Semiclassical treatment of quantum chaotic transport with a tunnel barrier
SSemiclassical treatment of quantum chaotic transportwith a tunnel barrier
Pedro H. S. Bento, Marcel NovaesInstituto de F´ısica, Universidade Federal de UberlˆandiaUberlˆandia, MG, 38408-100, Brazil
Abstract
We consider the problem of a semiclassical description of quantum chaotic transport,when a tunnel barrier is present in one of the leads. Using a semiclassical approachformulated in terms of a matrix model, we obtain transport moments as power seriesin the reflection probability of the barrier, whose coefficients are rational functions ofthe number of open channels M . Our results are therefore valid in the quantum regimeand not only when M (cid:29)
1. The expressions we arrive at are not identical with thecorresponding predictions from random matrix theory, but are in fact much simpler.Both theories agree as far as we can test.
We consider quantum transport through systems with chaotic classical dynamics. Apossible setting would be a two dimensional electron gas, shaped as a mesoscopic cavityusing semiconductors, connected to source and drain by two attached leads and submit-ted to a small external voltage [1]. Assuming low temperature and a classical dwell timeinside the cavity which is much higher than the Ehrenfest time, the statistical propertiesof the electronic transport are remarkably universal, i.e. insensitive to system details anddepending only on the symmetry class of the problem (for spinless particles, this reduces topresence/absence of time-reversal symmetry) [2].The process of quantum scattering through a cavity with two leads is described by ascattering S matrix, of dimension equal to the number of open channels M , which relatesincoming to outgoing quantum amplitudes. Conservation of charge implies that S is unitary.Another possibility is to use the eigenvalues of the related transmission matrix T , definedbelow, which is hermitian. The measurable characteristics of the system, like average andvariance of conductance, shot-noise etc., are related to symmetric functions of the eigenvaluesof T [3, 4].When the corresponding classical dynamics is strongly chaotic and universality is ex-pected, it is fruitful to consider these matrices as random objects, taken from appropriateensembles. This is the random matrix theory (RMT) approach. We shall focus on systemswith broken time-reversal symmetry. When both leads are ideal, i.e. perfectly transmitting,the S is uniformly distributed in the unitary group with Haar measure, and the transmis-sion matrix has a Jacobi distribution [5]. Results valid for a large number of open channels, M (cid:29)
1, were obtained along the 1990’s [6, 7, 8] and were reviewed in [9]. Exact results,valid for arbitrary M , were produced in the 2000’s [10, 11, 12, 13, 14] after a connectionwith the Selberg integral was exploited [15].The semiclassical approximation is a different approach, starting from expressions formatrix elements of quantum observables in terms of the action and stability of classical tra-jectories [16, 17]. A stationary phase argument establishes that sets of trajectories interfereconstructively only if they are correlated, and this correlation is mediated by the existence ofclose encounters. According to the theory initially developed by Sieber and Richter [18, 19]1 a r X i v : . [ n li n . C D ] A ug nd later further developed by Haake and collaborators [20, 21, 22] (see also [23]), after someintegrations over phase space are performed, the calculation of transport moments can beformulated diagrammatically in terms of ribbon graphs, with simple rules determining thecontribution of each graph which, when both leads are ideal, are determined by its genus. Inthat case, it has been established by Berkolaiko and Kuipers [24, 25, 26] that this approachis equivalent to random matrix theory and provides a microscopic justification for it (thedemonstration of this equivalence was vastly simplified by the introduction