Hilbert-space geometry of random-matrix eigenstates
Alexander-Georg Penner, Felix von Oppen, Gergely Zarand, Martin R. Zirnbauer
HHilbert-space geometry of random-matrix eigenstates
Alexander-Georg Penner, Felix von Oppen, Gergely Zar´and,
2, 3 and Martin R. Zirnbauer Dahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universit¨at Berlin, 14195, Berlin, Germany Exotic Quantum Phases “Momentum” Research Group, Department of Theoretical Physics,Budapest University of Technology and Economics, 1111 Budapest, Budafoki ´ut 8, Hungary MTA-BME Quantum Correlations Group, Institute of Physics,Budapest University of Technology and Economics, 1111 Budapest, Budafoki ´ut 8, Hungary Institut f¨ur Theoretische Physik, Universit¨at zu K¨oln, Z¨ulpicher Straße 77a, 50937 K¨oln, Germany
The geometry of multi-parameter families of quantum states is important in numerous contexts,including adiabatic or nonadiabatic quantum dynamics, quantum quenches, and the characteri-zation of quantum critical points. Here, we discuss the Hilbert-space geometry of eigenstates ofparameter-dependent random-matrix ensembles, deriving the full probability distribution of thequantum geometric tensor for the Gaussian Unitary Ensemble. Our analytical results give the exactjoint distribution function of the Fubini-Study metric and the Berry curvature. We discuss relationsto Levy stable distributions and compare our results to numerical simulations of random-matrixensembles as well as electrons in a random magnetic field.
Introduction.—
The geometry underlying the eigen-states of parameter-dependent quantum Hamiltonians isconcisely described in terms of the quantum geometrictensor [1, 2]. Its symmetric part is the Fubini-Study met-ric, while its antisymmetric part is the Berry curvature[3]. Both contributions to the quantum geometric tensorhave important physical consequences, in particular inthe context of adiabatic quantum dynamics beyond theBorn-Oppenheimer approximation. When a slow systemis coupled to a fast one, the symmetric and antisymmet-ric parts of the quantum geometric tensor govern electricand magnetic gauge forces acting on the slow system. Animportant application of these ideas is to the semiclas-sical dynamics of Bloch electrons [4], where these gaugeforces are at the core of anomalous Hall effects, bothunquantized and quantized. In this case, each band de-fines a family of quantum states which is parametrizedby the Bloch momenta, and it is by now well understoodthat the physics of electronic systems is affected by thelocal geometry [4] as well as the global topology of thebands [5]. In disordered or interacting systems, the mag-netic fluxes threading the system in a real-space torusgeometry play a role which is quite analogous to that ofthe Bloch momenta of noninteracting clean systems [6].The corresponding boundary geometric tensor has beenshown to provide an appropriate scaling variable for An-derson transitions, and to assume a universal probabilitydistribution at the critical point [7]. More generally, thequantum geometric tensor is an important characteristicof quantum phase transitions [2, 8].Here, we derive the exact joint probability distribu-tion of the quantum geometric tensor for the GaussianUnitary Ensemble (GUE) of random-matrix theory. Theprobability distribution of the quantum geometric tensorfor random-matrix ensembles was recently introduced byBerry and Shukla [9], extending earlier work on the Berrycurvature [10–12]. Berry and Shukla base their discussionon analytical results for small random matrices, which is sufficient to obtain the correct asymptotics of the dis-tribution function, but fails to describe the bulk of thedistribution for generic systems. Here, we find the ex-act analytical distribution in the limit of large randommatrices. Large random matrices are a powerful toolto describe spectra and eigenstates of generic quantumsystems [13, 14] and are applicable to a remarkably di-verse set of systems, including nuclear spectra [15], quan-tum chromodynamics [16], few-body chaotic quantumsystems [17], disordered electron systems [18, 19], non-integrable many-body systems [20, 21], and many-bodylocalization [22–24]. Most recently, random-matrix the-ory was instrumental in claims that quantum processorshave reached the regime of quantum supremacy [25]. Acentral role in this argument was played by the Porter-Thomas distribution, one of only few distribution func-tions in random-matrix theory which are known exactlyand have a simple analytical form. In view of the scarcityof exact analytical distributions in random-matrix the-ory, it is quite remarkable that the characteristic functionof the joint distribution function of the quantum geomet-ric tensor can be obtained exactly.
