RKKY to Kondo crossover in helical edge of a topological insulator

Two spatially separated magnetic impurities coupled to itinerant electrons give rise to a dynamically generated exchange (RKKY) inter-impurity interaction that competes with the individual Kondo screening of the impurities. It has been recently shown by Yevtushenko and Yudson [Phys. Rev. Lett. 120, 147201 (2018)] that the RKKY interaction and the RKKY vs Kondo competition become nontrivial on helical edges of two-dimensional topological insulators where there is lock-in relation between the electron spin and its direction of motion. Kondo screening always takes over and dominates at large inter-impurity distances and it can also dominate all the way to short distances if the Kondo coupling is sufficiently large and anisotropic. In the present paper, we study the Kondo-RKKY competition in detail on a qualitative and quantitative level. For this we employ the numerically exact numerical renormalization group (NRG) for a broad parameter scan of two Kondo coupled impurities vs magnetic anisotropy, impurity distance, and temperature, and comment on the role of finite bandwidth. We give a pedagogical introduction on the the setup of the two-impurity setting within the NRG in the helical context. Overall we establish a plain crossover from RKKY to Kondo with increasing impurity distance, which permits an intuitive physical picture by simply comparing length scales set by the Kondo screening cloud vs the thermal length scale vs the impurity distance.

Figure 1

Helical edge mode of the one-dimensional edge in a 2D topological insulator, e.g., as it occurs in the Kane-Mele model [35]. Focusing on low energies, the dispersion vs momentum $k$ is given by ${\varepsilon }_{k\sigma }=\sigma vk$ with spin $\sigma \in \{\uparrow ,\downarrow \}\equiv \{+1,-1\}$ indicated by the blue (red) line, respectively, where $v$ denotes Fermi velocity. We assume a finite half-bandwidth $D$, throughout. This gives rise to an effective Brillouin zone with a discrete lattice spacing $a=\pi v/D$ [cf. Eq. (2)].

Figure 2

Eigenstates and eigenlevels of the RKKY Hamiltonian (20). Red lines and arrows show various decay channels of the singlet state (dashed lines) and up/down states (solid lines). These include spin flips (outer paths) or phase flip (center path), as probed by the transverse ${\langle S_{\mathcal{L}}^{+}\parallel S_{\mathcal{R}}^{-}\rangle }_{\omega }$ etc., and the longitudinal ${\langle S_{\mathcal{L}}^z\parallel S_{\mathcal{R}}^z\rangle }_{\omega }$ dynamical spin correlation function, respectively, and reflect backward and forward scattering in the helical context.

Figure 3

NRG analysis of the 2HKM for $J=0.1$ (i.e., $j_0=0.05$). Each row corresponds to a different integer impurity distance $x$ as indicated with the left panel, which increases exponentially from top to bottom. The vertical axis is the same for all panels, namely $J_z/J\equiv j_z/j_0$, whereas the horizontal axis represents energy in various forms. Here we adopt the NRG energy scale point of view, throughout, which starts at large energies at the left and then proceeds towards exponentially smaller energy scales towards the right. Rows (columns) are labeled by letters (numbers), respectively, e.g., having (a1) for the upper left panel. The vertical black dotted [solid blue] marker replicated in all panels indicate the coherence scale $E_x$ (38) [RKKY scale $E_{\mathrm{R}}$ (20)], respectively, as labeled in panel (a1), having $E_x/E_{\mathrm{R}}=1/\pi j_0^2=127.3$, throughout. Similarly, the brown solid curved line shows the analytical single-impurity Kondo temperature $T_{\mathrm{K}}(J,Jz)$ (Appendix pp2) for reference. This curve $T_{\mathrm{K}}$ is visually cutoff at $E_{\mathrm{R}}$, because $T_{\mathrm{K}}$ is irrelevant at lower temperatures. The brown marker in panel (h2) shows the finite intercept $T_{\mathrm{K}}(J=0.1,J_z)=4.31\times {10}^{-8}$ at $J_z=0$. Each column shows the quantity indicated above the top panel: Panels (:,1) [first column] show the static (equal-time) inter-impurity correlations $\langle S_z^{\mathcal{L}}{S}_z^{\mathcal{R}}\rangle$ vs temperature and $J_z$. Panels (:,2) [second column] gives a visual impression of the changes along the NRG energy flow diagram vs energy scale ${\omega }_n\sim {\mathrm{\Lambda }}^{-n/2}$ (for precise prefactors, see [46]). This encodes cumulative effective thermal weights in three arbitrary but fixed low-energy symmetry sectors into red-green-blue (RGB) colors for all odd Wilson shells $n$ (to avoid even/odd effects) based on an effective inverse temperature ${\beta }_n\equiv 4{\omega }_n$. The symmetry sectors chosen for this were $q\equiv (Q;S_z^{\mathrm{tot}})\in (0;0,1,-1)$ with $Q$ the total charge relative to half-filling. The remainder of the columns shows dynamical spin-spin correlation functions ${\langle {\widehat{S}}^{\eta }\parallel {\widehat{S}}^{{\eta }^{\prime }†}\rangle }_{\omega }$ as indicated at the top of each column at zero temperature ($T=1.8\times {10}^{-11}$). The solid-dotted lines shows the respective derived inverse static susceptibility $T_S^{\eta {\eta }^{\prime }}=1/4{\chi }_S^{\eta {\eta }^{\prime }}$ [cf. Eq. (34)]. Blue (red) indicates positive (negative) value, respectively. The number at the top right of each panel indicates the maximum absolute value the spectral data $A$ of the broadened NRG spectral data, and hence gives an impression of numerical range. Blue (red) shading in all panels except for the second column indicates positive (negative) values, respectively. The dashed lines in (a,c;3) and (a,c;5) represent exponential fits as indicated with (a5) of the maximum of the spectral data for $J/J_z\le 0.5$. NRG parameters (e.g., see [46] for detailed definitions): $\mathrm{\Lambda }=4$, truncation energy $E_{\mathrm{trunc}}=8$; $z$ averaged over $n_z=4$, with log-Gauss broadening $\sigma =0.3$ of the spectral data after $z$ averaging.