of semiclassicalmatrix models in [27]).In electronic systems, it is more realistic to assume the presence of tunnel barriers inthe leads [28, 29, 30, 31], so that channel i has an associated tunnel rate Γ i , with Γ i = 1being the ideal case. In the random matrix setting this is implemented by introducing theso-called Poisson kernel to model the statistical distribution of the S matrix [32, 33]. In theperturbative M (cid:29) T , valid for arbitrary M , were derived [38, 39] (see also [40]) interms of hypergeometric functions of matrix argument. This theory was then used in [41]to derive finite- M results for transport moments.Within the semiclassical theory, modified diagrammatic rules valid in the presence oftunnel barriers [42] were able to reproduce the average conductance and shot-noise to leadingorder in M − , in agreement with RMT. This was later taken further to compute the varianceof conductance [43]. These semiclassical investigations have even been capable of taking intoaccount effects that are not captured by random matrix theory, related to the existence of afinite Ehrenfest time (see also [44, 45, 46, 47], for example). They have also been modifiedin order to be applied to the statistics of time delay and to Andreev systems [48, 49].However, all these previous semiclassical efforts were restricted to the leading orders in M − . It should be possible to push this theory further, since there are still no semiclassicalresults that are valid in the presence of a tunnel barrier and in the truly quantum regime,i.e. for arbitrary values of M . The purpose of the present work is to fill this gap.We make use of a novel semiclassical approach which is based on a matrix integralrepresentation [27, 50]. The advantage of this method is that all diagrams are built into thetheory from the beginning and do not need to be explicitly constructed. This approach hasalso been used to treat energy-dependent statistics [51, 52]. By appropriately adapting it,we are able to treat the situation with a tunnel barrier. The results we find are in agreementwith the corresponding ones obtained within RMT [40, 41], but are in fact much simpler. If X is a N × N matrix with eigenvalues x j , 1 ≤ j ≤ N , then p λ ( X ) = (cid:96) ( λ ) (cid:89) i =1 Tr( X λ i ) = (cid:96) ( λ ) (cid:89) i =1 N (cid:88) j =1 x λ i j (1)is the power sum symmetric function, labelled by an integer partition, i.e. a non-decreasingsequence λ = ( λ , λ , . . . ) of (cid:96) ( λ ) positive integers. If (cid:80) i λ i = n we say λ partitions n anddenote this by λ (cid:96) n or | λ | = n . If π is some permutation with cycle type λ , then thisfunction can also be written as p λ ( X ) = (cid:88) i ,...i n n (cid:89) k =1 X i π ( k ) ,i k . (2)We assume a chaotic cavity with two leads, supporting N and N open channels. Thetotal number of channels is M = N + N . (3)2he S -matrix is given by S = (cid:18) r tt (cid:48) r (cid:48) (cid:19) , where r is a N × N reflection block and t is a N × N transmission block (and similarly for r (cid:48) and t (cid:48) ). The N × N transmission matrixis T = t † t . The dimensionless transport moments are the functions p λ ( T ): for instance, theconductance is p ( T ), while the shot-noise is p ( T ) − p ( T ). These moments are relatedto the statistical properties of the electric current in the system as a function of time:conductance and shot-noise, for instance, are related to average and variance of current.In a realistic system these transport moments are wildly fluctuating functions of theenergy and therefore have a random behaviour of their own, and we can talk about theirstatistical properties. An ensemble average (in random matrix theory) or a local energyaverage (in semiclassical theory) may be introduced. We denote both these averages by (cid:104) p λ ( T ) (cid:105) . The variance of conductance, for example, would be related to (cid:104) p (cid:105) − (cid:104) p (cid:105) . Noticethat p = p , . Transport moments associated with partitions with more