Tomographic collective modes in a magnetic field

Abstract

Two-dimensional Fermi liquids at low temperatures have been theoretically established to exhibit an odd-even effect in the collective quasiparticle relaxation rates where even-parity deformations of the Fermi surface decay at a much faster rate than odd-parity ones. A predicted consequence of this effect is a new tomographic transport regime that mixes hydrodynamic and collisionless transport. In the presence of a magnetic field, however, the tomographic regime is expected to evolve towards conventional transport regimes as soon as the cyclotron radius becomes smaller than the dominant odd-parity mean free path. In this work, we examine this transition from the point of view of collective modes, using a numerically exact solution of the linearized Boltzmann equation within a generalized relaxation time approximation for the odd-parity and even-parity modes. In the absence of a magnetic field, the transverse conductivity exhibits two diffusive tomographic collective modes, and we find that at a critical magnetic field one of these two tomographic modes disappears. Which tomographic mode persists depends on the Landau parameters, with the remaining mode becoming increasingly dominated by hydrodynamic modes at high fields. We corroborate our analysis using a variational approach for the Fermi surface deformation that captures the angular structure of the deformation and the critical magnetic field strength. The collective modes discussed here can in principle be observed by examining the damping of longitudinal and transverse current responses in finite magnetic fields.

Figures (7)

• Figure 1: Real part of the longitudinal conductivity ${\rm Re}[\sigma_L]$ (top row, panels (a)) and the transverse conductivity ${\rm Re}[\sigma_T]$ (bottom row. panels (b)) calculated with the odd-even staggered relaxation rate model in [Eq. \ref{['eq:constant_gammaodd']}]. The left column [panels (i)] shows the response in the absence of magnetic field, and the right column [panels (ii)] in the presence of a finite magnetic field with $\omega_c/\gamma_e = 1$. We set $F_1 =5$ and $\gamma_o/\gamma_e=10^{-2}$, and the unit of the conductivity is $ne^2\gamma_e/(m^*v_F^2)$ with $n = k_F^2/\pi$ the density.
• Figure 2: Collective mode spectrum for $F_1 = 5$ and $\gamma_o/\gamma_e = 10^{-2}$ in the (a) absence and (b) presence of a magnetic field of strength $\omega_c/\gamma_e = 10^{-1}$ (left and right columns). The top (bottom) row shows the (i) real and (ii) and imaginary parts. Red lines describe the longitudinal sound mode. The blue lines denote transverse sound modes, which become propagating at finite $k\xi$khoo_shear_2019. The green and purple lines denote upper and lower branches of the tomographic collective modes Eq. \ref{['eq:frequencies_exact']}. The inset in (a,ii) highlights the branch cut between these two modes. The black dashed and dotted lines show the analytical predictions for (a) the longitudinal and transverse sound modes and (b) the magnetoplasmon and magnetosound modes, Eqs. \ref{['eq:mp']}-\ref{['eq:ms']}.
• Figure 3: Spectrum of the two lowest diffusive collective modes of the transverse conductivity, $\omega_{tom}$, as a function of the magnetic field strength $\omega_c/\gamma_o$. (a)-(d) are obtained assuming a constant odd-parity relaxation rate [Eq. \ref{['eq:constant_gammaodd']}], for (a) $F_1=0.0$, (b) $F_1=2.0$, (c) $F_1=2.6$, and (d) $F_1=3.0$. We set $v_Fk/\gamma_e = 5\cdot10^{-3}$ and $\gamma_o/\gamma_e = 10^{-5}$, corresponding to $k\xi = 1.58$. The green and purple lines denote the upper and lower branches, which at zero-magnetic field coincide with the predictions of Eq. \ref{['eq:frequencies_exact']} [horizontal dashed lines]. The vertical black dashed (dotted) lines are estimates for the critical magnetic field obtained from Eq. \ref{['eq:critical_cond']}. (e)-(h) similar to (a-d) but using the $m$-dependent relaxation rates in Eq. \ref{['eq:m_dependent_gammaodd']}. We use $v_Fk/\gamma_e=10^{-2}$ and $\gamma_o/\gamma_e\approx 0.8 \times 10^{-5}$ with (e) $F_1=0$, (f) $F_1=2.0$, (g) $F_1=2.5$, and (h) $F_1=3.0$.
• Figure 4: Stability diagram indicating the presence of tomographic modes as a function of $F_1$ and the magnetic field for the linearized collision integral with constant odd-parity relaxation rates [Eq. \ref{['eq:constant_gammaodd']}]. For the green (purple) shaded region there exists an upper (lower) branch tomographic mode, with an overlap region in which both modes are present. The solid green and purple lines depict the critical values of $\omega_c/\gamma_o$. The black dotted and dashed lines denote the point $F_c$ at which the lower branch emerges, and the point $F_1^*$ at which the instability point of the two collective modes meet. We choose the same parameters as in Fig. \ref{['fig:3']} and set $v_Fk/\gamma_e = 5\cdot10^{-3}$ and $\gamma_o/\gamma_e = 10^{-5}$ ($k\xi \approx 1.58$).
• Figure 5: Fermi surface deformation $h(\theta_{\bf p})$ for different Landau parameters (i-v) $F_1= 0, \ 2.5, \ 3, \ 5, \ 10$ (left to right column) and magnetic fields (a-f) $\omega_c/\gamma_o = 10^{-2}, \ 10^{-1}, \ 10^{0}, \ 10^{1}, \ 10^{2}, \ 10^{5}$ (bottom to top row). We use the same parameters as in Fig. \ref{['fig:3']}, i.e. a constant odd-parity relaxation rate $\gamma_o/\gamma_e =10^{-5}$ and set $v_Fk/\gamma_e = 5 \cdot 10^{-3}$ ($k\xi = 1.58$). We normalize the amplitude of the deformation to be approximately $10\%$ of the Fermi surface. Green (purple) lines denote the upper (lower) branch deformation. The insets in each panel give the real-valued Fourier coefficients $a_m$ and $b_m$ [Eq. \ref{['eq:expansion_coefficients']}] in the $m$th angular momentum channel. We use the positive axis to denote cosine modes and the negative axis for the sine modes. We also indicate weight in the odd-parity (even-parity) sector by orange (blue) columns.