Quantum geometric tensor.—
We consider the eigen-states | ˜ n ( λ ) (cid:105) of a multi-parameter family of Hamiltoni-ans H ( λ ) with λ = ( λ , . . . , λ n ). A metric structure as-sociated with the parameter-dependent eigenstates canbe obtained by defining the distance in Hilbert space fortwo states with infinitesimally different parameters as ds = 1 − |(cid:104) ˜ n ( λ ) | ˜ n ( λ + d λ ) (cid:105)| = (cid:88) αβ Re g ( n ) αβ ( λ ) dλ α dλ β . (1)Explicitly expanding in d λ yields the Hermitian quantumgeometric tensor [1, 2] g ( n ) αβ = (cid:104) ∂ α ˜ n | ∂ β ˜ n (cid:105) − (cid:104) ∂ α ˜ n | ˜ n (cid:105)(cid:104) ˜ n | ∂ β ˜ n (cid:105) . (2)The distance ds is entirely determined by the real andsymmetric part, which is also known as the quantum a r X i v : . [ c ond - m a t . d i s - nn ] N ov metric tensor. The imaginary and antisymmetric part isreadily identified as the Berry curvature [3], which can benonzero for broken time-reversal symmetry. Equation (1)indicates that the quantum geometric tensor g ( n ) αβ quitegenerally governs the behavior of systems under quantumquenches which involve small changes of the parameters.Following Berry and Shukla [9], we consider a two-parameter family of Hermitian N × N Hamiltonians H = H + xH x + yH y , (3)which depend on the real parameters x and y . Evalu-ating the derivatives in Eq. (2) at x = y = 0, one canexpress the quantum geometric tensor in terms of theeigenenergies E n and eigenstates | n (cid:105) of H , g ( n ) αβ = (cid:88) m ( (cid:54) = n ) (cid:104) n | H α | m (cid:105)(cid:104) m | H β | n (cid:105) ( E n − E m ) (4)with α, β ∈ { x, y } .For orientation, we first consider the distribution func-tion of individual matrix elements of the quantum geo-metric tensor for an N × N matrix Hamiltonian H of anintegrable system whose energy eigenvalues are statisti-cally independent. In this case, the matrix elements ofthe quantum geometric tensor in Eq. (4) are sums over N − g ( n ) αβ = (cid:80) m ( (cid:54) = n ) x m ,and one expects their probability distributions P int ( g ) toconverge to a stable distribution in the limit N → ∞ . Inthe absence of correlations between the eigenvalues andthus of level repulsion, the distribution of the individualterms in the sum is readily seen to fall off as 1 / | x | / atlarge | x | [26], with large values of | x | originating fromnear degeneracies in the spectrum of H . Importantly,both the average and the variance diverge for this distri-bution. As a result, the sum (4) does not constitute astandard random walk, for which the central limit theo-rem predicts a normal distribution. Instead, the matrixelements g ( n ) αβ can be viewed as Levy flights and theirprobability distributions are Levy stable distributions.The terms in the sum have random signs for the real andimaginary parts of off-diagonal matrix elements, but arestrictly positive for diagonal elements, leading to differentstable distributions. For an asymptotic 1 / | x | / decay atlarge | x | , one finds distributions P int ( g ) = (cid:82) d ξ π e iξg ˜ P int ( ξ )with characteristic functions [26, 27]˜ P int ( ξ ) = (cid:40) e − √ γ | ξ | (1+ i sgn ξ ) diagonal e − √ γ | ξ | off-diagonal , (5)where γ controls the scale. Due to the (cid:112) | ξ | singularity ofthe characteristic function, the distributions P int ( g ) falloff as 1 / | g | / at large | g | , indicating that they are dom-inated by individual terms in the sum (4). Physically,this broad distribution is a direct consequence of the fact that the level spacing distribution of integrable systemsremains nonzero in the limit of zero spacing. Joint distribution function for the GUE.—
In genericsystems, level repulsion suppresses the likelihood of smallenergy denominators and the distribution of matrix ele-ments of the quantum geometric tensor decays faster.If we continue to assume that the matrix elements aredominated by individual terms in the sum (4), the tail ofthe distribution can be predicted on the basis of random2 × ∼ / | g | / [9]. In addition to suppressing the probabil-ity with which near degeneracies occur, level repulsionintroduces correlations between the terms in the sum inEq. (4). As a result, the distribution of the quantum ge-ometric tensor no longer belongs to the family of Levystable distributions. Remarkably, however, it can still becomputed exactly.We now focus on large random matrices drawnfrom the Gaussian Unitary Ensemble, which neitherobeys time-reversal symmetry nor imposes any other(anti)symmetry (symmetry class A in the Altland-Zirnbauer classification [28, 29]). The (Hermitian) ma-trices H , H x , and H y are drawn from three statisticallyindependent GUEs, P ( H , H x , H y )d H d H x d H y ∝ e − N tr( H + H x + H y ) (cid:89) i,j d ( H ) ij d ( H x ) ij d ( H y ) ij , (6)where the averages over H x and H y are introduced forconvenience. We comment below on the case when theaverage is over H only. Exploiting Hermiticity, weparametrize