Luttinger liquid physics and spin-flip scattering on helical edges

We investigate electronic correlation effects on edge states of quantum spin Hall insulators within the Kane-Mele-Hubbard model by means of quantum Monte Carlo simulations. Given the $U\left(1\right)$ spin symmetry and time-reversal invariance, the low-energy theory is the helical Tomanaga-Luttinger model, with forward scattering only. For weak to intermediate interactions, this model correctly describes equal-time spin and charge correlations, including their doping dependence. As apparent from the Drude weight, bulk states become relevant in the presence of electron-electron interactions, rendering the forward-scattering model incomplete. Strong correlations give rise to slowly decaying transverse spin fluctuations, and inelastic spin-flip scattering strongly modifies the single-particle spectrum, leading to graphenelike edge state signatures. The helical Tomanaga-Luttinger model is completely valid only asymptotically in the weak-coupling limit.

Figure 1

Projective CTQMC results for the effective model (1). (a) Real-space correlations $S^{xx}(r=L/2)$ as a function of system size $L$ at different $U/\lambda$, normalized to 1 at $r=1$ for comparison. The log-log plot reveals a power law with an interaction-dependent exponent $\eta =2K$. (b) $S^{xx}(q=0)/L$ as a function of $1/L$, demonstrating the absence of long-range order. Lines are fits to the form $S^{xx}(q=0)/L=b/L+c/L^{\eta }$, with $\eta$ taken from (a). Data in (a) and (b) are extrapolated to $\theta =\infty$. (c) Luttinger liquid parameters $K$ and $v$. $K$ is extracted from fits to the form $S^{xx}\left(r\right)=a/r^{\eta }$ shown as lines in (a). $v$ is estimated from the density structure factor $N(q=2\pi /L,\tau )$ and finite-size extrapolation; $v_{\mathrm{F}}=2\lambda$.

Figure 2

Doping dependence of spin and charge structure factors for $U/\lambda =8$ from grand-canonical CTQMC simulations ($L=32$, $\beta =60$). The $q=0$ points in (b) and (c) are linear extrapolations.

Figure 3

Single-particle spectral function $A_{\uparrow }(k,\omega )$ from projector-CTQMC simulations for $L=24$, $\theta =80$, and (a) $U/\lambda =8$, (b) $U/\lambda =40$.

Figure 4

Single-particle spectral function $A_{\uparrow }(k,\omega )$ from second-order perturbation theory. $U/t=2$.

Figure 5

Luttinger parameter $K$ (from Fig. 1) and spectral weight $Z$ from fits to the density structure factor $N(q,\tau )$ at $\theta =80$ and finite-size scaling. The Luttinger liquid result $K=Z$ is violated for a wide range of parameters.