Fidelity susceptibility in Gaussian Random Ensembles

TL;DR

This paper introduces the use of fidelity susceptibility as a dimensionless measure for complex quantum systems, providing analytical distributions for Gaussian ensembles and validating with numerical data.

Contribution

It analytically derives fidelity susceptibility distributions for Gaussian orthogonal and unitary ensembles, extending its application to complex quantum systems.

Findings

Analytical distributions match numerical data

Fidelity susceptibility serves as a useful measure for quantum systems

Applicable to Gaussian orthogonal and unitary classes

Abstract

The fidelity susceptibility measures sensitivity of eigenstates to a change of an external parameter. It has been fruitfully used to pin down quantum phase transitions when applied to ground states (with extensions to thermal states). Here we propose to use the fidelity susceptibility as a useful dimensionless measure for complex quantum systems. We find analytically the fidelity susceptibility distributions for Gaussian orthogonal and unitary universality classes for arbitrary system size. The results are verified by a comparison with numerical data.

Equations

$$\mathcal{F}(\hat{\rho }(0),\hat{\rho }(\lambda ))=1-\frac{1}{2}\chi {\lambda }^2+O({\lambda }^3),$$

$${\chi }_n=\sum _{m\ne n}\frac{|H_{1,nm}{|}^2}{(E_n-E_m{)}^2},$$

$$P(\chi ,E)=\frac{1}{N\rho (E)}⟨\sum _{n=1}^N\delta (\chi -{\chi }_n)\delta (E-E_n)⟩$$

$$P(H_a)\sim \exp \left(-\frac{\beta }{4J^2}\mathrm{Tr}{H}_a^2\right),H_a=[H_{a,nm}]$$

$$P(\chi )\sim {\int }_{-\infty }^{\infty }d\omega e^{-i\omega \chi }{⟨{\left[\frac{\det {\bar{H}}^2}{\det {\left({\bar{H}}^2-\frac{2i\omega J^2}{\beta }\right)}^{\frac{1}{2}}}\right]}^{\beta }⟩}_{N-1},$$

$$P(\chi )\sim {\int }_0^{\infty }dr{r}^s\delta \left(\chi -2J^2{r}^2/\beta \right){⟨\det {\bar{H}}^{2\beta }{\mathrm{e}}^{-r^{2}X}⟩}_{N-1},$$

$$P_N^O(\chi )=\frac{C_N^O}{\sqrt{\chi }}{\left(\frac{\chi }{1+\chi }\right)}^{\frac{N-2}{2}}{\left(\frac{1}{1+2\chi }\right)}^{\frac{1}{2}}$$

$$\left[\frac{1}{1+2\chi }+\frac{1}{2}{\left(\frac{1}{1+\chi }\right)}^2{\mathcal{I}}_{N-2}^{O,2}\right],$$

$${\mathcal{I}}_N^{O,2}=⟨\det V^2\left(2\mathrm{Tr}{V}^{-2}+{\left(\mathrm{Tr}{V}^{-1}\right)}^2\right){⟩}_N/⟨\det V^2{⟩}_N.$$

$$\left\langle\mathrm{det}\bar{H}^{2}\mathrm{e}^{-r^{2}X}\right\rangle_{N-1}\Big{|}_{r=0}=J^{2}\left\langle\det\bar{V}^{2}\right\rangle_{N-2}(\mathcal{I}^{O,2}_{N-2}+2),$$

$${\mathcal{I}}_N^{O,2}=\{\begin{array}{l}N\frac{N+2}{N+3/2},N\mathrm{even},\\ N+1/2,N\mathrm{odd}.\end{array}$$

$$P^O(x)=\frac{1}{6}\frac{1}{x^2}\left(1+\frac{1}{x}\right)\exp \left(-\frac{1}{2x}\right),$$

$$P_N^U(\chi )=C_N^U{\left(\frac{\chi }{1+\chi }\right)}^{N-2}{\left(\frac{1}{1+2\chi }\right)}^{\frac{1}{2}}\left[\frac{3}{4}{\left(\frac{1}{1+2\chi }\right)}^2+\frac{3}{2}\frac{1}{1+2\chi }{\left(\frac{1}{1+\chi }\right)}^2{\mathcal{I}}_{N-2}^{U,2}+\frac{1}{4}{\left(\frac{1}{1+\chi }\right)}^4{\mathcal{I}}_{N-2}^{U,4}\right],$$