the quantum geometric tensor g ( n ) as g ( n ) = g ( n )0 + g ( n ) · τ , where τ denotes the vector of Pauli matri-ces and g ( n )0 = tr g ( n ) , g ( n )1 = Re g ( n ) yx , g ( n )2 = Im g ( n ) yx , and g ( n )3 = tr( τ g ( n ) ). Notice that g ( n )2 measures the Berrycurvature and g ( n )0 , g ( n )1 , g ( n )3 parametrize the quantummetric tensor. We can then define the joint probabilitydistribution of the quantum geometric tensor through P ( g ) ∝ (cid:42)(cid:88) n δ ( E n ) δ ( g − g ( n )0 ) δ ( g − g ( n ) ) (cid:43) H ,H x ,H y . (7)Here, the brackets denote the random-matrix average andthe first δ -function ensures that we consider the quantumgeometric tensor for states which are at the center of thespectrum [30].The corresponding characteristic function defined via P ( g ) = (cid:82) dξ π d ξ (2 π ) e i [ ξ g + ξ · g ] ˜ P ( ξ , ξ ) takes the form˜ P ( ξ , ξ ) ∝ (cid:42)(cid:88) n δ ( E n ) e − i [ ξ g ( n )0 + ξ · g ( n ) ] (cid:43) H ,H x ,H y . (8)In the limit of N → ∞ , the random-matrix averages canbe performed explicitly. We defer technical details tofurther below and the supplemental material [26], andfocus first on discussing our results. Results.—
We find that the characteristic function for the quantum geometric tensor takes the exact form˜ P ( ξ , ξ ) = r ( X + , X − ) e − ( X + + X − ) , (9)where we defined X ± = (1 + i sgn ξ ± ) (cid:112) γ | ξ ± | in terms of ξ ± = ξ ± | ξ | and the rational function r ( a, b ) = 1 + ( a + b ) + 13 ( a + 3 ab + b ) + 124 a + 9 a b + 17 a b + 9 ab + b a + b + 1120 ab (5 a + 16 ab + 5 b )+ 1720 a b (13 a + 29 ab + 13 b ) a + b + 1240 a b + 11920 a b a + b + 134560 a b ( a + b ) . (10)For the specific scalings of the GUE matrices in Eq. (6),we find γ GUE = 4 N . Notice that ˜ P (0 , ) = 1, so that P ( g ) is normalized. Equations (9) and (10) give the exactcharacteristic function of the distribution of the quantumgeometric tensor for large GUE matrices, and are thecentral results of this paper.We first specify Eqs. (9) and (10) to the distributionof individual matrix elements of g . The characteristicfunction of the distribution of the diagonal elements g xx and g yy can be obtained by setting ξ = ± ξ = ξ and ξ = ξ = 0. Interestingly, the resulting exponentialfactor in Eq. (9) has just the same form as in Eq. (5).The same happens for the distributions of Re g xy and theBerry curvature Im g xy , which are obtained from Eq. (9)by setting ξ = ξ or ξ = ξ , respectively, with all other ξ j = 0. Thus, it is the rational prefactor in Eq. (9)that accounts for the spectral correlations introduced bythe GUE. Expanding the exponential in Eq. (9), we ob-serve that the leading nonanalyticity of ˜ P ( ξ , ξ ) is of theform | ξ | / , which contrasts with the leading | ξ | / sin-gularity of the characteristic function ˜ P int ( ξ ) in Eq. (5).This implies that for the GUE, the distribution func-tion of the quantum geometric tensor indeed falls off as P ( g ) ∝ / | g | / for large | g | and thus faster than the cor-responding distribution P int ( g ) ∝ / | g | / for integrablesystems, corroborating the expectation based on 2 × P ( g ) obtained by Fourier trans-forming Eq. (9) depends on the quantum geometric ten-sor only through its eigenvalues g ± = g ± | g | . Writ-ing g = U diag[ g + , g − ] U † , the distribution function is in-dependent of the diagonalizing unitary matrix U , andemploying a convenient redundancy of parametrization, we define the corresponding joint eigenvalue distribution p ( g + , g − ) through P ( g ) dg = p ( g + , g − ) dg + dg − dµ ( U ) , (11)where dµ ( U ) is the invariant measure of the unitarygroup, with the group volume normalized to unity. We g xx / N P ( g xx / N ) simulationtheory g xx (a) Re g xy / N P ( R e g xy / N ) simulationtheory(Re g xy ) (b) g + / N g / N p ( g + / N , g / N ) Theory (c) g + / N g / N p ( g + / N , g / N ) Simulation (d)
Figure 1. Distribution functions of (a) the diagonal and (b)the off-diagonal matrix element (real part) of the quantumgeometric tensor. Numerical data for large random matrices(blue lines) are compared to the Fourier transform of the ana-lytical result obtained from Eq. (9) (orange dots). (c) 3D plotof the distribution function p ( g + , g − ) based on the analyticalresult in Eq. (12). (d) Corresponding 3D plot obtained numer-ically for large random matrices, obtained by averaging over10 realizations of H in Eq. (3) with H , H x , and H y drawnindependently from the GUE with N = 100. Insets in (a)and (b): Log-log plots emphasizing the asymptotic 1 / | g | / decays (black line). find p ( g + , g − ) = − i ( g + − g − )32 π (cid:90) dξ + dξ − ( ξ + − ξ − ) × ˜ P ( ξ , ξ ) e i ( g + ξ + + g − ξ − ) (12)A 3D plot of this distribution is shown in Fig. 1(c) andcompared to a numerical histogram for GUE matrices inFig. 