GQ corrections in the circuit theory of quantum transport

G

_Q

corrections in the circuit theory of quantum transport

G. Campagnano and Yu. V. Nazarov

Kavli Institute of Nanoscience, Delft University of Technology, The Netherlands

共Received 18 May 2006; published 13 September 2006兲

We develop a finite-element technique that allows one to evaluate correction of the order of GQto various

transport characteristics of arbitrary nanostructures. Common examples of such corrections are the weak-localization effect on conductance and universal conductance fluctuations. Our approach, however, is not restricted to conductance only. It allows one in the same manner to evaluate corrections to the noise charac-teristics, superconducting properties, strongly nonequilibrium transport, and transmission distribution. To en-able such functionality, we consider Green’s functions of arbitrary matrix structure. We derive a finite-element technique from Cooperon and diffuson ladders for these Green’s functions. The derivation is supplemented with application examples. Those include transitions between ensembles and the Aharonov-Bohm effect. DOI:10.1103/PhysRevB.74.125307 PACS number共s兲: 73.20.Fz, 72.10.Bg, 72.15.Rn

I. INTRODUCTION

The theoretical predictions of weak localization1_and

uni-versal conductance fluctuations2 _{along with experimental}

discoveries in this direction3,4_{have laid the basis of the}

mod-ern understanding of quantum transport—transport in nanostructures—and have stimulated considerable interest in the topic. Early studies mostly concentrated on diffusive electron transport. Both effects arise from quantum interfer-ence, which is described in the language of slow modes: Cooperons and diffusons.5,6_{Each mode of this kind brings a}

quantum 共fluctuating兲 correction of the order of GQ ⬅e2_{/␲ប to the classical Drude conductance G of the sample.} This universal value sets an important division between clas-sical conductors 共GⰇGQ兲 where interference effects are small and quantum ones共G⯝GQ兲 where the transport is es-sentially quantum.

A complementary approach to GQcorrections comes from the random matrix theory 共RMT兲 of scattering.7,8 _This

ap-proach relates the statistical properties of the scattering ma-trix of a nanostructure to those of a certain ensemble of ran-dom matrices. GQcorrections are understood in terms of the fluctuations and rigidity of the spectral density of these ma-trices. Although the RMT approach can deal with diffusive systems, the most comprehensive setup includes the so-called quantum cavity—an element whose scattering matrix is presented by a completely random unitary matrix of a certain ensemble. The cavity can be seen as a region of space where electron motion is sufficiently chaotic共either ballistic or diffusive兲 and where electrons can get in and out through some constrictions.9 _{The transport is determined by the}

propagation in the constrictions while the random unitary matrix representing the cavity is responsible for “randomiza-tion” of the scattering. The RMT approach does not neces-sarily concentrate on the total conductance. One can work with the transmission distribution: the averaged density of the eigenvalues of the transmission matrix squared. This transmission distribution appears to be useful in a much broader physical context: it determines not only the conduc-tance of nanostructures, but also the noise, full counting sta-tistics of charge transfers, and properties of the same nano-structure with superconducting leads attached.8,10 _{It is a}

modern paradigm of quantum transport that an individual nanostructure is completely characterized by a set of trans-mission eigenvalues while the transtrans-mission distribution de-scribes the averaged properties of random nanostructures of the same design. This makes it relevant to study GQ correc-tions and fluctuacorrec-tions of the transmission distribution.14,15

The density of the transmission eigenvalues is of the order of

G / GQ, and GQcorrections are of the order of␦G / GQ⯝1. The microscopic Cooperon-diffuson description is equiva-lent to a proper RMT approach. This is best illustrated in the framework of a more general supersymmetric theory11 _that

allows for a nonperturbative treatment of fluctuations in quantum scattering. The Cooperons and diffusons in this theory are fluctuations of the supersymmetric field around the saddle point. For the quantum cavity, only a single mode of these fluctuations is relevant. The integration over these modes reproduces the RMT results.11–13

One can describe nanostructures in the framework of a simple finite-element approach usually termed “circuit theory.” The circuit theory has originated from attempts to find simple solutions of the Usadel equations in supercon-ducting heterostructures.16_{However, it has been quickly}

un-derstood that theories of the same structure can be useful in a much broader context: one can compute the transmission distribution,10,18 _{noise, and counting statistics}17 _and

investi-gate spin effects19 _{and nonequilibrium phenomena.}20 _In

cir-cuit theory, a nanostructure is presented in a language similar to that of electric circuits: It consists of nodes, reservoirs, and connectors. A node is in fact a quantum cavity; a connector can be of very different types—tunnel junction, ballistic con-tact, and diffusive wire—and is generally characterized by a set of transmission coefficients. In circuit theory, each node is described by a matrix related to the electron Green’s func-tion. In the limit GⰇGQ circuit theory provides a set of algebraic equations that allow one to express the matrices in the nodes in terms of fixed matrices in the reservoirs.

func-tions in the nodes with respect to self-energy parts. In terms of a supersymmetric␴ model, this corresponds to an expan-sion of the action up to quadratic terms. However, the for-mulation we present does not contain any anticommuting variables that complicate applications of the␴ models. The determinant is just that of a finite matrix; this facilitates the computation of GQcorrections for nanostructures of compli-cated design.

The structure of the article is as follows. To make it self-contained, we start with a short outline of the circuit theory of multicomponent Green’s functions adjusted for the pur-poses of further derivations. In Sec. III we derive micro-scopic expressions for GQ corrections and specify to finite elements in Sec. IV. Section V is devoted to a description of spin-orbit scattering and magnetic-field decoherence, which are used to describe transitions between different RMT en-sembles. Since the Aharonov-Bohm effect plays an important role in experimental observations of GQcorrections, we ex-plain how to incorporate it into our scheme in a short sepa-rate section VI. We illustsepa-rate the technique with several ex-amples共Sec. VII兲, concentrating on a simplest 2⫻2 matrix structure suitable to calculate GQcorrections to the transmis-sion distribution. The examples also involve the simplest cir-cuits: one with a single node and two arbitrary connectors, a chain of tunnel junctions, and two-node four-junction circuit to demonstrate the Aharonov-Bohm effect. We summarize in Sec. VIII.

II. FROM GREEN’S FUNCTIONS TO FINITE ELEMENTS

In this article, we consider Green’s functions of arbitrary additional matrix structure with Nch indices. We do this for the sake of generality: This allows for a description of super-conductivity, incorporating the Keldysh formalism and treat-ing nonequilibrium and time-dependent problems. This is also extremely convenient, since most relations in use do not depend on the “physical meaning” of the structure. We use a caret symbol for operators in coordinate space and an in-verted caret symbol for matrices in additional indices. The Green’s function thus reads Gˆ ⬅Gˇ共x,x

⬘

兲, where x stands for the 共three-dimensional兲 coordinate. The general Green’s function is defined as the solution of the following equation: 关− ⌺ˇ共x兲 − Hˆ兴Gˇ共x,x

⬘

兲 =␦共x − x

⬘

兲. 共1兲 All physical quantities of interest can be in principle cal-culated from Green’s functions. We address here the quan-tum transport of electrons in disordered media. In this case, one can work with the common Hamiltonian

Hˆ = Hˆ0+ u共x兲, Hˆ0= − 1 2mⵜx

2_{+ U共x兲. 共2兲}

Here U共x兲 describes the design of the nanostructure: poten-tial “walls” that determine its shape and form ballistic quan-tum point contacts, potential barriers in tunnel junctions, etc. The potential u共x兲 is random: it describes the random impu-rity potential responsible for the diffusive motion of elec-trons, isotropization of the electron distribution function, and, most importantly for this article, fluctuations of the

transport properties of the nanostructure. The physics at space scales exceeding the isotropization length共which is the mean free path in the case of diffusive transport兲 does not depend on a concrete model of randomness of this potential. The most convenient and widely used model assumes a nor-mal distribution of u共x兲 characterized by the correlator 具u共x兲u共x

⬘

兲典=w␦共x−x

⬘

兲. It is important for us that both Hˆ0 and u共x兲 are diagonal in check indices.

