arXiv:cond-mat/0303170v1 [cond-mat.mes-hall] 10 Mar 2003 Interplay of Short-Range Interactions and Quantum Interference Near the Integer Quantum Hall Transition V. M. Apalkov and M. E. Raikh Department of Physics, University of Utah, Salt Lake City, UT 84112, USA Short-range electron-electron interactions are incorporated into the network model of the integer quantum Hall effect. In the presence of interactions, the electrons, propagating along one link, experience exchange scattering off the Friedel oscillations of the density matrix of electrons on the neighboring links. As a result, the energy dependence of the transmission, T (ǫ), of the node, connecting the two links, develops an anomaly at the Fermi level, ǫ = ǫF . We show that this interaction-induced anomaly in T (ǫ) translates into the anomalous behavior of the Hall conductivity, σxy (ν), where ν is the filling factor (we assume that the electrons are spinless). At low temperatures, T → 0, the evolution of the quantized σxy with decreasing ν proceeds as 1 → 2 → 0, in apparent violation of the semicircle relation. The anomaly in T (ǫ) also affects the temperature dependence of the peak in the diagonal conductivity, σxx (ν, T ). In particular, unlike the case of noninteracting electrons, the maximum value of σxx stays at σxx = 0.5 within a wide temperature interval. I. INTRODUCTION Understanding of the integer quantum Hall transitions was strongly facilitated by the network model [1] introduced by Chalker and Coddington (CC). Within this model, delocalization of singleelectron states at certain discrete energies is governed by quantum interference of waves, propagating along the chiral links connecting the nodes of the network. In contrast to the original network model [2], the CC model emerges in a natural way from the microscopic consideration of an electron moving in a smooth potential, U(x, y), and a strong perpendicular magnetic field. Then the chiral links of the network are nothing but the equipotential lines, U(x, y) = const. This semiclassical picture corresponds to the case when the magnitude of the random potential, U0 , is much less than the cyclotron energy, while the correlation radius, Rc , is much bigger than the magnetic length, l. The interference effects become important within the energy interval l Rc Γ = U0 2 (1) around the center of the Landau band. Since Γ ≪ U0 , the “unit cell” of the CC network has a characteristic size Dc = Rc (U0 /Γ)4/3 , as follows from the percolation theory [3]. The actual lengths < of the equipotentials connecting the nodes, which are the saddle points of U(x, y) with heights ∼ Γ, 7/4 are much bigger than Dc , namely, Lc ∼ Rc (Dc /Rc ) [3]. On the quantitative level, the CC model yields an accurate value of the critical exponent νc ≈ 7/3 [4], which governs the divergence of the localization radius, ξ(ǫ), at small energies ǫ ≪ Γ, measured from the center of the band Γ |ǫ| ξ(ǫ) = Dc 1 7/3 . (2) The CC model also yields more delicate characteristics of the transition, such as multifractal exponent of the critical wave functions [5]. The effect of short-range electron-electron interactions on the quantum Hall transition was recently addressed in Ref. [6]. Previously, short-range interactions (e.g. screened by a gate) were demonstrated to be irrelevant on the basis of analysis of the renormalization dimension [7]. It was argued in Ref. [6] that, although short-range interactions do not affect the large-distance structure of the critical wave functions [8], they determine the conductance of the quantum Hall sample via the phase breaking time. The underlying physics of the interaction-induced phase breaking in the quantum Hall regime is the same as that for 2D disordered conductor in a zero field [9]. It was shown in Ref. [6] that the exponent, p, in power-law temperature dependence of the phase breaking length, Lϕ ∝ T −p/2 , is equal to p = 1.65 for the interactions screened by a gate. Ref. [6] contains a comprehensive analysis of the temperature and frequency scaling of the conductivity near the transition point. The length Lϕ enters into this analysis as a parameter. In short, the current understanding of the transition is based on the assumption that interference effects, responsible for localization, and short-range interactions are effectively decoupled from