Random Matrix Theory in Physics

RANDOM MATRIX THEORY IN PHYSICS

π π Thomas Guhr, Lunds Universitet, Lund, Sweden p(W)(s) = s exp s2 . (2) 2 − 4 As shown in Figure 2, the Wigner surmise excludes de- Introduction generacies, p(W)(0) = 0, the levels repel each other. This is only possible if they are correlated. Thus, the Poisson We wish to study energy correlations of quantum spec- law and the Wigner surmise reﬂect the absence or the tra. Suppose the spectrum of a quantum system has presence of energy correlations, respectively. been measured or calculated. All levels in the total spectrum having the same quantum numbers form one particular subspectrum. Its energy levels are at positions 1.0 xn, n = 1, 2, . . . , N, say. We assume that N, the number of levels in this subspectrum, is large. With a proper ) s

smoothing procedure, we obtain the level density R1(x), ( 0.5 i.e. the probability density of ﬁnding a level at the energy p x. As indicated in the top part of Figure 1, the level density R1(x) increases with x for most physics systems. 0.0 In the present context, however, we are not so interested 0 1 2 3 in the level density. We want to measure the spectral cor- s relations independently of it. Hence, we have to remove FIG. 2. Wigner surmise (solid) and Poisson law (dashed). the level density from the subspectrum. This is referred to as unfolding. We introduce a new dimensionless en- Now, the question arises: If these correlation patterns ergy scale ξ such that dξ = R (x)dx. By construction, 1 are so frequently found in physics, is there some simple, the resulting subspectrum in ξ has level density unity phenomenological model? — Yes, Random Matrix The- as shown schematically in the bottom part of Figure 1. ory (RMT) is precisely this. To describe the absence of It is always understood that the energy correlations are correlations, we choose, in view of what has been said analyzed in the unfolded subspectra. above, a diagonal Hamiltonian

H = diag (x1, . . . , xN ) , (3)

whose elements, the eigenvalues xn, are uncorrelated ran- ¡ dom numbers. To model the presence of correlations, we insert oﬀ–diagonal matrix elements,

H H N 11 · · · 1 . . ¢ H =  . .  . (4) HN HNN  1 · · ·  FIG. 1. Original (top) and unfolded (bottom) spectrum. We require symmetry H T = H. The independent elements Hnm are random numbers. The random matrix Surprisingly, a remarkable universality is found in the H is diagonalized to obtain the energy levels xn, n = spectral correlations of a huge class of systems, includ- 1, 2, . . . , N. Indeed, a numerical simulation shows that ing nuclei, atoms, molecules, quantum chaotic and dis- these two models yield, after unfolding, the Poisson law ordered systems and even quantum chromodynamics on and the Wigner surmise for large N, i.e. the absence or the lattice. Consider the nearest neighbor spacing distri- presence of correlations. This is the most important in- bution p(s). It is the probability density of finding two sight into the phenomenology of RMT. adjacent levels in the distance s. If the positions of the In this article, we setup RMT in a more formal way, levels are uncorrelated, the nearest neighbor spacing dis- we discuss analytical calculations of correlation functions, tribution can be shown to follow the Poisson law we demonstrate how this relates to supersymmetry and stochastic field theory, we show the connection to chaos, p(P)(s) = exp( s) . (1) we briefly sketch the numerous applications in many– − body physics, in disordered and mesoscopic systems, in While this is occasionally found, many more systems models for interacting fermions and in quantum chromo- show a rather different nearest neighbor spacing distri- dynamics. We also mention applications in other fields, bution, the Wigner surmise even beyond physics.

