Fidelity Susceptibility in Quantum Phase Transitions

• Fidelity susceptibility is a metric that quantifies the quadratic response of quantum state overlaps to infinitesimal Hamiltonian changes, detecting sudden ground-state reorganization at criticality.
• It leverages perturbation theory, finite-size scaling, and quantum Monte Carlo methods to extract universal critical exponents and diagnose various quantum phase transitions.
• Its connection to the quantum geometric tensor and linear response functions allows experimental access through measurable noise spectra, absorption, and quench dynamics.

Fidelity susceptibility quantifies the leading quadratic response of quantum state overlap—fidelity—under an infinitesimal change of a Hamiltonian control parameter. This metric captures the ground-state geometry and reveals universal scaling laws near critical points, making it a powerful, order-parameter–independent diagnostic of both conventional and topological quantum phase transitions. Fidelity susceptibility is rooted in the real part of the @@@@1@@@@ and can be connected to linear response, dynamical structure factors, and, in certain regimes, to measurable noise spectra. The concept extends from few-body models to disordered systems, random matrix ensembles, complex spin-orbit models, and admits both field-theoretic and holographic formulations.

1. Formal Definition and Fundamental Properties

Given a parameter-dependent Hamiltonian $H(\lambda)$ with nondegenerate ground state $|\Psi_0(\lambda)\rangle$, the fidelity between ground states at $\lambda$ and $\lambda+\delta\lambda$ is

$$F(\lambda, \lambda+\delta\lambda) = |\langle\Psi_0(\lambda)|\Psi_0(\lambda+\delta\lambda)\rangle|.$$

For small $\delta\lambda$, the leading quadratic term defines the fidelity susceptibility: $$F(\lambda, \lambda+\delta\lambda) \approx 1 - \tfrac{1}{2}\chi_F(\lambda)(\delta\lambda)^2 + \mathcal{O}((\delta\lambda)^3),$$

$$\chi_F(\lambda) = \lim_{\delta\lambda\to 0} \left[-2\ln F(\lambda,\lambda+\delta\lambda)\middle/(\delta\lambda)^2\right].$$

For general Hamiltonians $H(\lambda)=H_0 + \lambda H_1$, second-order perturbation theory yields the Lehmann representation

$$\chi_F(\lambda) = \sum_{n\neq 0} \frac{|\langle\Psi_n(\lambda)|H_1|\Psi_0(\lambda)\rangle|^2}{[E_n(\lambda)-E_0(\lambda)]^2}.$$

Geometrically, fidelity susceptibility is the metric component $g_{\lambda\lambda}$ of the quantum geometric tensor: $$\chi_F(\lambda) = \langle\partial_\lambda\Psi_0|\partial_\lambda\Psi_0\rangle - |\langle\Psi_0|\partial_\lambda\Psi_0\rangle|^2$$ (Wang et al., 2015, Zhang et al., 1 Sep 2025).

The fidelity susceptibility measures the infinitesimal “distance” in Hilbert space induced by perturbing the parameter $\lambda$, and detects abrupt reorganization of the ground-state manifold at quantum critical points.

2. Fidelity Susceptibility and Quantum Criticality

Fidelity susceptibility generically diverges or displays pronounced peaks at quantum critical points (QCPs), where the ground state changes nonanalytically. Its finite-size scaling encodes universal critical exponents, such as the correlation length exponent $\nu$, dynamical exponent $z$, and, in topological settings, the curvature exponents of the quantum metric tensor.

Finite-size scaling near a continuous QCP of linear extent $L$ in $d$ spatial dimensions follows: $$\chi_F(\lambda,L) \sim L^{2/\nu}f[(\lambda-\lambda_c)L^{1/\nu}]$$ with $\chi_{F,\,\rm max} \sim L^{2/\nu}$ at criticality (Wang et al., 2015, Wei et al., 2017, Wei, 2019, Yu et al., 2022, Yu et al., 2014).

In the quantum Rabi model (QRM), fidelity susceptibility sharply peaks at the critical coupling $g_c$: $\chi_F(g_c) \sim \eta^{4/3},$ where $\eta$ is the ratio of two-level atom and cavity frequencies. The scaling collapse

$$\chi_F(\eta,g)\,\eta^{-4/3} = f\bigl( (g-1)\eta^{2/3} \bigr)$$

establishes the universal scaling regime near criticality (Wei et al., 2017).

Topological and disordered systems: In Dirac models, the fidelity susceptibility density $\chi_F(k;M)$ coincides—up to normalization—with the square of the curvature function whose integral yields topological invariants. At topological phase transitions,

$\chi_F(k_{\mathrm{c}};M) \sim |M|^{-2\gamma},$

with critical exponents encoding the universality class (Panahiyan et al., 2020). In disordered models (Anderson, Aubry–André), critical exponents extracted from $\chi_F$ scaling distinguish localization-delocalization transitions and their universality classes (Wei, 2019). In one-dimensional models, twisted boundary conditions can restore divergences in $\chi_F$ even when conventional scaling arguments would predict absence thereof, enabling detection of exotic or hidden QCPs (Thakurathi et al., 2012).