Most evident choice of the self-energy matrix is ⌺ˇ共x兲= −⑀,⑀being the energy parameter of the Green’s function.⌺ˇ is more complicated in the theory of superconductivity. We will find it convenient, at least for derivation purposes, to work with arbitrary ⌺ˇ共x兲. We can also consider the more general situation with nonlocal⌺ˇ共x,x

⬘

兲.

Provided the conductance of the nanostructure is suffi-ciently high 共GⰇGQ兲, one can disregard quantum GQ cor-rections and work with semiclassical averaged Green’s func-tions. Closed equations for those are obtained with the noncrossing approximation.21_{They include the impurity}

self-energy

关− ⌺ˇ共x兲 − ⌺ˇimp− Hˆ0兴Gˇ共⑀;x,x

⬘

兲 =␦共x − x

⬘

兲, 共3兲

⌺ˇimp共x兲 = wGˇ共x,x兲. 共4兲

At space scales exceeding the mean free path, one can write closed equations for the Green’s function in the coinciding points.22,23 _{It is convenient to change notations, introducing}

dimensionless Gˇ 共x兲→G共x,x兲/i␲␯, with ␯ the density of states at the Fermi energy. For purely diffusive transport, one obtains the Usadel equations

⵱ · jˇ − i␲␯关⌺ˇ共x兲,Gˇ共x兲兴 = 0, jˇ = −␲D共x兲␯Gˇ ⵱ Gˇ, 共5兲 D being the diffusion coefficient. The solutions of the Usadel

equations are defined only if one takes into account boundary conditions at “infinity:” equilibrium Green’s functions in the macroscopic leads adjacent to the nanostructure. It turns out that Gˇ satisfies unity condition Gˇ2= 1ˇ. For situations where the transport is not entirely diffusive, one would supplement the Usadel equation with boundary conditions of various kinds共cf. Ref.24兲.

An alternative way to proceed is to notice that the Usadel equation is almost a conservation law for the matrix current

jˇ, which allows for a finite-element approach. This

conserva-tion law is exact at a space scale smaller than the coherence length estimated as

冑

D /⌺. It does not relay on an assumption

of diffusivity: It occurs because the Hamiltonian Hˆ com-mutes with the “check” structure. It is then natural to proceed to the separation of the nanostructure into regions where

Gˇ 共x兲 can be assumed constant. The next steps are the same

After the discretization of the nanostructure we can write a Kirchhoff-like equation

兺

_c Iˇc+

兺

␣ Iˇl,␣= 0, 共6兲

where the indices c and␣label the nodes and connectors of the nanostructure. The current through a connector c which connects nodes c1 and c2 reads

Iˇc= 1 4

兺

n Tn c_关Gˇ c1,Gˇc2兴 1 + Tn c_共Gˇ c1Gˇc2+ Gˇc2Gˇc1− 2兲/4 , 共7兲 兵Tn

c_{其 being the set of transmission eigenvalues of the} trans-mission matrix squared relative to the connector. Equation 共5兲 shows that the current jˇ is not fully conserved; to this aim

we included in Eq.共6兲 a leakage current Iˇ_l,␣= −i␲

␦␣关⌺ˇ␣

,Gˇ_␣兴, ␦_␣= 1

␯V_␣, 共8兲

where V_␣ is the volume of the node;␦_␣ can be easily recog-nized as the average level spacing in the node.

It is opportune to notice that the conservation law共6兲 can

be obtained by requiring that the values assumed by the ma-trix Green’s function in the nodes be such so as to minimize the following action:

S =

兺

c Sc+

兺

␣ S␣ , 共9兲 Sc= 1 2

兺

n Tr ln

冋

1 +Tn 4共Gˇc1G ˇ c2+ Gˇc2Gˇc1− 2兲

册

, 共10兲 S␣= −i_␦␲ ␣ Tr⌺ˇ_␣Gˇ_␣. 共11兲

The minimization of the action must be carried out provided that the Green’s functions in each node satisfy the normal-ization condition Gˇ_␣2= 1; this implies that the variation of the Green’s function␦Gˇ_␣ has to anticommute with Gˇ_␣ itself. A possible way to satisfy this condition is to write the variation as

␦Gˇ_␣= Gˇ_␣␦vˇ_␣−␦vˇ_␣Gˇ_␣, 共12兲 where no restriction is imposed on␦vˇ_␣. The Kirchhoff equa-tions are then rewritten as

␦S

␦vˇ_␣=

冉

Gˇ ,

␦S

␦Gˇ_␣

冊

= 0. 共13兲

The same action can be used at the microscopic level too, even before averaging over u共x兲. To do this, we note once again that⌺ˇ共x兲 form, at least formally, a set of parameters of our model. This can be straightforwardly extended to nonlo-cal operators ⌺共x,x

⬘

兲. To this end, we define the action in terms of the following variational formula:

␦S = −

冕

dxdx

⬘

Tr关Gˇ共x

⬘

,x兲␦⌺ˇ共x,x

⬘

兲兴, 共14兲

which is traditional in Green’s function applications.21,25

With making use of formal operator traces, this action can be written in terms of either⌺ˆ or Gˆ,

S = Tr ln共⌺ˆ + Hˆ兲, 共15兲

S = Tr关− ln共− Gˆ兲 − 共⌺ˆ + Hˆ兲Gˆ兴, 共16兲 apart from a constant. Equation共1兲 is reproduced by varying

either Eq.共15兲 or 共16兲 over ⌺ˆ or Gˆ, respectively. Since we

are not planning to treat GQcorrections beyond perturbation theory, we do not attempt to use the exponent of this action as an integrand in some path integral representation of a result of exact averaging over u共x兲. This has been done in Ref. 11 for supersymmetric matrices Qˇ and in Ref. 26 for Keldysh-based two-energy matrices Qˇ 共x,⑀,⑀

⬘

兲. We note, however, that the action used in Refs.11and26is equivalent to Eq.共9兲 if one substitutes Gˇ=Qˇ.

To this end, for averaged Green’s functions the action in use is defined simply as

␦S = −

冕

dxdx

⬘

Tr关具Gˇ共x

⬘

,x兲典␦⌺ˇ共x,x

⬘

兲兴. 共17兲 III. G_QCORRECTIONS TO MULTICOMPONENT

GREEN’S FUNCTIONS

In this section, we outline a microscopic approach to GQ corrections suitable for multicomponent Green’s functions. The main idea is the same as in Refs. 15 and 27 where a similar derivation has been done for the purely diffusive case and for a concrete 2⫻2 matrix structure. It is known5,6_that

the diffuson and Cooperon modes responsible for GQ correc-tions are presented as ladder diagrams made from averaged Green’s functions. In the usual technique, such diagrams have vertices corresponding to two current operators in the Kubo formula. One does not have to work with vertices: Instead, one considers Cooperon and diffuson contributions to the action that do not have any. The part of the action that presents GQcorrections is given by wheel diagrams. These wheels are made of either Cooperon or diffuson ladder sec-tions in a straightforward way共see Fig.1兲

We argue that the optimal way to present this part of the action is to double the existing check structure and to con-sider Eq. 共1兲 in a so-extended setup. Indeed, suppose we

would like to address the most general application of GQ corrections: parametric correlations.28 _{In this case, we start}

two worlds and gives the rungs of the ladder diagrams that involve white and black Green’s functions共Fig.1兲. This is to

obtain the diffuson ladder wheel. The Cooperon ladder wheel is obtained by inverting the direction of the Green’s function in one of the worlds. This corresponds to using transposed self-energies for this world,⌺ˇw→共⌺ˇw兲T. This guarantees that the corresponding Green’s functions are also transposed. Fluctuations at the same values of the parameters are natu-rally given by diagrams where⌺ˇband⌺ˇweither are the same in both worlds共diffusons兲 or mutually transposed 共Cooper-ons兲.

Finally, we note that the doubled structure is also useful for evaluation of the weak-localization correction. In this case, the last section of the Cooperon ladder is twisted before closing the wheel.

To proceed further, let us introduce the operators Kˆ pre-senting a section of a corresponding ladder,

Kˆ_diffab,cd共x,x

⬘

兲 = w共x兲具Gb ac_共x,x

_⬘

_兲典具G w db_共x,x

_⬘

_{兲典, 共18兲} Kˆ_Cooperab,cd 共x,x

⬘

兲 = w共x兲具Gb ac_共x,x

_⬘

_兲典具G w bd_共x,x

_⬘

_{兲典, 共19兲} where Latin letters represent check indices. As we have noted, the white Green’s function is transposed for the Coop-eron. Those are operators in the space spanned by the coor-dinates and the two check indices.