each other. In this paper we point out that, with short-range interactions, there exists another interactionrelated process, qualitatively different from the phase breaking, which is relevant for the quantum Hall transition, namely, the interaction-induced interference (III). This process was uncovered by Matveev, Yue, and Glazman [10] in course of the calculation of 1D electron scattering from the point-like impurity in the presence of the Fermi sea. In Ref. [10] a nontrivial interplay between short-range interactions and quantum interference was traced on the microscopic level. Here we generalize the consideration of Ref. [10] to the quantum Hall geometry by incorporating the III process into the CC model. We find that strong enough interactions result in the qualitative change in the behavior of the quantized Hall conductivity, σxy , with increasing magnetic field. In the spinless situation, instead of the transition from e2 /h to the insulator, σxy undergoes the transitions 1 → 2 → 0. The transition 2 → 0 is not accompanied by any change in the diagonal conductivity, σxx , which remains close to zero, in apparent violation of the semicircle relation [11,12]. We have also studied the impact of III on the evolution of the σxx peak with T . It turns out that III gives rise to an almost metallic behavior at the center of the Landau band, in the sense, that the peak value σxx = 0.5 remains temperature-independent within a wide interval of T . II. INTERACTION-INDUCED INTERFERENCE IN THE QUANTUM HALL REGIME The original scenario [10] of III in one dimension is the following. In the absence of interactions, the transformation of an incident wave into reflected and transmitted waves occurs in the vicinity of the impurity. In addition, the impurity causes the Friedel oscillations of the electron density and of the density matrix, which fall off slowly at large distances. As the electron-electron interactions are switched on, both perturbations, having the spatial structure with a period (2kF )−1 , where kF is the Fermi momentum, cause the additional coupling of the incident and reflected (from the impurity) electron waves. As a consequence of this coupling, the reflected wave transforms back into the incident wave at distances far away from the impurity, and, subsequently, gets transmitted. Interference of this “secondary” transmitted wave, which is due to interactions, with the zero-order transmitted wave is the III process. Compared to the zero-order transmitted wave, the secondary wave travels an additional closed path, first away from the impurity, and then towards the impurity. 2 Direct calculation [10] shows that, the III is constructive, if the coupling between the incident and reflective waves is mediated by the Friedel oscillations of the electron density, and destructive, if it is mediated by the Friedel oscillations of the density matrix. III is strong, when the energy, ǫ, of the incident electron is close to the Fermi level, ǫF . This is due to the Bragg-like resonant enhancement [10], which originates from the fact that the size of the region, where the scattering from the Friedel oscillations takes place, is ∝ |ǫ − ǫF |−1 . When this size is smaller than the length of the 1D channel, the amplitude of the interaction-induced transmitted wave diverges as ln |ǫ − ǫF |, indicating that the net transmission is strongly modified by the interactions. L1 L4 C B M interaction region A D L2 L3 FIG. 1. Schematic illustration a Larmour circle drifting along equipotentials L1 ...L4 separated by a saddle point; dotted lines illustrate exchange-induced scattering processes leading to III . This scattering takes place within dashed regions ∼ d away from the saddle point. Let us consider the network model of the quantum Hall effect from the point of view of III. Within the network model, the role of scatterers is played by the saddle points. As it is illustrated in Fig. 1, due to the structure of the saddle-point potential (maximum along the horizontal and minimum along the vertical direction), the electron seas are located only in the dashed regions. Scattering of the electron, drifting along the equipotential line, by the saddle point, differs from the 1D impurity scattering [10] in three respects (i) Equipotential branches, say, L1 , L2 diverge. Thus, the incident