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1. Restating the question in plain language

You have a family of Hamiltonians (or Boltzmann weights) that depend on a set of parameters
[ H_{\boldsymbol\theta}(x),\qquad \boldsymbol\theta=(\theta_1,\dots ,\theta_n) . ]

When we perform a real‑space renormalisation‑group (RG) step (e.g. Kadanoff block‑spin decimation) the parameters are transformed, [ \boldsymbol\theta\;\longrightarrow\;\boldsymbol\theta’ = \mathcal R(\boldsymbol\theta) . ]

Thus the RG is a vector field (or “flux”) on the parameter space: at each point we have a tangent vector that tells us how the parameters change under an infinitesimal RG step.

Independently, the Fisher–Rao (information) metric on the same parameter space is defined for the Boltzmann distribution [ p(x;\boldsymbol\theta)=\frac{e^{-H_{\boldsymbol\theta}(x)}}{Z_{\boldsymbol\theta}} ] by
[ g_{ij}(\boldsymbol\theta)=\Big\langle \partial_i\log p \,\partial_j\log p\Big\rangle =\Big\langle \bigl(\partial_i H_{\boldsymbol\theta} - \partial_i \log Z_{\boldsymbol\theta}\bigr) \bigl(\partial_j H_{\boldsymbol\theta} - \partial_j \log Z_{\boldsymbol\theta}\bigr)\Big\rangle . ]

The student asks:

  • What, if anything, does the Fisher–Rao metric tell us about the RG flow?
  • Do the metric’s geodesics have any relation to RG trajectories?
  • Why does the metric diverge at critical points, and can that be interpreted in the language of RG?

Below we answer these points step by step, making explicit the mathematical connections and the physical interpretation.

2. Detailed answer

2.1 The Fisher–Rao metric for a Boltzmann family

For a Boltzmann distribution the log‑likelihood is

[ \log p(x;\boldsymbol\theta)= - H_{\boldsymbol\theta}(x)-\log Z_{\boldsymbol\theta}. ]

Differentiating w.r.t. a parameter (\theta_i),

[ \partial_i\log p = -\partial_i H_{\boldsymbol\theta}(x)+\big\langle \partial_i H_{\boldsymbol\theta}\big\rangle, \qquad \big\langle \partial_i H_{\boldsymbol\theta}\big\rangle\equiv \partial_i\log Z_{\boldsymbol\theta}. ]

Hence the Fisher matrix becomes the covariance of the “score” operators:

[ \boxed{\,g_{ij}(\boldsymbol\theta)=\big\langle \big(\partial_i H_{\boldsymbol\theta}-\langle\partial_i H_{\boldsymbol\theta}\rangle\big) \big(\partial_j H_{\boldsymbol\theta}-\langle\partial_j H_{\boldsymbol\theta}\rangle\big) \big\rangle\,}. ]

If the Hamiltonian is linear in the parameters,
(H_{\boldsymbol\theta}(x)=\sum_\alpha \theta_\alpha\, O_\alpha(x)) (the usual “exponential family”), then

[ \partial_i H_{\boldsymbol\theta}=O_i(x),\qquad g_{ij}= \langle O_i O_j\rangle - \langle O_i\rangle\langle O_j\rangle . ]

Thus each entry of (g) is a connected two‑point correlation function (susceptibility, specific heat, …).

Key point: Near a continuous phase transition the long‑range fluctuations make these correlations diverge, and consequently the Fisher metric diverges.

2.2 RG flow as a vector field on the same manifold

Consider an infinitesimal RG step that rescales the lattice by a factor (b=1+\mathrm{d}\ell) and integrates out short‑wavelength modes. The effect on the couplings can be written as

[ \frac{\mathrm{d}\theta_i}{\mathrm{d}\ell}= \beta_i(\boldsymbol\theta), ]

where the beta functions (\beta_i) are the components of the RG vector field (\boldsymbol\beta).

The RG flow is not generated by the Fisher metric; it is defined by the coarse‑graining transformation. However, the metric provides a natural way to measure distances between two nearby Hamiltonians, and therefore to quantify how fast the flow moves in the information‑theoretic sense.

2.3 Relationship between the metric and the RG vector field

2.3.1 Gradient‑flow interpretation (when it holds)

In many simple models (e.g. Gaussian fixed point, free scalar field) the beta functions can be expressed as the gradient of a scalar potential with respect to the Fisher metric:

[ \beta_i(\boldsymbol\theta)= - g_{ij}(\boldsymbol\theta)\,\partial_j \Phi(\boldsymbol\theta). \tag{1} ]

Proof sketch.
For an exponential family the log‑partition function is the cumulant generating function, [ \Phi(\boldsymbol\theta)=\log Z_{\boldsymbol\theta}. ] Its gradient gives the expectation values of the observables: [ \partial_i \Phi = \langle O_i\rangle . ] In the linearized RG near a fixed point, the linear part of (\beta_i) is (\beta_i = y_i\theta_i) (with scaling dimensions (y_i)). The metric at the fixed point is constant (up to a rescaling), so (1) can be satisfied by choosing (\Phi = \frac12\sum_i y_i\,\theta_i^2). Hence the flow is a steepest descent of (\Phi) measured with the Fisher metric.

When (1) holds, the RG trajectories are geodesics of the “dual” connection (the Amari–Chentsov (\alpha)-connection with (\alpha= -1)). This is a well‑studied result in information geometry.

