cond-mat/9501025 Generalized circular ensemble of scattering matrices for a chaotic cavity with non-ideal leads P. W. Brouwer Instituut-Lorentz, University of Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands 5 9 9 Abstract 1 n a We consider the problem of the statistics of the scattering matrix S of J a chaotic cavity (quantum dot), which is coupled to the outside world by 9 non-ideal leads containing N scattering channels. The Hamiltonian H of 1 the quantum dot is assumed to be an M ×M hermitian matrix with prob- v 5 ability distribution P(H) ∝ det[λ2 + (H − ε)2]−(βM+2−β)/2, where λ and 2 ε are arbitrary coefficients and β = 1,2,4 depending on the presence or 0 1 absence of time-reversal and spin-rotation symmetry. We show that this 0 “Lorentzian ensemble” agrees with microscopic theory for an ensemble of 5 9 disordered metal particles in the limit M → ∞, and that for any M ≥ N it / implies P(S) ∝ |det(1 −S¯†S)|−(βM+2−β), where S¯ is the ensemble average t a of S. This “Poisson kernel” generalizes Dyson’s circular ensemble to the case m S¯ 6= 0 and was previously obtained from a maximum entropy approach. The - d present work gives a microscopic justification for the case that the chaotic n o motion in the quantum dot is due to impurity scattering. c : v i X PACS numbers: 05.45.+b, 72.10.Bg r a Typeset using REVTEX 1 I. INTRODUCTION Recent experiments1–4 onconductance fluctuationsandweak-localizationeffectsinquan- tum dots have stimulated theoretical work5–12 on phase-coherent conduction through cav- ities in which the classical electron motion can be regarded as chaotic. If the capacitance of the quantum dot is large enough, a description in terms of non-interacting electrons is appropriate (otherwise the Coulomb blockade becomes important5,7). For an isolated chaotic cavity, it has been conjectured and confirmed by many examples that the statistics of the Hamiltonian H agrees with that of the Gaussian ensemble of random-matrix theory.13,14 If the chaotic behavior is caused by impurity scattering, the agreement has been established by microscopic theory: Both the Gaussian ensemble and the ensemble of Hamiltonians with randomly placed impurities are equivalent to a certain non-linear σ-model.15,16 Transport properties can be computed by coupling M eigenstates of H to N scattering channels.16–19 Since N ≪ M this construction introduces a great number of coupling parameters, whereas only a few independent parameters determine the statistics of the scattering matrix S of the system.16 For transport properties at zero temperature and infinitesimal applied voltage, one only needs to know S at the Fermi energy E , and an approach which starts directly from the F ensemble of scattering matrices at a given energy is favorable. Following up on earlier work onchaotic scattering inbilliards,20 two recent papers8,9 have studied thetransportproperties of a quantum dot under the assumption that S is distributed according to Dyson’s circular ensemble.21,22 In Refs. 8 and9 the coupling of the quantum dot to the external reservoirs was assumedtooccurviaballisticpointcontacts(or“idealleads”). Theextensiontocouplingvia tunnel barriers (non-ideal leads) was considered in Ref. 11. In all cases complete agreement was obtained with results which were obtained from the Hamiltonian approach.7,17,18 This agreement calls for a general demonstration of the equivalence of the scattering matrix and the Hamiltonian approach, for arbitrary coupling of the quantum dot to the external reservoirs. It is the purpose of this paper to provide such a demonstration. A proof of the equivalence of the Gaussian and circular ensembles has been published by Lewenkopf and Weidenmu¨ller,19 for the special case of ideal leads. The present proof applies to non-ideal leads as well, and