1 Random Matrix Theory s. This also becomes obvious by working out the differ- ential probability P (β)(H)d[H] of the random matrices Classical Gaussian Ensembles N H in eigenvalue–angle coordinates x and U. Here, d[H] For now, we consider a system whose energy levels are is the invariant measure or volume element in the matrix correlated. The N N matrix H modeling it has no fixed space. When writing d[ ], we always mean the product of · zeros but random×entries everywhere. There are three all differentials of independent variables for the quantity possible symmetry classes of random matrices in stan- in the square brackets. Up to constants, we have dard Schrödinger quantum mechanics. They are labeled β d[H] = ∆N (x) d[x]dµ(U) , (6) by the Dyson index β. If the system is not time–reversal | | invariant, H has to be Hermitean and the random entries where dµ(U) is, apart from certain phase contribu- Hnm are complex (β = 2). If time–reversal invariance tions, the invariant or Haar measure on O(N), U(N) or holds, two possibilities must be distinguished: If either USp(2N), respectively. The Jacobian of the transforma- the system is rotational symmetric, or it has integer spin tion is the modulus of the Vandermonde determinant and rotational symmetry is broken, the Hamilton matrix H can be chosen real symmetric (β = 1). This is the ∆N (x) = (xn xm) (7) − case in eqn [4]. If, on the other hand, the system has n

(β) (β) β 2 Once more, we used rotational invariance which implies PN (H) = CN exp 2 tr H , (5) (β) −4v that PN (x) is invariant under permutation of the levels xn. Since the same then also holds for the correlation where the constant v sets the energy scale and the con- (β) functions [8], it is convenient to normalize them to the stant CN ensures normalization. The three symme- combinatorial factor in front of the integrals. A constant try classes together with the probability densities [5] de- (β) ensuring this has been absorbed into PN (x). fine the Gaussian ensembles: the Gaussian orthogonal Remarkably, the integrals in eqn [8] can be done in (GOE), unitary (GUE) and symplectic (GSE) ensemble closed form. The GUE case (β = 2) is mathematically for β = 1, 2, 4. the simplest and one finds the determinant structure The phenomenology of the three Gaussian ensembles differs considerably. The higher β, the stronger the level (2) (2) Rk (x1, . . . , xk) = det[KN (xp, xq)]p,q=1,...,k . (9) repulsion between the eigenvalues xn. Numerical simulation quickly shows that the nearest neighbor spacing All entries of the determinant can be expressed in terms (β) β (2) distribution behaves like p (s) s for small spacings of the kernel K (xp, xq) which depends on two energy ∼ N

2 arguments (xp, xq). Analogous but more complicated Statistical Observables formulae are valid for the GOE (β = 1) and the GSE (β = 4), involving quaternion determinants and integrals The unfolded correlation functions yield all statistical ob- and derivatives of the kernel. servables. The two–level correlation function X2(r) with r = ξ ξ is of particular interest in applications. If we As argued in the Introduction, we are interested in 1 − 2 the energy correlations on the unfolded energy scale. The do not write the superscript (β) or (P), we mean either (β) level density is formally the one–level correlation func- of the functions. For the Gaussian ensembles, X2 (r) is tion. For the three Gaussian ensembles it is, to leading shown in Figure 3. One often writes X2(r) = 1 Y2(r). − order in the level number N, the Wigner semicircle The two–level cluster function Y2(r) nicely measures the deviation from the uncorrelated Poisson case, where one 1 (P) (P) (β) 2 2 has X (r) = 1 and Y (r) = 0. R1 (x1) = 4Nv x1 (10) 2 2 2πv2 q − for x1 2√Nv and zero for x1 > 2√Nv. None of 1.5 the |common| ≤ systems in physics |has| such a level density. When unfolding, we also want to take the limit of in- ) 1.0 r (

finitely many levels N to remove cutoff effects due )

→ ∞ ( to the finite dimension of the random matrices. It suf- 2 fices to stay in the center of the semicircle where the mean X 0.5 (β) level spacing is D = 1/R1 (0) = πv/√N. We introduce 0.0 the dimensionless energies ξp = xp/D, p = 1, . . . , k which have to be held fixed when taking the limit N . The 0 1 2 3 unfolded correlation functions are given by → ∞ r FIG. 3. Two–level correlation function X (β)(r) for GOE (β) k (β) 2 X (ξ1, . . . , ξk) = lim D R (Dξ1, . . . , Dξk) . (11) (solid), GUE (dashed) and GSE (dotted). k N→∞ k

