A microscopic approach to nonlinear theory of spin-charge separation

Abstract

The fate of spin-charge separation beyond the low energy remains elusive up to now. Here we develop a microscopic theory of the correlation functions using the strong coupling expansion of the Hubbard model and demonstrate its validity down to the experimentally relevant $r_{\rm s}>1$. Evaluating the spectral function, we show the general stability of the nonlinear spin-charge modes in whole energy band and investigate all the nonlinear features systematically. We confirm the general prediction experimentally in semiconductor quantum wires. Furthermore, we observe a signal consistent with a continuum of the nonlinear excitations and with a final spectral density around the $3 k_{\rm F}$ point, indicating the robustness of the Hubbard model predictions for a finite range interaction.

Figures (3)

• Figure 1: A Occupation numbers $n_{k}$ evaluated using Eqs. (\ref{['eq:Tc']}--\ref{['eq:dZ']}) for $N=100$ particles. The dash-dotted line is the infinite-interaction limit $\gamma=\infty$, the solid line is at finite interaction $\gamma=10$, and the dashed line is the Fermi step of the free-particle limit $\gamma=0$. B and C is the derivative of $n_{k}$ in A with respect to $k$, exhibiting singularities around the $k_{\rm F}$ and $3k_{\rm F}$ points, respectively.
• Figure 2: A Spectral function $A\left(k,E\right)$ evaluated for $N=200$ particles and a finite interaction strength $\gamma=3$. The magenta and green dashed lines are the dispersions of the pure charge and spin modes evaluated using the expansion in Eqs. (\ref{['eq:k1']},\ref{['eq:q1']}). The green dotted line is the replica of the spin mode in the hole sector in the $E>\mu$ and $k>k_{\rm F}$ region. The black dashed line is the free-particle dispersion for $\gamma=0$. B Density of states $\rho\left(E\right)$ for $\gamma=3$ (solid black line) and $\gamma=\infty$ (dash-dotted black line). The magenta and green dashed lines mark the charge ($-\mu_{\rm c}$) and the spin ($-\mu_{\rm s}$) chemical potentials for $\gamma=3$ obtained as the minimum energy of the charge and spin dispersions w.r.t. the electron chemical potential $\mu$ in A. C Constant-momentum cuts of $A\left(k,E\right)$ in A around the Fermi point at $k=1.1k_{\mathbf{F}}$, of the nonlinear extension of the spin mode in the particle sector at $k=1.6k_{\rm F}$, and of the nonlinear charge mode above the $3k_{\rm F}$ point at $k=3.3k_{\rm F}$.
• Figure 3: A The conductance $G(B,V_{\rm dc})$ measured for $V_{\rm FG}=-664$ mV and presented as the $dG/dB$ derivative. The green and magenta dashed lines are the dispersions of spin and charge modes obtained from the full Lieb-Wu equations for $\gamma=1.25$, and corrected for capacitance using $c_{UL}=5.6\,{\rm mFm^{-2}}$ and $c_{UW}=4.7\,{\rm mFm^{-2}}$, see details in Vianez21. The upper horizontal axis is the linear transformation of $B$ using the two crossing points with $V_{\rm dc}=0$ line as $B_{\rm lo}=0.75\,{\rm T}$ is $-k_{\rm F}$ and $B_{\rm hi}=3.33\,{\rm T}$ is $k_{\rm F}$. The inset is a schematic of the cross-section of our device. B Open black circles are the conductance along the charge mode in the region marked by the olive-yellow dashed line in A, for which $E/\mu_s$ is obtained as $V_{\rm dc}$ divided by the voltage of the minimum of the dashed-green parabola in A. The magenta line is the maximum of $A(k,E)$ in Fig. \ref{['fig:spectral_function']}A along the charge mode in the particle sector. C Integration over $V_{\rm dc}$ of the conductance $G$ within the light-blue dashed rectangle of height $V_0=4$ mV in A as a function of $B$ for $V_{\rm FG}=-664$ mV (black circles) and $V_{\rm FG}=-693$ mV (blue stars).