Intrinsic Nonlinear Thermal Hall Effect

• Intrinsic Nonlinear Thermal Hall Effect is defined as a second-order transverse heat current response driven solely by band geometry and interband coherence.
• It relies on quantum metrics and Berry-connection polarizability to generate dissipationless signals in magnonic, electronic, and altermagnetic systems.
• Experimental approaches use temperature gradients and inversion-breaking techniques in systems like CrCl₃ and Mn₅Si₃ to isolate intrinsic nonlinear thermal responses.

Searching arXiv for the cited papers and closely related work on intrinsic nonlinear thermal Hall transport. Intrinsic nonlinear thermal Hall effect denotes a second-order transverse heat or energy-current response to a temperature gradient, $j^Q=j_Q^{(1)}+j_Q^{(2)}+\cdots$ with $j_Q^{(2)}\sim O[(\nabla T)^2]$, whose coefficient is set by band geometry rather than by impurity-controlled relaxation (Varshney et al., 2023). In the bosonic quantum-kinetic formulation, it appears as a $\tau^0$ contribution generated by interband coherence and the quantum metric; related electronic and planar variants are expressed through the thermal Berry-connection polarizability (TBCP) and its magnetic-field response (Varshney et al., 2023, Zhang et al., 8 Mar 2025, Barman, 3 Nov 2025). The subject now encompasses magnonic, electronic, and symmetry-selected altermagnetic realizations, and is distinguished from nonlinear anomalous, Drude, side-jump, and skew-scattering mechanisms by its intrinsic, band-geometric character (Varshney et al., 2023, Kim, 4 Mar 2026, Zhou et al., 2021).

1. Definition and constitutive structure

The nonlinear thermal Hall effect is the appearance of a transverse heat current at second order in a temperature gradient. In the quantum kinetic theory formulation for bosons, the second-order response is written as

$$J_a=-\kappa_{abc}(\tau)(\nabla_b T)(\nabla_c T)\equiv \kappa_{abc}(\tau)E_T^bE_T^c, \qquad E_T\equiv -\nabla T/T,$$

where $\kappa_{abc}(\tau)$ is a rank-3 nonlinear thermal conductivity tensor and $\tau$ is the relaxation time (Varshney et al., 2023). In two-dimensional settings, the Hall component is the transverse part of this tensor, such as $\kappa_{xyy}$ or $\kappa_{yxx}$, depending on geometry and symmetry (Li et al., 2024, Kim, 4 Mar 2026).

A broader planar variant has also been formulated for coplanar $\nabla T$ and $\mathbf B$, where the intrinsic nonlinear planar thermal Hall effect scales as $j_Q^{(2)}\sim O[(\nabla T)^2]$0. In that case the response is written as

$j_Q^{(2)}\sim O[(\nabla T)^2]$1

so the intrinsic coefficient is a rank-4 tensor rather than the rank-3 tensor of the zero-field nonlinear thermal Hall effect (Barman, 3 Nov 2025).

The literature uses “intrinsic” in a precise but not fully uniform way. In the magnonic quantum-kinetic and TBCP-based electronic formulations, intrinsic denotes a dissipationless or scattering-time-independent response controlled by band geometry (Varshney et al., 2023, Zhang et al., 8 Mar 2025). A semiclassical Boltzmann analysis of a tilted Dirac model also labels the anomalous-velocity contribution intrinsic, but emphasizes instead its low-temperature $j_Q^{(2)}\sim O[(\nabla T)^2]$2 scaling relative to side-jump and skew-scattering terms (Zhou et al., 2021). This terminological difference is part of the modern discussion of the field rather than an inconsistency in the existence of the effect.

2. Microscopic formulations and band-geometric origin

The magnonic quantum-kinetic theory starts from the density-matrix equation

$j_Q^{(2)}\sim O[(\nabla T)^2]$3

with the thermal driving term

$j_Q^{(2)}\sim O[(\nabla T)^2]$4

where $j_Q^{(2)}\sim O[(\nabla T)^2]$5 is the Berry-connection matrix (Varshney et al., 2023). Perturbatively expanding $j_Q^{(2)}\sim O[(\nabla T)^2]$6, the second-order heat current follows from

$j_Q^{(2)}\sim O[(\nabla T)^2]$7

Within this framework, the band-resolved quantum geometric tensor is

$j_Q^{(2)}\sim O[(\nabla T)^2]$8

with $j_Q^{(2)}\sim O[(\nabla T)^2]$9 the quantum metric and $\tau^0$0 the Berry curvature (Varshney et al., 2023).

