Temperature induced phase averaging in one-dimensional mesoscopic systems Severin G. Jakobs,1 Volker Meden,2 Herbert Schoeller,1 and Tilman Enss3 1 Institut f¨r Theoretische Physik A, RWTH Aachen, D-52056 Aachen, Germany u Institut f¨r Theoretische Physik, Universit¨t G¨ttingen, D-37077 G¨ttingen, Germany u a o o 3 Dipartimento di Fisica, Universit` di Roma “La Sapienza”, Piazzale Aldo Moro 2, I-00185 Roma, Italy a (Dated: April 22, 2018) arXiv:cond-mat/0606486v4 [cond-mat.mes-hall] 21 Feb 2007 2 We analyse phase averaging in one-dimensional interacting mesoscopic systems with several barriers and show that for incommensurate positions an independent average over several phases can be induced by finite temperature. For three strong barriers with conductances Gi and mutual dis√ tances larger than the thermal length, we obtain G ∼ G1 G2 G3 for the total conductance G. For an interacting wire, this implies power laws in G(T ) with novel exponents, which we propose as an experimental fingerprint to distinguish temperature induced phase averaging from dephasing. PACS numbers: 71.10.Pm, 72.10.Fk, 73.63.Nm I. INTRODUCTION Mesoscopic systems are characterized by spatial dimensions smaller than the phase breaking length Lϕ so that the phase of an electron is not destroyed by inelastic processes. Along their optical paths, electrons pick up phases from propagation and scattering. Even for negligible dephasing, certain circumstances can lead to an averaging of these phases, a phenomenon which has been analyzed in connection with localization, see e.g. Refs. 1,2. It remains however a fundamental task to compare in detail the effects of phase averaging and dephasing. For a noninteracting model system it was recently shown that a single one-channel dephasing probe gives the same full counting statistics as a single phase-averaging probe.3 However, it was noted that this result does no longer hold for several probes, i.e. phase averaging over many independent phases seems to be fundamentally different from dephasing. In the present paper we address this issue in detail. We discuss a generic one-dimensional interacting system coupled to two reservoirs with few barriers at arbitrary but fixed positions. We show that for incommensurate barrier positions an independent average over several phases can be induced by finite temperature. For more than two barriers we find that phase averaging drastically differs from dephasing and show that the commonly made assumption that barriers with mutual distances larger than the thermal length LT = vF /T can be treated independently4,5,6 (note that in experiment contacts to leads constitute additional barriers) does not hold in the absence of dephasing. This means that even in the high-temperature regime interference effects are still important, as also shown recently for quantum dots in the sequential tunneling regime.7,8 For three strong barriers with individual conductances Gi , we ob√ tain G ∼ G1 G2 G3 for the total conductance, in drastic contrast to the addition of resistances which follows from dephasing. For an interacting quantum wire (QW) this implies power-law scaling of G(T ) with novel exponents which can be used as an experimental fingerprint for phase averaging. In section II we introduce a noninteracting model for a quantum wire with barriers which allows a simple discussion of temperature induced phase averaging. In section III we study the influence of temperature induced phase averaging on the scaling beghaviour of interacting quantum wires and propose an experimental setup to distinguish phase averaging from dephasing. While sections II and III focus primarily on a wire with three barriers, section IV treats phase averaging for a wire with four barriers. II. NONINTERACTING WIRE For a discussion of the basic physical idea consider first the noninteracting case. We model the QW coupled to leads by an infinite tight-binding chain for spinless electrons at half-filling (µ = 0) p H=− c† cn n+1 n∈Z,n=ni − τi c† i +1 cni + h.c. n (1) i=1 with hopping matrix elements equal to 1 (defining the energy unit) except for n = ni , i = 1, . . . , p, where the barriers are situated. Without barriers the dispersion relation is ǫ = −2 cos (ka) with band width D = 4, where k is the momentum and a the lattice spacing. The distance between subsequent barriers is Li = aNi with Ni = ni+1 − ni . We characterize the barriers by their ± ± transmission ti (ǫ) and reflection ri (ǫ) = |ri (ǫ)|eiδi (ǫ) amplitudes for right- and left-running scattering waves. The linear conductance in units of e2 /h follows from9 G(T ) = dǫ − ∂f ∂ǫ 2 |t(ǫ)| . (2) where f = 1/(eǫ/T + 1), and t(ǫ) is the transmission amplitude. Thus, the effect of temperature is an aver2 age of the transmission probability |t| over an energy range ∆ǫ ∼ T . This is fundamentally different from dephasing which destroys the phase information for each 2 p−1 |t|2 = i=1 2π 1 2π 0 dϕi |t|2 (3) provided that vF /Li ≪ T ≪ D. Note that this result does not hold for commensurate lengths Li (where the path in phase space closes very quickly; see below) and becomes less relevant for a large number p of barriers (since Li /LT has to be chosen too large in order to cover a representative part of the high dimensional phase space). The phase average can be calculated easily for a small number of barriers. For two barriers we use t = t1 t2 /(1 − eiϕ1 |r1 r2 |) and for three barriers (forming two dots which we refer to as left and right) t = t1 t2 t3 1 − eiϕL |r1 r2 | 1 − eiϕR |r2 r3 | +ei(ϕL +ϕR ) |r1 r3 | |t2 | 2 −1 , (4) with ϕL,R ≡ ϕ1,2 . By accident, phase averaging yields the addition of resistances for two barriers. In contrast, for three barriers we obtain |t|2 = T1 T2 T3 ( i