than one part aresometimes called ‘nonlinear statistics’ (but we avoid this terminology).Transport moments can also be encoded in a different family of symmetric functionscalled Schur functions. These are given by s λ ( X ) = det (cid:16) x n + λ i − ii (cid:17) det (cid:0) x n − ii (cid:1) = det (cid:16) x n + λ i − ii (cid:17) ∆( X ) , (4)where n = | λ | and ∆( X ) = N (cid:89) i =1 N (cid:89) j = i +1 ( x j − x i ) (5)is called the Vandermonde of X . The set of Schur functions { s λ , λ (cid:96) n } spans the vectorspace of homogeneous symmetric polynomials of degree n . They are related to power sumsby p µ ( X ) = (cid:88) λ (cid:96)| µ | χ λ ( µ ) s λ ( X ) , (6)where χ λ ( µ ) are the irreducible characters of the permutation group.An important role is played by the value of the Schur function when all its argumentsare equal to 1. In that case s λ (1 N ) = d λ n ! [ N ] λ , (7)where d λ = χ λ (1 n ) is the number of standard Young tableaux of shape λ and [ N ] λ is ageneralization of the rising factorial given by[ N ] λ = (cid:96) ( λ ) (cid:89) i =1 ( N + λ i − i )!( N − i )! . (8)This can also be written as a product over the contents of the Young diagram (see AppendixA), [ N ] λ = (cid:89) ( i,j ) ∈ λ ( N + j − i ) . (9)Finally, let us mention that the product of two such functions can be written as a linearcombination of them according to s λ ( X ) s µ ( X ) = (cid:88) ν C νλµ s ν ( X ) . (10)The quantities C νλµ are called Littlewood-Richardson coefficients. They are different fromzero only if | ν | = | λ | + | µ | and additionally ν contains both λ and µ . We say ν contains µ , ν ⊃ µ , when ν i ≥ µ i for all i . 3 .2 Random matrix theory In the ideal case when there are no tunnel barriers, the joint probability distribution ofthe N eigenvalues of T is, assuming without loss of generality that N ≤ N , given by P ( T ) = 1 Z | ∆( T ) | N (cid:89) i =1 T N − N i , (11)where Z is a normalization constant. The calculation of the average value of any Schurfunction amounts to a Selberg-like integral [54, 55], (cid:104) s µ ( T ) (cid:105) = (cid:90) [0 , N s µ ( T ) P ( T ) dT = d µ n ! [ N ] µ [ N ] µ [ M ] µ . (12)In the non-ideal situation, Vidal and Kanzieper obtained the joint probability distribu-tion of reflection eigenvalues R i = 1 − T i , assuming only one of the leads contains a tunnelbarrier and time-reversal symmetry is broken. Assuming the N channels in the second leadare ideal, while in the first lead the tunnelling probabilities Γ i of each channel are collectedin the matrix Γ, their result is that P ( R ) ∝ det(Γ) M det( F ) ∆( R )∆(Γ) N (cid:89) i =1 (1 − R i ) N − N , (13)where F is a matrix with elements given in terms of a hypergeometric function, F ij = F ( N + 1 , N + 1; 1; (1 − Γ i ) R j ) . (14)The above result was used in [41] to obtain average transport moments. In terms ofSchur functions of reflection eigenvalues, it was shown that (cid:104) s λ ( R ) (cid:105) = det(Γ) M (cid:88) ρ s ρ (1 − Γ) [ M ] ρ [ N ] ρ (cid:88) ν C νλρ d ν | ν | ! [ N ] ν [ M ] ν , (15)where the infinite sum over ρ includes all possible partitions and the quantities C νλρ are theLittlewood-Richardson coefficients. In the regime of weakly non-ideal leads, Γ i ≈
1, thisresult can be seen as a perturbative expansion in the small variable 1 − Γ.In this work we shall further assume that in the first lead all tunnelling probabilities areequal, and we will express them in terms of an opacity parameter γ , which is the reflectionprobability of the barrier, Γ i = 1 − γ . Then we can use the relation s ρ ( γ ) = γ | ρ | d ρ [ N ] ρ | ρ | ! (16)to write (cid:104) s λ ( R ) (cid:105) = (1 − γ ) N M (cid:88) ρ γ | ρ | d ρ | ρ | ! [ M ] ρ [ N ] ρ (cid:88) ν C νλρ d ν | ν | ! [ N ] ν [ M ] ν . (17)The average value of a Schur function of the transmission eigenvalues can be obtainedfrom the above equation by using the binomial-like theorem [56] s µ ( T ) = s µ (1 − R ) = (cid:88) λ ⊂ µ ( − | λ | B µ,λ ( N ) s λ ( R ) , (18)in which B µ,λ ( N ) = det (cid:18)(cid:18) N + µ i − iN + λ j − j (cid:19)(cid:19) = [ N ] µ [ N ] λ d µ/λ ( | µ | − | λ | )! , (19)4ith d µ/λ = ( | µ | − | λ | )! det (cid:18) µ i − i − λ j + j )! (cid:19) (20)being the number of standard Young tableaux of shape µ/λ .This leads to a rather cumbersome final expression for (cid:104) s µ ( T ) (cid:105) , that we have not beenable to simplify. Our initial aim was to reproduce this result using the semiclassical approx-imation. However we shall see that in fact that theory leads to a much cleaner formula. The leads that connect the chaotic cavity to the outside world have widths W and W .These must be classically small in order to ensure that the dwell time τ D be long enough forthe dynamics to be strongly chaotic. On the other hand, in the semiclassical regime of small (cid:126) the number of open channels N i ∼ W i / (cid:126) will be large (the total energy of the electrons isfixed).The semiclassical approximation starts by writing the elements of the S matrix as sumsover classical trajectories: S oi = 1 √ M τ D (cid:88) α : i → o A α e i S α / (cid:126) , (21)where each trajectory α starts at channel i and ends at channel o , having action S α (theprefactor A α is related to the trajectory’s stability).The calculation of a transport moment like p λ ( T ) = N (cid:88) (cid:126)i =1 n (cid:89) k =1 ( t † t ) i π ( k ) ,i k = N (cid:88) (cid:126)i =1 N (cid:88) (cid:126)o =1 n (cid:89) k =1 t † i π ( k ) ,o k t o k ,i k , (22)with π being any permutation that has cycle type λ , requires multiple sums over trajectories, p λ ( T ) = 1 M n τ nD n (cid:89) k =1 (cid:88) i k ,o k (cid:88) α k ,σ k A α A ∗ σ e i ( S α −S σ ) / (cid:126) , (23)with the understanding that α k goes from i k to o k , while σ k goes from i π ( k ) to o k . Thequantity A α = (cid:81) k A α k is a collective stability, while S α = (cid:80) k S α k is the collective actionof the α trajectories, and analogously for σ .The transport moment (23) is a strongly fluctuating function of the energy. Its localaverage value can be computed under a stationary phase approximation, which leads to thecondition that the set of α trajectories has almost the same collective action as the set of σ trajectories. These so-called action correlations exist when the α ’s and σ ’s are piecewisealmost equal, except in small regions where an encounter takes place. A q -encounter isa region where q pieces of trajectories run nearly parallel and the σ ’s are permuted withrespect to the α ’s (we are considering only systems with broken time-reversal symmetry, so σ trajectories never run opposite to α trajectories).As illustration, we present in Figure 1 two contributions to the simplest transport mo-ment, the average conductance (cid:104) p ( T ) (cid:105) . Trajectory α is depicted in solid line, while σ is indashed line. In panel a) we have a 3-encounter, while in panel b) we have two 2-encounters.For the sake of visual clarity, the encounters are greatly magnified so that their internalstructure is visible. Also we do not try to reproduce the actual trajectories which would beextremely convoluted and chaotic. For more details regarding this theory, and plenty morefigures, we refer the reader to previous works.When a tunnel barrier is present, say in the left lead for instance, action correlationsmay be of a slightly different nature, as trajectories that hit the barrier from the inside may5ail to tunnel out and, instead, may be reflected back into the cavity [42, 48]. When thishappens a trajectory will be composed of two or more parts, corresponding to its excursionsbetween hits in the barrier. Two trajectories may then differ in the order of these excursions,while still having the same action. This is illustrated in Figure 2. In panel a) α and σ hitthe left lead twice before leaving the cavity; they differ in order they traverse those two‘loops’. In panel b) α and σ hit the left lead once and, in addition, there is a 2-encounter.These special situations may be interpreted in