Formulation of intrinsic nonlinear thermal conductivity for bosonic systems using quantum kinetic equation

Abstract

Nonlinear responses in transport phenomena have attracted significant attention because they can arise even when linear responses are forbidden by symmetry, with the quantum geometry of Bloch wave functions playing an essential role. While such effects have been extensively studied in electric transport, similar quantum-geometric mechanisms are also expected to govern nonlinear thermal transport. In particular, thermal responses are crucial in bosonic systems such as magnons and phonons, which are charge-neutral quasiparticles. However, a consistent theoretical description of nonlinear thermal transport remains challenging because of the difficulty in the treatment of energy magnetization in higher-order responses with Luttinger's gravitational potential method. Here, we formulate the intrinsic nonlinear thermal conductivity of bosonic systems using a quantum kinetic equation approach that avoids Luttinger's method and naturally incorporates contributions from energy magnetization. We identify three distinct contributions to the nonlinear thermal conductivity: two expressed in terms of quantum-geometric quantities, namely the quantum metric and the thermal Berry-connection polarizability (TBCP), and a third determined solely by the band dispersions. Applying our formalism to a specific quantum spin model within linear spin-wave theory, we show that the TBCP term dominates the nonlinear thermal Hall effect in the absence of threefold symmetry. Our results differ quantitatively from those obtained using semiclassical theory, thereby highlighting the importance of quantum corrections beyond the semiclassical picture. These findings establish a general framework for intrinsic nonlinear thermal responses in bosonic systems and reveal quantum-geometric mechanisms underlying thermal transport beyond linear response theory.

Figures (8)

• Figure 1: Schematic illustration of $\chi(x,y)$ introduced in Sec. \ref{['subsec:thermal_current']}. The $L\times L$ square enclosed by the black solid line denotes region $D$, where $\chi(x,y)=1$. Outside this region, $\chi(x,y)$ decays to zero over a length scale of $\mathcal{O}(L^0)$. Here, $\chi(x,y)$ is plotted over its support $D'$.
• Figure 2: Schematic illustration of a spin model on a honeycomb lattice, where $\bm a_1$ and $\bm a_2$ are primitive translation vectors, and $\bm \delta_1$, $\bm \delta_2$, and $\bm \delta_3$ are vectors connecting nearest-neighbor sites. The honeycomb lattice consists of two sublattices, A and B, shown by red and blue circles, respectively. The green arrows indicate the directions associated with the DM interaction; we define $\nu_{ij}=-\nu_{ji}=+1$ when the direction of the vector from site $i$ to site $j$ is the same as that of the corresponding green arrow.
• Figure 3: Magnon band structure of the isotropic ferromagnetic honeycomb lattice model with $J_1=J_2=J_3$ under a staggered magnetic field. (a) Momentum dependence of the magnon bands along high-symmetry lines in the first Brillouin zone. (b), (c) Contour maps of the magnon band dispersions for the (b) low-energy and (c) high-energy bands. In panels (b) and (c), the dashed hexagons indicate the first Brillouin zone. The positions of the high-symmetry points shown in panel (a) within the first Brillouin zone are indicated in panels (b) and (c).
• Figure 4: Momentum dependence of the (a) $xx$, (b) $xy$, and (c) $yy$ components of TBCP and the (d) $xx$, (e) $xy$, and (f) $yy$ components of the quantum metric for the low-energy band in the isotropic ferromagnetic honeycomb lattice model with $J_1=J_2=J_3$ under a staggered magnetic field. The dashed hexagons indicate the first Brillouin zone.
• Figure 5: (a) Temperature dependence of the nonlinear thermal Hall conductivity $\kappa_{y;xx}$ in the isotropic ferromagnetic honeycomb lattice model with $J_1=J_2=J_3$ under a staggered magnetic field. The solid line represents the conductivity obtained from our theory, and the dashed lines represent those obtained by Li and Zhu (2024) using the semiclassical approach Li2024 and by Varshney et al. (2023) using the quantum kinetic theory Varshney2023. (b) Decomposition of $\kappa_{y;xx}$ into three contributions: $\kappa^{\mathrm{TBCP}}$, $\kappa^{\mathrm{QM}}$, and $\kappa^{\mathrm{disp}}$, which are introduced in Eqs. \ref{['eq:kappa_ijk^TBCP-k']}, \ref{['eq:kappa_ijk^QM-k']}, and \ref{['eq:kappa_ijk^disp-k']}, respectively.