Correlations of quantum curvature and variance of Chern numbers
SSciPost Physics Submission
Correlations of quantum curvature and variance of Chernnumbers
Omri Gat , and Michael Wilkinson † Racah institute of Physics, Hebrew University, Jerusalem 91904, Israel Chan Zuckerberg Biohub, 499 Illinois Street, San Francisco, CA 94158, USA School of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes,MK7 6AA, England*[email protected] † [email protected] 8, 2020 Abstract
We analyse the correlation function of the quantum curvature in complex quan-tum systems, using a random matrix model to provide an exemplar of a universalcorrelation function. We show that the correlation function diverges as the in-verse of the distance at small separations. We also define and analyse a correlationfunction of mixed states, showing that it is finite but singular at small separa-tions. A scaling hypothesis on a universal form for both types of correlations issupported by Monte-Carlo simulations. We relate the correlation function of thecurvature to the variance of Chern integers which can describe quantised Hallconductance.
Contents a r X i v : . [ c ond - m a t . d i s - nn ] D ec ciPost Physics Submission6 Conclusion 22References 23 The quantum curvature Ω n of an eigenstate of a quantum system (with index n ) is an ob-ject which characterises the sensitivity of the eigenfunction to variation of parameters of theHamiltonian. It plays an important role in the the dynamics of quantum systems [1–4]. Inthis paper we characterise fluctuations of the quantum curvature in generic complex quantumsystems (which have many energy levels and no constants of motion or Anderson localisationeffects). We analyse correlations of the quantum curvature in parameter space using randommatrix models [5, 6], which are applicable to generic complex quantum systems upon applica-tion of a scaling transformation. We relate the correlation function to statistics of the Chernnumbers, which arise in the analysis of quantised conductance phenomena.The quantum curvature is defined for a system with a Hamiltonian ˆ H , which depends uponat least two parameters (with the position in parameter space being denoted by X = ( X , X )).It may be defined for a non-degenerate level by writingΩ n d X ∧ d X = − i tr (cid:104) ˆ P n d ˆ P n ∧ d ˆ P n (cid:105) , (1)where ˆ P n = | φ n (cid:105)(cid:104) φ n | is the projection onto the eigenstate | φ n (cid:105) of ˆ H with index n . Severaldynamical applications of Ω n have been discovered. Mead and Truhlar [1] showed that when X is varied slowly, there is a component of the Born-Oppenheimer reaction force which isproportional to the product of Ω n and to the rate of change of parameters, ˙ X . Related appli-cations arise in solid-state physics [2, 3]. Berry [4] emphasised that the integral of Ω n over anarbitrary surface is proportional to a ‘geometric phase’ which appears in adiabatic approx-imations to the wavefunction, and this is our motivation for referring to Ω n as a ‘quantumcurvature’. Note, however, that Ω n is identically zero if the Hamiltonian can be representedby a real-valued matrix.In the applications considered in [1, 3, 4], the parameter X is varied slowly as a function oftime. This can result in transitions between energy levels, so that the system will evolve toa mixed state. In particular, near-degeneracies of levels will allow Landau-Zener transitionsbetween states [7], which results in a diffusive spread of the probability of a given level beingoccupied [8]. For this reason we shall also consider a ‘smoothed’ curvature, ¯Ω ε ( E ), involvinga weighted average of Ω n over an energy interval of length ε centred at E :¯Ω ε ( E, X ) = (cid:88) n Ω n ( X ) w ε ( E − E n ( X )) w ε ( E ) = 1 √ πε exp( − E / ε ) . (2)A Gaussian smoothing is preferred here because this is the kernel for diffusive spread overenergy levels. We assume ρε (cid:29)
1, so that many levels are included in the average, but that ε ciPost Physics Submission is small compared to other energy scales in the system. Another motivation for considering¯Ω ε is that we shall see that the dependence of its statistics upon ε allows inference aboutcorrelation of the Ω n between different values of the level index, n .It is known that quantum systems with many energy levels may exhibit universal behaviourif there are no constants of motion other than the Hamiltonian, and no Anderson localisationeffects . These universal properties are most conveniently computed using random matrixensembles [5, 6, 9]. We shall discuss the use random matrix models in section 2. There wereview how random matrix approaches have been extended to systems where the Hamiltoniandepends smoothly on a parameter [10–15], and introduce (section 2.4) a hypothesis on theuniversal form of the correlation functions of the quantum curvature. In order to compute thisuniversal correlation function, we consider a random matrix model in which the Hamiltoniandepends smoothly upon two parameters. It suffices to consider a model (introduced in section2.5) in which (cid:104) Ω (cid:105) = 0, and for which the statistics are homogeneous and isotropic in parameterspace. For this model we investigate two correlation functions C nm ( X ) = (cid:104) Ω n ( X , m (0 , (cid:105) , C (∆ E, X ) = (cid:104) ¯Ω ε ( E + ∆ E, X ) ¯Ω ε ( E , (cid:105) (3)where X = | X | (angle brackets denote expectation values throughout).Thouless et al. [2] showed that Ω n arises in an evaluation of the Hall conductance via theKubo formula, and that the the Hall conductance of a filled band is quantised by arguing that N n = 12 π (cid:90) BZ d X Ω n ( X ) (4)takes integer values (where, in this case, the parameter X is a Bloch wavevector and where theintegral runs over the Brillouin zone). This topological invariant, known as the Chern index[16]. The integral of Ω n / π over any closed two-dimensional manifold is also an integer-valuedtopological invariant. Later Thouless extended these results to show quantised conductancein ‘sliding’ periodic potentials [3], using adiabatic approximations, akin to those in [1], ratherthan the Kubo formula. We shall use our results on the correlation function C ( X ) to computethe variance Var( N n ) of the Chern integers in our model. We also argue that the correlationfunction of the Chern integers in complex quantum systems is well-approximated by: (cid:104) N n N m (cid:105) − (cid:104) N n (cid:105)(cid:104) N n (cid:105) = 12 Var( N n ) [2 δ nm − δ n,m +1 − δ n,m − ] (5)(we have (cid:104) N n (cid:105) = 0 for our random matrix model). The Chern integers can change by ± ciPost Physics Submission and Chern numbers in random matrix fields, arguing that they are universal, and develop-ing a scaling theory for them. Berry and Shukla [22–24] studied the single-point probabilitydensity function p (Ω) of the curvature, and showed that the distribution has a power lawdecay p (Ω) ∼ | Ω | − / for | Ω | large. The tails of the curvature distribution are dominatedby near-degeneracy events, and the decay exponent, determined by the codimension of thedegeneracies, is small enough that the variance of the single-level curvature (cid:104) Ω (cid:105) is infinite,while the expectation value (cid:104) Ω (cid:105) = 0 converges due to symmetry.As a consequence of the broad distribution of Ω n , the single level correlation functions C nm ( X ), m = n, n ±
