Unusual quasiparticles and tunneling conductance in quantum point contacts in $\nu=2/3$ fractional quantum Hall systems

Quantum Hall effect (QHE) emerges in quantizing magnetic fields in a two-dimensional electron gas. It exhibits a zero longitudinal resistance and the Hall conductance quantized in terms of the number of conductance quanta $e^{2}/2\pi\hbar$, where $e$ is the electron charge and $\hbar$ is the Planck constant. When electron correlations become important, the fractional quantum Hall effect (FQHE) exhibits a fractionally quantized conductance. FQHE excitations are fractionally charged quasiparticles that obey abelian or non-abelian anyon statistics Leinaas and Murheim (1977); Wilczek (1982a, b); Wilczek and Zee (1983); Halperin (1984); Arovas et al. (1984); Read and Green (2000). FQHE with its edge states is the remarkable breeding ground for the observation of topological phenomena and the creation of qubits for topological quantum computing Nayak et al. (2008); Lindner et al. (2012); Clarke et al. (2012); Vaezi (2013); Mong et al. (2014); Vaezi (2014); Simion et al. (2018); Liang et al. (2019); Hutzel et al. (2019); Tylan-Tyler and Lyanda-Geller (2015).

QHE conductance quantization in many cases stems from the edge state picture Halperin (1982) in the Landauer- Buttiker approach Büttiker (1988). For the FQHE, Wen Wen (1995) found that at filling factors of Laughlin Laughlin (1983) states $\nu=n\Phi_{0}/B=1/m$, where $m$ is odd, $n$ is the electron density, $\mathbf{B}$ is the external magnetic field, and $\Phi_{0}=2\pi\hbar c/e$ is the flux quanta, the excitations on the edge of the 2D electron system can be described as the chiral Luttinger liquids. While in quantum wires Ponomarenko (1995); Maslov and Stone (1995); Safi and Schulz (1995) conductance does not depend on the Luttinger liquid parameters, the picture of chiral Luttinger liquids allowed to explain the quantized Hall conductance Wen (1995); Kane and Fisher (1992) and tunneling of $\nu=1/m$ edge states through the quantum point contact (QPC) Fendley et al. (1995a, b); Sandler et al. (1998). Furthermore, in Fendley et al. (1995a) the thermodynamic Bethe Anzatz and the integrability of the Luttinger model with impurity interaction were used to calculate the non-equilibrium temperature-dependent conductance and noise.

A more complex situation emerges for hierarchical FQHE states Haldane (1983); Girvin (1984); Halperin (1984), such as $\nu=2/3$ state, for which several models were proposed. Since this state is a hole-conjugate of the $\nu=1/3$ state, it was suggested in MacDonald (1990); Johnson and MacDonald (1991) that its edge state consists of an outer downstream integer mode (from the underlying $\nu=1$ state) and an inner, counterpropagating -1/3 edge mode. Consideration of a bilayer system in McDonald and Haldane (1996) discussed this conjugated state and two other models, two independent $\nu=1/3$ edge channels and a pseudospin singlet state. The importance of disorder scattering between edge modes was underscored in Kane et al. (1994); Kane and Fisher (1997), where it was shown that the scattering results in two decoupled modes, a single downstream charge mode with conductance $2e^{2}/3h$ and a counterpropagating neutral mode. Soft confining potential leads to a reconstructed edge involving the formation of $\nu=1/3$ strip and two downstream charge modes with conductance $e^{2}/3h$ and counterpropagating neutral modes that were considered in Meir (1994); Wang et al. (2013). With different approaches to the structure of the edge, there have also been two approaches to transport. Conductance quantization implies equilibration between the chemical potentials of the reservoirs and the outgoing edges. The first approach, incoherent transport model, assumes a short coherent length and suppression of quantum interference between the channels Kane and Fisher (1995); Chamon and Fradkin (1997); Protopopov et al. (2017); Nosiglia et al. (2018); Spanslatt et al. (2020); the second approach is quantum solution taking interference into account Fendley et al. (1995a); Ponomarenko and Averin (2003). Experimentally, studies of tunneling through quantum point contacts (QPC) at $\nu=2/3$ in Bid et al. (2009); Baer et al. (2014); Sabo et al. (2017) have shown a conductance plateau at $e^{2}/3h$, however, recent work Nakamura et al. (2023) demonstrated its appearance within $2\%$ of $e^{2}/2h$.

