Universality of S-matrix correlations for deterministic plus random Hamiltonians N. Mae and S. Iida Faculty of Science and Technology, Ryukoku University, Otsu 520-2194, Japan (October 21, 2000) We study S-matrix correlations for random matrix ensembles with a Hamiltonian H =H0+ϕ, 1 inwhichH0 isadeterministicN×N matrixandϕbelongstoaGaussianrandommatrixensemble. Using Efetov’s supersymmetry formalism, we show that in the limit N → ∞ correlation functions 0 0 of S-matrix elements are universal on the scale of the local mean level spacing: the dependence of 2 H0 enters into these correlation functions only through the average S-matrix and theaverage level density. This statement applies to each of the three symmetry classes (unitary, orthogonal, and n symplectic). a J PACS number(s): 05.40.-a, 05.30.-d, 05.60.Gg 8 ] n n I. INTRODUCTION - s i The energy levels and/or the scattering matrices of a variety of physical systems with randomness (e.g., complex d nuclei, disordered conductors, classically chaotic systems, etc.) exhibit universal behavior: the statistical properties . t oftheobservablescanbeseparatedintotheuniversalpartsandthenon-universalpartsspecifictoindividualsystems. a m There have been increasing evidences that the universal parts depend only on the fundamental symmetries of the underlyingHamiltonianandarewelldescribedbyarandommatrixensemblewithaGaussiandistribution(seeRef.[1] - d fora review). Accordingtothe fundamentalsymmetries,there arethreeclassicalrandommatrixensembles: Systems n with broken time-reversal symmetry are described by the unitary ensemble and time-reversal invariant systems by o either the symplectic or the orthogonal ensemble depending on whether spin-orbit coupling is present or not [2]. In c spiteoftheirmanysuccessfulapplications,randommatrixmodelslackfirmfoundation. Especially,theGaussianform [ of the probability distribution is used for mathematical convenience and is not motivated by physicalprinciples. It is 2 therefore necessaryandimportantto investigatewhether statisticalproperties areidentical for more generalforms of v the probability distribution consistent with fundamental symmetries. 7 Therehasbeenseveralworkalongthisdirection. HackenbroichandWeidenmu¨ller[3]consideredanon-Gaussianand 2 unitary invariant probability distribution: P(H) ∝ exp[−NtrV(H)], where N is the dimension of the Hamiltonian 0 8 matrix H and V(H) is independent of N and arbitrary provided it confines the spectrum to some finite interval and 0 generates a smooth mean level density, in the limit N →∞. For each of the three symmetry classes, using Efetov’s 0 supersymmetry formalism, they showed that both energy level correlation functions and correlation functions of S- 0 matrixelements areindependent ofP(H) andhence universalif the argumentsofthe correlatorsarescaledcorrectly. / t For realistic situations, it is likely that the Hamiltonian is not completely randombut contains some regularparts. a m If the total Hamiltonian H = H0 + ϕ where H0 is a deterministic part and ϕ is a random one, the probability distribution takes a unitary non-invariant form: - d P(H)∝exp[−NtrV(ϕ)]=exp[−NtrV(H −H0)]. (1.1) n o For the unitary ensemble, Br´ezin et al. [4] discussed the universality of 2-point energy level correlations for V(ϕ) = c ϕ2/2+gϕ4. General n-point energy level correlation functions were shown to be universal by Br´ezin and Hikami [5] v: for V(ϕ) ∝ ϕ2. (The other type of unitary non-invariant distrubution P(H) ∝ exp{−Ntr[V(H)−HH0]} was also i considered by Zinn-Justin [6].) X Recently, we [7] numerically found the same universality of the S-matrix correlations for the distribution function r Eq.