$${\mathcal{I}}_N^{U,K}=\int d{Z}_{K,N}{⟨\det H^4{\left(\sum _{j,k}{z}_j{H}_{jk}^{-1}{z}_k^{\ast }\right)}^K⟩}_N$$

$${\mathcal{I}}_N^{U,2}=J^2\frac{{⟨\det H^4\left(\mathrm{Tr}(H^{-2})+(\mathrm{Tr}H{)}^{-2}\right)⟩}_N}{{⟨\det H^4⟩}_N}.$$

$$Z_N(j_1,j_2)={⟨\det H^2\det (H-j_1)\det (H-j_2)⟩}_N,$$

$${\mathcal{I}}_N^{U,2}=\frac{J^2}{Z_N(0,0)}\left(2\frac{{\partial }^2}{\partial j_1\partial j_2}{Z}_N(0,0)-\frac{{\partial }^2}{\partial j_1^2}{Z}_N(0,0)\right).$$

$$Z_N(j_1,j_2)$$

$$W_{N+2}(\lambda ,\mu )=\frac{H_{N+2}(\lambda )H_{N+1}(\mu )-H_{N+2}(\mu )H_{N+1}(\lambda )}{\lambda -\mu }.$$

$${\mathcal{I}}_N^{U,2}=\{\begin{array}{l}\frac{1}{3}N,N\mathrm{even},\\ \frac{1}{3}(N+1),N\mathrm{odd}.\end{array}$$

$${\mathcal{I}}_N^{U,4}=\{\begin{array}{l}{N}^2+2N,N\mathrm{even},\\ N^2+4N+3,N\mathrm{odd}.\end{array}$$

$$P^U(x)=\frac{1}{3\sqrt{\pi }}\frac{1}{x^{5/2}}\left(\frac{3}{4}+\frac{1}{x}+\frac{1}{x^2}\right)\exp \left(-\frac{1}{x}\right)$$

$$P(\chi ,E)=\frac{1}{2\pi N\rho (E)}\sum _{n=1}^N{\int }_{-\infty }^{\infty }d\omega e^{-i\omega \chi }⟨\delta (E-E_n)\exp \left(i\omega \sum _{m\ne n}\frac{|H_{1,nm}{|}^2}{(E_n-E_m{)}^2}\right)⟩.$$

$$P(\chi ,E)=\frac{1}{2\pi N\rho (E)}\sum _{n=1}^N{\int }_{-\infty }^{\infty }d\omega e^{-i\omega \chi }⟨\delta (E-E_n)\prod _{m\ne n}{\left(1-\frac{2i\omega J^2}{\beta (E_n-E_m{)}^2}\right)}^{-\frac{\beta }{2}}⟩.$$

$$P(\chi ,E)\sim \sum _{n=1}^N{\int }_{-\infty }^{\infty }d\omega e^{-i\omega \chi }\int \prod _{j=1}^{N}d{E}_j\delta (E-E_n)\prod _{k<l}{\left|E_k-E_l\right|}^{\beta }{\mathrm{e}}^{-\frac{\beta }{4J^2}{\sum }_k{E}_k^2}\prod _{m\ne n}{\left(1-\frac{2i\omega J^2}{\beta (E_n-E_m{)}^2}\right)}^{-\frac{\beta }{2}}.$$

$$\int d{E}_1\cdots d{E}_N\delta (E-E_1)\prod _{k<l}{\left|E_k-E_l\right|}^{\beta }\exp \left(-\frac{\beta }{4J^2}\sum _k{E}_k^2\right)\prod _{m\ne n}{\left(1-\frac{2i\omega J^2}{\beta (E_n-E_m{)}^2}\right)}^{-\frac{\beta }{2}}=$$

$$\prod _{m=2}{\left[\frac{\left|E_m\right|}{{\left(1-\frac{2i\omega J^2}{\beta E_m^2}\right)}^{\frac{1}{2}}}\right]}^{\beta }={\left[\frac{\det {\bar{H}}^2}{\det {\left({\bar{H}}^2-\frac{2i\omega J^2}{\beta }\right)}^{\frac{1}{2}}}\right]}^{\beta }$$

