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Approximate half-integer quantization in anomalous planar transport in $d$-wave altermagnets

Srimayi Korrapati, Snehasish Nandy, Sumanta Tewari

TL;DR

This work shows that 2D $d_{x^2-y^2}$-wave altermagnets with substrate-induced Rashba SOC exhibit nearly half-quantized anomalous planar Hall and planar thermal Hall effects when an in-plane Zeeman field breaks the symmetry $\hat{\mathcal{C}}_{4z}\hat{\mathcal{T}}$. A shifted Dirac point under the Zeeman field opens a gap and creates a Berry-curvature monopole with BCM $\nu=\pm 1/2$, yielding plateaus in $\sigma_{xy}$ and $\kappa_{xy}$ when the chemical potential lies inside the gap; the effect shows a $\cos2\phi$ angular dependence and undergoes topological transitions as field orientation changes. The planar Nernst conductivity $\alpha_{xy}$ peaks outside the gap and follows the Mott relation at low temperature, while the Wiedemann–Franz law breaks down at higher $T$. These findings offer experimental benchmarks for detecting Berry-curvature–driven planar transport in altermagnets and show that half-quantized transverse responses can arise without Majorana modes.

Abstract

We investigate anomalous planar transport phenomena in a recently identified class of collinear magnetic materials known as $d$-wave altermagnets. The anomalous planar effects manifest in a configuration when the applied electric field/temperature gradient, magnetic field, and the measured Hall voltage are all co-planar, but the planar magnetic field is instrumental in breaking $\hat{C}_{4z}\hat{\mathcal{T}}$ symmetry of the $d$-wave altermagnet, where $\hat{\mathcal{T}}$ is the time reversal operator, resulting in a Zeeman gap at a shifted Dirac node and a nonzero Berry curvature monopole. We demonstrate that these systems exhibit nearly half-quantized anomalous planar Hall and planar thermal Hall effects at low temperatures that persist over a range of magnetic fields. The angular dependence of the planar transport reveals a $\cos2φ$ dependence on the magnetic field direction, where $φ$ is the azimuthal angle made by the magnetic field. We also discuss the anomalous planar Nernst effect, or transverse thermopower, and demonstrate that the Nernst conductivity peaks when the chemical potential lies just outside the induced Zeeman gap and vanishes within the gap. We further explore the dependence of all three coefficients on the polar and the azimuthal angle of the magnetic field when it is rotated in the full $3D$ space. Our results reveal the presence of approximately half-quantized anomalous planar thermal Hall plateau for a range of in-plane magnetic fields without requiring topological superconductivity and conducting Majorana modes, and can be probed in experiments in $d$-wave altermagnets.

Approximate half-integer quantization in anomalous planar transport in $d$-wave altermagnets

TL;DR

This work shows that 2D -wave altermagnets with substrate-induced Rashba SOC exhibit nearly half-quantized anomalous planar Hall and planar thermal Hall effects when an in-plane Zeeman field breaks the symmetry . A shifted Dirac point under the Zeeman field opens a gap and creates a Berry-curvature monopole with BCM , yielding plateaus in and when the chemical potential lies inside the gap; the effect shows a angular dependence and undergoes topological transitions as field orientation changes. The planar Nernst conductivity peaks outside the gap and follows the Mott relation at low temperature, while the Wiedemann–Franz law breaks down at higher . These findings offer experimental benchmarks for detecting Berry-curvature–driven planar transport in altermagnets and show that half-quantized transverse responses can arise without Majorana modes.

