Magnetic Wiedemann–Franz Law

• Magnetic Wiedemann–Franz law is a framework that extends the classical relationship between thermal and electrical conductivities by incorporating magnetic fields, Berry curvature, and magnon transport.
• The law reveals how quasiparticle integrity and scattering processes in heavy-fermion metals, graphene, and Weyl semimetals shape both longitudinal and transverse heat and charge transport.
• In insulating magnets, bosonic formulations show that magnons obey universal ratios linking thermal and spin conductance, providing insights into magnetic heat transport mechanisms.

The magnetic Wiedemann–Franz law denotes a family of extensions of the ordinary Wiedemann–Franz relation to settings in which magnetic field, magnetic order, Berry curvature, or magnon transport qualitatively reshape the connection between heat and charge transport. In its conventional metallic form, the law states that

$$L \equiv \frac{\kappa}{\sigma T} = L_0,\qquad L_0=\frac{\pi^2}{3}\left(\frac{k_B}{e}\right)^2$$

as $T\to 0$, when the same Landau quasiparticles carry both charge and heat and elastic scattering dominates. In magnetic contexts, the same question reappears in longitudinal, Hall, tensorial, and even bosonic forms: whether heat and charge remain locked by the Sommerfeld value, whether transverse coefficients satisfy an analogous law, and whether magnetic excitations such as magnons obey their own universal Lorenz ratio (Reid et al., 2013, Qiang et al., 2023, Nakata et al., 2015).

1. Definitions and principal formulations

A standard experimental reformulation, especially in correlated metals, introduces the electrical resistivity $\rho(T)=1/\sigma(T)$ and the thermal resistivity

$w(T)\equiv \frac{L_0T}{\kappa(T)}.$

In this notation, the zero-temperature Wiedemann–Franz law becomes

$w(0)=\rho(0),$

and the difference

$\delta(T)\equiv w(T)-\rho(T)$

directly measures how differently inelastic scattering degrades heat and charge currents (Reid et al., 2013).

Once a magnetic field is present, the conductivities become tensors, and the Lorenz ratio is naturally defined componentwise. In weak-field graphene, for example,

$$L_{xx}=\frac{\kappa_{xx}}{\sigma_{xx}T},\qquad L_{xy}=\frac{\kappa_{xy}}{\sigma_{xy}T},$$

and the low-temperature Fermi-liquid expectation is that both $L_{xx}$ and $L_{xy}$ tend to $L_0$ (Tu et al., 2023). In magnetic and topological conductors, the transverse Hall Lorenz ratio

$T\to 0$0

is the central object of the Hall Wiedemann–Franz law (Qiang et al., 2023).

A distinct but related meaning appears in insulating magnets, where charge transport is absent and magnons carry spin and heat. There the relevant ratio is not $T\to 0$1 but the ratio of magnonic thermal conductance $T\to 0$2 to magnon spin conductance $T\to 0$3,

$T\to 0$4

in the low-temperature limit. This defines a magnetic Lorenz number

$T\to 0$5

for bosonic spin transport (Nakata et al., 2015).

2. Field-tuned quantum critical metals

In heavy-fermion metals, the magnetic Wiedemann–Franz law is often posed as a zero-temperature test of whether quasiparticles survive a field-tuned magnetic quantum critical point. YbRh$T\to 0$6Si$T\to 0$7 is a canonical example. It orders antiferromagnetically at $T\to 0$8, and a small field suppresses $T\to 0$9 to zero at $\rho(T)=1/\sigma(T)$0 for $\rho(T)=1/\sigma(T)$1 and $\rho(T)=1/\sigma(T)$2 for $\rho(T)=1/\sigma(T)$3. Near that QCP, both $\rho(T)=1/\sigma(T)$4 and $\rho(T)=1/\sigma(T)$5 are linear in $\rho(T)=1/\sigma(T)$6 below $\rho(T)=1/\sigma(T)$7, but the linearity is cut off below a finite scale $\rho(T)=1/\sigma(T)$8, below which both quantities bend downward and converge as $\rho(T)=1/\sigma(T)$9 (Reid et al., 2013).