As we are dealing with probability densities, the Jaco- By construction, the average level number in an inter- bians dxp/dξp enter the reformulation in the new energy val of length L in the unfolded spectrum is L. The level k variables. This explains the factor D . Unfolding makes number variance Σ2(L) is shown to be an average over the correlation functions translation invariant, they de- the two–level cluster function, pend only on the diﬀerences ξp ξq. The unfolded corre- − lation functions can be written in a rather compact form. L For the GUE (β = 2) they read 2 Σ (L) = L 2 (L r)Y2(r)dr . (15) − Z − (2) sin π(ξp ξq) 0 Xk (ξ1, . . . , ξk) = det − . (12) π(ξp ξq) − p,q=1,...,k We ﬁnd L Σ2(L) levels in an interval of length L. There are similar, but more complicated formulae for the 2(P) GOE (β = 1) and the GSE (β = 4). By construction, In the uncorrelatedp Poisson case, one has Σ (L) = L. This is just Poisson’s error law. For the Gaussian ensem- one has X(β)(ξ ) = 1. 1 1 bles Σ2(β)(L) behaves logarithmically for large L. The It is useful to formulate the case where correlations are spectrum is said to be more rigid than in the Poisson absent, i.e. the Poisson case, accordingly. The level den- (P) case. As Figure 4 shows, the level number variance sity R1 (x1) is simply N times the (smooth) probability probes longer distances in the spectrum, in contrast to density chosen for the entries in the diagonal matrix [4]. the nearest neighbor spacing distribution. Lack of correlations means that the k–level correlation function only involves one–level correlations, 2 k (P) N! (P)

Rk (x1, . . . , xk) = k R1 (xp) . (13) ) (N k)!N L p=1 ( 1 − Y 2 The combinatorial factor is important, since we always normalize to N!/(N k)!. Hence, one ﬁnds − 0 0 1 2 3 (P) Xk (ξ1, . . . , ξk) = 1 (14) L FIG. 4. Level number variance Σ2(L) for GOE (solid) and for all unfolded correlation functions. Poisson case (dashed).

