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arXiv:2512.09045

On modeling quantum point contacts in quantum Hall systems

Prasoon Kumar, Kyrylo Snizhko

cond-mat.mes-hallcond-mat.str-elquant-phSubmitted Dec 9, 2025• Announced Dec 11, 2025
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Abstract

Quantum point contacts (QPC) are a key instrument in investigating the physics of edge excitations in the quantum Hall effect. However, at not-so-high bias voltage values, the predictions of the conventional point QPC model often deviate from the experimental data both in the integer and (more prominently) in the fractional quantum Hall regime. One of the possible explanations for such behaviors is the dependence of the tunneling between the edges on energy, an effect not present in the conventional model. Here we introduce two models that take QPC spatial extension into account: wide-QPC model that accounts for the distance along which the edges are in contact; long-QPC model accounts for the fact that the tunneling amplitude originates from a finite bulk gap and a finite distance between the two edges. We investigate the predictions of these two models in the integer quantum Hall regime for the energy dependence of the tunneling amplitude. We find that these two models predict opposite dependences: the amplitude decreasing or increasing away from the Fermi level. We thus elucidate the effect of the QPC geometry on the energy dependence of the tunneling amplitude and investigate its implications for transport observables.

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Quantum point contacts (QPC) are a key instrument in investigating the physics of edge excitations in the quantum Hall effect. However, at not-so-high bias voltage values, the predictions of the conventional point QPC model often deviate from the experimental data both in the integer and (more prominently) in the fractional quantum Hall regime. One of the possible explanations for such behaviors is the dependence of the tunneling between the edges on energy, an effect not present in the conventional model. Here we introduce two models that take QPC spatial extension into account: wide-QPC model that accounts for the distance along which the edges are in contact; long-QPC model accounts for the fact that the tunneling amplitude originates from a finite bulk gap and a finite distance between the two edges. We investigate the predictions of these two models in the integer quantum Hall regime for the energy dependence of the tunneling amplitude. We find that these two models predict opposite dependences: the amplitude decreasing or increasing away from the Fermi level. We thus elucidate the effect of the QPC geometry on the energy dependence of the tunneling amplitude and investigate its implications for transport observables.
Motivated by surprises in recent experimental findings, we study transport in a model of a quantum Hall edge system with a gate-voltage controlled constriction. A finite backscattered current at finite edge bias is explained from a Landauer-Buttiker analysis as arising from the splitting of edge current caused by the difference in the filling fractions of the bulk $({\ensuremath{\nu}}_{1})$ and constriction $({\ensuremath{\nu}}_{2})$ quantum Hall fluid regions. We develop a hydrodynamic theory for bosonic edge modes inspired by this model. The constriction region splits the incident long-wavelength chiral edge density-wave excitations among the transmitting and reflecting edge states encircling it. The competition between two interedge tunneling processes taking place inside the constriction, related by a quasiparticle-quasihole (qp-qh) symmetry, is accounted for by computing the boundary theories of the system. This competition is found to determine the strong coupling configuration of the system. A separatrix of qp-qh symmetric gapless critical states is found to lie between the relevant renormalization group (RG) flows to a metallic and an insulating configuration of the constriction system. This constitutes an interesting generalization of the Kane-Fisher quantum impurity model. The features of the RG phase diagram are also confirmed by computing various correlators and chiral linear conductances of the system. In this way, our results find excellent agreement with many recent puzzling experimental results for the cases of ${\ensuremath{\nu}}_{1}=1∕3$ and 1. We also discuss and make predictions for the case of a constriction system with ${\ensuremath{\nu}}_{2}=5∕2$.
In fractional quantum Hall systems, quasiparticles of fractional charge can tunnel between the edges at a quantum point contact. Such tunneling (or backscattering) processes contribute to charge transport and provide information on both the charge and statistics of the quasiparticles involved. Here, we study quasiparticle tunneling in the Moore-Read state, in which quasiparticles of charges e/4e/4e/4 (non-Abelian) and e/2e/2e/2 (Abelian) may coexist and both contribute to edge transport. On a disk geometry, we calculate the matrix elements for e/2e/2e/2 and e/4e/4e/4 quasiholes to tunnel through the bulk of the Moore-Read state, in an attempt to understand their relative importance. We find that the tunneling amplitude for charge e/2e/2e/2 quasihole is exponentially smaller than that for charge e/4e/4e/4 quasihole, and the ratio between them can be (partially) attributed to their charge difference. We find that including long-range Coulomb interaction only has a weak effect on the ratio. We discuss briefly the relevance of these results to recent tunneling and interferometry experiments at filling factor $\ensuremath{\nu}=5/2$.