Abstract

We investigate anomalous planar transport phenomena in a recently identified class of collinear magnetic materials known as -wave altermagnets. The anomalous planar effects manifest in a configuration when the applied electric field/temperature gradient, magnetic field, and the measured Hall voltage are all co-planar, but the planar magnetic field is instrumental in breaking symmetry of the -wave altermagnet, where is the time reversal operator, resulting in a Zeeman gap at a shifted Dirac node and a nonzero Berry curvature monopole. We demonstrate that these systems exhibit nearly half-quantized anomalous planar Hall and planar thermal Hall effects at low temperatures that persist over a range of magnetic fields. The angular dependence of the planar transport reveals a dependence on the magnetic field direction, where is the azimuthal angle made by the magnetic field. We also discuss the anomalous planar Nernst effect, or transverse thermopower, and demonstrate that the Nernst conductivity peaks when the chemical potential lies just outside the induced Zeeman gap and vanishes within the gap. We further explore the dependence of all three coefficients on the polar and the azimuthal angle of the magnetic field when it is rotated in the full space. Our results reveal the presence of approximately half-quantized anomalous planar thermal Hall plateau for a range of in-plane magnetic fields without requiring topological superconductivity and conducting Majorana modes, and can be probed in experiments in -wave altermagnets.

Paper Structure

This paper contains 6 sections, 18 equations, 7 figures.

Table of Contents

  1. INTRODUCTION
  2. Planar Anomalous Transport
  3. Model Hamiltonian
  4. Results
  5. Conclusions
  6. Acknowledgments

Figures (7)