1, which are finite for X (cid:54) = 0, diverge as X →
0. The smoothed curvaturecorrelation function C ε (∆ E, X ) is finite for all X , but fluctuations due to near degeneraciesmake it singular at short separations with a discontinuous derivative at X = 0. We calculatethe contribution of near-degeneracy fluctuations to the two-point correlation functions, whichtogether with the one-point correlation function of the smoothed curvature completely de-termines the short-separation behaviour of both the single-level and the smoothed curvaturecorrelation functions. This is the first main theoretical result of this paper; the other mainresult is the scaling forms of the two types of curvature correlation function that are conjec-tured to be universal. Both the short-distance and the scaling of the correlations are comparedwith comprehensive Monte-Carlo simulations, that support the theoretical prediction in thelarge-matrix-size limit.In section 2 we describe and motivate the random matrix models that we use. Section3 discusses our theoretical and numerical results on the correlation functions of the single-level curvature Ω n . The analogous discussion for the correlation function of the smoothedcurvature ¯Ω ε is the subject of section 4. We consider the implications for Chern numbers insection 5, estimating their variance and presenting an argument in support of equation (5).Finally, section 6 discusses our conclusions and prospects for further studies. There is ample evidence for universality of the properties of complex quantum systems (looselydefined as systems with many energy levels, which do not have Anderson localisation effectsor constants of motion which are independent of the Hamiltonian) [5, 9]. The universal prop-erties are manifest in spectral properties which involve small energy scales, or equivalentlyin dynamical behaviour on long timescales. Hermitean random matrix models of complexquantum systems, and have the attractive feature that they may be used to compute theuniversal properties analytically [6].Consider a Hamiltonian depending upon two parameters, X , and X (write X = ( X , X )).The quantum curvature Ω n is a fundamental characterisation of the sensitivity to parametersof the projection ˆ P n onto the level with index n . Following [4], we can use perturbation theoryto express Ω n in terms of matrix elements of derivatives of the Hamiltonian, and energy levels.This leads to the expressionΩ n = Im (cid:88) m (cid:54) = n ∂ H nm ∂ H mn − ∂ H nm ∂ H mn ( E n − E m ) = − i (cid:88) m (cid:54) = n ∂ H nm ∂ H mn − ∂ H nm ∂ H mn ( E n − E m ) . (6)4 ciPost Physics Submission Here E n are eigenvalues of the Hamiltonian ˆ H ( X , X ) with eigenvectors | φ n (cid:105) and ∂ i H nm arematrix elements of derivatives of the Hamiltonian in its eigenbasis: ∂ i H nm = (cid:104) φ n | ∂ ˆ H∂X i | φ m (cid:105) . (7)Equation (6) will be the basis for our calculations of the statistics of the curvatures, Ω n . Inorder to evaluate (6) we require information about statistics of both energy levels and matrixelements. The statistics of the energy levels E n for complex quantum systems have been very extensivelystudied [5, 6, 9]. It is hypothesised that short-ranged statistical properties of the spectrum,such as the probability distribution of the spacing of adjacent levels, are universal once theenergy levels are transformed to levels with unit mean spacing. If N ( E ) is a smooth functionrepresenting the mean integrated density of states, the transformed levels are e n = N ( E n ).In many cases the complex system is close to a classical limit, and the integrated density ofstates can be derived from the Weyl rule [19]. There are three universality classes of bulklevel statistics, which are exemplified by three Gaussian random matrix ensembles. Individualmatrices of our model have the Gaussian unitary ensemble (GUE) statistics, because thecurvature is odd under time reversal, and therefore zero in the other Gaussian ensembles(orthogonal and symplectic) that obey time-reversal symmetry. Equation (6) shows that Ω n diverges if E n approaches degeneracy with either the level above or below. For this reasonthe probability distribution function of the separation of two levels will play a central role inour analysis. If ρ ( E ) = d N/ d E is the mean density of states, then the PDF of the normalisedseparation S = ( E n +1 − E n ) ρ is well approximated by the Wigner surmise: for the GUE thistakes the form P ( S ) = 32 π S exp( − S /π ) . (8)The exact form of the distribution is complicated but when the matrix size is large it tendsto a universal limit, which for S (cid:28) P ( S ) ∼ π S . (9) In order to compute the statistics of the Ω n , we also need information about the statisticsof the matrix elements of derivatives of the Hamiltonian with respect to its parameters. Incomplex quantum systems, theoretical arguments and numerical experiments [10] support theuse of a model where the off-diagonal matrix elements (7) are statistically independent of eachother, independent of the energy levels, and approximately Gaussian distributed, with meanvalue equal to zero. To complete the characterisation of the distribution of these elements,we must specify their variance. The variance is a function of the energies of the two states,5 ciPost Physics Submission and we define σ ij ( E, ∆ E ) = 1 ρ ( E + ∆ E/ ρ ( E − ∆ E/ × (cid:88) n (cid:88) m ∂ i H nm ∂ j H mn w ε ( E − ( E n + E m ) / w ε (∆ E − ( E n − E m )) (10)(where the energy window function w ε is used instead of a Dirac delta function, so that σ ij has a smooth dependence upon its arguments). If the complex quantum system has agood classical limit, the covariance σ ij ( E, ∆ E ) can be calculated using the method describedin [25]. Because (6) implies that small energy separations dominate the sum, it is the value of σ ij ( E, ∆ E ) with ∆ E → n . We can alwaysmake a locally linear transformation of the coordinates ( X , X ) so that the covariance σ ij is amultiple of the identity, with diagonal elements denoted by σ . For convenience, the universalform for the correlation functions that we consider in this work will be computed in such anisotropic coordinate system. However, for the purposes of understanding the dimensions ofexpressions it is convenient to distinguish between derivatives with respect to X and X . Forthis reason we shall express the statistics of Ω n in terms of two variances σ i = (cid:104)| ∂ i H n +1 ,n | (cid:105) . (11)where the angular brackets indicate an average over n : in terms of equation (10), we identify σ i = σ ii ( E, In the case where two levels become nearly degenerate, we can approximate Ω n by a projectiononto a two-level subspace: in section 3 this approach will be used to determine the behaviourof C ( X ) analytically in the limit X →