3 Many more observables, also sensitive to higher order U = [u u uN ] are these eigenvectors. The proba- 1 2 · · · k > 2 correlations, have been defined. In practice, how- bility density of the components unm of the eigenvector ever, one is often restricted to analyzing two–level cor- un can be calculated rather easily. For large N it ap- relations. An exception is, to some extent, the nearest proaches a Gaussian. This is equivalent to the Porter– neighbor spacing distribution p(s). It is the two–level Thomas distribution. While wavefunctions are often not correlation function with the additional requirement that accessible in an experiment, one can measure transition the two levels in question are adjacent, i.e. that there are amplitudes and widths, giving information about the ma- no levels between them. Thus, all correlation functions trix elements of a transition operator and a projection of are needed if one wishes to calculate the exact nearest the wavefunctions onto a certain state in Hilbert space. neighbor spacing distribution p(β)(s) for the Gaussian If the latter are represented by a fixed matrix A or a fixed ensembles. These considerations explain that we have vector a, respectively, one can calculate the RMT predic- (β) (β) (β) X2 (s) p (s) for small s. But while X2 (s) satu- tion for the probability densities of the matrix elements ' (β) † † rates for large s, p (s) quickly goes to zero in a Gaus- unAum or the widths a un from the probability density sian fashion. Thus, although the nearest neighbor spac- of the eigenvectors. ing distribution mathematically involves all correlations, Scattering systems it makes in practice only a meaningful statement about the two–level correlations. Luckily, p(β)(s) differs only It is important that RMT can be used as a powerful tool very slightly from the heuristic Wigner surmise [2] (cor- in scattering theory, because the major part of the exper- responding to β = 1), respectively from its extensions imental information about quantum systems comes from (corresponding to β = 2 and β = 4). scattering experiments. Consider an example from compound nucleus scattering. In an accelerator, a proton is Ergodicity and Universality shot on a nucleus, with which it forms a compound nu- We constructed the correlation functions as averages over cleus. This then decays by emitting a neutron. More gen- an ensemble of random matrices. But this is not how we erally, the ingoing channel ν (the proton in our example) proceeded in the data analysis sketched in the Introduc- connects to the interaction region (the nucleus), which tion. There, we started from one single spectrum with also connects to an outgoing channel µ (the neutron). very many levels and obtained the statistical observable There are Λ channels with channel wavefunctions which just by sampling and, if necessary, smoothing. Do these are labeled ν = 1, . . . , Λ. The interaction region is de- ensemble average spectral av- scribed by an N N Hamiltonian matrix H whose eigen- two averages, the and the × erage, yield the same? — Indeed, one can show that values xn are bound state energies labeled n = 1, . . . , N. the answer is affirmative, if the level number N goes to The dimension N is a cutoff which has to be taken to ergodicity infinity at the end of a calculation. The Λ Λ scattering infinity. This is referred to as in RMT. × Moreover, as already briefly indicated in the Intro- matrix S contains the information about how the ingo- duction, very many systems from different areas of ing channels are transformed into the outgoing channels. physics are well described by RMT. This seems to be The scattering matrix S is unitary. Under certain and at odds with the Gaussian assumption [5]. There is often justified assumptions, a scattering matrix element hardly any system whose Hamilton matrix elements fol- can be cast into the form low a Gaussian probability density. The solution for this † −1 Sνµ = δνµ i2πW G Wµ . (16) puzzle lies in the unfolding. Indeed, it has been shown − ν that almost all functional forms of the probability density The couplings Wnν between the bound states n and the (β) PN (H) yield the same unfolded correlation functions, if channels ν are collected in the N Λ matrix W , Wν is no new scale comparable to the mean level spacing is its νth column. The propagator G×−1 is the inverse of present in P (β)(H). This is the mathematical side of the N † universality G = z1N H + iπ Wν Wν . (17) empirically found . − Ergodicity and universality are of crucial importance ν Xopen for the applicability of RMT in data analysis. Here, z is the scattering energy and the summation is only over channels which are open, i.e. accessible. For- Wavefunctions mula [16] has a clear intuitive interpretation. The scat- By modeling the Hamiltonian of a system with a ran- tering region is entered through channel ν, the bound dom matrix H, we do not only make an assumption states of H become resonances in the scattering pro- about the statistics of the energies, but also about those cess according to eqn [17], the interaction region is left of the wavefunctions. Because of the eigenvalue equa- through channel µ. This formulation applies in many tion Hun = xnun, n = 1, . . . , N, the wavefunction be- areas of physics. All observables such as transmission longing to the eigenenergy xn is modeled by the eigen- coefficients, cross sections and others can be calculated vector un. The columns of the diagonalizing matrix from the scattering matrix S.

4 We have not made any statistical assumptions yet. Of- which depends on the energies and k new source variables ten, one can understand generic features of a scattering Jp, p = 1, . . . , k ordered in 2k 2k diagonal matrices system by assuming that the Hamiltonian H is a random × matrix, taken from one of the three classical ensembles. x = diag (x1, x1, . . . , xk, xk)

