Systematic approach to statistics of conductance and shot-noise in chaotic cavities
Applying random matrix theory to quantum transport in chaotic cavities, we develop a novel approach to computation of the moments of the conductance and shot-noise (including their joint moments) of arbitrary order and at any number of open channels. The method is based on the Selberg integral theory combined with the theory of symmetric functions and is applicable equally well for systems with and without time-reversal symmetry. We also compute higher-order cumulants and perform their detailed analysis. In particular, we establish an explicit form of the leading asymptotic of the cumulants in the limit of the large channel numbers. We derive further a general Pfaffian representation for the corresponding distribution functions. The Edgeworth expansion based on the first four cumulants is found to reproduce fairly accurately the distribution functions in the bulk even for a small number of channels. As the latter increases, the distributions become Gaussian-like in the bulk but are always characterized by a power-law dependence near their edges of support. Such asymptotics are determined exactly up to linear order in distances from the edges, including the corresponding constants.
pacs:73.23.-b, 73.50.Td, 05.45.Mt, 73.63.Kv
Quantum transport in mesoscopic systems has been a subject of an intense study during the last decade, see reviews Beenakker (1997); Blanter and Büttiker (2000). Traditionally, the focus of interest has been on statistical properties of the conductance, , and shot-noise, . For noninteracting electrons, the Landauer-Büttiker scattering formalism Landauer () relates these quantities (expressed in their natural units) to the so-called transmission eigenvalues of the conductor:
Here, , where are the number of propagating modes (channels) in the two attached leads. The are the eigenvalues of the matrix , with being matrix of transmission amplitudes from the entrance to exit channels. They are mutually correlated random numbers, , whose distribution depends on the type of the conductor.
Below, we consider chaotic cavities (open quantum dots). In this case, random matrix theory (RMT) has proved to be successful in describing universal fluctuations in transport through such systems. Beenakker (1997); Alhassid (2000) Within this RMT approach, the joint probability density function (JPDF), , of the transmission eigenvalues is induced by the random scattering matrix drawn from one of Dyson’s circular ensembles Mehta (1991), according to the global symmetries present in the system. The exact expression for this JPDF is known Beenakker (1997); Baranger and Mello (1994); Jalabert et al. (1994) to have the following simple form:
where and denotes the Vandermonde determinant. The Dyson’s symmetry index depends on the presence () or absence () of time-reversal symmetry or that of spin-flip symmetry () in the system, thus distinguishing between the three canonical RMT ensembles (orthogonal, unitary or symplectic, respectively). Mehta (1991) The normalization constant is given by
Presently, there is a substantial progress in the understanding of statistics of various transport observables in chaotic cavities that is due to the recent developments of new analytical methods in the theory. Among them, the Selberg integral theory plays a special and important role, see the recent review Forrester and Warnaar (2008) for its current status. In the present context, it has been initially applied by two of us Savin and Sommers (2006) to find the average value of shot-noise and hence the Fano factor exactly. This approach has then been developed further to study full counting statistics of charge transfer Novaes (2007) as well as to obtain exact explicit expressions for the shot-noise variance and for the skewness and kurtosis of the charge and conductance distributions Sommers et al. (2007); Savin et al. (2008) (see also Refs. Brouwer and Beenakker (1996); Araújo and Macêdo (1998); Blanter et al. (2001); Bulashenko (2005); Bulgakov et al. (2006); Gopar et al. (2006); Béri and Cserti (2007); Vivo and Vivo (2008) for other RMT results on the relevant statistics in chaotic cavities). Since enters the Selberg integral as a continuous parameter, this method allows us to treat all the three ensembles on equal footing, thus giving a powerful alternative to diagrammatic Brouwer and Beenakker (1996); Bulgakov et al. (2006) or orthogonal polynomial Araújo and Macêdo (1998); Vivo and Vivo (2008) approaches, especially when the channel numbers are small.
A completely alternative treatment has been recently undertaken within the semiclassical approach Richter and Sieber (2002); Braun et al. (2006) which represents quantum transport in terms of classical trajectories connecting the leads. By constructing asymptotic semiclassical expansions for transport observables, this approach successfully accounts for both system-specific and universal (RMT) features, see Ref. Müller et al. (2007) for a review.