1(d), again showing excellent agreement.The GUE averages over the perturbations H x and H y are actually redundant in the limit of N → ∞ consideredabove. In [26], we show both analytically and numericallythat one obtains the same distribution (9) when averag-ing over the unperturbed GUE Hamiltonian H only. Random-flux model .—The distribution function of thequantum geometric tensor is thus not very sensitive tothe particular nature of the pertubation. This suggeststhat it applies to the large class of physical models whichhave been shown to display GUE random-matrix corre-lations. Here, we illustrate this broad applicability bysimulations for an appropriate Anderson model. Specifi-cally, we consider a tight-binding model H = (cid:88) (cid:104) ij (cid:105) t ij c † i c j + (cid:88) j (cid:15) j c † j c j (13)with random site energies (cid:15) j drawn from the interval[ − W, W ] and hopping amplitudes t ij = e iφ ij for the di-rected nearest-neighbor bonds (cid:104) ij (cid:105) with random phases φ ij = − φ ji . The random phases break time-reversal sym-metry, so that the model falls into the unitary symmetryclass. Placing the lattice on a torus, we thread the in-dependent loops of the torus by fluxes φ x and φ y . Wethen compute the corresponding quantum geometric ten-sor g αβ by explicitly constructing the current operators, J x = ∂ φ x H and J y = ∂ φ y H and evaluating the expressionin Eq. (4). Figure (2) shows the distribution functions of g xx and Re g xy for a 3D cubic lattice, where we filter theeigenstates at the center of the band and consider param-eters well inside the metallic phase (moderate disorder),such that states at the band center are extended and theelastic mean free path is small compared to the systemsize L . The results are indeed in good agreement withthe exact random-matrix distribution. We observe nu-merically that the off-diagonal elements converge fasterto the universal distribution than the diagonal elements.This difference persists for simulations of the correspond-ing Levy flights and is even more pronounced in simula-tions of a 2D random flux model. We also confirmed thatthe Berry curvature Im g xy has the same distribution asRe g xy in the random flux model. Derivation.—
We briefly sketch the derivation of ourcentral result in Eq. (9), with details relegated to [26].The averages over H x and H y in Eq. (8) reduce to Gaus-sian integrals and can be readily performed,˜ P ( ξ , ξ ) ∝ (cid:42) δ ( E N ) N − (cid:89) m =1 E m ( E m + iξ N ) + | ξ | N (cid:43) H . (14) g xx / G P ( g xx / G ) -0.4 -0.2 0 0.2 0.4 0.6 Re g xy / G P ( g xy / G ) W = 3; L = 12W = 2; L = 12W = 5; L = 12W = 4; L = 9Theory g xx P ( g xx ) (a) (b) Figure 2. Distribution functions of (a) the diagonal and (b)the off-diagonal matrix element (real part) of the quantumgeometric tensor of the 3D random flux model. Numericaldata [symbols; see legend in panel (b)] are compared to theanalytical result obtained from Eq. (9) (full line). The inset inpanel (a) shows unscaled data for g xx . The data in the mainpanels were scaled to collapse onto a universal curve using thesame set of scaling factors G for g xx in (a) and Re[ g xy ] in(b), namely G = 3 . W = 3; L = 12, G = 3 . W = 2; L = 12, G = 1 . W = 5; L = 12, and G = 1 . W = 4; L = 9. We reinterpret this as an average over an ( N − × ( N −
1) random matrix ˜ H with eigenvalues E m and m = 1 , . . . , N −
1, using the joint eigenvalue distributionof the GUE [31, 32]. This yields˜ P ( ξ , ξ ) ∝ (cid:42) (det ˜ H ) (cid:81) j =1 det (cid:16) ˜ H + ia j (cid:17) (cid:43) ˜ H , (15)where the parameters a j with j = 1 , . . . , a j = i ( ξ ± | ξ | ) / N .Equation (15) is now amenable to supersymmetrymethods (see also Ref. [33] for a general discussion ofspectral determinants in random-matrix theory). Onerewrites the determinants as Gaussian integrals over( N − H , and employs superbosonization [34, 35] to reducethe integration over the vectors to a finite-dimensional in-tegral. Computing this integral exactly in the N → ∞ limit by the saddle-point method yields Eqs. (9) and 10),see [26] for further details. Conclusion .—We have used supersymmetry techniquesto derive the exact distribution function of the quantumgeometric tensor for random matrices in the GaussianUnitary Ensemble and confirmed that it applies to phys-ical models of noninteracting electrons. The matrix ele-ments of the quantum geometric tensor can be thought ofas Levy flights with correlations, and some aspects of theresulting distribution resemble corresponding Levy stabledistributions. The quantum geometric tensor comprisesboth the Fubini-Study metric and the Berry curvature.Thus, it plays a central role in semiclassical transport ofelectrons where it governs gauge forces [4], in the theoryof topological phases [5] where it underlies the defini-tion of Chern numbers, and in the theory of disorderedsystems where it provides natural scaling variables to un-derstand Anderson localization transitions [7]. This wideapplicability promises numerous applications and exten-sions of our results to specific physical systems.