Summing up all diagrams, we find the formal operator expressions for contributions to the action. For fluctuations, we have

SG_Q= Tr关ln共1 − KˆCooper兲兴 + Tr关ln共1 − Kˆdiff兲兴, 共20兲 where the diffuson and Cooperon contributions are given by

Kˆdiffand KˆCooperrespectively. For the weak-localization cor-rection, one has to account for the fact that the last ladder section is twisted. We do this by introducing the permutation operator Pˇ , which exchange check indices,

Pˆ Kˆab,cd= Kˆba,cd. 共21兲 The contribution to the action becomes

Swl= 1

2Tr关Pˆ ln共1 − KˆCooper兲兴. 共22兲 The factor of 1₂ is included in the last formula to take into account that black and white Green’s functions, in the case of

weak localization, are not anymore independent. We note that Kˆ for the Cooperon is symmetric with respect to index exchange, so that Kˆ and Pˆ commute. The eigenfunctions and eigenvalues of K are therefore either symmetric共K+_{兲 or} an-tisymmetric 共K−_{兲 with respect to permutations. We can} re-write the last expression as a sum over these eigenvalues:

Swl= 1 2

兺

n

关ln共1 − Kn+兲 − ln共1 − Kn−兲兴. 共23兲 It is clear from the previous discussion that, in order to calculate the GQcorrections, one has to evaluate the eigen-values of the ladder section Kˇ , both for Cooperons and dif-fusons. We introduce now a method to compute this matrix easily. The observation is that the ladders under consider-ation are not specific for GQcorrections: The same ladders determine the response of semiclassical Green’s functions upon variation of⌺ˇ.29

To see this, let us go back to nonaveraged Green’s func-tions. We keep in mind that we have doubled the check space to include white and black sectors. We add by hand a source term: the self-energy which mixes up black and white Green’s functions,␦⌺ˇbw共x兲. This source term will give rise to a correction to the Green’s function in the same black-white sector. In the first order, we have

␦Gˇbw共x,x

⬘

兲 = −

冕

dx1dx2Gˇb共x,x1兲␦⌺ˇbw共x1,x2兲Gˇw共x1,x

⬘

兲, 共24兲 which is best illustrated by the diagram in Fig.2. The next step is to include the effect of the random potential u共x兲. We average Eq.共24兲, limiting ourselves to the noncrossing

ap-proximation and obtaining a set of ladder diagrams共Fig.2.兲

By summing up all the contributions we obtain the correction—taken in coinciding points—to the Green’s func-tion: 具␦Gˇbw共x,x兲典 = 1 w共x兲 Kˆ_diff 1 − Kˆdiff ␦⌺ˇbw共x兲. 共25兲 Equation 共25兲 is very valuable: it demonstrates that the

re-sponse of the Green’s function to the source term ␦⌺ˇbw is determined by the same ladder operator Kˆ , which we need to compute GQcorrections.

At the space scale of isotropization length, Kˆ ⬃1. Usually one is interested in the contribution arising from the larger space scale where the Cooperon-diffuson approximation is FIG. 1. Wheel diagrams that determine G_Q corrections to the

action. A single共double兲 line represents the Green’s functions from the black共white) block while dashed lines represent the averaging over disorder. The diffuson共left兲 and Cooperon 共right兲 wheels differ by mutual orientation of the lines.

FIG. 2. The response of the Green’s function␦Gˇbwon the

valid. At this scale, the eigenvalues of Kˆ are either 0 or very close to 1. To see this, we cite the results for the homoge-neous case with⌺ˇ=const⫻共x兲. A convenient basis in check space is one where⌺ˇ is diagonal, the eigenvalues being ⌺n. The Green’s function is diagonal in this basis as well, Gn = sa⬅sgn Im共⌺a兲. Owing to homogeneity, the section opera-tor is diagonal in the wave-vecopera-tor representation, its eigen-values being Knm_{共q兲. A direct calculation similar to that in} Ref.6gives Knm_{共q兲=0 if s}

n= sm. If sn⫽sm,

1 − Kab共q兲 ⬇␶„isb共⌺a−⌺b兲 + Dq2… + ¯ 共26兲 for ⌺␶, qlⰆ1 共␶= 2␲w␯ is the isotropization time兲. This

equation makes the relation between our technique and the common technique for Cooperons and diffusons in homoge-neous media. Usually, the self-energy⌺ has an equal number of eigenvalues with positive and negative imaginary parts. In this case, at each q, Kˆ has Nch

2

/ 2 zero and Nch 2

/ 2 nonzero eigenvalues.

Now we note that the zero eigenvalues contribute neither to Eq.共20兲 nor to Eq. 共25兲. As to those close to 1, we may

replace Kˆ by 1 in the numerator of Eq. 共25兲. We also note

that␦Gˇ can be presented as the derivative of the action 关cf.

Eq.共14兲兴. Therefore, we can write the GQcorrections due to the diffuson modes to the action in terms of a determinant made of derivatives of the semiclassical action with respect to⌺ˇbwand⌺ˇwb:

SG_Q,diff= − ln det

⬘

冉

− w共x兲

␦2_S

␦⌺ˇwb␦⌺ˇbw

冊

. 共27兲

The prime on the determinant signals that the zero eigenval-ues shall be excluded: det

⬘

is defined as the product of all nonzero eigenvalues. Indeed, as we have seen, some varia-tions of self-energies do not change the Green’s funcvaria-tions, giving rise to zero eigenvalues. We also note that the GQ corrections are not affected by the concrete form of w共x兲: Since the determinant of the matrix product is a product of their determinants, this matrix gives a constant contribution to the action which does not affect any physical quantities.

IV. METHOD

Let us now adapt the microscopic relation 共27兲 to the

finite-element approach outlined in the Sec. II.

With all previous derivations, this step is easy. We just replace the actual x-dependent Gˇ and ⌺ˇ by constants in each node. To get the action in these terms, one integrates over the volume of each node so that the formula共14兲 reads

␦S = −

兺

␣

i␲

␦␣Tr关Gˇ␣␦⌺ˇ␣兴, 共28兲 where the summation is over the nodes. The discrete analog of the determinant relation共27兲 is now

SG_Q,diff= − ln det

⬘

冉

− w_␣ V_␣ ␦2_S ␦⌺ˇwb␦⌺ˇbw

冊

− ln det

⬘

冉

␲ 2␶_␣␦_␣ ␦2_S ␦␩wb␦␩bw

冊

= const − ln det

⬘

冉

␦ 2_S ␦␩ˇwb␦␩ˇbw

冊

共29兲 where we have introduced the dimensionless response matrix

␩ˇ_␣⬅i␲⌺/␦_␣ and noticed that the matrix ⬀w brings a con-stant contribution to the action. The response matrix is deter-mined from the solution of the Kirchhoff equations at van-ishing source term ␩ˇ_␣bw

. It has Nnodes⫻Nch2 / 2 nonzero eigenvalues and the same number of zero ones. We observe that at⌺_w,b= 0 the eigenvalues of this matrix do not depend on the volume of the nodes; they are determined by the trans-mission eigenvalues of the connectors only and are of the order of G / GQ. Since rescaling of all conductances gives only an irrelevant constant contribution to the action, the GQ corrections depend only on the ratios of conductances of the connectors: This manifests the universality of these correc-tions.

The circuit theory action共9兲 is given in terms of Gˇ_␣. It is advantageous to present the answer forSG_Q in terms of the expansion coefficients of the action around the saddle point: the solution of semiclassical circuit-theory equations—that is, to use␦2_S/␦Gˇ

wb␦Gˇbwinstead␦2S/␦␩ˇwb␦␩ˇbw. If the latter matrix were invertible, we would make use of the fact that

␦2_S/␦_Gˇ

wb␦Gˇbw=共␦2S/␦␩ˇwb␦␩ˇbw兲−1. In fact, owing to the constraint Gˇ2_{= 1, there is a large number of zero eigenvalues} in the response matrix. So the task in hand is not completely trivial.

