Staggered repulsion of transmission eigenvalues in symmetric open mesoscopic systems
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A ug Staggered repulsion of transmission eigenvaluesin symmetric open mesoscopic systems
Marten Kopp and Henning Schomerus
Department of Physics, Lancaster University, Lancaster, LA1 4YB, United Kingdom
Stefan Rotter
Department of Applied Physics, Yale University, New Haven, CT 06520, USA (Dated: October 24, 2018)Quantum systems with discrete symmetries can usually be desymmetrized, but this strategyfails when considering transport in open systems with a symmetry that maps different openingsonto each other. We investigate the joint probability density of transmission eigenvalues for suchsystems in random-matrix theory. In the orthogonal symmetry class we show that the eigenvaluestatistics manifests level repulsion between only every second transmission eigenvalue. This finds itsnatural statistical interpretation as a staggered superposition of two eigenvalue sequences. For a largenumber of channels, the statistics for a system with a lead-transposing symmetry approaches that ofa superposition of two uncorrelated sets of eigenvalues as in systems with a lead-preserving symmetry(which can be desymmetrized). These predictions are confirmed by numerical computations of thetransmission-eigenvalue spacing distribution for quantum billiards and for the open kicked rotator.
PACS numbers: 05.45.Mt, 05.60.Gg, 73.23.-b
I. INTRODUCTION
Mesoscopic systems exhibit variations in their phase-coherent electronic transport properties that are con-veniently characterized via statistical approaches. Ge-ometries that classically give rise to chaotic motion typ-ically display universal fluctuations which can be cap-tured using ensembles of random scattering matrices [1].For normal conductors the universal properties fall intoDyson’s three universality classes with symmetry index β = 1 , , T n . Theseeigenvalues determine fundamental transport propertiessuch as the conductance G or the shot-noise Fano fac-tor F [1, 4]. In the Dyson ensembles, the probabilitydensity to find two closely spaced adjacent transmissioneigenvalues with small distance s = T n +1 − T n is sup-pressed as P ( s ) ∝ s β [5, 6]. This introduces a stiffnessin the transmission-eigenvalue sequence which suppressesthe fluctuations of the conductance and of the Fano factorwhen compared to the case of uncorrelated transmissioneigenvalues (the latter being characteristic for classicallyintegrable systems with a complete set of good quantumnumbers) [1, 4].From the investigation of closed systems it is wellknown that discrete symmetries result in a reduction oflevel repulsion. In such systems, desymmetrization deliv-ers independent variants of the system which differ by theboundary conditions on the symmetry lines (e.g., Dirich-let and Neumann boundary conditions for eigenfunctionsof odd and even parity, respectively). The statistics of thedesymmetrized versions can depend on the dimensional-ity of the irreducible representation [7], but still remain within the conventional universality classes. The com-bined level statistics is then built by superimposing theindependent level sequences of the desymmetrized vari-ants [5]. In open systems, this concept of desymmetriza-tion can be directly applied as long as the symmetry inquestion preserves the shape and position of the leads[8, 9].This paper is motivated by the observation that sys-tems with a lead-transposing symmetry (which maps dif-ferent openings onto each other while leaving the dynam-ics in the system unchanged) exhibit transport propertiesthat can only be understood as collective features of thedesymmetrized variants of the system [8, 9]. An obvi-ous indication of this complication is the fact that thesymmetry-reduced variants only possess a reduced num-ber of leads (we concentrate on systems with two leads,for which the desymmetrized variants only possess a sin-gle lead). We demonstrate that such systems exhibit nev-ertheless a reduced repulsion of transmission eigenvalueswhich is similar to that for systems with a lead-preservingsymmetry. For a large number N of transport chan-nels, the local statistical fluctuations in the eigenvaluesequence indeed become indistinguishable for both typesof symmetry. However, for a small numbers of channels,the statistics differ from each other, which can be tracedback to the absence or presence of 1 /N corrections inthese ensembles.In the specific case of β = 1, we derive exact closed ex-pressions for the joint probability density of transmissioneigenvalues thereby gaining detailed insight into thesestatistical features. In particular, we find for both thelead-preserving and the lead-transposing symmetry classthat level repulsion occurs only between every secondtransmission eigenvalue. The fluctuations in the trans-mission eigenvalue sequence hence find their most natu-ral statistical interpretation in a staggered superpositionof two independent level sequences. In such a superposi-tion, the transmission eigenvalues alternate between thetwo sequences when they are ordered by magnitude.The exact expressions for the joint probability densitywith β = 1 are different for the two types of symme-try. Hence, the details of the transport statistics for alead-transposing symmetry deviate from those for a lead-preserving symmetry. We show that these deviations aremost significant for a small number of channels, while fora large number of channels the local eigenvalue statisticsdo indeed converge onto each other.Previous studies of open systems with lead-transposingor lead-preserving symmetries have derived the distribu-tion of transmission eigenvalues for one or two open chan-nels and the one-point density for arbitrary numbers ofchannels [8, 9, 10, 11]. For time-reversal symmetric sys-tems with β = 1, a key observation of these works was anenhancement of universal fluctuations for both types ofsymmetry (when compared to asymmetric systems). Forsystems with a lead-transposing symmetry it was foundthat the weak localization correction is vanishing, lead-ing to ensemble averaged expressions for the conductanceand for the shot noise Fano factor which are entirely in-dependent of the channel number N [11]. The underlyingstaggered level statistics embodied in the joint distribu-tion of transmission eigenvalues provides a unifying ex-planation for all of these observations. We verify ourpredictions by numerical computations for quantum bil-liards [12, 13] and for the open kicked rotator [14, 15, 16].This paper is organized as follows. Section II providesbackground information on the scattering approach totransport and on standard random-matrix theory. In Sec.III we revisit the case of systems with a lead-preservingsymmetry and provide the exact reformulation of theeigenvalue statistics in the orthogonal symmetry class( β = 1) as a staggered superposition of two eigenvaluesequences. Section IV concerns systems with a lead-transposing symmetry. In particular, for β = 1 we derivethe exact joint probability density of transmission eigen-values for arbitrary N , and show that this again takes theform of a staggered eigenvalue sequence. We also describethe convergence of the local statistics for both types ofsymmetry, which emerges in the limit N → ∞ . SectionV provides numerical results that illustrate the similari-ties and differences of the random-matrix ensembles forthe two symmetry classes. This section also contains thecomparison to specific model systems. Section VI pro-vides a summary and discussion of our main results. II. BASIC CONCEPTSA. Scattering approach to transport