Acknowledgement.—
We thank Alex Altland,Christophe Mora, and Miklos Werner for insightfuldiscussions. This work has been supported by CRC 910of Deutsche Forschungsgemeinschaft, by the NationalResearch, Development and Innovation Office (NKFIH)through the Hungarian Quantum Technology NationalExcellence Program, project no. 2017-1.2.1- NKP-2017-00001, and by the Fund (TKP2020 IES,Grant No.BME-IE-NAT), under the auspices of the Ministry forInnovation and Technology. [1] J. Provost and G. Vallee, Commun.Math. Phys. , 289(1980).[2] L. Campos Venuti and P. Zanardi, Phys. Rev. Lett. ,095701 (2007).[3] M. V. Berry, Proc. Roy. Soc. Lond. A , 45 (1984).[4] D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. ,1959 (2010).[5] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. , 1057(2011).[6] Q. Niu, D. J. Thouless, and Y.-S. Wu, Phys. Rev. B ,3372 (1985).[7] M. A. Werner, A. Brataas, F. von Oppen, andG. Zar´and, Phys. Rev. Lett. , 106601 (2019).[8] A. Carollo, D. Valenti, and B. Spagnolo, Phys. Rep. ,1 (2020).[9] M. V. Berry and P. Shukla, Journal of Physics A: Math-ematical and Theoretical , 275202 (2020).[10] A. Steuwer and B. D. Simons, Phys. Rev. B , 9186(1998).[11] M. V. Berry and P. Shukla, J. Phys. A Math. Gen. ,475101 (2018).[12] M. V. Berry and P. Shukla, J. Stat. Phys. (2019),10.1007/s10955-019-02400-6.[13] F. J. Dyson, J. Math. Phys. , 140 (1962).[14] T. Guhr, A. M¨uller–Groeling, and H. A. Weidenm¨uller,Phys. Rep. , 189 (1998).[15] T. A. Brody, J. Flores, J. B. French, P. A. Mello,A. Pandey, and S. S. M. Wong, Rev. Mod. Phys. ,385 (1981).[16] J. Verbaarschot and T. Wettig, Ann. Rev. Nucl. Part. Science , 343 (2000).[17] O. Bohigas, M. J. Giannoni, and C. Schmit, Phys. Rev.Lett. , 1 (1984).[18] K. Efetov, Adv. Phys. , 53 (1983).[19] C. W. J. Beenakker, Rev. Mod. Phys. , 731 (1997).[20] D. Poilblanc, T. Ziman, J. Bellissard, F. Mila, andG. Montambaux, Europhys. Lett. , 537 (1993).[21] L. F. Santos and M. Rigol, Phys. Rev. E , 036206(2010).[22] A. Pal and D. A. Huse, Phys. Rev. B , 174411 (2010).[23] M. Serbyn and J. E. Moore, Phys. Rev. B (2016).[24] M. Filippone, P. W. Brouwer, J. Eisert, and F. von Op-pen, Phys. Rev. B , 201112 (2016).[25] F. Arute, K. Arya, R. Babbush, D. Bacon, J. C.Bardin, R. Barends, R. Biswas, S. Boixo, F. G. S. L.Brandao, D. A. Buell, B. Burkett, Y. Chen, Z. Chen,B. Chiaro, R. Collins, W. Courtney, A. Dunsworth,E. Farhi, B. Foxen, A. Fowler, C. Gidney, M. Giustina,R. Graff, K. Guerin, S. Habegger, M. P. Harrigan,M. J. Hartmann, A. Ho, M. Hoffmann, T. Huang,T. S. Humble, S. V. Isakov, E. Jeffrey, Z. Jiang,D. Kafri, K. Kechedzhi, J. Kelly, P. V. Klimov, S. Knysh,A. Korotkov, F. Kostritsa, D. Landhuis, M. Lind-mark, E. Lucero, D. Lyakh, S. Mandr`a, J. R. Mc-Clean, M. McEwen, A. Megrant, X. Mi, K. Michielsen,M. Mohseni, J. Mutus, O. Naaman, M. Neeley, C. Neill,M. Y. Niu, E. Ostby, A. Petukhov, J. C. Platt, C. Quin-tana, E. G. Rieffel, P. Roushan, N. C. Rubin, D. Sank,K. J. Satzinger, V. Smelyanskiy, K. J. Sung, M. D. Tre-vithick, A. Vainsencher, B. Villalonga, T. White, Z. J.Yao, P. Yeh, A. Zalcman, H. Neven, and J. M. Martinis,Nature (London) , 505 (2019).[26] A.-G. Penner, F. von Oppen, G. Zar´and, and M. R.Zirnbauer, Supplemental material.[27] J.-P. Bouchaud and A. Georges, Phys. Rep. , 127(1990).[28] M. R. Zirnbauer, J. Math. Phys. , 4986 (1996).[29] A. Altland and M. R. Zirnbauer, Phys. Rev. B , 1142(1997).[30] We systematically ignore prefactors which can be re-stored from the normalization condition at the end ofthe calculation.[31] F. von Oppen, Phys. Rev. Lett. , 798 (1994).[32] F. von Oppen, Phys. Rev. E , 2647 (1995).[33] A. V. Andreev and B. D. Simons, Phys. Rev. Lett. ,2304 (1995).[34] J. E. Bunder, K. B. Efetov, V. E. Kravtsov, O. M. Yev-tushenko, and M. R. Zirnbauer, J. Stat. Phys. , 809(2007).[35] P. Littelmann, H. J. Sommers, and M. R. Zirnbauer,Comm. Math. Phys. , 343 (2008).[36] M. R. Zirnbauer, “Another critique of the replica trick,”(1999), arXiv:cond-mat/9903338.[37] We have performed this calculation using Mathematica. SUPPLEMENTAL MATERIAL
I. DERIVATIONA. Quantum geometric tensor
Inserting a complete set of states into Eq. (2) of the main text, one obtains the expression g ( n ) αβ = (cid:88) m ( (cid:54) = n ) (cid:104) ∂ α ˜ n | ˜ m (cid:105)(cid:104) ˜ m | ∂ β ˜ n (cid:105) (S1)for the quantum geometric tensor. Differentiating (cid:104) ˜ m | H | ˜ n (cid:105) = 0 and using H | m (cid:105) = E m | m (cid:105) gives E n (cid:104) ∂ α m | n (cid:105) + E m (cid:104) m | ∂ α n (cid:105) + (cid:104) m | ∂ α H | n (cid:105) = 0 , (S2)where we specialized to x = y = 0. Finally using that (cid:104) ˜ m | ˜ n (cid:105) = 0 implies (cid:104) ∂ α m | n (cid:105) + (cid:104) m | ∂ α n (cid:105) = 0, one finds (cid:104) m | ∂ α n (cid:105) = (cid:104) m | ∂ α H | n (cid:105) E n − E m . (S3)Inserting this into Eq. (S1) gives Eq. (4) of the main text. B. Integrable systems