We proceed as follows. We expand the action by replacing each Gˇ in each node by

Gˇ = Gˇ0+ gˇ − Gˇ0gˇ2/2 +¯ 共30兲 and collecting terms of the second order in gˇ ,␩ˇ . The form

共30兲 satisfies the constraint Gˇ2= 1 up to second-order terms provided gˇGˇ0+ Gˇ0gˇ = 0. Let us work in the 共Nch

2_N

node兲-dimensional space indexed with the barred index

a

¯ composed of two check indices and one node index, a¯

⬅共a,b,␣兲. We present the result of the expansion as

␦S = ga¯ wb M¯ba¯g_b¯ bw −␩¯aga¯ bw . 共31兲

The variation of Eq.共31兲 under the constraint gˇGˇ0+ Gˇ0gˇ = 0 gives the response matrix␦2_S/␦␩

a

¯␦␩¯a. Next we consider the matrix⌸¯ba¯ defined through the following relation:

an-ticommuting subspace. Applying this projector to Eq.共31兲,

we show that the projected matrix⌸¯ba¯Mb¯c¯⌸¯bc¯ is an inverse of the response matrix within the anticommuting subspace:

SG_Q= ln det

⬘

关共⌸¯ba¯M¯c¯b⌸¯bc¯兲兴 = ln det共⌸a¯b¯M¯c¯b⌸¯bc¯+␦¯ba¯

−⌸¯ba¯兲. 共33兲

In the last equation we add the matrix 1 −⌸ˆ. This procedure replaces all zero eigenvalues with 1, so one can evaluate a usual determinant.

We recall that as far as fluctuations are concerned, there are two contributions of this kind coming from diffuson and Cooperon ladders, respectively. The weak-localization cor-rection involves a permutation operator that sorts out eigen-values involved according to Eq.共23兲. With this, Eqs. 共33兲

and 共29兲 give the GQ corrections in an arbitrary circuit-theory setup in the most general form.

V. DECOHERENCE AND ENSEMBLES

Until now we have assumed that the Hamiltonian com-mutes with the check structure and is invariant with respect to time reversal. This implies strict coherence of waves with a different check index which propagates in the disordered media described by this Hamiltonian. Even small check-dependent perturbations of the symmetric Hamiltonian give accumulating phase shifts to these waves and may signifi-cantly change their interference patterns at long distances. Due to their random nature, such phase shifts can be re-garded as decoherence although this should not be confused with a real decoherence coming from interaction-driven in-elastic processes.30

In real experimental situations, two sources of such deco-herence are usually of importance: spin-orbit scattering and magnetic fields. Already early studies of GQ corrections1,4 have revealed their significant dependence on these two fac-tors in the regime where those are too weak to affect the semiclassical transport. From the RMT point of view, these factors, upon increasing their strength, provide transitions form orthogonal ensembles共symmetric Hamiltonian兲 to two different ensembles: symplectic 共spin-orbit interaction兲 and unitary共magnetic field兲.11

We show in this section how to incorporate spin-orbit scattering and magnetic fields into our scheme. The most convenient way is to present them as perturbative corrections to the Gˇ -dependent action 共Fig.3兲.

The spin-orbit scattering enters the Hamiltonian in the form Hˆso=␴ˇaHa共x,x

⬘

兲, Ha共x,x

⬘

兲=−Ha共x

⬘

, x兲, ␴ˇa represent-ing spin Pauli matrices in check space. In the second order in

Ha_{the averaging gives共Fig.3兲}

Sso=

冕

dx

␲␯

8␶so共x兲

Tr关Gˇ共x兲␴aGˇ 共x兲␴a兴. 共34兲 At the level of the microscopic approach, the spin-orbit scat-tering takes place anywhere in the nanostructure. In the finite-element approach, it is advantageous to ascribe spin-orbit scattering to nodes rather than to connectors. This is consistent with the main idea of our scheme: Random phase

shifts take place in the nodes. The spin-orbit contribution in each node ␣ is obtained by integrating Eq. 共34兲 over the

node, Sso= ␩so 4 Tr关Gˇ␣␴ˇaG ˇ ␣␴ˇa兴, 共35兲 where␩so⬅␲/共2␶_␣共so兲␦␣兲.

The magnetic field is incorporated into the Hamiltonian through modification of the derivative,

i⵱ → i ⵱ − e cប⌺ˇHA,

where A is the vector potential and⌺ˇH共⌺ˇH2= 1兲 describes the interaction of different check waves with the magnetic field. In its simplest form, ⌺ˇH is the matrix in the white-black structure introduced such that ⌺ˇ_Hb= 1, ⌺ˇ_Hw= −1 provided we describe a Cooperon. This is consistent with the requirement that one of the Hamiltonians must be transposed to describe a Cooperon ladder. This is not the only plausible form of this matrix. For instance, in nonequilibrium superconductivity⌺ˇH involves electron-hole Nambu structure.

The magnetic-field decoherence contribution can also be assigned to a node and reads

SH=

␩H

2 Tr关Gˇ␣⌺ˇHG

ˇ

␣⌺ˇH兴, 共36兲

where␩H=␲/共␶H␦␣兲 and␶Hcorresponds to Cooperon mag-netic decoherence time in a common theory. The latter is known to depend on the geometry of the node and its characteristics.31_{If the transport within the node is diffusive,}

1 /␶H= 4共e/ប兲2D具A2典, where 具¯典 denotes averaging over the volume of the node. The vector potential is taken in the gauge where it is orthogonal to the boundaries of the node.

We have an order of magnitude, 1 /␶H␦

⯝共⌽/⌽0兲2_共G

node/ GQ兲, ⌽ being the magnetic flux through the node,⌽0⬅␲ប/e being the flux quantum, and Gnodebeing a typical conductance of the node. The latter is limited by its Sharvin value in the ballistic regime where the isotropization length is of the order of the node size.

The magnetic field produces not only random but also deterministic phase shifts. This gives rise to the Aharonov-Bohm effect discussed in the next section.

To find the effect of decoherence terms共35兲 and 共36兲 on

the eigenvalues forming the localization correction, we ex-pand the action as done to obtain Eq.共31兲. The decoherence

contribution to Mˆ is diagonal in the node index and can be made diagonal in a barred index by the proper choice of basis in check space. For instance, if no external spin polar-ization is present in the structure, the spin-orbit contribution is diagonal in the basis made of singlets and triplets in spin space. The simple realization of⌺Hmentioned is automati-cally diagonal. If in addition this diagonal contribution is the same in all nodes, both decoherence effects just shift the eigenvalues of Mˆ corresponding to the symmetric Hamil-tonian. This gives an extremely convenient model of deco-herece effects.

The action for fluctuations is modified as follows:

Sdiff=

兺

n ln共Mn兲 + 3 ln共Mn+␩so兲, 共37兲 SCooper=

兺

n ln共Mn+␩H兲 + 3 ln共Mn+␩so+␩H兲, 共38兲 where the summation goes over nondegenerate eigenvalues of Mˆ and the factor of 3 comes from the threefold degen-eracy of the triplet. To derive the modification for the weak-localization contribution, we note that singlets and triplets are, respectively, antisymmetric and symmetric with respect to permutations. Therefore, triplet extensions of symmetric and antisymmetric eigenvalues are, respectively, symmetric and antisymmetric. The weak-localization correction thus reads Swl =1 2

兺

n lnMn − +␩H Mn + +␩H +3 2 ln Mn + +␩so+␩H Mn − +␩so+␩H , 共39兲

M+共−兲 being共anti兲symmetric eigenvalues of Mˆ.

Since eigenvalues of Mˆ are of the order of G/GQ, the

decoherence effects become important at

␩so,␩H⯝G/GQ—that is, when inverse decoherence times match the Thouless energy Eth=共G/GQ兲␦ of the node, 1 /␶so, 1 /␶H⯝Eth.

VI. AHRONOV-BOHM EFFECT

The Aharonov-Bohm共AB兲 effect plays a crucial role in experimental observations and identification of GQ correc-tions共see, e.g., Ref. 32兲. Therefore it must be incorporated

into our scheme, and in this section we explain how to do this. This extends the results of Ref.33where the AB effect was considered in superconducting circuit theory. In the fol-lowing, we do not consider any orbital effects of the mag-netic field but just the topological one.

