Quantum metric statistics for random-matrix families

The quantum metric tensor G ij for parameterised families of quantum states, in particular the trace G = trG ij , depends on the symmetry of the system (e.g. time-reversal), and the dimension N of the underlying matrices. Modelling the families by the stationary Gaussian ensembles of random-matrix, theory, we calculate the probability distribution of G, exactly for N = 2, and approximately for N = 3 and N → ∞. Codimension arguments establish the scalings of the distributions near the singularities at G → ∞ and G = 0, near which asymptotics gives the explicit analytic behaviour. Numerical simulations support the theory.


Introduction
There is continuing interest, experimental [1] and theoretical [2][3][4][5][6], in the metric describing distances between quantum states [7]. For a family of states labelled by a parameter r, the distance between neighbouring states is A physical example of G is its appearance in the electric gauge force, beyond Born-Oppenheimer, on a slow system coupled to a fast one [8][9][10]. 3 Author to whom any correspondence should be addressed.
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Here we study the typical behaviour of G ij , by calculating the probability distribution P G (G) of its trace G = G ii . We will emphasise the asymptotic behaviour near the singularities at G = ∞, where the distance between states is infinitely sensitive to the parameters, and G = 0, where the distance between states is insensitive to the parameters. The statistics discussed here correspond to continuously parameterised families of eigenstates of N × N matrices drawn from the Gaussian orthogonal ensemble (GOE), where we consider two parameters (x, y), and the Gaussian unitary ensemble (GUE), where we consider three (x, y, z).
The paper is organised as follows. In section 2, we specify the random-matrix models of parameterised Hamiltonians for which we will calculate P G (G). For consistent comparison between different cases, in particular different N, we will scale the metric trace by G = gf (N ), in which f (N ) depends on the underlying symmetry, and calculate the scaled distribution P g (g); the scaling is explained in section 2. We will illustrate the behaviour of G(r) for typical members of the ensembles, and, for the GUE, compare this with the related geometric phase curvature [11,12], whose statistics we have recently explored [13,14]. In section 3 we employ codimension arguments to anticipate the limiting behaviour of P G (G) for large and small G. The heart of the paper is sections 4 and 5. Section 4 gives the detailed calculations of P G (G) and P g (g) for the GOE, including the N dependence; for N > 2 this involves some approximations, so we also compare the theoretical distibutions with numerical simulations. Section 5 describes the corresponding calculations and simulations for the GUE. The appendices A (GOE) and B (GUE) derive the detailed asymptotics of P g (g) for large and small g.
Throughout the paper we will indicate the r dependence explicitly only where necessary for clarity.
First, some preliminaries. The metric we will study corresponds to the eigenstates |n (r) of a parameterised N × N Hamiltonian matrix: (1.2) G ij (r) is the symmetric part of the quantum geometric tensor T ij [9]. For the state |n (r) , this is (1.3) (Here and hereafter, the index n is implied unless stated otherwise, and ∞ m =n=1 · · · denotes the double sum ∞ m=1 ∞ n=1,m =n · · ·). The antisymmetric part C ij is the geometric phase curvature, whose flux through a circuit determines the geometric phase [11], and which also corresponds to the post-Born-Oppenheimer gauge force of geometric magnetism acting on a slow system coupled to a fast one [9,15,16].
Differentiating the eigenequation (1.2) leads to an expression for the metric trace in terms of the matrix elements of ∇H and the energy separations S nm : The sum of squares in the numerator, and the square in the denominator, reflect the fact that G ij is a positive (semi)definite matrix (i.e. a i G ij a j 0 for any vector a j ). The sum of squares will lead to a characteristic N dependence of P G (G) for small G, different for the GOE and GUE. The corresponding geometric phase curvature quantity is the length of the vector represented by the components of the antisymmetric tensor C ij : This vanishes when H is a real symmetric matrix-e.g. for GOE systems, where there is timereversal symmetry. The curvature does not vanish for GUE systems, but its small C behaviour [13,14] is very different from what we will find for G.