  • Figure 1: Schematic diagram of the experimental setup for measuring anomalous planar Hall, planar thermal Hall and planar Nernst effects in altermagnets. The magnetic ordering on the square lattice with $\hat{\mathcal{C}_{4z}}\hat{\mathcal{T}}$ symmetry is indicated by the spin-up sublattice in blue and spin-down sublattice in red. Here, the Néel vector is along the out-of-plane $\hat{z}$ direction. The spin-up and spin-down sublattices are connected via a $\pi/2$ rotation about the center. An in-plane magnetic field $\bm{B_{\parallel}}$ modifies the Berry curvature ($\bm{\Omega} = \Omega_z$) to produce a non-zero Berry curvature Monopole (see Fig. \ref{['fig:hamiltonian']}) under the presence of which (a) a longitudinal electric field induces a transverse potential difference ($\Delta V$) in the Hall effect, (b) a longitudinal temperature gradient ($-\bm{\nabla} T$) gives rise to a transverse thermal gradient ($\Delta T$) in the thermal Hall effect and (c) a longitudinal temperature gradient gives rise to a transverse potential difference ($\Delta V$) in the Nernst effect.
  • Figure 2: Panel (a) shows the energy dispersion of the Hamiltonian (\ref{['eq:hamLat']}) in the absence of an applied magnetic field with the inset showing the gapless Dirac cone at the $\Gamma$ point. In panel (d), an applied magnetic field $B_x$ shifts the Dirac point from $\Gamma$-point to $\bm{k}^*$ and opens up a gap of $2|\Delta^*|$ (see Eq. (\ref{['eq:gap']})). Panels (b) and (e) show the spin textures of the single anisotropic spin-split Fermi surfaces when the chemical potential is set at the band-touching point in the absence of $\bm{B_\parallel}$ and at center of the Dirac energy gap in the presence of $\bm{B_\parallel} = (7\text{ T},0)$, respectively. The black arrows indicate in-plane spin-polarization while the red-blue color bar represents the out-of-plane spin-polarization. Panel (c) shows the momentum-resolved Berry curvature distribution of the lower band having a quadrupole nature in the absence of a magnetic field (where the positive and negative lobes of the distribution exactly cancel each other out, rendering the BCM zero) while panel (f) reveals the effective monopole characteristic induced by an in-plane magnetic field $B_x$ (where there is no longer an equal magnitude of positive and negative regions, yielding a finite BCM). The parameters used here are $t=0.5$ eV, $\lambda =0.1$ eV, $t_{\text{AM}} =0.25$ eV, comparable to parameters used in Refs. Rao_2024Yan_2023.
  • Figure 3: Numerically calculated amplitudes for intrinsic anomalous planar charge and thermal Hall responses for the model Hamiltonian (\ref{['eq:hamLat']}) using the same parameters as those mentioned in the caption of Fig. \ref{['fig:hamiltonian']}. Panel (a) and (b) show the magnitude of anomalous planar Hall conductance $\sigma_{xy}$ and thermal Hall conductance $\kappa_{xy}$ as a function of shifted chemical potential $\tilde{\mu}=\mu-\epsilon^*_{7\text{T}}$ with $\epsilon^* = -t(4-\bm{k^{*2}})$, respectively, at various temperatures for a planar magnetic field $B_y = 7$ T. (c) The magnitude of both $\sigma_{xy}$ and $\kappa_{xy}$ at the shifted chemical potential $\tilde{\mu} = 0$ are plotted as a function of applied planar magnetic field $B_{\parallel}$ at a fixed in-plane angle $\phi = \pi/2$ with $T = 0.01$ K, showing approximately half-integer quantized anomalous planar thermal Hall plateau in $d$-wave altermagnets. Here, $\sigma_0 = e^2/h$ and $\kappa_0 = (\pi^2/3)k_B^2T/h$.
  • Figure 4: Phase diagram for the $\mathcal{C}_{4z}\mathcal{T}$ breaking $d$-wave altermagnet model given in Eq. (\ref{['eq:hamLat']}). Panel (a) shows the Dirac mass term $\Delta^{*}$ (see Eq. (\ref{['eq:gap']})) as a function of $B_x$ and $B_y$ for a purely in-plane magnetic field and describes the dependence of the topological phase on the in-plane angle $\phi=\arctan(B_y/B_x)$. The period of $\pi$ emerges from the $\cos(2\phi)$ dependence of $\Delta^{*}$. The white dashed lines show the band gap closing as a result of $\hat{\mathcal{M}}_{x=\pm y}$ symmetry being restored and indicate a phase transition. The sign of the BCM$\nu$ is specified for each quadrant, revealing a period of $\pi$. In panel (b), $\Delta^*$ is shown as a function of in-plane component $B_y$ and out-of-plane component $B_z$. The dashed white curve shows the band gap closing and separates the two phases with different signs for the conductivities. In panels (c) and (d), the distinct signs of the two topological phases becomes manifest via the signs of the nearly half-quantized anomalous planar charge Hall and thermal Hall responses respectively at $T = 0.01$ K. The insets of panels (c) and (d) show that the topological phase boundary at any given $\phi$ does not lie exactly on the equator but is rather dictated by the critical angle $\theta_c$ given by Eq. (\ref{['eq:critang']}). The planar anomalous charge conductivity $\sigma_{xy}$ and thermal Hall conductivity $\kappa_{xy}$ depicted in panels (e) and (f), respectively where the magnetic field is restricted to the $xy$-plane but the azimuthal angle is varied at $T=0.01$ K. Panels (g) and (h) show the variation of $\sigma_{xy}$ and $\kappa_{xy}$ as an out-of-plane component of the magnetic field is introduced by varying the polar angle $\theta$. The parameters used are the same as those mentioned in the caption of Fig. \ref{['fig:hamiltonian']}. Here, $\sigma_0 = e^2/h$ and $\kappa_0 = (\pi^2/3)k_B^2T/h$
  • Figure 5: The variation of $\sigma_{xy}$ and $\kappa_{xy}$ with temperature at three different chemical potentials for planar magnetic field $B_x = 7$ T is depicted in panels (a) and (b) respectively. The chemical potentials considered here correspond to the center of the Dirac band gap and to points symmetrically located in the upper and lower bands, represented by the shifted chemical potentials $\tilde{\mu}=0,\pm2|\Delta^*|$. Panel (c) shows the validity of the Wiedemann Franz law at low temperatures for all the chemical potentials and a rapid departure from it at temperatures higher than $0.1$ K, i.e. the temperature corresponding to the magnitude of the Dirac band gap energy. All the other parameters are same as Fig. \ref{['fig:hamiltonian']}. Here, $\sigma_0 = e^2/h$ and $\kappa_0 = (\pi^2/3)k_B^2T/h$.
  • ...and 2 more figures