$$\det {\left({\bar{H}}^2-\frac{2i\omega J^2}{\beta }\right)}^{-\frac{\beta }{2}}\sim \int d\mathbf{z}\exp \left[-{\mathbf{z}}^{†}\left({\bar{H}}^2-\frac{2i\omega J^2}{\beta }\right)\mathbf{z}\right]=\int d\mathbf{z}\exp \left(-{\mathbf{z}}^{†}{\bar{H}}^2\mathbf{z}\right)e^{\frac{2i\omega J^2|\mathbf{z}{|}^2}{\beta }},$$

$$P(\chi )\sim {\int }_{-\infty }^{\infty }d\omega e^{-i\omega \chi }⟨\det {\bar{H}}^2\int d\mathbf{z}\exp \left(-|\mathbf{z}{|}^{2}X\right)e^{\frac{2i\omega J^2|\mathbf{z}{|}^2}{\beta }}⟩,$$

$${⟨\det {\bar{H}}^2{\mathrm{e}}^{-r^{2}X}⟩}_{N-1}=\int d{\bar{H}}_{11}{\mathrm{e}}^{-A{\bar{H}}_{11}^2}\prod _{j=2}^{N-1}d{\bar{H}}_{1j}{\mathrm{e}}^{-B{\sum }_{j=2}^{N-1}{\bar{H}}_{1j}^2}\det {\bar{H}}^2{D}^{N-2}V,$$

$$\bar{H}=\left[\begin{array}[]{cc}H_{11}&H_{1j}\\ H_{1k}&V\end{array}\right].$$

Full text

The discovery of many body localization (MBL) phenomenon resulting in non-ergodicity of the dynamics in many body systems Basko et al. (2006) restored also the interest in purely ergodic phenomena modeled by Gaussian random ensembles (GRE) Mehta (1990) and in possible measures to characterize them. The gap ratio between adjacent level spacings Oganesyan and Huse (2007) was introduced precisely for that purpose as it does not involve the so called unfolding Haake (2010) necessary for meaningful studies of level spacing distributions and yet often leading to spurious results Gómez et al. (2002). Still, the level spacing distribution belongs to the most popular statistical measures used for single particle quantum chaos studies Bohigas et al. (1984, 1993); Stöckmann (1999); Guhr et al. (1998) and also in the transition to MBL Serbyn and Moore (2016); Bertrand and García-García (2016); Sierant and Zakrzewski (2019). A particular place among different measures was taken by those characterizing level dynamics for a Hamiltonian $H(\lambda)$ dependent on some parameter $\lambda$. In Pechukas-Yukawa formulation Pechukas (1983); Yukawa (1985) energy levels are positions of a fictitious gas particles, derivatives with respect to the fictitious time $\lambda$ are velocities (level slopes), the second derivatives describe curvatures of the levels (accelerations). Simons and Altschuler Simons and Altshuler (1993) put forward a proposition that the variance of velocities distribution is an important parameter characterizing universality of level dynamics. This led to predictions for distributions of avoided crossings Zakrzewski et al. (1993) and, importantly, curvature distributions postulated first on the basis of numerical data for GRE Zakrzewski and Delande (1993) and then derived analytically via supersymmetric method by von Oppen von Oppen (1994, 1995) (for alternative techniques see Fyodorov and Sommers (1995); V Fyodorov (2011)). Curvature distributions were recently addressed in MBL studies Filippone et al. (2016); Monthus (2017).

Apart from quantum chaos studies in the eighties and nineties of the last millennium, another “level dynamics” tool has been introduced in the quantum information area, i.e. the fidelity, ${\cal F}$ Uhlmann (1976). It compares two close (possibly mixed) quantum states $\rho(\lambda_{1})$ and $\rho(\lambda_{2})$ for different values of the parameter, $\lambda$. For pure states, as considered below, and for $\lambda_{1}=0$ we adopt the following definition ${\cal F}=|\langle\psi(0)|\psi(\lambda)\rangle|$ Zanardi and Paunković (2006) (note that sometimes fidelity is defined as a square of ${\cal F}$, such an overlap was considered in the context of parametric dynamics of eigenvectors in Alhassid and Attias (1995)). For sufficiently small difference of parameter values, $\lambda$ it is customary to introduce a fidelity susceptibility $\chi$ via Taylor series expansion

[TABLE]

(with linear term vanishing due to wavefunction normalization condition). Fidelity susceptibility is directly related to the quantum Fisher information (QFI), $G$, being directly proportional to the Bures distance between density matrices at slightly differing values of $\lambda$ Hübner (1993); Invernizzi et al. (2008); Braun et al. (2018), with $G(\lambda)=4\chi$.

