Angular-Momentum-Resolved Heat Flux Density

• Angular-momentum-resolved heat flux density is the decomposition of thermal energy flow into channels defined by the angular momentum carried by excitations or modes.
• The framework is applied in atomistic phonon transport, spin caloritronics, and radiative systems, enabling analysis via local flux densities and transverse Hall-like responses.
• Quantitative models across scales—from crystalline lattices to turbulent flows—offer insights into nonequilibrium energy conversion and angular momentum accumulation.

Searching arXiv for papers directly relevant to angular-momentum-resolved heat flux density and adjacent formulations. Angular-momentum-resolved heat flux density denotes a description of energy transport in which the heat current is decomposed according to the angular momentum carried by the transporting excitations or field modes. In current usage the term is not a single universally standardized object: in atomistic phonon transport it is realized through local phonon angular momentum densities and bond- or site-resolved angular momentum currents; in spin-orbit caloritronics it appears as a decomposition of interfacial spin Peltier heat flow into spin- and orbital-current-mediated parts; and in fluctuational electrodynamics it can appear explicitly as a modal quantity $\Phi_n(\omega)$ whose photons carry energy $\hbar\omega$ and angular momentum $\hbar n$ (Lopez et al., 2 Apr 2026, Park et al., 11 Jun 2026, Shah et al., 23 Jun 2026). This suggests that the expression functions less as a unique operator than as a general transport principle: heat flow is “resolved” by an angular momentum label, whether orbital, spin, phononic, hydrodynamic, or radiative.

1. Conceptual scope and formal meaning

In the narrowest microscopic sense, angular-momentum-resolved heat flux density is a local flux density of a specified angular momentum component, defined alongside the ordinary heat or energy flux density. In the broadest sense, it is any formulation in which the total heat current is decomposed into contributions associated with distinct angular momentum channels. The relevant angular momentum may be the orbital angular momentum of nuclei, spin and orbital currents in spin caloritronics, the azimuthal quantum number of a radiative mode, or the angular momentum transported by coherent structures in a flow.

Two structures recur across otherwise dissimilar theories. First, there is an ordinary energy or heat flux density, such as a kinetic-energy current, a Poynting vector, or a convective heat flux. Second, there is a companion angular momentum current or angular momentum flux tensor carrying a label that specifies which component or channel is being tracked. The ratio or joint constitutive response of these two quantities then measures how efficiently a thermal drive converts energy flow into angular momentum flow.

A frequent source of confusion is that “resolved” does not always mean a decomposition into orthogonal eigenchannels in the strict spectral sense. In some works it means site resolution or bond resolution in real space; in others it means separation into spin and orbital driving channels; in still others it means decomposition by the integer angular-momentum index $n$ of a cylindrical electromagnetic mode. The technical implementation is therefore field-specific, but the organizing idea is stable.

2. Atomistic phonon formulation in crystalline solids

The most explicit atomistic realization is the phonon angular momentum Hall framework. A finite harmonic crystal with equilibrium positions $\mathbf r_s$, masses $m_s$, displacements $\mathbf u_s(t)$, velocities $\dot{\mathbf u}_s(t)$, mass matrix $\mathbf M$, and force-constant matrix $\mathbf K$ is driven out of equilibrium by Langevin baths with nonuniform temperatures. In that setting the local phonon angular momentum density at site $\hbar\omega$0 is defined as the orbital angular momentum of the nuclei about their equilibrium positions,

$\hbar\omega$1

with steady-state component

$\hbar\omega$2

Here $\hbar\omega$3 is the antisymmetric generator of rotations about axis $\hbar\omega$4. In this formulation, nonzero $\hbar\omega$5 requires a nonuniform temperature profile, because the relevant displacement–velocity covariance vanishes for a uniform equilibrium temperature (Lopez et al., 2 Apr 2026).

The corresponding bond-resolved phonon angular momentum current from site $\hbar\omega$6 to site $\hbar\omega$7 is

$\hbar\omega$8

where $\hbar\omega$9 is the bond force-constant tensor. The site-resolved current vector is then

$\hbar n$0

This $\hbar n$1 is exactly an angular-momentum-resolved flux density at the scale of a lattice site. In parallel, the bond-resolved kinetic-energy current is

$\hbar n$2

with site current vector

$\hbar n$3

The pairing $\hbar n$4 and $\hbar n$5 is the central atomistic embodiment of angular-momentum-resolved heat transport.

The local balance equation makes the interpretation precise: $\hbar n$6 Steady-state divergence of the angular momentum current is therefore balanced by damping. At boundaries where transverse current cannot continue, this produces edge accumulation of phonon angular momentum.