and reflected waves do not interfere far away from the saddle point. > (ii) Starting from short distances, L1 , L2 ∼ l, the electron wave functions, Φµ,ǫ (L), taken at L = L1 and L = L2 do not overlap. This is because Φµ,ǫ (L) describe the Landau orbits, drifting along the equipotentials U(x, y) = ǫ; the spatial extent of these functions in the direction normal to the equipotential is ∼ l. Here the index µ = L, R labels the wave functions incident from the left and from the right, respectively (Fig. 1). (iii) interaction of an electron incident along L1 with both reflected (drifting along L2 ) and transmitted (drifting along L3 ) electrons must be taken into account. The immediate consequence of (ii) is complete absence of the Friedel oscillations of the electron density. Thus, the Hartree contribution to the Fermi edge anomaly in 1D impurity scattering, caused by III, is absent in the problem of the reflection from the saddle point. Our prime observation is 3 that, the contribution, caused by the scattering from the Friedel oscillations of the density matrix (exchange contribution), survives in the quantum Hall geometry. To illustrate this, consider the density matrix of electrons drifting along L1 and L2 Φ∗ (L2 )Φµ,ǫ (L1 ), µ,ǫ ρ (L2 , L1 ) = (3) µ ǫ<ǫF where, for simplicity, we neglect the spin. If the short-range character of interactions is due to the gate electrode at distance, d, from the 2D plane, then the interaction potential has the form V (ρ) = e2 1 1 . −√ 2 κ ρ ρ + 4d2 (4) The exchange Hamiltonian, acting on the wave function of the electron drifting, say, along equipotential L2 , creates an electron at the point L = L1 of the opposite branch, according to the rule ˆ Hex ΦL,ǫ | L=L1 =− dL2 V (L, L2 )ρ (L2 , L)ΦL,ǫ (L2 ), (5) where V (L1 , L2 ) is the interaction energy of two electrons located at points L1 and L2 on different branches. Below we assume that d is much smaller than the period, Dc , of the CC network. Therefore, the contribution to the integral (5) comes only from the piece of equipotential L2 which lies within the distance ∼ d from the saddle point (the actual length of this piece is much bigger than d, see Fig. 1). However, the most important feature of the integral (5) is that, in contrast to the Hartree contribution, the integrand does not contain the product of the wave functions from different equipotential branches taken at the same point. In other words, the structure of the integrand does not restrict the relative positions of the points L and L2 within the magnetic length. In short, the Larmour motion drops out from the exchange contribution, thus making it similar to the 1D case. The integrand in Eq. (5) describes the additional closed-loop contour A → B → M → A, traversed by the secondary wave, as illustrated in Fig. 1. Thus we conclude, that III in the quantum Hall geometry is exclusively due to the indistinguishability of the electrons. In contrast to the 1D case, where it diverges logarithmically, the integral (5) converges. The reason is that, on average, the branches L1 , L2 depart from each other at large distances. With converging integral (5), there is no need to consider the higher order (in the interaction strength) contributions to the III. By virtue of analogy to the 1D case, the further calculation of the III correction to the transmission coefficient of the saddle point is straightforward. III. CALCULATION OF THE III CORRECTION Following Ref. [10], we use Eq. (5) to write down the correction to the wave function, describing the transmitted wave, i.e., the wave propagating along the equipotential L3 in Fig. 1 δΦL,ǫ (L3 ) = δΦ21 (L3 ) + δΦ34 (L3 ) + δΦ31 (L3 ) + δΦ24 (L3 ), (6) where each term in the r.h.s. corresponds to a certain closed-loop path for the secondary wave. In contrast to the 1D case, where only two paths contribute to the III, there are four possible paths in the quantum Hall geometry (see Fig. 1). The contribution from the path that includes two neighboring equipotentials Li , Lj has the form δΦij (L3 ) = −iπ Li dLi Φ∗,ǫ (Li ) L Lj dLj V (Li , Lj )ρ (Li , Lj ) 4 µ Φ∗ (Lj )Φµ,ǫ (L3 ) µ,ǫ (7) To proceed further, we have to specify the scattering properties of the saddle point (i) without the loss of generality, we assume that for the function ΦL,ǫ of the state, in which the wave, incident along L1 , is reflected into L2 and transmitted into L3 , the reflection coefficient [13] 1 r0 (ǫ) = π(ǫ − W ) Γ 1 + exp 1/2 , (8) 1/2 2 < and the transmission coefficient, t0 (ǫ) = [1 − r0 (ǫ)] , are both real. Here W ∼ Γ is the saddle point height. (ii) away from the saddle point the functional form of the incident wave is v(L1 )−1/2 exp [iǫψ(L1 )], where ψ(L) is related to the drift velocity v(L) along the equipotential as dψ(L)/dL = v −1 (L) and ψ|L = 0 = 0. Substituting this form into Eqs. (3), (7), and then Eq. (7) into Eq. (6), we obtain the following expression for the III correction to the transmission coefficient δt(ǫ) = −r0 (ǫ) [t0 (ǫ)r0 (ǫF ) (I21 + I34 ) − t0 (ǫF )r0 (ǫ) (I31 + I24 )] , (9) where the functions Iij (ǫ) are defined as Iij = 1 π ∞ ∞ 0 d Li 0 d Lj V (Li , Lj ) v(Li )v(Lj ) cos { (ǫ − ǫF ) [ψ(Li ) + ψ(Lj )]} . ψ(Li ) + ψ(Lj ) (10) Note that the last two terms in Eq. (9), originating from the contours C → B → M → C and A → D → M → A, that are specific for the quantum Hall geometry, enter with the sign opposite to that of the “conventional” contributions from A → B → M → A and C → D → M → C. This fact reflects the unitarity of the scattering matrix of the saddle point. In order to incorporate the correction Eq. (9) into the scattering matrix, we introduce the effective energy according to the rule T (ǫ) = |t0 (ǫ) + δt(ǫ)|2 = T0 (˜) = 1 + exp − ǫ π ǫ−W ˜ Γ −1 . (11) The meaning of the effective energy, ǫ(ǫ, W ) is the following. The distribution of the saddle-point ˜ heights is even, so that W = 0. The fact that without interactions delocalization occurs at ǫ = 0 can be formally expressed by the condition (ǫ − W ) = 0. Correspondingly, with interactions, the condition for delocalization takes the form ǫ(ǫ, W ) − W = ǫ(ǫ, W ) = 0. From Eq. (11) we obtain ˜ ˜ for the effective energy 1/2 cosh π (ǫ − W )  Γ ǫ= ǫ−  ˜ π cosh π (ǫF − W ) Γ Γ  (I21 + I34 ) exp π (ǫ − ǫF ) 2Γ − (I31 + I24 ) exp π (ǫF − ǫ) 2Γ . (12) Disorder averaging should be performed over the saddle points and over the equipotentials. For energies close to the Fermi level, |ǫ − ǫF | ≪ Γ we have I21 = I34 = I31 = I24 = I (ǫ − ǫF ), so that Eq. (12) takes the form ǫ(ǫ) = ǫ − (ǫ − ǫF ) I(ǫ − ǫF ) , ˜ (13) where I(ǫ − ǫF ) is the even function of (ǫ − ǫF ) defined as e2 I(ǫ − ǫF ) = κ ∞ 0 ∞ dψ1 0  cos { (ǫ − ǫF ) [ψ1 + ψ2 ]}  1 dψ2 − ψ1 + ψ2 D(ψ1 , ψ2 ) 5 1 D(ψ1 , ψ2 )2 + 4d2  . (14) Eq. (14) emerges upon substituting of the interaction potential (4) into Eq. (10) and introducing the function D(ψ1 , ψ2 ), which is the distance between the points, located at neighboring equipotentials, and corresponding to the accumulated phases ψ1 and ψ2 , respectively. It is easy to see that the integral Eq. (14) contains a contribution from short distances (small ψ), which changes slowly with (ǫ − ǫF ), (characteristic scale ∼ Γ) and a contribution from large distances which is a sharp function of energy. The energy scale, ǫ0 , of this contribution can be estimated from the condition ǫ0 ∼ ψ −1 , where ψ is the characteristic phase, for which the distance D is of the order of the distance to the gate, d. Taking into account that D scales with the length, L, along the equipotential as L4/7 , we can present the dependence D(ψ) as D ∼ Rc (vψ/Rc )4/7 , where v is the characteristic drift velocity. This velocity can be expressed through the parameters of the random potential, i.e., v ∼ U0 l2 /Rc , so that v/Rc ∼ Γ [see Eq. (1)]. Then the condition D(ǫ−1 ) = d yields 0 ǫ0 = Γ Rc d 7/4 ≪ Γ. (15) The characteristic magnitude of I follows from Eq. (14) taking into account that the contribution to the integral comes from ψ1 ∼ ψ2 ∼ ǫ−1 . As a result, Eq. (14) can be presented as 0 ǫ − ǫF I(ǫ − ǫF ) = α I 0 + F ǫ0 , e2 α= κdΓ d Rc 7/4 . (16) The contribution I 0 ∼ ln (Rc /l) comes from small distances, while the asymptotic behavior of the function F is the following F (u)|u≪1 = F 0 − F 1 |u|5/7 , F (u)|u≫1 = F∞ |u|−5/7 . (17) With the use of Eq. (16) the expression (13) for the average effective energy takes the form ǫ(ǫ) = ǫF + (ǫ − ǫF ) (1 − αI 0 ) − αF ˜ ǫ − ǫF ǫ0 . (18) We see that at small |ǫ − ǫF | ≪ ǫ0 the slope of ǫ(ǫ) can be either positive or negative depending ˜ on the sign of [1 − α (I 0 + F 0 )]. The consequences of this fact are discussed in the next Section. IV. IMPLICATIONS A. T → 0 As was discussed