We report on an analytical study of the statistics of conductance, $g$, and
shot-noise power, $p$, for a chaotic cavity with arbitrary numbers $N_{1,2}$ of
channels in two leads and symmetry parameter $\beta = 1,2,4$. With the theory
of Selberg's integral the first four cumulants of $g$ and first two cumulants
of $p$ are calculated explicitly. We give analytical expressions for the
conductance and shot-noise distributions and determine their exact asymptotics
near the edges up to linear order in distances from the edges. For $0<g<1$ a
power law for the conductance distribution is exact. All results are also
consistent with numerical simulations.Comment: 7 pages, 3 figures. Proc. of the 3rd Workshop on Quantum Chaos and
  Localisation Phenomena, Warsaw, Poland, May 25-27, 200

Savin, D. V.

Sommers, H. -J.

Wieczorek, W.

Acta Physica Polonica A

English

arXiv

We report on an analytical study of the statistics of conductance, g, and shot-noise power, p, for a chaotic cavity with arbitrary numbers N$\text{}_{1,2}$ of channels in two leads and symmetry parameterβ = 1, 2, 4. With the theory of Selberg's integral the first four cumulants of g and first two cumulants of p are calculated explicitly. We give analytical expressions for the conductance and shot-noise distributions and determine their exact asymptotics near the edges up to linear order in distances from the edges. For 0 < g < 1 a power law for the conductance distribution is exact. All results are also consistent with numerical simulations

Sommers, H.-J.

Biblioteka Nauki - repozytorium artykuÅÃ³w

Statistics of Conductance and Shot-Noise Power for Chaotic Cavities

H.-J Sommers

W. Wieczorek

D.V. Savin

Crossref

We report on an analytical study of the statistics of conductance, g, and shot-noise power, p, for a chaotic cavity with arbitrary numbers N₁,₂ of channels in two leads and symmetry parameter β = 1,2,4. With the theory of Selberg's integral the first four cumulants of g and first two cumulants of p are calculated explicitly. We give analytical expressions for the conductance and shot-noise distributions and determine their exact asymptotics near the edges up to linear order in distances from the edges. For 0<g<1 a power law for the conductance distribution is exact. All results are also consistent with numerical simulations