corrects a subtle flaw in the proof of Ref. 19 for the ideal case. The circular ensemble ofscattering matrices ischaracterized bya probabilitydistribution P(S) which is constant, that is to say, each unitary matrix S is equally probable. As a consequence, the ensemble average S¯ is zero. This is appropriate for ideal leads. A generalization of the circular ensemble which allows for non-zero S¯ (and can therefore be applied to non-ideal leads) has been derived by Mello, Pereyra, and Seligman,23,24 using a maximum entropy principle. The distribution function in this generalized circular ensemble is known in the mathematical literature25 as the Poisson kernel, P(S) ∝ det(1−S¯†S) −(βN+2−β). (1.1) (cid:12) (cid:12) (cid:12) (cid:12) Here β ∈ {1,2,4} is the symmetry(cid:12)index of the(cid:12)ensemble of scattering matrices: β = 1 or 2 in the absence or presence of a time-reversal-symmetry breaking magnetic field; β = 4 in zero magnetic field with strong spin-orbit scattering. (In Refs. 23 and 24 only the case β = 1 was considered.) One verifies that P(S) = constant for S¯ = 0. Eq. (1.1) was first 2 recognized as a possible generalization of the circular ensemble by Krieger,26 for the special case that S¯ is proportional to the unit matrix. In this paper we present a microscopic justification of the Poisson kernel, by deriving it from an ensemble of random Hamiltonians which is equivalent to an ensemble of disordered metal grains. For the Hamiltonian ensemble we can use the Gaussian ensemble, or any other ensemble to which it is equivalent in the limit M → ∞.27 (The microscopic justification of the Gaussian ensemble only holds for M → ∞.) For technical reasons, we use a Lorentzian distribution for the Hamiltonian ensemble, which in the limit M → ∞ can be shown to be equivalent to the usual Gaussian distribution. The technical advantage of the Lorentzian ensemble over the Gaussian ensemble is that the equivalence to the Poisson kernel holds for arbitrary M ≥ N, and does not require taking the limit M → ∞. The outline of this paper is as follows: In Sec. II the usual Hamiltonian approach is summarized, following Ref. 16. In Sec. III, the Lorentzian ensemble is introduced. The eigenvalue and eigenvector statistics of the Lorentzian ensemble are shown to agree with the Gaussian ensemble in the limit M → ∞. In Sec. IV we then compute the entire distribution function P(S) of the scattering matrix from the Lorentzian ensemble of Hamiltonians, and show that it agrees with the Poisson kernel (1.1) for arbitrary M ≥ N. In Sec. V the Poisson kernel is shown to describe a quantum dot which is coupled to the reservoirs by means of tunnel barriers. We conclude in Sec. VI. II. HAMILTONIAN APPROACH The Hamiltonian approach16,19,28 starts with a formal division of the system into two parts, the leads and the cavity (see Fig. 1a). The Hamiltonian of the total system is repre- sented in the following way: Let the set {|ai} represent a basis of scattering states in the lead at the Fermi energy E (a = 1,...,N), with N the number of propagating modes at F E . The set of bound states in the cavity is denoted by {|µi} (µ = 1,...,M). We assume F M ≥ N. The Hamiltonian H is then given by16 H = |aiE ha|+ |µiH hν|+ |µiW ha|+|aiW∗ hµ| . (2.1) F µν µa µa a µ,ν µ,a X X X(cid:16) (cid:17) The matrix elements H form a hermitian M ×M matrix H, with real (β = 1), complex µν (β = 2), or real quaternion (β = 4) elements. The coupling constants W form a real µa (complex, realquaternion)M×N matrixW. TheN×N scattering matrixS(E )associated F with this Hamiltonian is given by S(E ) = 1−2πiW†(E −H +iπWW†)−1W. (2.2) F F Forβ = 1,2,4the matrixS is respectively unitary symmetric, unitary, andunitary self-dual. Usually one assumes that H is distributed according to the Gaussian ensemble, 