terms of ‘encounters in the lead’. Fig.2ais then viewed as a degenerate case of Fig.1a, in which the 3-encounter happens in the lead,namely, it is replaced by reflections. Analogously, Fig.2b is viewed as a degenerate case ofFig.1b, in which one of the 2-encounters happens in the lead while the other one remainsinside the cavity. Sets of action-correlated trajectories can be represented by diagrams which are ribbongraphs. A q -encounter becomes a vertex of valence 2 q . The pieces of trajectories betweenvertices become oriented ribbons, bordered by one of the α trajectories on one side and oneof the σ trajectories on the other. We show in Figure 3 the ribbon graphs corresponding tothe trajectories shown in Figure 1.In the ideal case when there are no tunnel barriers, after the appropriate phase-spaceintegrals are performed the semiclassical theory boils down to summing over diagrams, withdiagrammatic rules that are as follows: the contribution of a diagram is multiplied by • M − for each ribbon, • − M for each vertex, • N for each channel where a trajectory begins, • N for each channel where a trajectory ends.For example, the leading contribution to the conductance comes from the trivial diagramwith no encounters and identical trajectories, α = σ . This gives N N /M . The nextcontributions are sketched in Figure 3. Panel a) has four ribbons and one 3-vertex, giving − N N /M ; Panel b) has five ribbons and two 2-vertices, giving N N /M .As discussed in [42, 48], in the non-ideal case when tunnel barriers are present, thesemiclassical diagrammatic rules must be modified. Assuming that the second lead is idealand that in the first lead all tunnelling probabilities are equal, Γ i = 1 − γ , the contributionsbecome • ( N (1 − γ ) + N ) − = ( M − N γ ) − for each ribbon, • − N (1 − γ q ) − N = − M + N γ q for each vertex of valence 2 q , • N (1 − γ ) for each channel where a trajectory begins, (a) (b) Figure 1: Action correlated trajectories (in solid and dashed lines) that contribute to thesemiclassical evaluation of the conductance. a) Trajectories differ by a 3-encounter; b)Trajectories differ by two 2-encounters. Black rectangles represent the leads.6 a) (b) Figure 2: Action correlated trajectories (in solid and dashed lines) that contribute to thesemiclassical evaluation of the conductance when a tunnel barrier is present in the left lead.They may be seen as degenerate cases of the ones in Figure 1, in which the encounter takesplace in the lead. Black rectangles represent the leads. • N for each channel where a trajectory ends. • γ for each encounter happening at the first lead.The leading order contribution to the calculation of the conductance, for example, be-comes N N (1 − γ ) / ( M − N γ ). The diagrams in Fig.3a) and Fig.3b), on the other hand,now give N N (1 − γ )( − M + N γ )( M − N γ ) and N N (1 − γ )( − M + N γ ) ( M − N γ ) , (24)respectively. Moreover, we must now allow encounters in the lead (these do not countas vertices in the diagrammatic theory, however). The trajectories in Figure 2 and theirdiagrams in Figure 4 thus come into play. Fig.4a has three ribbons, no vertices and tworeflections; Fig.4b has four ribbons, one 2-vertex and one reflection. Their contributions are N N (1 − γ ) γ ( M − N γ ) and N N (1 − γ ) γ ( − M + N γ )( M − N γ ) , (25)respectively. In the ideal case when there are no tunnel barriers, the diagrammatic rules can beimplemented by means of the matrix integral (cid:42) n (cid:89) k =1 t † i π ( k ) ,o k t o k ,i k (cid:43) = lim N → Z (cid:90) e − M (cid:80) ∞ q =1 1 q Tr( Z † Z ) q n (cid:89) k =1 Z † i π ( k ) ,o k Z o k ,i k dZ, (26) a) b) Figure 3: Diagrams that represent the trajectories in Fig.1. Encounters are depicted asvertices, arcs of trajectories are depicted as ribbons, bordered by an α trajectory on oneside (solid line) and a σ trajectory on the other (dashed line). Black rectangles representthe leads. 