matter. The system will then be forced to choose one by spontaneously breaking P-H symmetry. On the other hand, the ubiquity of LL-mixing provides a P-H symmetry breaking field. A measure of the mixing strength is given by the ratio of typical Coulomb energy $\varepsilon=e^{2}/4\pi\epsilon_{0}\ell$ to the cyclotron energy $\kappa=\varepsilon/\hbar\omega$, where $\ell$ is the magnetic length. In experiments, $\kappa$ varies from 0.8 to 2.8 and either the Pfaffian (Pf) or the anti-Pfaffian (aPf) will be favored as the ground state. Until now, however, which one has not been definitively determined.

To address the effects of P-H symmetry-breaking, several studies have been carried out using different approximations. Simon and the present authorRezayi and Simon (2011) employed a truncated model of LL-mixing by keeping only 3 LLs (3-LL model). Additionally we controlled the number of particle and hole excitations that result from inter-LL transitions. In a subsequent work by Zaletel et. al.Zaletel et al. (2015) this last restriction was completely removed. The authors employed an infinite density matrix renormalization group (iDMRG)Zaletel et al. (2013). The method incorporates the matrix product formZaletel and Mong (2012); Estienne et al. (2013) of FQHE states on a cylinder of infinite length but finite radius. The iDMRG also kept a finite number of LLs (3-5), but could reach cylinder circumferences of 20 magnetic lengths. A square torus of this dimension would include about 30 electrons, which is a considerable increase in system size over previous studies. Both approaches predict the aPf to be favored irrespective of the strength of $\kappa$.

On the other hand, Wojs *et. al.Wójs et al. (2010); [ThisworkwasbasedontheresultsofBisharaandNayak; whichdidnotincludethe$M=9$3-bodypseudo-potentials;thetwo-bodypseudo-potentialsweresubsequentlyrevisedinRef.\cite[cite]{@@bibref{AuthorsPhrase1YearPhrase2}{Peterson-Nayak}{@@citephrase{(}}{@@citephrase{)}}}bynormalorderingofthe3-bodyinteractions.Interestingly; theoverlapofthegroundstatewiththeaPfonlyimprovesif$H(M=9)$isaddedtotheBishara-NayakHamiltonian;]Bishara-Nayak and more recently Pakrouski et. al.Pakrouski et al. (2015) have concluded that the Pf is the ground state. The authors used the effective 2-bodyHaldane (1983) and 3-bodySimon et al. (2007) pseudo-potentialsSimon and Rezayi (2013); Peterson and Nayak (2013); Sodemann and MacDonald (2013) that include LL-mixing corrections to the lowest order in $\kappa$. At first sight, these calculations have cast doubt on the results of the 3-LL model.

In this letter the issue is revisited in order to resolve this discrepancy and to compare the 3-LL model predictions to perturbation theory results. The recent calculations of infinite-size pseudo-potentials produced a 3-way agreementSimon and Rezayi (2013); Peterson and Nayak (2013); Sodemann and MacDonald (2013) on their values and are not the cause of the discrepancy.

Our approach to calculating the effective Hamiltonian closely follows the previous methods, but uses the torus geometry for finite-size systems. This is the only compact geometry that avoids the shift111The shift S is defined by the flux-charge equation $N_{\phi}=\nu_{v}^{-1}N_{e}-S$, where $\nu_{v}$ is the filling factor of the valence electrons. On the sphere S=3 and -1 for the Pf and aPf respectively. and facilitates the comparison between Pfaffian and anti-Pfaffian. A study on the sphere is also reported in this letter which corroborates the torus results. For this and subsequent torus results we will follow the method of Wojs et. al. and Pakrouski et. al. and use infinite-size perturbation theory (PT) values of the 2-body and the 3-body pseudo-potentials.

The Hamiltonian allowing inter-LL transitions is

[TABLE]

where $m$ is the combined label of LL index $n$, linear momentum $k$, and spin. $V(r)$ is the Coulomb interaction. We will consider the electron layer to have zero width. The matrix elements in H, which are independent of spin, are calculated self-consistently for up to 31 LLs for each size that we studied. Another important difference with infinite-size calculations is that the 3-body pseudo-potential corrections are not singled out by their relative angular momentum but are automatically included in their entirety.

The effective Hamiltonian, which includes the lowest order LL-mixing corrections can be written as:

[TABLE]

where $H_{1}$ is the Hamiltonian for electrons in the partially filled 1LL and $\lvert p\rangle$ is an intermediate state with kinetic energy $E_{p}$, which can have at most two electrons in the excited LLs with index $n>1$. $E_{0}$ is the kinetic energy of the appropriate basis states (described below) and is dropped from $H_{\text{eff}}$. The prime on the sum restricts $p$ so that $E_{p}\neq E_{0}$. In Eq. 2, the Hamiltonians are expressed in units of $\varepsilon$ and kinetic energies are given in units of the cyclotron energy.