The essential decomposition is into three contributions distinguished by their $\tau^0$1 scaling.

Contribution	Scaling in $\tau^0$2	Origin
Intrinsic nonlinear thermal Hall current	$\tau^0$3	Interband coherence and quantum metric
Nonlinear anomalous thermal Hall current	$\tau^0$4	Berry curvature and magnetic-moment terms
Nonlinear Drude current	$\tau^0$5	Band dispersion and distribution functions

The compact $\tau^0$6 result in the bosonic theory depends solely on the quantum metric $\tau^0$7 and remains finite even when the Berry curvature vanishes (Varshney et al., 2023). This is the defining feature of the intrinsic nonlinear thermal Hall effect in that formulation: it is a band-geometric response controlled by interband coherence rather than by the Berry-curvature dipole or by semiclassical drift.

A complementary formulation uses the TBCP. In the magnonic thermal scalar-potential and thermal vector-potential approaches, the temperature gradient modifies the Berry connection as

$\tau^0$8

with

$\tau^0$9

The intrinsic second-order magnon thermal Hall conductivity is then

$$J_a=-\kappa_{abc}(\tau)(\nabla_b T)(\nabla_c T)\equiv \kappa_{abc}(\tau)E_T^bE_T^c, \qquad E_T\equiv -\nabla T/T,$$0

which makes the TBCP the central geometric quantity of the intrinsic effect (Li et al., 2024).

In electronic PT-symmetric systems, the low-temperature intrinsic second-order thermal Hall conductivity is likewise written as a Fermi-surface functional of the TBCP,

$$J_a=-\kappa_{abc}(\tau)(\nabla_b T)(\nabla_c T)\equiv \kappa_{abc}(\tau)E_T^bE_T^c, \qquad E_T\equiv -\nabla T/T,$$1

while the corresponding intrinsic electrical coefficient is determined by the Berry-connection polarizability. This leads to a second-order intrinsic Wiedemann–Franz law, with $$J_a=-\kappa_{abc}(\tau)(\nabla_b T)(\nabla_c T)\equiv \kappa_{abc}(\tau)E_T^bE_T^c, \qquad E_T\equiv -\nabla T/T,$$2 linearly proportional to $$J_a=-\kappa_{abc}(\tau)(\nabla_b T)(\nabla_c T)\equiv \kappa_{abc}(\tau)E_T^bE_T^c, \qquad E_T\equiv -\nabla T/T,$$3 through a chemical-potential-dependent prefactor (Zhang et al., 8 Mar 2025).

3. Symmetry constraints and selection rules

Symmetry restrictions are mechanism-dependent. In the bosonic quantum-kinetic theory, time-reversal symmetry implies $$J_a=-\kappa_{abc}(\tau)(\nabla_b T)(\nabla_c T)\equiv \kappa_{abc}(\tau)E_T^bE_T^c, \qquad E_T\equiv -\nabla T/T,$$4 while $$J_a=-\kappa_{abc}(\tau)(\nabla_b T)(\nabla_c T)\equiv \kappa_{abc}(\tau)E_T^bE_T^c, \qquad E_T\equiv -\nabla T/T,$$5 is invariant, so $$J_a=-\kappa_{abc}(\tau)(\nabla_b T)(\nabla_c T)\equiv \kappa_{abc}(\tau)E_T^bE_T^c, \qquad E_T\equiv -\nabla T/T,$$6 and $$J_a=-\kappa_{abc}(\tau)(\nabla_b T)(\nabla_c T)\equiv \kappa_{abc}(\tau)E_T^bE_T^c, \qquad E_T\equiv -\nabla T/T,$$7 vanish in time-reversal-preserving systems. Inversion symmetry must also be broken for a nonzero Brillouin-zone integral of the second-order response (Varshney et al., 2023). At the same time, systems that break $$J_a=-\kappa_{abc}(\tau)(\nabla_b T)(\nabla_c T)\equiv \kappa_{abc}(\tau)E_T^bE_T^c, \qquad E_T\equiv -\nabla T/T,$$8 and $$J_a=-\kappa_{abc}(\tau)(\nabla_b T)(\nabla_c T)\equiv \kappa_{abc}(\tau)E_T^bE_T^c, \qquad E_T\equiv -\nabla T/T,$$9 individually but preserve $\kappa_{abc}(\tau)$0 satisfy $\kappa_{abc}(\tau)$1, which eliminates the Berry-curvature-driven nonlinear anomalous term while allowing the quantum-metric intrinsic term and the Drude term to remain finite (Varshney et al., 2023).