1 P(H) = exp −1βMλ−2trH2 , (2.3) V 4 (cid:16) (cid:17) with V a normalization constant and λ an arbitrary coefficient which determines the density of states at E . The coupling matrix W is fixed. Notice that P(H) is invariant under F 3 transformations H → UHU† where U is orthogonal (β = 1), unitary (β = 2), or symplectic (β = 4). This implies that P(S) is invariant under transformations W → UW, so that it can only depend on the invariant W†W. The ensemble-averaged scattering matrix S¯ can be calculated analytically in the limit M → ∞, at fixed N, E , and fixed mean level spacing F ∆. The result is16 1−πW†W/λ S¯ = . (2.4) 1+πW†W/λ ItispossibletoextendtheHamiltonian(2.1)toincludea“background”scatteringmatrix S which does not couple to the cavity.29 The matrix S is symmetric for β = 1 and can be 0 0 decomposed as S = Oe2iΦOT, where the matrix O is orthogonal and Φ is real and diagonal. 0 In the limit M → ∞, the average scattering matrix S¯ is now given by29 1−πW†W/λ S¯ = OeiΦ eiΦOT. (2.5) 1+πW†W/λ Lewenkopf and Weidenmu¨ller19 used this extended version of the theory to relate the ¯ Gaussian and circular ensembles, for β = 1 and S = 0. Their argument is based on the assumption that Eq. (2.5) can be inverted, to yield W†W and S as a function of S¯. Then 0 P(S) = P (S) is fully determined by S¯ (and does not require separate knowledge of W†W S¯ and S ). Under the transformation S → USUT (with U an arbitrary unitary matrix), S¯ is 0 mapped to US¯UT, which implies P (S) = P (USUT). (2.6) S¯ US¯UT For S¯ = 0 one finds that P(S) is invariant under transformations S → USUT, so that P(S) must be constant (circular ensemble). There is, however, a weak spot in this argument: Equation (2.5) can not beinverted for the crucial case S¯ = 0. It is only possible to determine W†W, not S . This is a serious objection, since S is not invariant under the transformation 0 0 S → USUT, and one can not conclude that P(S) = constant for S¯ = 0. We have not succeeded in repairing the proof of Ref. 19 for S¯ = 0, and instead present in the following ¯ sections a different proof (which moreover can be extended to non-zero S). A situation in which the cavity is coupled to n reservoirs by n leads, having N scat- j tering channels (j = 1,...,n) each, can be described in the framework presented above by combining the n leads formally into a single lead with N = n N scattering channels. j=1 j Scattering matrix elements between channels in the same lead correspond to reflection from P the cavity, elements between channels in different leads correspond to transmission. In this notation, the Landauer formula for the conductance G of a cavity with two leads (Fig. 1b) takes the form 2e2 N1 N1+N2 G = |S |2. (2.7) ij h Xi=1j=XN1+1 4 III. LORENTZIAN ENSEMBLE For technical reasons we wish to replace the Gaussian distribution (2.3) of the Hamilto- nians by a Lorentzian distribution, 1 −(βM+2−β)/2 P(H) = λM(βM+2−β)/2det λ2 +(H −ε)2 , (3.1) V (cid:16) (cid:17) where λ and ε are parameters describing the width and center of the distribution, and V is a normalization constant independent of λ and ε. The symmetry parameter β ∈ {1,2,4} indicates whether thematrixelements ofH arereal[β = 1, LorentzianOrthogonalEnsemble (ΛOE)], complex [β = 2, Lorentzian Unitary Ensemble (ΛUE)], or real quaternion [β = 4, Lorentzian Symplectic Ensemble (ΛSE)]. (We abbreviate “Lorentzian” by a capital lambda, because the letter L is commonly used to denote the Laguerre ensemble.) The replacement of (2.3) by (3.1) is allowed because the eigenvector and eigenvalue distributions of the Gaussian and the Lorentzian ensemble are equal on a fixed energy scale, in the limit M → ∞ at a fixed mean level spacing ∆. The equivalence of the eigenvector distributions is obvious: The distribution of H depends solely on the