7here we integrate over N × N complex matrices Z . The quantity Z = (cid:90) e − M Tr( ZZ † ) dZ = N ! M N N − (cid:89) j =1 j ! (27)is a normalization constant.The diagrammatic approach to (26) proceeds from keeping e − M Tr( Z † Z ) as a Gaussianmeasure and expanding the remaining exponentials as power series. The integral is thenperformed using the well known Wick’s rule [27, 51, 57, 58]. The product of Z † and Z matrix elements represent the channels. The diagrams thus produced are exactly like thesemiclassical ribbon graphs, with the same diagrammatic rules, M − for each ribbon, − M for each vertex ( N and N do not appear yet since we are keeping fixed the indices i and o ). However, when producing all possible connections as per Wick’s rule, summation overfree indices in the traces will produce powers in the dimension N . These are closed cyclesthat correspond to periodic orbits. Since the semiclassical sum does not include such orbits,we let N → i from 1to N and over o from 1 to N , and write it in the form (cid:104) p λ ( T ) (cid:105) = lim N → Z (cid:90) det(1 − Z † Z ) M p λ ( Z † Q ZQ ) dZ, (28)where λ is the cycle type of π and Q = (cid:18) N
00 0 N (cid:19) , Q = (cid:18) N
00 1 N (cid:19) (29)are projectors, with 1 N and 0 N being the identity and the null matrix in dimension N .Then, expand the power sum into Schur functions and use the fact that (see equation (18)in [53]) (cid:90) det(1 − Z † Z ) M − N s µ ( Z † Q ZQ ) dZ = s µ ( Q ) s µ ( Q ) s µ (1 M ) . (30)Finally, since s µ ( Q i ) = s µ (1 N i ) = d µ n ! [ N i ] µ , we obtain (cid:104) p λ ( T ) (cid:105) = 1 n ! (cid:88) µ (cid:96) n χ µ ( λ ) d µ [ N ] µ [ N ] µ [ M ] µ . (31)The above calculation is the semiclassical derivation of transport moments in the ideal case,which agrees exactly with RMT (12). a) b) Figure 4: Diagrams that represent the trajectories in Fig.2. Encounters may now happen‘in’ the first lead, as trajectories may be reflected by the tunnel barrier back inside thecavity. Black rectangles represent the leads. 8 .2 Including the tunnel barrier
In the non-ideal case when a tunnel barrier is present in the first lead, we have differentdiagrammatic rules to implement. If there were no encounters in the lead, we could proposethe modified matrix modellim N → (1 − γ ) n Z (cid:90) exp (cid:34) − ∞ (cid:88) q =1 ( M − N γ q ) q Tr( Z † Z ) q (cid:35) n (cid:89) k =1 Z † o π ( k ) ,i k Z o k ,i k dZ, (32)where Z = (cid:90) e − ( M − N γ )Tr( ZZ † ) dZ = N !( M − N γ ) N N − (cid:89) j =1 j ! . (33)The prefactor (1 − γ ) n corresponds to all the trajectories entering the cavity through thebarrier. The Gaussian measure e − ( M − N γ )Tr( Z † Z ) produces ( M − N γ ) − for each ribbon;expanding the rest of the exponential produces − M + N γ q for each q -encounter, so thediagrammatic rules would be indeed correct.However, we must also incorporate encounters in the first lead. Information aboutwhat happens at the leads must be contained in the factor (cid:81) nk =1 Z † o π ( k ) ,i k Z o k ,i k , becausediagrammatically each matrix element from Z is a pair of trajectories entering the cavity,while each matrix element from Z † is a pair of trajectories leaving the cavity. So we mustintroduce some modification to this term.When an encounter happens in the lead, a vertex-like structure is produced. In Fig.4athis pseudo-vertex has valence 5, while in Fig.4b it has valence 3. It is as if a matrix elementfrom Z got replaced by a matrix element from ( ZZ † ZZ † Z ) in the first case and ( ZZ † Z ) inthe second. To produce a pseudo-vertex of valence 2 m +1 we should replace Z by Z ( Z † Z ) m .This must be accompanied by γ m according to the diagrammatic rules. Encounters in thelead can thus be implemented by means of a geometric series, and we therefore postulatethe integral (cid:104) p λ ( T ) (cid:105) = lim N → Z (cid:90) e (cid:104) − (cid:80) ∞ q =1 ( M − N γq ) q Tr( Z † Z ) q (cid:105) n (cid:89) k =1 Z † i π ( k ) ,o k (cid:18) Z − γZ † Z (cid:19) o k ,i k dZ. (34) The integral in (34) is more complicated than the one in (26). In order to compute