above, the position of a delocalized state can be found from the condition ǫ(ǫ) = 0, where ǫ is defined by Eq. (18). Interactions enter into this equation via the parameter ˜ ˜ α. It is easy to see that for α < (I 0 + F 0 )−1 , i.e., for weak enough interaction strength, the square bracket in the r.h.s of Eq. (18) is always positive. Then it can be easily demonstrated that at any ǫF the equation ǫ(ǫ) = 0 has only one solution, so that the interactions do not change the scenario ˜ of the quantum Hall transition qualitatively. The situation changes as the interaction parameter exceeds the critical value αc = (I 0 + F 0 )−1 . For small (α − αc ) ≪ αc the equation ǫ(ǫ) = 0 can be ˜ analyzed analytically, since the function F in the r.h.s of (18) can be replaced by its small-argument asymptotics Eq. (17). Then we have ǫ(ǫ) = ǫF + (ǫ − ǫF ) ˜ 1− α ǫ − ǫF + αc F 1 αc ǫ0 6 5/7 = 0. (19) T 1.0 ε ε 0 0.5 (a) 0 T 1.0 ε ε (1) F ε 0 ε 0.5 (b) ε ε (2) F T 1.0 ε 0 ε 0.5 (c) ε ε (3) 0 F FIG. 2. Energy dependence of the power transmission coefficient, T (ǫ), for the interaction strength exceeding (2) (1) the critical value is shown schematically for three positions of the Fermi level, ǫF . (a) ǫF ≫ ǫ0 ; (b) ǫF ∼ ǫ0 ; (c) (3) (3) ǫF < 0, |ǫF | ≫ ǫ0 . Insets show the corresponding energy dependencies of the effective energy, ǫ(ǫ), defined by Eq. ˜ (11). Upon introducing a dimensionless variable, Z= Eq. (19) takes the form 2 αc F 1 α − αc 7/5 7 (ǫ − ǫF ) , ǫ0 (20) 7/5 Z |Z| 5/7 −1 =− 19/5 αc F 1 (α − αc ) 12/5 ǫF ǫ0 . (21) The l.h.s of Eq. (21) is an odd function of Z and has extrema at Z = ± (7/12)7/5 . Therefore, for ǫF within the interval 5 7 ǫF < ǫ0 7 12 7/5 (α − αc )12/5 19/5 αc (22) 7/5 F1 Eq. (21) has three solutions. These solutions correspond either to two delocalized states below the Fermi level and one delocalized state above the Fermi level (for ǫF < 0) or to two delocalized states above the Fermi level and one delocalized state below the Fermi level (for ǫF > 0). This is quite an unusual situation, since we consider spinless electrons with a filling factor between ν = 0 and ν = 1. Conventially, as it is the case α < αc , in this situation there is only a single delocalized state. Fig. 2 helps to trace the evolution of the delocalized states with increasing magnetic field. Note first, that, as it follows from Eqs. (11) and (18), for α > αc the energy dependence of the transmission (1) coefficient, T (ǫ), has a region with a negative slope. At low magnetic fields, (ǫF ≫ ǫ0 , Fig. 2a) in this region we have T (ǫ) > 1/2. Thus, there is a single delocalized state below the Fermi level. This corresponds to the Hall conductivity σxy = 1, as in the absence of interactions. The effect of (3) (3) interactions is also negligible at strong magnetic fields (ǫF < 0, |ǫF | ≫ ǫ0 , Fig. 2c). In this case, the region with negative slope occurs at T (ǫ) < 1/2. A single delocalized state lies above the Fermi level, i.e., σxy = 0. A nontrivial evolution with magnetic field takes place at small ǫF , Fig. 2b. Namely, as Fig. 2a transforms to Fig. 2c with increasing magnetic field, first, in addition to a delocalized state below the Fermi level, two delocalized states emerge above the Fermi level. With further increasing magnetic field, the “upper” of two new delocalized states remains above the Fermi level, while the “lower” one moves below the Fermi level. At the critical field when the “lower” delocalized state crosses the Fermi level the diagonal conductivity, σxx , exhibits a sharp peak, accompanied by a jump of σxy from σxy = 1 to σxy = 2. Two delocalized states below the Fermi level persist within a certain range of magnetic fields, and then disappear abruptly, so that the “normal” arrangement, Fig. 2c, is reinstated. This abrupt disappearance of two delocalized states below the Fermi level manifests itself as a second jump of σxy from σxy = 2 to σxy = 0. Note, that σxx , which is determined by the states in the immediate vicinity of the Fermi level, remains zero as σxy experiences a second jump. The evolution of σxy and σxx with magnetic field (inverse filling factor) is illustrated in Fig. 3. σ 2 σxy 1 σxx ν -1 FIG. 3. The components of the conductivity tensor at low temperatures, T ≪ ǫ0 , are shown schematically vs. inverse filling factor for the interaction strength exceeding the critical value. Note, that the quantized values of σxy correspond to spinless electrons. 