Let us suppose that the nanostructure presents a closed ring threaded by a magnetic flux⌽. As explained above, in the presence of a magnetic field the momentum operator has to be modified according to

i⵱ → i ⵱ − e/ប⌺ˇHA,

where A is the vector potential. Neglecting orbital effects, one can get rid of the vector potential in the Schrödinger

equation by a gauge transformation. Let us have an ideal cut in the nanostructure that breaks the loop共Fig.4.兲 The

topo-logical effect of the flux can be incorporated into a boundary condition for the wave function␺L,Ron two sides of the cut,

␺L= exp共i␾AB⌺H兲␺R. The phase of the wave function there-fore presents a discontinuity at the cut that is equal to ±␾AB,␾AB=␲⌽/⌽0. Since the transformation does not ex-plicitly depend on x, it can be immediately extended to semi-classical Green’s functions, so that those functions at two sides are related by

GˇL= exp共i␾AB⌺ˇH兲GˇRexp共− i␾AB⌺ˇH兲. 共40兲 This solves the problem at the microscopic level. Once the nanostructure has been discretized to finite elements, we note that the cut always occurs between a connector and a node. The most convenient way to deal with the gauge trans-formation共40兲 is to put it into the action of the

correspond-ing connector. To do this, we observe that the Green’s func-tion at the right end of the connector is not GˇR of the node anymore: since the cut is crossed, it is eventually GˇL given by Eq.共40兲. The connector action in the presence of flux is

therefore Sc= 1 2

兺

n Tr ln

冋

1 +Tn 4 共Gˇc1G ˇ c2共␾AB兲 + Gˇc2共␾AB兲Gˇc1− 2兲

册

, 共41兲 where Gˇ 共␾AB兲 = exp共i␾AB⌺ˇH兲Gˇ exp共− i␾AB⌺ˇH兲. 共42兲 One checks that the variation of the so-modified action re-produces the Kirchhoff laws for matrix current given in Ref.

33. Owing to global gauge invariance, it does not matter to which connector and to which end of the connector the Aharonov-Bohm phase is ascribed. If there are more loops in the nanostructure, more connector actions have to be modi-fied in such a way.

VII. EXAMPLES

In this section, we will give a set of examples to illustrate concrete applications of the technique developed. For the sake of simplicity, we choose the simplest matrix structure that gives a sensible circuit theory. We consider 2⫻2 matrix Green’s functions whose values in two terminals can be pa-rametrized by a single parameter␾,

Gˇ 共␾兲 =

冉

0 e

−i␾

ei␾ 0

冊

. 共43兲

The use of these matrix structures is that they give access to a fundamental quantity in quantum transport: the transmis-sion distribution of transmistransmis-sion eigenvalues of two-terminal nanostructure. We have to explain this relation before going on to concrete examples. The averaged transmission distri-bution is defined as

␳共T兲 =

兺

n

具␦共T − Tn兲典, 共44兲

where the sum is done over all the transport channels and the average is, in principle, to be intended over an ensemble of nanostructures of the same design. For GⰇGQthe transmis-sion eigenvalues are dense in the interval 关0,1兴 and self-averaging takes place.

Let us take a connector and set the Green’s functions at its ends to Gˇ 共␾1兲 and Gˇ共␾2兲. By virtue of Eq. 共10兲 the connector

action reads S共␾1−␾2兲 =

兺

n ln

冋

1 − Tnsin2 ␾1−␾2 2

册

. 共45兲

The trick is to regard the whole nanostructure as a single complex connector between left and right reservoirs and set the Green’s functions in the reservoirs to Gˇ 共0兲 and Gˇ共␾兲. The total action共9兲 becomes now the connector action of the

whole nanostructure and defines the transmission distribution in question,

S共␾兲 =

冕

dT␳共T兲ln

冋

1 − T sin2 ␾

2

册

. 共46兲

If one computes the␾dependence ofS, the transmission distribution can be extracted from its analytic continuation on complex␾共Ref. 10兲: ␳共T兲 = − 1 ␲T

冑

1 − TRe

冋

⳵S ⳵␾

冉

␲+ 2i cosh−1 1

冑

T− 0 +

冊

册

_. 共47兲 The circuit theory of Sec. II gives the answer in the limit

GⰇGQ. The weak-localization contributionSwlgives the GQ correction to the transmission distribution. The fluctuation contributionSG_Q=Sdiff+SCooperthat depends on two param-eters␾w,bgives correlations of transmission distributions共cf. Ref.15兲: SG_Q共␾b,␾w兲 =

冕

dTdT

⬘

具具␳共T兲␳共T

⬘

兲典典ln

冋

1 − T sin2 ␾b 2

册

⫻ln

冋

1 − T

⬘

sin2␾w 2

册

. 共48兲

A simple application of the above formulas is the GQ correc-tions to the conductance. Those are given by the derivatives of corresponding actions at␾b,w= 0, ␦Gwl GQ = − 2

冏

⳵ 2_S wl共␾兲 ⳵␾2

冏

␾=0, 共49兲 具具GbGw典典 GQ 2 = 4

冏

⳵4_S G_Q共␾b,␾w兲 ⳵␾b 2_⳵␾ w 2

冏

␾b_,␾w₌₀ . 共50兲 A. Junction chain

The first example is a chain of tunnel junctions. We will study GQcorrections for a chain of N identical junctions that connects two reservoirs共Fig.5兲. The connector action for a

tunnel junction assumes a very simple form Sc =共GT/ 4GQ兲Tr共Gˇ1cGˇ2c兲, GTbeing the conductance of the tun-nel junction. Several tuntun-nel junctions in series, however, pro-vide a good approximate for diffusive wire. Therefore, in the limit N→⬁ we can compare GQcorrections with the known results14,15_{for corrections to the transmission distribution of}

a one-dimensional diffusive conductor.

We set the Green’s functions in the reservoirs on the left and on the right to Gˇ 共0兲 and Gˇ共␾兲, 关cf. Eq. 共43兲兴,

respec-tively. The semiclassical action for the system reads

S = GT 4GQ

兺

k=0 N−1 TrGˇkGˇk+1− i␲ ␦S

兺

k=1 N−1 Tr⌺ˇkGˇk. 共51兲

Here k = 1 , . . . , N − 1 labels the nodes, while k = 0 and k = N identify left and right reservoirs, respectively. All the nodes are assumed to be identical with the same␦S.

Since all the junctions are identical, the semiclassical so-lution is easy to find: the “phase” ␾ drops by the same amount at each junction, and the solution reads Gˇk = Gˇ 共k␾/ N兲, provided ⌺ˇk⬀1ˇ within each 2⫻2 block. This gives the optimal value of the action,

FIG. 5. The chain of tunnels junctions of the same conductance

GT. In our finite-element approach, there are two Cooperon and two

S =NGT 2GQ

cos

冉

␾

N

冊

, 共52兲

from which one can evaluate the semiclassical transmission distribution by using relation共47兲. In the limit N→⬁, S=

−共GD/ 4GQ兲␾2, GD⬅GT/ N being the conductance of the whole chain. This gives the well-known transmission

distri-bution for the diffusive conductor, ␳共T兲

=共GD/ 2GQ兲/T

冑

1 − T.10

As explained above, to calculate GQcorrections we aug-ment the check dimension of the Green’s functions by intro-ducing the black and white structure. Consequently, the pa-rameter ␾ gets a “color” index b or w. The semiclassical solution for resulting 4⫻4 matrix is nonzero in bb and ww blocks, Gˇk 0 =

冉

Gˇ 共k␾b/N兲 0 0 Gˇ 共k␾w/N兲

冊

. 共53兲

Now we shall derive the matrix Mˆ eigenvalues which de-termine GQ corrections. It is advantageous to use a param-etrization of the deviations from the semiclassical solution,

gˇ, which automatically satisfy gˇGˇ +Gˇgˇ=0 in each node. To

this end, we rewrite the action 共51兲 in a special basis: that

one where Gˇk0is diagonal in each node,

Gˇk 0 =

冢

1 0 0 0 0 − 1 0 0 0 0 1 0 0 0 0 − 1

冣

. 共54兲

Then the deviation of the form

gˇk=

冢

0 0 0 gk,p bw 0 0 g_k,mbw 0 0 g_k,pwb 0 0 g_k,mwb 0 0 0

冣

共55兲

satisfies the condition mentioned.