it,we introduce the singular value decomposition Z = U DV and perform first the integrationof the angular variables U and V over the unitary group U ( N ). This is done in [27] and theresult is that the integral N (cid:88) (cid:126)i =1 N (cid:88) (cid:126)o =1 (cid:90) dU dV ( V † DU † ) i π ( k ) ,o k (cid:18) U DV − γX (cid:19) o k ,i k (35)equals (cid:88) µ (cid:96) n [ N ] µ [ N ] µ [ N ] µ χ µ ( λ ) s µ (cid:18) X − γX (cid:19) , (36)where X = D .Let us mention in passing that the function s µ (cid:16) X − γX (cid:17) is a particular case of the canon-ical stable Grothendieck functions studied in [59].We now turn to the radial integral, i.e. the integral over the diagonal matrix X . Summingthe series in the exponent we find that this is (cid:90) dX | ∆( X ) | det(1 − X ) M det(1 − γX ) − N s µ (cid:18) X − γX (cid:19) , (37)9here | ∆( X ) | is the Jacobian of the singular value decomposition. In order to makeprogress, we must consider this integral in the form of a power series in γ . To expressthe second determinant, we resort to the well known Cauchy identity,det(1 − γX ) − N = (cid:88) ω s ω ( γ ) s ω ( X ) , (38)where the infinite sum is over all possible partitions and the first Schur function has N variables equal to γ . On the other hand, the Schur function with the awkward argumentcan be written as s µ (cid:18) X − γX (cid:19) = (cid:88) ρ ⊃ µ γ | ρ |−| µ | A µρ s ρ ( X ) , (39)with the coefficients being given in terms of a determinant with binomial elements: A µρ = det (cid:18)(cid:18) ρ i − iµ j − j (cid:19)(cid:19) . (40)We derive expansion (39) in Appendix B.Littlewood-Richardson coefficients allow us to write s ω ( X ) s ρ ( X ) = (cid:88) ν C νω,ρ s ν ( X ) (41)and arrive at a well known Selberg-like integral [54, 55],1 Z (cid:90) det(1 − X ) M s ν ( X ) | ∆( X ) | dX = ( M − γN ) N d ν [ N ] ν | ν | ![ M ] ν N − (cid:89) j =0 ( M + j )!( M + N + j )! . (42)Having computed all the integrals, this is the time to consider the limit N →
0. First,( M − γN ) N →
1. Also, N − (cid:89) j =0 ( M + j )!( M + N + j )! → N − (cid:89) j =0 ( M + j )!( M + j )! → . (43)Finally, we need to deal with lim N → [ N ] ν [ N ] µ . (44)From the expression of [ N ] ν in terms of contents, Eq. (9), we know that, for small N ,[ N ] ν = t ν N D ( ν ) + O ( N D ( ν )+1 ) , (45)where t ν is the product of all non-zero contents, t ν = (cid:89) ( i,j ) ∈ νi (cid:54) = j ( j − i ) (46)and D ( ν ) is the size of the Durfee square of ν , i.e. the side length of the largest squarediagram contained in ν . Since ν ⊃ ρ because of (41) and ρ ⊃ µ because of (40), we have D ( ν ) ≥ D ( µ ), so the limit (44) exists and is different from zero only if D ( ν ) = D ( µ ). Inthis case it equals t ν /t µ , which we can also write as t ν/µ (see Appendix A for more details).Collecting all the terms, what we have for (cid:104) p λ ( T ) (cid:105) is(1 − γ ) n (cid:88) µ (cid:96) n [ N ] µ [ N ] µ χ µ ( λ ) (cid:88) ω s ω ( γ ) (cid:88) ρ ⊃ µ γ | ρ |− n A µρ (cid:88) ν ⊃ µD ( ν )= D ( µ ) C νω,ρ d ν | ν | ! 1[ M ] ν t ν/µ . (47)10 .4 Simplification Expression (47) can be simplified if we notice that (cid:88) ω s ω ( γ ) C νω,ρ = s ν/ρ ( γ ) = γ | ν |−| ρ | s ν/ρ (1 N ) (48)is a skew-Schur function. From the Jacobi-Trudi expression in terms of complete symmetricfunctions we can see that this is s ν/ρ (1 N ) = det (cid:18)(cid:18) N + ν i − i − ρ j + j − ν i − i − ρ j + j (cid:19)(cid:19) . (49)Then, we get (cid:104) p λ ( T ) (cid:105) = (1 − γ ) n (cid:88) µ (cid:96) n [ N ] µ [ N ] µ χ µ ( λ ) (cid:88) ν ⊃ µD ( ν )= D ( µ ) γ | ν |− n E µν ( N ) d ν | ν | ! 