Figure 1 depicts open two-dimensional quantum bil-liards representing mesoscopic systems with two attachedleads (L – left and R – right), each carrying N in-coming and N outgoing modes. The systems in Fig. ( a ) ( b )( e ) ( f )( d )( c )L R FIG. 1: (Color online) Sketches of quantum billiards (a,b)without any spatial symmetry, (c) with a lead-preservingsymmetry, (d) with both a lead-preserving as well as a lead-transposing symmetry, (e) with a lead-transposing reflectionsymmetry, and (f) with a lead-transposing inversion symme-try. The inversion symmetry in panel (f) survives in the pres-ence of a finite magnetic field, as is indicated by a symmetricpair of trajectories. V , one solvesthe Schr¨odinger equation for fixed values of the 2 N am-plitudes a = [ a ( L ) n , a ( R ) n ] T in the incoming modes. Thisresults in linear relations b = S a for the 2 N amplitudes b = [ b ( L ) n , b ( R ) n ] T in the outgoing modes, which delivers a2 N × N -dimensional scattering matrix of the form S = (cid:18) r t ′ t r ′ (cid:19) . (1)Here r, r ′ , t, t ′ are N × N -dimensional matrices describingreflection at each lead and transmission from one lead tothe other, respectively.The scattering matrix is unitary, and its structure isfurther constrained by symmetries of the system. Thethree main universality classes arise for systems withtime-reversal and spin-rotation symmetry (orthogonalsymmetry class with S = S T , symmetry index β = 1),systems without time-reversal symmetry (unitary sym-metry class with no constraints on S , β = 2), and sys-tems with time-reversal but broken spin-rotation sym-metry (symplectic symmetry class composed of self-dualmatrices S = S R , β = 4). Spatial symmetries entail ad-ditional constraints on the scattering matrix, which aredetailed in Secs. III and IV.The transmission eigenvalues T n are defined as theeigenvalues of the hermitian matrix tt † . In the case ofspin-independent transport or Kramers degeneracy (thelatter occurs for β = 4), the transmission eigenvalues aretwofold degenerate. We then only account for each pairof eigenvalues once and introduce a spin-degeneracy fac-tor α = 2. When the two-fold degeneracy is lifted then α = 1. From here on, N refers to the number of distincttransmission eigenvalues (ignoring accidental degenera-cies). Furthermore we will assume that the transmissioneigenvalues are ordered by magnitude, T ≤ T ≤ T ≤ . . . ≤ T N , (2)as this results in a number of technical simplifications.The conductance quantum is defined as G = αe /h .With these conventions, the transmission eigenvaluesdetermine fundamental transport properties such as theconductance via G = G N X n =1 T n (3)and the shot-noise power via P = 2 G eV N X n =1 T n (1 − T n ) . (4)Here V is the bias voltage, which is assumed to be small. B. Dyson’s circular ensembles
Random-matrix theory delivers a statistical descrip-tion of transport by drawing the scattering matrices fromensembles of unitary matrices which obey the constraintsof the given universality class. For the three main univer-sality classes with β = 1, 2, or 4, random-matrix theoryis based on Dyson’s circular ensembles, for which theprobability measure is given by the Haar measure of uni-tary symmetric, unitary, or unitary self-dual matrices,respectively. The joint probability density of transmis-sion eigenvalues then takes the form [1] P ( { T n } ) ∝ Y m>n ( T m − T n ) β Y l T − β/ l . (5)The first product in Eq. (5) involves pairs of transmis-sion eigenvalues and favors sequences in which neighbor-ing transmission eigenvalues do not approach each otherclosely. (As we have ordered the transmission eigenval-ues by magnitude, all differences T m − T n are positive.)This suppresses fluctuations in the eigenvalue sequence and ultimately results in conductance fluctuations of theorder of a single conductance quantum, which for large N approach the asymptotic valuevar G/G = 18 β . (6)For large N , the one-point probability density of trans-mission eigenvalues approaches P ( T ) = 1 π p T (1 − T ) . (7)The second product in Eq. (5) induces an asymmetry intothis bi-modal distribution, which for large N results inthe weak-localization correction h G i − N G = G (cid:18) − β (cid:19) (8)of the ensemble-averaged conductance.An insightful quantity derived from the joint proba-bility density P ( { T n } ) is the distribution P ( s ) of spac-ings s = T n +1 − T n between neighboring transmissioneigenvalues. For uncorrelated eigenvalues with averagespacing ¯ s one would expect a Poisson distribution, P ( s ) = ¯ s − e − s/ ¯ s , (9)while for the circular ensembles and N ≫
1, the spac-ing distribution can be well approximated by the Wignerdistributions [5, 6], P ( s ) = π s s exp (cid:16) − πs s (cid:17) β = 1 π ¯ s s exp (cid:16) − s π ¯ s (cid:17) β = 2 π ¯ s s exp (cid:16) − s π ¯ s (cid:17) β = 4 . (10)Lead-preserving and lead-transposing symmetries en-tail further constraints on the scattering matrix. Theconsequences of these constraints for the transmissioneigenvalue statistics are explored in the remainder of thispaper. III. LEAD-PRESERVING SYMMETRIES
A useful reference point for our subsequent investiga-tion of systems with a lead-transposing symmetry (in Sec.IV) are open systems with a lead- preserving symmetry,to which one can directly apply the standard ideas ofdesymmetrization. The goal of the present section is toreformulate the resulting random-matrix statistics for thecase of a lead-preserving symmetry with β = 1 as a stag-gered level repulsion, as this will allow us to establish aconnection to the case of a lead-transposing symmetry. ( a ) ( b ) D i r i c h l e t N e u m a n n ( c )( d ) T n T n o d d p a r i t ye v e n p a r i t ye v e n i n d e xo d d i n d e x FIG. 2: (Color online) (a,b) Desymmetrization of the quan-tum billiard with a lead-preserving reflection symmetry,shown in Fig. 1(c). (c) Sketch of the individual transmis-sion eigenvalue sequences of fixed parity. (d) Reorganizationas a staggered level sequence, where transmission eigenvaluesalternate after ordering by magnitude.