Random-matrix models
The statistics of G depend on the number of parameters r in the families of Hamiltonians, but the principles underlying the calculations do not. For definiteness, and bearing in mind the energy denominators in the sum (1.4), we choose the number of parameters as the codimension of degeneracies in the parameter space: two for real symmetric Hamiltonians and three for complex Hermitian Hamiltonians [17]. For simplicity of presentation, we choose families in which the parameter dependence is linear. Thus, for real symmetric Hamiltonians, the ensemble of families we consider, and the corresponding G(r), are involving four N × N matrices H 0 , H 1 , H 2 , H 3 drawn independently from the GUE. The matrix elements are now complex, so the geometric phase curvature is not zero: We will denote ensemble averages by · · · ; the similar bra-ket notation for quantum expectation values should not cause confusion. To scale the metric traces, it would seem natural to use the ensemble mean value G . But, as we will see in section 3, this is infinite for the GOE (as a consequence of level repulsion, itself a consequence of the codimension of degeneracies). However, the fractional moments G s (0 < s < 1) are finite, and we choose s = 1/2, i.e. the mean √ G , with the scaling.
This stratagem is not needed for the GUE, because G is finite, so the corresponding scaling is obvious: For all cases, we also calculate the cumulative metric trace distribution: This is convenient for comparing theory with numerical simulation, because it does not require collecting the data into bins. It is helpful to illustrate the systems we are studying by displaying an individual family from each of the two ensembles. Figure 1 shows G(x, y) for a GOE family. The infinite spikes correspond to degeneracies: zeros of the denominator m = n ± 1 in (2.1), which have codimension two. Figure 2(a) shows the analogous G(x, y, z 0 ) for a GUE family, that is, as a function of x and y for a fixed value z 0 . In contrary to the GOE case, there are no infinite spikes, because degeneracies have codimension 3 so they typically do not occur with only two parameters. But we expect, and have found numerically, that there are degeneracies for z values close to z 0 , and these zeros of the denominators in (2.2) are responsible for the peaks in figure 2(a): ghosts of nearby degeneracies. Figure 2(b) shows the geometric phase curvature (2.3) for the same family (i.e. the same four GUE matrices in (2.2)). It looks very similar to figure 2(a), because C(r) contains the same denominators. But as the sectional graph in figure 2(c) illustrates, the two functions are significantly different; in particular, the small values of G are much larger than those of C, for a fundamental reason, to be explained in the next section (below equation (3.4)).
Although the parameters r do not appear explicitly in the expressions (2.1) and (2.2), they are hidden in the eigenstates |n , |m , and the eigenvalues E n , E m , because these constitute the spectrum of the total Hamiltonians H in (2.1) and (2.2), which depends on r. But in calculating the statistics we will henceforth set r = 0, and then the eigenstates and eigenvalues are those of H 0 . This is an important simplification. If we were to reinstate the parameters, this would affect the scaling of the probability distributions but not their form, as was discussed for the geometric phase curvature [13].

Codimensions and scaling
Parameter values for which G = ∞ or G = 0 correspond to singularities of the metric tensor. When G = ∞, the quantum state is infinitely sensitive to changes in the parameters, and when G = 0 the state is locally insensitive to such changes. The two singularities give rise to characteristic asymptotic behaviour of P G (G), different for the GOE and the GUE. Near the different singularities, we will estimate P G (G) by choosing local parameter-space coordinates whose number equals the codimension. The resulting behaviour will provide a check on the explicit calculations in sections 4 and 5.

Large G
In (1.4) or (2.1), G = ∞ corresponds to zeros of S n,n±1 , i.e. degeneracies. For the GOE, degeneracies have codimension 2, and G diverges quadratically. Therefore we can choose local coordinates ξ 1 , ξ 2 , and estimate P G (G) as follows: It follows that, as claimed, the mean value of G is infinite but the mean value of √ G is finite, justifying the scaling (2.4). For the GUE, degeneracies have codimension 3, and the analogous calculation based on (2.2) gives the estimate Thus the mean value of G is finite, justifying the scaling (2.5). The power-law (3.2) is the same as for the geometric phase curvature C [13]; for the GOE there is no such analogy because C = 0.

Small G
All matrix elements in (1.4) contribute positively so they must all vanish in order for G to vanish. For the GOE, each of the N − 1 terms in the sum in (2.1) contains two positive contributions, from x and y, so the codimension of zeros of G is 2(N − 1). At each such zero, G vanishes quadratically, so For the GUE, each of the N -1 terms contains six contributions: two from each of x, y and z, from the real and imaginary parts of the complex matrix elements. Therefore the codimension of zeros is 6(N -1), and the analogous argument gives Both N dependences for small G are consequences of the sums of squares in the numerators for the formulas (2.1) and (2.2); this behaviour of the metric trace is very different from the curvature C, whose corresponding numerator in (2.3) does not involve sums of squares. The expressions (3.3) and (3.4) indicate that in the limit of infinite matrices P G (G) must vanish faster than any power of G, i.e. G = 0 is an essential singularity, whose explicit forms for GOE and GUE we derive in appendices A and B.

