Integrable theory of quantum transport in chaotic cavities
The problem of quantum transport in chaotic cavities with broken time-reversal symmetry is shown to be completely integrable in the universal limit. This observation is utilised to determine the cumulants and the distribution function of conductance …
Authors: Vladimir Al. Osipov, Eugene Kanzieper
In tegrable theory of quan tum transport in c haotic ca vities Vladimir Al. Osip ov 1 , 2 and Eug ene K anziep er 1 1 Dep art ment of Applie d Mathematics, H. I.T.—Holon Institute of T e chnolo gy, Holon 58102, I sr ael 2 F achb er eich Physik, Universit¨ at Duisbur g-Essen, D-47057 Duisbur g, Germany (Dated: June 17, 2008) The p roblem of qu an tum transp ort in c haotic ca v ities with brok en time-reversal symmetry is sho wn to b e completely integra ble in the universal limit. This observ ation is utilised to determine the cumulan ts and the distribution function of conductance for a cavit y with ideal leads su p p orting an arbitrary num b er n of propagating mo des. Expressed in terms of solutions to the fifth Pa inlev´ e transcendent and/or th e T oda lattice equation, the condu ct ance d istribution is further analysed in the large- n limit that reveals long exp onential tails in t he otherwise Gaussian curve. P ACS n umbers: 73.23.–b, 05.45.M t, 02.30.Ik Intr o duction. —The low temper ature e le ctronic co n- duction through a ca vity exhibiting chaotic classical dy- namics is governed b y qua n tum phas e-coherence effects [1, 2]. In the absence of electr on-electron interactions [3, 4, 5], the most comprehensive theore tical framew ork by which the phase coherent electron tra nspo rt can b e explored is provided by the scattering S -matrix approach pioneered by Landauer [6]. There ex ist tw o different, though mutually overlapping, scatter ing -matrix descrip- tions [7 ] of qua n tum transp ort. A semiclassical formulation [8 ] of the S -matrix ap- proach is tailor-made to the analysis of energy-averaged charge co nduction [9] through a n individual cavit y . Rep- resenting quan tum transpo r t observ ables (such as con- ductance, shot-noise p ow er, transferred ch arg e etc.) in terms o f cla ssical tra jectories connecting the leads a t- tached to a ca vity , the s emiclassical approach [10] effi- ciently acco un ts for sys tem-spec ific features [1 1] o f the quantum transp ort. Besides, it also covers the long- time scale univ ersal transpo rt regime [12] emerging in the limit [13] τ D ≫ τ E , where τ D is the average electron dwell time and τ E is the Ehr e nfest time (the time scale where quan- tum e ffects set in). The latter universal r e gime [14] can alter na tiv ely be studied w ithin a sto chastic approach [4, 15] based on a random matrix description [16] of electron dynamics in a cavit y . Modelling a single electron Hamiltonian by an M × M random matrix H of prop er s ymmetry , the sto chastic appro ach starts with the Hamiltonian H tot of the total system compris e d by the cavit y and the lea ds: H tot = M X k,ℓ =1 ψ † k H kℓ ψ ℓ + N L + N R X α =1 χ † α ε F χ α + M X k =1 N L + N R X α =1 ψ † k W kα χ α + χ † α W ∗ kα ψ k . (1) Here, ψ k and χ α are the annihilation opera tors o f elec- trons in the cavit y and in the lea ds, r espe c tively . In- dices k and ℓ enumerate electr on states in the c avity: 1 ≤ k , ℓ ≤ M , with M → ∞ . Index α coun ts propa- gating mo des in the left (1 ≤ α ≤ N L ) and the right ( N L + 1 ≤ α ≤ N ) le a d. The M × N matrix W describ es the coupling of electro n s tates with the F ermi energy ε F in the cavit y to those in the leads; N = N L + N R is the total num ber of pro pa gating mo des (channels). Since in Landauer-type theor ies the trans p or t obs erv ables ar e ex- pressed in ter ms of the N × N sca ttering matrix [5] S ( ε F ) = 1 1 − 2 iπ W † ( ε F − H + iπ WW † ) − 1 W , (2) the knowledge of its distribution is central