Figure 4

NRG spectral data of the 2HKM for $J=0.1, J_z=0, x=10$ for the dynamical correlations functions as partly indicated with the left axis [transverse in the upper panels, longitudinal in the lower panels; intra-impurity for the left panels, whereas in the right panels also includes the inter-impurity correlation for comparison; note the sign in the legend with panel (d)]. Left panels: Individual spectral data for $z$ shifted logarithmic discretization ($n_z=8$ curves with $z\in [0,1[$, having $\mathrm{\Lambda }=4$, broadening $\sigma =0.1$), and the corresponding $z$ averaged data in the right panel. The horizontal and vertical axis are globally scaled by the $z$ averaged low-energy scale $T_{S_z}=1.4\times {10}^{-4}=0.556E_{\mathrm{R}}$ only, so with spectral sum rules in mind, the data shown is of order one.

Figure 5

Exactly the same bare data as in Fig. 4, except that for each $z$ shift the frequency scale is first scaled by $T_{S_{\alpha }}^{\eta {\eta }^{\prime }}/T_{S_{\alpha }}^{\eta {\eta }^{\prime }}\left(z\right)$ (with $T_{S_{\alpha }}^{\eta {\eta }^{\prime }}\equiv {\langle T_{S_{\alpha }}^{\eta {\eta }^{\prime }}\left(z\right)\rangle }_z$ the $z$ averaged value, and $\alpha \in \{z,\pm \}$ chosen, respectively, for each curve), and then combined as in Fig. 4. The $z$ shift specific $T_{S_z}\left(z\right)$ is also determined within the NRG in the same calculation as part of the post analysis [cf. Eq. (34)]. Overall, this procedure leads to a significantly improved quality of peak shape that is significantly narrower, as compared to the spurious spread of spectral peaks in the right panels in Fig. 4.

Figure 6

Same analysis as in Fig. 6 except for an increased impurity distance $x=1000$, leading to the reduced value for $T_{S_z}=3.0\times {10}^{-6}=1.19E_{\mathrm{R}}$. The broadening was also increase to $\sigma =0.15$.

Figure 7

Exactly the same bare data as in Fig. 6, except that for each $z$ shift the frequency scale is first scaled by $T_{S_{\alpha }}^{\eta {\eta }^{\prime }}/T_{S_{\alpha }}^{\eta {\eta }^{\prime }}\left(z\right)$ (similar to Fig. 5).

Figure 8

Low-energy scales vs distance from inverse static susceptibilities [cf. Eq. (34)] as indicated in the legend of panel (a) for $J_z=0$, throughout. Left panels ($J=0.05$) [right panels ($J=0.10$)] have temperature $T<{10}^{-10}$ [the Kondo temperature $T_{\mathrm{K}}\sim 4\times {10}^{-8}]$ as low-energy cutoff, respectively. The blue vertical markers translate the low-energy scale to distance, i.e., $x_0\equiv j_0^2/\max (T,T_{\mathrm{K}})$, which also corresponds to the crossing point of $E_{\mathrm{R}}$ (black dashed) with $\max (T,T_{\mathrm{K}})$ (horizontal marker). The lower panels redraw the data in the upper panels, but vertically scaled by $T_{\mathrm{S}}$. The shading indicates the standard deviation of the color matched energy scale due to $z$ shifts in the NRG discretization.