3. Connection to Linear Response and Dynamical Probes

Fidelity susceptibility is directly related to dynamical susceptibilities and linear response. Defining the dynamical structure factor for the driving operator $H_1$: $$S_{H_1}(\omega) = 2\pi \sum_{n\neq 0} |\langle\Psi_n|H_1|\Psi_0\rangle|^2\,\delta(\omega - [E_n - E_0]),$$ there holds: $$\chi_F = \int_0^\infty \frac{S_{H_1}(\omega)}{\omega^2}d\omega,$$ and, more generally,

$$\chi_{2r+2} = \int_0^\infty \frac{S_{H_1}(\omega)}{\omega^{2r+2}}d\omega$$

for higher-order adiabatic susceptibilities (Wei et al., 2017, Wei, 2019). These quantities encode the moments of the noise spectrum and may be extracted from energy absorption/emission measurements in cavity- or circuit-QED setups. In time-dependent (ramp or quench) protocols, the excitation probability on traversing a QCP is proportional to the generalized susceptibility, $P_{\rm ex}\propto b^2\chi_{2r+2}(g_c)$, connecting $\chi_F$ to nonadiabatic response.

4. Numerical Computation and Monte Carlo Estimators

Fidelity susceptibility can be computed efficiently in large-scale simulations using universal estimators embedded in modern quantum Monte Carlo (QMC) algorithms. In QMC expansions sampling $k$ insertions of $H_1$, the covariance of left and right half-segments gives: $$\chi_F^{T>0} = \frac{\langle k_L k_R\rangle-\langle k_L\rangle\langle k_R\rangle}{2\lambda^2}.$$ This estimator applies in continuous-time worldline, diagrammatic, and stochastic series expansion (SSE) QMC, and remains valid at finite and zero temperature (Wang et al., 2015).

For large systems or critical models with diverging correlation length, path-integral and transfer-matrix DMRG methods yield $\chi_F$ from overlaps between ground states at nearby parameters without requiring explicit calculation of order parameters. DMRG extraction of $\chi_F$ enables high-precision diagnosis of commensurate-incommensurate, chiral, floating, and Potts-type transitions in quantum simulators (Yu et al., 2022).

Quantum algorithms based on block-encoding and quantum singular value transformation can evaluate $\chi_F$ efficiently, bypassing direct summations over exponentially many excited states (Zhang et al., 1 Sep 2025).

5. Extensions: Disordered, Topological, Non-Hermitian, and Holographic Systems

Disordered Systems: In models with true-random or quasi-periodic disorder (Anderson, Aubry–André), $\chi_F$ shows maxima at QCPs, and its finite-size scaling yields distinct universality classes (e.g., $\nu=2/3$, $z=2$ for Anderson; $\nu=1$, $z\approx2.37$ for Aubry–André) (Wei, 2019).

Topological Transitions: Fidelity susceptibility is a sensitive probe of topological phase transitions, as in the Kitaev chain, where it sharply peaks at the TQPT point indicating the emergence of Majorana end modes. Its robustness to disorder is established by the preservation of peak features under local potential, hopping, and even pairing-disorder perturbations (Wang et al., 2013).

Non-Hermitian Systems: By extending the inner-product structure to the biorthogonal metric, fidelity susceptibility identifies exceptional points (EPs) in non-Hermitian Hamiltonians, with a divergence to $-\infty$ distinguishing EPs from Hermitian QCPs ($+\infty$) (Tzeng et al., 2020).

Holography and Information Complexity: AdS/CFT correspondence links fidelity susceptibility to the regularized volume of maximal slices in AdS, giving a geometric dual that encodes $PV$-criticality and universal scaling. This relationship provides a framework for studying the interplay between information-theoretic, thermodynamic, and geometric order parameters in strongly correlated and nonrelativistic systems (Momeni et al., 2016, Momeni et al., 2017).

6. Model Applications and Summary Table

Fidelity susceptibility has been broadly utilized across multiple models. The following table summarizes select models and $\chi_F$'s role:

Model/Class	Use of $\chi_F$	Critical Scaling/Exponent(s)
Quantum Rabi Model	QPT diagnosis, universal finite-size scaling	$\chi_F \sim \eta^{4/3}$, $\nu=3/2$, $z=1/3$ at $g_c=1$ (Wei et al., 2017)
Disordered (Anderson/AA)	Locate localization transitions, class fingerpint	$\nu=2/3$ (Anderson), $\nu=1$ (AA), $z=2,2.37$ (Wei, 2019)
XXZ/Hubbard-like models	Detect phase boundaries (BKT, CDW, SDW, BOW, etc.)	Various, e.g. $\ln N$, $N^{0.3784}$, or $N^2$ divergence (Yu et al., 2014)
Topological SC (Kitaev)	Identify TQPT/Majorana edge, disorder stability	$\chi_F \sim N^2$ at TQPT (Wang et al., 2013)
Rydberg Chains (DMRG)	Distinguish Ising, Potts, chiral, floating QPTs	$\chi_F \sim L^{2/\nu}$, with $\nu$ class-dependent (Yu et al., 2022)
Dirac/topological insulators	Map to curvature function ($\chi_F \sim F^2$)	$\chi_F \sim |M|^{-2\gamma}$, exponents topologically fixed (Panahiyan et al., 2020)

7. Thermodynamic Connections and Experimental Access

The fidelity susceptibility connects to thermodynamic susceptibilities through upper/lower bounds. In the commuting case, $\chi_F$ reduces to variance-driven quantities such as magnetic susceptibility or specific heat; in noncommuting cases, correction terms involving operator commutators appear but diverging behavior at criticality remains equivalent (Brankov et al., 2011). Experimental determination is feasible via absorption/emission spectra, Bragg spectroscopy, or quench dynamics in cold-atom systems; finite-temperature and non-Hermitian extensions demonstrate its broad applicability.

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In conclusion, fidelity susceptibility provides a universal, information-geometric diagnostic of quantum phase transitions, sensitive not only to symmetry-breaking but also topological and localization transitions, robust to disorder, and directly linked to dynamical response. Its scaling behavior classifies universality, and its relation to the quantum geometric tensor situates it at the intersection of quantum information, condensed matter theory, and emerging holographic dualities.