Fidelity Susceptibility and Relevant Deformations

Fidelity susceptibility is the quadratic loss of overlap along a normalized state family. It enhances small-gap contributions and can expose a critical crossover, but its raw value can also be dominated by volume, ultraviolet cutoff, boundary conditions, or even a missing wavefunction normalization. A defensible QFT claim therefore combines a fixed fidelity convention with finite-size and cutoff tests.

Required background. State perturbations and susceptibility provide normalized state tangents and the rule for separating local regulator terms from a response coefficient.

Helpful background. Bures and Kubo–Mori metrics provide the factor-of-four convention and explain why fidelity and relative-entropy Hessians differ for noncommuting mixed states.

The chapter’s differentiable-family starting point fixes the tangent normalization, its comparison of information geometries fixes the fidelity convention, and its independent validity gates organize the support and regulator checks below.

Ground-state fidelity susceptibility

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

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

Then

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

For $V=\partial_gH$ and a discrete, nondegenerate spectrum,

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

This is the real part of the ground-state quantum geometric tensor Venuti and Zanardi 2007, Eqs. (1) and (2). Exact degeneracy requires a ground-state projector and non-Abelian transport; the single-vector formula is then not valid.

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

$$\chi_F =\int_0^\infty d\tau\;\tau \left[\langle V(\tau)V(0)\rangle-\langle V\rangle^2\right].$$

The spectral and imaginary-time forms, including their gap assumptions, appear in You, Li, and Gu 2007, Eqs. (4)–(11), pp. 022101-1–022101-2. In continuum QFT, the integral is distributional near $\tau=0$; local contact terms and source counterterms can shift its ultraviolet part.

QFT application: a regulated free-field mass deformation

Put a real free scalar in a periodic box of side $L$ and retain modes below $\Lambda$. Let $\mathcal K_{\Lambda,L}$ label an orthonormal basis of independent real normal coordinates, one per physical oscillator:

$$H_\Lambda(r) =\frac12\sum_{k\in\mathcal K_{\Lambda,L}} \left[\pi_k^2+(k^2+r)\phi_k^2\right], \qquad \omega_k(r)=\sqrt{k^2+r}, \qquad r=m^2>0.$$

For periodic boundary conditions, the real-mode multiplicities equal those of the full momentum lattice; no additional $k\leftrightarrow-k$ factor is inserted. The strict inequality $r>0$ is important because the periodic zero mode has no normalizable oscillator vacuum at $r=0$.

One normalized oscillator ground state is

$$\psi_\omega(q)=\left(\frac{\omega}{\pi}\right)^{1/4}e^{-\omega q^2/2}.$$

Its exact overlap at two frequencies is

$$f_k(\omega,\omega') =\left(\frac{2\sqrt{\omega\omega'}}{\omega+\omega'}\right)^{1/2}.$$

Using $\partial_r\omega_k=1/(2\omega_k)$ and expanding the normalized overlap gives the finite-regulator benchmark

$$\boxed{ \chi_F(L,r;\Lambda) =\frac1{32} \sum_{k\in\mathcal K_{\Lambda,L}} \frac{1}{(k^2+r)^2} }.$$

This is an exact analytic result at fixed $(L,r,\Lambda)$, not a continuum answer. Its controls are the spatial dimension $d_s$, $L$, $\Lambda$, $r$, boundary conditions, real-mode multiplicities, and the normalization of the coupling $r$. A numerical overlap also needs a parameter step $\delta r$, precision control, and logarithmic accumulation of the mode product to avoid underflow.