The decisive result is that $w(T)\equiv \frac{L_0T}{\kappa(T)}.$0 at low temperature for all fields studied, including the critical field. The associated $w(T)\equiv \frac{L_0T}{\kappa(T)}.$1 becomes nearly linear above $w(T)\equiv \frac{L_0T}{\kappa(T)}.$2 and then drops sharply toward zero below that scale. Comparison of cleaner and dirtier samples shows that $w(T)\equiv \frac{L_0T}{\kappa(T)}.$3 is essentially identical within error bars, which was taken as strong evidence that $w(T)\equiv \frac{L_0T}{\kappa(T)}.$4 and hence that the Wiedemann–Franz law holds in YbRh$w(T)\equiv \frac{L_0T}{\kappa(T)}.$5Si$w(T)\equiv \frac{L_0T}{\kappa(T)}.$6, even at the field-tuned QCP (Reid et al., 2013). In that interpretation, the non-Fermi-liquid linear-$w(T)\equiv \frac{L_0T}{\kappa(T)}.$7 regime is truncated by a finite crossover scale associated with short-range magnetic correlations rather than extending to $w(T)\equiv \frac{L_0T}{\kappa(T)}.$8.

The same work uses CeCoIn$w(T)\equiv \frac{L_0T}{\kappa(T)}.$9 as a comparison case and shows that the outcome can be anisotropic. At its field-tuned QCP, CeCoIn$w(0)=\rho(0),$0 with $w(0)=\rho(0),$1 shows a finite $w(0)=\rho(0),$2, a low-temperature drop in $w(0)=\rho(0),$3, and recovery of the Wiedemann–Franz law, whereas for $w(0)=\rho(0),$4, $w(0)=\rho(0),$5 and $w(0)=\rho(0),$6 remain perfectly linear down to $w(0)=\rho(0),$7, $w(0)=\rho(0),$8 extrapolates to a finite $w(0)=\rho(0),$9, and the law is violated in that direction (Reid et al., 2013). The magnetic Wiedemann–Franz law in correlated metals is therefore not solely a question of whether a magnetic QCP exists, but also of whether finite crossover scales and strong $\delta(T)\equiv w(T)-\rho(T)$0-space anisotropies survive down to the lowest temperatures.

A related Hall-channel test appears in underdoped YBa$\delta(T)\equiv w(T)-\rho(T)$1Cu$\delta(T)\equiv w(T)-\rho(T)$2O$\delta(T)\equiv w(T)-\rho(T)$3, where the electrical and thermal Hall conductivities were measured immediately above the vortex-melting field $\delta(T)\equiv w(T)-\rho(T)$4. There the $\delta(T)\equiv w(T)-\rho(T)$5 Hall relation $\delta(T)\equiv w(T)-\rho(T)$6 is satisfied just above $\delta(T)\equiv w(T)-\rho(T)$7, which was used to rule out a vortex liquid at $\delta(T)\equiv w(T)-\rho(T)$8 in that high-field regime (Grissonnanche et al., 2015).

3. Hall and anomalous Hall formulations in magnetic topological matter

In magnetic and topological conductors, the most incisive form of the magnetic Wiedemann–Franz law is often transverse. The relevant coefficients are the anomalous Hall conductivity $\delta(T)\equiv w(T)-\rho(T)$9, thermoelectric Hall conductivity $$L_{xx}=\frac{\kappa_{xx}}{\sigma_{xx}T},\qquad L_{xy}=\frac{\kappa_{xy}}{\sigma_{xy}T},$$0, and thermal Hall conductivity $$L_{xx}=\frac{\kappa_{xx}}{\sigma_{xx}T},\qquad L_{xy}=\frac{\kappa_{xy}}{\sigma_{xy}T},$$1. A Hall Wiedemann–Franz law means $$L_{xx}=\frac{\kappa_{xx}}{\sigma_{xx}T},\qquad L_{xy}=\frac{\kappa_{xy}}{\sigma_{xy}T},$$2 as $$L_{xx}=\frac{\kappa_{xx}}{\sigma_{xx}T},\qquad L_{xy}=\frac{\kappa_{xy}}{\sigma_{xy}T},$$3, while finite-temperature deviations encode the energy dependence of Berry curvature and disorder corrections (Qiang et al., 2023).