GOE calculations
From (2.1), and the representation of a probability distribution as the average of a δ-function, we have The ensemble average is over the GOE matrix elements and eigenvalue separations; these are statistically independent; moreover, in H 1 , nm and H 2 , nm the GOE matrices are independent of the eigenstates n and m of H 0 . Therefore the averages can be evaluated separately, starting with the matrix elements. These are real and Gauss-distributed, and we can normalise them arbitrarily because we will scale P G (G) according to (2.4). Thus we take The average over matrix elements in (4. There are only two eigenvalues, so the product in (4.3) has only one factor, involving the spacing S = S 2 − S 1 , whose distribution, with the mean value of S chosen as unity, has the well known Wigner form [19] Thus the metric trace distribution can be evaluated by integrating over t first and then averaging over S: The average required for scaling is so the scaled distribution, with √ g = 1, is P g (g) = 4 This result is exact. The associated cumulative distribution, also exact, is (4.8)

GOE theory, N=3
Now there are three states 1, 2, 3, and we calculate P G (G) for the state 2, so there are two spacings. Thus the distribution involving the required average (4.3), is Evaluating the t integral gives (4.10) The neighbouring spacings S 1 and S 2 are correlated [19], but the correlation is weak ( S 1 S 2 / S 1 2 − 1 ≈ −0.069), and, as in our geometric curvature theory [13], we neglect it. For these spacings, we use the Wigner distribution (4.4), which is a close approximation for the N = 3 eigenvalue spacings [19]. The integrals over S 1 and S 2 can be evaluated explicitly, with the result again agreeing with the limits (3.1) and (3.3). The required average is from which the scaled P g (g), with √ g = 1, follows from (2.4). The cumulative distribution

GOE theory, N → ∞
Now there are infinitely many eigenvalue separations. For the two closest, m = n ± 1, we again use the uncorrelated Wigner distibutions (4.4). For the non-nearest-neighbour separations m = n ± k, k 2, the rigidity of the spectrum suggests approximating the separations by their mean values: S n,n±k ≈ ±k, (k 2) , (4.14) in which we have incorporated the mean eigenvalue spacing unity. Although this approximation exaggerates the correlations between distant eigenvalues, it agrees well with numerics for large N; we could incorporate the known distant-level correlation [19], but then subsequent integrations would be intractable. Thus the metric trace distribution is The infinite product can be evaluated exactly: The integral over t could be evaluated in terms of the residues at the infinite string of poles at t = -iS 2 and t = -ik 2 , but the resulting expressions are cumbersome, and the subsequent integrations over the separations intractable. Therefore we evaluate the two S averages first, using Thus (4.15) can be written in terms of a single integral: It does not seem possible to evaluate this in closed form, but it can be computed numerically. For the average required for scaling, it is more efficient to integrate over G first, leaving the following limit of the integral over t:  The cumulative distribution is, similarly, (4.20) The large N asymptotic behaviour of (4.18) for large and small g, derived in appendix A, is These results are consistent with the codimension arguments in section 3. Figure 3 shows the scaled distributions, and the cumulative distributions, for the GOE, for N = 2, N = 3 and N = ∞.

GOE numerical simulations
We computed G from (2.1) for 20 000 samples of the three GOE matrices, for several values of N, and the corresponding values of √ G and hence g. P g (g) can be computed from the Metric tensor GOE trace distributions P g (g) (left panels) and cumulative distributions Q g (g) (right panels), for the indicated values of N. For P g (g), the simulations are the histograms, and the theory is the full red curves; for Q g (g), the simulations are the full black curves, and the theory is the dashed red curves.
histogram of values of g. Q g (g) can be obtained by plotting the ordered list of g values. Figure 4 shows the results. The agreement with theory is excellent. For N = 2 this is not surprising, because the formulas (4.7) and (4.8) are exact. The agreement for N = 3, and the convergence onto the N → ∞ theory for N = 5, 10, 50, justify the approximations for the eigenvalue separations.

GUE calculations
For the GUE, the calculations leading to P g (g) are slightly more complicated than for GOE but the principles are the same. Therefore we state the results and only outline the derivations. Instead of (4.1), and using (2.2), the distribution of the unscaled G is Now the matrix elements are complex, with their real and imaginary parts Gauss-distributed, so we can take, by analogy with (4.2),

(5.2)
For each m in the product, there are now six real matrix elements, so the average over them gives

GUE theory, N = 2
There is only one separation, namely the spacing, S = S 2 − S 1 , whose GUE distribution, replacing (4.4), is This agrees with the limits in (3.2) and (3.4). Now the average over G converges, and can be used for scaling: so the scaled distribution, with g = 1, is, exactly, The associated cumulative distribution is

GUE theory, N → ∞
Again using the approximation (4.14) that the non-nearest-neighbour separations can be replaced by their means, we obtain the GUE counterpart of (4.15): The product is the cube of h(t) in (4.16). The average over the two nearest spacings is Thus the GUE metric trace distribution, the counterpart of (4.18), is As with the GOE distribution, this integral can be evaluated numerically. The average required for scaling is The cumulative distribution is  The large N asymptotic behaviour of (5.17) for large and small g, derived in appendix B, is These results are consistent with the codimension arguments in section 3. Figure 5 shows the theoretical GUE distributions for N = 2, N = 3 and N = ∞-the counterpart of the GOE distributions in figure 3.