This is one RMT approach used in scattering theory. J = diag (+J1, J1, . . . , +Jk, Jk) . (21) Another RMT approach is based on the scattering ma- − − trix itself, S is modeled by a Λ Λ unitary random ma- (2) × We notice the normalization Zk (x) = 1 at J = 0. The trix. Taking into account additional symmetries, one ar- generating function [20] is an integral over an ordinary rives at the three circular ensembles, circular orthogonal N N matrix H. It can be exactly rewritten as an inte- (COE), unitary (CUE) and symplectic (CSE). They cor- gral×over a 2k 2k supermatrix σ containing commuting respond to the three classical Gaussian ensembles and are and anticomm×uting variables, also labeled with the Dyson index β = 1, 2, 4. The eigen- phases of the random scattering matrix correspond to the (2) (2) Z (x + J) = Q (σ)sdet −N (x + J σ)d[σ] . (22) eigenvalues of the random Hamiltonian matrix. The un- k Z k − folded correlation functions of the circular ensembles are identical to those of the Gaussian ensembles. The integrals over the commuting variables are of the ordinary Riemann–Stiltjes type, while those over the anti- Supersymmetry commuting variables are Berezin integrals. The Gaussian probability density [5] is mapped onto its counterpart in Apart from the symmetries, random matrices contain superspace nothing but random numbers. Thus a certain type of redundancy is present in RMT. Remarkably, this redun- (2) (2) 1 dancy can be removed, without losing any piece of infor- Q (σ) = c exp str σ2 , (23) k k −2v2 mation by using supersymmetry, i.e. by a reformulation of the random matrix model involving commuting and where c(2) is a normalization constant. The supertrace anticommuting variables. For the sake of simplicity, we k str and the superdeterminant sdet generalize the cor- sketch the main ideas for the GUE, but they apply to the responding invariants for ordinary matrices. The total GOE and the GSE accordingly. number of integrations in eqn [22] is drastically reduced One defines the k–level correlation functions by using as compared to eqn [20]. Importantly, it is independent the resolvent of the Schrödinger equation, of the level number N which now only appears as the neg- (2) ative power of the superdeterminant in eqn [22], i.e. as an Rk (x1, . . . , xk) = k explicit parameter. This most convenient feature makes b 1 1 P (2)(H) tr d[H] . (18) it possible to take the limit of infinitely many levels by πk Z N x H means of a saddle point approximation to the generating pY=1 p − function. The energies carry an imaginary increment x = xp iε Loosely speaking, the supersymmetric formulation can p and the limit ε 0 has to be taken at the end be viewed as an irreducible representation of RMT which → of the calculation. The k–level correlation functions yields a clearer insight into the mathematical structures. (2) Rk (x1, . . . , xk) as defined in eqn [8] can always be ob- The same is true for applications in scattering theory and tained from the functions [18] by constructing a linear in models for crossover transitions to be discussed be- (2) low. This explains why supersymmetry is so often used combination of the Rk (x1, . . . , xk) in which the signs of the imaginary increments are chosen such that only the in RMT calculations. b imaginary parts of the traces contribute. Some trivial δ– It should be emphasized that the rôle of supersym- distributions have to be removed. The k–level correlation metry in RMT is quite different from the one in high functions [18] can be written as the k–fold derivative energy physics where the commuting and anticommuting variables represent physical particles, bosons and k (2) 1 ∂ (2) fermions, respectively. This is not so in the RMT con- R (x1, . . . , xk) = Z (x + J) k (2π)k k k text. The commuting and anticommuting variables have p=1 ∂Jp J=0 b no direct physics interpretation, they appear simply as Q (19) helpful mathematical devices to cast the RMT model into of the generating function an often much more convenient form.

k Crossover Transitions (2) (2) det(xp + Jp H) Zk (x + J) = PN (H) − d[H] , Z det(xp Jp H) The RMT models discussed up to now describe four ex- pY=1 − − treme situations, the absence of correlations in the Pois- (20) son case and the presence of correlations as in the three

