Quantum Hall Plateaus Research Articles

Universal Critical Behavior of the Quantum Hall Plateau Transition

Universal Critical Behavior of the Quantum Hall Plateau Transition

Conformal invariance, multifractality, and finite-size scaling at Anderson localization transitions in two dimensions

We generalize universal relations between the multifractal exponent \alpha_0 for the scaling of the typical wave function magnitude at a (Anderson) localization-delocalization transition in two dimensions and the corresponding critical finite size scaling (FSS) amplitude \Lambda_c of the typical localization length in quasi-one-dimensional (Q1D) geometry: (i) When open boundary conditions are imposed in the transverse direction of Q1D samples (strip geometry), we show that the corresponding critical FSS amplitude \Lambda_c^o is universally related to the boundary multifractal exponent \alpha_0^s for the typical wave function amplitude along a straight boundary (surface). (ii) We further propose a generalization of these universal relations to those symmetry classes whose density of states vanishes at the transition. (iii) We verify our generalized relations [Eqs. (6) and (7)] numerically for the following four types of two-dimensional Anderson transitions: (a) the metal-to-(ordinary insulator) transition in the spin-orbit (symplectic) symmetry class, (b) the metal-to-(Z_2 topological insulator) transition which is also in the spin-orbit (symplectic) class, (c) the integer quantum Hall plateau transition, and (d) the spin quantum Hall plateau transition.

Photoinduced Transition between Conventional and Topological Insulators in Two-Dimensional Electronic Systems

Manipulating the topological properties of insulators, encoded in invariants such as the Chern number and its generalizations, is now a major issue for realizing novel charge or spin responses in electron systems. We propose that a simple optical means, subjecting to a driving laser field with circular polarization, can be fruitfully incorporated to this end. Taking as a prototypical example the two-band insulator first considered by Haldane, we show how the electron system can be tuned through phases associated with different Chern numbers as the laser intensity is adiabatically swept, i.e., a photoinduced analog of the quantum Hall plateau transition. The implications of our findings include the possibility of laser tuning a conventional insulator into a quantum spin Hall system.

QUANTUM NETWORK MODELS AND CLASSICAL LOCALIZATION PROBLEMS

A review is given of quantum network models in class C which, on a suitable 2d lattice, describe the spin quantum Hall plateau transition. On a general class of graphs, however, many observables of such models can be mapped to those of a classical walk in a random environment, thus relating questions of quantum and classical localization. In many cases it is possible to make rigorous statements about the latter through the relation to associated percolation problems, in both two and three dimensions.

Crossover from the nonuniversal scaling regime to the universal scaling regime in quantum Hall plateau transitions

We present in this Brief Report our experimental results on the quantum Hall plateau-to-plateau transition in long-range Coulombic disordered two-dimensional electron systems embedded in the ${\text{Al}}_{x}{\text{Ga}}_{1\ensuremath{-}x}{\text{As-Al}}_{0.32}{\text{Ga}}_{0.68}\text{As}$ heterostructures (with $x=0\mathrm{%}$ and 0.21%) in a large temperature range from 1.2 K down to 1 mK. In these samples a crossover behavior is observed from the high-temperature, nonuniversal scaling regime to the low-temperature, universal scaling regime, with the temperature exponent $\ensuremath{\kappa}$ changing from $\ensuremath{\kappa}=0.58$ to 0.42, respectively. The crossover temperature increases with increasing $x$, from 120 mK for $x=0\mathrm{%}$ to 250 mK for $x=0.21\mathrm{%}$. When the Al concentration reaches $x=0.85\mathrm{%}$ at which the short-range random alloy potential dominates the disorder, the crossover temperature is beyond 1.2 K and the universal scaling is observed over two decades of temperature.