0. Writeˆ H ( X ) = ˆ H + (cid:88) i =1 , ∂ ˆ H∂X i X i (12)and the matrix elements are H nm = E n δ nm + (cid:88) i =1 , ∂ i H nm X i (13)(where the states | φ n (cid:105) are eigenstates at X = ). Assume that the levels n , n + 1 are nearlydegenerate at X = , with the separation E n +1 − E n being much smaller than other gaps inthe spectrum. In this case the curvature close to X = is determined by the projection ofthe Hamiltonian into the two-level subspace spanned by | φ n (cid:105) and | φ n +1 (cid:105) . The projection ofthe Hamiltonian into this subspace is represented by a 2 × H ( X , X ) = (cid:88) i =0 h i ( X, Y ) τ i (14)where the ˜ σ i are Pauli matrices , τ = (cid:18) (cid:19) , τ = (cid:18) − ii 0 (cid:19) , τ = (cid:18) − (cid:19) (15)6 ciPost Physics Submission with τ equal to the 2 × h = 0. Close to the origin the projected Hamiltonian is then˜ H = (cid:15)τ + (cid:88) i =1 (cid:88) j =1 , W ij τ i X j (16)Here (cid:15) = E n +1 − E n , W ij = ∂h i ∂X j (cid:12)(cid:12)(cid:12)(cid:12) X = X =0 . (17)The W ij are related to the matrix elements of the derivatives as follows: W ,j = Re[ ∂ j H n +1 ,n ] , W ,j = Im[ ∂ j H n +1 ,n ] , W ,j = ∂ j H n +1 ,n +1 − ∂ j H n,n . (18)In a complex system, the matrix elements ∂ j H nm appear random. For a system with acomplex Hermitean Hamiltonian, we expect that Re ∂ i H n +1 ,n , Im ∂ i H n +1 ,n are independentGaussian variables, with mean equal to zero and variance σ i /
2. The diagonal matrix ele-ments need not have a mean value equal to zero (as evidenced, for example, by semiclassicalcalculations on chaotic quantum systems, presented in [14, 26]). However, using argumentsabout unitary invariance of the ensemble of Hamiltonians, it is argued that the variance ofthe diagonal elements is Var[ ∂ i H n +1 ,n +1 ] = Var[ ∂ i H n,n ] = σ i [10, 26]. Because these elementsare independent, W ,i = [ ∂ i H n =1 ,n +1 − ∂ i H n,n ] / σ i /
2. We conclude that the W i,j are Gaussian random variables with mean value zero and variance (cid:104) W i,j (cid:105) = σ j . (19) The universality hypothesis is most extensively supported for energy level statistics [5, 9], butthere is also strong evidence that it holds for parametric dependence of energy levels [10–15],and by extension it should also hold for dynamical properties [8].In the case of a system which depends upon a single parameter X , it is argued [10] thatthe eigenfunctions depend very sensitively upon parameters, so that correlation functionsdecay on a characteristic length scale ∆ X upon which the eigenfunction lose their identity.Furthermore, perturbation theory indicates that (cid:104) φ n | ∂ ˆ H/∂X | φ n +1 (cid:105) ∆ X ∼ ∆ E , where ∆ E is the typical separation of energy levels. Because the typical size of the matrix elementis (cid:104) φ n | ∂ ˆ H/∂X | φ n +1 (cid:105) ∼ σ , and the typical spacing of levels is ∆ E ∼ ρ − , we expect thatcorrelation functions will be functions of the dimensionless variable ρσ ∆ X , and this is inaccord with numerical investigations [10, 12].In order to define the quantum curvature, however, we must consider a Hamiltonianwhich depends upon more than one parameter. Let us assume that our system has twoparameters, Y = ( Y , Y ) say, and that the matrix elements of derivatives with respect to the Y i variables have a covariance Σ ij (defined by analogy with equation (10)). We can apply asmooth transformation of the parameter space to produce a set of transformed coordinates X = ( X , X ), so that small displacements in parameter space close to Y are described by aunimodular 2 × M : δ X = ˜ M δ Y , det( ˜ M ) = 1 . (20)7 ciPost Physics Submission We shall calculate the correlation functions in these transformed coordinates, X = ( X , X ).We choose the transformation matrix ˜ M so that the covariance matrix ˜ σ (with elements σ ij )is a multiple of the identity matrix, with diagonal elements equal to σ ). If these diagonalelements are denoted by σ , then ˜ M satisfies˜Σ = ˜ M ˜ σ ˜ M T = σ ˜ M ˜ M T , σ = det( ˜Σ ) . (21)Now consider the form of the correlation function in the isotropic coordinates, C nn ( X ), whichmust be a function of σ , σ and ρ . Dimensional considerations imply that C nn is proportionalto σ σ ρ . In terms of the transformed variables, in which the covariances are diagonal( σ ij = σ δ ij ), the correlation function takes the form C ( X ) = σ ρ f ( ρσX ) (22)where f ( · ) is a universal function. We shall determine f ( x ) numerically, and compute itsasymptotic behaviour as x → C ( Y ) = det( ˜Σ ) ρ f (cid:16) ρ [det( ˜Σ )] / | ˜ M Y | (cid:17) . (23)The arguments leading to (22) are immediately applicable to off-diagonal correlation functions C n,n + s with fixed s , so that C n,n + s ( X ) = σ ρ f s ( ρσX ) , (24)with a set of universal scaling functions f s ( x ).The smoothed curvature correlation function depends on the energy separation ∆ E inaddition to the parameter separation X . In section 5 we show that C is proportional to σ σ ρ /ε and argue that its scaling form is C (∆ E, X ) = π / σ ρ ε g ( ρσX, ∆ E/ε ) (25)in the isotropic coordinates (the dimensionless coefficient is chosen so that g (0 ,
0) = 1) and cal-culate explicitly the small- x asymptotics of g ( x, y ) for any y . Furthermore we shall determine g ( x, y ) numerically for all x and y , and confirm that it is indeed universal. We performed our numerical studies on a field of M × M random matrices taking valueson the two-sphere. At each point, the statistics of the matrix field are representative of theGaussian unitary ensemble (GUE), as defined in [6]. By choosing the distribution that ishomogeneous and isotropic, the model is fully specified by (cid:104) H (cid:105) = 0 and the two-point matrixelement correlation function (cid:104) H ij ( X ) H ∗ i (cid:48) j (cid:48) ( X (cid:48) ) (cid:105) = c ( θ ) δ ii (cid:48) δ jj (cid:48) (26)where θ is the angle subtended by the points X and X (cid:48) on the sphere; c is a smooth function of θ with c (0) = 1 and c (cid:48) (0) = 0, making the random matrix field realisations smooth functionson the sphere with variance equal to unity. 8 ciPost Physics Submission The simulation results shown below were all obtained for a Gaussian correlation function c ( θ ) = exp( − θ / θ ), where ˜ θ is a parameter of the model. For this model, the covariancecoefficients σ ij of the matrix element variances form a diagonal matrix, so that the coefficientsin equation (11) are σ = σ = 1 / ˜ θ . The single point distribution implied by (26) is standardGUE, so that when M is large the mean density of states is well-approximated by Wigner’s‘semicircle law’ [6], ρ ( E ) = √ M − E π , | E | ≤ M , (27)and zero otherwise.