eigenvalues for both the Lorentzian and the Gaussian ensemble, so that the eigenvector distribution is uniform for both ensembles. In order to prove the equivalence of the distribution of the eigenvalues E ,E ,...,E (energy levels), we compare the n-level cluster functions T (E ,E ,...,E ) 1 2 M n 1 2 n for both ensembles. The general definition of the T ’s is given in Ref. 22. The first two T ’s n n are defined by M T (E) = h δ(E −E )i (3.2a) 1 i i=1 X M M M T (E ,E ) = h δ(E −E )δ(E −E )i−h δ(E −E )ih δ(E −E )i. (3.2b) 2 1 2 1 i 2 j 1 i 2 j i,j=1 i=1 j=1 X X X The brackets h...i denote an average over the ensemble. The cluster functions in the Gaus- sian ensemble are known for arbitrary n,22 for the Lorentzian ensemble we compute them below. From Eq. (3.1) one obtains the joint probability distribution function of the eigenvalues, 1 −(βM+2−β)/2 P({E }) = λM(βM+2−β)/2 |E −E |β λ2 +(E −ε)2) . (3.3) j i j i V iY<j Yi (cid:16) (cid:17) We first consider the case λ = 1, ε = 0. We make the transformation 1+iH S = . (3.4) 1−iH The eigenvalues eiφj of the unitary matrix S are related to the energy levels E by j 1+iE eiφj = j ⇐⇒ φ = 2arctanE . (3.5) j j 1−iE j 5 The probability distribution of the eigenphases follows from Eqs. (3.3) and (3.5), 1 P({φ }) = 2−M(βM+2−β)/2 |eiφi −eiφj|β. (3.6) j V i<j Y This is precisely the distribution of the eigenphases in the circular ensemble. The cluster functions in the circular ensemble are known.21,22 The n-level cluster functions TΛ in the n Lorentzian ensemble are thus related to the n-level cluster functions TC in the circular n ensemble by n 2 TΛ(E ,...,E ) = TC(2arctanE ,...,2arctanE ) . (3.7) n 1 n n 1 n 1+E2 j=1 j Y For n = 1 one finds the level density M ρ(E) = , (3.8) π(1+E2) independent of β. For n = 2 one finds the pair-correlation function 4 sin2(M arctanE −M arctanE ) TΛ(E ,E ) = 1 2 . (3.9) 2 1 2 (1+E2)(1+E2) sin2(arctanE −arctanE ) 1 2 1 2 Eq. (3.9) holds for β = 2. The expressions for β = 1,4 are more complicated. The n-level cluster functions for arbitrary λ and ε can be found after a proper rescaling of the energies. Eq. (3.7) generalizes to E −ε E −ε n 2λ TΛ(E ,...,E ) = TC 2arctan 1 ,...,2arctan n . (3.10) n 1 n n λ λ λ2 +(E −ε)2 (cid:18) (cid:19)j=1 j Y The large-M limit of the T ’s is defined as n Y (ξ ,...,ξ ) = lim ∆nT (ξ ∆,...,ξ ∆). (3.11) n 1 n n 1 n M→∞ For both the Gaussian and the Lorentzian ensembles, the mean level spacing ∆ at the center of the spectrum in the limit M → ∞ is given by ∆ = λπ/M. Therefore, the relevant limit M → ∞ at fixed level spacing is given by M → ∞, λ → ∞, ∆ = λπ/M fixed for both ensembles. Equation (3.10) allows us to relate the Y ’s in the Lorentzian and circular n ensembles, YΛ(ξ ,...,ξ ) = lim (2π/M)nTC(2arctan(πξ /M),...,2arctan(πξ /M)) n 1 n n 1 n M→∞ = lim (2π/M)nTC(2πξ /M,...,2πξ /M) n 1 n M→∞ = YC(ξ ,...,ξ ). (3.12) n 1 n It is known that the cluster functions YC in the circular ensemble are equal to the clus- n ter functions YG in the Gaussian ensemble.22 Equation (3.12) therefore shows that the n 6 Lorentzian and the Gaussian ensembles have the same cluster functions in the large-M limit. The technical reason for working with the Lorentzian ensemble instead of with the Gaus- sian ensemble is that the Lorentzian ensemble has two properties which make it particularly easy to compute the distribution of the scattering matrix. The two properties are: Property 1: If H is distributed according to a Lorentzian ensemble with width λ and center ε, then H−1 is again distributed according to a Lorentzian ensemble, with width λ˜ = λ/(λ2 +ε2) and center ε˜= ε/(λ2 +ε2). Property 2: If the M × M matrix H is distributed according to a Lorentzian ensemble, then every N×N submatrix of H obtained by omitting M−N rows and the corresponding columns is again distributed