Finite-size collapse and its domain

Set

$$y=L\sqrt r.$$

For the periodic momentum grid $k=2\pi\mathbf n/L$, the exact benchmark becomes

$$\frac{\chi_F(L,r;\Lambda)}{L^4} =\frac1{32} \sum_{\substack{\mathbf n\in\mathbb Z^{d_s}\\ 2\pi\lvert\mathbf n\rvert\leq\Lambda L}} \frac{1}{\left(4\pi^2\lvert\mathbf n\rvert^2+y^2\right)^2}.$$

At fixed $y>0$, the omitted tail scales as

$$\delta\!\left(L^{-4}\chi_F\right) =O\!\left((\Lambda L)^{d_s-4}\right)$$

for $d_s<4$. Hence $L^{-4}\chi_F$ approaches a cutoff-independent scaling function in those dimensions, with a known convergence test. At $d_s=4$ the raw sum has logarithmic ultraviolet growth; for $d_s>4$ it has power growth. Those cases require subtraction or renormalization of a local extensive background before any universal scaling function is claimed.

The general critical-theory check reaches the same conclusion. For

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

with dynamic exponent $z=1$, an infrared-dominated total susceptibility scales as

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

This follows from the integrated correlator and is conditional on infrared dominance Venuti and Zanardi 2007, Eqs. (5)–(8). For the free mass deformation, $\mathcal O=\phi^2/2$ has $\Delta_{\mathcal O}=d_s-1$, so the predicted singular power is $L^4$. The explicit mode sum shows exactly why this power survives only without an overriding ultraviolet term.

At fixed nonzero $r$ and large $L$, a useful decomposition is

$$\chi_F(L,r;\Lambda) =a_{\rm UV}(r,\Lambda)L^{d_s} +\chi_F^{\rm sing}(L,r)+\text{boundary and finite-size terms}.$$

Neither $a_{\rm UV}$ nor a finite constant is generally universal. A coupling redefinition $\widetilde r=a r$ also transforms the component as $\chi_{\widetilde r}=\chi_r/a^2$; dimensions and operator normalization belong in every comparison.

Adversarial test: cutoff and state normalization

The exact single-mode overlap contains the normalized prefactor $(\omega/\pi)^{1/4}$. Replacing it by an arbitrary positive prefactor $A(\omega)$ gives

$$\widetilde f_k =A(\omega)A(\omega') \sqrt{\frac{2\pi}{\omega+\omega'}},$$

which need not equal one at coincident parameters and can contain spurious linear and quadratic terms. Across many modes these arbitrary factors become volume- and cutoff-dependent contributions. The first adversarial check is therefore brutally simple: normalize every state before taking an overlap and require $f(g,g)=1$ to numerical precision.

The second check changes $\Lambda$ at fixed $y=L\sqrt r$:

• for $d_s<4$, the collapsed curves must approach one another with residual $O((\Lambda L)^{d_s-4})$;
• for $d_s=4$, a logarithmic local term must be isolated;
• for $d_s>4$, the leading power background $a_{\rm UV}\propto\Lambda^{d_s-4}$ must be removed under a declared scheme.

Only a peak location, singular exponent, logarithmic coefficient, or collapse that survives both normalization and the predicted cutoff subtraction is a candidate for universal information. In this free benchmark the susceptibility grows monotonically as $r\downarrow0$; a finite-$r$ “critical peak” created by unnormalized states, a coarse step, or one cutoff is an artifact.

For numerical overlaps with $r>\delta r/2$, use the centered estimator

$$\chi_F^{(\delta r)}(r) =-\frac{2}{\delta r^2} \log f\!\left(r-\frac{\delta r}{2},r+\frac{\delta r}{2}\right) =\chi_F(r)+O(\delta r^2),$$

and repeat at smaller $\delta r$. The total error budget contains this step error, floating-point summation or overlap error, finite-volume corrections, and the explicit cutoff tail. At $y=0$ the periodic free-field state itself fails to exist, so no extrapolation can turn that endpoint into a normalized finite-volume vacuum.

For mixed states, the purification-independent object is Uhlmann root fidelity, not the overlap of one arbitrarily chosen purification Uhlmann 1976, Eq. (1), p. 273. Its infinitesimal metric is geometric Bures, $F_{\rm SLD}/4$, and generally differs from the BKM relative-entropy Hessian. For a reduced QFT state, hold the region and algebra fixed or else shape response is mixed into the state deformation.