For integrable systems, the eigenvalues E n can be taken as statistically independent and the spacings | E n − E m | inEq. (4) obey a Poisson distribution. Thus, the distribution p s ( s ) of the spacings remains constant in the limit s → p s ( s ) in the limit s →
0, the terms x ∼ /s in the sum in Eq. (4) have a probability distribution p x ( x ), which decays at large x as p x ( x ) = p s ( s ) (cid:12)(cid:12)(cid:12)(cid:12) dsdx (cid:12)(cid:12)(cid:12)(cid:12) ∼ | x | / . (S4)For this asymptotic decay of p x ( x ), both the average and the variance of x diverge. By consequence, in the limit oflarge N , the distribution function of the entire sum in Eq. (4) converges to an appropriate Levy stable distributionwith the same asymptotic decay [27]. The stable distribution depends on whether the signs of the terms in the sumare random (off-diagonal element of the quantum geometric tensor) or not (diagonal element). The characteristicfunctions of the corresponding stable distributions are given in Eq. (5) in the main text.We include a heuristic argument yielding Eq. (5) for the distribution of the diagonal elements of the quantumgeometric tensor. Assuming the existence of a stable distribution, we can choose a convenient distribution p x ( x ) forthe individual terms in the sum in Eq. (4), with the only requirement that the distribution fall off as 1 / | x | / atlarge | x | . Such a choice is a Gaussian distribution for the spacings s , with the numerators in Eq. (4) simply takenas fixed. As we saw above, the fact that p s ( s ) ∼ exp (cid:8) − γ s / N (cid:9) remains nonzero in the limit s → p x ( x ) ∼ / | x | / . With this choice, we find p x ( x ) ∼ (cid:90) ∞ ds e − γ N s δ ( x − s ) . (S5)Here, we focused on the diagonal element of the quantum geometric tensor, for which all terms in the sum in Eq. (4)are positive. We also made the dependence on the matrix size N explicit, choosing the same scalings as for the GUE.Using the Fourier representation of the δ -function, the corresponding characteristic function takes the form˜ p x ( ξ ) ∼ (cid:90) ∞ ds exp (cid:18) − γ N s − iξs (cid:19) . (S6)Here, ξ should be taken to have an infinitesimal negative imaginary part. This integral can be performed and yields˜ p ( ξ ) = e − √ γ N | ξ | (1+ i sgn ξ ) (S7)Due to statistical independence, the characteristic function ˜ P ( ξ ) of the entire sum in Eq. (4) is simply given by˜ P ( ξ ) = [˜ p ( ξ )] N = e − (cid:113) Nγ | ξ | (1+ i sgn ξ ) . (S8)This is just a rescaled version of the characteristic function for the distribution of an individual term in Eq. (4) [whosedistribution is thus already equal to the Levy stable distribution for our choice of p s ( s )] and coincides with Eq. (5) inthe main text with the identification γ = N γ . C. GUE average
Following Refs. [31, 32], we perform the average in Eq. (14) of the main text using the joint eigenvalue distributionfor H , p N ( E , . . . , E N ) ∝ (cid:89) i
0, which we denote as a and a , and two roots with Re a j <
0, which we denote as a and a . We also introduced parameters b j with j = 1 , . . . ,
6. The b j need to be set to zero at the end, but it turns out to be convenient to retain them at intermediatesteps of the calculation.We represent the determinants as Gaussian integrals. The determinants in the denominator are written as integralsover complex variables z, ¯ z (with Einstein’s summation convention in force)det − ( ˜ H + i a ) = (cid:90) z, ¯ z e ± i¯ z k ( ˜ H +i a ) kl z l . (S13)For convergence, we choose the upper sign when Re a > a and a , and thelower sign when Re a < a and a . The determinants in the numerator are written as integrals overGrassmann variables ζ, ¯ ζ , det[i( ˜ H + i b )] = (cid:90) ζ, ¯ ζ e − i¯ ζ k ( ˜ H +i b ) kl ζ l , (S14)where we note that (cid:81) j =1 det( ˜ H + i b j ) = ( − N (cid:81) j =1 det[i( ˜ H + i b j )].We now collect the random factors into X ≡ exp (cid:110) i ˜ H kl (cid:16) z l ¯ z k − z l ¯ z k + z l ¯ z k − z l ¯ z k + ζ lf ¯ ζ fk (cid:17)(cid:111) , (S15)where f = 1 , . . . ,
6, and introduce supervectors { Ψ lµ } = (cid:0) z l , z l , z l , z l , ζ l , ζ l , ζ l , ζ l , ζ l , ζ l (cid:1) (S16)to abbreviate the notation. Then, we haveΨ lµ ( s ¯Ψ) µk = z l ¯ z k − z l ¯ z k + z l ¯ z k − z l ¯ z k + ζ lf ¯ ζ fk (S17)with s = diag(1 , − , , − , , , , , , . (S18)Taking the GUE expectation value has now been reduced to a Gaussian integral, which yields E GUE ( X ) = E GUE (cid:16) e i ˜ H kl (Ψ s ¯Ψ) l k (cid:17) = e ( − λ / N )(Ψ s ¯Ψ) l k (Ψ s ¯Ψ) kl . (S19)Using the cyclicity of trace and supertrace, the exponent on the right hand side can be written as a supertrace, E GUE ( X ) = e − ( λ / N ) tr(Ψ s ¯Ψ) = e − ( λ / N ) STr( ¯ΨΨ s ) . (S20)Here, λ denotes the disorder strength parameter of the GUE, which was set to λ = 1 in the main text. D. Superbosonization step