In this paper, we find an exact quantum mechanical solution to the model of tunneling through QPC in several $\nu=2/3$ edge configurations. It exhibits the fractional charge $e/2$ of backscattering quasiparticles, and $e^{2}/2h$ conductance plateau. One case when the model is realized is associated with the transition between spin-polarized and unpolarized $\nu=2/3$ FQH phases Eisenstein et al. (1990); Kraus et al. (2002); Wu et al. (2012) which occurs due to the crossing of the two composite fermion (CF) energy states Jain (1989, 2007) with opposite spin polarization Ronen et al. (2018); Kukushkin et al. (1999); Liang et al. (2019). It was recently shown Kazakov et al. (2016, 2017); Wu et al. (2018); Wang et al. (2021) that the two different spin phases can be induced in the neighboring regions of the FQH liquid by electrostatic gates. The properties of a contact between the regions naturally lead to the model with two CF edge modes of the spin-polarized phase coupled by electron tunneling through QPC at low voltage and temperature to one CF edge mode of the same spin in the unpolarized phase. The same model also describes tunneling through QPC between the FQH regions in the identical spin phases when two CF edge modes in one of those regions are well separated in space due to its potential profile so that only one mode can tunnel. The exact solution to the model shows that with an increase of the applied voltage or temperature, the QPC conductance grows from zero and saturates at $e^{2}/2h$ while a weak electron tunneling between the edge modes with the same spin transforms into the backscattering carried by the charge $q=e/2$ quasiparticles. This confirms the $e^{2}/2h$ conductance earlier found by us for the contact between spin-polarized and unpolarized regions using a strong coupling boundary conditions approach Wang et al. (2021). We show that the quasiparticle $e/2$ charge is also observable in the shot noise. Testing the predicted non-linear applied voltage dependence of the tunneling current, temperature dependence, and properties of the shot noise can distinguish coherent and incoherent transport models and uncover the physics of edges in both polarized and unpolarized FQHE at $\nu=2/3$.

Model for edge states. The key point of our approach is the assumption of separation of the charge and neutral/spin edge modes due to the long-range Coulomb interaction, discussed in Wen (1995). We treat polarized and unpolarized phases of $\nu=2/3$ FQHE on equal footing describing edge states as chiral Luttinger liquids with action

where $\varphi_{c}$ and $\varphi_{n}$ are bosonic operators and $v_{c}$ and $v_{n}$ are velocities of the charge and the neutral modes, correspondingly. Their relation to electron edge modes corresponding to the two CF $\Lambda-$ levels and to quasiparticle bosonic fields, properties of these operators, transformations between different representations, and underlying assumptions are discussed in the supplementary materials. The neutral operators describe the difference in the occupation numbers of $\Lambda$-levels in the polarized phase and the spin density in the unpolarized phase. We will use the notation $\varphi_{pc}$ and $\varphi_{uc}$ for charge modes, $\varphi_{pn}$ and $\varphi_{us}$ for neutral (spin) modes in the corresponding phases. We assume that no edge reconstruction takes place.

Tunneling and charge current. We begin with tunneling through a QPC between the polarized and unpolarized phases. In the weak coupling limit, it is carried by the whole electrons through the allowed spin-conserving processes involving the edge modes of the two CF $\Lambda$ levels in each phase. The point contact (junction) electron tunneling at $x=0$ between polarized and unpolarized phases is described by tunneling Hamiltonian

Here $\{\xi_{b},\xi_{c}\}=2\delta_{b,c}$, $\xi$’s are Majorana fermions accounting for the fermion statistics of different CF edge modes and composing the corresponding Klein factors, and

where $\Phi_{pa}$ and $\Phi_{u1}$ are the CF fields, the tunneling amplitudes $U_{a}$ can be chosen real and positive, and $V$ is the applied voltage. We assume coherent propagation of charge and neutral (spin) modes along the edge and allow for the possibility of interference. The same tunneling Hamiltonian Eq. (2) can also describe tunneling between the two spin-polarized phases, with a single edge channel $u=p^{\prime}$ participating in tunneling, if the potential profile ensures that the second $\Lambda$-level edge is located too far from the $p^{\prime}$ edge. Thus we study the QPC tunneling of two modes into one mode; tunneling of one mode into one mode will emerge for tunneling amplitude $U_{2}=0$.