8 B. “High” T According to Ref. [6] the temperature dependence of the diagonal conductivity, σxx of a macroscopic sample can be expressed through the power transmission coefficient, G(ǫ, Lφ ), of the square of a size Lφ calculated for electron with energy ǫ as follows σxx = 1 2T dǫ cosh ǫ − ǫF 2T −2 G ǫ, Lφ (T ) , (23) where cosh−2 comes from the derivative of the Fermi function. The point of Ref. [6] is that the 1/νc function G(ǫ, Lφ ) is, in fact, some universal function, G0 (X), of the argument X = Lφ /ξ (ǫ) . The scaling function, G0 (X), satisfies two conditions (i) G0 (0) = 0.5, so that at T → 0 Eq. (23) yields σxx = 0.5 (in the units of e2 /h). (ii) G0 (X) ∼ exp (−c|X|νc ) for |X| ≫ 1, where c ∼ 1 is a numerical factor. The latter condition expresses the fact that at short enough ξ the transmission is determined by tunneling, i.e., | ln G(ǫ, Lφ )| ∼ Lφ /ξ(ǫ). It is convenient to parameterize the temperature dependence of Lφ by introducing the characteristic temperature, T0 , via the relation T T0 Lφ (T ) = Dc −p/2 . (24) Using Eq. (24), the condition of applicability of the network model, Lφ ≫ Dc , can be presented as T ≪ T0 . The energies of electrons, contributing to σxx , are ∼ T . Therefore, the condition that the network model is adequate within the energy strip, ǫ ∼ Γ, where the quantum interference is important, reads T0 ≫ Γ. In the numerical results reported below we measure energies and temperatures in the units of Γ. With localization length given by Eq. (2), the dimensionless argument X takes the form T0 X= Γ p/2νc T Γ −p/2νc ǫ . Γ (25) Consideration in Sect. III suggests that, in the presence of interactions, the energy ǫ in Eq. (25) should be replaced by the effective energy ǫ(ǫ) determined by Eq. (13). Below we study how this ˜ replacement affects the temperature and magnetic field dependences of σxx calculated from Eq. (23). For numerical calculations we have chosen the following form of the function G0 G0 = 1 exp 1 − 1 + c2/νc X 2 2 νc /2 . (26) We have also checked that different forms of G0 , that satisfy the conditions (i) and (ii), change the numerical results only weakly. Upon substituting ǫ into Eq. (25), then Eq. (25) into Eq. ˜ (26), and finally Eq. (26) into Eq. (23), we find that σxx contains only one unknown parameter, c2/νc (T0 /Γ)p/νc , which we have set equal to 2. In Fig. 4 the calculated dimensionless σxx as a function of dimensionless magnetic field ǫF /Γ is shown for several dimensionless temperatures T /Γ. In Fig. 4a the effect of III is neglected, i.e., ǫ = ǫ. In Fig. 4b σxx is calculated in the presence of III for ǫ(ǫ) dependence shown in the inset. ˜ ˜ We see that in the absence of III the broadening of the σxx peak with increasing T is accompanied by a rapid decrease of the maximal value, σxx (0) = σxx |ǫF =0 . In particular, at T /Γ = 0.05, σxx (0) decreases by 20 percent. At the same time, with III the decrease of σxx (0) at T /Γ = 0.05 is only 2 percent. The significant drop of σxx (0) by 20 percent occurs only at rather “high” temperature 9 T = 0.2Γ. Note, that ˜(ǫ) dependence in Fig. 4b corresponds to the interaction parameter I(0) = 0.8 ǫ well below the critical value. The drop in σxx (0) with T might seem counterintuitive. It might be argued that the decrease of Lφ seems to drive the transport towards fully incoherent regime [11,19], in which the σxx (0) = 0.5. However, as it was pointed out in Ref. [6], as a result of the broadening of the Fermi distribution with T , the transmission of the square Lφ is almost zero for most of electrons involved in transport. sxx (a) 0,4 0,2 0,0 sxx e ~ (b) 0,4 -0.5 e 0 -0.5 0,2 0,0 -0,4 -0,2 0,0 0,2 eF 0,4 G FIG. 4. Diagonal conductivity (in the units e2 /h), calculated from Eqs (23) and (26), is plotted vs. the dimensionless magnetic field ǫF /Γ for (a) ǫ = ǫ (noninteracting electrons) and (b) for ˜(ǫ) shown in the inset. The curves are ˜ ǫ calculated for T /Γ = 10−3 (lowest T ), T /Γ = 1.2 · 10−2 , 2.5 · 10−2 , 3.7 · 10−2 , 5 · 10−2 , respectively. In the quantum Hall geometry, experimentally measured characteristics are the components, ρxx and ρxy , of the resistivity tensor, rather than σxx , σxy . The question how the dependence of σxx on the dimensionless magnetic field in Fig. 4 