In PT-symmetric electronic systems, the logic is different. PT symmetry forces the Berry curvature to vanish, so linear anomalous Hall and Berry-curvature-dipole mechanisms are absent, yet the BCP and TBCP remain symmetry-allowed. This is why intrinsic second-order Hall-like electrical and thermal responses can survive in PT-symmetric Dirac antiferromagnets and related models (Zhang et al., 8 Mar 2025).

The intrinsic nonlinear planar thermal Hall effect obeys sharper crystallographic constraints. Its tensor is antisymmetric in the first two indices, $\kappa_{abc}(\tau)$2, and a finite response requires broken inversion symmetry together with the absence of a horizontal mirror $\kappa_{abc}(\tau)$3. The permitted point groups include $\kappa_{abc}(\tau)$4, $\kappa_{abc}(\tau)$5, $\kappa_{abc}(\tau)$6, $\kappa_{abc}(\tau)$7, and $\kappa_{abc}(\tau)$8, while $\kappa_{abc}(\tau)$9 and $\tau$0 are excluded because of $\tau$1 (Barman, 3 Nov 2025).

For altermagnets, the selection rules have been reduced to three necessary conditions for a nonvanishing intrinsic $\tau$2: a nontrivial quantum metric, broken mirror symmetry $\tau$3, and broken twofold rotational symmetry $\tau$4. In the square-lattice two-band formulation,

$\tau$5

with $\tau$6 proportional to the quantum metric. If the Hamiltonian is $\tau$7-symmetric, the Brillouin-zone integrand is point-antisymmetric and the integral vanishes identically (Kim, 4 Mar 2026).

These selection rules are not interchangeable. A nonzero intrinsic nonlinear thermal Hall response in one platform does not imply that the same symmetry pattern suffices in another; the allowed tensor structure depends on whether the microscopic mechanism is the magnonic quantum metric, the TBCP, or the thermal-field-corrected Berry curvature.

4. Canonical models and representative results

The first explicit magnonic demonstration was obtained for a two-dimensional ferromagnetic honeycomb lattice with nearest-neighbor and next-nearest-neighbor exchange, next-nearest-neighbor Dzyaloshinskii–Moriya interaction (DMI), and a Zeeman field,

$\tau$8

Linear spin-wave theory gives a $\tau$9 bosonic Hamiltonian $\kappa_{xyy}$0, and near $\kappa_{xyy}$1 the low-energy Dirac theory reads

$\kappa_{xyy}$2

with $\kappa_{xyy}$3 (Varshney et al., 2023). In this model, the Berry curvature vanishes at $\kappa_{xyy}$4, while the quantum metric remains finite and peaks near the band edges as $\kappa_{xyy}$5. The low-energy scaling $\kappa_{xyy}$6 near band edges therefore enhances the intrinsic response as the magnon gap decreases, whereas $\kappa_{xyy}$7 suppresses Berry-curvature-driven terms in the same limit (Varshney et al., 2023).

A crucial construction is inversion breaking without DMI. Adding a tilt term $\kappa_{xyy}$8 breaks inversion and valley symmetry but does not affect Berry curvature or quantum metric because those depend only on $\kappa_{xyy}$9, not on $\kappa_{yxx}$0. This permits a finite intrinsic nonlinear thermal Hall effect even with $\kappa_{yxx}$1 and $\kappa_{yxx}$2 (Varshney et al., 2023).

