The nuts and bolts of gauge invariance of heat transport

In this work I revisit the notion of gauge invariance in thermal transport and show, in the simplest and most general possibile terms, why heat conductivity is unaffected by the specific choice of energy density. I provide the minimal and general conditions under which any two energy densities, though differing locally, lead to the same heat conductivity within the Green-Kubo framework. The relevance of gauge invariance in heat-transport simulations performed with machine-trained neural-network potentials is also briefly highlighted.

💡 Research Summary

In this paper the author revisits the long‑standing issue of gauge freedom in the definition of the microscopic energy density and its impact on the calculation of thermal conductivity via the Green‑Kubo formalism. The starting point is the standard Green‑Kubo expression for the thermal conductivity of a classical system, κ = (k_B β²/V)∫₀^∞⟨J_E(t)·J_E(0)⟩dt, where J_E is the volume integral of the microscopic energy current j_E(r). The microscopic energy density ε(r|R,V) is split into a kinetic part (½ ∑_I M_I v_I² δ(r−r_I)) and a potential part w(r|R) that does not depend on velocities. Because w(r|R) can be represented in many ways (e.g., as a sum of atomic contributions or as a continuous field), the energy density is intrinsically ill‑defined; however, any two representations that give the same total energy and the same forces on the atoms are physically equivalent.

The core question addressed is whether two such “energy gauges” also lead to identical thermal conductivities. Let w₁ and w₂ be two admissible potential‑energy densities and define w′ = w₂ − w₁. The associated force‑density difference φ′_I(r|R) = −∂w′/∂r_I must integrate to zero (∫φ′_I dr = 0) to guarantee identical atomic forces. The author shows that this condition alone is not sufficient; an additional short‑range (or “short‑sighted”) assumption is required: φ′_I(r|R) = 0 for |r − r_I| > R_c. Under this assumption the difference between the two energy currents, J′ = J₂ − J₁, can be written as J′ = ∑_I D′_I(R)·v_I with D′_I(R) = −∫ r⊗φ′_I(r|R) dr. The vector‑valued differential form ω(R) = ∑_I D′_I(R)·dr_I is exact, which implies that the time integral of J′ (the Helfand moment) depends only on the initial and final atomic configurations and is bounded for any trajectory in phase space. Consequently, the contribution of J′ to the Green‑Kubo integral vanishes, and the thermal conductivity is invariant under the change of energy gauge.

In a single‑component system the only possible convective term arising from J′ is proportional to the total mass current, which is conserved and therefore does not affect transport coefficients. For multi‑component systems the analysis is extended by considering both the energy flux J_E and the particle flux J_N. The Onsager matrix Λ_ij is introduced, and the thermal conductivity is expressed as the Schur complement κ = Λ_EE − (Λ_EN)²/Λ_NN. The author demonstrates that adding any linear combination of the particle flux to the energy flux (J_E → J_E + c J_N) leaves κ unchanged, confirming gauge invariance in the multivariate case.

The theoretical results are illustrated with first‑principles molecular dynamics of the solid electrolyte Li₃PS₄. Different empirical potentials produce distinct energy‑current power spectra Λ_EE(ω), yet the derived κ(ω) (the Schur complement of the convective block) converges to the same zero‑frequency value, confirming that the Onsager coefficient Λ_EE is gauge‑dependent while the thermal conductivity is not.

Finally, the paper discusses practical implications for simulations using machine‑learned neural‑network potentials. Adding species‑dependent constant shifts to atomic energies changes the energy density but leaves forces unchanged; the resulting shift in the energy current is purely convective. By the proven convective invariance, the Green‑Kubo thermal conductivity computed from such potentials is unaffected. This provides a robust theoretical foundation for flexible energy‑gauge choices in modern atomistic simulations.

In summary, the author provides a minimal and general proof that any two short‑ranged, locally defined energy densities yielding identical forces produce the same thermal conductivity. The proof clarifies that the previously used sufficient condition—boundedness of the Helfand moment difference—is in fact a special case of a more general convective invariance. The work solidifies the theoretical basis of gauge invariance in heat transport and opens the way for its application to more complex interactions, including long‑range forces, in future studies.