2.3.2 General case: metric as a “speed” measure

Even if the flow is not a pure gradient, we can always decompose the beta vector into components parallel and orthogonal to the metric:

[ \beta_i = \underbrace{g_{ij} v^j}{\text{metric‑dual of a vector }v} \;+\; \underbrace{w_i}{\text{orthogonal to }g}. ]

The scalar quantity

[ \boxed{\,|\boldsymbol\beta|^2_g \equiv g^{ij}\beta_i\beta_j\,} ]

is the information‑geometric speed of the RG flow. Near a critical point the metric diverges, but the beta functions typically vanish (the fixed point). The product (|\boldsymbol\beta|_g) can stay finite or go to zero, giving a precise way to say that the flow slows down in the information sense as the system approaches criticality.

2.3.3 Fixed points as “centers of curvature”

Because (g_{ij}) diverges as correlation length (\xi\to\infty), the Riemann curvature built from (g) also diverges. In information geometry this signals a singular statistical manifold: two distinct Hamiltonians become indistinguishable (their KL‑divergence tends to zero) because long‑range fluctuations dominate. Consequently, RG fixed points appear as boundary points of the statistical manifold where the metric is ill‑conditioned.

2.4 Do Fisher‑Rao geodesics coincide with RG trajectories?

Answer: Not in general, but under special circumstances they do.

Situation Geodesic = RG? Reason
Linear (Gaussian) theory, near a free fixed point Yes (up to reparametrisation) The beta functions are linear, the Fisher metric is flat, and the flow is a straight line, i.e. a geodesic.
Exponential family with a potential (\Phi) such that (1) holds Yes (gradient flow) The RG is the steepest‑descent curve of (\Phi) w.r.t. the Fisher metric, which coincides with the (\alpha=-1) geodesics.
Interacting critical points (e.g. 2‑D Ising) No The beta functions contain non‑gradient pieces; the manifold is curved; RG trajectories curve away from metric geodesics.
Non‑equilibrium or non‑canonical ensembles Usually no The Fisher metric is defined, but the RG transformation is not a simple coarse‑graining of the same Boltzmann weight.

Thus, geodesics are useful as a benchmark: if you compute the Fisher metric for a given model and find that the RG flow follows a geodesic, you have learned that the flow is a pure gradient with respect to the information metric. If not, the deviation tells you about the presence of non‑gradient (rotational) components in the beta functions.

2.5 Interpretation of the divergence at criticality

  1. Statistical‑information view:
    The Fisher metric is the Hessian of the relative entropy (Kullback–Leibler divergence) between neighboring distributions. At a critical point, the susceptibility (\chi_{ij} = \partial_i\partial_j \log Z) diverges, meaning that an infinitesimal change in (\boldsymbol\theta) produces an infinitely large change in the distribution as measured by KL. Hence the distance (metric) blows up.

  2. RG view:
    The RG vector field (\boldsymbol\beta) vanishes at a fixed point. The combination (|\boldsymbol\beta|_g) therefore tends to zero more strongly than the metric itself diverges, reflecting the critical slowing down: the system changes only logarithmically with the RG “time” (\ell).

  3. Geometric picture:
    The statistical manifold develops a conical singularity at the critical point. Approaching the tip (the fixed point) the metric eigenvalues go to infinity, while the flow lines (RG trajectories) become tangent to the cone’s axis. This picture underlies the “information‑theoretic renormalisation” proposals (e.g. Refs. [Mehta & Schwab 2014], [Bény & Osborne 2015]).

2.6 Summary of the concrete connections

Concept Fisher–Rao metric RG flow
Definition Covariance of (\partial_i H) (connected correlations) (\dot\theta_i = \beta_i(\theta))
Critical behavior Divergent eigenvalues (\sim) susceptibilities, specific heat (\beta_i\to 0) (fixed point) but linearised flow (\dot\theta_i = y_i\theta_i) with scaling dimensions (y_i)
Distance (ds^2 = g_{ij}\,d\theta_i d\theta_j) = KL‑second order expansion No intrinsic metric, but one can define “RG speed” (|\beta|_g)
Geodesics vs RG Geodesics are curves that extremise information distance RG trajectories are integral curves of (\beta); coincide with geodesics only when (\beta) is a gradient of a potential w.r.t. (g)
Useful quantity Fisher information matrix = susceptibility matrix → tells which direction in parameter space is most “stiff” Eigenvectors of Jacobian (\partial_j\beta_i) at a fixed point give relevant/irrelevant directions; the Fisher metric aligns with these eigenvectors at a Gaussian fixed point
Practical use Compute (g) to identify the most sensitive coupling; to design optimal RG schemes (e.g. information‑preserving coarse graining) Use (g) to weight the RG step so that the information loss per RG iteration is minimal (principle of maximal fidelity).

3. Final answer

  • The Fisher–Rao metric on the space of Boltzmann distributions is precisely the matrix of connected correlation functions (susceptibilities).
  • Near a critical point these correlations diverge, so the metric blows up. This reflects the fact that an infinitesimal change of the couplings leads to a macroscopic change of the probability distribution.
  • The RG transformation defines a vector field (\boldsymbol\beta(\boldsymbol\theta)) on the same manifold. The metric does not dictate the flow, but it provides a natural way to measure the speed and direction of the flow in an information‑theoretic sense: [ |\boldsymbol\beta|_g^2 = g^{ij}\beta_i\beta_j . ]
  • When the beta functions can be written as the gradient of a scalar potential with respect to the Fisher metric, [ \beta_i

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