5 fully rotational invariant models GOE, GUE and GSE. simplified form. It has a rather general meaning for A real physics system, however, is often between these harmonic analysis on symmetric spaces, connecting to extreme situations. The corresponding RMT models can the spherical functions of Gelfand and Harish–Chandra, vary considerably, depending on the specific situation. Itzykson–Zuber integrals and to Calogero–Sutherland Nevertheless, those models in which the random matrices models of interacting particles. All this generalizes to for two extreme situations are simply added with some superspace. In the supersymmetric version of Dyson’s weight are useful in so many applications that they ac- Brownian motion the generating function of the correla- quired a rather generic standing. One writes tion functions is propagated, H(α) = H(0) + αH(β) , (24) 4 ∂ (0) ∆ Z (s, t) = Z (s, t) , (27) where H is a random matrix drawn from an en- s k β ∂t k semble with a completely arbitrary probability density P (0)(H(0)). The case of a fixed matrix is included, be- N (0) cause one may choose a product of δ–distributions for where the initial condition Zk(s, 0) = Zk (s) is the gen- the probability density. The matrix H (β) is random and erating function of the correlation functions for the arbi- drawn from the classical Gaussian ensembles with prob- trary ensemble. Here, s denotes the eigenvalues of some (β) (β) supermatrices, not to be confused with the spacing be- ability density PN (H ) for β = 1, 2, 4. One requires that the group diagonalizing H (0) is a subgroup of the one tween adjacent levels. Since the Laplacean ∆s lives in diagonalizing H(β). The model [24] describes a crossover this curved eigenvalue space, this diffusion process es- transition. The weight α is referred to as transition pa- tablishes an intimate connection to harmonic analysis on rameter. It is useful to choose the spectral support of superspaces. Advantageously, the diffusion [27] is the H(0) and H(β) equal. One can then view α as the root– same on the original and on the unfolded energy scales. mean–square matrix element of H (β). At α = 0, one has the arbitrary ensemble. The Gaussian ensembles are for- Fields of Application mally recovered in the limit α , to be taken in a proper way such that the energies→remain∞ finite. Many–Body Systems We are always interested in the unfolded correlation functions. Thus, α has to be measured in units of the Numerous studies apply RMT to nuclear physics which is mean level spacing D such that λ = α/D is the physi- also the field of its origin. If the total number of nucleons, cally relevant transition parameter. It means that, de- i.e. protons and neutrons, is not too small, nuclei show pending on the numerical value of D, even a small effect single–particle and collective motion. Roughly speaking, on the original energy scale can have sizeable impact on the former is decoherent out–of–phase motion of the nu- the spectral statistics. This is referred to as statistical cleons confined in the nucleus, while the latter is coherent enhancement. The nearest neighbor spacing distribution in–phase motion of all nucleons or of large groups of them is already very close to p(β)(s) for the Gaussian ensembles such that any additional individual motion of the nucle- if λ is larger than 0.5 or so. In the long range observables ons becomes largely irrelevant. It has been shown em- such as the level number variance Σ2(L), the deviation pirically that the single–particle excitations lead to GOE from the Gaussian ensemble statistics becomes visible at statistics, while collective excitations produce different interval lengths L comparable to λ. statistics, often of the Poisson type. Mixed statistics as Crossover transitions can be interpreted as diffusion described by crossover transitions are then of particular processes. With the fictitious time t = α2/2, the prob- interest to investigate the character of excitations. For example, one applies the model [24] with H (0) drawn ability density PN (x, t) of the eigenvalues x of the total (β) Hamilton matrix H = H(t) = H(α) satisfies the diffusion from a Poisson ensemble and H from a GOE. Another equation application of crossover transitions is breaking of time– reversal invariance in nuclei. Here, H (0) is from a GOE 4 ∂ and H(β) from a GUE. Indeed, a fit of spectral data to ∆xPN (x, t) = PN (x, t) , (25) β ∂t this model yields an upper bound for the time–reversal where the probability density for the arbitrary ensem- invariance violating root–mean–square matrix element in (0) nuclei. Yet another application is breaking of symmetries ble is the initial condition PN (x, 0) = PN (x). The Laplacean such as parity or isospin. In the case of two quantum numbers, positive and negative parity, say, one chooses N ∂2 β ∂ ∂ H(0) = diag (H(+), H(−)) block–diagonal with H (+) and (−) (β) ∆x = 2 + (26) H drawn from two uncorrelated GOE and H from ∂xn xn xm ∂xn − ∂xm nX=1 n