A. Constraints on the scattering matrix
An example of a system with a lead-preserving reflec-tion symmetry is shown in Fig. 1 (c). Figure 2 (a,b)shows the desymmetrized version of the system, whichis halved at the symmetry line. Dirichlet boundary con-ditions on the line of symmetry select scattering wavefunctions with an odd parity, while Neumann boundaryconditions yield even parity. Consequently, the transmis-sion matrix t assumes a block structure where each blockcorresponds to a given parity. As dictated by the one-dimensional transverse-mode quantization in the leads,the block of even parity has dimension N ≡ [( N + 1) / N ≡ [ N/ · ] denotes the integer part of a number). Hence,both blocks have either the same size (when N = N + N is even), or the block with even parity is by one largerthan the block with odd parity (when N is odd).The total transmission-eigenvalue sequence is thereforeobtained from a superposition of two sequences of size N and N [for illustration see Fig. 2(c)]. In order tofix the way we address the elements of this superposi-tion, we impose the ordering of Eq. (2) and denote by P the set of all strictly increasing sequences of indices I n ∈ { , , , . . . , N } , where each sequence is of length N . Such sequences are of the form I = ( I , I , . . . , I N ),where 1 ≤ I < I < I < . . . < I N ≤ N . Foreach sequence we also define a complementary sequence¯ I = ( ¯ I , ¯ I , ¯ I , . . . , ¯ I N ), which consists of the indices1 ≤ ¯ I < ¯ I < ¯ I < . . . < ¯ I N ≤ N not contained in I . This partition delivers two ordered subsequences T I n and T ¯ I n . B. Conventional random-matrix theory
Within random-matrix theory, the joint probabilitydistribution of the total transmission-eigenvalue sequenceis the sum of the corresponding probabilities for each wayto distribute the transmission eigenvalues into two setscontaining N and N eigenvalues. With each sequenceobeying the statistics of the appropriate Dyson ensembleone finds with Eq. (5) P ( { T n } ) ∝ X I ∈P Y m>n ( T I m − T I n ) β Y m>n (cid:0) T ¯ I m − T ¯ I n (cid:1) β × N Y l =1 T − β/ l . (11)For large N , the separation into two effectively inde-pendent systems with Dirichlet and Neumann boundaryconditions naturally results in a doubling of the con-ductance fluctuations (6) and a doubling of the weak-localization correction (8). Moreover, level repulsion isonly effective for transmission eigenvalues which are partof the same sequence. This modifies the spacing proba-bility density, which can be calculated from the generalexpression [5] P ( s ) = d ds Y i Z ∞ Z ∞ p i ( ρ i ρ s + y + z ) dydz (12)for multiple sequences i , where p i ( s ) is the spacing proba-bility densities of each sequence, while ρ i ρ is the associatedfractional eigenvalue density.For two sequences following the Wigner distribution(10), the resulting spacing probability densities is (¯ s ≡ P β =1 ( s ) = e − x √ π xe − x E ( x ) , x = √ πs P β =2 ( s ) = 6 x e − x π + 2 x − x √ π e − x E ( x ) + E ( x )2 ,x = s √ π (13b) P β =4 ( s ) = x √ π (6 + 4 x − x ) e − x E ( x ) + E ( x )2+ 2 x π (9 + 28 x + 8 x ) e − x , x = 4 s √ π , (13c)where E ( x ) = erfc ( x ) denotes the complementary errorfunction. C. β = 1 : Reformulation as a staggered eigenvaluesequence In most situations encountered in random-matrix the-ory, the combinatorial sum over partitions involved in thesuperposition of eigenvalue sequences ([as in Eq. (11)]cannot be performed explicitly. For the specific case β = 1, however, the combinatorial sum over I in Eq.(11) can be carried out (see below), which then yields aclosed-form expression P ( { T n } ) ∝ Y m>n, both odd ( T m − T n ) Y m>n, both even ( T m − T n ) Y l √ T l . (14)(A similar simplification does not present itself in thecases β = 2 and β = 4.) This result finds its nat-ural statistical interpretation as a staggered superposi-tion of two sequences, which is illustrated in Fig. 2(d).In such a superposition, the transmission eigenvalues ineach sequence are not distinguished by the parity ofthe associated wavefunction under the symmetry oper-ation. Instead, the transmission eigenvalues are orderedby magnitude (irrespective of parity), and one sequenceis composed of all odd-indexed transmission eigenvalues(of which there are N ) while the other sequence is com-posed of all even-indexed transmission eigenvalues (ofwhich there are N ). Compared to the original super-position of two independent sequences, this differs by theadditional constraint T I ≤ T ¯ I ≤ T I ≤ T ¯ I ≤ T I ≤ T ¯ I . . . (15)(which is satisfied when all the ordered indices I n are oddwhile the indices ¯ I n are all even).In order to demonstrate the equivalence of Eq. (11)(for β = 1) and Eq. (14) we have to show that the level-repulsion terms are proportional to each other (both ex-pressions share the same product of one-point weights Q l T − / l , and the proportionality constant is fixed bynormalization). We set out to work towards this goal bydefining a matrix M = (cid:18) − v v − v v − v v . . . ( − N v N w w w w w w . . . w N (cid:19) , (16)which is composed of column vectors v n = (1 , T n , T n , . . . , T N − n ) T , (17) w n = (1 , T n , T n , . . . , T N − n ) T . (18)The determinant det M can be evaluated in two dif-ferent ways. In the first way, we expand it in terms ofsubdeterminants with N vectors v n from the first N