Fidelity susceptibility emerged as a useful tool to study quantum phase transitions as at the transition point the ground state changes rapidly leading to the enhancement of $\chi$ Zanardi and Paunković (2006); You et al. (2007); Zanardi et al. (2008); Invernizzi et al. (2008); Salvatori et al. (2014); Bina et al. (2016); Boyajian et al. (2016); Mehboudi et al. (2016). All of these studies were restricted to ground state properties while MBL considers the bulk of excited states (for a discussion of thermal states see Zanardi et al. (2007); Quan and Cucchietti (2009); Sirker (2010); Rams et al. (2018)). In the context of MBL we are aware of a single study which considered the mean fidelity susceptibility across the MBL transition Hu et al. (2016). In particular, nobody addressed the issue of fidelity susceptibility behavior for GRE. The aim of this letter is to fill this gap and to provide analytic results for the fidelity susceptibility distributions for the most important physically, orthogonal and unitary ensembles. This provides novel characteristics of GRE as well as a starting point for the study of fidelity susceptibility in the transition to and within the MBL domain Maksymov et al. (2019).

Consider $H=H_{0}+\lambda H_{1}$ with $H_{0},H_{1}$ corresponding to the orthogonal (unitary) class of GRE i.e., Gaussian Orthogonal Ensemble (GOE) corresponding to level repulsion parameter $\beta=1$ or Unitary Ensemble (GUE) with $\beta=2$. For such a Hamiltonian one may easily prove that fidelity susceptibility of $n$-th eigenstate of $H_{0}$ is given by

[TABLE]

with $E_{n}$ being the $n$-th eigenvalue of $H_{0}$. We aim at calculating the probability distribution of the fidelity susceptibility

[TABLE]

at the energy $E$. The averaging is over two, independent GRE ($\beta=1,2$)

[TABLE]

with $a=0,1$. Using Fourier representation for $\delta(\chi-\chi_{n})$, the average over $H_{1}$ reduces to calculation of Gaussian integrals. Since the formula (2) involves only the eigenvalues of $H_{0}$, the averaging over $H_{0}$ can be expressed as an average over the well-known joint probability density of eigenvalues Haake (2010) for a suitable GRE. At the center of the spectrum ($E=0$), after straightforward algebraic manipulations (see Sup for details) we get

[TABLE]

where the averaging is now over $(N-1)\times(N-1)$ matrix $\bar{H}$ from an appropriate Gaussian ensemble. Similar averages have been considered in studies of curvature distributions von Oppen (1994, 1995); Fyodorov and Sommers (1995), nonorthogonality effects in weakly open systems Poli et al. (2009); Fyodorov and Savin (2012) and considered in a more general fashion for the GOE case in Fyodorov and Nock (2015).

To perform the average in (5) we employ technique developed in Fyodorov and Sommers (1995) and express the denominator as a Gaussian integral over a vector $\textbf{z}\in\mathbb{R}^{N-1}$ for $\beta=1$ or $\textbf{z}\in\mathbb{C}^{N-1}$ for $\beta=2$. Employing the invariance of GRE with respect to an adequate class (orthogonal or unitary) of transformations allows us to choose $\textbf{z}=r[1,0,\ldots,0]^{T}$, hence we arrive at

[TABLE]

where $X=\sum_{j=1}^{N-1}|\bar{H}_{1j}|^{2}$ depends on the first row of $\bar{H}$ only, and $s=\beta(N-1)-1$. After calculating the ensuing Gaussian integrals over $\bar{H}_{1j}$ we can reduce the averaging to one over $(N-2)\times(N-2)$ block of $\bar{H}$, $V_{ij}=\bar{H}_{i+1,j+1}$ for $1\leq i,j\leq N-2$), using the expression $\det\bar{H}=\det V(\bar{H}_{11}-\sum_{j,k=2}^{N-1}\bar{H}_{1j}V^{-1}_{jk}\bar{H}_{1k}^{*})$ for a determinant of a block matrix.