Consider the form of the correlation function C ( X ) in the limit as X →
0. In this limit thecorrelation function diverges, due to near-degeneracies, and we can calculate its form usingthe projection into a two-dimensional sub-space, as considered in subsection 2.3.The quantum curvature 2-form, denoted by ˜Ω, is described by a single coefficient Ω n whenexpressed in terms of the coordinates ( X , X ):˜Ω = Ω n d X ∧ d X . (28)We can also write ˜Ω using the coefficients h i (defined in equation (16)) as coordinates, inwhich case it is expressed in terms of components Ω jk ,˜Ω = (cid:88) j,kj 0) is then Ω n ( X ) = ( (cid:15) + W X )Θ − W X Θ + W X Θ (cid:2) ( (cid:15) + W X ) + W X + W X (cid:3) / . (36)We now wish to compute the correlation function C nn ( X ) = (cid:104) Ω n (0)Ω n ( X ) (cid:105) where the expec-tation value averages over the distributions of the W ij and the (cid:15) . We shall average over the W i, , then over the W i, , and finally over (cid:15) . We find the following results for averages over W i : (cid:104) Θ Θ (cid:105) W i = − σ W W , (cid:104) Θ Θ (cid:105) W i = σ W W , (cid:104) Θ (cid:105) W i = σ W + W ] . (37)At this stage it is convenient to change the W i variables to polar coordinates ( R, θ, φ ) W = R cos θ , W = R sin θ cos φ , W = R sin θ sin φ (38)so that, noting that the W ij are independent Gaussian distributed variables with zero meanand variance σ j / 2, the probability element for these variables isd P = 1( πσ ) / exp[ − ( W + W + W ) /σ ]d W d W d W = 1 π / σ R exp( − R /σ ) sin θ d R d θ d φ . (39)Now consider the average of Ω n ( X )Ω n (0), evaluated using (36). First we average over W i using equation (37): (cid:104) Ω n (0)Ω n ( X ) (cid:105) W i = σ (cid:15) W [( (cid:15) + W X ) − W X ][ (cid:15) + 2 (cid:15)RX cos θ + R X ] / + σ (cid:15) W [( (cid:15) + W X ) − W X ][ (cid:15) + 2 (cid:15)RX cos θ + R X ] / = σ (cid:15) R sin θ(cid:15) [ (cid:15) + 2 (cid:15)RX cos θ + R X ] / . (40)Now introduce a dimensionless parameter λ = RX(cid:15) (41)and compute the average of equation (40) over the W i : (cid:104) Ω n (0)Ω n ( X ) (cid:105) W ij ∼ √ π σ (cid:15) σ (cid:90) ∞ d R R exp( − R /σ ) F ( λ ) (42)10 ciPost Physics Submission where F ( λ ) = (cid:90) π d θ sin θ [1 + 2 λ cos θ + λ ] / . (43)Introducing another dimensionless variable µ = (cid:15)σ X (44)we have (cid:104) Ω n (0)Ω n ( X ) (cid:105) W ij = 14 √ π σ (cid:15)σ X G ( µ ) (45)where G ( µ ) = (cid:90) ∞ d λ λ exp( − µ λ ) F ( λ ) . (46)Finally, we average over (cid:15) , and multiply by a factor of two because the near-degeneracy canbe with either a level above or one below. Hence the contribution to the correlation functionfrom nearly degenerate levels is (cid:104) Ω n ( X )Ω n (0) (cid:105) ∼ π / ρ σ σ X (cid:90) ∞ d µ µ G ( µ ) . (47)This is the dominant contribution to the curvature correlation as X → 0. Evaluating theintegrals, we find F ( λ ) = (cid:90) π d θ sin θ [1 + 2 λ cos θ + λ ] / = (cid:26) < λ < λ λ ≥ µ then λ gives C nn ( X ) ∼ π / ρ σ σ X . (49)This is consistent with the expected universal scaling form, equation (22), with f ( x ) ∼ π / / x (50)as x → C nn and C n − ,n are dominated byevents of near degeneracy of E n − and E n , but since Ω n − and Ω n are anti correlated duringthese events, C n − ,n must have the opposite sign, and since C nn receives an independent equalcontribution from near degeneracies of E n and E n +1 while C n − ,n does not, the latter shouldbe also be smaller by a factor of two in absolute value. It follows that C n − ,n ( X ) ∼ − π / ρ σ σ X , (51)and therefore f ( x ) ∼ − π / / x in the limit as x → f s ( x ) was defined in equation(24)). 11 ciPost Physics Submission M = = 50 M = = x - - xf s ( x ) Figure 1: Plot of xf s ( x ), obtained by the scaling transformation (24) of the shifted single-levelcorrelation function C n,n + s ( θ ), s = 0 , 1, calculated numerically by Monte-Carlo simulations forthe Gaussian random matrix field model with Gaussian matrix element correlation functionsand correlation length ˜ θ = 1 for several matrix sizes M and energy level groups. Each data setshows the average of xf s ( x ) over a range of four ( M = 30) to twenty ( M = 150) consecutiveenergy levels as a function of x . Positive (negative) values correspond to s = 0 ( s = 1),respectively, and s = 1 data points are shown in lighter hue. The colours next to eachvalue of M represent, from bottom to top, the following energy-level intervals: 16 ≤ n ≤ ≤ n ≤ 20, 22 ≤ n ≤ 23, 25 ≤ n ≤ 26, for M = 30; 26 ≤ n ≤ 28, 30 ≤ n ≤ 32, 34 ≤ n ≤ ≤ n ≤ 40, for M = 50; 51 ≤ n ≤ 55, 58 ≤ n ≤ 62, 65 ≤ n ≤ 69, 72 ≤ n ≤ 76, for M = 100; and 76 ≤ n ≤ 84, 89 ≤ n ≤ 96, 101 ≤ n ≤ ≤ n ≤ M = 150.Each energy-level interval is averaged with the corresponding levels below the midpoint of thespectrum. We investigated the correlation function C ( X ) numerically for our M × M GUE random matrixfield defined on a unit 2-sphere, as described in subsection 2.5. For this purpose we sampledthe joint probability distribution of two matrices ˆ H ( X ), ˆ H ( X ) subtending angle θ on thesphere, as well as their X derivatives. Since different matrix elements of ˆ H ( X ) are independent(except for those related by hermiticity), it is sufficient to sample independent realisationsof the six-variable joint Gaussian distribution for H ( X ) jk , H ( X ) jk ∂ α H ( X ) jk , ∂ β H ( X ) jk ( α, β = 1 , ≤ j ≤ k ≤ M to sample a single realisation of Ω( X ) and Ω( X ). Thesix-by-six covariance matrix of the matrix elements and their derivatives is straightforwardlydetermined from the matrix-element correlation function c ( θ ) and its derivatives.This process was repeated for a number n θ of equally spaced angular separations betweenzero (exclusive) and θ m . The respective values of n θ and θ m were 120 and 0 . π for M = 30,12 ciPost Physics Submission M = = x - - xf s ( x ) Figure 2: Points show scaled diagonal and nearest neighbour single-level correlation functions,as described in figure 1, except that data are averaged only over the central range of energylevels (as detailed in