The current polarization rectification of the integer quantized Hall effect

We report on our theoretical investigation considering the widths of quantized Hall plateaus (QHPs) depending on the density asymmetry induced by the large current within the out-of-linear response regime. We solve the Schrödinger equation within the Hartree type mean field approximation using Thomas–Fermi–Poisson nonlinear screening theory. We observe that the two-dimensional electron system splits into compressible and incompressible regions for certain magnetic field intervals, where the Hall resistance is quantized and the longitudinal resistance vanishes, if an external current is imposed. We found that the strong current imposed induces an asymmetry on the widths of the incompressible strips (ISs) depending linearly on the current intensity and can be balanced by an inhomogeneous donor distribution.

Quantum Hall Plateau Transition in Graphene with Spatially Correlated Random Hopping

We investigate how the criticality of the quantum Hall plateau transition in disordered graphene differs from those in the ordinary quantum Hall systems, based on the honeycomb lattice with ripples modeled as random hoppings. The criticality of the graphene-specific n = 0 Landau level is found to change dramatically to an anomalous, almost exact fixed point as soon as we make the random hopping spatially correlated over a few bond lengths. We attribute this to the preserved chiral symmetry and suppressed scattering between K and K' points in the Brillouin zone. The results suggest that a fixed point for random Dirac fermions with chiral symmetry can be realized in freestanding, clean graphene with ripples.

Interferometric signature of non-Abelian anyons

We consider the tunneling current through a double point-contact Fabry-P\'erot interferometer such as used in recent experimental studies of the fractional quantum Hall plateau at filling fraction $\ensuremath{\nu}=5/2$. We compare the predictions of several different models of the state of the electrons at this plateau: the Moore-Read, anti-Pfaffian, $\text{SU}{(2)}_{2}$ NAF, $K=8$ strong pairing, and (3,3,1) states. All of these predict the existence of charge $e/4$ quasiparticles, but the first three are non-Abelian while the last two are Abelian. We give explicit formulas for the scaling of charge $e/2$ and charge $e/4$ quasiparticle contributions to the current as a function of temperature, gate voltage, and distance between the two point contacts for all three models. Based on these, we analyze several possible explanations of two phenomena reported for recent experiments by Willett et al., namely, halving of the period of the observed resistance oscillations with rising temperature and alternation between the same two observed periods at low temperatures as the area of the interference loop is varied with a side gate. We conclude that the most likely explanation is that the observed alternation is due to switching between even and odd numbers of charge $e/4$ quasiparticles enclosed within the loop as a function of side-gate voltage, which is a clear signature of the presence of non-Abelian anyons. However, there are important features of the data which do not have a simple explanation within this picture. We suggest further experiments which could help rule out some possible scenarios. We make the corresponding predictions for future tunneling and interference experiments at the other observed second Landau level fractional quantum Hall states.

ABJM with flavors and FQHE

We add fundamental matters to the N=6 Chern-Simons theory (ABJM theory), and show that D6-branes wrapped over AdS_4 x S^3/Z_2 in type IIA superstring theory on AdS_4 x CP^3 give its dual description with N=3 supersymmetry. We confirm this by the arguments based on R-symmetry, supersymmetry, and brane configuration of ABJM theory. We also analyze the fluctuations of the D6-brane and compute the conformal dimensions of dual operators. In the presence of fractional branes, the ABJM theory can model the fractional quantum Hall effect (FQHE), with RR-fields regarded as the external electric-magnetic field. We show that an addition of the flavor D6-brane describes a class of fractional quantum Hall plateau transition.

Scaling in Plateau-to-Plateau Transition: A Direct Connection of Quantum Hall Systems with the Anderson Localization Model

The quantum Hall-plateau transition was studied at temperatures down to 1 mK in a random alloy disordered high mobility two-dimensional electron gas. A perfect power-law scaling with kappa=0.42 was observed from 1.2 K down to 12 mK. This perfect scaling terminates sharply at a saturation temperature of Ts approximately 10 mK. The saturation is identified as a finite-size effect when the quantum phase coherence length (Lphi proportional, T(-p/2)) reaches the sample size (W) of millimeter scale. From a size dependent study, Ts proportional, W(-1) was observed and p=2 was obtained. The exponent of the localization length, determined directly from the measured kappa and p, is nu=2.38, and the dynamic critical exponent z=1.