Integrating (6) over $r$ we find (details described in Sup ) that the desired fidelity susceptibility distribution $P^{O}_{N}(\chi)$ for GOE reads

[TABLE]

where $C^{O}_{N}$ is a normalization constant and

[TABLE]

The form of (8) is suited for a random matrix theory calculation of $\mathcal{I}^{O,2}_{N}$. However, to obtain $\mathcal{I}^{O,2}_{N}$ it suffices to note that our calculation implies that

[TABLE]

showing that $\mathcal{I}^{O,2}_{N}$ is actually determined by the second moments of determinants of matrices of appropriate sizes from GOE. Moments as well as the full probability distribution of determinant of GOE matrices were obtained in Delannay and Le Caër (2000) for arbitrary $N$. Using the expression for the second moment in (9) we get

[TABLE]

The formula (10) is exact for arbitrary $N\geq 0$. Inserting appropriate values of $\mathcal{I}^{O,2}_{N}$ into (7) we obtain an exact formula for the fidelity susceptibility distribution $P^{O}_{N}(\chi)$ for GOE matrix of arbitrary size $N$. Comparison of the resulting distribution $P^{O}_{N}(\chi)$ with numerically generated fidelity susceptibility distributions for small matrix sizes $N\leq 20$ is shown in Fig. 1. However, it is the large $N$ regime which is interesting from the point of view of potential applications. For $N\gg 1$ the $\mathcal{I}^{O,2}$ increases linearly $\mathcal{I}^{O,2}_{N}=N$ with the matrix size $N$. This, together with the form of $P^{O}_{N}(\chi)$ implies that $P^{O}_{\alpha N}(\alpha\chi)\approx P^{O}_{N}(\chi)$. Indeed, the distribution $P(\chi)$ scales linearly with $N$ as visible in Fig. 2. The linear in $N$ scaling of $\chi$ suggests to introduce scaled fidelity susceptibility, $x={\chi}/{N}$. Inserting it into (7) and taking $N\rightarrow\infty$ limit one obtains

[TABLE]

which is the final, simple, analytic result for a large size GOE matrix. It performs remarkably well also for modest size matrices e.g. $N=200$ – compare Fig. 3. For smaller matrices – for instance for $N=20$, the rescaled distribution $P(x)$ has a correct large $x$ tail and a nonzero slope at $x=0$ as compared to nonanalytic behavior of $P^{O}(x)$ at $x=0$ in (11). Observe also that the mean scaled fidelity susceptibility does not exist as the corresponding integral diverges logarithmically showing the importance of the heavy tail of the distribution. The expression (11) was also obtained in study of the so called complexness parameter Poli et al. (2009).

Starting from (6) for GUE ($\beta=2$), after a few technical steps (described in detail in Sup ) we obtain the following, exact for arbitrary $N$, expression for the fidelity susceptibility distribution:

[TABLE]

where $C^{U}_{N}$ is a normalization constant. $P_{U}(\chi)$ for GUE depends on two $N$-dependent factors $\mathcal{I}^{U,2}_{N-2}$ and $\mathcal{I}^{U,4}_{N-2}$ that remain to be determined. They take the form Sup

[TABLE]

where $dZ_{K,N}=(\pi J)^{K}\int\prod_{j}\mathrm{d}^{2}z_{j}\mathrm{e}^{-\pi\sum_{j}|z_{j}|^{2}}/\left\langle\mathrm{det}H^{4}\right\rangle_{N}$, $K=2,4$ and $H$ is $N\times N$ GUE matrix. Performing the integration in (13) we find that $\mathcal{I}^{U,2}_{N}$ can be expressed in the following way

[TABLE]

Introducing the following generating function

[TABLE]

we immediately verify that

[TABLE]

The generating function $Z_{N}(j_{1},j_{2})$ is actually a correlation function of a characteristic polynomial of the $H$ matrix. It was shown in Brézin and Hikami (2000); Fyodorov and Strahov (2003) that such quantities can be calculated exactly as determinants of appropriate orthogonal polynomials. A kernel structure of those expressions has been identified in Strahov and Fyodorov (2003) leading to formulas most convenient in our calculation of $Z_{N}(j_{1},j_{2})$. The generating function $Z(j_{1},j_{2})$ is given by

[TABLE]

with the kernel $W_{N+2}(\lambda,\mu)$ defined as

[TABLE]

The Hermite polynomials $H_{N}(\lambda)$ are orthogonal with respect to the measure $\mathrm{e}^{-\frac{1}{2J^{2}}x^{2}}\mathrm{d}x$ and normalized in such a way that the coefficient in front of $\lambda^{N}$ is equal to unity.