figure 1), but for different matrix element correlation lengths, confirminguniversality. The colours next to each value of M represent, from bottom to top, ˜ θ = 1 / , M = 100) and ˜ θ = 1 , M = 150). The dashed lines and solid curves have the same meaningas in figure 1.120 and 0 . π for M = 50, 100 and 0 . π for M = 100, and 80 and 0 . π for M = 150. Thecurvature correlation functions reported here were calculated by averaging the product of thecurvatures of matrices randomly sampled in this manner. We used 10 realisations of 30 × × realisations of 50 × 50, 10 of 100 × × realisations of 150 × f and f (defined as in equations (22) and(24)) as a function of the scaled separation x . Different colours correspond to different choicesof M , ˜ θ , and energy level range. In figure 1 we vary the energy interval of the spectrum, andin figure 2 we show data for two different values of the correlation length ˜ θ , combining datafor different values of the matrix dimension M in each plot. The quality of the data collapseis a strong indication that the functions f ( x ) and f ( x ) are universal, and the short distanceasymptotics, equations (49) and (51), are confirmed by the matching of the dashed horizontallines at 4 π / / − π / / x calculations of xf ( x ) and xf ( x ) (respectively).The solid curves are quadratic-exponential fits xf ( x ) ≈ (4 π / / 3) exp[ − ( ax + bx )] , xf ( x ) ≈ ( − π / / 3) exp[ − ( a x + b x )] , (52)with a = 3 . b = 2 . a = 3 . b = 3 . 55; we use the fits to estimate the value of integralswhich will play a role in section 5: I = (cid:90) ∞ d x x f ( x ) ≈ . , I = − (cid:90) ∞ d x x f ( x ) ≈ . . (53)13 ciPost Physics Submission We present analytical results on the correlation function of the smoothed curvature, C (∆ E, X ),in the cases where X = 0 (subsection 4.1) and X nonzero but small (subsection 4.2), beforepresenting our numerical results on this correlation function in subsection 4.3. We defined¯Ω ε ( E, X ) as a local, smoothly weighted average of the Ω n in an interval of width ε centredon E , by equation (2), its correlation function C (∆ E, X ) by (3). We expect the dependenceof C on the energy base point E is weak, and only through the mean density of states in theuniversal part of the smoothed curvature correlation function. Unlike the single-level curvature correlations, the correlations of the smoothed curvature donot diverge as X → 0, but degeneracies do play a significant role by causing the X -dependenceof the correlation function to have a discontinuous derivative. First we consider the correlationfunction at X = 0, before looking at its behaviour for small X in section 4.2.In this subsection we calculate C (∆ E, n is real, we have C (∆ E, 0) = (cid:28) (cid:88) n (cid:88) n (cid:48) w ε ( E + ∆ E − E n ) w ε ( E − E n (cid:48) ) (cid:88) m (cid:54) = n (cid:88) m (cid:48) (cid:54) = n (cid:48) K nmn (cid:48) m (cid:48) ( E n − E m ) ( E n (cid:48) − E m (cid:48) ) (cid:29) (54)where K nmkl = [ ∂ H nm ∂ H mn − ∂ H nm ∂ H mn ][ ∂ H kl ∂ H lk − ∂ H kl ∂ H lk ] ∗ . (55)Now consider how to compute (54) in random matrix theory. Note that ˆ H , ∂ ˆ H and ∂ ˆ H are independent GUE matrices. Because ˆ H is statistically independent from ∂ i ˆ H , and GUEis invariant under unitary transformations, the matrix elements ∂ i H nm in the eigenbasis ofˆ H have standard GUE statistics with variances σ i = (cid:104)| ∂ i H nm | (cid:105) . Furthermore, averagingover ∂ i H nm is independent of the average over ˆ H , which is implemented as an average of theeigenvalues, E n . The expectation value of K nmlk for the GUE model is (cid:104) K nmkl (cid:105) = 2 σ σ [ δ nk δ ml − δ nl δ mk ] (56)so that C (∆ E, 0) = 2 σ σ (cid:28) (cid:88) n (cid:88) n (cid:48) w ε ( E + ∆ E − E n ) w ε ( E − E n (cid:48) ) × (cid:88) m (cid:54) = n (cid:88) m (cid:48) (cid:54) = n (cid:48) δ nn (cid:48) δ mm (cid:48) − δ nm (cid:48) δ mn (cid:48) ( E n − E m ) ( E n (cid:48) − E m (cid:48) ) (cid:29) = 2 σ σ (cid:28) (cid:88) n w ε ( E + ∆ E − E n ) (cid:88) m (cid:54) = n [ w ε ( E − E n ) − w ε ( E − E m )] 1( E n − E m ) (cid:29) . (57)The largest terms in the m sum are those with m close to n . For such m we can approximatethe difference of the window functions by its Taylor series w ε ( E − E n ) − w ε ( E − E m ) = w (cid:48) ε ( E − E n )( E n − E m ) − w (cid:48)(cid:48) ε ( E − E n )( E n − E m ) + · · · (58)14 ciPost Physics Submission since pairs of terms with m = n ± ˜ m cancel, the sum is dominated by terms of O ( (cid:104) ( E m − E n ) − (cid:105) m , where (cid:104)(cid:105) m stands for averaging over the distribution of E m with E n fixed. Thisexpectation value is finite because level repulsion implies that the probability that | E m − E n | <(cid:15) is ∼ (cid:15) for (cid:15) small. The fast decay of (cid:104) ( E m − E n ) − (cid:105) m as | m − n | increases makes the termswith m close to n dominant, so that the higher order terms in (58) negligible, so that C (∆ E, 0) = − σ σ (cid:88) n (cid:28) w ε ( E + ∆ E − E n ) w (cid:48)(cid:48) ε ( E − E n ) S n (cid:29) (59)where we define S n = (cid:88) m (cid:54) = n (cid:28) E n − E m ) (cid:29) m . (60)Since S n is dominated by the smallest separations of energy levels, we expect that S n ∼ Aρ ( E n ) where A is a dimensionless constant. The value of A can be deduced from a ‘virialrelation’ derived by Dyson (see discussion in [6]), who showed that the eigenvalues of a M × M GUE matrix satisfy M (cid:88) n =1 (cid:88) m (cid:54) = n (cid:104) ( E n − E m ) − (cid:105) = M ( M − . (61)Combining this with Wigner’s semicircle law (27) for the mean density of states we find S n ∼ π ρ ( E n )] . (62)Hence, in the limit where ρε (cid:29) C (∆ E, ∼ − π σ σ ρ (cid:90) ∞−∞ d E w ε ( E + ∆ E ) w (cid:48)(cid:48) ε ( E )= π / ρ σ σ ε (cid:34) − (cid:18) ∆ Eε (cid:19) (cid:35) exp (cid:20) − ∆ E ε (cid:21) . (63)Note that this is consistent with the universal scaling form, equation (25), with g (0 , y ) = (cid:34) − (cid:18) ∆ Eε (cid:19) (cid:35) exp (cid:20) − ∆ E ε (cid:21) . (64) We can also consider the parameter dependence