GUE numerical simulations
We computed G from (2.2) for 20 000 samples of the four GUE matrices, for several values of N, and the corresponding values of G and hence g. Again P g (g) can be computed from the histogram of values of g, and Q g (g) by plotting the ordered list of g values. Figure 6 shows the results-the counterpart of figure 4 for the GOE. Again the agreement with theory is excellent. For N = 2 this is not surprising, because the formulas (4.7) and (5.8) are exact. The agreement for N = 3, and the convergence onto the N → ∞ theory for N = 5, 10, 50, justify the approximations for the eigenvalue separations. Figure 6. Metric tensor GUE trace distributions P g (g) (left panels) and cumulative distributions Q g (g) (right panels), for the indicated values of N. For P g (g), the simulations are the histograms, and the theory is the full red curves; for Q g (g), the simulations are the full black curves, and the theory is the dashed red curves.

Concluding remarks
The main results of our study are formulas for the scaled probability distributions P g (g) of the quantum metric tensor trace G, for eigenstates of N × N matrices depending on two parameters (GOE, section 4) and three parameters (GUE, section 5), choices reflecting the codimensions of eigenvalue degeneracies. These are fat-tailed distribution; for the GOE, the mean G is infinite, but √ G is finite, leading to the scale-independent trace g = G/ √ G 2 ; for the GUE, the variance G 2 is infinite, but G is finite, so the scaled trace isg = G/ G . Codimension arguments in section 3 established the scaling behaviour of P g (g) near the singularities at G = ∞ and G = 0, with explicit coefficients derived in appendix A (GOE) and appendix B (GUE). These results do not exhaust the statistics of the quantum metric tensor. Many extensions can be envisaged. For the explicit random-matrix models considered, these include • the distribution of the invariant detG; • the distributions of the three eigenvalues of the tensor G ij .
More elaborate random-matrix models could include • different numbers of parameters, beyond our two (GOE) and three (GUE); • combinations of GOE and GUE matrices in the same ensemble, e.g. sparse or structured ensembles representing localised or multifractal eigenstates; • ensembles where the matrix elements are correlated; • calculations of statistics of the Riemannian curvature tensor in parameter space, associated with the quantum metric (different from the geometric phase curvature in (1.3) and (1.5)); this depends on derivatives of G ij , and could be calculated for parameters r = 0, or families depending nonlinearly on the parameters.

Appendix A. GOE metric tensor trace asymptotics for large and small G
(i). GOE N → ∞, large g For large G or g, the integral (4.18) is dominated by the behaviour of the nonexponential factor in the integrand for small t. Using (4.17), this is where a 1 and a 2 are constants. The first two terms contribute zero to the integral (4.18), because the contour can be shifted to Im t = −∞, where exp −i G gt vanishes. The large g asymptotics of P g (g) is determined by the third term, in which the t contour can be deformed to surround the cut on the negative imaginary axis, where log t contributes its discontinuity 2πi. Elementary manipulations then give Figure 7(a) illustrates the convergence to this limiting form.
(ii). GOE N 1, small g The asymptotics of (4.17) will depend in the behaviour of the integrand at a saddlepoint high in the upper t half-plane, avoiding the simple poles of 1/sin 2 π √ it (cf (4.16)), which lie at t = -in 2 (n integer). In this region 1/sin 2 π √ it is dominated by one of its two exponentials. We will be interested in Im t 1, for which This integral can be evaluated exactly, but to get the leading-order asymptotics it is simpler to use the steepest-descent approximation [20], based on the saddle-point in the exponent, at Straightforward application of this technique determines the essential singularity at g = 0: .
(A.6) Figure 7(b) illustrates the convergence to this limiting form.

Appendix B. GUE metric tensor trace asymptotics for large and small G
(i). GUE N → ∞, large g For large G or g, the integral (5.17) is dominated by the behaviour of the nonexponential factor in the integrand for small t. Using (5.16), this is h 3 (t) f 1 (t) = −1 − i 48 − 6π + π 3 π t + 140 √ i π t 3/2 + · · · (B.1) The first two terms contribute zero to the integral (5.17), because the contour can be shifted to Im t = −∞, where exp −i G gt vanishes. The large g asymptotics of P g (g) is determined by the third term, in which the t contour can be deformed to surround the cut on the negative imaginary axis. Contour integration then gives P g (g) ≈ 105 π G 3/2 g 5/2 = 0.223 g 5/2 (g >> 1) . (B.2) Figure 8(a) illustrates the convergence to this limiting form.
(B.6) Figure 8(b) illustrates the convergence to this limiting form.