translates into the behavior of ρxx is by no means trivial. The common way to relate σxx and ρxx is to employ the semicircle law [11], i.e., ρxx = 2 1/2 2σxx / 1 ± (1 − 4σxx ) . The underlying physics of the semicircle law is that the system is strongly inhomogeneous and represents a random mixture of the domains with quantized σxy and σxx = 0, while the macroscopic σxx the portion of the domains with σxy = 1 (left half of the semicircle) and σxy = 0 (right half of the semicircle). Therefore, in order for the semicircle relation to apply, we have to assume that σxx (ǫF ), calculated above, represents, in fact, the average of the following distribution f (σ) = σxx (ǫF )δ (σ) + (1 − σxx (ǫF )) δ(1 − σ). Obviously, Eq. (23) is valid for a homogeneous system. More specifically it implies that system parameters, e.g. the electron concentration, do not change within the spatial scale, Lφ . Formally this corresponds to the distribution f (0) (σ) = δ(σ − σxx ). Both distributions f and f (0) have the same average. Whether or not macroscopic inhomogeneities (with a scale much bigger than Lφ ) transform f (0) (σ) into f (σ) depends on their magnitude. This magnitude should be neither very low (otherwise, they will only slightly smear f (0) (σ)) nor very high (otherwise the distribution f will correspond to σxx = 0.5 with no sensitivity to ǫF ). Assuming that the conditions for the semicircle rule are met, we immediately conclude that nondiagonal resistivity, 10 ρxy , is quantized at ρxy = h/e2 for all σxx (ǫF ) curves shown in Fig. 4b. This is the so called quantum Hall insulator behavior [18]. On the other hand, in the absence of III, σxx = 0.5 only for the lowest temperature, as shown in Fig. 4a. Hence, no quantization of ρxy . In this sense, III “reinforce” the metallic regime, and thus extend the quantum Hall insulator behavior to higher temperatures. Indeed, the dependence ρxx on the dimensionless magnetic field, calculated from Fig. 4b using the semicircle relation, and shown in Fig. 5 is close to the straight line in the logarithmic scale. The slope at high enough temperatures changes linearly with T and extrapolates to a nonzero value. However, we would like to point out that the temperature interval, in which this behavior takes place (10−2Γ < T < 5 · 10−2Γ) is much narrower than in the experiment [17], where this behavior persisted as the temperature was raised by a factor of 30. ln rxx 0,5 (a) 0,0 -0,5 -0,05 dr 0,00 0,05 eF (b) G 0,10 0,05 0,00 0,00 0,02 0,04 TG FIG. 5. (a) Diagonal resistivity is plotted in logarithmic scale as a function of dimensionless magnetic field, ǫF /Γ. The curves are calculated for T /Γ = 10−3 , 0.012, 0.025, 0.037, and 0.05. (b) The derivative δρ = d ln ρxx /d(ǫF /Γ) at point ǫF = 0 vs. dimensionless temperature, T /Γ. V. Concluding Remarks The most sound result of the present paper is the magnetic field dependence of σxy and σxx at low temperatures and strong enough interactions, Fig. 3. In particular, the rise of σxy from σxy = 1 to σxy = 2 (for spinless electrons) with increasing magnetic field seems counterintuitive. Indeed, σxy is the measure of the number of delocalized states below the Fermi level. Increasing magnetic field is supposed to facilitate localization. At the same time, the behavior of σxy shown in Fig. 3 reflects the fact that the III allows the emergence of additional delocalized states below the Fermi level as magnetic field increases. The reason for this counterintuitive behavior is the following. Exchange interactions modify, via III, the strength of the electron scattering from the disorder potential. This modification is sensitive to the position of ǫF . As the Fermi level moves with magnetic field, the scattering of an electron with a given energy, ǫ, below ǫF changes, so that, at certain position of 11 ǫF , the delocalization condition, i.e., equal overall deflection to the left and to the right, is met for the energy ǫ. We are not aware of any experiment in which the behavior depicted in Fig. 3 was observed directly. Recent observation [22] of a maximum in σxy at low magnetic fields can be loosely interpreted as interaction-induced creation of the additional delocalized states below ǫF . Experimentally observed anomalous behavior of the components of the resistivity tensor in strong magnetic fields (quantum