of the correlation function of the smoothedcurvature, namely C (∆ E, X ), following a similar approach to that leading to equation (49).The value of ¯Ω ε ( E ) diverges at degeneracies, but (cid:104) ¯Ω ε (cid:105) is finite. The change in the cor-relation function close to X = 0 is determined by nearly-degenerate levels. If E n is close to E n +1 , the change in Ω ε due to varying the parameters by a small displacement ( X, 0) is∆ ¯Ω ε ( X ) = w (cid:48) ε ( E − E n ) (∆ E ( X )Ω n ( X ) − ∆ E (0)Ω n (0)) (65)15 ciPost Physics Submission M = = = = y - - g ( y ) Figure 3: Plot of the numerically calculated (dots) scaled one-point smoothed-curvaturecorrelation function g (0 , y ), obtained by (25) from C (0 , ∆ E ), as a function of y = ∆ E/ε ,compared with the exact large- M asymptotic (64) (solid curve). Dots of different colourscorrespond to different matrix sizes M , and several energy window widths ε , all centered at E = 0. The colours next to each value of M represent, from bottom top, data for ε/π =0 . , . , . , . 67 ( M = 30), 0 . , . , . , . 63 ( M = 50), 0 . , . , . , . M = 100), and0 . , . , . , . 69 ( M = 150).where ∆ E ( X ) = E n +1 ( X ) − E n ( X ), and where Ω n ( X ) is given by equation (36), which wewrite in the form Ω n ( X ) = 4 ( (cid:15) + W X )Θ − W X Θ + W X Θ [∆ E ( X )] , (66)where Θ i were defined in equation (33), and∆ E ( X ) = 2[( (cid:15) + W X ) + W X + W X ] / . (67)We shall consider the quantity ¯Ω ε ( E + ∆ E, ε ( E, X ) − ¯Ω ε ( E, ≡ ¯Ω ε ∆ ¯Ω ε . This is¯Ω ε ∆ ¯Ω ε = w (cid:48) ( E + ∆ E − E n ) w (cid:48) ( E − E n ) Θ (cid:15) (cid:20) ( (cid:15) + W X )Θ − W X Θ + W X Θ ( (cid:15) + W X ) + W X + W X − Θ (cid:15) (cid:21) . (68)Taking the expectation value of Ω ε ∆Ω ε ( X ), using the same approach and notations as before16 ciPost Physics Submission M = = = = y - - g ( y ) Figure 4: Same as figure 3 but with correlation functions calculated numerically for energywindows centered at E / √ M = 0 , / , , / ε = π/ (cid:104) ∆Ω ε ( X )Ω ε (cid:105) W i = w (cid:48) ε ( E + ∆ E − E n ) w (cid:48) ε ( E − E n ) σ (cid:20) R sin θ(cid:15) + 2 RX(cid:15) cos θ + R X − R sin θ(cid:15) (cid:21) (cid:104) ∆Ω ε ( X )Ω ε (cid:105) W ij = w (cid:48) ε ( E + ∆ E − E n ) w (cid:48) ε ( E − E n ) σ √ πσ (cid:15) (cid:90) ∞ d R R exp( − R /σ ) × (cid:90) π sin θ (cid:20) 11 + 2 λ cos θ + λ − (cid:21) (cid:104) ∆Ω ε ( X )Ω ε (cid:105) = w (cid:48) ε ( E + ∆ E − E n ) w (cid:48) ε ( E − E n ) 8 π / ρ σ σ X × (cid:90) ∞ d µ µ (cid:90) ∞ d λ λ exp( − λ µ ) F ( λ ) (69)where we have taken expectation values with respect to the W i , then W i then (cid:15) (using thesame polar coordinates for the W i , the same definitions of λ and µ as section 3), and F ( λ ) = (cid:90) π d θ sin θ (cid:20) 11 + 2 λ cos θ + λ − (cid:21) . (70)This yields (cid:104) ∆Ω ε ( X )Ω ε (cid:105) = 8 π / A w (cid:48) ε ( E − E n )] ρ σ σ X (71)where A = (cid:90) ∞ d µ µ (cid:90) ∞ d λ λ exp( − λ µ ) F ( λ ) = − π . (72)17 ciPost Physics Submission Finally, we multiply by two, to account for near degeneracies with the level below as well asthe level above, and sum over energy levels. Noting that (cid:88) n w (cid:48) ε ( E + ∆ E − E n ) w (cid:48) ε ( E − E n ) ∼ ρ πε (cid:90) ∞−∞ d E E exp (cid:16) − E ε (cid:17) exp (cid:20) − ( E + ∆ E ) ε (cid:21) = ρ √ π ε (cid:34) − (cid:18) ∆ Eε (cid:19) (cid:35) exp (cid:20) − ∆ E ε (cid:21) (73)We then have C (∆ E, − C (∆ E, X ) ∼ π ρ σ σ ε ρσ X (cid:34) − (cid:18) ∆ Eε (cid:19) (cid:35) exp (cid:20) − ∆ E ε (cid:21) . (74)This is consistent with C (∆ E, X ) having the universal scaling form (25) where the scalingfunction g ( x, y ) satisfies g ( x, y ) ∼ x (cid:28) (cid:18) − y (cid:19) exp( − y / (cid:32) − π / | x | + O ( x ) (cid:33) . (75)If the Ω n were statistically independent, we would expect to find C ∼ ε − . The fact that C ∼ ε − is indicative of cancellation effects due to correlations between the Ω n , as describedby equation (51). We used the data from the Monte-Carlo simulations described in subsection 3.2 to evaluatethe smoothed curvature correlation function C (∆ E, X ) for the parametric GUE model definedin subsection 2.4. We examined the scaling of the correlation function as we varied severalparamters: the matrix dimension M , the width ε of the energy interval, the position in thespectrum of the states included in the averaging (which affects the density of states, ρ ), andthe correlation length ˜ θ of the random matrix model.The scaled numerically calculated single-point correlation function g (0 , y ) (where y =∆ E/ε ) is shown in figures 3 and 4 overlaid with the large- M exact asympototics (63). Thenumerical results indeed approach the universal correlations when M increases, but the con-vergence is slow, with a few percent deviation even for M = 150. In figure 3 we vary the widthof energy interval, ε , confining the average to states close to the centre of the spectrum. Infigure 4 we vary the position of the averaging interval within the spectrum (keeping ε = 1 / M increases is also observed in figures 5, 6, and 7, where thenumerically calculated g ( x, y ) is plotted as a function of x = σρX for a few values of y = ∆ E/ε .All of these figures show data for a wide range of different values of M : in figure 5 we vary ε (keeping close to the centre of the band), in figure 6 we vary the energy interval (keeping ε fixed), and in figure 7 we compare results for different values of ˜ θ . The slow convergenceas M increases obscures the scaling collapse of the discontinuity of slope of g ( x, y ) at x = 0.In order to illustrate the validity of (75), the slowly converging part is removed from thecorrelation function in figures 8, 9, and 10. These show the subtracted correlation function g ( x, y ) − g (0 , y ), with slopes at x = 0 that agree well with the small- x singularity of (75), andexhibiting a very good data collapse confirming the universality of the scaling function g .18 ciPost Physics Submission y = = = = = = = = x - g ( x,y ) Figure 5: Plot of the numerically calculated scaled smoothed-curvature correlation function g ( x, y ), obtained by (25) from C ( X, ∆ E ), as a function of x = ρσX , for a few fixed values of y = ∆ E/ε . Horizontal dashed lines show the exact large- M asymptotic (64) of g (0 , y ) for thecorresponding y , and also serve to label the data sets. Curves of different colours correspondto different matrix sizes M , and energy window widths ε , all centered at E = 0 with fixedcorrelation length ˜ θ = 1. The gaps at x = 0 between the data curves and the dashed linesdecrease for larger M , as seen in figure 3. Collapse of the data curves confirms two-variablescaling and universality. The colours next to each value of M represent, from bottom to top,data for ε/π = 0 . , . , . , . 