Quantum Hall effect in a high-mobility two-dimensional electron gas on the surface of a cylinder

The quantum Hall effect is investigated in a high-mobility two-dimensional electron gas on the surface of a cylinder. The novel topology leads to a spatially varying filling factor along the current path. The resulting inhomogeneous current-density distribution gives rise to additional features in the magneto-transport, such as resistance asymmetry and modified longitudinal resistances. We experimentally demonstrate that the asymmetry relations satisfied in the integer filling factor regime are valid also in the transition regime to non-integer filling factors, thereby suggesting a more general form of these asymmetry relations. A model is developed based on the screening theory of the integer quantum Hall effect that allows the self-consistent calculation of the local electron density and thereby the local current density including the current along incompressible stripes. The model, which also includes the so-called `static skin effect' to account for the current density distribution in the compressible regions, is capable of explaining the main experimental observations. Due to the existence of an incompressible-compressible transition in the bulk, the system behaves always metal-like in contrast to the conventional Landauer-Buettiker description, in which the bulk remains completely insulating throughout the quantized Hall plateau regime.

DYNAMICAL GAPS AND QUANTUM HALL EFFECT IN GRAPHENE

We analyze the gap equation for Dirac quasiparticles in graphene in a magnetic field using a low-energy effective model with a contact interaction. It is found that the order parameters connected with the quantum Hall (QH) ferromagnetism and the magnetic catalysis scenarios necessarily coexist. The ground-state solutions of the gap equation describe all the recently discovered novel QH plateaus in graphene in strong magnetic fields.

Super Universality of the Quantum Hall Effect and the “Large N Picture” of the ϑ Angle

It is shown that the "massless chiral edge excitations" are an integral and universal aspect of the low energy dynamics of the $\vartheta$ vacuum that has historically gone unnoticed. Within the $SU(M+N)/S(U(M) \times U(N))$ non-linear sigma model we introduce an effective theory of "edge excitations" that fundamentally explains the quantum Hall effect. In sharp contrast to the common beliefs in the field our results indicate that this macroscopic quantization phenomenon is, in fact, a {\em super universal} strong coupling feature of the $\vartheta$ angle with the replica limit $M=N=0$ only playing a role of secondary importance. To demonstrate super universality we revisit the large $N$ expansion of the $CP^{N-1}$ model. We obtain, for the first time, explicit scaling results for the quantum Hall effect including quantum criticality of the quantum Hall plateau transition. Consequently a scaling diagram is obtained describing the cross-over between the weak coupling "instanton phase" and the strong coupling "quantum Hall phase" of the large $N$ theory. Our results are in accordance with the "instanton picture" of the $\vartheta$ angle but fundamentally invalidate all the ideas, expectations and conjectures that are based on the historical "large $N$ picture."

Low-field quantum Hall transport in an electron Fabry-Perot interferometer: Dependence of constriction filling on front-gate voltage

We report systematic quantum Hall transport experiments on Fabry-Perot electron interferometers at ultralow temperatures. The GaAs/AlGaAs heterostructure devices consist of two constrictions defined by etch trenches in the two-dimensional electron layer, enclosing an approximately circular island. Front gates deposited in etch trenches allow one to change the constriction filling, relative to the bulk. A systematic variation of the front-gate voltage affects the constriction and the island electron density, while the bulk density remains unaffected. This results in quantized plateaus in longitudinal resistance, while the Hall resistance is dominated by the low-density, low-filling constriction. At lower fields, when the quantum Hall plateaus fail to develop, we observe bulk Shubnikov-de Haas oscillations in series corresponding to an integer filling of the magnetoelectric subbands in the constrictions. This shows that the whole interferometer region is still quantum coherent at these lower fields at 10 mK. Analyzing the data within a Fock-Darwin model, we obtain the constriction electron density as a function of the front-gate bias and, extrapolating to the zero field, the number of electric subbands (conductance channels) resulting from the electron confinement in the constrictions.