For the parameter set $\kappa_{yxx}$3, $\kappa_{yxx}$4 in one figure set and $\kappa_{yxx}$5 in another, the predicted magnitudes at $\kappa_{yxx}$6, $\kappa_{yxx}$7, and $\kappa_{yxx}$8 are explicit (Varshney et al., 2023).

• For $\kappa_{yxx}$9: the linear Hall term is $\nabla T$0, the nonlinear intrinsic term is $\nabla T$1, the nonlinear anomalous term is $\nabla T$2, and the nonlinear Drude term is $\nabla T$3.
• For $\nabla T$4: the linear Hall term is $\nabla T$5, the nonlinear intrinsic term is $\nabla T$6, the nonlinear anomalous term is $\nabla T$7, and the nonlinear Drude term is $\nabla T$8.

A separate intrinsic magnon formulation based on TSP and TVP applies the theory to a monolayer ferromagnetic hexagonal lattice with DMI, sublattice-staggered Zeeman field, and uniaxial strain. There, $\nabla T$9 when the strain parameter is $\mathbf B$0, while finite strain gives $\mathbf B$1, with opposite signs for tension $\mathbf B$2 and compression $\mathbf B$3; the magnitude increases with DMI strength $\mathbf B$4 (Li et al., 2024).

On the electronic side, a four-band PT-symmetric Dirac model,

$\mathbf B$5

has $\mathbf B$6 by PT symmetry but nonzero BCP and TBCP if inversion is broken by the tilt $\mathbf B$7. In the isotropic case $\mathbf B$8, the intrinsic second-order coefficients are

$\mathbf B$9

$j_Q^{(2)}\sim O[(\nabla T)^2]$00

with $j_Q^{(2)}\sim O[(\nabla T)^2]$01, and

$j_Q^{(2)}\sim O[(\nabla T)^2]$02

which is the model-specific second-order intrinsic Wiedemann–Franz law (Zhang et al., 8 Mar 2025).

The intrinsic nonlinear planar thermal Hall effect uses a tilted Dirac model

$j_Q^{(2)}\sim O[(\nabla T)^2]$03

and yields explicit low-temperature coefficients such as

$j_Q^{(2)}\sim O[(\nabla T)^2]$04

$j_Q^{(2)}\sim O[(\nabla T)^2]$05

so the response vanishes in the untilted limit and is enhanced near band edges (Barman, 3 Nov 2025).

For altermagnets, tight-binding square-lattice models separate $j_Q^{(2)}\sim O[(\nabla T)^2]$06-wave and $j_Q^{(2)}\sim O[(\nabla T)^2]$07-wave cases. The $j_Q^{(2)}\sim O[(\nabla T)^2]$08-wave model,

$j_Q^{(2)}\sim O[(\nabla T)^2]$09

breaks $j_Q^{(2)}\sim O[(\nabla T)^2]$10 and supports $j_Q^{(2)}\sim O[(\nabla T)^2]$11, whereas the $j_Q^{(2)}\sim O[(\nabla T)^2]$12-wave model,

$j_Q^{(2)}\sim O[(\nabla T)^2]$13

preserves $j_Q^{(2)}\sim O[(\nabla T)^2]$14 and yields $j_Q^{(2)}\sim O[(\nabla T)^2]$15 to numerical accuracy $j_Q^{(2)}\sim O[(\nabla T)^2]$16 until an even-parity $j_Q^{(2)}\sim O[(\nabla T)^2]$17-breaking term is introduced (Kim, 4 Mar 2026).

5. Relation to extrinsic mechanisms and neighboring phenomena

The intrinsic nonlinear thermal Hall effect is often discussed alongside nonlinear anomalous, Drude, side-jump, and skew-scattering terms, but these are distinct mechanisms. In the magnonic quantum-kinetic theory, the $j_Q^{(2)}\sim O[(\nabla T)^2]$18 nonlinear anomalous thermal Hall current is the thermal analogue of the electronic Berry-curvature-dipole nonlinear Hall effect and vanishes when $j_Q^{(2)}\sim O[(\nabla T)^2]$19; the $j_Q^{(2)}\sim O[(\nabla T)^2]$20 Drude term is geometry-free and depends only on band dispersion and distribution functions (Varshney et al., 2023). In the tilted Dirac Boltzmann theory, side-jump and skew-scattering dominate away from the Dirac point and are two to three orders of magnitude larger than the intrinsic contribution at higher Fermi energy, while the intrinsic term alone survives exactly at the Dirac point because the extrinsic terms carry the factor $j_Q^{(2)}\sim O[(\nabla T)^2]$21 (Zhou et al., 2021).