(1.1)withV(ϕ)∝ϕ2 fortheorthogonalensemble,i.e.,withtheaverageS-matrixS beentakenastheparameters, a the correlations are independent of H0 while S depends on H0. Our purpose of the present article is to analytically show this universality in any of the three symmetry classes. More precisely, we show m n ω ki ω lj S E− S∗ E+ =f (ωρ(E),S(E)), (1.2) aibi 2 cjdj 2 β iY=1jY=1h (cid:16) (cid:17)i h (cid:16) (cid:17)i where m, n, k , l are non negative integers, the bar denotes ensemble average. The universal functions f depend i j β on the symmetry classes (β =1,2, and 4 for orthogonal,unitary, and symplectic classes) and are independent of H0, except for the indices {a ,b ,k ,c ,d ,l }, while the average local level density ρ and the average S-matrix S depend i i i j j j on H0. 1 II. THE MODEL Following the approach of Ref. [8], we write the scattering matrix S(E) as S (E)=δ −2iπ W† D(E)−1 W , (2.1a) ab ab aµ νb µν Xµ,ν h i in which D(E)=E+i0+−H +iπWW†, (2.1b) E is the energy, 0+ is positive infinitesimal, H represents the projection of the full Hamiltonian onto the interaction region, and W describes the coupling between the eigenstates of the interaction region and the scattering states in the free-propagationregion. The indices a, b refer to the physical scattering channels, and µ, ν refer to the complete orthonormal states characterizing the interaction region. We assume that N ×N matrix H can be written as H =H0+ϕ, (2.2) where H0 is a given, nonrandom, Hermitian matrix, and ϕ is a member of the Gaussian ensemble. The symmetry property of H0 is the same as ϕ. The independent elements of the matrix ϕ are uncorrelated random variables with a Gaussian probability distribution centered at zero. The second moments for the unitary ensemble are given by λ2 ϕµνϕµ′ν′ = δµν′δνµ′. (2.3) N (See Ref. [9] for the orthogonal and the symplectic cases.) Here, λ is a strength parameter. III. DERIVATION For definiteness, we show the derivation for the unitary ensemble and m,n ≤ 2 in Eq. (1.2). The generalization to the other symmetry classes and/or higher values of m and n is straightforward and commented upon in Sec. IV. The derivation is based on the use of Efetov’s supersymmetry method [8,10]. We take the notation from Ref. [8] and use the [1, 2] block notationfor the matrix representationin which 1 and 2 refer to the retardedandadvanced block, respectively. Consider the following generating function: det D (E )+2πWJ (F)W† p p p Z(J)= , (3.1) det[D (E )−2πWJ (B)W†] (cid:2) p p p (cid:3) whereDp(Ep)=diag D(E1),D†(E2) ,Jp(F)=diag[J1(F),J2(F)],andJp(B)=diag[J1(B),J2(B)]. Thescattering matrix can be generated from Z(J) as follows: (cid:2) (cid:3) ∂Z(J) ∂Z(J) S (E ) =δ −i L=δ −i L, (3.2) p p ab ab ∂J (B) ab ∂J (F) p ba(cid:12)J=0 p ba(cid:12)J=0 (cid:12) (cid:12) (cid:12) (cid:12) where Sp(Ep) = diag S(E1),S†(E2) and L = diag(cid:12)(1,−1). Using standard (cid:12)procedure [8], we can represent the average of Z(J) as an integral over a 4×4 graded matrix field σ: (cid:2) (cid:3) Z(J)= d[σ] exp{L(σ)}, (3.3a) Z where N ω− L(σ)=− trg σ2 −trgln D(σ)− L+iπWW†L−2πL WJ (g)W† . (3.3b) 2λ2 2 g p (cid:20) (cid:21) (cid:0) (cid:1) Here,trg denotesthe gradedtrace,D(σ)=E−σ−H0,ω− =ω−i0+, Jp(g)=diag[J1(B),J1(F),J2(B),J2(F)],and Lg =diag(1,−1,1,−1). We have defined E =(E1+E2)/2 and ω =E2−E1. 2 In the limit N →∞, this integral can be done with the use of the saddle-point approximation. We are interested in correlations involving energy differences ω of the order of the mean level spacing ∼ O(N−1). Hence, we expand L(σ) in powers of ω: N L(σ)≈− trg σ2 −trgln[D(σ)]−trgln 1+iπD(σ)−1WW†L 2λ2 −trgln 1(cid:0)−2(cid:1)π D(σ)+iπWW†L −1hL WJ (g)W† + ω−itrg D(σ)−1L . (3.4) g p 2 n (cid:2) (cid:3) o h i It shouldbe notedsuchanexpansionis notpossible for WW† because W†W ∼O(1). Ofthe five terms in expression (3.4) the last three terms are O(1). The first two terms are O(N) and determine the saddle-point σsp. To derive the saddle-pointequationwe write H0 and σsp in the formsH0 =U−1diag(ǫ1,...,ǫN)U andσsp =T−1σDspT,where σDsp is diagonal and T has the form (1+t12t21)1/2 it12 T = . (3.5) −it21 (1+t21t12)1/2 ! The saddle-point equation reads λ2 N 1 σsp = . (3.6) D N E−σsp−ǫ µ=1 D µ X Forordinaryvariables(ratherthanmatrices),Eq.