67, ( M = 30), 0 . , . , . , . 63 ( M = 50), 0 . , . , . , . M = 100), and 0 . , . , . , . 85 ( M = 150) except that ε/π = 0 . , . , . , . 54 for y = 1 . , M = 150, and that for y = 3 . ε/π = 0 . , . 25, ( M = 30), 0 . , . , . , . M = 50), 0 . , . , . , . M = 100), and 0 . , . , . 31 ( M = 150). Finally, we show how our results on the correlation function of the quantum curvature can beused to make deductions about statistical fluctuations of Chern numbers. The Chern numbercan be expressed as an integral of the quantum curvature: see equation (4)First, let us estimate the variance of N n . In our random matrix model it is clear that (cid:104) N n (cid:105) = 0. We consider the case where the parameter space is isotropic, so that the correlationfunction C ( X ) is independent of the direction of X . In this case, we write σ = σ ≡ σ . Takingthe second moment of (4), and using the fact that when M (cid:29) (cid:104) N n (cid:105) ∼ π A σ ρ I , I = (cid:90) ∞ dx x f ( x ) (76)where A is the area of the closed surface of the parameter space, and f ( x ) is the functiondefined in equation (23). Numerical evaluation of the integral in (76) (quoted in equation(53)) gives I ≈ . 69. This result is compatible with the results of [21], (based upon data19 ciPost Physics Submission y = = = = = = = = x - g ( x,y ) Figure 6: Numerical data and horizontal lines as in 5, except that data are shown for energywindows based at E / √ M = 0 , / , M and y ; the energy windowwidth is equal to the second in the list of ε values shown in figure 5 for the corresponding M and y .obtained with less powerful computers) which suggest that I ≈ . N ε ( E ) = (cid:88) n N n w ε ( E − ¯ E n ) (77)where ¯ E n is an average of E n ( X ) over the Brillouin zone. We can express the variance of thesmoothed Chern number in two ways. First, express this in terms of the correlation functionof Ω ε ( E, X ): (cid:104) N ε (cid:105) = 1(2 π ) (cid:90) d X (cid:90) d X (cid:48) (cid:104) Ω ε ( E, X )Ω ε ( E, X (cid:48) ) (cid:105)∼ A (2 π ) (cid:90) d X C (0 , | X | ) (78)where A is the area of the Brillouin zone, and in the final step we assume that the correlationis homogeneous, isotropic and short-ranged. The scaling form for the correlation function C ,equation (25), indicates that (cid:104) N ε (cid:105) ∼ √ πκ A ρσ ε (79)where κ is an integral of the scaling function: κ = (cid:90) ∞ d x x g ( x, . (80)Alternatively, we can compute the variance of the smoothed Chern number directly, if weassume that the correlation function of Chern numbers is given by (5). (This hypothesis is20 ciPost Physics Submission y = = = = = = x - g ( x,y ) Figure 7: Numerical data and horizontal lines as in 5, except that data are shown forenergy single energy window based at E = 0, but for different correlation lengths ˜ θ = 1 / , M = 100) and ˜ θ = 1 , M = 150). The energy window widths are equal, respectively foreach M and y , to the second in the list of ε values shown in figure 5.equivalent to assuming that the Chern number increments associated with gaps are uncorre-lated). Using (5) we infer that (cid:104) N ε (cid:105) = (cid:88) n (cid:88) m w ε ( E − E n ) w ε ( E − E m ) (cid:104) N n N m (cid:105)∼ Var( N n ) (cid:88) n w ε ( E − E n ) (cid:20) w ε ( E − E n ) − w ε ( E − E n − ) − w ε ( E − E n +1 ) (cid:21) . (81)Expanding the term in square brackets about E − E n , we have: (cid:104) N ε (cid:105) ∼ (cid:104) N n (cid:105) (cid:88) n w ε ( E − E n ) (cid:20) w (cid:48) ε ( E − E n )( E n +1 + E n − − E n ) − w (cid:48)(cid:48) ε ( E − E n )[( E n − E n ) + ( E n − − E n ) ] (cid:21) . (82)The terms E n +1 + E n − − E n fluctuate in sign so that the sum containing w (cid:48) ε ( E − E n ) as afactor vanishes. The remaining term gives (cid:104) N ε (cid:105) ∼ − (cid:104) N n (cid:105)(cid:104) ∆ E (cid:105) ρ (cid:90) ∞−∞ d E w ε ( E ) w (cid:48)(cid:48) ε ( E ) = (cid:104) N n (cid:105)(cid:104) ∆ E (cid:105) ρ √ πε (83)where (cid:104) ∆ E (cid:105) is the mean-squared nearest neighbour spacing. On the basis of the univer-sality hypothesis discussed in section 2, we expect (cid:104) ∆ E (cid:105) = γ/ρ , where γ is a universaldimensionless constant. Using the ‘Wigner surmise’ distribution for ∆ E , equation (8), yields γ = (cid:104) S (cid:105) = 3 π/ (cid:104) N ε (cid:105) = 3 I √ π A ρσ ε . (84)21 ciPost Physics Submission y = = = = = = = = x - - - - g ( x,y ) Figure 8: Plot of the same data as in figure 5, showing differences between scaled smoothedcurvature correlation function g ( x, y ) at different points, and the correlation function g (0 , y )at the same point, as a function of x for several fixed values of y . Straight dashed lines showthe small- x asymptotic (75) of g ( x, y ) for the corresponding y , and also serve to label the datasets. Compared to figure 5 the curves exhibits significantly better data collapse, and goodagreement with the slopes of the dashed lines.which is consistent with equation (79). The fact that (cid:104) N ε (cid:105) is proportional to ε − is, therefore,an indication that the fluctuations of Chern numbers on successive levels are anticorrelated,as described by equation (5). We have analysed the universal fluctuations of the adiabatic curvature Ω n for complex quan-tum systems, as exemplified by a parametric GUE model. We find that the correlation func-tion C ( X ) of Ω n has a X − divergence as X → 0, which is