rows and N vectors w m from the remaining rows. Inother words, we sum over all determinants of the formdet (cid:18) − v v v . . . w w w . . . (cid:19) , (19)etc., where the indices of the vectors v I n form an or-dered subsequence I and the indices of the vectors w ¯ I n are given by the complementary subsequence ¯ I . The al-ternating signs in front of the vectors v I n can be pulledout of the determinant at the cost of an overall fac-tor ( − I + I + ... + I N . Next, we use permutations of neighboring rows to bring all vectors v I n to the left(into row n ). This results in an additional sign factor( − ( I − I − ... +( I N − N ) . The determinant of theresulting block matrix factorizes. Overall, this expansionyieldsdet M = ( − N ( N +1) / × X I ∈P det( v I , v I , . . . , v I N ) det( w ¯ I , w ¯ I , . . . , w ¯ I N ) . (20)Each subdeterminant is of the form of a Vandermondedeterminant, and thereforedet M = ( − N ( N +1) / × X I ∈P Y m>n ( T I m − T I n ) Y m>n (cid:0) T ¯ I m − T ¯ I n (cid:1) . (21)Secondly, the determinant det M can be evaluated byadding in Eq. (16) the first N rows to the last N rows.This yieldsdet M = det (cid:18) − v v − v v − v v . . . w w w . . . (cid:19) . (22)Proceeding again with the evaluation of subdeterminantswe are left with a single choice, namely, to select vectors v n with odd index and vectors w n with even index. Ac-counting for all signs and now also factors of two, thisresults indet M = ( − N ( N +1) / N × det( v , v , v , . . . ) det( w , w , w , . . . ) . (23)As this again involves Vandermonde determinants, wefinddet M = ( − N ( N +1) / N × Y m>n, both odd ( T m − T n ) Y m>n, both even ( T m − T n ) . (24)The two results Eqs. (21) and (24) deliver the remark-able identity X I ∈P Y m>n ( T I m − T I n ) Y m>n (cid:0) T ¯ I m − T ¯ I n (cid:1) = 2 N Y m>n, both odd ( T m − T n ) Y m>n, both even ( T m − T n ) . (25)An equivalent identity has been derived for superposi-tions of energy eigenvalue sequences with a length differ-ence of at most one which are distributed according to theGaussian orthogonal ensemble [17]. Relation (25) showsthat the level-repulsion term in Eq. (11) is indeed propor-tional to the level-repulsion term in Eq. (14). As alreadymentioned, the one-point product Q l T − / l in both ex-pressions is identical, and the proportionality constant is Dirichlet Neumann S - S + ( a ) ( b )( c ) T n u p p e r a r c l o w e r a r c ( d ) Q FIG. 3: (Color online) (a,b) Desymmetrization of the quan-tum billiard with a lead-transposing reflection symmetry,shown in Fig. 1(e). (c,d) Eigenphases Θ n of the matrix Q on the unit circle, and their projection Eq. (28) which deliv-ers the transmission eigenvalues. fixed by normalization. It follows that for β = 1, the in-dependent superposition of two transmission-eigenvaluesequences with N and N levels (with N and N con-strained to differ at most by one) is identical to a stag-gered superposition of two transmission-eigenvalue se-quences with N and N levels, which are correlated bythe ordering requirement (15). IV. LEAD-TRANSPOSING SYMMETRIES
Systems with a lead-transposing symmetry require aseparate treatment since the symmetry operation onlycommutes with the Hamiltonian, but not with the cur-rent operator (which changes its sign). In the presenceof an applied bias, the symmetry operation exchangesthe electronic source and drain reservoirs. An obvioussymptom of this complication is the fact that the desym-metrized system only possesses a single lead (see Fig.3). Mathematically, the transmission matrix does notassume a block structure but remains full. We will firstadapt the concept of desymmetrization to derive the con-straints of the scattering matrix, and then turn to thejoint probability density of the transmission eigenvaluesin random-matrix theory. Just as in the previous section,we then focus on the orthogonal symmetry class ( β = 1)and derive a closed expression for the joint probabilitydensity, which again assumes the form of a staggered levelrepulsion. A. Constraints on the scattering matrix
The presence of a lead-transposing symmetry immedi-ately results in the constraint r = r ′ , t = t ′ (when time-reversal symmetry is broken by a magnetic field, this canbe achieved by an inversion symmetry but not by a re-flection symmetry). In order to further exploit the con-sequences of the symmetry, let us inspect a time-reversalsymmetric system with a reflection symmetry, as shownin Fig. 1(e). As shown in Fig. 3(a,b), the desymmetrizedversions are cut at the symmetry line, where they areequipped with Dirichlet or Neumann boundary condi-tions for wavefunctions of odd ( − ) or even parity (+),respectively. Such wave functions are readily constructedstarting from the original system when one chooses in-coming amplitudes of the form a ( R ) = ± a ( L ) . The outgo-ing amplitudes are then given by b ( L ) = ( r ± t ) a ( L ) . Con-sequently, the scattering matrices of the desymmetrizedsystems are given by S ± = r ± t. (26)The desymmetrized systems only possess a single open-ing. In order to revert to the scattering matrix of theoriginal system we invert Eq. (26). The transport in theoriginal system is therefore described by the transmissionmatrix t = ( S + − S − ), which gives tt † = 14 (2 − S + S †− − S − S † + ) . (27)The properties of this matrix—and especially, of itseigenvalues T n —are not separable and depend on the in-terplay of both desymmetrized variants. B. Conventional random-matrix theory