The tunneling charge current, from Eq. (2) is given by

where $\rho_{pc}=\frac{1}{\sqrt{2}\pi}\partial_{x}\varphi_{pc}$ is the charge density, e=1. Its average and fluctuations are defined by the operators $\Psi_{a}(t),\ a=1,2$, whose evolution in the absence of tunneling is characterized by their second-order correlators. The latter are found from the Gaussian action Eq.(1) for $\varphi_{pc}$, $\varphi_{pn}$ and similarly for $\varphi_{uc}$ and $\varphi_{ps}$ in the form

$g(t)$ is the single point correlator at $x=0$ of a normalized chiral boson field, with action as in Eq.(1) for $\varphi_{pn}$,

Here $\alpha=1/D$, $D$ is the energy cut-off in both edges, and $\delta\rightarrow 0$ should be taken in the final results. In terms of the normalized chiral boson fields $\phi_{j}(x,t),j=0,a$, the operators $\Psi_{a}(t),\ a=1,2$ can also be represented as

which after fermionization

allows to re-write the tunneling Hamiltonian (2) as

leading to its interpretation as the tunneling process of electrons from the two non-interacting chiral channels into the FQHE edge of $\nu=1/3$ filling factor. Combining these interfering channels into a single tunneling channel and applying bosonization of its chiral fermion field

we obtain the tunneling Hamiltonian ${\cal H}_{T}$ and current $J_{T}$:

where $\tilde{t}^{2}=2\sum_{a}U_{a}^{2}$, and the single Klein factor is omitted since its drops out from any perturbative order in $\tilde{t}$ due to the charge conservation.

The electron tunneling described by Eqs. (11,12) can be viewed as a single point electron tunneling between the two counterpropagating primary one-component edges of equal filling factor $\nu=1/2$ with densities $\rho_{d,n}(x,t)=\pm\frac{1}{2\pi}\sqrt{\nu}\partial_{x}\phi_{d,n}(x,t)$ and the corresponding electron annihilation operators $e^{i\frac{1}{\sqrt{\nu}}\phi_{d,n}(x,t)}$. Here $\phi_{d,n}$ are normalized right and left moving chiral fields. Thus, the problem described by Eq.(2) maps onto and is equivalent to the problem of tunneling between the one-component edges $\phi_{d}(x,t)$ and $\phi_{n}(x,t)$ as described by Hamiltonian

The tunneling current $J_{T}$ in the latter problem coincides with $J_{T}$ in Eqs. (4,12). Its average was calculated in Fendley et al. (1995a). At finite temperatures $T$ this average is given by

where $f(\frac{\epsilon}{T})$ is the Fermi distribution function, and $\Gamma=2D^{2}/(\pi\tilde{t})$ Weiss (1996). The integral in (14) is evaluated in terms of the imaginary part of the digamma-function $\psi(x)$ Weiss et al. (1991):

describing the dependence of tunneling current on the applied voltage and temperature in the whole range of parameters. In Fig.1 conductance $\langle J_{T}\rangle/V$ is plotted as function of dimensionless variables $1/\tilde{V}=\Gamma/V$ and $1/\tilde{T}=\Gamma/2\pi T$. At $T=0$ $\langle J_{T}\rangle$ is given by

Tunneling current in various regimes. In order to elucidate different contributions to the tunneling current, we consider limiting cases of Eq. (16). In the regime of low $V$ , $V\ll\Gamma$, of electron tunneling described by Eq. (13), the current $\langle J_{T}^{l}\rangle$ is cubic in $V$:

Therefore, we observe that in the coherent approach to transport, the behavior of tunneling current and conductance at small voltages is nonlinear in the applied voltage for $T\ll V$, as opposed to the picture of incoherent transport Protopopov et al. (2017); Nosiglia et al. (2018); Spanslatt et al. (2020) that results in Landauer-Buettiker current linear in voltage and voltage-independent conductance.

In the regime of high voltages, $V\gg\Gamma$, the exact Eq.(16) gives the following result for the average current, which we will denote $J_{T}^{h}$:

The first term in this expression defines conductance $G=e^{2}/2h$. The second term corresponds to the reduction of the tunneling current $\langle J_{T}\rangle$ defined by by the quasiparticle backscattering current $J_{bsc}$

where $\langle J_{T}\rangle=V/4\pi$.

In the supplement, we show an alternative way to calculate the tunneling currents using the strong coupling boundary conditions. We calculate the quasiparticle charge, and neutral and spin tunneling currents for tunneling through QPCs between polarized and unpolarized phases or between two polarized FQH liquids with one edge channel precluded from tunneling. The fermionization method above shows that tunneling of two interfering edge channels from one QPC side into a single channel on the other side is equivalent to a single tunneling channel with a renormalized tunneling amplitude.

We note that QPC conductance $G=e^{2}/2h$ has been also discussed Nakamura et al. (2023) in terms of the incoherent equilibration model. However, neither the nonlinear current-voltage characteristics nor the temperature dependence following from Eq. (15) arise in that approach. $G=e^{2}/2h$ has been also discussed for a special tunneling configuration, in which tunneling between two $\nu=2/3$ regions proceeds through the quantum dot with $\nu=1$ Lai and Yang (2013).