In this basis共see the Appendix for details兲 the action reads

S = GT 2GQ

兺

k=0 N−1 TrGˇkLˇGˇk+1Lˇ−1− i␲ ␦S

兺

k=1 N−1 Tr⌺ˇkGˇk, 共56兲 where the bb共ww兲 block of Lˇ is given by

Lbb共ww兲=

冉

cos共␾

b共w兲_{/N兲 i sin共␾}b共w兲_/N兲

i sin共␾b共w兲/N兲 cos共␾b共w兲/N兲

冊

. 共57兲

We expand the Green’s matrices according to Eq.共30兲, write

the quadratic form in terms of gˇ, and diagonalize it共see the Appendix兲 to find the following set of eigenvalues 共l = 1 , . . . , N − 1兲: 4GQ GT Ml±共␾ b ,␾w兲 = 2 cos ␾ b 2Ncos ␾w 2Ncos ␲l N − cos␾ b N − cos ␾w N ⫿

冑

4 sin2 ␾ b 2Nsin 2 ␾ w 2Ncos 2 ␲l N +⑀ 2_, 共58兲 where⑀⬅2␲GQ共⌺b−⌺w兲/GTi␦S measures the difference of the Green’s function energy parameter in bb and ww blocks in units of a single-node Thouless energy. To obtain eigen-values that determine the weak-localization contribution, we set␾w= −␾b=␾,⑀= 0. This yields

M_wl,l+ 共␾兲 = GT 4GQ cos

冉

␾ N

冊

冋

cos ␲l N − 1

册

, 共59兲 M_wl,l− 共␾兲 = GT 4GQ

冋

cos␲l N − cos ␾ N

册

. 共60兲

Let us discuss the weak-localization correction first. If we neglect decoherence factors, we can sum up over l to find a compact analytical expression

Swl= 共N − 2兲 2 ln

冉

cos ␾ N

冊

+ 1 2 ln

冢

sin2␾ N sin␾

冣

. 共61兲 In the limit N→⬁ this reproduces the known correction for a one-dimensional diffusive wire,15

Swl= 1 2ln

冉

␾

sin␾

冊

. 共62兲

It is interesting to note that the weak-localization correction is absent for N = 2. We will see below that this is a general property of a single-node tunnel-junction system. It was ob-served in Ref.15that a part of the weak-localization correc-tion in diffusive conductors is universal: It depends neither on the shape nor on the dimensionality of the conductor. The universal part is concentrated near transmissions close to 1 and is given by

␦␳wl共T兲 = − 1

4␦共T − 1兲, 共63兲

while the nonuniversal part is a smooth function of T. The relation共61兲 possesses this property at any N, since the

uni-versal part comes from the divergency in Eq. 共61兲 at␾=␲ where the eigenvalue M_wl,1− goes to zero. Our approach proves that this correction is universal for a large class of the nanostructures, not limited to diffusive ones, for any nano-structure where transmission eigenvalues approach 1. This is guaranteed by the logarithmic form of the action. If M_wl,1− ⬀共␲−␾兲 at␾→␲, the correction is given by Eq.共63兲

irre-spective of the proportionality coefficient.

Expanding Eq.共61兲 at␾→0 we find the correction to the

␦Gwl GQ = −1 3 共N − 1兲共N − 2兲 N2 . 共64兲

This is written for an orthogonal ensemble; a well-known factor 共1−2/␤兲 defines the correction for other pure en-sembles. The effect of spin and magnetic decoherence can be taken into account by shifting the eigenvalues共59兲 and 共60兲

according to Eq.共39兲 since the decoherence factors in each

node are the same,␨H,so⬅␩H,so共4GQ/ GT兲.

The Swl is still given by an analytical although lengthy expression 共see the Appendix兲. The correction to the trans-mission distribution corresponding to this expression is plot-ted in Fig.6for different strengths of the spin-orbit coupling to illustrate the transition between the orthogonal and sim-plectic ensembles. The correction to the conductance is given by

␦Gwl

GQ

= 3F共N,␨H+␨so兲 − F共N,␨H兲, 共65兲 where we define an auxiliary function F共x,N兲:

F共x,N兲 = −共N − 1兲 N2 − 1 N2 共2 + x兲2 x共4 + x兲 − 1 N

冑

x共4 + x兲 22N_{+关2 + x +}

冑

_{x共4 + x兲兴}2N 22N−关2 + x +

冑

x共4 + x兲兴2N.

Let us discuss the parametric correlations. Without deco-herence factors and at the same energy共⑀= 0兲 one can still sum up over the modes to obtain an analytical expression

Sdiff=SCooperon=共N − 1兲ln

冉

cos

␾b N + cos ␾w N

冊

+

兺

± ln

冢

sin ␾b±␾w 2 sin ␾b±␾w 2N

冣

. 共66兲

The fluctuation of conductance obtained with Eq.共50兲 reads

具具␦G2典典 GQ2 = 2 15 N4+ 15N − 16 N4 共67兲

and converges to the known expression for a quasi-one-dimensional diffusive conductor at N→⬁. We notice that this convergence is rather quick; the fluctuation at N = 5 dif-fers from the asymptotic value by 10% only. We see thus that the diffusive wire, which in principle contains an infinite number of Cooperon and diffuson modes, can be, with suf-ficient accuracy, described by the finite-element technique even at a low number of elements.

Another point to discuss concerns the correlations of the transmission eigenvalues Tn; those can be obtained by ana-lytic calculation of Eq. 共48兲. It is instructive to concentrate

on the relatively small eigenvalue separations, those are much smaller than 1, but still exceeding an average spacing ⯝GQ/ G between the eigenvalues, GQ/ GⰆ兩T−T

⬘

兩Ⰶ1. We observe that the correlation in this case is determined by the divergence ofS at␾b−␾w→ ±2␲. Indeed, M1

−

approaches 0

in this limit. This again suggest the universality of these correlations. Indeed, as shown in Ref.15for diffusive con-ductors, the correlations in this parameter range are deter-mined by universal Wigner-Dyson statistics and reduce to

具具␳共T兲␳共T

⬘

兲典典 = − 2

␲2_␤Re

1

共T − T

⬘

+ i0+_兲2. 共68兲 Since the conductance fluctuations are contributed by corre-lations of Tn at scale⬃1 as well, they are not universal. We plot in Fig.7 the correlator of conductance fluctuations as a function of the energy difference at several N.

B. AB ring

In this subsection we exemplify evaluation of the AB ef-fect within our scheme. We concentrate on the simple circuit presented in Fig.8. It contains four tunnel junctions and two nodes labeled A and B. The conductances of the junctions are chosen to reuse the results of the previous section for a chain of three tunnel junctions: The solution of semiclassical cir-cuit theory equations is given by Eq. 共53兲 for N=3. The

action reads

FIG. 6. Weak-localization correction to the transmission distri-bution of a system of four identical junctions at different values of the spin-orbit parameter ␨so. We plot here cumulate correction X共T兲⬅兰_T1dT⬘T⬘␦␳共T⬘兲. X共1兲 represents the universal singular part

of the correction关cf. Eq. 共63兲兴 while X共0兲 gives a correction to the

conductance. The lowest curve corresponds to strictly zero␨soand

therefore represents a pure orthogonal ensemble. Its negative value at T = 1 is partially compensated for by the positive nonuniversal contribution coming from T⯝1 so that the resulting correction to the conductance,␦Gwl/ GQ= X共0兲⬇0.2. The two higher curves

S = GT 4GQ Tr

再

GˇLGˇA+ 1 2G ˇ AGˇB+ 1 2G ˇ AGˇB共␾AB兲 + GˇBGˇR

冎

− i␲ ␦i=A,B

兺

Tr⌺ˇiGˇi. 共69兲

where GˇL,R are Green’s functions in the reservoirs and

GˇB共␾AB兲 is modified according to Eq. 共40兲. To study the cor-relation of the conductance fluctuations, we consider differ-ent Green’s functions for white and black blocks and sub-jected to different fluxes ␾AB

b _⫽␾ AB w

. For the weak-localization correction, we set␾AB

b

= −␾AB w

=␾AB

To calculate the matrix Mˆ we use again the basis where

G_i共0兲are diagonal and the parametrization for gˇ introduced in the previous subsection. It reads

M =

冉

Md Mod Mod* Md

冊

, where 2⫻2 blocks Md,odare given:

GTMd 4GQ =

冉

− cos␸ b − cos␸w+⑀ 0 0 − cos␸b− cos␸w−⑀

冊

GTMod 4GQ =1 + e i共␾_ABb −␾_ABw_兲 2

冢

cos␸ b 2 cos ␸w 2 sin ␸b 2 sin ␸w 2 sin␸ b 2 sin ␸w 2 cos ␸b 2 cos ␸w 2

冣

.