6 Nuclear excitation spectra are extracted from scatter- A wealth of such empirical studies led to the Bohigas– ing experiments. An analysis as described above is only Giannoni–Schmit conjecture. We state it here not in its possible if the resonances are isolated. Often, this is not original, but in a frequently used form: Spectra of systems the case and the resonance widths are comparable to or whose classical analogues are fully chaotic show correla- even much larger than the mean level spacing, making tion properties as modeled by the Gaussian ensembles. it impossible to obtain the excitation energies directly The Berry–Tabor conjecture is complementary: Spec- from the cross sections. One then analyzes the latter and tra of systems whose classical analogues are fully reg- their fluctuations as measured and applies the concepts ular show correlation properties which are often those sketched above for scattering systems. This approach has of the Poisson type. As far as concrete physics appli- also been successful for crossover transitions. cations are concerned, these conjectures are well–posed. Due to the complexity of the nuclear many–body prob- From a strict mathematical viewpoint, they have to be lem, one has to use effective or phenomenological interac- supplemented with certain conditions to exclude excep- tions when calculating spectra. Hence, one often studies tions such as Artin’s billiard. Due to the definition of whether the statistical features found in the experimen- this system on the hyperbolic plane, its quantum version tal data are also present in the calculated spectra which shows Poisson–like statistics, although the classical dy- result from the various models for nuclei. namics is chaotic. Up to now, no general and mathemat- Other many–body systems, such as complex atoms and ically rigorous proofs of the conjectures could be given. molecules, have also been studied with RMT concepts, However, semiclassical reasoning involving periodic orbit but the main focus has always been on nuclei. theory and, in particular, the Gutzwiller trace formula, yields at least a heuristic understanding. Quantum Chaos Quantum chaos has been studied in numerous systems. Originally, RMT was intended for modeling systems with An especially prominent example is the Hydrogen atom many degrees of freedom such as nuclei. Surprisingly, put in a strong magnetic field which breaks the integra- RMT proved useful for systems with few degrees of free- bility and drives the correlations towards the GOE limit. dom as well. Most of these studied aim at establishing a Disordered and Mesoscopic Systems link between RMT and classical chaos. Consider as an example the classical motion of a point–like particle in a An electron moving in a probe, a piece of wire, say, is rectangle billiard. Ideal reflection at the boundaries and many times scattered at impurities in the material. This absence of friction are assumed, implying that the parti- renders the motion diffusive. In a statistical model, one cle is reflected infinitely many times. A second billiard is writes the Hamilton operator as a sum of the kinetic part, built by taking a rectangle and replacing one corner with i.e. the Laplacean, and a white–noise disorder potential a quarter circle as shown in Figure 5. The motion of V (~r) with second moment the particle in this Sinai billiard is very different from the 0 (d) 0 one in the rectangle. The quarter circle acts like a convex V (~r)V (~r ) = cV δ (~r ~r ) . (28) mirror which spreads out the rays of light upon reflection. h i − This effect accumulates, because the vast majority of the Here, ~r is the position vector in d dimensions. The con- possible trajectories hit the quarter circle infinitely many stant cV determines the mean free time between two scat- times under different angles. This makes the motion in tering processes in relation to the density of states. It is the Sinai billiard classically chaotic, while the one in the assumed that phase coherence is present such that quan- rectangle is classically regular. The rectangle is separa- tum effects are still significant. This defines the meso- ble and integrable, while this feature is destroyed in the scopic regime. The average over the disorder potential Sinai billiard. One now quantizes these billiard systems, can be done with supersymmetry. In fact, this is the calculates the spectra and analyzes their statistics. Up context in which supersymmetric techniques in statisti- to certain scales, the rectangle (for irrational squared ra- cal physics were developed, before they were applied to tio of the side lengths) shows Poisson behavior, the Sinai RMT models. In the case of weak disorder, the result- billiard yields GOE statistics. ing field theory in superspace for two–level correlations acquires the form

dµ(Q)f(Q) exp ( S(Q)) (29) Z −

where f(Q) projects out the observable under considera- tion and where S(Q) is the eﬀective Lagrangean