The case of (broken time-reversal symmetry) is known for several reasons to be the special one in RMT. For the problem in question, further progress in this case has been made very recently along the following two directions. Novaes Novaes (2008) combined the Selberg integral with facts from the theory of symmetric functions to compute non-perturbatively moments of both the transmitted charge and conductance but not those of shot-noise. Alternatively, Osipov and Kanzieper Osipov and Kanzieper (2008) combined the theory of integrable systems with RMT, as given by (2), bringing out an effective formalism for calculating the cumulants of the conductance and shot-noise. Osipov and Kanzieper (2009) However, the relevant consideration for the systems with preserved time-reversal symmetry, , is still lacking.
The distribution functions of the conductance and shot-noise are also studied quite intensively on their own. However, no explicit expressions have been reported so far except for a few cases, namely, for the conductance distribution at Baranger and Mello (1994); Jalabert et al. (1994); García-Martín and Sáenz (2001); Bulgakov et al. (2006) and for the shot-noise distribution at . Pedersen et al. (1998) Asymptotic analysis of the both distributions at has been performed very recently in Ref. Vivo et al. (2008), see also Ref. Osipov and Kanzieper (2008). To the best of our knowledge, no general results valid at arbitrary and are available thus far. Meanwhile, the conductance distribution with dephasing Brouwer and Beenakker (1997) has been directly measured in open quantum dots Huibers et al. (1998) and in microwave billiards.Hemmady et al. (2006) The shot-noise power in chaotic cavities has been recently studied experimentally. Oberholzer et al. (2001) Counting electrons in quantum dots is also experimentally accessible. Gustavsson et al. (2009) All this provides an additional motivation for the present study.
In this work we explore further the direction along the lines of Novaes’s work Novaes (2008) and develop a systematic approach for computing the moments of linear statistics in transmission eigenvalues for the systems with both preserved and broken time-reversal symmetry. This approach yields the moments of the conductance and shot-noise of arbitrary order, including their joint moments and cumulants. In the next section we present the detailed exposition of the method used, including the relevant facts from the theory of symmetric functions. This method is then applied in Sec. III to derive expressions for the moments and cumulants of the conductance and shot-noise in a closed form. Sections IV and V complement this study by investigating the corresponding distribution functions and their asymptotic behaviour. Our main findings are summarized and discussed in the concluding section VI.
Ii The method
The method is based on expanding powers of the conductance or shot-noise (or any other linear statistic) in Schur functions . These functions are symmetric polynomials in the transmission eigenvalues indexed by partitions . In the group representation theory the Schur functions are the irreducible characters of the unitary group and hence are orthogonal. This orthogonality is quite useful since it means that the coefficients in Schur function expansions are just “Fourier coefficients”, and, hence, can be found by integration over the unitary group. It gives an efficient way of calculating the expansion coefficients explicitly, the fact that we exploit in our approach. The Schur functions can be then averaged over the JPDF (2),
with the help of integration formulas due to Hua Hua (1963). The Schur function expansions and Hua’s integration formulas provide us with the necessary ingredients to compute all moments (or cumulants) of the conductance and/or shot-noise, see Sec. III for the detailed analysis.
In this section, we first give a brief summary of the required facts about partitions and Schur functions, MacDonald (1998) then develop the systematic way of performing the expansion over Schur functions and finally determine Schur function averages.
ii.1 Partitions and Schur functions
A partition is a finite sequence of non-negative integers (called parts) in decreasing order . The weight of a partition, , is the sum of its parts, , and the length, , is the number of its non-zero parts. No distinction is made between partitions which differ only by the number of zero parts. Different partitions of weight represent different ways to write as the sum of positive integers and can be graphically visualized through the Young diagrams. For example, one has only one partition in the trivial case of ; two partitions for ; three partitions for , etc.
For any partition of length , one can define a symmetric polynomial in variables as follows:
The denominator here is nothing else but the Vandermonde determinant . It divides the corresponding factor in the nominator, leaving the quotient as a homogeneous polynomial in the ’s of degree . These polynomials are called the Schur functions. For one-part partitions, , Schur functions are just the complete symmetric functions, , while for partitions which have no parts other than zero or one, , the Schur functions are the elementary symmetric functions . This can be verified directly from (5). It should be noted that the Schur functions corresponding to the partitions of form a basis in the space of homogeneous symmetric polynomials of degree , so that any homogeneous symmetric polynomial can be written as a linear combination of Schur functions.