Consider the composite object (with k = 1 , , . . . , N for N × N GUE matrices) M µν = N − ¯Ψ µk Ψ kν . (S21)This is a supermatrix of dimension (4 | × (4 | z, ¯ z and ζ, ¯ ζ of integration to supermatrices M as new integration variables. In the fermion-bosonblock decomposition, M = (cid:18) M BB M BF M FB M FF (cid:19) , (S22)the block M BB is a positive Hermitian 4 × M BB = N − ¯ z k z k . . . ¯ z k z k ... . . . ...¯ z k z k . . . ¯ z k z k , (S23)while M FF , M FF = N − ¯ ζ k ζ k . . . ¯ ζ k ζ k ... . . . ...¯ ζ k ζ k . . . ¯ ζ k ζ k , (S24)turns into a unitary 6 × M BF and M FB are Grassmann variables. The change of variablesis carried out by using the superbosonization identity (cid:90) z, ¯ z (cid:90) ζ, ¯ ζ F (cid:0) M (¯ z, z, ¯ ζ, ζ ) (cid:1) = (cid:90) D M SDet N ( M ) F ( M ) , (S25)where a normalization constant is absorbed into the new integration measure, D M . The new measure is scale invariantand, up to a constant, uniquely determined by the symmetries of the problem. E. Saddle-point approximation
After superbosonization, we have˜ P ( ξ , ξ ) = (cid:90) D M SDet N ( M ) e − ( Nλ /
2) STr( Ms ) − N STr( smM ) (S26)with m = diag( a , a , a , a , b , . . . , b ) as defined in the main text. In the limit of large random matrices, N → ∞ , theintegral can now be performed by saddle-point integration. Since m ∼ N − , the corresponding term can be neglectedin determining the saddle-point manifold, and the saddle-point equation becomes M − − λ sM s = 0 . (S27)This has the supermanifold of dominant (for N → ∞ ) solutions M s = λ − Q, Q = T Σ T − , (S28)where Σ = diag(1 , − , , − , , − , , − , , − , T ∈ U(2 , | . (S29)Thus, saddle-point integration yields ˜ P ( ξ , ξ ) = (cid:90) DQ e − ( N/λ ) STr( Qm ) , (S30)where DQ is the invariant measure on U(2 , | / U(2 | × U(2 | F. Semiclassical exactness
Our integral representation for ˜ P is semiclassically exact, c.f., [36], which significantly simplifies the calculation.The principle of semiclassical exactness is easiest to apply if the critical points of the integrand are isolated. In thepresent case, that is not the case once we set b j →
0. It is for this reason that we introduced the b j at all intermediatestages of the calculation and take the limit b j → Q crit = diag(+1 , − , +1 , − , s , s , s , s , s , s ) (S31)where s f ∈ {± } and (cid:80) f s f = 0. There exist 6! / (3!3!) = 20 critical points, namely Q crit = Σ and 19 more.Then, the value of the integral (S30) is a sum of 20 terms (one for each critical point) and each term contributesby the value of the integral at the critical point times a factor originating from the corresponding fluctuation integralin Gaussian approximation. The contribution from the critical point Q crit = Σ takes the form˜ P ( ξ , ξ ) Σ = λN ∆( ξ , ξ ) e − ( N/λ ) STr(Σ m ) , (S32)where ∆( ξ , ξ ) is given by ∆( ξ , ξ ) = (cid:81) i =1 (cid:81) j =2 , , ( a i − b j ) (cid:81) i =3 (cid:81) j =1 , , ( a i − b j ) (cid:81) i =1 (cid:81) j =3 ( a i − a j ) (cid:81) i =1 , , (cid:81) j =2 , , ( b i − b j ) . (S33)The contributions from the other 19 critical points Q crit are obtained by applying to [ b , b , b , b , b , b ] the samepermutation that turns Σ into the given Q crit , and ˜ P ( ξ , ξ ) follows by summing over the contributions of all criticalpoints.The denominator of Eq. (S33) is singular in the limit b j →
0. However, after summing over all critical pointsone finds that there is a compensating factor in the numerator and the limit becomes well defined. Performing thiscalculation [37] gives Eqs. (9) and (10).0 g xx / N P ( g xx / s N ) simulationtheory(s=0.98) g xx Re g xy / N P ( R e g xy / s N ) simulationtheory(s=1.02)(Re g xy ) s n xx ( s ) s n xy ( s ) Figure S1. Top panels: Distribution functions of matrix elements of the quantum geometric tensor (left: g xx ; right: R eg xy ),obtained by sampling 10 realizations of H in Eq. (3) with H drawn from the Gaussian Unitary Ensemble with N = 100.The sampling is performed for fixed perturbation matrices H x and H y (chosen as matrices drawn independently from theGaussian Unitary Ensemble). The insets show a corresponding log-log plot, emphasizing the asymptotic 1 / | g | / decay. A plotof f ( g ) ∝ / | g | / is shown for comparison. Numerical data (blue) are compared to the analytical prediction (orange dots)given in Eq. (9) in the main text with γ = sγ GUE and s = 0 .
