Superdiffusive transport protected by topology and symmetry in all dimensions

TL;DR

This work identifies a nodal-hybridization mechanism that yields superdiffusive transport (0<γ<1) in all dimensions by exploiting symmetry and topology to protect nodes in the c–f hybridization amplitude. It introduces a physically grounded two-component model ( itinerant c-electrons coupled to disordered or interacting f-electrons) in which a nodal manifold S_node in momentum space controls quasiparticle lifetimes, leading to universal conductance scaling G ∼ L^{−γ} with γ determined by nodal geometry and node order. The authors derive γ_high and γ_low using codimension arguments and validate them across graphene and multi-Weyl semimetal models, predicting experimental signatures such as linear-in-T resistivity, divergent low-frequency σ(ω), and momentum-resolved ARPES line widths. They further show how non-nodal perturbations induce a controlled crossover to diffusion and present a universal scaling collapse for disorder-driven crossover, offering a practical route to observe and engineer anomalous transport via nodal structures. Overall, the work links band topology and crystal symmetries to tunable anomalous transport in quantum materials, with clear experimental probes.

Abstract

Superdiffusion is an anomalous transport behavior. Recently, a new mechanism, termed the ``nodal mechanism," has been proposed to induce superdiffusion in quantum models. However, existing realizations of the nodal mechanism have so far been proposed on fine-tuned, artificial Hamiltonians, posing a significant challenge for experimental observation. In this work, we propose a broad class of models for generating superdiffusion potentially realizable in condensed matter systems across different spatial dimensions. A robust nodal structure emerges from the hybridization between the itinerant electrons and the local impurity orbitals, protected by the intrinsic symmetry and topology of the electronic band. We derive a universal scaling law for the conductance, $G \sim L^{-γ}$, revealing how the exponent is dictated by the dimensionality of the nodal structure ($D_{\text{node}}$) and its order $n$, and the dimensionality of the system $(D)$ at high temperatures or that of the Fermi surface ($D^F$) at low temperatures. Through numerical simulations, we validate these scaling relations at zero temperature for various models, including those based on graphene and multi-Weyl semimetals, finding excellent agreement between our theory and the computed exponents. Beyond the scaling of conductance, our framework predicts a suite of experimentally verifiable signatures, notably a new mechanism for linear-in-temperature resistivity ($ρ\sim T$) and a divergent low-frequency optical conductivity ($σ(ω) \sim ω^{γ-1}$), establishing a practical route to discovering and engineering anomalous transport in quantum materials.

Figures (7)

