Protection of a non-Fermi liquid by spin-orbit interaction

We show that a thermoelectric transport through a quantum dot–single-mode quantum point contact nanodevice demonstrating pronounced fingerprints of nonFermi liquid (NFL) behavior in the absence of external magnetic field is protected from magnetic field NFL destruction by strong spin-orbit interaction (SOI). The mechanism of protection is associated with the appearance of additional scattering processes due to lack of spin conservation in the presence of both SOI and small Zeeman field. The interplay between in-plane magnetic field $\vec{B}$ and SOI is controlled by the angle between $\vec{B}$ and ${\vec{B}}_{\mathrm{SOI}}$. We predict strong dependence of the thermoelectric coefficients on the orientation of the magnetic field and discuss a window of parameters for experimental observation of NFL effects.

Figure 1

Typical setup for thermoelectric measurements: “cold” contact (light orange area) at reference temperature $T$, quantum dot–quantum point contact electrostatically defined by gates (blue boxes), separated by a tunnel barrier from a “hot” (deep orange) contact at temperature $T+\mathrm{\Delta }T$. The voltage $V$ is applied across the device for zero current measurements (see text for the details).

Figure 2

(Top panels) Two sub-band spectra: (left) Zeeman splitting in the absence of SOI; (center) $\vec{B}\perp {\vec{B}}_{\mathrm{SOI}}$, angle between magnetic field and $y$ direction $\phi =0$; (right) arbitrary angle $-\pi /2<\phi <\pi /2$. (Bottom panels) Magnetic field dependence of reflection amplitudes $|r_{\mu \nu }|$ for the spectra shown in top panels. For illustration we performed all calculations with model barrier $V\left(y\right)=V_{0}exp(-|y|/L_{\mathrm{QPC}})$, $k_{0F}{L}_{\mathrm{QPC}}=3.6$, and height of the barrier $V_0$ is tuned to get $r_0^2=0.1$ (see details in the text). Insert shows relative orientation of $\vec{B}$ and ${\vec{B}}_{\mathrm{SOI}}$.

Figure 3

(Left panel) Gate voltage and magnetic field dependence of ${\mathrm{\Gamma }}_{\alpha }/E_c$: red, ${\mathrm{\Gamma }}_A$; blue, ${\mathrm{\Gamma }}_B$. (Right panel) A “phase diagram” as follows: Four main regimes of the thermoelectric transport are inside the green, red, blue, and magenta domains; A, perturbative NFL, B, weak partial NFL, C, strong partial NFL, D, nonperturbative FL (see details in Sec. 7). Domain boundaries are defined by the crossover condition ${\mathrm{\Gamma }}_A(B,N)=T$ and ${\mathrm{\Gamma }}_B(B,N)=T$. For all plots ${\alpha }_R=0.15v_F$, $\phi =\pi /4$, $r_0^2=0.1$, $k_{0F}{L}_{\mathrm{QPC}}=3.6$, $T=0.05E_c$.

Figure 4

(Top left) $e{S}_{\max }$ as a function of $B/B_{\mathrm{SOI}}$ for different angles $\phi$ at $T=0.001E_c$ and ${\alpha }_R=0.15v_F$, from top to bottom $\phi =\pi /12,\pi /6,\pi /3,5\pi /12$. (Insert) $eS\left(N\right)$ for $\phi =5\pi /12$ for $B=0.5B_{\mathrm{SOI}}$ (black), $B=B_{\mathrm{SOI}}$ (red), $B=1.5B_{\mathrm{SOI}}$ (green). (Top right) $e{S}_{\max }$ as a function of the angle $\phi$ for different amplitudes of $B/B_{\mathrm{SOI}}$ at ${\alpha }_R=0.15v_F$ and $T=0.001E_c$ from top to bottom: $B/B_{\mathrm{SOI}}=0.6,0.8,1.0,1.2.$ (Insert) $eS\left(N\right)$ for $\phi =5\pi /12$, $B=0.5B_{\mathrm{SOI}}$, $T=0.001$ (black), $0.01$ (red), $0.1$ (blue). (Bottom left) $e{S}_{\max }$ as a function of $B/B_{\mathrm{SOI}}$ for different $T/E_c$ at ${\alpha }_R=0.15v_F$ and $\phi =\pi /6$; $T/E_c={10}^{-1}$, blue (regime $A$); ${10}^{-2}$, green (crossover $A\to C$); ${10}^{-3}$, red ($B\to D\to C$); ${10}^{-4}$, black ($B\to D$). (Bottom right) $e{S}_{\max }$ as a function of $T/{\mathrm{\Gamma }}_B^{\min }[{\mathrm{\Gamma }}_B^{\min }={\mathrm{\Gamma }}_B(1/2)]$ with ${\alpha }_R=0.15v_F$, $B=0.5B_{\mathrm{SOI}}$, for $\phi =5\pi /12$ (black), $\phi =\pi /3$ (red), $\phi =\pi /6$ (green), $\phi =\pi /12$ (blue), ($r_0^2=0.1$ and $k_{0F}{L}_{\mathrm{QPC}}=3.6$).