The Schur functions of matrix argument that we shall use below are defined by the rhs in (5) evaluated at the eigenvalues of the matrix. Taking as an example the matrix of transmission probabilities, one has
where are exactly the transmission eigenvalues that appear in (2). Although not apparent from this definition, the Schur functions of matrix argument are polynomials in the matrix entries Jucobi-Trudi () and, obviously, for any non-degenerate matrix .
ii.2 Schur function expansions
In order to determine the moments of the conductance and shot noise along the lines explained above, one needs to expand the powers of these quantities in Schur functions. To this end, it is more convenient to work with the corresponding generating functions or . These functions belong to the general class of multiplicative symmetric functions, where the coefficients of the Schur function expansion
can be determined explicitly provided that the function is analytic in a neighborhood of in the complex -plane, see, e.g., Appendix in Ref. Fyodorov and Khoruzhenko (2007). Indeed, thinking of the ’s as of the eigenvalues of a unitary matrix , one can write
The main advantage of going unitary is the orthogonality of Schur functions ( is the normalized Haar measure):
which is a fact from the theory of group representations. One now recognizes a “Fourier series” in (7) and, hence,
The integral on the rhs in (9) is a standard one in RMT. To evaluate it, one first transforms it to the eigenvalues of the unitary matrix . The corresponding Jacobian is , canceling the denominator in the Schur function . The resulting integral can then be evaluated with the help of the Andrejeff identity, Pólya and Szegö (1976) yielding
ii.3 Schur function averages
Hua in his bookHua (1963) evaluated many useful matrix integrals. The following two are relevant in the context of our work:
where the average is over the JPDF (2), as in Eq. (4). If then and both integrals follow from the Selberg integral Mehta (1991). It should be noted that the rhs in (12) is exactly Selberg’s expression and the rhs in (13) can be manipulated to the one obtained by Selberg with the help of the duplication formula for the Gamma function. For non-empty partitions , the integral in (12) is a particular case of the Kadell-Kaneko-Yan generalization Kadel (1997); Kaneko (1993); Yan (1992) of the Selberg integral. However, the integral in (13) is different, as the Kadell-Kaneko-Yan generalization of the Selberg integral for involves zonal polynomials.
Hua’s identities are valid for arbitrary continuous . Specifying further to our case of and , we arrive after a simple algebra at
where we have introduced the coefficient clambda ()
We note that the obtained expressions for in terms of and yield zero if the length of the partition is greater than so when averaging Schur function expansions one need not bother about the restriction .
The Schur function average for in terms of the channel numbers and , Eq. (14), has a simple structure, being a ratio of polynomials
where the ’s are integers. Expression (17) for is less revealing. We found it useful to have the Schur function averages tabulated, see Table I for averages corresponding to partitions of , . This table suggests that
Iii Moments and cumulants of the
conductance and shot-noise
We now apply the results obtained in the previous two sections to calculate the moments of the conductance and shot-noise in a closed form. The final expressions involve summation over all partitions of in the case of conductance and in the case of shot-noise, with being the order of the moment. The cumulants can be obtained from the moments with the help of the well-known recursion
This method is well suited for analytic computations of lower order cumulants and also can be straightforwardly implemented in a computer algebra system for computations of higher order cumulants symbolically. Since a number of the partitions of the given grows only asymptotically at (i.e. slower than pure exponential), our method is very efficient for computing the cumulants up to reasonably large orders, as discussed below.
The moments of the conductance can be obtained from the generating function
The desired Schur function expansion for the exponential function can be read from (11) by choosing there. Throughout this paper we use the convention that for . The factorial determinant in (11) can be evaluated by elementary transformations on its rows or columns and the answer turns out to be exactly the coefficient introduced in (15). Recalling that the Schur functions are homogeneous, thus , one turns the Schur function expansion of the moment generating function into the following series in powers of :
The second sum on the right is over all partitions of . From (21) one easily obtains all moments of the conductance:
With Eq. (22) in hand, one can obtain cumulants by making use of Eq. (19). On this way we have successfully reproduced the first four cumulants which have been obtained before (exactly for any ). For the reference purpose, we state explicitly the conductance variance Brouwer and Beenakker (1996)
with being the conductance average, and the third cumulant Savin et al. (2008)
which is a measure of the skewness of the probability distribution. An explicit expression for the fourth cumulant is quite lengthy for arbitrary and the corresponding large expansion can be found in Ref. Savin et al. (2008) (see also below). However, in the particular case of , it can be simplified further to the following compact form:
Let us now discuss higher cumulants of the conductance, . Their explicit expressions are cumbersome and we consider mainly the physically interesting cases of the small or large channel numbers. In the quantum regime of a few open channels, we have found that these cumulants do not show a pronounced decay with increasing , see Fig. 1. In this case, the distribution function is strongly non-Gaussian. However, as the number of channels in the both leads increases, the system approaches the semiclassical (‘metalic’) regime where one should expect Altshuler et al. (1986) the following dependence of the cumulants on the total number of channels, :
The same scaling is generally applicable to any linear statistic on transmission eigenvalues (e.g., shot-noise), implying a Gaussian distribution in the limit Politzer (1989); Beenakker (1993) (see, however, the next section for discussion).