Random-matrix ensembles for systems with lead-transposing symmetry can be obtained by assumingthat the scattering matrices S + and S − of the desym-metrized variants are statistically independent realiza-tions of the appropriate standard circular ensemble. Theresulting ensembles are identical to those introduced byBaranger and Mello [9], who based their considerationson a maximal-entropy principle.Earlier works have addressed isolated aspects of theseensembles, but not the complete transmission-eigenvaluestatistics. For instance, it has been observed that a lead-transposing symmetry increases the conductance fluctu-ations [8, 11] but eliminates the weak-localization cor-rection [11]. For large N , the conductance fluctuationsdouble, just as is the case for lead-preserving symmetries.We now provide a complete explanation of these observa-tions on the basis of the joint probability density of thetransmission eigenvalues.The starting point of these considerations is the rela-tion T n = sin (Θ n /
2) = 12 (1 − cos Θ n ) (28)between the transmission eigenvalues T n and the eigen-phases Θ n of the unitary matrix Q ≡ S + S †− , which fol-lows from Eq. (27). As illustrated in Fig. 3(c,d), thestatistics of transmission eigenvalues is hence directly im-posed by the statistics of the real parts cos Θ n of theunimodular eigenvalues e i Θ n of Q .In random-matrix theory, the eigenphases Θ n followthe statistics of the associated circular ensemble. Thisis evident for the unitary ensemble ( β = 2), which is in-variant under the multiplication of an arbitrary fixed ma-trix (it hence suffices, e.g., to assume that S + is randomwhile S − is fixed, or vice versa). In the orthogonal case( β = 1), the unitary transformation Q ′ = S − / − QS / − = S − / − S + S − / − results in a symmetric matrix with identi-cal eigenvalues. Their circular statistics then follows fromthe fact that the circular orthogonal ensemble is invariantunder the symmetric involution with any fixed symmet-ric matrix (here, S − / − ). The same transformation alsosucceeds in the case of self-dual matrices ( β = 4).Because of the uniform distribution of eigenphases inthe circular ensemble [5], the one-point probability den-sity P ( T n ) is given by Eq. (7) for any finite N (i.e., notonly in the limit N → ∞ ) [11]. The joint probabilitydensity of the eigenphases Θ n is given by [5] P Θ ( { Θ n } ) ∝ Y m>n (cid:20) σ m sin Θ m − Θ n (cid:21) β . (29)Here we ordered the eigenphases by their moduli,0 ≤ | Θ | ≤ | Θ | ≤ | Θ | ≤ . . . ≤ | Θ N | ≤ π, (30)and denoted σ n = sgn Θ n . Since Eq. (28) does not dis-criminate the sign of Θ n we proceed to the distributionof the moduli θ n = | Θ n | , P θ ( { θ n } ) = X { σ n } P Θ ( { σ n θ n } ) . (31)With the help of the relationssin( θ n /
2) = p T n , cos( θ n /
2) = p − T n , (32)and also accounting for the Jacobian dθ n dT n = 1 p T n (1 − T n ) , (33)this yields the joint probability density [11] P ( { T n } ) ∝ Y l p T l (1 − T l ) × X { σ n } Y m>n hp T n (1 − T m ) − σ m σ n p T m (1 − T n ) i β . (34)This expression is symmetric under the replacement T n → − T n , which explains the absence of weak-localization corrections to the conductance. Moreover, transmission eigenvalues do not repel each other when σ n = − σ m , i.e., when the underlying eigenphases Θ n lieon the opposite (upper and lower) arcs of the unit cir-cle [see again Fig. 3(c)]. As the sets of eigenphases onboth arcs is only weakly cross-correlated, this explainsthe doubling of the conductance fluctuations for large N . C. Staggered level repulsion for β = 1 While the general conclusions of the previous sectioncan be drawn for any β , it should be noted that Eq.(34) still involves a combinatorial sum, and hence is sim-ilar in status as expression (11) for systems with a lead-preserving symmetry. We now show that a much moredetailed insight is possible for the orthogonal symmetryclass ( β = 1), where the combinatorial sum in Eq. (34)can be carried out explicitly (see below). The result-ing statistics again assume the form of a staggered levelrepulsion, but are not identical to Eq. (14) (which wasderived from the superposition of two independent levelsequences): For N an odd integer, we find P ( { T n } ) ∝ Y m>n, both odd ( T m − T n ) Y l odd p T l (1 − T l ) × Y m>n, both even ( T m − T n ) , (35a)while for even NP ( { T n } ) ∝ Y m>n, both odd ( T m − T n ) Y l odd √ T l × Y m>n, both even ( T m − T n ) Y l even √ − T l . (35b)Similar to Eq. (14), the joint probability density againseparates into two factors, each involving only every sec-ond eigenvalue. In particular, neighboring levels are notprohibited to approach each other closely, and statisti-cal fluctuations of observables are enhanced, as has beenearlier observed for the conductance and the Fano fac-tor [8, 9, 10, 11]. The correlation between the two levelsequences is again imposed only indirectly by the require-ment that the sequences are staggered. This ordering re-quirement is independent of the parity of the wave func-tion – indeed, in the present case, parity is not well de-fined as the transmission eigenvalues arise from the com-bined properties of S + and S − .In order to derive Eq. (35), let us first inspect Eq. (29).Because of the ordering (30), each factor σ m appears m − P Θ ( { Θ n } ) ∝ Y l even σ l Y m>n sin Θ m − Θ n . (36)We next pass over to the joint distribution of moduli(31). In order to evaluate the combinatorial sum over the σ n we express the factor of sine functions in Eq. (36) asa Vandermonde determinant, Y m>n sin Θ m − Θ n − i ) N ( N − / det B ( { σ n θ n } ) , (37)where B ml ( { Θ n } ) = exp( i Θ m l ), m = 1 , , , . . . , N , whilethe index l runs in integer steps from − ( N − / N − /