These distinctions have direct diagnostic value. One strategy is to vary $j_Q^{(2)}\sim O[(\nabla T)^2]$22 indirectly through temperature, strain, or disorder and compare the transverse nonlinear signal with the longitudinal conductivity: in the magnonic proposal, the intrinsic piece scales as $j_Q^{(2)}\sim O[(\nabla T)^2]$23, the $j_Q^{(2)}\sim O[(\nabla T)^2]$24 anomalous term scales linearly, and the Drude term scales as $j_Q^{(2)}\sim O[(\nabla T)^2]$25 (Varshney et al., 2023). In the electronic Boltzmann analysis, the low-temperature fit

$j_Q^{(2)}\sim O[(\nabla T)^2]$26

isolates the intrinsic slope $j_Q^{(2)}\sim O[(\nabla T)^2]$27 from the extrinsic intercept $j_Q^{(2)}\sim O[(\nabla T)^2]$28 (Zhou et al., 2021).

The intrinsic nonlinear thermal Hall effect should also be distinguished from several adjacent phenomena. The work on phonon Hall viscosity in $j_Q^{(2)}\sim O[(\nabla T)^2]$29-RuCl$j_Q^{(2)}\sim O[(\nabla T)^2]$30 demonstrates an intrinsic phonon thermal Hall conductivity through Hall viscosity and the acoustic Faraday effect, but explicitly states that its theory and measurements are strictly in linear response and contain no explicit nonlinear thermal Hall derivation (Shragai et al., 7 Oct 2025). Likewise, the cubic optical-phonon study exhibits an intrinsic thermal Hall conductivity with a nonlinear $j_Q^{(2)}\sim O[(\nabla T)^2]$31 magnetic-field dependence, but that nonlinearity is in magnetic field rather than in $j_Q^{(2)}\sim O[(\nabla T)^2]$32 (Hu et al., 5 Jan 2025). A further related direction is intrinsic second-order thermal Hall noise in time-reversal-invariant conductors, where “thermal” refers to Johnson–Nyquist noise of the charge current rather than to the heat-current operator; the intrinsic part of that second-order noise is governed by the quantum fluctuation of the quantum geometric tensor (Wei et al., 2023).

A plausible implication is that the modern subject is best understood as a family of second-order thermal Hall responses rather than a single universal formula. What unifies the family is the central role of band geometry—quantum metric, TBCP, Berry-connection polarizability, or thermal-field-corrected Berry curvature—together with the requirement that symmetry not force the Brillouin-zone integral to cancel.

6. Experimental routes, candidate materials, and open questions

The basic detection geometry for the magnonic nonlinear thermal Hall effect is to apply $j_Q^{(2)}\sim O[(\nabla T)^2]$33 along $j_Q^{(2)}\sim O[(\nabla T)^2]$34 and measure a transverse $j_Q^{(2)}\sim O[(\nabla T)^2]$35, so that the nonlinear signal is isolated by its quadratic dependence on the temperature gradient (Varshney et al., 2023). In the planar case, one rotates the in-plane $j_Q^{(2)}\sim O[(\nabla T)^2]$36 and $j_Q^{(2)}\sim O[(\nabla T)^2]$37 to test angular forms such as

$j_Q^{(2)}\sim O[(\nabla T)^2]$38

for $j_Q^{(2)}\sim O[(\nabla T)^2]$39 with $j_Q^{(2)}\sim O[(\nabla T)^2]$40, or

$j_Q^{(2)}\sim O[(\nabla T)^2]$41

for $j_Q^{(2)}\sim O[(\nabla T)^2]$42 with $j_Q^{(2)}\sim O[(\nabla T)^2]$43. These sinusoidal modulations are proposed as symmetry fingerprints of the intrinsic nonlinear planar response (Barman, 3 Nov 2025).