(3.6)has the N+1solutions. The N−1ofwhicharereal,andthe remaining two may have non-zero imaginary parts according to the values of E. Taking the two complex solutions (r ±i∆) [5], we obtain σsp = r −i∆L. The explicit expressions of r and ∆ are not available because Eq. (3.6) D becomes in general an (N + 1)-th polynomial. Several references discussed the properties of Eq. (3.6) (see, e.g., [4,11]). Hereafter we consider the case where ∆ ∼O(1). From the relation between ∆ and the average level density ρ (Eq. (3.8a)), this means E lies far awayfrom the edge of the spectrum. Substituting σsp for σ in Eq. (3.4), we find ω− L(σsp)≈−trgln 1+iπD(σsp)−1WW†L + trg D(σsp)−1L 2 −trglnh1−2πW† D(σsp)+iπiWW†L −1hWL J (g) i. (3.7) g p n (cid:2) (cid:3) o The one-point functions ρ(E) and S(E) are evaluated at the saddle-point. We thus have N∆ ρ(E)= (3.8a) πλ2 and S (E)=1−2iπW† D(σsp)+iπWW†L −1WL. (3.8b) p D (cid:2) (cid:3) Using these one-point functions ρ(E) and S(E) we can write each term of Eq. (3.7) as follows: trgln 1+iπD(σsp)−1WW†L =trgln 1− S (E)−1 LM , (3.9a) p h i (cid:8) (cid:2) (cid:3) (cid:9) ω− trg D(σsp)−1L =−2iπω−ρ(E)trg(t12t21), (3.9b) 2 h i and 2πW† D(σsp)+iπWW†L −1W =i T−1L S (E)−1 −1T −T−1MT −1. (3.9c) p (cid:2) (cid:3) n (cid:2) (cid:3) o Here, t12t21 −it12(1+t21t12)1/2 M = (3.10) −it21(1+t12t21)1/2 −t21t12 ! 3 and we used the property TLT−1 = L+2M. More explicitly, Eqs. (3.9a) and (3.9c) can be expressed with the use of t12 and t21 as follows: R.H.S. of Eq. (3.9a)=trgln(1+T12t12t21) =trgln(1+T21t12t21) (3.11a) and S(E)(1+T21t12t21)−1−1 −it12(1+t21t12)1/2 T12−1+t21t12 −1 R.H.S. of Eq. (3.9c)=i , −it21(1+t12t21)1/2 T21−1+t12t21 −1 1−S†(E)(1+(cid:0)T12t21t12)−1 (cid:1) ! (cid:0) (cid:1) (3.11b) where T12 =1−S(E)S†(E) andT21 =1−S†(E)S(E)[12]. (The derivationis giveninAppendix.) Thus wefind that all the dependence of Eq. (3.7) on W and H0 is completely absorbed in ρ(E) and S(E). Equations (3.9a), (3.9b), and (3.9c) show universality. The explicit forms of correlation functions f are identical with those for the Gaussian β ensemble and are found in appropriate references (see, e.g., [1,13]). IV. SUMMARY Forthesakeofsimplicity,wepresentedthederivationforonlytheunitaryensemble. Ineitheroftheorthogonaland the symplectic ensemble, the internal structures of t12 and t21 differ from the unitary case. However, our derivaton is completely independent of such structures and applies equally to the orthogonal and the symplectic ensembles. Taking the generating function: max{m,n}det D (E )+WJq(F)W† p p p Z(J)= , (4.1) det[D (E )−WJq(B)W†] q=1 (cid:2) p p p (cid:3) Y we can show Eq. (1.2) for m>2 or n>2 along exactly parallel lines. Insummary,wehaveshownthatthelocaluniversalityinthebulkscalinglimitstillholdsfortheS-matrixcorrelation functions even though unitary invariance is broken by the addition of a deterministic matrix to the ensemble. The startingrandommatrixmodelcontainsparametersW andH0whicharespecifictoindividualsystems. Afterensemble averaging, these original parameters are completely absorbed into much fewer parameters S(E) and ρ(E). Thus the S-matrix correlaton functions of the type Eq. (1.2) have universal forms which are independent of H0 but for S(E) and ρ(E) and are determined only by the symmetry of the ensemble. This holds for all the three symmetry classes (orthogonal, unitary, and symplectic). The derivation can be similarly applied to the spectral correlation functions. Thuswe haveextendedthe previousresultsby Br´ezinandHikami [5]to the orthogonalandthe symplectic ensembles though only two-point functions are considered. The present results were derived under the restrictions that the correlation functions contain only two values of energy,E1 andE2,andthatV(ϕ)hasaGaussianform. ItisanaturalconjecturethattheuniversalityoftheS-matrix correlationfunctionsholdsevenifthesetworestrictionsareremoved. Theincreaseofthenumberofenergyarguments makes the structure of σsp more complicated. With this point taken properly into account, the similar derivation D is probable. The extension to the general form of V(ϕ) seems less trivial because we are no longer able to use a Hubbard-Stratonovichtransformationin orderto introduce a gradedmatrix σ. The simlar procedureused in Ref. [3] may be incorporated into present derivation. ACKNOWLEDGMENTS Stimulating discussions with K. Nohara and K. Takahashi are appreciated with thanks. APPENDIX: DERIVATION OF EQS. (3.11a) AND (3.11b) To derive Eqs. (3.11a) and (3.11b), we use the fact that for any analytic function F, we have t12F(t21t12) = F(t12t21)t12. Using the identity 4 a b trgln =trgln a−bd−1c +trgln(d), (A1) c d (cid:18) (cid:19) (cid:0) (cid:1) we obtain Eq. (3.11a). For the abbreviation Y ≡ T−1L S (E)−1 −1T −T−1MT −1, using the property T−1 = p LTL, we get the following equation n (cid:2) (cid:3) o S(E)−1 −1+t12t21A it12(1+t21t12)1/2A −1 Y = −1 , (A2)  (cid:2)it21(1+(cid:3)t12t21)1/2A − S†(E)−1 −t21t12A  h i  −1 −1 where A= S(E)−1 + S†(E)−1 +1. With the use of the formula (cid:2) (cid:3) h i a b −1 a−bd−1c −1 −a−1b d−ca−1b −1 = , (A3) (cid:18) c d(cid:19) −d−(cid:0)1c a−bd−(cid:1)1c −1 d−(cid:0)ca−1b −1(cid:1) ! (cid:0) (cid:1) (cid:0) (cid:1) the [1, 1] block of Y can be written as follows Y11 =A−1 S†(E)−1 −1+t12t21A S(E)−1 −1A−1 S†(E)−1 −1−t12t21 −1. (A4) (cid:26)h i (cid:27)(cid:26)(cid:2) (cid:3) h i (cid:27) Using the property −1 −1 A= S†(E)−1 S†(E)S(E)−1 S(E)−1 h −i1 h i(cid:2) (cid:3)−1 = S(E)−1 S(E)S†(E)−1 S†(E)−1 , (A5) (cid:2) (cid:3) h ih i we obtain the [1, 1] block of Eq. (3.11b). Similarly the [2, 1] block of Y can be written as follows −1 −1 Y21 =it21(1+t12t21)1/2 S†(E)−1 +t12t21A AY11. (A6) (cid:26)h i (cid:27) Substituting Eq. (A4) for Y11 in Eq. (A6), we obtain the [2, 1] block of Eq. (3.11b). The other blocks are obtained along exactly parallel lines. [1] T. Guhr, A.Mu¨ller-Groeling, and H.A. Weidenmu¨ller, Phys.Rep.299, 189 (1998). [2] In this article, we will restrict our attention to the correlations on the scale of the local mean level spacing in the bulk of the spectrum. For the universalities on the other regions see, e.g., E. Kanzieper and V. Freilikher, in Diffuse Waves in Complex Media, edited by J. P. Fouque,NATOASI Series C 531 (Kluwer, Dordrecht, 1999), p. 165. [3] G. Hackenbroich and H.A. Weidenmu¨ller, Phys.Rev. Lett.74, 4118 (1995). [4] E. Br´ezin, S. Hikami, and A. Zee, Phys.Rev. E 51, 5442 (1995). [5] E. Br´ezin and S. Hikami, Nucl. Phys.B 479, 697 (1996); Phys.Rev.E 55, 4067 (1997); 56, 264 (1997). [6] P.Zinn-Justin, Commun. Math. Phys. 194, 631 (1998). [7] N.Mae and S. Iida(unpublished). [8] J. J. M. Verbaarschot, H.A. Weidenmu¨ller, and M. R. Zirnbauer, Phys.Rep.129, 367 (1985). [9] M. L. Mehta, Random Matrices (Academic Press, New York,1991). [10] K.B. Efetov, Supersymmetry in Disorder and Chaos (Cambridge University Press, 1997). [11] A.Zee, Nucl. Phys. B 474, 726 (1996). [12] Thetransmission coefficients(orstickingprobabilities) inRef.[8]correspond totheeigenvaluesofT12 (whichareequalto those of T21). [13] C. W. J. Beenakker, Rev.Mod. Phys. 69, 731 (1997). 5