Bosonization Solution to Spin-Valley Kondo Problem: Finite-Size Spectrum and Renormalization Group Analysis

Abstract

Spin-valley Anderson impurities (SVAIM) with (anti-)Hund's splitting provide a natural explanation to the origin of pairing potential and pseudogap in the magic-angle graphene. In this work, we derive and analytically solve the low-energy Kondo theories for SVAIM at half-filling, with especial focus on the two anti-Hund's regimes: the impurity is either dominated by a valley doublet, or a trivial singlet. In the doublet regime, we reveal that a novel pair Kondo scattering $λ_x$ is required to flip the valley doublet, which involves a quartic operator of bath electrons. Our renormalization group (RG) calculation based on the Coulomb gas analog shows $λ_x$ drives a phase transition of the Berezinskii-Kosterlitz-Thouless type. One side of the transition is an anisotropic doublet phase, characterized by non-universal phase shifts of bath electrons and non-analytic impurity susceptibilities, while the other is a Fermi liquid formed by pair-Kondo resonance. The finite-size many-body spectrum, thermodynamic quantities, and correlation functions for both phases are analytically solved. Remarkably, the solution in the pair-Kondo Fermi liquid is achieved via the constructive approach of bosonization-refermionization along a solvable fixed line, where the many-body interaction $λ_x$ is mapped into a pseudo-fermion bilinear in a rigorous manner. Finally, we also apply the RG analysis to the singlet regime, and identify a second-order phase transition between the Kondo Fermi liquid and a local singlet phase.

Figures (5)

• Figure 1: A schematic phase diagram of the spin-valley Anderson impurity model (SVAIM) at half-filling. AD, FL, LS stands for anisotropic doublet, (Kondo) Fermi liquid, and local singlet, respectively.
• Figure 2: RG flow of the pair-Kondo model in the doublet regime. Red and blue fixed lines represent the anisotropic doublet (AD) and Kondo Fermi liquid (FL) phases, respectively.
• Figure 3: Phase shift of energy levels of the pair-Kondo model at the Fermi liquid fixed point $\rho^\star=\frac{1}{4}$. Black vertical dashed lines indicate $\epsilon=\frac{2\pi}{L} \left( n-\frac{1}{2} \right)$ ($n\in \mathbb{Z}$), and yellow vertical dashed lines represent $\varepsilon_f$, which is $\frac{2\pi}{L} n_0$ and $\frac{2\pi}{L}\left( n_0 + \frac{1}{2} \right)$ in (a) and (b), respectively. Red dots indicate $\epsilon=\frac{2\pi}{L}n$. The blue lines are $\frac{\epsilon-\varepsilon_f}{\Gamma}$. The black curves are the function $-\tan\left( \frac{L}{2} \epsilon \right)$, their crossings with the blue lines give the eigenvalues $\epsilon_n = \frac{2\pi}{L}\left( n - \frac{1}{2} + \delta_n \right)$. $\frac{2\pi}{L} \delta_n$ is shown by the gray arrows.
• Figure 4: RG flow in the singlet regime. (a) The vertical dashed line indicates $\varepsilon_D = 2\alpha$, and the the dashed curve indicates $\zeta_x = \sqrt{\frac{\varepsilon_D}{4(e^{2\varepsilon_D}-1)}}$. The red dot is the critical point. Here $\alpha=2\rho_z-2\rho_z^2$. (b) The critical exponent $\nu$ as a function of $\alpha$.
• Figure 5: Contour integral about $f(z)=(-\mathrm{i} z + 0^+)^{\alpha-1}$. The bold red line represents the branch-cut of $f(z)$. The red numbers $1$, $e^{\pm (\alpha-1) \frac{\pi}{2}}$, $e^{\pm (\alpha-1)\pi}$ represent $f(z)/|f(z)|$ in at $z=\mathrm{i} y$, $\mp x$, $\mp 0^+ -\mathrm{i} y$, respectively, where $x,y>0$.