Novel Finite Temperature Conductivity in Quantum Hall Systems

We study quantum Hall systems (mainly the integer case) at finite temperatures and show that there is a novel temperature dependence even for a pure system, thanks to the ‘anomalous’ nature of generators of translation. The deviation of Hall conductivity from its zero temperature value is controlled by a parameter T0 = πρ/m N which is sample specific and hence the universality of quantization is lost at finite temperatures. PACS numbers: 73.40.Hm, 11.15.Bt Typeset using REVTEX 1 Quantum Hall (QH) systems have proved to be a rich source of exploring many interesting and unexpected features of quantized gauge theories. The Hall conductivity which is quantized is topological [1] in nature; it appears (in the integer system that we mainly study here) as the coefficient of Chern-Simons (CS) term which is induced by first order quantum corrections. Further it is exact [2] – with no higher order corrections, and it does not suffer any renormalization. It is also known that the Hall conductivity can be looked upon as a manifestation of chiral anomaly [3,4,5,6,7], inherited by the effectively planar system from the parent three dimensional system. Recall that in the Landau level problem at hand, the gauge transformations get mixed up with the Euclidean transformations in such a manner that the associated group is no more the (2+1) Euclidean group E3 , as one would normally have for a pure system. Rather, it is the group M3 of magnetic translations which is a proper subgroup of E3. It is recognized that the transition from E3 → M3 is crucial. The generators of translation (or equivalently, the operators for the centre of the orbit) do not commute [8]. In his studies on the closely related CS superconductivity (CSS), Fradkin [9] has designated this feature as ‘anomalous’ and has drawn detailed and explicit comparision with the well-known Schwinger – Anderson mechanism [10,11] which is a proper field theoretic anomaly. Here we do not attempt to rewrite the above mentioned non-commutativity in the standard language of field theoretic anomaly. However, we do believe in the essential correctness of Fradkin’s analogy, and as an explicit consequence we shall show that such an ‘anomaly’ is responsible for a novel temperature evolution of Hall conductivity σH even for a pure system. In this context we may recall that it is standard lore [12] that the presence of impurities, apart from its crucial role in stabilizing the quantization (in form of plateaus) is further required to destroy translational invariance in the system. It is believed that without such a breaking QH effect (QHE) would be trivial. Note that according to this argument, a uniform distribution of impurities would still be insufficient to give temperature dependence to Hall conductivity. We show that the Maxwell gauge interactions that are at play here belie such a naive expectation. 2 Such a temperature dependence has been noticed in the allied albeit rather academic example of CSS [13,14]. Even for QH systems, Bellisard et. al. [15], have made rough estimates of the temperature dependence of σH for a pure system. We believe that this paper presents, for the first time, a complete finite temperature (FT) analysis. Further, we also hope that the results obtained here will be verified experimentally. Consider a system of (weakly) interacting electrons in two space dimensions in the presence of a uniform external magnetic field of strength B, confined to the direction perpendicular to the plane. The strength is fine tuned such that N Landau levels (LL) are exactly filled. In the presence of sufficiently high magnetic field (as is relevant to our case), the spins of the fermions would be ‘frozen’ in the direction of magnetic field. Therefore, one may treat the fermions as spinless. The study of such a spinless system can be accomplished with the Lagrangian density, L = ψ∗iD0ψ − 1 2m |Dkψ| 2 + ψμψ − eAin 0 ρ+ 1 2 ∫ d3x′Ain 0 (x)V (x− x′)Ain 0 (x ) . (1) Here Dν = ∂ν − ie(Aν + A in 0 δν,0) (where Aν is the external Maxwell gauge field and A in 0 is identified as internal scalar potential), μ is the chemical potential, and m and ρ are the effective mass and the mean density of electrons respectively. The fourth term in Eq.(1) describes the charge neutrality of the system. Finally, V (x− x) represents the inverse of the instantaneous charge interaction potential (in the operator sense). The above Lagrangian density is equivalent to the usual interaction term with quartic form of fermi fields, which can be obtained by an integration of Ain 0 field in Eq. (1). Note also that the electrons interact with each other via 1/r or some other short range potential, i.e., the internal dynamics is governed by the (3+1)-dimensional Maxwell Lagrangian as is appropriate for the medium. The procedure for evaluating the FT properties of the system with the above Lagrangian density is standard. We do not discuss the details here since they have been presented in the allied context of CSS elegantly by Randjbar-Daemi, Salam and Strathdee [13], and has been extensively used [14]; in brief, we construct the partition function (β = 1/T being the inverse temperature), 3