We will generally consider the matrix elements $\langle i\lvert H_{\text{eff}}\rvert j\rangle$, where the set {$\lvert i\rangle$} is a relevant Slater determinant basis for $N_{e}$ valance electrons in the 1LL and includes a lowest LL filled with both spins. We will choose $N_{e}=2$ or 3 when calculating pseudo-potentials. This form is also readily applicable when the degeneracy is not completely lifted after the action of $H_{1}$, which is the case for certain odd electron numbers on a hexagonal torusPapić et al. (2012). In this case, the ground state that represents the basis {$\lvert i\rangle$} is a doublet. Extra degeneracies of this type result from the combination of anti-unitary P-H conjugation and unitary discrete rotational symmetriesTinkham (1964).

It can be seen from the definition of the basis states that the two creation operators of the right ${\cal{H}}$ acting on $\lvert i\rangle$ (and destruction operators of the left ${\cal{H}}$ acting on $\langle j\rvert$) will have LL indices of either 0 or 1. It is therefore convenient to organize ${\cal{H}}$ in terms of the number of LLL holes (0,1,2) in $\lvert p\rangle$. For example, in the case of two holes, one can ignore the intermediate states $\lvert n_{1},n_{2}\rangle$, where $n_{i}>1$ are excited LL indices, since these terms do not depend on the specifics of $\lvert i\rangle$ and $\lvert j\rangle$ and amount to an overall constant in Eq. 2. The last term of $H_{\text{eff}}$ can be represented by Feynman diagrams, which have already appeared in printPeterson and Nayak (2013); Sodemann and MacDonald (2013) and will not be repeated here. It is then a bookkeeping exercise; some relevant details can be found in the appendices B and C of Sodemann and MacDonald. The results are 1, 2, and 3-body effective interactions for electrons in the 1LL. However, the one-body potentials, by translational symmetry, are independent of orbital index. Since they only serve to modify the chemical potential, they will also be ignored. It will be assumed throughout that the states in the 1LL are fully spin-polarized, which is consistent with recent experimentsEisenstein et al. (2017); Tiemann et al. (2012) as well as with previous calculationsMorf (1998); Feiguin et al. (2009); Rezayi and Simon (2011). However, virtual reversed spin excitations from the LLL are allowed. Even these make a negligible difference and can be ignored.

In the case of a doublet ground state of the $H_{1}$, we follow Landau and LifshitzLandau and Lifshitz (1977) and diagonalize the LL-mixing part of the $H$ in the 2-dimensional Hilbert space spanned by the doublet, which separates the Pf and the aPf components without mixing-in any other state of the $H_{1}$. Fig. 1 shows the difference in the ground state energy difference $E_{\text{Pf}}-E_{\text{aPf}}=\Delta_{0}$ as a function of the number of LLs included in the sum of Eq. 2. As observed in the 3-LL model, the aPf is favored unless only the lowest 2 LLs are kept. The inset shows $\Delta_{0}/N_{e}$ as a function of $1/N_{e}$. While the linear regime has not quite been reached and it is difficult to extrapolate to large sizes, the results are consistent with a “gap” that is extensiveZaletel et al. (2015).

2-body and 3-body pseudo-potentials.- A very practical way of presenting the electronic interaction potential for quantum Hall states is to express them in terms of the energy (or pseudo-potential) for $n$-electrons in a state of fixed relative angular momentum. It was first introduced by HaldaneHaldane (1983) for the 2-body case and later extended to the $n$-body case by Simon et. al.Simon et al. (2007). Fig. 2 gives the values of the 3-body pseudo-potentials $V_{M}$ (in units of $\varepsilon\kappa$), in an ascending order. The first 5 are labeled by their relative angular momenta $M$. The inset gives the results of Sodemann and MacDonald, who crucially calculated $V_{9}$. In both cases the most important pseudo-potentials have relative angular momenta that are multiples of 3222It can be seen that finite-size effects become larger as $M$ is increased. For larger $M$s the average relative distance between the particles increases and may become comparable to the linear dimension of the system. As a result the $M=12$ may be off by 15-20% of its infinite-size value. The main difference between the present effective Hamitonian and those of Wojs *et. al. *and Pakrouski et. al. is the $M$=9 3-body pseudo-potential. It will be shown that $V_{9}$ has a critical role in determining the nature of the ground state. If included, the ground state reverts to the aPf. Table 1 gives the differences of $V_{M}$ from the $M=3$ value for finite-size systems in the present study and compares them with their infinite-size values. Table 2 is the same as Table 1 but for 2-body pseudo-potentials, which already are very close to their infinite size values.