For topological kagome magnets, a useful unified representation is

$$L_{xx}=\frac{\kappa_{xx}}{\sigma_{xx}T},\qquad L_{xy}=\frac{\kappa_{xy}}{\sigma_{xy}T},$$4

with $$L_{xx}=\frac{\kappa_{xx}}{\sigma_{xx}T},\qquad L_{xy}=\frac{\kappa_{xy}}{\sigma_{xy}T},$$5 defined by energy moments of a kernel $$L_{xx}=\frac{\kappa_{xx}}{\sigma_{xx}T},\qquad L_{xy}=\frac{\kappa_{xy}}{\sigma_{xy}T},$$6 that may represent intrinsic Berry-curvature physics or disorder terms (Qiang et al., 2023). In a massive Dirac description of TbMn$$L_{xx}=\frac{\kappa_{xx}}{\sigma_{xx}T},\qquad L_{xy}=\frac{\kappa_{xy}}{\sigma_{xy}T},$$7Sn$$L_{xx}=\frac{\kappa_{xx}}{\sigma_{xx}T},\qquad L_{xy}=\frac{\kappa_{xy}}{\sigma_{xy}T},$$8 and Mn$$L_{xx}=\frac{\kappa_{xx}}{\sigma_{xx}T},\qquad L_{xy}=\frac{\kappa_{xy}}{\sigma_{xy}T},$$9Ge, the intrinsic Hall Lorenz ratio obeys a closed analytic formula depending only on $L_{xx}$0 and $L_{xx}$1, with a critical chemical potential

$L_{xx}$2

For $L_{xx}$3, the intrinsic Hall ratio is above $L_{xx}$4; for $L_{xx}$5, it is below $L_{xx}$6; and at $L_{xx}$7, it recovers the classical value exactly (Qiang et al., 2023). In this framework, disorder corrections restore Hall Wiedemann–Franz and Mott relations at leading order, but the experimentally relevant deviations in TbMn$L_{xx}$8Sn$L_{xx}$9 and Mn$L_{xy}$0Ge are dominated by intrinsic topological terms rather than disorder (Qiang et al., 2023).

A different Hall violation mechanism was identified in Mn$L_{xy}$1Ge. There the anomalous Lorenz ratio remains close to $L_{xy}$2 up to $L_{xy}$3, but above that it deviates because the thermal and electrical Berry-curvature sums weight the energy dependence of $L_{xy}$4 differently. The finite-temperature violation is therefore attributed not to inelastic scattering, as in ordinary metals, but to a mismatch between the thermal and electrical summations of Berry curvature (Xu et al., 2018). This makes the anomalous Lorenz ratio an unusually sensitive probe of the Berry spectrum near the chemical potential.

NdAlSi illustrates yet another Hall-sector outcome. In this magnetic Weyl semimetal, $L_{xy}$5 becomes strongly nonmonotonic in temperature and field and reaches a maximum close to $L_{xy}$6 in an intermediate regime. Analysis excludes charge-neutral excitations as the origin of the enhanced Hall Lorenz number and instead attributes $L_{xy}$7 to Kondo-type elastic scattering off localized 4$L_{xy}$8 electrons, which produces a peculiar energy dependence of the quasiparticle relaxation time (Zhang et al., 2024). Here the magnetic Wiedemann–Franz law is not simply broken by topology alone, but by the interplay of local moments, Weyl carriers, and elastic spin-dependent scattering.