We have found a closed formula for the generating function $Z_{N}(j_{1},j_{2})$ (see Sup for details). Calculating the derivatives in (16) and taking the limits $j_{1}\rightarrow 0$ and $j_{2}\rightarrow 0$ we obtain

[TABLE]

The next step is to use the idea analogous to the argument with ratio of second moments of determinants of GOE matrices which allowed us to obtain the exact expression for $\mathcal{I}^{O,2}_{N}$ (9). Employing formulas for the fourth moment of determinant of GUE matrix Mehta and Normand (1998); Cicuta and Mehta (2000) and taking into account the expression for $\mathcal{I}^{U,2}_{N}$ we obtain

[TABLE]

The distribution (12) together with expressions (21), (22) for $\mathcal{I}^{U,4}_{N}$ and $\mathcal{I}^{U,2}_{N}$ is the exact fidelity susceptibility distribution for GUE for arbitrary $N$. As shown in Fig. 4 the expression (12) is confirmed by numerical data for different system sizes $N\gg 1$. Similar, perfect agreement of our formula $P^{U}_{N}(\chi)$ with numerically generated data is obtained for small $N\geq 2$ (data not shown). Moreover, similarly to the GOE case, $P^{U}_{N}(\chi)$ scales linearly with increasing $N$. Therefore, considering again the distribution of scaled fidelity susceptibility $x=\chi/N$ we arrive at the large $N$ limit of the simple form

[TABLE]

which works well for GUE data as shown in Fig. 5.

Remarkably, the obtained distributions of fidelity susceptibility both for GOE (7) and GUE (12) are exact for arbitrary $N\geq 2$. This is unusual situation, even for GRE – for instance, the simple analytic form of the level spacing distribution $P(s)$ for $N=2$ becomes more complicated for larger $N$ Mehta (1990). We study thus the onset of universal large $N$ behavior of the rescaled fidelity susceptibility distribution $P(x)$. The results are shown in Fig. 6. Clearly, the power-law tail of the distributions for GOE (GUE) is observed for all $N$. This power-law tail arises in instances when the sum for $\chi_{n}$ (2) is dominated by a single term with small energy denominator. The algebraic decay $x^{-2}$ $(x^{-5/2})$ for GOE (GUE) can be derived from the small $s$ behavior of level spacing distribution $P(s)$ Gaspard et al. (1990); Monthus (2017). The approach to the limiting $N\rightarrow\infty$ distributions $P^{O,U}(x)$ is associated with decreasing number of instances of very small fidelity susceptibility.

To conclude, we have derived closed formulae for fidelity susceptibility distributions corresponding to level dynamics for both the orthogonal and the unitary class of Gaussian random ensembles. Particularly simple analytic expressions are found in the large $N$ limit. The fidelity susceptibility distributions obtained for quantally chaotic systems may be compared with the results found for GOE (GUE) in order to characterize the degree to which a given system is faithful to random matrix predictions. The obtained distributions also open a way to address level dynamics in the transition between delocalized – ergodic and many-body localized regimes Maksymov et al. (2019).

As a last touch let us mention that fidelity susceptibility is experimentally accessible by Bragg spectroscopy Gu and Yu (2014), e.g. in ultra-cold atomic systems Ernst et al. (2010); Clément et al. (2010) or by a direct measurement of many-body wave functions overlap either in a NMR setting Zhang et al. (2008) or a system of ultra-cold bosons Islam et al. (2015). That paves a way for comparing experimental measurements with universal features of fidelity susceptibility distribution provided in this work.