In the remainder of this letter the effective Hamiltonian for the infinite system (see Table 2 and the inset of Fig. 2) will be used. While the first five pseudo-potentials (in order of their relative angular momenta) are unique, the $M=9$ has two states and a choice of basis is necessary. The Hamiltonian matrix for $M=9$ and its corresponding basis wave functions are described by Sodemann and MacDonaldSodemann and MacDonald (2013) and LaughlinLaughlin (1983b) respectively. It is:

[TABLE]

where the states $\bm{\lvert l,m\rangle}$ have relative angular momentum $M=2l+3m$. The relevant pseudo-potentials and their projection operators are obtained by finding the eigenvalues and eigenvectors of Eq. 3. However, $H(M=9)$ is well approximated by just the first term, which is the most dominant by far.

The effective Hamitonian, including the $M=9$ term, is then diagonalized. In general, the eigenvalues and eigenvectors will have a non-linear dependence on $\kappa$. However, the linear regime will suffice for our conclusions.

The majority of numerical calculations on the 5/2 state have been carried out in the spherical geometryHaldane (1983) Because of the different shifts for Pf and aPf on the sphere, P-H symmetry is not applicable. Comparing the Pf with $N^{P}_{e}$ electrons to the aPf with an equal number of holes $N^{A}_{h}=N^{P}_{e}$, forces the number of electrons to be related by $N^{A}_{e}=N^{P}_{e}-2$. This difference in system sizes excludes any meaningful energetic comparison between Pf and aPf.

As in the Pakrouski et. al. paper, we compare overlaps and gaps between the ground and the first excited state. As noted by these authors, the gap in the spectrum tracks the respective overlaps and may provide another diagnostic tool. Fig. 3 shows the results for $N_{e}=16$ for the Pf and $N_{e}=14$ for the aPf. It can be seen that now the aPf has the larger overlaps and gaps.

Transition to the Pfaffian.- Clearly, there will be a phase transition as the $V_{9}$ pseudo-potential approaches zero. Making it less attractive weakens the aPf; at $V_{9}\approx-0.0011$ a first order transition to the Pf is observed. The results for the $N_{e}=15$ doublet on the torus is shown in Fig. 4. In this case the aPf-Pf energy difference $\Delta_{0}$ displays a perfectly linear dependence on $V_{9}\ (=-0.0088+\delta V_{9})$. It is noteworthy that to reach the point of transition the magnitude of $V_{9}$ has to be reduced by approximately 90% of its nominal value, which is a measure of how robust the ground state is. While the effect of a finite thickness of the 2-D layer has been ignored here, it seems unlikely that it could have a qualitative effect on the properties of the ground state.

The linear dependence on $V_{9}$ is also seen on the sphere for both gaps and overlaps (Fig. 5). The trend of the gaps, shown in the inset, track those of the corresponding overlaps. The Pf and the aPf have opposing dependence on $V_{9}$. It is difficult to pin-point the transition point with any accuracy here, but it is somewhere between the crossing points of overlaps and the gaps. This is just a finite-size effect due to the size mismatch between the Pf and aPf, which prevents the two states from competing for the same ground state.

Since there is no shift on the torus, comparison of the two states is straightforward. Fig. 6 shows the overlaps as a function of $\kappa$ for a series of $\delta V_{9}$ values. The dependence on $V_{9}$ for each $\kappa$ is also linear (not shown). In contrast to the sphere results for small $\kappa$, there is opposing dependence of the overlaps on $\kappa$, with the aPf increasing and Pf decreasing. Increasing $\delta V_{9}$ past $0.008$ reverses the trends and the overlap with the Pf increases with $\kappa$.

Discussion.– Entanglement properties rather than high overlaps are a better indicator of which topological phase of matter a particular state may belong to. These have already been reported for the aPf ground state of the Coulomb potential, both on the cylinderZaletel et al. (2015) and on the spherePakrouski et al. (2015). However, very high overlaps can be reached (97%, $\kappa=2.1$ for $N_{e}=14$ on the sphere and 96%, $\kappa=1.5$ for $N_{e}=16$ on the hexagonal torus) by adiabatically varying the effective Hamiltonian to include only three ($V_{3}$, $V_{6}$, and $V_{9}$) of the 3-body pseudo-potentials and the Bishara-Nayak 2-body pseudo-potentials used by Woj’s et. al. This can be done without encountering a phase transition. While these parameters may seem unjustified or unphysical they do establish a broader phase diagram for the aPf ground state.

We have shown that under plausible experimental conditions the aPf is found to be favored for small LL-mixing parameters. The only exception is when just the lowest two LLs are kept, then the Pf is preferred. A reversal to the Pf at very large sizes cannot be ruled out by the present study. However, the aPf ground state appears to be robust, particularly on the torus, making such a reversal unlikely.