On the Theory of Electric DC‐Conductivity: Linear and Non‐linear Microscopic Evolution and Macroscopic Behavior (original) (raw)

Quantum Effect in D. C. and Hall Conductivities

Progress of Theoretical Physics, 1973

The Wigner representation formalism is applied to investigating the effect of the momentum-coordinate commutation relation ~n the d.c. and Hall conductivities of a system of noninteracting electrons moving in a• potential :field of randomly distributed impurities. The conductivities are expanded in powers of A and the second-and fourth-order terms are shown to vanish within the Born approximation, as far as the expansion is reasonable. This situation is discussed in comparison with the result of the kinetic theory.

Activated Conductivity in the Quantum Hall Effect

Physical Review Letters, 1994

Activated dissipative conductivity ¢==o-*~exp(-A/T) and the activated deviation of the Hall conductivity from the precise quanfizafion &r~v=~-ie2/hf~exp(-A/T) are studied in a plateau range of the quantum Hall effect. The prefactors cr*~ and o*~ are calculated for the case of a long-range random potential in the fxa~ework of a classical theory. There is a range of temperatures Tx << T<< T2 where ¢r*~ = e2/h. In this range ~ ~ (e2/h)(T/Ta)S°/21<< o'*~. At large T>> T2. on the other hand, a~ = e2/h and ~ = (ea/h)(Ta/T) I°/ts << a~,. Similar results are valid for a fractional plateau near the lining factor p/q if charge e is replaced by e/q.

Phenomenological Model for Frequency-Related Dissipation in the Quantized Hall Resistance

IEEE Transactions on Instrumentation and Measurement, 2007

Recently, ac measurements of the quantized Hall resistance have shown a linear relationship between the deviation of the Hall resistance from the perfectly quantized value ∆R H and the dissipation in the system represented by the longitudinal resistivity ρ ac xx . In this paper, we present a phenomenological model based on electrodynamic arguments that model this relation. The dissipation is due to the displacement current flowing in the system. All the microscopic features are included in a complex tensorial dielectric susceptibility which remains to be investigated if more physical insight in the ac transport properties of the 2-D electron gas is desired.

Quantum electron transport beyond linear response

Physica A-statistical Mechanics and Its Applications, 2002

After a brief historical survey and summary of previous work involving the linear response theory (LRT) Hamiltonian, H =H (system)−AF(t), where the latter part refers to the e ect of an applied ÿeld F(t) as in Kubo's theory, we deal with the nonlinear problem in which the applied electric ÿeld of arbitrary strength, as well as a possible magnetic ÿeld, are included in the system Hamiltonian, H = H [system(E; B)]. In Part A of this study we deal with the general formalism on the many-body level. Projection operators are applied to the Von Neumann equation in the interaction picture, H (system) = H 0 + V , which after the Van Hove limit, → 0; t → ∞; 2 t ÿnite, leads to the master equation for @ =@t, containing both the Pauli-Van Hove diagonal part involving the transitions W , and a nondiagonal quantum-interference part, as obtained by us previously. Likewise, the full many-body current operator JA is obtained by manipulation of the Heisenberg equation of motion for dA=dt in the interaction picture.

Dynamic Conductance in Quantum Hall Systems

1996

In the framework of the edge-channel picture and the scattering approach to conduction, we discuss the low frequency admittance of quantized Hall samples up to second order in frequency. The first-order term gives the leading order phase-shift between current and voltage and is associated with the displacement current. It is determined by the emittance which is a capacitance in a capacitive arrangement of edge channels but which is inductive-like if edge channels predominate which transmit charge between different reservoirs. The second-order term is associated with the charge relaxation. We apply our results to a Corbino disc and to two-and four-terminal quantum Hall bars, and we discuss the symmetry properties of the current response. In particular, we calculate the longitudinal resistance and the Hall resistance as a function of frequency.

Low-frequency anomalies and scaling of the dynamic conductivity in the quantum Hall effect

Physical Review B, 1996

A numerical study of the dynamic conductivity xx () in the lowest Landau level for a quantum Hall system with short-range and long-range disorder potentials is performed. In the latter case two distinct types of low-frequency anomalies are observed: a scaling regime with an anomalous diffusion exponent of ϭ0.36Ϯ0.06 independent of the potential correlation range and a semiclassical regime giving evidence of the existence of long time tails in the velocity correlation decaying proportional to t Ϫ2. The range of validity of this behavior increases with increasing. The universal value of the critical conductivity is xx c ϭ(0.5Ϯ0.02)e 2 /h for ϭ0 to 2 magnetic lengths. ͓S0163-1829͑96͒00720-5͔ PHYSICAL REVIEW B

Quantization of the Hall conductivity in the Harper-Hofstadter model

Physical Review B

We study the robustness of the quantization of the Hall conductivity in the Harper-Hofstadter model towards the details of the protocol with which a longitudinal uniform driving force Fx(t) is turned on. In the vector potential gauge, through Peierls substitution, this involves the switchingon of complex time-dependent hopping amplitudes e − i Ax(t) in thex-direction such that ∂tAx(t) = Fx(t). The switching-on can be sudden, Fx(t) = θ(t)F , where F is the steady driving force, or more generally smooth Fx(t) = f (t/t0)F , where f (t/t0) is such that f (0) = 0 and f (1) = 1. We investigate how the time-averaged (steady-state) particle current density jy in theŷ-direction deviates from the quantized value jy h/F = n due to the finite value of F and the details of the switching-on protocol. Exploiting the time-periodicity of the HamiltonianĤ(t), we use Floquet techniques to study this problem. In this picture the (Kubo) linear response F → 0 regime corresponds to the adiabatic limit forĤ(t). In the case of a sudden quench jy h/F shows F 2 corrections to the perfectly quantized limit. When the switching-on is smooth, the result depends on the switch-on time t0: for a fixed t0 we observe a crossover force F * between a quadratic regime for F < F * and a non-analytic exponential e −γ/|F | for F > F * . The crossover F * decreases as t0 increases, eventually recovering the topological robustness. These effects are in principle amenable to experimental tests in optical lattice cold atomic systems with synthetic gauge fields. arXiv:1809.05562v1 [cond-mat.quant-gas]

Theory of the quantized Hall effect (III)

Nuclear Physics B, 1984

In the previous paper, we have demonstrated the need for a phase transition as a function of 0 in the non-linear o-model describing the quantized Hall effect. In this work, we present arguments for the occurrence of exactly such a transition. We make use of a dilutc gas instanton approximation as well as present a more rigorous duality argument to show that the usual scaling of the conductivity_ to zero at large distances i~ altered whenever o~l~ u~-2te-/t,~ n integer. This then completes our theo~ of the quantized Hall effect,

Universal Prefactor of Activated Conductivity in the Quantum Hall Effect

Physical Review Letters, 1995

The prefactor of the activated dissipative conductivity in a plateau range of the quantum Hall effect is studied in the case of a long-range random potential. It is shown that due to long time it takes for an electron to drift along the perimeter of a large percolation cluster, phonons are able to maintain quasiequilibrium inside the cluster. The saddle points separating such clusters may then be viewed as ballistic point contacts between electron reservoirs with different electrochemical potentials. The network of ballistic conductances is shown to determine the conductivity. The prefactor is universal and equal to 2e 2 /h at an integer filling factor ν and to 2e 2 /q 2 h at ν = p/q.