Hall insulator [18]) has spurred theoretical attempts to modify the standard microscopic description [1] of the quantum Hall effect. However, in previous considerations [19–21] additional delocalized states never emerged abruptly below ǫF . This is because these considerations assumed electrons to be either noninteractingzulicke or incoherent [19,20]. As we have demonstrated, there is a fundamental difference between the energy dependence of the III correction in one dimension [10] and in the quantum Hall geometry. In 1D, this correction is even in (ǫ − ǫF ), while in quantum Hall geometry it is an odd function of (ǫ − ǫF ). The underlying reason is the electron-hole symmetry, which we discuss below in more detail. (e) (i) The Hall conductivity can be calculated using the language of either electrons, σxy = σxy , or (h) holes, σxy = σxy . The filling factor for holes is 1 − ν, where ν is the filling factor for electrons. Even (e) (h) in the presence of interactions the condition σxy (ν) = σxy (1 − ν) should be obeyed. The fact that this condition is indeed obeyed within our consideration can be seen from Eqs. (20), (21). Changing ν by (1 − ν) corresponds in to the replacement of ǫF by −ǫF . Upon this replacement we get from Eq. (21) Z → −Z. Then from Eq. (20) we get for the position of the delocalized state ǫ → −ǫ. Thus, counting the delocalized states with the Fermi level −ǫF , using the language of holes, yields the same number as for the Fermi level ǫF , using the language of electrons. (ii) We turn to the different aspect of the e-h symmetry. It is generally assumed that there is a relation between the Hall conductivities of the 2D electron system in the lowest Landau level at filling factors ν and (1 − ν) e2 σxy (ν) + σxy (1 − ν) = . (27) h This equation follows from the electron-hole symmetry and from the fact that the Hall conductivity of completely occupied Landau level is equal to e2 /h. In other words, Eq. (27) implies that there is only one delocalized state at any position of the Fermi level. It is easy to see that this relation is violated in our case. Indeed, at a magnetic field, at which there are two delocalized state below the Fermi level and one delocalized state above the Fermi level, the Hall conductivity of electrons is σxy (ν) = 2e2 /h, while the Hall conductivity of holes is σxy (1−ν) = e2 /h. Therefore, the sum of these two values yields 3e2 /h, in contradiction to the r.h.s. of Eq. (27). Such inconsistency does not mean that our results violate the electron-hole symmetry. The reason for this is that r.h.s. of Eq. (27) should be calculated more carefully. Without interactions it is simply the conductivity of completely occupied Landau level, where the single-particle states, which are occupied by electrons, do not depend on the filling factor ν that enters Eq. (27). In the presence of interactions the situation is completely different . Again, the r.h.s. of Eq.(27) is the conductivity of completely occupied Landau level, but the single-particle states, which should be occupied in this case, strongly depend on the filling factor, ν. Strictly speaking, we should take the single-particle states of the electron system at filling factor ν and then, considering them fixed, add the electrons to all unoccupied states. Since in the example considered above, the “completely” occupied Landau band contains three delocalized states, such a counting yields the Hall conductivity 3e2 /h. This suggests that Eq. (27) should be modified in the following way σxy (ν) + σxy (1 − ν) = Nν 12 e2 , h (28) where Nν is the number of delocalized states in the lowest Landau level, when the electron filling factor is equal to ν. Note finally, that the anomalous dependence T (ǫ) in Fig. 2 emerges for the dimensionless interaction strength, α, defined by Eq. (16), exceeding the critical value. This result was obtained in the first order in α. If one attempts to incorporate higher order terms in α in the spirit of Ref. [10], then the second term in the r.h.s. of Eq. (13) would become singular in (ǫ − ǫF ). As a result the anomalous behavior of T (ǫ) would persist for arbitrary small α. We acknowledge the support of the NSF under grant No. INT-0231010. Interesting discussion with L.I. Glazman is gratefully acknowledged. 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