Figure 9

Schematic representation of the two-impurity Wilson setup of the bath giving rise to two intercoupled chains (for a quantitative example, see Fig. 10). The system moves to small energies towards the right with the Wilson shells $n$ having energy ${\varepsilon }_n\sim {\mathrm{\Lambda }}^{-n/2}$. At high energies $\varepsilon \gg E_x\equiv \frac{v}{x}$ [Eq. (A9)] one may think of the bath states coupled to their respective impurity. Therefore, the physics represents two independent copies of one-impurity problems down to energies $E_x$. Around the energy scale of $E_x$ the impurities start to coherently interact with each other (if the impurities are, for example, already Kondo screened above the energy $E_x$, then the two impurities remain decoupled down to zero energy). The two-impurity physics takes place at energies $\varepsilon \lesssim E_x$ where the relevant description of the bath effectively changes from the local $(\mathcal{R},\mathcal{L})$ to the symmetrized $(+,-)$ representation. The antisymmetric channel ($\eta =-$) starts to smoothly decouple, but stays in the system as a passive spectating bath space. The symmetric channel ($\eta =+$) is the one that remains fully coupled to the impurities. While the representation of the bath in the main text is always in the symmetric/antisymmetric configuration [cf. Eq. (A42), except for a final rotation of ${\widehat{f}}_0$ back to the local representation]. If the same unitary mixing of modes were to be applied for subsequent Wilson shells still, the property of having two independent copies of bath modes remains intact down to $E_x$.

Figure 10

Typical Wilson ladder for the two-impurity helical system as described by Eq. (A33) for an impurity distance $x={10}^6$. The labels on top indicate the Wilson shell $n$ (based on $\mathrm{\Lambda }=2$). The widths of the bonds are proportional to the hopping amplitudes rescaled by ${\mathrm{\Lambda }}^{-n/2}$. They are all real-valued, where black (red) color shows positive (negative) hopping amplitudes, respectively. Around the energy $1/x$ (shell $n\sim 40$), the structure of the Wilson ladder changes, as qualitatively already argued in Fig. 9. The upper leg corresponds to the symmetric ($\eta =+$), and the lower leg to the antisymmetric channel ($\eta =-$). The rescaled Creutz couplings have amplitudes below double precision accuracy, hence are absent. (b) Same as in (a), except that starting from the position “blocktrafo” towards the right a nearest-rung shell-mixing numerically determined block transformation was performed on top of (a). This shows that at low energies the system can be exponentially decoupled towards later shells into two independent channels.

Figure 11

Static $\langle S_L^z{S}_R^z\rangle$ impurity correlations for $T_{\mathrm{K}}\ll E_{\mathrm{R}}\sim \frac{J^2}{x}$ obtained by NRG towards the wide-band limit $D\gg J(=J_x=J_y)$ for $J_z=0$. (a) Deviations of the computed $\langle S_L^z{S}_R^z\rangle$ from the expected RKKY value of $-1/4$ on a loglog plot. Data is shown for various impurity distances $x$, with the polynomial fit obtained for $x=10$ showing approximate quadratic behavior (red dashed line). (b) Comparison of energy scales for the same parameter range as in (a) where the temperature $T\sim {10}^{-9}$ used in NRG simulations in (a) is indicated by the horizontal gray line. The data for the Kondo scale $T_{\mathrm{K}}$ was obtained via poor-man's scaling [cf. Appendix pp2]. Red line shows a simple estimate for the RKKY energy at $x=10$, demonstrating that (a) is deeply in the RKKY regime, i.e., $E_{\mathrm{R}}\gg T_{\mathrm{K}},T$.

Figure 12

Kondo scale of the anisotropic Kondo model from poor-man's scaling [Eqs. (B4) and (B5)]. $J_x<0$ is physically equivalent to $J_x>0$, such that the lower half of the panel is a mirror image of the upper half. The thick black horizontal line ($J_z<0$ for $J_x=0$) represents a stable line of fixed points without a Kondo scale. The entire white region below the diagonal lines to the left flows towards it. The black dotted horizontal line ($J_z>0$ for $J_x=0$) represents a line of unstable fixed points that flow to strong coupling for any small but finite $J_x$ (strictly at $J_x=0$ one has $T_{\mathrm{K}}=0$, represented by white color). The gray lines represent RG flow contours as in Eq. (B3). The Kondo temperature is defined by the starting point $(x_0,z_0)$ on such a contour. The Kondo scale increases monotonically from left to right, and also with increasing $|J_x|$. The green contours show lines of constant $T_K(J,J_z)-{\left({\rho }_{0}J\right)}^2\le 0$. The thick green contour to the right describes $T_K(J,J_z)={\left({\rho }_{0}J\right)}^2=E_{\mathrm{R}}(x=1)$, which represents the largest RKKY energy possible by putting two impurities right next to each other, with the closest distance being one “lattice spacing” [cf. Eq. (20b)].