Effects of interaction on field-induced resonances in a confined Fermi liquid

We consider the two-dimensional electron gas confined laterally to a narrow channel by a harmonic potential. As the Zeeman splitting matches the intersubband separation, the nonlocal spin polarization develops a minimum as reported by Frolov et al. [Nature (London) 458, 868 (2009)]. This phenomenon, termed ballistic spin resonance, is due to the degeneracy between the nearest oppositely polarized subbands that is lifted by spin-orbit coupling. We showed that the resonance survives the weak and short-range interaction. The latter detunes it, and as a result shifts the Zeeman splitting at which the minimum in spin polarization occurs. Here this shift is attributed to the absence of the Kohn theorem for the spin-sloshing collective mode. We characterized the shift due to weak interaction qualitatively by analyzing the spin-sloshing mode within the Fermi liquid phenomenology.

Figure 1

(a) The spins shown as (red) straight arrows injected into the channel via the quantum point contact denoted by a (black) square are polarized along the applied magnetic field, $\mathit{B}\parallel \widehat{z}$. The channel is running parallel to the $x$ axis. Normally, they maintain their polarization in the course of propagation along the channel. (b) At the BSR, $E_Z\approx {\omega }_c$, the spins are still injected with the initial polarization along $\mathit{B}$. Their state of polarization, however, is modified as they move along the channel. (c) In the absence of a magnetic field, the energy levels form subbands due to the quantization along $\widehat{z}$. For the parabolic confinement, the subband splitting, ${\omega }_c$, is equal to the angular frequency of oscillations across the channel. (d) At the BSR, $E_Z\approx {\omega }_c$, the adjacent subbands with opposite spin polarization are nearly degenerate. The spin-flip intersubband $p\text{−}h$ excitations shown by thick round (blue) arrows become soft at the BSR, and they play a central role in our analysis.

Figure 2

The two Fermi surfaces located at $z$ and $z+dz$ shown as solid circles. As the confining potential $U\left(z\right)$ decreases with $z$, the Fermi momenta at the two points satisfy $p_F(z+dz)>p_F\left(z\right)$. When a quasiparticle is transferred from $z$ to $z+dz$, its energy changes partly due to the interaction with the quasiparticles in the annulus $p_F\left(z\right)<p<p_F(z+dz)$ (gray) of a radius $d{p}_F$. This interaction is an ultimate cause of the force renormalization, Eq. (10). For the quasiparticle transferred along the path A, the energy change is zero. The translation along path B with momentum kept constant is used to derive Eq. (10).

Figure 3

The representation of the spin-sloshing mode. The magnetic field $\mathit{B}$ is assumed to point in the $z$ direction perpendicular to the channel running along the $x$ direction. Thick arrows (red) represents the spin polarization changing across the channel, and vanishing at its center, $z=0$. The profile is translationally invariant along the channel, and all the spins collectively precess at frequency ${\omega }_s$. (a) The snapshot of spin polarization at an instant such that the polarization is perpendicular to the $xz$ plane. (b) In a quarter period of precession, $\pi /2{\omega }_c$, the spin polarization starting in a configuration shown in (a) rotates by ${90}^{\circ }$ and becomes parallel to the $xz$ plane.

Figure 4

(a) The scattering process that leads to the most singular contributions to the correlation functions has both the initial and final states at resonance with the external frequency. To the leading order, this amplitudes contains the direct and exchange terms. (b) The Dyson equation for the dressed Green function, Eq. (44), shown by the thick arrowed line combining the most singular contributions at the mass shell. The thin line stands for the Green function of free electrons. (c) The Dyson equation for the matrix correlation function $\widehat{\Gamma }$, Eq. (39), shown by the shaded area. For the free electron gas, $\widehat{\Gamma }$ is given by the polarization operator, Eq. (43).