• Figure 1: The nodal hybridization mechanism for anomalous transport. (a)A schematic of the two-component model. It consists of itinerant $c$-electrons (which may possess internal degrees of freedom like sublattice or orbital, represented by dots within circles) forming a conduction band ($\hat{H}_c$) and the $f$-electrons (black circles) as the scattering source ($\hat{H}_f$). The crucial element is the hybridization term ($\hat{H}_{\text{hyb}}$) that couples the two systems. (b) The consequence of nodal structure in momentum space. The specific form of the hybridization creates "nodal manifold" (thick black line) where the coupling vanishes. Quasiparticles on or near this manifold (red shaded area) have anomalously long lifetimes and dominate the transport. A state exactly on the node is ballistic ($\tau \to \infty$), while a state far from the node, represented by the blurred circle, is strongly scattered (finite $\tau$).
• Figure 2: Motion of nodal quasiparticle under different transverse boundary condtions (a) Periodic boundary condition (PBC) along $y$: transverse momentum is conserved, and a quasiparticle in a protected nodal state $\ket{\psi_{\mathbf{k}_0}}$ propagates ballistically, wrapping around the cylinder without changing its transverse momentum, allowing superdiffusive behavior to manifest. (b) Open boundary condition (OBC) along $y$: reflection at the boundaries does not conse rve transverse momentum, causing boundary-induced channel mixing, scattering a protected nodal quasiparticle $\ket{\psi_{\mathbf{k}_0}}$ to an unprotected state $\ket{\psi_{\mathbf{k}'}}$ with finite mean free path. Superdiffusive transport is therefore more robust in systems with $L_y \gg L_x$ where bulk properties dominate.
• Figure 3: Symmetry protected nodal structures and their transport signatures in 2D and 3D models. This figure provides an overview of the symmetry selection rule, from the real-space geometry to the momentum-space nodal structure and its direct consequences for zero-temperature transport. (a) Two-Dimensional Square Lattice (a1) The real-space geometry: an $f$-electron (red) at the plaquette center, hybridizing with its four neighboring $c$-electron (black). (a2) The resulting momentum-space structure. The hybridization vanishes on a nodal manifold (red lines) at the Brillouin zone boundaries. The panel shows two representative Fermi surfaces. The $M$-pocket Fermi surface (blue solid line) with energy $E_F = 1$ centered at the $(\pi, \pi)$ point and intersects with nodal manifold and the $\Gamma$-pocket Fermi surface (green dash line) with energy $E_F = -1$ centered at $(0,0)$ does not. (a3) Numerical simulation of the transport. The conductance $G_{xx}$ is calculated for the two Fermi energies shown in (a2). For energy $E_F = 1$ corresponding to the $M$-pocket Fermi surface intersecting with nodes (blue triangles), the transport is superdiffusive with $G \sim L_x^{-1/2} (\gamma = 1/2)$. For energy $E_F = -1$ corresponding to the non-intersecting $\Gamma$-pocket (green triangles), the conductance shows exponential decay and indicates localization. The case with hybridization orbital $\ket{d}$ breaking the mirror symmetry with Fermi energy $E_F = 1$ (gray circles) is also localization. The inset highlights the exponential decay on a semi-log plot. (b) Three-Dimensional Simple Cubic Lattice (b1) The real-space model: an $f$-electron at the body center hybridizing with its eight neighboring $c$-electrons. (b2) The corresponding nodal structure, consisting of nodal planes (semi-transparent red) at Brillouin zone boundaries. Two possible Fermi surfaces are shown: the $M$-pocket Fermi surface centered at $(\pi,\pi,\pi)$ with $E_F = 3$ (blue) and the $\Gamma$-pocket Fermi surface centered at the origin with $E_F = -3$ (green). (b3) Numerical simulation on the 3D model. The conductance $G_{zz}$ is computed at the Fermi energies corresponding to the surfaces in (b2). Superdiffusion with $\gamma = 1/2$ (blue triangles) is only observed with the Fermi surface and the nodal planes intersected. The cases where the intersection does not appear (green triangles) or when the $f$-orbitals are not mirror symmetric show exponential decay. The numerical result perfectly match the low-temperature prediction $\gamma_{\text{low}} = (D^F-D^F_{\text{node}})/(2n) = 1/2$ for the 2D and the 3D system. The transverse size of the system is fixed when doing the numerical simulation and $L_y =202$ for the 2D square lattice and $L_x = L_y = 21$ for the simple cubic lattice system. Results are averaged over at least 700 disorder configurations for the 2D model and at least 300 disorder configurations for the 3D model. The $f$-electron energies $\epsilon_{\mathbf{R}}$ are drawn form a uniform distribution in the range $[-W,W]$, where $W_{2D} = 0.8$ and $W_{3D} = 2.0$ (for the $\Gamma$-pocket Fermi surface, $W$ is taken as $0.5$ to enhance the readibility of the data). The hybridization strength is set to be $V_{2D} =1$ and $V_{3D} = \sqrt{2}$.
