## Universal Conductance Fluctuations

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This definition has been adapted from [1].
In the 1960's, Wigner, Dyson, Mehta, and others discovered that the fluctuations in the energy level density are governed by level repulsion and therefore take a universal form [2]. The universality of the level fluctuations is expressed by the Dyson-Mehta formula [3] for the variance of a linear statistic (The quantity $A$ is called a linear statistic because products of different $E_{n}$'s do not appear, but the function $a(E)$ may well depend non-linearly on $E$.)$A=\sum _{n}a(E_{n})$ on the energy levels $E_{n}$. The Dyson-Mehta formula reads
 (1)
where $a(k)=\int _{-\infty }^{\infty }\!dE\,{\rm e}^{{\rm i}kE}a(E)$ is the Fourier transform of $a(E)$. Eq. (1) shows that: 1. The variance is independent of microscopic parameters; 2. The variance has a universal $1/\beta$-dependence on the symmetry index.
Figure 1: Fluctuations as a function of perpendicular magnetic field of the conductance of a 310 nm long and 25 nm wide Au wire at 10 mK. The trace appears random, but is completely reproducible from one measurement to the next. The root-mean-square of the fluctuations is $0.3\,e^{2}/h$, which is not far from the theoretical result $\sqrt {1/15}\,e^{2}/h$ [See Sec. II of [1]]. Adapted from Ref. [4].

In a pair of seminal 1986-papers [5, 6], Imry and Altshuler and Shklovkskiı̆ proposed to apply random matrix theory to the phenomenon of universal conductance fluctuations (UCF) in metals, which was discovered using diagrammatic perturbation theory by Altshuler [7] and Lee and Stone [8]. UCF is the occurrence of sample-to-sample fluctuations in the conductance which are of order $e^{2}/h$ at zero temperature, independent of the size of the sample or the degree of disorder — as long as the conductor remains in the diffusive metallic regime (size $L$ large compared to the mean free path $l$, but small compared to the localization length $\xi$). An example is shown in Fig. 1.
The similarity between the statistics of energy levels measured in nuclear reactions on the one hand, and the statistics of conductance fluctuations measured in transport experiments on the other hand, was used by Stone et al.[9, 10] to construct a random matrix theory of quantum transport in metal wires. The random matrix is now not the Hamiltonian $H$, but the transmission matrix $t$, which determines the conductance through the Landauer formula
 (2)
The conductance quantum is $G_{0}=2e^{2}/h$, with a factor of two to account for spin degeneracy. Instead of repulsion of energy levels, one now has repulsion of the transmission eigenvalues $T_{n}$, which are the eigenvalues of the transmission matrix product $tt^{\dagger }$. In a wire of cross-sectional area ${\cal A}$ and Fermi wave length $\lambda _{F}$, there are of order $N\simeq A/\lambda _{F}^{2}$ propagating modes, so $t$ has dimension $N\times N$ and there are $N$ transmission eigenvalues. The phenomenon of UCF applies to the regime $N\gg 1$, typical for metal wires.
Random matrix theory is based on the fundamental assumption that all correlations between the eigenvalues are due to the Jacobian $J=\prod _{i from matrix elements to eigenvalues. If all correlations are due to the Jacobian, then the probability distribution $P(T_{1},T_{2},\ldots T_{N})$ of the $T_{n}$'s should have the form $P\propto J\prod _{i}p(T_{i})$, or equivalently,
 (3) (4)
with $V=-\beta ^{-1}\ln p$. Eq. (3) has the form of a Gibbs distribution at temperature $\beta ^{-1}$ for a fictitious system of classical particles on a line in an external potential $V$, with a logarithmically repulsive interaction $u$. All microscopic parameters are contained in the single function $V(T)$. The logarithmic repulsion is independent of microscopic parameters, because of its geometric origin.
Unlike the RMT of energy levels, the correlation function of the $T_{n}$'s is not translationally invariant, due to the constraint $0\leq T_{n}\leq 1$ imposed by unitarity of the scattering matrix. Because of this constraint, the Dyson-Mehta formula (1) needs to be modified, as shown in Ref. [11]. In the large-$N$ limit, the variance of a linear statistic $A=\sum _{n}f(T_{n})$ on the transmission eigenvalues is given by
 (5)
The function $F(k)$ is defined in terms of the function $f(T)$ by the transform
 (6)
The formula (5) demonstrates that the universality which was the hallmark of UCF is generic for a whole class of transport properties, viz. those which are linear statistics on the transmission eigenvalues. Examples, reviewed in Ref. [12], are the critical-current fluctuations in Josephson junctions, conductance fluctuations at normal-superconductor interfaces, and fluctuations in the shot-noise power of metals.
For more details see [1], Sec. II.
References:
• [1]^abc C.W.J. Beenakker 0904.1432 [cond-mat.mes-hall]
• [2]^ C.E. Porter, ed., Statistical Theories of Spectra: Fluctuations, Academic Press, New York 1965.
• [3]^ F.J. Dyson and M.L. Mehta, J. Math. Phys. 4 (1963) 701.
• [4]^ S. Washburn and R.A. Webb, Adv. Phys. 35 (1986) 375.
• [5]^ Y. Imry, Europhys. Lett. 1 (1986) 249.
• [6]^ B.L. Altshuler and B.I. Shklovskiı̆, Sov. Phys. JETP 64 (1986) 127.
• [7]^ B.L. Altshuler, JETP Lett. 41 (1985) 648.
• [8]^ P.A. Lee and A.D. Stone, Phys. Rev. Lett. 55 (1985) 1622.
• [9]^ K.A. Muttalib, J.-L. Pichard, and A.D. Stone, Phys. Rev. Lett. 59 (1987) 2475.
• [10]^ A.D. Stone, P.A. Mello, K.A. Muttalib, and J.-L. Pichard, in: Mesoscopic Phenomena in Solids, ed. by B.L. Altshuler, P.A. Lee, and R.A. Webb, North-Holland, Amsterdam 1991.
• [11]^ C.W.J. Beenakker, Phys. Rev. Lett. 70 (1993) 1155.
• [12]^ C.W.J. Beenakker, Rev. Mod. Phys. 69 (1997) 731.