Figure 5

The dotted (blue) lines show the spectrum of collective modes that are superpositions of $p\text{−}h$ excitations between the subbands $|n+1,+1\rangle$ and $|n,-1\rangle$ as a function of the interaction parameter $\overline{V}\nu =(V_{\rho }-3V_s)\nu$. The solid (red) line is the frequency of the spin-sloshing mode as given by Eq. (56) obtained for the weakly interacting Fermi liquid. The horizontal dashed lines indicate the noninteracting excitation energy in units of ${\omega }_c$, $|{\omega }_c-E_Z|/{\omega }_c$. The parameters used are (arb. units) $E_F=50$, $m=1$ and (a) $E_Z=-0.5{\omega }_c$, ${\omega }_c=10$, giving $N_t=5$ according to Eq. (40); (b) $E_Z=-0.5{\omega }_c$, ${\omega }_c=1$, resulting in $N_t=50$ modes; (c) $E_Z=0.1{\omega }_c$, ${\omega }_c=5$, giving $N_t=10$ modes. The spin-sloshing mode corresponds to the lowest energy mode of collective excitations.

Figure 6

The dotted (blue) lines show the spectrum of $p\text{−}h$ excitations between the subbands $|n+1,+1\rangle$ and $|n,-1\rangle$ as a function of the interaction parameter $\overline{V}\nu =(V_{\rho }-3V_s)\nu$. The solid (red) line is the frequency of the spin-sloshing mode as given by Eq. (56). (a) At $E_Z={\omega }_c$, the system is at resonance and goes off resonance at finite interaction. (b) At $E_Z=0.9{\omega }_c$ the resonance is achieved for $\overline{V}\nu =40/3\approx 1.33$, as follows from Eq. (2). The horizontal dashed lines indicate the noninteracting excitation energy $|{\omega }_c-E_Z|/{\omega }_c$. The parameters used are (arb. units) $E_F=50$, $m=1$, ${\omega }_c=1$ corresponding to $N_t=50$. The spin-sloshing mode follows the continuum.

Figure 7

The definition of quasiparticle momenta and spin indices of the scattering amplitude, Eq. (A1).

Figure 8

The three lowest energy subbands are $|0,+1\rangle$, $|0,-1\rangle$, and $|1,+1\rangle$. The spin polarization is denoted by straight thin (red) arrows. Only the $|0,+1\rangle$ crosses the Fermi level, $E_F$. The sloshing mode is the superposition of $p\text{−}h$ excitations from $|0,+1\rangle$ to $|1,+1\rangle$ subbands as indicated by a round thick (blue) arrow.

Figure 9

The dotted (blue) lines show the spectrum of collective modes that are the superpositions of $p\text{−}h$ excitations between the subbands $|n+1,\pm 1\rangle$ and $|n,\pm 1\rangle$ as a function of the interaction parameter $\overline{V}\nu =(V_{\rho }-3V_s)\nu$. The parameters used are (arb. units) $E_F=50$, $m=1$, $E_Z=0.5{\omega }_c$, and ${\omega }_c=1$ corresponding to $N_t=50$. The horizontal dotted line at $\omega /{\omega }_c=1$ demonstrates the Kohn theorem for the sloshing mode.

Figure 10

The same three lowest energy subbands, $|0,+1\rangle$, $|0,-1\rangle$, and $|1,+1\rangle$ as in Fig. 8. The spin polarization is denoted by straight thin (red) arrows. The subbands $|0,+1\rangle$ and $|0,-1\rangle$ cross the Fermi level, $E_F$. The spin-precession mode is the superposition of $p\text{−}h$ excitations from $|0,+1\rangle$ to $|0,-1\rangle$ subbands as indicated by a round thick (blue) arrow.

Figure 11

The dotted (blue) lines show the spectrum of collective modes that are the superpositions of $p\text{−}h$ excitations between the subbands $|n,+1\rangle$ and $|n,-1\rangle$ as a function of the interaction parameter $\overline{V}\nu =(V_{\rho }-3V_s)\nu$. The parameters used are (arb. units) $E_F=50$, $m=1$, $E_Z=0.5{\omega }_c$, and ${\omega }_c=1$ corresponding to $N_t=50$. The horizontal dotted line at $\omega /{\omega }_c=({\omega }_c-E_Z)/{\omega }_c=0.5{\omega }_c$ demonstrates the Kohn theorem for spin-precession mode.