S(Q) = str ( Q(~r))2 + i2rMQ(~r) ddr . (30) FIG. 5. The Sinai billiard. − Z D ∇

7 This is the supersymmetric non–linear σ model. It is RMT for scattering systems could be applied and led to used to study level correlations, but also to obtain infor- numerous new results. mation about the conductance and conductance fluctuations when the probe is coupled to external leads. The Quantum Chromodynamics supermatrix field Q(~r) is the remainder of the disorder Quarks interact by exchanging gluons. In quantum chro- average, its matrix dimension is four or eight, depending modynamics, the gluons are described by gauge fields. on the symmetry class. This field is a Goldstone mode. Relativistic quantum mechanics has to be used. Ana- It does not directly represent a particle as often the case lytical calculations are only possible after some drastic in high energy physics. The matrix Q(~r) lives in a coset assumptions and one must resort to lattice gauge theory, space of certain supergroups. A tensor M appears in the i.e. to demanding numerics, to study the full problem. calculation, and r is the energy difference on the unfolded The massless Dirac operator has chiral symmetry, scale, not to be confused with the position vector ~r. implying that all non–zero eigenvalues come in pairs The first term in the effective Lagrangean involving a ( λn, +λn) symmetrically around zero. In chiral RMT, − gradient squared is the kinetic term, it stems from the the Dirac operator is replaced with block off–diagonal Laplacean in the Hamiltonian. The constant is the matrices classical diffusion constant for the motion of theD elec- 0 Wb tron through the probe. The second term is the ergodic W = † , (31) W 0 term. In the limit of zero dimensions, d 0, the ki- b → netic term vanishes and the remaining ergodic term yields where Wb is a random matrix without further symme- precisely the unfolded two–level correlations of the Gaus- tries. By construction, W has chiral symmetry. The as- sian ensembles. Thus, RMT can be viewed as the zero– sumption underlying chiral RMT is that the gauge fields dimensional limit of field theory for disordered systems. effectively randomize the motion of the quark. Indeed, For d > 0, there is a competition between the two terms. this simple schematic model correctly reproduces low– The diffusion constant and the system size determine energy sum rules and spectral statistics of lattice gauge D an energy scale, the Thouless energy Ec, within which calculations. Near the center of the spectrum, there is the spectral statistics is of the Gaussian ensemble type a direct connection to the partition function of quan- and beyond which it approaches the Poisson limit. In tum chromodynamics. Furthermore, a similarity to dis- Figure 6, this is schematically shown for the level num- ordered systems exists and an analogue of the Thouless ber variance Σ2(L) which bends from Gaussian ensemble energy could be found. to Poisson behavior when L > Ec. This relates to the crossover transitions in RMT. Gaussian ensemble statis- Other Fields tics means that the electron states extend over the probe, Of the wealth of further investigations, we can mention while Poisson statistics implies their spatial localization. but a few. RMT is in general useful for wave phenomena Hence, the Thouless energy is directly the dimensionless of all kinds, including classical ones. This has been shown conductance. for elastomechanical and electromagnetic resonances. An important field of application is quantum gravity 2.0 and matrix model aspects of string theory. We decided

1.5 not to go into this, because the reason for the emergence

) of RMT concepts there is very diﬀerent from everything L

( 1.0 2 else discussed above. 0.5 RMT is also successful beyond physics. Not surprisingly, it always received interest in mathematical statis- 0.0 tics, but, as already said, it also relates to harmonic anal- 0 20 40 60 80 L ysis. A connection to number theory exists as well. The FIG. 6. Level number variance Σ2(L). In this example, high–lying zeros of the Riemann ζ function follow the the Thouless energy is Ec ≈ 10 on the unfolded scale. The GUE predictions over certain interval lengths. Unfortu- Gaussian ensemble behavior is dashed. nately, a deeper understanding is still lacking. As the interest in statistical concepts grows, RMT keeps ﬁnding new applications. Recently, one even A large number of issues in disordered and mesoscopic started using RMT for risk management in ﬁnance. systems has been studied with the supersymmetric non– linear σ model. Most results have been derived for quasi– See also one–dimensional systems. Through a proper discretiza- tion, a link is established to models involving chains of 318 — Classical mechanics random matrices. As the conductance can be formulated 390 — Quantum mechanics in terms of the scattering matrix, the experience with 453 — Quantum mechanical scattering theory

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