according to a Lorentzian ensemble, with the same width and center. The proofs of both properties are essentially contained in Ref. 25. In order to make this paper self-contained, we briefly give the proofs in the appendix. IV. SCATTERING MATRIX DISTRIBUTION FOR THE LORENTZIAN ENSEMBLE The general relation between the Hamiltonian H and the scattering matrix S is given by Eq. (2.2). After some matrix manipulations, it can be written as −1 S = 1+iπW†(H −E )−1W 1−iπW†(H −E )−1W . (4.1) F F (cid:16) (cid:17)(cid:16) (cid:17) We can write the coupling matrix W as ˜ W = UQW, (4.2) where U is an M ×M orthogonal (β = 1), unitary (β = 2), or symplectic (β = 4) matrix, W˜ is an N ×N matrix, and Q is an M ×N matrix with all elements zero except Q = 1, nn 1 ≤ n ≤ N. Substitution into Eq. (2.2) gives S = 1+iπW˜ †H˜W˜ 1−iπW˜ †H˜W˜ −1, (4.3) (cid:16) (cid:17)(cid:16) (cid:17) where we have defined H˜ ≡ QTU†(H −E )−1UQ. F We assume that H is a member of the Lorentzian ensemble, with width λ and center 0. Then the matrix H −E is also a member of the Lorentzian ensemble, with width λ and F center E . Property 1 implies that (H − E )−1 is distributed according to a Lorentzian F F ensemble with width λ˜ = λ/(λ2 +E2) and center ε˜= E /(λ2 +E2). Orthogonal (unitary, F F F symplectic) invariance of the Lorentzian ensemble implies that U†(H − E )−1U has the F same distribution as (H −E )−1. Using property 2 we then find that H˜ [being an N ×N F submatrix of U†(H − E )−1U] is distributed according to the same Lorentzian ensemble F ˜ (width λ and center ε˜). We now compute the distribution of the scattering matrix, first for a special coupling, then for the general case. 7 A. Special coupling matrix First we will consider the special case that W˜ = π−1/2δ (4.4) nm is proportional to the unit matrix. The relation (4.3) between the S and H˜ is then ˜ 1+iH S = . (4.5) 1−iH˜ Thus the eigenvalues E˜ of H˜ and eiφj of S are related via j 1+iE˜ eiφj = j ⇐⇒ φ = 2arctanE˜ . (4.6) 1−iE˜ j j j Since transformations H˜ → UH˜U† (with arbitrary orthogonal, unitary, or symplectic N ×N matrix U) leave P(H˜) invariant, P(S) is also invariant under S → USU†. So P(S) can only depend on the eigenvalues eiφj of S. The distribution of the E˜’s is [cf. Eq. (3.3)] P({E˜ }) = 1 λN(βN+2−β)/2 |E˜ −E˜ |β λ˜2 +(E˜ −ε˜)2 −(βN+2−β)/2. (4.7) j j k j V jY<k Yj (cid:16) (cid:17) From Eqs. (4.6) and (4.7) we obtain the probability distribution of the φ’s, 1 1−σσ∗ N(βN+2−β)/2 −(βN+2−β) P({φ }) = |eiφj −eiφk|β 1−σ∗eiφj , (4.8a) j V 2 (cid:18) (cid:19) jY<k Yj (cid:12) (cid:12) (cid:12) (cid:12) 1−λ˜ −iε˜ λ2 +E2 −λ−iE (cid:12) (cid:12) σ = = F F. (4.8b) 1+λ˜ +iε˜ λ2 +E2 +λ+iE F F Eq. (4.8) implies that P(S) has the form of a Poisson kernel, P(S) = det(1−S¯S¯†)(βN+2−β)/2 det(1−S¯†S) −(βN+2−β), (4.9) 2N(βN+2−β)/2V (cid:12) (cid:12) (cid:12) (cid:12) the average scattering matrix S¯ being given by(cid:12) (cid:12) S¯ = σδ . (4.10) nm nm B. Arbitrary coupling matrix Now we turn to the case of arbitrary coupling matrix W˜ . We denote the scattering matrix at coupling W˜ by S, and denote the scattering matrix at the special coupling (4.4) by S . The relation between S and S is 0 0 S = r +t′S (1−r′S )−1t ⇐⇒ S = (t′)−1(S −r)(1−r†S)−1t†, (4.11) 0 0 0 8 where we abbreviated r = (1−πW˜ †W˜ )(1+πW˜ †W˜ )−1, (4.12a) r′ = −W˜ (1−πW˜ †W˜ )(1+πW˜ †W˜ )−1W˜ −1, (4.12b) t = 2π1/2W˜ (1+πW˜ †W˜ )−1, (4.12c) t′ = 2π1/2(1+πW˜ †W˜ )−1W˜ †. (4.12d) The symmetry of the coupling matrix W˜ is reflected in the symmetry of the 2N×2N matrix r t′ S = , (4.13) 1 t r′ ! which is unitary symmetric (β = 1), unitary (β = 2) or unitary self-dual (β = 4). The probability distribution P of S is given by Eq. (4.9). The distribution P of S 0 0 follows from dS 0 P(S) = P (S ) , (4.14a) 0 0 dS where the Jacobian dS /dS is the ratio of infinitesimal