The parameter ⑀ which characterizes the energy difference between black and white Green’s function is defined as in the previous subsection. At⑀= 0 we obtain an explicit expression for the diffuson eigenvalues共the Cooperon ones are obtained by␾w_→−␾w_and␾ AB w _→−␾ AB w _兲, 4GQ GT M_1,2+ = − cos␾b 3 − cos␾w 3 ± cos

冉

␾AB b −␾AB w 2

冊

cos

冉

␾b−␾w 6

冊

, 共70兲 4GQ GT M_1,2− = − cos␾b 3 − cos␾w 3 ± cos

冉

␾AB b −␾AB w 2

冊

cos

冉

␾b+␾w 6

冊

. 共71兲 The weak-localization correction to the action reads

Swl共␾兲 = 1 2 ln

冉

cos2共␾/3兲关4 − cos2共␾AB兲兴 4 cos2共␾/3兲 − cos2共␾AB兲

冊

, 共72兲

from which we calculate the correction to conductance as a function of the flux:

␦Gwl GQ = − 4 cos 2_共␾ AB兲 9关7 − cos共2␾AB兲兴 . 共73兲

We see that the weak-localization correction cancels at half-integer flux ␾AB=␲. This is because the junctions forming the loop are taken to be identical. The flux dependence ex-hibits higher harmonics, indicating semiclassical orbits that encircle the flux more than once.

For the correlator of conductance fluctuations we obtain 具␦G2典 GQ 2 =

兺

± 259 − 4 cos共␾AB w ±␾AB b _{兲 + cos 2共} ␾AB w ±␾AB b _兲 81关cos共␾AB w ±␾AB b _{兲 − 7兴}2 , 共74兲 where plus and minus signs indicate Cooperon and diffuson contributions, respectively. The higher harmonics are present as well.

FIG. 7. Energy dependence of the correlator of the conductance fluctuations 具具G共0兲G共E兲典典 for chains with a different number of junctions N. The energy difference is normalized to the Thouless energy of the whole chain, E_th⬅␦_sG_T/ 2␲G_QN2_{. Note the fast} con-vergence of the correlator to that of a diffusive wire for large N and negative correlations at large E.

C. Two connectors and one node

Probably the simplest system to be considered by circuit-theory methods consists of a single node and two connectors 共Fig.9兲. Since in this case there are only Nch eigenvalues, one can straightforwardly elaborate on complicated arbitrary connectors. For this setup we are still able to find an analyti-cal expression for Cooperon and diffusion eigenvalues. This allows us to get an expression for the weak-localization cor-rection to the conductance which was vanishing in the case

of two tunnel junctions. Each connector is in principle char-acterized by the distribution of transmission coefficients兵Tn

R_其 and兵Tn

L_{其 or, equivalently, by the functional form of the} con-nector action given by Eq.共10兲, SLandSR. The action for the whole system reads

S = SL共GˇL,Gˇ 兲 + SR共Gˇ,GˇR兲 − i

␲ ␦s

Tr⌺ˇGˇ. 共75兲

Gˇ being the Green’s function of the node. We employ 2⫻2

matrices parametrized by Eq.共43兲 and set the Green’s

func-tions in the left and right reservoirs to Gˇ 共−␾/ 2兲 and Gˇ共␾/ 2兲, respectively. The saddle-point value of Gˇ is given by the phase␹ and for a general choice ofSL andSRdoes depend on␾,␹⬅␹共␾兲. The total action in the saddle point is there-fore S共␾兲=SL共␹+␾/ 2兲+SR共␹−␾/ 2兲. We expand the Green’s function according to Eq. 共30兲. The second-order

correction to the action in this case reads

S共2兲_{= − i}␲ ␦s Tr⌺ˇgˇ −1 2

兺

n Tr

冦

Tn L 4

冋

1 +Tn L 4 共兵Gˇ 0_,Gˇ L其 − 2兲

册

gˇ2Gˇ0GˇL+

冤

Tn L 4

冋

1 +Tn L 4共兵Gˇ 0_,Gˇ L其 − 2兲

册

冥

2 共gˇGˇLgˇGˇL+ gˇ2兲 + 共L ↔ R兲

冧

. 共76兲 Acting like in the previous subsections, we find the two following diffuson eigenvalues共the Cooperon ones are obtained by the substitution␾w→−␾w兲: M±共␾b,␾w兲 =

兺

i=b,w I共␾i兲

冋

cot

冉

␹共␾i兲 −␾ i 2

冊

− cot

冉

␹共␾ i_{兲 +}␾ i 2

冊

册

− 1 − cos

冋

␹共␾b兲 +␾ b 2 ⫿␹共␾ w_{兲 ⫿}␾ w 2

册

2 _i=b,w

兺

再

1 sin2关␹共␾i兲 +␾i/2兴

冋

I

⬘

共␾i兲 ␹

⬘

共␾i_{兲 + 1/2}− I共␾ i_兲cot

冉

_␹共␾i_{兲 +}␾ i 2

冊

册

冎

+ 1 − cos

冋

␹共␾b_{兲 −}␾ b 2 ⫿␹共␾ w_{兲 ±}␾ w 2

册

2 _i=b,w

兺

再

1 sin2关␹共␾i兲 −␾i/2兴

冋

I

⬘

共␾i_兲 ␹

⬘

共␾i_{兲 − 1/2}− I共␾ i_兲cot

冉

_␹共␾i_{兲 −}␾ i 2

冊

册

冎

, 共77兲

Here we introduce I共␾兲⬅⳵S/⳵␾ to characterize the deriva-tive of the total semiclassical action. We see that M− ap-proaches zero in the limit␾b,␾w→ ±␲,⫿␲ provided I共␾兲 stays finite. As discussed, this divergence guarantees the uni-versality of the correlations of transmission eigenvalues.

Below we specify to three different cases.

1. Symmetric setup

If we set SR=SL, ␹共␾b共w兲兲 is zero regardless of the con-crete form SL. The total action therefore reads S共␾兲 = 2SL共␾/ 2兲. The eigenvalues 共77兲 take a simpler form. To

compute the weak-localization correction to the action we set

␾b_{= −␾}w_{=␾to find} Swl= 1 2 ln

冉

I

⬘

共␾兲 I共␾兲 tan ␾ 2

冊

+ const. 共78兲

For tunnel junctions, I共␾兲⬀sin共␾/ 2兲 and the correction dis-appears.

Expanding Eq.共78兲 in Taylor series near␾→0 we

repro-duce the well-known result of Refs.13and34for the weak-localization correction to the conductance. To comply with FIG. 9. The simplest possible circuit comprises one node, two

the notations used there, we characterize connectors with tp =兺nTn p : ␦Gwl GQ = − t2 4t1 . 共79兲

A similar expansion of the diffusion and Cooperon eigenval-ues reproduces the result of Refs.13and34for conductance fluctuations, 具共␦G兲2典 GQ =3t2 2 + 2t₁2− 2t1共t2+ t3兲 8t₁2 . 共80兲 2. Diffusive connectors

It is instructive for understanding the circuit theory of GQ corrections to specify the relation 共78兲 to diffusive

connec-tors. Since in this case I共␾兲⬀␾, we obtain

Swl,node= 1

2 ln关tan共␾/2兲/␾兴. 共81兲 A two-connector, single-node situation can be easily realized in a quasi-one-dimensional wire with inhomogeneous resis-tivity distribution along the wire. A low-resisresis-tivity region would make a node if bounded by two shorter resistive re-gions that would make the connectors. On the other hand, it has been proven in Ref.27that the weak-localization correc-tion in inhomogeneous wires does not depend on the resis-tivity distribution. Therefore, it has to be universally given by Eq.共62兲, Swl,1d=共1/2兲ln共␾/ sin␾兲⫽Swl,node. How to un-derstand this apparent discrepancy?