4. Dirac, graphene, and Weyl semimetals in magnetic field

In weak-field graphene, the magnetic Wiedemann–Franz problem is naturally tensorial. Within a semiclassical Boltzmann framework including bipolar diffusion, energy-dependent scattering, and a possible substrate-induced gap $L_{xy}$9, both $L_0$0 and $L_0$1 recover $L_0$2 as $L_0$3, but at finite temperature $L_0$4 develops a sharp peak that is enhanced by $L_0$5 while remaining at nearly the same temperature. The transverse ratio $L_0$6 can be either positive or negative, depending on $L_0$7, $L_0$8, $L_0$9, and $T\to 0$00, because electrons and holes contribute with opposite signs to Hall-like responses (Tu et al., 2023). In that sense, a weak magnetic field does not destroy the low-temperature law, but it amplifies finite-temperature departures created by bipolar diffusion.

The hydrodynamic Dirac-fluid regime leads to a very different conclusion. For a two-dimensional Dirac electron liquid in graphene under isothermal conditions, the Wiedemann–Franz law does not hold, and the diagonal and Hall Lorenz ratios generally differ from one another. The breakdown is most pronounced near charge neutrality, where the Lorenz ratio can exceed its nominal Sommerfeld value by an order of magnitude, and the field dependence is nonmonotonic because of an emergent magnetic friction specific to non-Galilean liquids with finite intrinsic conductivity (Levchenko, 24 Mar 2025). The phrase magnetic Wiedemann–Franz law in this context therefore refers not to a preserved universal ratio but to a hydrodynamic breakdown of the relation between heat and charge magnetotransport.

Weyl and Dirac semimetals furnish both preserved and violated cases. In compressively strained HgTe, magneto-thermal conductance in the Weyl regime increases with field and matches the electrical conductance according to the Wiedemann–Franz law, with $T\to 0$01 in the Weyl and $T\to 0$02-type regimes at $T\to 0$03 (Aravindnath et al., 14 Oct 2025). By contrast, Cd$T\to 0$04As$T\to 0$05 shows a field-induced drastic violation: the Lorenz number drops from $T\to 0$06 at $T\to 0$07 and zero field to $T\to 0$08 at $T\to 0$09, and from $T\to 0$10 at $T\to 0$11 and zero field to $T\to 0$12 at $T\to 0$13. That suppression was interpreted as a magnetic-field-driven breakdown of Landau quasiparticles and as evidence for a field-induced quantum critical point (Pariari et al., 2015).

Ferromagnetic Weyl semimetals can also violate the law strongly in the longitudinal channel. In Co$T\to 0$14MnAl, after separating the phonon and electronic contributions to $T\to 0$15, the extracted Lorenz ratio is below $T\to 0$16 throughout $T\to 0$17–$T\to 0$18, with a maximum near $T\to 0$19 around $T\to 0$20–$T\to 0$21 and values tending toward zero near $T\to 0$22 and $T\to 0$23 (Robinson et al., 2023). The paper discusses the result in terms of non-parabolic Weyl-like bands, slow electron–hole recombination, and possible hydrodynamic transport.

5. Magnonic realizations in insulating magnets

In insulating ferromagnets, the magnetic Wiedemann–Franz law becomes a genuinely bosonic transport law. For a junction of two ferromagnetic insulators coupled weakly by interfacial exchange, magnons are the only carriers of both magnetic moment and heat. In linear response, the magnon current and heat current obey an Onsager matrix with $T\to 0$24, the spin conductance is $T\to 0$25, and the thermal conductance at zero magnon current is

$T\to 0$26

In the regime $T\to 0$27, the ratio becomes universal: $T\to 0$28 while the magnonic Seebeck and Peltier coefficients approach

$T\to 0$29

These quantities are independent of microscopic parameters such as $T\to 0$30, $T\to 0$31, $T\to 0$32, and $T\to 0$33 in the low-temperature limit (Nakata et al., 2015).