2. The multilinearity of the determinant thenyields P θ ( { θ n } ) ∝ ( − i/ N ( N − / det C, (38)where C ml = 2 cos( θ m l ) for odd m and C ml = 2 i sin( θ m l )for even m .For every l > l th column in C to the − l th column, which cancels all sine terms in the lattercolumns. The determinant det C = det D det E then fac-torizes, where D ml = cos θ m l , m odd, and E ml = sin θ m l , m even. If N is even, the index l is now restricted to l = 1 / , / , . . . , ( N − /
2. For odd N , this index isrestricted to l = 0 , , , . . . , ( N − / D ,and to l = 1 , , . . . , ( N − / E .For odd N we can write D ml as a polynomial of de-gree l in cos θ m , and E ml as sin θ m times a polynomial ofdegree l − θ m . We only need to keep the high-est monomial, as the other terms are linear combinationsof the rows of lower index l . This leaves us again withVandermonde determinants,det D ∝ Y m>n, both odd (cos θ n − cos θ m ) , (39)det E ∝ Y l even sin θ l Y m>n, both even (cos θ n − cos θ m ) . (40)For even N , the index l is half-integer, and the elementsof D can now be written as cos( θ m /
2) times a polynomialin cos( θ m ), while those of E can be written as sin( θ m / D ∝ Y l odd cos( θ l / Y m>n, both odd (cos θ n − cos θ m ) , (41)det E ∝ Y l even sin( θ l / Y m>n, both even (cos θ n − cos θ m ) . (42)The joint probability density (35) follows by transformingfrom θ n to T n , where the Jacobian is given by Eq. (33),while the factors in the expressions for D and E canbe rewritten with the help of Eq. (32) and the relationcos θ n − cos θ m = 2( T m − T n ). D. Large- N asymptotics It is natural to ask whether the similarity of Eq. (35)to Eq. (14) indicates a possible interpretation as a super-position of two independent level sequences [from whichEq. (14) was derived]. In Eq. (35), however, this inter-pretation is prevented by the different one-point weight terms associated to the even and odd indexed eigenval-ues. A symptom of this difference is the fact that Eq.(14) implies finite- N weak-localization corrections to theconductance, while Eq. (35) delivers the absence of suchcorrections, in agreement with the general conclusions inSec. IV B. Hence, the statistics of systems with a lead-transposing and a lead-preserving symmetry (with β = 1)only find a common ground when both are interpreted asa staggered level sequence.For the case of a lead-preserving symmetry, the frame-work of superpositions of independent level sequencesof course provides a powerful tool for the derivation oflow-point correlation functions and local statistics [suchas the two-point correlation function, or the level spac-ing distribution (13)]. We now argue that in the limit N → ∞ , this framework can also be adopted for systemswith a lead-transposing symmetry.In this limit, the transmission eigenvalues form a quasi-continuum, and the asymptotical statistics follow fromthe formal analogy to the statistics of coordinates ofa dense set of parallel line charges in one dimension(the Coulomb gas), which exhibit a logarithmic repul-sion [1, 5]. In leading order, the weight terms enter theanalysis of the statistical fluctuations only via the one-point function P ( T ): For fixed index n , the transmis-sion eigenvalue T n are confined to a small neighborhoodaround a nominal equilibrium position ¯ T n , which is givenby the implicit equation n − / N R ¯ T n P ( T ) dT . Sub-sequently, the weight terms can be approximated by aconstant (with all the T n fixed to ¯ T n ), while the fluc-tuations are exclusively governed by the level-repulsionfactors of the joint probability distribution. As the level-repulsion factors are identical in Eqs. (14) and (35) oneconcludes that the local statistics in both ensembles be-come indistinguishable in the limit of N → ∞ .We therefore obtain the following remarkable result ofpurely statistical origin: For a lead-transposing symme-try, as N is sent to infinity the local statistics (embodiedin low-point correlation functions) converges to that of asuperposition of two independent level sequences. Thisis the case even though a classification of transmissioneigenvalues by parity is not possible. In particular, we ar-rive at the prediction that in this limit, the level-spacingdistribution is well approximated by Eq. (13). V. NUMERICAL INVESTIGATIONS