a consequence of near-degeneracies(equations (49), (50)). We also investigated the correlation function numerically, and foundthat it is consistent with the scaling hypothesis of parametric random matrix theory (equation(22)), as illustrated by figures 3–4.Because of Landau-Zener transitions these near-degeneracies spread the density matrixover a range of eigenstates, implying that we should also consider a smoothed curvature,¯Ω ε . We find that the correlation function C of ¯Ω ε scales as ε − (equation (63)), and has adiscontinuous first derivative at X = 0, described by equations (74) and (75). The numericalevaluation of the smoothed correlation function is illustrated in figures 5–10.We used these results to analyse the variance of the Chern integers. Their variance isgiven by (76), which is consistent with the surmise made in [21], and we present evidence thattheir correlation function is described by (5).22 ciPost Physics Submission y = = = = = = = = x - - - - g ( x,y ) - g ( ) Figure 9: Plot of the same data as in figure 6, subtracted as explained in figure 8. Straightdashed lines have the same significance as in figure 8. Acknowledgements MW is grateful for the generous support of the Racah Institute, who funded a visit to Israel.Both authors are grateful to the Heilbronn Institute and Prof. Jonathan Robbins at the Uni-versity of Bristol, who organised a workshop where this research was initiated. OG benefittedfrom helpful discussions with Thomas Guhr and Boris Gutkin. Funding information OG thanks the German-Israeli Foundation for financial supportunder grant number GIF I-1499-303.7/2019. References [1] C. A. Mead and D. G. Truhlar, Detrermination of Born-Oppenheimer Nuclear MotionWave-functions including Complications due to Conical Intersections and Identical Nu-clei , J. Chem. Phys. , , 2284-96, (1979), doi: 10.1063/1.437734.[2] D. J. Thouless, M. Kohmoto, M. P. Nightingale and M. den Nijs M, Quantised Hallconductance in a two-dimensional periodic potential , Phys. Rev. Lett. , , 405-8, (1982),doi: 10.1103/PhysRevLett.49.405.[3] D. J. Thouless, Quantisation of Particle Transport , Phys. Rev. B , , 6083-87, (1983),doi: 10.1103/PhysRevB.27.6083.[4] M. V. Berry, Quantal phase factors accompanying adiabatic changes , Proc. Roy. Soc.Lond , A392 , 45-57, (1984), doi: 10.1098/rspa.1984.0023.23 ciPost Physics Submission y = = = = = = x - - - - g ( x,y ) - g ( ) Figure 10: Plot of the same data as in figure 7, subtracted as explained in figure 8. Straightdashed lines have the same significance as in figure 8.[5] C. E. Porter, editor, Fluctuations of Quantal Spectra, Statistical Theories of Spectra:Fluctuations , Academic Press, New York, (1965), ISBN-13 : 978-0125623506.[6] M. L. Mehta, Random Matrices , 2nd ed. Academic Press, New York, (1991), ISBN:9781483295954.[7] C. Zener, Non-adiabatic crossing of energy levels , Proc. Roy. Soc. Lond. A , , 696-703,(1932), doi: 10.1098/rspa.1932.0165.[8] M. Wilkinson, Statistical Aspects of Dissipation by Landau-Zener Transitions , J. Phys.A , , 4021-4037, (1988), doi: 10.1088/0305-4470/21/21/011.[9] A. Mirlin, Statistics of energy levels and eigenfunctions in disordered systems, PhysicsReports, , 259–382 (2000), doi: 10.1016/S0370-1573(99)00091-5.[10] E. J. Austin and M. Wilkinson, Statistical Properties of Parameter-Dependent ChaoticQuantum Systems , Nonlinearity , , 1137-50, (1992), doi: 10.1088/0951-7715/5/5/006.[11] B. D. Simons and B. L. Altshuler, Universal velocity correlations in disordered and chaoticsystems, Phys. Rev. Lett., , 4063–4066 (1993), doi: 10.1103/PhysRevLett.70.4063.[12] A. Szafer and B. L. Altshuler, Universal correlation in the spectra of disordered systemswith an Aharonov-Bohm flux, Phys. Rev. Lett., , 587–590, (1993), doi: 10.1103/Phys-RevLett.70.587.[13] C. W. J. Beenakker and B. Rejaei, Random-matrix theory of parametric correlations inthe spectra of disordered metals and chaotic billiards, Physica A: Statistical Mechanicsand its Applications, , 61–90, (1994), doi: 10.1016/0378-4371(94)90032-9.24 ciPost Physics Submission [14] M. Wilkinson and P. N. Walker, A Brownian Motion Model for the Parameter De-pendence of Matrix Elements , J. Phys. A: Math. Gen. , , 6143-60, (1995), doi:10.1088/0305-4470/28/21/017.[15] H. Attias and Y. Alhassid, Gaussian random-matrix process and universal parametriccorrelations in complex systems, Phys. Rev. E , 4776–4792, (1995), doi: 10.1103/Phys-RevE.52.4776.[16] S-S. Chern and J. Simons, Characteristic forms and geometric invariants , Ann. Math. , , 48-69, (1974), doi:10.2307/1971013. JSTOR 1971013.[17] B. Simon, Holonomy, the quantum adiabatic theorem, and Berry phase , Phys. Rev. Lett. , , 2167-70, (1983), doi: 10.1103/PhysRevLett.51.2167.[18] M. V. Berry and J. M. Robbins, The geometric phase for chaotic systems , Proc. Roy.Soc. Lond. A , , 631-61, (1992), doi: 10.1098/rspa.1992.0039.[19] M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics , Springer, New York,(1990), ISBN-13: 978-0387971735.[20] J. M. Robbins, The Geometric Phase for Chaotic Unitary Families , J. Phys. A: Math.Gen. , , 1179-89, (1994), doi: 10.1088/0305-4470/27/4/013.[21] P. N. Walker and M. Wilkinson, Universal Fluctuations of Chern Integers , Phys. Rev.Lett. , , 4055-8, (1995), doi: 10.1103/PhysRevLett.74.4055.[22] M. V. Berry and P. Shukla, Geometric phase curvature for random states , J. Phys. A:Math. Theor. , , 475101, (2018), doi: 10.1088/1751-8121/aae5dd.[23] M. V. Berry and P. Shukla, Geometry of 3D Monochromatic Light: Local Wavevectors,Phases, Curl Forces, and Superoscillations , J. Opt. , , 064002 (2019), doi: 10.1088/2040-8986/ab14c4.[24] M. V. Berry and P. Shukla, Quantum metric statistics for random-matrix families , J.Phys. A: Math. Theor. , , 275202, (2020), doi: 10.1088/1751-8121/ab91d6[25] M. Wilkinson, A Semiclassical Sum Rule for Matrix Elements of Classically Chaotic Sys-tems , J. Phys. A: Math. Gen. , , 2415-2423, (1987), doi: 10.1088/0305-4470/20/9/028.[26] M. Wilkinson, Random matrix theory in semiclassical quantum mechanics of chaoticquantum systems , J. Phys. A: Math. Gen. ,21