Prominent magnonic material candidates are two-dimensional ferromagnetic honeycomb chromium trihalides. CrCl$j_Q^{(2)}\sim O[(\nabla T)^2]$44 is highlighted because its magnons are gapless and DMI is negligible, so $j_Q^{(2)}\sim O[(\nabla T)^2]$45 while the quantum metric remains finite near $j_Q^{(2)}\sim O[(\nabla T)^2]$46; inversion breaking by strain, substrate engineering, or an Aharonov–Casher phase is therefore expected to expose the intrinsic nonlinear effect. CrBr$j_Q^{(2)}\sim O[(\nabla T)^2]$47 and CrI$j_Q^{(2)}\sim O[(\nabla T)^2]$48 provide gapped counterparts (Varshney et al., 2023). For altermagnets, $j_Q^{(2)}\sim O[(\nabla T)^2]$49-wave systems such as orthorhombic Mn$j_Q^{(2)}\sim O[(\nabla T)^2]$50Si$j_Q^{(2)}\sim O[(\nabla T)^2]$51 are proposed as promising because parity-mixing hybridizations naturally break $j_Q^{(2)}\sim O[(\nabla T)^2]$52, whereas ideal $j_Q^{(2)}\sim O[(\nabla T)^2]$53-wave systems such as CrSb and NiS should exhibit vanishing intrinsic $j_Q^{(2)}\sim O[(\nabla T)^2]$54 unless additional symmetry breaking is present (Kim, 4 Mar 2026).

Experimental magnitudes remain small but not parametrically negligible. For monolayers at $j_Q^{(2)}\sim O[(\nabla T)^2]$55, $j_Q^{(2)}\sim O[(\nabla T)^2]$56, and $j_Q^{(2)}\sim O[(\nabla T)^2]$57, intrinsic magnonic nonlinear thermal Hall currents of order $j_Q^{(2)}\sim O[(\nabla T)^2]$58–$j_Q^{(2)}\sim O[(\nabla T)^2]$59 are predicted, with Drude terms comparable but symmetry-controllable (Varshney et al., 2023). In the planar electronic setting, the conductivity is evaluated in units of $j_Q^{(2)}\sim O[(\nabla T)^2]$60 for parameters $j_Q^{(2)}\sim O[(\nabla T)^2]$61, $j_Q^{(2)}\sim O[(\nabla T)^2]$62, $j_Q^{(2)}\sim O[(\nabla T)^2]$63, and $j_Q^{(2)}\sim O[(\nabla T)^2]$64, where clear angular modulations appear (Barman, 3 Nov 2025).

The main practical difficulties are also clearly identified: establishing stable nanoscale temperature gradients without spurious phonon currents, maintaining magnon lifetimes in the picosecond range, breaking inversion without introducing unwanted Berry-curvature channels when the goal is to isolate the quantum-metric response, and remaining in the weak-disorder regime where the quasiparticle picture and crystal momentum are meaningful (Varshney et al., 2023). Open theoretical questions include field-induced magnetization corrections to the second-order current, the role of magnon–magnon interactions at elevated temperature, and extensions beyond the weak-disorder limit; in the phonon Hall-viscosity context, nonlinear generalizations are explicitly proposed but not developed (Varshney et al., 2023, Shragai et al., 7 Oct 2025).

Taken together, the present literature establishes the intrinsic nonlinear thermal Hall effect as a band-geometric second-order transport phenomenon with several concrete realizations. In magnons it can persist even when Berry curvature vanishes; in PT-symmetric electronic systems it survives where Berry-curvature-dipole physics is forbidden; in planar geometries it acquires characteristic $j_Q^{(2)}\sim O[(\nabla T)^2]$65 scaling and angular fingerprints; and in altermagnets it is sharply filtered by $j_Q^{(2)}\sim O[(\nabla T)^2]$66 and $j_Q^{(2)}\sim O[(\nabla T)^2]$67 selection rules (Varshney et al., 2023, Zhang et al., 8 Mar 2025, Barman, 3 Nov 2025, Kim, 4 Mar 2026).