3. Conductivity tensors, Hall response, and edge accumulation

The atomistic currents become transport coefficients once the bath temperatures $\hbar n$7 are decomposed into a uniform part plus a gradient-induced perturbation. The phonon angular momentum current at site $\hbar n$8 can be written as

$\hbar n$9

and the kinetic-energy current similarly as

$n$0

For a macroscopic uniform temperature gradient,

$n$1

these kernels yield local conductivity tensors

$n$2

Bulk averages over interior sites define macroscopic tensors $n$3 and $n$4, which are the natural analogues of Hall and longitudinal conductivities for phonon angular momentum and energy transport (Lopez et al., 2 Apr 2026).

In a two-dimensional geometry with $n$5, the transverse phonon-angular-momentum response is encoded in $n$6. The phonon angular momentum deflection angle is defined as

$n$7

and a mixed Hall-like angle comparing transverse angular momentum flow to longitudinal heat flow is

$n$8

This is the closest thing in that framework to a global angular-momentum-resolved heat conductivity.

The physical picture is a nonequilibrium Hall-like conversion: a longitudinal temperature gradient generates a longitudinal energy current and, through thermally induced mixing of polarized vibrational motion, a transverse current of phonon angular momentum. In square- and honeycomb-lattice models, and in example materials including graphene, Si, MgO, and BaTiO$n$9, the bulk angular momentum current is predominantly transverse and the finite-sample response is an edge accumulation of $\mathbf r_s$0 with opposite signs on opposite edges. For the finite samples considered, the accumulated phonon angular momentum is of order $\mathbf r_s$1 per atom (Lopez et al., 2 Apr 2026).

A central misconception is that such a phonon angular momentum Hall effect must rely on broken inversion symmetry, chirality, or Berry curvature in direct analogy with electronic Hall transport. The atomistic theory states the opposite: it does not require broken inversion or chirality, exists already at zero magnetic field and in centrosymmetric lattices, and arises from nonequilibrium correlations under $\mathbf r_s$2, not from Berry curvature. What is required is a nonequilibrium temperature profile and polarization mixing in the force-constant network.

4. Spin- and orbital-channel decompositions in spin caloritronics

In spin Peltier systems the same general idea appears as a channel decomposition of interfacial heat flux density. In yttrium iron garnet/Pt/CuO$\mathbf r_s$3 heterostructures, the measured spin Peltier temperature modulation can be written conceptually as

$\mathbf r_s$4

where the spin part is driven by a spin current generated in Pt via the spin Hall effect and the orbital part is driven by an orbital current generated at the Cu/CuO$\mathbf r_s$5 interface via the orbital Rashba–Edelstein effect and then converted into spin current in Pt through spin–orbit coupling. The experiment emphasizes that the orbital channel does not directly heat or cool the magnetic interface; it provides an additional route to generate spin current, which then produces the spin Peltier heat flux (Park et al., 11 Jun 2026).

The measured observable is the field-odd lock-in thermography amplitude $\mathbf r_s$6, extracted from the complex temperature oscillation under square-wave AC current. Because the spin Peltier effect is odd in magnetization, $\mathbf r_s$7 is proportional to the SPE-driven interfacial heat flux density. In the wedge geometry, where the effective CuO$\mathbf r_s$8 thickness varies continuously from $\mathbf r_s$9 to $m_s$0 nm, the spatial dependence of $m_s$1 maps the thickness-dependent angular-momentum-resolved heat flux density.

Quantitative separation of spin and orbital contributions is achieved by modeling current partition between Pt and CuO$m_s$2, subtracting the Pt-only contribution scaled by the Pt current fraction, and defining a normalized CuO$m_s$3 charge-to-heat conversion efficiency

$m_s$4

This quantity is proportional to an effective orbital Peltier-like coefficient that includes orbital-to-spin conversion in Pt. The extracted $m_s$5 exhibits a pronounced maximum at $m_s$6 nm, with

$m_s$7

compared with about $m_s$8 for the conventional spin-Hall SPE in the same device, giving a ratio $m_s$9. For $\mathbf u_s(t)$0 nm the decay is approximately exponential with characteristic length

$\mathbf u_s(t)$1

Control experiments with Al-capped and Al-inserted structures identify the Cu/CuO$\mathbf u_s(t)$2 interface as the essential source of the orbital contribution (Park et al., 11 Jun 2026).

5. Radiative and fluctuational-electrodynamic formulations

In radiative transport, angular-momentum-resolved heat flux density can be defined spectrally or mode by mode. For a two-dimensional topological sheet such as the Haldane model, the far-field spectral heat flux density is

$\mathbf u_s(t)$3

while the spectral angular momentum flux density is

$\mathbf u_s(t)$4

Energy radiation is therefore governed by $\mathbf u_s(t)$5, whereas angular momentum radiation is governed by $\mathbf u_s(t)$6, tying net angular momentum emission directly to the Hall or magneto-optical response and hence to broken time-reversal symmetry (Zhang et al., 2020).