Sommers, H

Wieczorek, W

Savin, DV

Brunel University Research Archive

Vol. 112 (2007) ACTA PHYSICA POLONICA A No. 4Proceedings of the 3rd Workshop on Quantum Chaos and Localisation PhenomenaWarsaw, Poland, May 25–27, 2007Statistics of Conductance and Shot-NoisePower for Chaotic CavitiesH.-J. Sommersa, W. Wieczoreka and D.V. SavinbaFachbereich Physik, Universita¨t Duisburg-Essen, 47048 Duisburg, GermanybDepartment of Mathematical Sciences, Brunel UniversityUxbridge, UB8 3PH, UKWe report on an analytical study of the statistics of conductance, g,and shot-noise power, p, for a chaotic cavity with arbitrary numbers N1,2 ofchannels in two leads and symmetry parameter β = 1, 2, 4. With the theoryof Selberg’s integral the first four cumulants of g and first two cumulants of pare calculated explicitly. We give analytical expressions for the conductanceand shot-noise distributions and determine their exact asymptotics near theedges up to linear order in distances from the edges. For 0 < g < 1 a powerlaw for the conductance distribution is exact. All results are also consistentwith numerical simulations.PACS numbers: 73.23.–b, 73.50.Td, 05.45.Mt, 73.63.Kv1. IntroductionOur goal is to discuss the statistics of conductance and shot- noise power forchaotic cavities using essentially the properties of Selberg’s integral. The staticconductance G relates linearly the time averaged current I(t0) = G · V to theexternal voltage V between two electron reservoirs. Fluctuations of the currentaround its mean value are conventionally described by the spectral noise powerP = 2∫∞0dtδI(t+ t0)δI(t0), with δI = I − I. As temperature goes to zero, theonly source of noise that remains non-vanishing is related to the discreteness ofthe electric charge carriers, the so-called shot-noise. For mesoscopic conductors,an adequate framework for the problem is based on random-matrix theory (RMT)approach to quantum transport, we refer to [1, 2] for reviews.We consider a chaotic cavity with two attached leads supporting N1 and N2channels, respectively, and coupled perfectly to the interior of the cavity. Accord-ing to the Landauer–Bu¨ttiker formalism, the conductance G and the shot-noisepower P can be expressed in terms of the transmission eigenvalues Ti as follows:G = G0n∑i=1Ti ≡ G0 · g (1)(691)692 H.-J. Sommers, W. Wieczorek, D.V. SavinandP = P0n∑i=1Ti(1− Ti) ≡ P0 · p, (2)where G0 = 2e2/h is the conductance quantum and P0 = 2eV G0. The positivenumbers Ti ≤ 1 are the n ≡ min(N1, N2) non-zero eigenvalues of the matrix tt†,with the transmission matrix t being the N1×N2 submatrix of the full scatteringmatrix and consisting of transition amplitudes from N1 left to N2 right channels.For chaotic cavities, universal fluctuations of Ti can be described by RMT [1].Considering the moments of g and p, the exact (RMT) results valid at ar-bitrary channel numbers N1,2 and repulsion parameter β were reported in theliterature only for the average and variance of the conductance [3, 4, 1]:〈g〉 = N1N2N − 1 + 2β, N ≡ N1 +N2, (3)var(g) =2βN1N2(N1 − 1 + 2β )(N2 − 1 + 2β )(N − 2 + 2β )(N − 1 + 2β )2(N − 1 + 4β ), (4)and very recently for the average shot-noise power [5]:〈p〉 =N1N2(N1 − 1 + 2β )(N2 − 1 + 2β )(N − 2 + 2β )(N − 1 + 2β )(N − 1 + 4β )= N1N2β2var(g)〈g〉 . (5)To derive these results in a uniform way, it is convenient to use the known expres-sion for the joint probability density of transmission eigenvalues Ti [5, 1]:P(T1, T2, . . . , Tn) = N−1βn∏i=1Tα−1i∏j<k|Tj − Tk|β ,α ≡ β2(|N1 −N2|+ 1), (6)to perform the corresponding integrations on Eqs. (3)–(5). The normalization con-stant Nβ above is given byNβ =n−1∏j=1Γ (1 + β2 (1 + j))Γ (α+β2 j)Γ (1 +β2 j)Γ (1 + β2 )Γ (1 + α+β2 (n+ j − 1))(7)and assures that (6) is a probability density. It is known for discrete positive nand continuous α and β as Selberg’s integral [6].As to the distribution functions, simple closed expressions can be obtained forthe conductance distribution at n = 1, 2 [3, 4, 7] and for the shot-noise distributionat N1 = N2 = 1 only [8]. To the best of our knowledge, no general results valid atarbitrary N1,2 and β have been presented thus far.2. CumulantsTo study the cumulants of g and p, one needs to know what are the moments〈Tn11 · · ·Tnkk 〉, with 〈. . .〉 standing for the integration over the joint probabilityStatistics of Conductance and Shot-Noise Power . . . 