1[ M ] ν t ν/µ , (50)where E µν ( N ) = (cid:88) µ ⊂ ρ ⊂ ν A µρ s ν/ρ (1 N ) . (51)The calculation of E µν is possible by using Lemma 9.1 from [59], a version of the Cauchy-Binet formula which states that, if H νµ = (cid:80) ρ F νρ G ρµ with F νρ = det( f ν i − i,ρ j − j ) and G ρµ = det( g ρ i − i,µ j − j ), then H νµ = det (cid:32)(cid:88) k f ν i − i,k g k,µ j − j (cid:33) . (52)Applied to our problem, this Lemma gives E µν ( N ) = det (cid:32)(cid:88) k (cid:18) N + ν i − i − k − ν i − i − k (cid:19)(cid:18) kµ j − j (cid:19)(cid:33) , (53)= det (cid:18)(cid:18) N + ν i − iN + µ j − j (cid:19)(cid:19) = [ N ] ν [ N ] µ d ν/µ ( | ν | − n )! . (54)If, instead of computing (cid:104) p λ ( T ) (cid:105) , we choose to write the average value of a Schur function,then for µ (cid:96) n we get (cid:104) s µ ( T ) (cid:105) = (1 − γ ) n [ N ] µ ∞ (cid:88) m =0 γ m m ! (cid:88) ν (cid:96) n + mν ⊃ µD ( ν )= D ( µ ) [ N ] ν [ M ] ν d ν d ν/µ | ν | ! t ν/µ . (55)The m = 0 term is given by ν = µ and indeed coincides with the result for the ideal case.The semiclassical result in (55) is actually much simpler than the cumbersome expressionfrom random matrix theory, obtained by combining (17) and (18). That these two results,derived from different theories using different methods, are in fact identical is not obviousat all, but we have checked that this is indeed true for all partitions up to n = 5 and allorders in γ up to 6.In particular, we can write a rather simple formula for the average conductance. When µ = 1 we have that ν must be a hook, ν = ( m + 1 − k, k ), and d ν/ = d ν = (cid:18) mk (cid:19) . (56)The total content is t ν/ = ( m − k )! k !, so that (cid:104) s ( T ) (cid:105) = (1 − γ ) N N M ∞ (cid:88) m =0 γ m m + 1 m (cid:88) k =0 ( N + 1) m − k ( N − k ( M + 1) m − k ( M − k , (57)where ( M ) k and ( M ) k are the usual rising and falling factorials.11 Figure 5: Left: Diagram of the partition (4 , , , , , , , , / (2 , ,
1) and its contents.
By using a formulation in terms of matrix integrals, we developed a semiclassical ap-proach to quantum chaotic transport that is able to describe systems with a tunnel barrierin one of the leads. Our results incorporate the barrier in a perturbative way, but are exactin the number of channels, i.e. there is no large- M expansion.Exact agreement was found, as far as can be computed, with corresponding results fromrandom matrix theory. However, it came as a surprise that the semiclassical expression fortransport moments is actually much simpler than the RMT one, which is not very explicit asit depends on Littlewood-Richardson coefficients. In particular, we found a nice semiclassicalformula for the average conductance.Let us mention that, by adapting the diagrammatic rules, it would also be possibleto incorporte an energy dependence into the problem and compute the average value ofquantities containing the S matrix at energy E and its adjoint S † at energy E + (cid:15) . All thatis required is to replace M − N γ q by M (1 − iq(cid:15) ) − N γ q in the exponent of (34) and thecalculation would proceed similarly.Another extension of the present work could be the treatment of time-reversal invariantsystems. As discussed in [50, 52] the matrix model approach can be used in that case byreplacing complex matrices with real ones and Schur polynomials with zonal polynomials.That topic deserves further exploration. Acknowledgments
We thank Jack Kuipers for helping us understand diagrams with encounters in thelead. We thank user61318 of MathOverflow for bringing reference [59] to our attention intheir answer to question 364518. Financial support from CAPES and from CNPq, grant306765/2018-7, are gratefully acknowledged.
Appendix A
A partition λ (cid:96) n can be represented by a diagram, which is a left-justified collectionof boxes containing λ i boxes in line i . In Figure 5 we show the diagram associated with λ = (4 , , , , j th box in line i is given by j − i . These contents arealso shown in Figure 5. The Durfee square is highlighted in grey, its diagonal contains theboxes with zero content. In this case we have[ N ] (4 , , , , = ( N − N − ( N − ( N + 4) N = 576 N + O ( N ) . (58)Notice that the product of all non-zero contents is t λ = 576.When λ ⊃ µ , the skew shape λ/µ exists and is represented by the diagram of λ withthe boxes contained in the diagram of µ being removed. An example is shown Figure 5 inwhich λ = (4 , , , ,