We have performed the asymptotic analysis of our exact RMT expressions in the limit when both . It suggests that the leading order term in the expansion of the -th cumulant, , has the following general structure:
where is an independent of homogeneous symmetric polynomial of order , see Table II for the first four ones. The expression on the rhs in (27) is of the order of , being in agreement with the above estimate (26) obtained within a different approach (weak localisation diagrammatics). The next to leading order term in the expansion of , the so-called weak localization correction, is of the order of . It vanishes for systems with broken time-reversal symmetry (). Further terms in this expansion can be easily computed as well if necessary.
In the special case of symmetric cavities, , the leading term (27) in the expansion of the cumulants vanishes for all and so does the next-to-leading term of the expansion of any odd cummulant (independently of , it contains a factor explicitly). This indicates that the Gaussian distribution is approached in this case much faster as compared to (26) – (27).
Generally, we note that in the symmetric case all odd cumulants at must vanish identically, as it follows by the simple symmetry argument Savin et al. (2008) (indeed, the joint distribution (2) becomes then symmetric under the change of all implying the symmetry of the conductance distribution about its mean ). It has been recently checked Novaes (2008) that representation (22) at satisfies this property. For the even cumulants, the expansion of our exact expressions gives the following leading term at ():
that agrees with the recent result by Osipov and Kanzieper Osipov and Kanzieper (2008) obtained by a completely different method. In the case of , we have found with the help of symbolic computations in Mathematica that
for . Correspondingly, we put forward the conjecture that Eq. (29) holds for all .
Having an aim to find also the joint moments of the conductance and shot-noise, we consider the generating function for the moments of :
The moment generating function of shot-noise is then simply given by whereas that of the conductance follows as . At finite , the quantity has a physical meaning of the total noise including both thermal and shot-noise contributions, with being then the known function of the temperature and applied voltage.Blanter and Büttiker (2000)
where are polynomials in and ,
In order to extract from this the moments of one needs to expand the coefficients in powers of . After some algebra, see Appendix A for details, one arrives at the desired expansion
The determinant on the rhs (34) can be evaluated in terms of the partition leading to an explicit expression for the coefficients , see Eq. (58). In the particular case of , is just the coefficient given by (15). We note that depend only on and and not on . The summation indices in (34) run over all integers from 0 to and are not subject to any ordering. From expansion (33), one easily finds that the -th moment of the total noise reads as follows:
where the second sum is over all partitions of . The joint moment of the conductance and shot-noise is then given by Eq. (35), with being replaced by the binomial coefficient . It is interesting to note that by setting , one also obtains the moments of the sum of squares of the transmission coefficients:
To the best of our knowledge the above formulas have not been reported in the literature before.
and for the shot-noise varianceSavin et al. (2008). The explicit expression for the later is rather lengthy (see Ref. Savin et al. (2008) for the corresponding large expansion) but turns out to be quite compact in the particular case of =n:
Higher cumulants of shot-noise, , similarly to those of conductance, are non-vanishing when the number of channels is small, implying a strongly non-Gaussian distribution also in this case. In the opposite limit of the large number of channels, , we have found the leading term of the expansion to have the following structure:
with being an independent of homogeneous polynomial of order , see Table III. The next order term of the expansion has been found to have similar structure to that of the conductance (explicit expressions being, of course, different), thus the same conclusions apply for this term, too.
We consider now the particular case of . In contrast to the conductance, both even and odd cumulants of shot-noise are non-vanishing at finite , even for systems with broken time reversal symmetry (). In the limit of , the expansion of our exact expressions suggests the following asymptotic behavior of the -th cumulant of the shot-noise ():
Equation (39) agrees with the very recent result Osipov and Kanzieper (2009) obtained by a different method. We have been able to verify by symbolic computations in Mathematica that Eq. (40) holds up to the 8-th cumulant and, thus, conjecture it to hold for any .
Iv Distribution functions
We consider now the distribution function of the conductance, , and that of shot-noise,