693density (6) and ni ≥ 0. Moments with all ni = 1 as well as 〈T 21 〉 can be foundfrom recursion relations already given in Mehta’s book [6] that come from theSelberg integral theory. The latter can be successfully applied [5] to derive results(1), (2). This approach was recently extended further to find 〈T 31 〉 and 〈T1(1−T1)× T2(1 − T2)〉 exactly [9]. Here, we have calculated all moments with∑i ni ≤ 4by using tricks of partial integrations to reduce all moments to forms of Selberg’sintegral. We will report on that in more detail elsewhere [10]. By this method weare able to calculate the so-called skewness (third cumulant) of the conductance,which we represent in the following compact form:〈〈g3〉〉 ≡ 〈(g − 〈g〉)3〉= var(g)4[(1− 2β )2 − (N1 −N2)2]β(N − 3 + 2β )(N − 1 + 2β )(N − 1 + 6β ). (8)It is worth noting that the skewness vanishes for symmetric cavities (N1 = N2)at β = 2. This also holds for any odd cumulant of the conductance, as theconductance distribution becomes symmetric around n/2 in this case∗.We have also calculated the kurtosis 〈〈g4〉〉 of the conductance (which is thefourth cumulant of g) and the variance 〈〈p2〉〉 = var(p) of the shot-noise power(the second cumulant). These expressions are given explicitly but are too lengthyto be reported here. In the case of the single-mode leads, N1 = N2 = 1, onegets 〈〈g4〉〉 = − 324725 ,− 1120 ,− 1540 and var(p) = 4525 , 1180 , 1180 at the values of β = 1,2, 4, respectively. In the opposite semiclassical limit of large channel numbers,N1,2 À 1, we write the results as the following 1/N expansions:〈〈g4〉〉var(g)=24β2N6[(N1 −N2)2(N21 +N22 − 4N1N2) +β − 2βN(12(N41 +N42 )−64N1N2(N21 +N22 ) + 105N21N22)]+O(1/N4), (9)var(p)〈p〉 =2βN5[N41 +N42 − 4N1N2(N1 −N2)2 +β − 2βN(9(N41 +N42 )−42N1N2(N21 +N22 ) + 70N21N21)]+O(1/N3). (10)One can readily see that higher cumulants contribute at least in the next orderof 1/N , so that the full distributions get more Gaussian-like as N grows. Thistendency becomes even stronger for the conductance distribution in symmetriccavities at β = 2, as 〈〈g4〉〉 vanishes then in the leading and next-to-leading orders.In this case of symmetric cavities, N1,2 = nÀ 1, we can further find〈〈g4〉〉 = 3128β3n3[1− 2β+(β + 2)22β2n]+O(1n5), (11)∗This can be seen from the symmetry of the integral kernel (6) at α = 1 by the changeof all Ti → 1− Ti in Pg(g) = 〈δ(g −∑i Ti)〉.694 H.-J. Sommers, W. Wieczorek, D.V. Savinvar(p) =164β[1 +β − 2βn+4 + β(β − 2)2β2n2]+O(1n3). (12)3. DistributionsFinally, we discuss shortly the distribution of the conductance, Pg(g) =〈δ(g −∑i Ti)〉 and that of the shot-noise power, Pp(p) = 〈δ(p −∑i Ti(1 − Ti)〉,deferring a detailed consideration to a separate publication [11]. Writing the func-tions Pg(g) and Pp(p) as the Fourier series over the interval of support, we obtain,e.g., for the conductance distribution the following representation:P (β)g (g) = n!∞∑m=12nsin(mpign)A(β)(m), (13)where A(β)(m) is known at β = 1, 2, 4. For example,A(2)(m) = C2ImdetB(2)kl (m) (14)is given by the imaginary part of the determinant of the following matrix:B(2)kl (m) =∫ 10dT Tα+k+l−3eimpiT/n, for k, l = 1, 2, . . . , n. (15)In a similar way we may write P (1)g (g) and P(4)g (g) as imaginary parts of certainPfaffians. The sum (13) can be done numerically where care has to be taken forthe Pfaffians, which may equivalently be written as quaternion determinants ofcertain self-dual antisymmetric matrices, which are easier to be calculated.Analogous expressions are also obtained for P (β)p (p).Fig. 1. The conductance distribution at β = 1, N1 = 4 and N2 = 1, 2, 3, 4, 5, 7, 9(from left to right).It is worth noting that at any finite n these distributions are continuous butnot analytic functions everywhere. Nonanalyticity in the distribution of conduc-tances in quasi-one-dimensional wires (the cusp point at g = 1) has been recentlyreported in [7, 12], see also [3]. In the present context of chaotic cavities, it canStatistics of Conductance and Shot-Noise Power . . . 695be understood from the following geometrical consideration. In the case of Pg(g),calculating the average amounts to an integration over an (n − 1)-dimensionalplane g =∑i Ti cut by the n-dimensional cube 0 ≤ Ti ≤ 1, such that thereappear (sometimes weak) singularities at all points g = M with an integer M ,0 ≤M ≤ n. A similar situation appears also for Pp(p). Here, we integrate over the(n − 1)-dimensional sphere n4− p = ∑i(Ti − 12 )2 and singularities appear when-ever the sphere touches one edge or surface of the cube, that is for p = M4 with0 ≤M ≤ n.Figure 1 illustrates the conductance distribution at β = 1 keeping N1 = 4fixed and varying N2 from 1 to 9. In Fig. 2 we plot the distribution of the shot--noise power at β = 2, keeping N1 = 4 fixed and varying N2 from 2 to 14. In bothcases one can see the tendency of the distributions to get