I Supplementary material to

“Fidelity susceptibility in Gaussian Random Ensembles”

I.1 Derivation of formulas (5) and (6)

To obtain the equation (5) we use Fourier representation for $\delta(\chi-\chi_{n})$ rewriting (3) as

[TABLE]

The averaging over $H_{1}$ with the probability density (4) reduces to a Gaussian integral and gives,

[TABLE]

The remaining averaging over the distribution $P(H_{0})$ reduces to average over eigenvalues $E_{1},\ldots,E_{N}$ of $H_{0}$

[TABLE]

Now we can perform the integral over $E_{n}$. There are $N$ such integrals due to the summation from $n=1$ to $n=N$ at the beginning of the formula. So let’s take $E_{n}=E_{1}$. Due to the delta function we can substitute $E_{1}=E$ and rewrite the averaging over the eigenvalues as

[TABLE]

where the averaging goes over the joint probability of the remaining eigenvalues $E_{2},\ldots,E_{n}$.

At the center of the spectrum $E=0$ the averaged quantity reads

[TABLE]

Plugging (S.5) into (S.4), we finally arrive at (5).

The denominator in (5) can be expressed in the form of a Gaussian integral

[TABLE]

where $\mathbf{z}$ is a $N-1$-dimensional vector, real for $\beta=1$ and complex for $\beta=2$. Due to the invariance of the ensembles with respect to appropriate ($O(N-1)$ or $U(N-1)$) rotations the average does not depend on the direction of $\mathbf{z}$, but only on its norm $|\mathbf{z}|^{2}$

[TABLE]

where $X$ is some quadratic form in the elements of $\bar{H}$ specified below. In the spherical coordinates $d\mathbf{z}\sim drr^{\beta(N-1)\beta-1}$ (where $r:=|\mathbf{z}|$), integrating over $\omega$ results in $\delta\left(\chi-2J^{2}r^{2}/\beta\right)$ and thus we arrive at (6).

I.2 Fidelity susceptibility distribution for GOE

For GOE ($\beta=1$), choosing $\mathbf{z}=r[1,0,0..]^{T}$ we rewrite the average in (6) as

[TABLE]

with $A=\frac{1}{4J^{2}}+r^{2}$, $B=\frac{1}{2J^{2}}+r^{2}$, $X=\sum_{j=1}^{N-1}|\bar{H}_{1j}|^{2}$ and

[TABLE]

The block $V$ is itself a $(N-2)\times(N-2)$ GOE matrix (with the GOE density $\mathrm{D}^{N-2}V=\prod_{k<j}dV_{kj}\exp(-\frac{1}{4J^{2}}\operatorname{Tr}V^{2})$). Using the general formula for the determinant of a block matrix

[TABLE]

we get (since the upper diagonal block is in fact one-dimensional, $\mathbf{A}=H_{11}$),

[TABLE]

Thus, (S.8) becomes

[TABLE]

where the average is now taken over the matrix $V$. Changing variables $\bar{H}_{1j}=(\frac{\pi}{B})^{\frac{1}{2}}y_{j}$ (only terms with even powers of $\bar{H}_{11}$ survive the integration over $\bar{H}_{11}$)

[TABLE]

Denote

[TABLE]

Changing the order of integration and averaging in (S.14), integration over $y_{j}$ can be done in the following way

[TABLE]

where a change of variables $z_{j}=O\xi_{j}$ such that $O^{T}V^{-1}O=\mathrm{diag}\left(E^{-1}_{2},\ldots,E^{-1}_{N-1}\right)$ was performed. Thus

[TABLE]

which is precisely the form of (8). The averages in (S.16) contain functions of eigenvalues of $V$ – therefore this formula is suited for averaging over joint probability distribution of eigenvalues for GOE. However, we can proceed in an easier way. Plugging in definitions of $A$ and $B$, (S.13) becomes

[TABLE]

where all of the normalization constants are kept. Putting $r=0$ in this formula we arrive at (9) which allows for straightforward (and exact) calculation of $\mathcal{I}^{O,2}_{N}$. Moreover, using (S.17) in (6), remembering that $\delta\left(\chi-2J^{2}r^{2}\right)\propto\left(\frac{1}{\chi}\right)^{\frac{1}{2}}\delta\left(r-\left(\frac{\chi}{2J^{2}}\right)^{\frac{1}{2}}\right)$ we obtain the fidelity susceptibility distribution for GOE (7).

We finally note that the form (11) of $P^{O}(x)$ is such that distribution of $t=\frac{1}{x}$ is many aspects simpler:

[TABLE]

which suggests that further inquires of properties of fidelity susceptibility outside the realm of GRE could be done for $t=\frac{N}{\chi}$ variable.