For the three standard Dyson ensembles of random-matrix theory, the joint probability density (5) manifeststhe celebrated repulsion between neighboring eigenval-ues, since the probability to find two closely spaced ad-jacent eigenvalues is suppressed as ( T n +1 − T n ) β . In con-trast, the joint densities (14) and (35) (both derived for β = 1) describe sequences of reduced stiffness, where onlyevery second level is subject to mutual level repulsion. Asargued before, as long as N takes on moderate values,the latter joint densities imply quantitative differences in P ( s ) s β =1 β =2 β =4 0 1 0 1 2 3Poisson β =1 β =2 β =4 FIG. 4: (Color online) Probability density P ( s ) oftransmission-eigenvalue spacings for systems with a lead-transposing symmetry, obtained from 10 random matriceswith N = 50. Smooth curves: Spacing probability den-sity (13) for superpositions of eigenvalues of two independentsequences from the standard circular ensembles. The insetshows the Wigner distributions (10) from standard random-matrix theory and the Poisson distribution (9) for uncorre-lated eigenvalues. the transmission eigenvalue statistics for lead-preservingand lead-transposing symmetries, while for large N thesestatistics should converge onto each other.In this section we illustrate the differences and simi-larities between these scenarios for all three main sym-metry classes ( β = 1 , ,
4) via numerical sampling of therandom-matrix ensembles, and also compare to realis-tic model systems. For convenient characterization ofthe eigenvalue repulsion we employ the nearest-neighborspacing distribution P ( s ), as well as spacing distribu-tions to more distant neighbors. As we will see, the lo-cal statistics of systems with a lead-transposing symme-try actually show a much weaker N dependence than forsystems with a lead-preserving symmetry. This featurecould be anticipated by (but also goes beyond) the ab-sence of weak localization corrections in the one-pointfunction (discussed in Sec. IV B). A. Random-matrix theory
We start with the characterization of the level statisticswithin the various random-matrix ensembles. Let us firstconsider the case of a lead-transposing symmetry with arelatively large number of transport channels, for whichwe expect that the local statistics is close to that of a su-perposition of two independent level sequences. Startingpoint of the numerical computations is Eq. (27), wherethe matrix Q = S + S †− is drawn from the appropriateDyson ensemble. In order to obtain the nearest-neighborspacing distribution P ( s ), we unfold the eigenvalue se-quences to a mean local spacing ¯ s ≡ N = 50. For P ( s n ) N=4, β =1(a) 0 0.2 0.4 0.6 0.8 P ( s n ) N=4, β =2(c) 0 0.2 0.4 0.6 0.8 0 1 2 3 4 P ( s n ) s n N=4, β =4(e) N=100, β =1(b) n=1n=2n=3 N=100, β =2(d) 0 1 2 3 4 s n N=100, β =4(f) FIG. 5: (Color online) Probability densities of spacings s n to the first, second and third neighboring transmission eigen-value for the random-matrix ensembles of systems with a lead-transposing symmetry (solid curves) or a lead-preserving sym-metry (dashed curves). In the left panels the number of trans-port channels N = 4, while in the right panels N = 100. Toppanels: orthogonal symmetry class ( β = 1). Middle panels:unitary symmetry class ( β = 2). Bottom panels: symplec-tic symmetry class ( β = 4). For each ensemble, the resultsrepresent a sample of 10 realizations. this large number of open channels we find that the nu-merical histograms indeed match the predictions from thesuperposition of two independent level sequences [solidcurves; see Eq. (13)].For comparison, the inset in Fig. 4 shows the stan-dard Wigner distributions (10), as well as the Poissondistribution (9). In the Poisson distribution the eigen-value spacing density is maximal at s = 0; for larger s the probability density decreases monotonically. For theWigner distributions the most likely eigenvalue spacingoccurs at a finite value of s ; for s →
0, the distributionsdecay algebraically ∝ s β , while for s → ∞ they decay as aGaussian. The distributions in the main panel combinethe partial absence of level repulsion for small s [with0 P ( s ) s quantum billiards(a) asymmetricsymmetric 0 1 2 3 4 s open kicked rotators(b) asymmetricsymmetric FIG. 6: (Color online) (a) Nearest-neighbor spacing distri-bution P ( s ) for the lead-asymmetric stadium billiard of Fig.1(b), averaged over energies in the range N = 5 −
14, andthe lead-transposing symmetric stadium billiard of Fig. 1(e),with N = 5 ,
6. (b) The same for open quantum kicked ro-tators with N = 12. In both panels, the solid curves showthe Wigner distribution (10) with β = 1 and the predictionof random-matrix theory for systems with a lead-transposingsymmetry [which can be safely approximated by Eq. (13a)]. P ( s = 0) = 1 /
2] with the Gaussian decay of the Wignerdistributions for large s .For large N , virtually identical results are obtainedfor the conventional case of a lead-preserving symmetry.This is demonstrated in detail in Fig. 5, which also showsthe spacing distributions to the second and third-nearestneighbor. Here, solid curves are for a lead-transposingsymmetry, and dashed curves are for a lead-preservingsymmetry (corresponding to a superposition of indepen-dent level sequences from the appropriate Dyson ensem-ble). For N = 100 (right panels), dashed and solid curveslie on top of each other and are practically indistinguish-able. This clearly supports the convergence of the localstatistics of both cases for large N .The left panels in Fig. 5 show the level-spacing dis-tributions for N = 4. In this case, the results fora lead-transposing symmetry are distinctively differentfrom those for a lead-preserving symmetry. Interest-ingly, the nearest-neighbor spacing distribution for alead-transposing symmetry is very similar for small andlarge N ; the distribution for N = 4 is already well ap-proximated by Eq. (13). In comparison, the nearest-neighbor spacing distribution for a lead-preserving sym-metry shows a much stronger N -dependence. B. Comparison to model systems