to the sto chas- tic approach. (Tw o suc h observ ables – the conductance G = tr ( C 1 SC 2 S † ) and the shot noise p ow er P = tr ( C 1 SC 2 S † ) − tr ( C 1 SC 2 S † ) 2 measured in prop er dimen- sionless units [4] – ar e o f most interest. Here, C 1 = diag( 1 1 N L , 0 N R ) and C 2 = diag(0 N L , 1 1 N R ) are the pr o- jection matrices.) F or random ma trices H drawn from r otationally inv ari- ant Ga us sian ensembles [17], the distribution of S ( ε F ) is describ ed [1 5] by the Poisson kernel [18, 19, 2 0] P ( S ) ∝ det 1 1 − ¯ SS † det 1 1 − S ¯ S † β / 2 − 1 − β N / 2 . (3) Here, β is the Dyson index [17] a ccommo dating sys- tem symmetries ( β = 1 , 2 , and 4) whilst ¯ S is the av- erage scattering ma tr ix [4], ¯ S = V † diag( p 1 − Γ j ) V , that c hara c ter ises co uplings betw een the cavit y and the leads in terms of tunnel probabilities [21] Γ j of j -th mo de in the leads (1 ≤ j ≤ N ); the matrix V is V ∈ G ( N ) /G ( N L ) × G ( N R ) where G stands for orthog- onal ( β = 1), unitary ( β = 2) or symplectic ( β = 4) group. The ab ov e description be c omes pa rticularly simple for chaotic cavities that coupled to the leads throug h ballis- tic p oint contacts (“ideal” leads, Γ j = 1). Indeed, unifor - mit y of P ( S ) ov er G ( N ) implies that scatter ing ma trices S b elong [22] to one of the three Dyson c ir cular ensembles [17] ab out whic h virtually everything is known. Not with- standing this remark a ble simplicity , av a ilable analytic r e - sults for statistics of electron tr ansp ort are quite limited [4, 23]. In particular, distribution functions of co nduc- tance and shot nois e power, a s well a s their higher o rder cum ulants, are largely unknown for an arbitr ary num ber 2 of propag ating mo des, N L and N R , and th us do no t c a tc h up with existing exp erimental capabilities [24]. In this Letter, w e com bine a stochastic version of the S -matrix approach with ideas o f integrability [25, 26] to show that the pro blem of universal quantum transp ort in chaotic cavities with broken time-reversal symmetry ( β = 2) is co mpletely integrable. Although o ur theory applies [27] to a v ariety of tr ansp ort observ ables, the fur- ther discussio n is purp ose ly res tricted to the statistics o f Landauer conductance. This will help us keep the pre- sentation as transpar en t as p ossible. Conductanc e distribution. —In order to des c ribe fluc- tuations of the conductance G = tr ( C 1 SC 2 S † ) in an ade- quate wa y , one needs to know its entire distribution func- tion. T o determine the latter, we define the moment gen- erating function F n ( z ) = h exp ( − z G ) i S ∈ CUE(2 n + ν ) (4) which, in a ccordance with the ab ov e discussion, involv es av eraging over scattering matrice s S ∈ CUE(2 n + ν ) drawn from the Dyson circ ula r unitary ensem ble [1 7]. F or the sake o f conv enience, we hav e intro duced the notation n = min( N L , N R ) and ν = | N L − N R | so tha t the to- tal num b er N L + N R of pro pagating mo des in t wo leads equals 2 n + ν . While the averaging in Eq. (4) can explicitly b e p er- formed with the help of the Itzykson-Zub er formula [28], a hig h sp ectral degener acy o f the pro jection ma tr ices C 1 and C 2 makes this calculation quite tedious. T o av oid unnecessary technical complications , it is bene fic ia l to employ a p ola r decompo s ition [19] of the scattering ma- trix. This bring s into play a set of n transmissio n eigen- v alues T = ( T 1 , · · · , T n ) ∈ (0 , 1) n which characterise the conductance [6] in a particularly simple manner, G ( T ) = P n j =1 T j . The uniformity of the sca ttering S - ma trix distr ibution gives rise to a nontrivial joint probability density function of transmissio n eigenv alues in the for m [29, 30] P n ( T ) = c − 1 n ∆ 2 n ( T ) n Y j =1 T ν j . (5) Here ∆ n ( T ) = Q j
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