A more explicit modal construction appears for concentric nonreciprocal cylinders, where heat transfer and torque are written in terms of an angular-momentum-resolved heat flux density $\mathbf u_s(t)$7. Each channel $\mathbf u_s(t)$8 carries photons with energy $\mathbf u_s(t)$9 and angular momentum $\dot{\mathbf u}_s(t)$0, and the energy and torque transferred from one cylinder to the other are

$\dot{\mathbf u}_s(t)$1

In reciprocal media, contributions from $\dot{\mathbf u}_s(t)$2 and $\dot{\mathbf u}_s(t)$3 cancel, so there is no net torque. Nonreciprocity breaks that symmetry and allows thermal fluctuations to generate mechanical rotation; the same $\dot{\mathbf u}_s(t)$4 also governs fluctuation-induced friction and the steady-state rotation of the inner cylinder in the proposed contactless heat engine (Shah et al., 23 Jun 2026).

A planar thin-film analogue expresses the radiated energy and angular momentum per mode $\dot{\mathbf u}_s(t)$5 through generalized Fresnel coefficients. The corresponding phase-space densities are

$\dot{\mathbf u}_s(t)$6

so that $\dot{\mathbf u}_s(t)$7 acts as an energy emissivity and $\dot{\mathbf u}_s(t)$8 as an angular-momentum emissivity. This suggests a natural mode-resolved definition of angular-momentum-resolved radiative heat flux density as the pair $\dot{\mathbf u}_s(t)$9 or, equivalently, the energy flux density together with the angular momentum carried per emitted photon (Zhang et al., 1 Jan 2026).

6. Continuum, hydrodynamic, and nonequilibrium generalizations

The same logic extends beyond phonons and radiation. In a hydrodynamic electron fluid in a Corbino disk under perpendicular magnetic field, the heat flux is decomposed into radial and tangential components,

$\mathbf M$0

Because the geometry is rotationally symmetric, a nonzero $\mathbf M$1 implies orbital angular momentum of the heat flow about the disk center, with local density naturally proportional to $\mathbf M$2. The heat flux deflection is suppressed by momentum-relaxing scattering and promoted by momentum-conserving scattering; moreover, when an electric potential gradient or a temperature gradient in the same direction is applied separately, the direction of heat flux is reversed in the electron hydrodynamic regime (Zhang et al., 16 Apr 2026).

In turbulent Taylor–Couette flow, the local radial heat flux density

$\mathbf M$3

and the local radial angular-velocity flux density

$\mathbf M$4

show very similar spatial patterns. The strongest heat-transporting regions and the strongest angular-momentum-transporting regions overlap in mutual plume structures, and the global transport scalings in the turbulent Taylor-vortex regime are close: $\mathbf M$5

$\mathbf M$6

$\mathbf M$7

This is not a spectral resolution by angular momentum quantum number, but it is an angular-momentum-resolved view of heat transport in the sense that extreme heat flux is localized within structures that also transport angular momentum intensely (Leng et al., 2021).

In stellar interiors, the Transformed Eulerian Mean formalism makes the coupling even more explicit. The wave heat flux $\mathbf M$8 enters the mean entropy equation and, through the residual circulation, contributes to the effective wave angular momentum flux. In shellular rotation the radial wave flux contains a term proportional to $\mathbf M$9, so latitudinal wave heat transport directly enters radial angular momentum redistribution. For mixed modes in an evolved $\mathbf K$0 model, the influence of the wave heat flux on the mean angular momentum is not negligible and even dominant when considering the sum of prograde and retrograde modes (Belkacem et al., 2015).

Quantum-coherent transport theories reinforce the same point. In chiral phonon systems, the temperature-gradient-induced mean phonon angular momentum contains both diagonal and off-diagonal contributions in the phonon density matrix, with the off-diagonal term becoming dominant when phonon scattering is strong enough (Zhong et al., 2022). In nonequilibrium photon transport, a unified Meir–Wingreen formalism expresses emitted energy, force, and torque through the same photon Green’s functions and the operators $\mathbf K$1, $\mathbf K$2, and $\mathbf K$3, providing a direct route to angular-momentum-resolved heat flux once the field is expanded in angular-momentum eigenmodes (Zhang et al., 2021).

Across these settings, angular-momentum-resolved heat flux density serves as a unifying descriptor for thermal transport processes in which energy flow cannot be characterized solely by magnitude and direction. It additionally carries a rotational label—local, modal, channel-specific, or structural—that determines how thermal driving redistributes angular momentum, generates edge accumulation or torque, and couples heat transport to spin, orbital, radiative, hydrodynamic, or astrophysical dynamics.