In order to validate that realistic quantum systems canindeed be described by random matrix theory (on whichall previous considerations are based), we compare ourpredictions with numerical results for such systems. Inparticular, we present results of numerical computations for quantum billiards, which model a lateral quantumdot, and for the open kicked rotator, which is based onan efficient quantum map. We focus on systems in theorthogonal symmetry class ( β = 1) and contrast sys-tems with a lead-transposing symmetry to systems with-out any spatial symmetry.The quantum billiards are derived from the stadiumgeometry, with leads positioned to either break or con-serve the reflection symmetry about the vertical centerline [see Figs. 1(b,e)]. The computations are performedusing a modular recursive Green’s function method [12,13], with energies that permit 5 ≤ N ≤
14 open chan-nels in each of the two leads. As shown in Fig. 6(a), theeigenvalue spacing distribution agrees well with the pre-dictions of random-matrix theory, both in presence andin absence of a lead-transposing symmetry.The open quantum kicked rotator [14, 15, 16] is definedby the scattering matrix S = P [ e − iε − F (1 − P T P )] − F P T , (43)where ε is the quasi-energy, F nm = ( iM ) − / e iπM ( m − n ) − iMK π (cos πnM +cos πmM ) (44)is the M × M -dimensional Floquet operator of the kickedrotator, and P is an 2 N × M -dimensional matrix whichprojects the internal Hilbert space onto the openings.We assume that M is even and M ≫ N . The reflec-tion symmetry of the closed system is manifested in thesymmetry F nm = F M − n,M − m , and the lead-transposingsymmetry of the open system is present when in addition P nm = P N − n,M − m .Figure 6(b) shows the spacing distributions obtainedfor kicked rotators with symmetrical and asymmetricallead placement and N = 12. The data represents 6600 re-alizations which are generated by varying the quasienergy ε ∈ [0 , π ), the kicking strength K ∈ [10 , M ∈ [498 , β = 1). It is worth emphasiz-ing that the applicability of this statistical concept [em-bodied in the random-matrix results Eq. (35)] does notrely on any pre- or postprocessing or -selection of thetransmission eigenvalues in the model systems (as thereis no intrinsic property of the transmission eigenvaluesor their associated scattering wave functions – such as aparity – that could be used to divide these eigenvaluesinto two sets). VI. SUMMARY AND CONCLUSIONS
We analyzed the transport in open systems with a lead-transposing or a lead-preserving symmetry via the com-plete joint probability density of transmission eigenval-ues, obtained in random-matrix theory.1For a lead-preserving symmetry, the standard conceptof desymmetrization reduces the problem to the inves-tigation of independent non-symmetric variants of thesystem. For a lead-transposing symmetry, however, thetransport characteristics only arise as a collective prop-erty of the symmetry-reduced variants of the system. Westill found that both types of symmetry result in a similarreduction of level repulsion, so that transmission eigen-values can approach each other closely. For a large num-ber of transport channels N , the local eigenvalue statis-tics for both types of symmetry indeed become indistin-guishable.Our main analytical results concern a detailed expla-nation of these features for systems which also exhibittime-reversal and spin-rotation invariance (the orthogo-nal symmetry class, with symmetry index β = 1). Inthis case, the transmission eigenvalue statistics of sys-tems with a lead-transposing or lead-preserving symme-try find a common natural interpretation as a staggeredsuperposition of two independent level sequences. In sucha superposition the eigenvalues alternate between the se-quences when they are ordered by magnitude. The jointprobability densities for the two types of symmetry onlydiffer in one-point weight factors. For lead-preserving symmetries these weight factor incorporate 1 /N correc-tions for quantities such as the ensemble-averaged con-ductance, while these corrections are absent for a lead-transposing symmetry. This results in differences of thelocal eigenvalue statistics when N is small, but becomesinsignificant when N is large.While we concentrated on systems with discrete spatialsymmetries, our results can also be applied for discretesymmetries of different origin (e.g., arising from internaldegrees of freedom) that yield equivalent constraints onthe scattering matrix. Acknowledgments
We gratefully acknowledge assistance with the billiardcomputations by Florian Aigner, as well as useful dis-cussions with Eugene Bogomolny, Piet Brouwer, Vic-tor Gopar, Jon Keating, and Martin Zirnbauer. Thiswork was supported by the European Commission, MarieCurie Excellence Grant MEXT-CT-2005-023778. S.R.wishes to thank the Max-Kade foundation and the W.M.Keck foundation for support. [1] C. W. J. Beenakker, Rev. Mod. Phys. , 731 (1997).[2] F. J. Dyson, J. Math. Phys. , 140 (1962).[3] A. Altland and M. R. Zirnbauer, Phys. Rev. B , 1142(1997).[4] Y. M. Blanter and M. B¨uttiker, Phys. Rep. , 1 (2000).[5] M. L. Mehta, Random Matrices , 3rd ed. (Elsevier, 2004).[6] F. Haake,