Quasiparticle charge. The reduction of tunneling current due to quasiparticle backscattering is described and their charge is determined by taking into account a sudden change of strong coupling boundary conditions at $t=t_{0}$. For tunneling, e.g., between $p1$ edge mode in the polarized region and $u1$ edge mode with the same spin in the unpolarized region through the QPC at $x=0$ the strong coupling boundary conditions $\tilde{t}\rightarrow\infty$ are given by

Here both $\Phi_{\alpha,1}(x,t),\alpha=p,u$ fields are extended to finite $x$ from their $x=0$ expressions in Eq. (3), as chiral right moving fields. A sudden variation of the boundary conditions changes $n=0$ to $n=1$; both $n$ minimize $-\tilde{t}\cos{(\Phi_{p1}(0,t)-\Phi_{u1}(0,t))}$. The boundary conditions also must keep continuous the two dual fields

where $\theta(x)$ is the Heaviside step function. The jump from $n=0$ to $n=1$ leads to a jump in the dual fields

Since $\left(\eta_{1}(x,t_{0})+\eta_{2}(x,t_{0})\right)|_{x=-0}^{+0}=0,$ the corresponding jumps in these fields are given by

Then we get the change

Using the continuity condition

we find

Then the changes in the charge density and the density of the neutral mode are giiven by

In the supplement, we evaluate changes in tunneling densities of the spin mode in the unpolarized region and demonstrate relations between changes of densities for charge, neural and spin modes on both sides of the QPC.

As was shown in Ponomarenko and Averin (2004), the calculated transferred charge (28) enables us to identify the charge $q$ of backscattering quasiparticles. Here $q=e/2$. Surprisingly, this charge is different from $q=e/3$ that one expects from Kane et al. (1994). This difference stems from the absence of tunneling between two edge states of opposite spin polarizations in the polarized and unpolarized regions, or, for tunneling between the regions with the same spin polarization, from a single channel participating in tunneling in at least one of the regions, due to a potential profile.

Shot noise. The experimental determination of the charge of the tunneling quasiparticles can be achieved by measurements of the shot nonequilibrium noise Saminadayar et al. (1997); De-Picciotto et al. (1997). The shot noise of the tunneling current Kane and Fisher (1994); Fendley et al. (1995a); Chamon et al. (1996); Lesage and Saleur (1997); Sandler et al. (1999) is given by

From Fendley et al. (1995a) it follows that

At small voltages and weak backscattering $V\ll\Gamma$

At large voltages $V\gg\Gamma$

The Schottky formula gives the charge of tunneling quasiparticles $q$ in the limits of weak electron tunneling and weak quasiparticle backscattering as the ratio of the corresponding values of noise to the currents, $S^{l}/\langle J_{T}\rangle$ and $S^{h}/\langle J_{bsc}\rangle$. The resulting charge (writing e explicitly) is

Thus, the result for shot noise confirms the one obtained from the analysis of the sudden change in the boundary conditions: for tunneling at a filling factor $\nu=2/3$ through the QPC in our model, the quasiparticle charge is $e/2$ rather than the expected $e/3$. We note that the noise signal arising in our model is much stronger than the noise in the incoherent equilibration model.

Discussion and conclusion. We have shown that the coherent quantum-mechanical model of transport in the $\nu=2/3$ FQHE, that potentially involves interference of chiral Luttinger liquid edge modes, leads to an exact solution of the problem of tunneling through the QPC between $\nu=2/3$ FQHE regions with different or similar spin phases. With the increase of the applied voltage, the QPC conductance grows from zero and saturates at $e^{2}/2h$ while a weak electron tunneling through the QPC between the same spin edge modes transforms into the backscattering carried by the fractional charge $q=e/2$ quasiparticles. Unusual new quasiparticles and fractional conductance emerge in the QPC with one or two CF modes scattering into one mode as is the case in tunneling between polarized and unpolarized FQHE phases and can occur in tunneling between similar phases as a result of engineering the potential profile. Using the fermionization method, we have shown that tunneling of the two CF modes from one of the sides of the QPC is renormalized with the account of interference between the two modes, and is equivalent to tunneling of a single mode.

Recent experiments on QPC tunneling at $\nu=2/3$ have shown signatures of $e^{2}/2h$ plateau. Besides our model this conductance can be discussed using the incoherent equilibration transport approach. However, then the non-linear current-voltage characteristics and temperature dependence of tunneling current and noise, which are predicted in the present work and can be tested experimentally, do not emerge in the incoherent model.

Acknowledgement. We are grateful to M. Manfra and L. Rokhinson for helpful discussions. YLG was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award DE-SC0010544.

Supplementary Materials