We study a quantum point contact in the fractional quantum Hall regime at Landau level filling factors $\ensuremath{\nu}=1/3$ and 5/25/25/2. By using exact diagonalizations in the cylinder geometry, we identify the edge modes in the presence of a parabolic confining potential. By changing the sign of the potential, we can access both the tunneling through the bulk of the fluid and the tunneling between spatially separated droplets. This geometry is realized in the quantum point contact geometry for two-dimensional electron gases. In the case of the model Moore-Read Pfaffian state at filling factor $\ensuremath{\nu}=5/2$, we identify the conformal towers of many-body eigenstates including the non-Abelian sector. By a Monte-Carlo technique, we compute the various scaling exponents that characterize the edge modes. In the case of hard-core interactions whose ground states are exact model wave functions, we find equality of neutral and charged velocities, both bosonic and fermionic, for the Pfaffian state.
We study equilibration of quantum Hall edge states at integer filling factors, motivated by experiments involving point contacts at finite bias. Idealizing the experimental situation and extending the notion of a quantum quench, we consider time evolution from an initial nonequilibrium state in a translationally invariant system. We show that electron interactions bring the system into a steady state at long times. Strikingly, this state is not a thermal one: Its properties depend on the full functional form of the initial electron distribution and not simply on the initial energy density. Further, we demonstrate that measurements of the tunneling density of states at long times can yield either an overestimate or an underestimate of the energy density, depending on details of the analysis, and discuss this finding in connection with an apparent energy loss observed experimentally. More specifically, we treat several separate cases: for filling factor $\ensuremath{\nu}=1$ we discuss relaxation due to finite-range or Coulomb interactions between electrons in the same channel, and for filling factor $\ensuremath{\nu}=2$ we examine relaxation due to contact interactions between electrons in different channels. In both instances we calculate analytically the long-time asymptotics of the single-particle correlation function. These results are supported by an exact solution at arbitrary time for the problem of relaxation at $\ensuremath{\nu}=2$ from an initial state in which the two channels have electron distributions that are both thermal but with unequal temperatures, for which we also examine the tunneling density of states.
The elementary low energy excitations in the fractional quantum Hall (FQH) regime are known to be fractionally charged quasiparticles and quasiholes. This work focusses on quasiholes in a finite system of a few electrons treated by exact numerical diagonalization. Recent experimental [Chung et al., PRB 67, 201104 (2003)] and theoretical [Kane, Fisher, PRB 67, 045307 (2003)] work on quantum point contacts in the FQH regime showed strong evidence for single quasiparticle tunneling to be forbidden at low temperature. We want to gain insight into such constricted FQH systems by a numerical approach. Initially the basics of numerically treating a system in rectangular geometry and procedures to insert a quasihole excitation are presented. The excitations' stability and fractional charge is confirmed for different interactions. The question of quasihole tunneling is approached in two different systems: 1. A system posessing bound, localized states of a quasihole shows evidence for tunnel coupling between these states to exist. 2. Different potential forms to model a quantum point contact are investigated and the injection of a single quasihole near the constriction is surveyed in the system's time evolution.
We have recently calculated exact nonequilibrium quantum transport properties through a point contact in a Luttinger liquid. Using a particular quasiparticle basis of the Hilbert space dictated by integrability, we here compute explicitly the exact I(V) characteristic and conductance out of equilibrium as a function of driving voltage V and temperature T. These are described by universal scaling functions of two variables, the scaled point-contact interaction strength, and V/T. The differential-conductance curve as a function of the interaction strength broadens significantly as V/T is increased, and develops a pronounced maximum at a (universal) critical value (eV/kB{\mathit{k}}_{\mathit{B}}kB​T)=7.188 68. . .. In addition, we derive an exact duality between strong and weak backscattering. The theory presented here has recently been realized experimentally in resonant tunneling-transport experiments between edge states in fractional quantum Hall effect devices. In this context the exact duality is between electron tunneling and Laughlin-quasiparticle tunneling.