Stacking faults, bound states, and quantum Hall plateaus in crystalline graphite

We analyze the electronic properties of a simple stacking defect in Bernal graphite. We show that a bound state forms, which disperses as ${|\mathbit{k}\ensuremath{-}\mathbit{K}|}^{3}$, in the vicinity of either of the two inequivalent zone corners $\mathbit{K}$. In the presence of a strong $c$-axis magnetic field, this bound state develops a Landau-level structure which for low energies behaves as ${E}_{n}\ensuremath{\propto}{|nB|}^{3/2}$. We show that buried stacking faults have observable consequences for surface spectroscopy, and we discuss the implications for the three-dimensional quantum Hall effect (3DQHE). We also analyze the Landau-level structure and chiral surface states of rhombohedral graphite and show that, when doped, it should exhibit multiple 3DQHE plateaus at modest fields.

Boundary criticality and multifractality at the two-dimensional spin quantum Hall transition

Multifractal scaling of critical wave functions at a disorder-driven (Anderson) localization transition is modified near boundaries of a sample. Here this effect is studied for the example of the spin quantum Hall plateau transition using the supersymmetry technique for disorder averaging. Upon mapping the spin quantum Hall transition to the classical percolation problem with reflecting boundaries, a number of multifractal exponents governing wave-function scaling near a boundary are obtained exactly. Moreover, additional exact boundary scaling exponents of the localization problem are extracted, and the problem is analyzed in other geometries.

Gravity dual of a quantum Hall plateau transition

We show how to model the transition between distinct quantum Hall plateaus in terms of D-branes in string theory. A low energy theory of 2+1 dimensional fermions is obtained by considering the D3-D7 system, and the plateau transition corresponds to moving the branes through one another. We study the transition at strong coupling using gauge/gravity duality and the probe approximation. Strong coupling leads to a novel kind of plateau transition: at low temperatures the transition remains discontinuous due to the effects of dynamical symmetry breaking and mass generation, and at high temperatures is only partially smoothed out.

Charge-statistics separation and probing non-Abelian states

Several states were proposed as candidates for the $\ensuremath{\nu}=5/2$ quantum Hall plateau. We suggest an experiment which can determine the physical state. The proposal involves transport measurements in the geometry with three quantum Hall edges connected by two quantum point contacts. In contrast to interference experiments, this approach can distinguish the Pfaffian and anti-Pfaffian states as well as different states with identical Pfaffian or anti-Pfaffian statistics. The transport is not sensitive to the fluctuations of the number of the quasiparticles trapped in the system.

Resistance Fluctuations and Aharonov–Bohm-Type Oscillations in Antidot Arrays in the Quantum Hall Regime

Resistance fluctuation phenomenon in antidot lattices in the quantum Hall regime are studied. Magnetoresistance of finite antidot array systems in the quantum Hall plateau transition regime exhibits two types of oscillatory effect. One is the aperiodic resistance fluctuations (RFs) and the other is the Aharonov-Bohm (AB)-type oscillations. Their dependences on the magnetic field and the gate voltage are quite distinct. While the aperiodic RFs are attributed to the complex evolution of the conducting network of compressible channels, the AB-type oscillations are interpreted in terms of edge states formed around individual antidots. The self-consistent screening effect is important for the both phenomenon, whereas, the single electron charging effect plays a minor role in the present case.

Boundary Multifractality at the Integer Quantum Hall Plateau Transition: Implications for the Critical Theory

We study multifractal spectra of critical wave functions at the integer quantum Hall plateau transition using the Chalker-Coddington network model. Our numerical results provide important new constraints which any critical theory for the transition will have to satisfy. We find a nonparabolic multifractal spectrum and determine the ratio of boundary to bulk multifractal exponents. Our results rule out an exactly parabolic spectrum that has been the centerpiece in a number of proposals for critical field theories of the transition. In addition, we demonstrate analytically exact parabolicity of the related boundary spectra in the two-dimensional chiral orthogonal "Gade-Wegner" symmetry class.