I.3 Calculation and results for GUE

Writing (6) for GUE - $\beta=2$, one gets choosing $\mathbf{z}=r[1,0,0..]^{T}$

[TABLE]

with $A=\frac{1}{2J^{2}}+r^{2}$ and $B=\frac{1}{J^{2}}+r^{2}$. Changing variables: $\bar{H}_{1j}=\bar{H}^{R}_{1j}+i\bar{H}^{I}_{1j}=(\frac{\pi}{B})^{\frac{1}{2}}(x_{j}+\mathrm{i}y_{j})=(\frac{\pi}{B})^{\frac{1}{2}}z_{j}$ and using the formula for determinant of block matrix one gets

[TABLE]

[TABLE]

Denote

[TABLE]

and

[TABLE]

Expressing $A$ and $B$ in terms of $J^{2}$ and $r^{2}$ results in

[TABLE]

[TABLE]

First of all, this equation used in (6) implies the form of the fidelity susceptibility distribution for GUE (12). Moreover, taking $r=0$ in (S.23) and using expression for the fourth moment of determinant of GUE matrix from Mehta and Normand (1998); Cicuta and Mehta (2000) we get that

[TABLE]

which, together with the exact result for $\mathcal{I}^{U,2}_{N}$ obtained below (equations (S.37), (S.38)) is equivalent to (22). To complete the derivation of fidelity susceptibility we need to address the task of calculating $\mathcal{I}^{U,2}_{N}$ to which we turn now.

Let us start by expressing $\mathcal{I}^{U,2}_{N}$ in terms of invariants ($H$ is now $N\times N$ GUE matrix),

[TABLE]

Substituting $z_{i}=U\xi_{i}$ with $U$ such that $UH^{-1}U^{{\dagger}}=\mathrm{diag}\left(E^{-1}_{1},...,E^{-1}_{N}\right)$ and then putting $\xi_{i}=r_{i}\mathrm{e}^{\mathrm{i}\phi_{i}}$ one gets

[TABLE]

One can integrate over the phases $\phi_{j}$, resulting in a factor $(2\pi)^{N}$ which cancels out with $1/(2\pi)^{N}$ arising in substitution $t_{i}=\pi r_{i}^{2}$ so that the integral becomes

[TABLE]

where $m_{2}$ and $m_{1}$ are the second and the first moments of $\mathrm{e}^{-t}$ distribution. Using (S.27) in (S.25), remembering that $m_{2}=2$ and $m_{1}=1$ one obtains the following expression

[TABLE]

demonstrating validity of (13).

I.4 The generating function

Consider the generating function (15)

[TABLE]

Using the equality

[TABLE]

we verify that (16) indeed holds. Moreover, as a side product one gets

[TABLE]

which can be used as a validation of our calculation by comparison of the result with (S.24).

I.5 Calculation of generating function

Formulas best suited for our task of finding $Z(j_{1},j_{2})$ are worked out in Strahov and Fyodorov (2003):

[TABLE]

where $\Delta(\lambda_{1},...,\lambda_{K})$ is Vandermonde determinant and the kernel $W_{N+K}$ reads

[TABLE]

where $\Pi_{M}(\lambda)$ are monic polynomials orthogonal with respect to a measure $\mathrm{e}^{-V(x)}\mathrm{d}x$ and $C_{N,K}$ is a constant. For the GUE case $V(x)=\frac{1}{2J^{2}}x^{2}$. Using the equations (19), (20) – we obtain the following closed analytical expression for the generating function $Z(j_{1},j_{2})$

[TABLE]

[TABLE]

for even $N$. Expression for odd $N$ can be analogously derived.

It is interesting to note that von Oppen, during his calculation of distribution of curvatures for GUE von Oppen (1994) calculated

[TABLE]

using technique of supersymmetric integrals (for a pedagogical introduction of this technique see Haake (2010)). The formula for $Z(j_{1},j_{2})$ derived by us is an extension of the above expression– one can show that in the limit $\lim_{j_{1}\rightarrow 0}Z_{N}(j_{1},j_{2})$ one recovers the von Oppen’s formula (S.35) for $N\gg 1$. Calculating the derivatives one readily obtains:

[TABLE]

and

[TABLE]

which via (16) implies that

[TABLE]