volume elements around S and S. 0 0 This Jacobian is known,25,30 dS det(1−r†r) (βN+2−β)/2 0 = . (4.14b) dS |det(1−r†S)|2! After expressing S in terms of S by means of Eq. (4.11), we find that P(S) is given by the 0 same Poisson kernel as Eq. (4.9), but with a different S¯, 1−π(λ˜ +iε˜)W†W S¯ = . (4.15) 1+π(λ˜ +iε˜)W†W In the limit M → ∞ at fixed level spacing ∆ = λπ/M, Eq. (4.15) simplifies to ∆M −π2W†W ¯ S = . (4.16) ∆M +π2W†W The extended version of the Hamiltonian approach which includes a background scat- tering matrix S can be mapped to the case without background scattering matrix by a 0 transformation S → S′ = USUT (β = 1), S → S′ = USV (β = 2), or S → S′ = USUR (β = 4), where U and V are unitary matrices.29 (UT is the transposed of U, UR is the dual of U.) The Poisson kernel is covariant under such transformations,23 i.e. it maps to a Poisson kernel with S¯′ = US¯UT (β = 1), S¯′ = US¯V (β = 2), or S¯′ = US¯UR (β = 4). As a consequence, the distribution of S is given by the Poisson kernel for arbitrary coupling matrix W and background scattering matrix S . This proves the general equivalence of the 0 Poisson kernel and the Lorentzian ensemble of Hamiltonians. 9 V. IDEAL VERSUS NON-IDEAL LEADS The circular ensemble of scattering matrices is appropriate for a chaotic cavity which is coupled to the leads by means of ballistic point contacts (“ideal” leads). In this section we will demonstrate that the generalized circular ensemble described by the Poisson kernel is the appropriate ensemble for a chaotic cavity which is coupled to the leads by means of tunnel barriers (“non-ideal” leads). The system considered is shown schematically in Fig. 2. We assume that the segment of the lead between the tunnel barrier and the cavity is long enough, so that both the N ×N scattering matrix S of the cavity andthe 2N×2N scattering matrix S of the tunnel barrier 0 1 are well-defined. The scattering matrix S has probability distribution P = constant of the 0 0 circular ensemble, whereas the scattering matrix S is kept fixed. 1 We decompose S in terms of N ×N reflection and transmission matrices, 1 r t′ S = 1 1 . (5.1) 1 t r′ 1 1 ! The N ×N scattering matrix S of the total system is related to S and S by 0 1 S = r +t′(1−S r′)−1S t . (5.2) 1 1 0 1 0 1 This relation has the same form as Eq. (4.11). We can therefore directly apply Eq. (4.14), which yields P(S) ∝ |det(1−r†S)|−(βN+2−β). (5.3) 1 ¯ Hence S is distributed according to a Poisson kernel, with S = r . 1 VI. CONCLUSION In conclusion we have established by explicit computation the equivalence for M ≥ N of a generalized circular ensemble of scattering matrices (described by a Poisson kernel) and an ensemble of M ×M Hamiltonians with a Lorentzian distribution. The Lorentzian and Gaussian distributions are equivalent in the large-M limit. Moreover, the Gaussian Hamiltonian ensemble and the microscopic theory of a metal particle with randomly placed impurities give rise to the same non-linear σ-model.15,16 Altogether, this provides a micro- scopic justification of the Poisson kernel in the case that the chaotic motion in the cavity is caused by impurity scattering. For the case of a ballistic chaotic cavity, a microscopic justification is still lacking. The equivalence of the Poisson kernel and an arbitrary Hamiltonian ensemble can be reformulated in terms of a central limit theorem: The distribution of a submatrix of H−1 of fixed size N tends to a Lorentzian distribution when M → ∞, independent of the details of the distribution of H. A central limit theorem of this kind for N = 1 has previously been formulated and proved by Mello.24 This work was motivated by a series of lectures by P. A. Mello at the summerschool in Les Houches on “Mesoscopic Quantum Physics”. Discussions with C. W. J. Beenakker, K. 10