Fidelity Susceptibility and Relevant Deformations

Fidelity susceptibility measures the quadratic loss of overlap under a parameter deformation. It can reveal a critical crossover because low-energy states with small gaps receive enhanced weight, but it also contains volume, boundary, and ultraviolet contributions. A universal diagnosis requires a specified state family, fidelity convention, deformation operator, and finite-size or cutoff analysis.

Required background. State perturbations and susceptibility supply normalized tangents.

Helpful background. Bures and Kubo–Mori metrics distinguish overlap geometry from relative-entropy geometry.

Ground-state fidelity susceptibility

Let $H(g)$ have a nondegenerate normalized ground state $\lvert0(g)\rangle$. Define

$$\lvert\langle0(g)\mid0(g+\delta g)\rangle\rvert =1-\frac12\chi_F(g)\,\delta g^2+O(\delta g^3).$$

Choosing the parallel-transport phase $\langle0\mid\partial_g0\rangle=0$ gives

$$\chi_F =\langle\partial_g0\mid\partial_g0\rangle -\lvert\langle0\mid\partial_g0\rangle\rvert^2.$$

For $V=\partial_g H$ and a discrete nondegenerate spectrum, the spectral representation used by Venuti and Zanardi 2007, pp. 1–3 is

$$\chi_F =\sum_{n\ne0} \frac{\lvert\langle n\lvert V\rvert0\rangle\rvert^2} {(E_n-E_0)^2}.$$

The squared gap in the denominator explains sensitivity to criticality. It also makes the formula unstable near exact degeneracy, where one must use a ground-state subspace and non-Abelian transport instead of a single vector.

With $V(\tau)=e^{\tau H}Ve^{-\tau H}$,

$$\chi_F =\int_0^\infty d\tau\;\tau\, \langle V(\tau)V(0)\rangle_c.$$

This imaginary-time representation assumes convergence and the subtraction of $\langle V\rangle^2$. Contact terms at $\tau=0$ can contribute regulator-dependent local pieces.

The structural map places Fidelity Susceptibility and Relevant Deformations along the perturbative sequence from normalized state or shape variations to information metrics and modular transport.

At fixed comparison algebra, normalization removes the linear term in relative entropy. The second response supports several inequivalent constructions: state and shape tangents, stress-tensor kernels, monotone metrics, and zero-mode-projected transport. The diagram is schematic and not to scale.

Relevant deformations in QFT

Consider a Lorentz-invariant critical theory in $d_s$ spatial dimensions deformed by

$$H(g)=H_*+g\int d^{d_s}x\;\mathcal O(x),$$

where $\mathcal O$ has scaling dimension $\Delta_{\mathcal O}$. At the fixed point and for dynamic exponent $z=1$, naive infrared scaling gives

$$\chi_F(L)\sim L^{\,2d_s+2-2\Delta_{\mathcal O}}$$

when the integrated correlator is infrared dominated and no logarithm or boundary term intervenes. More generally, the scaling variable is set by the correlation-length exponent and dynamic exponent.

This power law is conditional. If the short-distance part dominates, $\chi_F$ instead has an extensive cutoff-dependent term. At the marginal exponent a logarithm can appear. Boundaries and the entangling surface introduce additional area-like terms. The robust analysis fits several sizes and cutoffs to

$$\chi_F(L,\epsilon) =a_{\rm UV}(\epsilon)L^{d_s} +a_{\partial}(\epsilon)L^{d_s-1} +\chi_F^{\rm sing}(L) +\cdots,$$

rather than calling the leading coefficient universal.

Mixed and reduced states

For mixed states, fidelity susceptibility is the infinitesimal Bures metric once rooted-versus-squared fidelity conventions are matched:

$$D_B^2(\rho_g,\rho_{g+dg}) =\frac14F_Q(g)\,dg^2.$$

This quantity differs from the Kubo–Mori or relative-entropy susceptibility for noncommuting tangents. At finite temperature, the change of eigenvalues and the rotation of eigenvectors both contribute. For reduced QFT states, one must also hold the region and algebra fixed; otherwise shape response is mixed into the state-family metric.

An overlap computed from a particular purification is not automatically the Uhlmann fidelity. Optimizing over purifications or using the operator formula is necessary for a purification-independent mixed-state result.

Finite-size crossover rather than a single peak

A peak in $\chi_F(g,L)$ can drift with $L$, broaden at finite temperature, or arise from an avoided crossing unrelated to a thermodynamic transition. A credible critical analysis reports:

• the peak position and width versus $L$;
• collapse using independently justified critical exponents;
• the regular extensive and boundary backgrounds;
• gap and symmetry-sector information;
• cutoff or bond-dimension convergence;
• comparison with at least one independent observable.

Fidelity susceptibility is a diagnostic, not a phase label by itself.

Deformation normalization and contact terms

Rescaling $g\mapsto ag$ rescales $\chi_F\mapsto\chi_F/a^2$. The operator normalization and coupling units are therefore part of the result. In continuum perturbation theory, local counterterms quadratic in the source can shift the integrated two-point function. Universal logarithmic coefficients or scaling exponents may survive, while the finite constant need not.

For a spacetime-dependent source $g(x)$, susceptibility is a kernel,

$$\delta^2\!\bigl[-\log f\bigr] =\frac12\int d^dx\,d^dy\; \delta g(x)\,\mathcal G_F(x,y)\,\delta g(y),$$

with distributional contacts. Quoting a single number requires specifying the source profile.

Common pitfalls

Calling the largest peak a critical point without scaling. Track its drift, background, gap, and collapse across system sizes.

Comparing susceptibilities with differently normalized couplings. The metric transforms covariantly under reparametrization. State the operator normalization and units.

Equating fidelity and relative-entropy Hessians. They agree classically but differ along noncommuting mixed-state directions.

Before interpreting this response coefficient, use the validity map to check normalization, tangent domains, contact and regulator terms, metric choice, and analyticity independently.

Normalization, domain, contact-term, metric, and analyticity checks are independent. A failed gate narrows or withdraws the physical claim; it is not a change of notation. The decision map is schematic and not to scale.

References

• Venuti, Lorenzo Campos, and Paolo Zanardi. “Quantum Critical Scaling of the Geometric Tensors.” Physical Review Letters 99 (2007): 095701. DOI; arXiv.

Further reading

• Gu, Shi-Jian. “Fidelity Approach to Quantum Phase Transitions.” International Journal of Modern Physics B 24 (2010): 4371–4458. DOI; arXiv.