A Hall version exists for two-dimensional clean insulating magnets with magnon Landau levels generated through the Aharonov–Casher effect. In the magnonic quantum Hall regime, an appropriately defined thermal Hall conductance $T\to 0$34 and magnon Hall conductance $T\to 0$35 satisfy

$T\to 0$36

for an almost flat lowest band carrying nonzero Chern number (Nakata et al., 2016). The form mirrors the fermionic Hall Wiedemann–Franz law, but the carriers are bosonic spin-wave excitations rather than electrons.

The same framework extends to fluctuations. For magnonic spin and heat currents in a ferromagnetic insulating junction, the zero-frequency noises satisfy universal relations tied to the magnonic Wiedemann–Franz law, and the magnonic spin-Fano factor reduces to the universal value $T\to 0$37 in the low-temperature limit, remaining valid even beyond linear response in the shot-noise regime (Nakata et al., 2018).

The universality is, however, a linear-response statement. In topologically trivial insulating magnets driven into the strong nonlinear regime, the ratio of the nonlinear thermal transport coefficient to the spin transport coefficient no longer scales linearly with $T\to 0$38. Instead,

$T\to 0$39

with a nonuniversal coefficient $T\to 0$40, and the magnonic Wiedemann–Franz law breaks down (Nakata et al., 2022).

6. Theoretical constraints, extensions, and scope

Across these realizations, the magnetic Wiedemann–Franz law functions less as a single theorem than as a diagnostic principle. When it holds, the implication is that the low-temperature carriers of heat and charge remain the same objects—charged quasiparticles in metals, or magnons in insulating magnets—and that elastic processes dominate the asymptotic transport. When it fails, the form of the failure differentiates among several mechanisms: short-range magnetic correlations truncating non-Fermi-liquid regimes, direction-dependent hot-spot physics, Berry-curvature energy structure, bipolar diffusion, hydrodynamic flow, Kondo-type spin scattering, or genuinely bosonic nonlinear response (Reid et al., 2013, Xu et al., 2018, Nakata et al., 2022).

Recent exact Boltzmann treatments of semimetals with impurity and electron–electron scattering in a magnetic field sharpen this point. In both two-carrier semimetals and the Baber-scattering limit, the longitudinal and transverse laws $T\to 0$41 hold at zero temperature, but electron–electron scattering produces finite-temperature deviations in field. To organize the magnetotransport, that work introduced resistivity-based Lorenz ratios $T\to 0$42 and $T\to 0$43, and in weak field found, for Baber scattering,

$T\to 0$44

showing that the Hall sector can amplify longitudinal deviations (Takahashi et al., 8 Oct 2025).

A further extension is nonlinear and intrinsic. In PT-symmetric antiferromagnets, ordinary Berry curvature vanishes, but Berry connection polarizability and thermal Berry connection polarizability remain allowed and generate second-order intrinsic Hall and thermal Hall currents. In that setting,

$T\to 0$45

which defines a second-order intrinsic Wiedemann–Franz law characterized by the chemical potential in the low-temperature regime (Zhang et al., 8 Mar 2025). This construction shows that magnetic Wiedemann–Franz relations can survive even when the relevant transport coefficients are nonlinear, Hall-like, and entirely geometric.

Taken together, these works establish a broad taxonomy. In field-tuned heavy-fermion metals such as YbRh$T\to 0$46Si$T\to 0$47, the law can survive magnetic quantum criticality and constrain non-quasiparticle theories (Reid et al., 2013). In topological magnets, Hall-channel deviations encode Berry curvature or Kondo-type scattering (Qiang et al., 2023, Zhang et al., 2024). In graphene and some Weyl systems, weak-field tensorial laws may hold at low $T\to 0$48 while finite-temperature or hydrodynamic effects produce large departures (Tu et al., 2023, Levchenko, 24 Mar 2025). In insulating magnets, the law acquires a bosonic form with the magnetic moment $T\to 0$49 replacing the electric charge $T\to 0$50 (Nakata et al., 2015). The unifying idea is that magnetism does not merely perturb the Wiedemann–Franz law: it turns the relation between heat and charge transport into a sensitive probe of quasiparticle integrity, scattering kinematics, and band geometry.