• Figure 4: Topologically protected nodal arcs and superdiffusive transport in graphene. This figure provides an overview of the topological selection rule in graphene, from real-space geometries to momentum space nodal structures and their validation via zero-temperature transport simulations (a) Real-space illustration of two distinct, $PT-$symmetric $f$-electron placements on the honeycomb lattice: at the plaquette center and the bond center (indicated by blue circles). And the dashed loops enclose the $c$-electron orbitals that hybridize within them. (b) The quasi-one-dimensional stripe geometry used for two-terminal conductance calculation in Kwant. The centeral scattering region (black) has length $L_x$ and fixed width $L_y = 201$ (in the unit of lattice constant). The gray shaded area stands for the two semi-infinite clean leads connected to the scattering region. (c1) The nodal structrue when $f$-electrons are at the bond center. The color map shows the squared hybridiztion amplitude, which vanishes along nodal arcs (red lines) connecting Dirac points ($\mathbf{K},\mathbf{K}'$). The dashed magenta circles represented the Fermi surface corresponding to $E_F = 0.5$, which intersects the nodal arcs at discrete points. (c2) Conductance $G_{xx}$ versus length $L_x$ for the bond-center model. The $PT-$symmetric case (blue triangles) with Fermi energy $E_F = 0.5$ shows a power-law decay $G_{xx} \sim L_x^{-1/2}$, confirming the superdiffusive transport exponent $\gamma= 1/2$. (d1) Nodal structure when $f$-electrons are at the plaquette center. While the real-space geometry is different, topology again dictates the formation of nodal arcs connecting the Dirac points. (d2) The conductance scaling for the plaquette-center model is identical, with $\gamma = 1/2$. This confirms that the superdiffusive exponent is related to the topology, independent of specific hybridization geometry. In both (c2) and (d2), if $f$-electron breaks the $PT-$symmetry, the nodal structure will be destroyed and lead to conventional localization behavior (gray circles). The numerical results match the low-temperature prediction $\gamma_{\text{low}} = (D^F -D^F_{\text{node}})/(2n) =1/2$, where the 1D Fermi surface intersects the 1D nodal manifold at discrete points. Results are averaged over at 400 disorder configurations. The $f$-electron energies are drawn uniformly in $[-W,W]$ with disorder strength $W =1.65$ for the bond-center model and $W = 1.0$ for the plaquette-center model. The hybridization strength is set as $V=\sqrt{2}$ for both models.
• Figure 5: Higher-order nodal arcs and symmetry-switched transport regimes in multi-Weyl semimetals A demonstration of the interplay between topology and symmetry and the direct numerical verification of the resulting transport exponents. All simulations are performed at zero temperature. (a) Double-Weyl Semimetal($|C|=2$). (a1) Conceptual schematic of a stable, higher-order ($n=2$) nodal arc (blue line) pinned at the rotation axis connecting double-Weyl points of charge $\pm 2$, protected by the $C_4$ rotation symmetry. The dashed circle stands for the Fermi surface. (a2) When the hybridization orbital breaks the rotation symmetry, the higher-order arc splits into two first order ($n=1$) arcs (orange lines). (a3) The resulting transport regimes. The second-order node leads to the superdiffusion with $G_{zz} \sim L_z^{-1/2}$ (blue triangles), matching the predicted exponent at low temperature $\gamma = (D^F - D^F_{\text{node}})/(2n) = 1/n = 1/2$. The split first-order nodes lead diffusion with $G_{zz} \sim L_z^{-1}$ (red diamonds). Nodes are absent for a scalar potential without any internal structures, for example, $\hat{V}_{\mathbf{R}} \propto \hat{I}$ and $\hat{I}$ is the identity operator. This will lead to localization behavior (gray circles). (b) Triple-Weyl Semimetal($|C|=3$). (b1-b2) The same principle of symmetry-protected fusion and splitting applies to a triple-Weyl point. (b3) The resulting transport regimes. The third-order node leads to the superdiffusion with $G_{zz} \sim L_z^{-1/3}$ (blue triangles), matching the predicted exponent at low temperature $\gamma = 1/n = 1/3$. For split first order nodes and absence of nodes, we again have diffusion with $G_{zz} \sim L_z^{-1}$ (red diamonds) and localization (gray circles). We adapt a two-band effective lattice model to simulate the conductance and we leave the detailed setup in the Appendix \ref{['app:numerics']}. The Fermi energy is set to be $E_F = 2$ corresponding to the Fermi surface (dashed circle) in (a1-a2) and (b1-b2). The system size along transverse directions are fixed: $L_x = L_y = 20$. The $f$-electron energies are drawn from a unifrom distribution $[-W,W]$, where $W=3.5$ for the double-Weyl case and $W=2.5$ for the triple Weyl case. Results are averaged over at least $100$ disorder configurations.