Common pitfalls

Treating a raw maximum as a phase transition. Track its drift, width, cutoff dependence, gap, and independently motivated finite-size collapse.

Forgetting that susceptibility is a metric component. Rescaling a dimensionful coupling rescales $\chi_F$. State the deformation operator and parameter units.

Multiplying tiny overlaps directly. In a field theory the product can underflow long before the physics is singular. Sum $\log f_k$ and verify the result against the exact mode formula.

Exercises

1. Derive the oscillator overlap and the coefficient $\chi_F^{(k)}=1/(32\omega_k^4)$ for the mass-squared parameter $r$.

Solution

Gaussian integration gives

$$\int dq\,\psi_\omega(q)\psi_{\omega'}(q) =\left(\frac{2\sqrt{\omega\omega'}}{\omega+\omega'}\right)^{1/2}.$$

Set $\omega'=\omega+d\omega$. Expanding its logarithm gives

$$f_k=1-\frac1{16}\left(\frac{d\omega}{\omega}\right)^2 +O\!\left((d\omega/\omega)^3\right).$$

Since $d\omega=dr/(2\omega)$,

$$f_k=1-\frac{dr^2}{64\omega^4}+O(dr^3) =1-\frac12\chi_F^{(k)}dr^2+O(dr^3),$$

so $\chi_F^{(k)}=1/(32\omega^4)$.

2. Use continuum power counting to classify the ultraviolet behavior of the free-field sum in $d_s=3,4,5$. What should be plotted for a finite-size collapse?

Solution

At large momentum,

$$\chi_F\sim L^{d_s}\int^{\Lambda}dk\,k^{d_s-5}.$$

The integral converges in $d_s=3$, grows as $\log\Lambda$ in $d_s=4$, and grows as $\Lambda$ in $d_s=5$. At fixed $y=L\sqrt r$, the singular contribution is $L^4$ in all three cases. Thus one plots $L^{-4}\chi_F$ against $y$ directly in $d_s=3$. In $d_s=4$ or $5$, first subtract the declared logarithmic or power-divergent extensive term, then plot the subtracted quantity divided by $L^4$. Collapse of the raw $d_s=5$ data would be spoiled by the $L^5\Lambda$ background.

3. Suppose a code uses $\widetilde\psi_r(q)=e^{br}\psi_r(q)$ instead of a normalized state. Diagnose the centered overlap estimator and give a validation protocol.

Solution

The overlap is multiplied by $e^{b(r-\delta r/2)}e^{b(r+\delta r/2)}=e^{2br}$, independent of the step but not equal to one at coincident parameters. Therefore

$$-\frac{2}{\delta r^2}\log\widetilde f =\chi_F^{(\delta r)}-\frac{4br}{\delta r^2},$$

which diverges as the step is refined and can mimic an arbitrarily large feature. Before extracting curvature, normalize each vector, test $f(r,r)=1$, compare the numerical answer with $\frac1{32}\sum_k(k^2+r)^{-2}$, halve $\delta r$ to check the $O(\delta r^2)$ remainder, and vary $\Lambda$ according to the predicted tail law.

References

• Uhlmann, Armin. “The ‘Transition Probability’ in the State Space of a *-Algebra.” Reports on Mathematical Physics 9 (1976): 273–279. DOI; Open PDF.
• Venuti, Lorenzo Campos, and Paolo Zanardi. “Quantum Critical Scaling of the Geometric Tensors.” Physical Review Letters 99 (2007): 095701. DOI; arXiv.
• You, Wen-Long, Ying-Wai Li, and Shi-Jian Gu. “Fidelity, Dynamic Structure Factor, and Susceptibility in Critical Phenomena.” Physical Review E 76 (2007): 022101. DOI; APS PDF.