978 (left) and s = 1 .
018 (right). Bottom panels: Distribution ofscaling factors as defined in Eq. (S37). The scale factors describe the fits of the distributions of the quantum geometric tensorto our analytical result in Eq. (9) and are obtained by sampling and fitting the distributions of g xx and Re g xy for 600 sets ofrandom, but fixed perturbation matrices drawn from the GUE. II. AVERAGING OVER H ONLY
In the main text, we assume that the two parameters x and y couple to independent random matrices, i.e., weaverage over both the unperturbed Hamiltonian H and the perturbations H x and H y . This assumption can berelaxed. Averaging only over the unpertubed Hamiltonian H , the matrix elements in the numerator of Eq. (4) arestill random variables as they involve the eigenvectors of the GUE matrix H . In the limit N → ∞ , the matrixelements of the perturbation matrices in the eigenbasis of H become Gaussian random variables with zero mean andcovariance E GUE {(cid:104) n | H α | m (cid:105)(cid:104) m | H β | n (cid:105)} = 1 N tr H α H β (S34) E GUE {(cid:104) n | H α | m (cid:105)(cid:104) n | H β | m (cid:105)} = 0 . ( m (cid:54) = n ) (S35)As long as we consider perturbations H x and H y such that, to leading order in the large- N limit, the covariancematrix C αβ = 1 N tr H α H β (S36)for H α is proportional to the unit matrix, the calculations can now proceed exactly as in the case discussed in thebulk of this paper, in which one averages over the perturbations H x and H y .This situation occurs when the perturbations are drawn independently from a GUE, but then held fixed whileaveraging over H . The resulting distributions are in excellent agreement with our analytical result. A comparisonbetween the numerical results and the exact distribution of the quantum geometric tensor in Eq. (9) is shown inFig. S1 (top panels). The random fluctuations of the strength of the perturbation matrices across the GUE can be1accounted for by introducing a scale factor s through γ = sγ GUE , (S37)relative to the GUE result γ GUE = 4 N . By fitting the numerical results to Eq. (9) for different GUE matrices H x and H y , we can numerically obtain the corresponding distributions of scaling factors as shown in Fig. S1 (bottom panels).In accordance with random-matrix estimates, the deviation of the scale factor from unity is of order 1 /N .We note that our approach to computing the joint distribution function for the quantum geometric tensor can alsobe extended to the case of a general covariance matrix. Then, we first define new perturbations H α and parameters r = ( x, y ) through r = D r (S38) H α = (cid:88) β D αβ H β , (S39)where we choose the orthogonal matrix D such that the covariance matrix becomes diagonal. We then have to extendthe calculation to situations in which the effective averages over H x and H y are still GUE-like, albeit with differentdisorder parameters λ x and λ y . Performing the average over the eigenvectors of the unperturbed Hamiltonian willthen result in Eq. (S12) with a j = i N [ ξ ( λ x + λ y ) + ξ ( λ x − λ y )] ± i N (cid:114)
14 [ ξ ( λ x − λ y ) + ξ ( λ x + λ y )] + λ x λ y ( ξ + ξ ) . (S40)We first consider the distributions of the diagonal and off-diagonal elements of the quantum geometric tensor. Toobtain the distribution of the off-diagonal elements, we set ξ = ξ = 0. In this case, the product λ x λ y simply rescalesthe otherwise unchanged distribution function. To obtain the distribution functions of the diagonal elements, we set ξ = ± ξ = ξ and ξ = ξ = 0. Again, the distribution functions are merely rescaled, though differently for g xx and g yy . Finally, the joint distribution function follows by setting ξ ± = 12 [ ξ ( λ x + λ y ) + ξ ( λ x − λ y )] ± (cid:114)
14 [ ξ ( λ x − λ y ) + ξ ( λ x + λ y )] + λ x λ y ( ξ + ξ ) (S41)in the characteristic function in Eq. (9), Fourier transforming, and reverting to the quantum geometric tensor withrespect to the original parameters x and yy