peaked around the meanvalues, so that in the bulk they can be effectively described by a Gaussian withthe known mean and variance, see Eqs. (3–5).Fig. 2. Distributions of the shot-noise power at β = 2, N1 = 4 and N2 = 2, 3, 4, 6, 8(left plot) and at β = 2, N1 = 4 and N2 = 8, 10, 12, 14 (right plot).4. AsymptoticsWe are able to give some exact asymptotics near the edges of support ofthe distributions expressed in terms of exact integrals, which are related to somegeneral forms of Selberg’s integral. For example, for 0 < g < 1 we findP (β)g (g) = const× gαn+β2 n(n−1)−1, (16)where the proportionality factor is exactly known. This result follows easily fromcalculating the powers of g in 〈δ(g −∑i Ti)〉 under scaling of all Ti → gT˜i andnoticing that the upper integration limit of T˜i remains unchanged (= 1) as longas g < 1. Similarly, P (β)g (g) behaves near the upper edge as follows:P (β)g (g) ∝ (n− g)(n−1)(1+β2 n), n− 1 < g < n. (17)For the distribution of the shot-noise power, one finds696 H.-J. Sommers, W. Wieczorek, D.V. SavinP (β)p (p) ∝(n4− p)n2 [1+β2 (n−1)]−1,n− 14< p <n4, (18)whereas near the lower edge (0 < p < 14 ) the asymptotics is a bit more com-plicated, since the contribution of the corners of the cube of integration becomedisconnected. These expressions can be extended outside the regions specifiedabove, being valid then in a linear order in distances from the edges. The expan-sions can even be improved and all constants can be given explicitly as expressionscontaining products of gamma functions from Selberg’s integral.Fig. 3. The distribution of the shot-noise power (dashed line) and correspondingasymptotics (full lines) at β = 4 and N1 = 4, N2 = 2.In Fig. 3 we show as an example the distribution of the shot-noise power atβ = 4, N1 = 4, N2 = 2 (dashed line) with calculated asymptotics near the edges(full lines). Indicated as dots are the average 〈p〉 and 〈p〉±√var(p) known exactly.We have also compared (and found a perfect agreement) all the expressions withnumerical simulations of the RMT statistics of Ti and resulting distributions for gand p.5. ConclusionsIn summary, we have applied essentially the theory of Selberg’s integral to theproblems of quantum transport in chaotic cavities. The cumulants up to the fourthorder for the conductance and up to the second order for the shot-noise power havebeen calculated exactly at arbitrary channel numbers and repulsion parameter β.We have also given the conductance and shot-noise distributions in closed formsuitable for subsequent analytic analysis (e.g., finding asymptotics) as well as fornumerical implementations. It would be desirable to compare our findings withrelevant experimental results, e.g., in microwave cavities which became availablerecently [13]. This could, however, be not so straightforward, as it requires takinginto account effects of dephasing [14] and absorption [15] as well. Further work inthis direction is needed.Statistics of Conductance and Shot-Noise Power . . . 697AcknowledgmentsSupport by SFB/TR12 of the DFG is acknowledged with thanks.References[1] C.W.J. Beenakker, Rev. Mod. Phys. 69, 731 (1997).[2] Ya.M. Blanter, M. Bu¨ttiker, Phys. Rep. 336, 1 (2000).[3] H.U. Baranger, P.A. Mello, Phys. Rev. Lett. 73, 142 (1994).[4] R.A. Jalabert, J.-L. Pichard, C.W.J. Beenakker, Europhys. Lett. 27, 255 (1994).[5] D.V. Savin, H.-J. Sommers, Phys. Rev. B 73, 081307 (2006).[6] M.L. Mehta, Random Matrices, 2nd ed., Academic Press, New York 1991.[7] A. Garc´ıa-Mart´ın, J.J. Sa´enz, Phys. Rev. Lett. 87, 116603 (2001).[8] M.H. Pedersen, S.A. van Langen, M. Bu¨ttiker, Phys. Rev. B 57, 1838 (1998).[9] M. Novaes, Phys. Rev. B 75, 073304 (2007).[10] D.V. Savin, H.-J. Sommers, W. Wieczorek, in preparation.[11] W. Wieczorek, D.V. Savin, H.-J. Sommers, in preparation.[12] K.A. Muttalib, P. Wo¨lfle, A. Garc´ıa-Mart´ın, V.A. Gopar, Europhys. Lett. 61, 95(2003).[13] S. Hemmady, J. Hart, X. Zheng, T.M. Antonsen, E. Ott, S.M. Anlage, Phys. Rev.B 74, 195326 (2006).[14] P.W. Brouwer, C.W.J. Beenakker, Phys. Rev. B 55, 4695 (1997).[15] Y.V. Fyodorov, D.V. Savin, H.-J. Sommers, J. Phys. A 38, 10731 (2005).

Statistics of conductance and shot-noise power for chaotic cavities

Statistics of conductance and shot-noise power for chaotic cavities

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Statistics of conductance and shot-noise power for chaotic cavities

Abstract

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Biblioteka Nauki - repozytorium artykuÅÃ³w

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Biblioteka Nauki - repozytorium artykuÅÃ³w