This illustrates a very general point: GQcorrections may be accumulated at various space scales ranging from the mean free path to sample size. The experimental observation of the corrections relies on the ability to separate the contri-butions coming from different scales—e.g., by changing the magnetic field.4 _{With our approach, we evaluate the part}

coming from interference at the scale of the node. The part coming from interference at a shorter scale associated with the connectors is assumed to be included in the transmission distribution of these connectors.

For our particular setup, this extra contribution comes from two identical connectors. Since only half of the phase␾ drops at each connector, the contribution equals 2Swl,1d共␾/ 2兲. Summing up both contributions, we obtain

Swl,node+ 2Swl,1d共␾/2兲 = Swl,tot=Swl,1d共␾兲.

That is, the weak localization correction in this case remains universal provided the contribution of the node is augmented by the contributions of two connectors.

3. Nonideal quantum point contact

The transmission distribution of an ideal multimode quan-tum point contact共QPC兲 with conductance GBⰇGQis very degenerate since all Tn= 1 or 0. This degeneracy is lifted if the QPC is adjacent to a disordered region, even if the scat-tering in this region is weak. This can be modeled as a con-nector with conductance GDⰇGBin series.

The weak-localization correction to the conductance was calculated a while ago.35_{In the relevant limit, it is}

parametri-cally small in comparison with GQ,␦Gwl= −GQ共GB/ GD兲. The usual way to verify the applicability of the semiclassical ap-proach to quantum transport is to compare the conductance of a nanostructure with the weak-localization correction to it. For generic nanostructure, this gives GⰇGQ. However, for our particular example ␦GwlⰆGB even for a few-channel QPC where GB⯝GQ. So the question is, is semiclassical approach really valid at GB⯝GQ?

To answer this question, we compute the weak-localization correction to the transmission distribution. Since the system is not symmetric, we make use of the full expres-sion共77兲. In the limit of GDⰇGB, the relevant values of␾ are close to ␲. We stress it by shifting the phase ␮=␲ −␾,兩␮兩 Ⰶ1.

The circuit-theory analysis in the semiclassical limit gives36 I共␮兲 = GD

冉

− ␮ 2 +

冑

␮2 4 + Rc

冊

, ␹= ␲ 2 +

冑

␮2 4 + Rc, where Rc⬅4GB/ GDⰆ1. This gives the following distribu-tion of reflecdistribu-tion coefficients关cf. Eq. 共47兲兴:

␳共R兲 = GD 2␲GQ

⌰共Rc− R兲

冑

Rc

R − 1.

We use the above relations with Eq.共77兲 to find the

Coop-eron eigenvalues, M+共␮兲 = − GD

冢

−1 2+ ␮/4

冑

␮2 4 + Rc

冣

␮2_{/4 + R} c Rc/2 , 共82兲 M−共␮兲 = GD

冉

− ␮ 2 +

冑

␮2 4 + Rc

冊

␮ 2Rc . 共83兲

This yields the weak-localization correction to the current,

Iwl共␮兲 =

2Rc

␮共␮2_{+ 4R} c兲

.

The resulting correction to the transmission distribution con-sists of two ␦-functional peaks of opposite sign, those lo-cated at the edges of the semiclassical distribution,

␦␳wl共R兲 = 1

4关␦共R − Rc兲 −␦共R兲兴. 共84兲 To estimate the conditions of applicability, we smooth the correction at the scale of Rc. This gives兩␦␳兩 /␳⯝GQ/ GBand the semiclassical approach does not work at GB⯝GQ. This agrees with RMT arguments given in Ref.36. The correction to the conductance calculated with Eq.共84兲 agrees with the

VIII. CONCLUSIONS

We present a finite-element method to evaluate quantum corrections—typically of the order of GQ—to transport char-acteristics of arbitrary nanostructures. This includes univer-sal conductance fluctuation and weak localization. We work with matrix Green’s functions of arbitrary structure to treat a wider class of problems that includes superconductivity, full counting statistics, and nonequlibrium transport. At the mi-croscopic level, the corrections are expressed in terms of diffuson and Cooperon modes of continuous Green’s func-tions. We employ a variational method based on an action to formulate a consistent finite-element approach.

We illustrate the method with a set of simple and physi-cally interesting examples. All examples are based on 2⫻2 matrices; this suffices to calculate the transmission distribu-tion of two-terminal structures. We show how a chain of tunnel junctions approaches the diffusive wire upon increas-ing the number of junctions and study transitions between ideal RMT ensembles in the chain. We consider the simplest finite-element system that exhibits Ahronov-Bohm effect. We obtain general results for a single-node system with two ar-bitrary connectors and check their consistency, with the well-known results for quantum cavity. This allows us to improve our understanding of quantum interference in inhomoge-neous diffusive wires and nonideal quantum point contacts.

ACKNOWLEDGMENTS

We appreciate useful discussions with Ya. M. Blanter, P.W. Brouwer, and M.R. Zirnbauer. This work was supported by the Netherlands Foundation for Fundamental Research on Matter共FOM兲.

APPENDIX: JUNCTION CHAIN

In this appendix we illustrate how to find the eigenvalues of the matrix M defined in Sec. IV for the chain of tunnel junctions introduced. We consider the action in the rotated basis presented in the text

S = GT 4GQ

兺

i=0 N TrGˇiLˇGˇi+1Lˇ−1− i␲ ␦s

兺

i=1 N−1 Tr⌺ˇiGˇi. 共A1兲 We expand the Green’s matrices according to Eq.共30兲

The second-order correction to the action reads共the first order is zero because we perform the expansion around the stationary point兲 S共2兲₌ GT 4GQ

兺

i=0 N Tr

再

gˇiLˇgˇi+1Lˇ−1− 1 2gˇiLˇG ˇ i+1 0 _Lˇ−1 −1 2G ˇ i 0_Lˇgˇ i+1Lˇ−1

冎

− i␲ ␦s

兺

i=1 N−1 Tr

再

␦⌺ˇigˇi− 1 2⌺ˇigˇi 2_Gˇ i 0

冎

_. 共A2兲 Taking the variation respect to g_p,kwb and g_m,kwb leads to the eigenvalue equations the matrix M:

sin ␾ b 2Nsin ␾w 2N共gp,k+1 wb + g_p,k−1wb 兲 + cos␾ b 2Ncos ␾w 2N共gm,k+1 wb + g_m,k−1wb 兲 −

冉

cos␾ b N + cos ␾w N

冊

gm,k wb − 1 2␰共⌺ b −⌺w兲g_m,kwb −4GQ GT Mg_m,kwb = 0, 共A3兲 sin ␾ b 2Nsin ␾w 2N共gm,k+1 wb + gm,k−1 wb _{兲 + cos} ␾ b 2Ncos ␾w 2N共gp,k+1 wb + g_p,k−1wb 兲 −

冉

cos␾ b N + cos ␾w N

冊

gp,k wb + 1 2␰共⌺ b −⌺w兲g_p,kwb −4GQ GT Mg_p,kwb= 0, 共A4兲

where ␰= i␦sGT/共4␲GQ兲. To solve the system of coupled equations it is convenient to write

g_p,kwb= gp⬎eiqk+ gp⬍e−iqk 共A5兲 and a similar equation for the m component. The coefficient

q is to be determined from the boundary conditions g_p,0wb= 0,

g_p,Nwb = 0.

From the boundary conditions we have gk⬎= −gk⬍ and q =␲l / N with l = 1 , . . . . , N − 1. We substitute the expressions

共A5兲 into the equations for the eigenvalues and get

4GQ GT Ml ±_共␾b ,␾w兲 = 2 cos ␾ b 2Ncos ␾w 2Ncos

冉

␲l N

冊

− cos␾ b N − cos ␾w N ⫿

冑

4 sin2 ␾ b 2Nsin 2 ␾ w 2 cos 2

冉

␲l N

冊

+⑀ 2_, 共A6兲 the quantity⑀being共⌺b_−⌺w_兲/2␰.

Here we report the correction to the action when decoher-ence and spin-orbit are present as generalization of Eq.共61兲.

Let us define the functions