Phase diagram of the Kane-Mele-Coulomb model

We determine the phase diagram of the Kane-Mele model with a long-range Coulomb interaction using an exact quantum Monte Carlo method. Long-range interactions are expected to play a role in honeycomb materials because the vanishing density of states in the semimetallic weak-coupling phase suppresses screening. According to our results, the Kane-Mele-Coulomb model supports the same phases as the Kane-Mele-Hubbard model. The nonlocal part of the interaction promotes short-range sublattice charge fluctuations, which compete with antiferromagnetic order driven by the onsite repulsion. Consequently, the critical interaction for the magnetic transition is significantly larger than for the purely local Hubbard repulsion. Our numerical data are consistent with $\mathrm{SU}\left(2\right)$ Gross-Neveu universality for the semimetal to antiferromagnet transition, and with 3D XY universality for the quantum spin Hall to antiferromagnet transition.

Figure 1

Acceptance rate $R_{\mathrm{acc}}$ for Monte Carlo updates and spin structure factor $S_{\mathrm{AF}}$ [Eq. (20)] as a function of the parameter $P$. Here, $U/t=4,\alpha =1,L=6$.

Figure 2

(a) Phase diagram of the KMC model with $\alpha =1$. The phases correspond to a semimetal (SM) which exists for $\lambda =0$, a quantum spin Hall insulator (QSHI), and an antiferromagnetic Mott insulator (AFMI). (b) Phase diagram of the KMH model (corresponding to $\alpha =0)$ based on previous simulations [10, 17, 30, 41] (see also Ref. [42]).

Figure 3

SM-AFMI quantum phase transition. (a) Finite-size scaling of the magnetization $m$ using quadratic fits. (b) Scaling intersection using the critical exponents for the Gross-Neveu universality class from the $\epsilon$ expansion [10, 24]. (c) Scaling collapse using the critical value $U_{\mathrm{c}}^{\left(\epsilon \right)}/t=5.45$. Here, $\lambda =0,\alpha =1$.

Figure 4

(a) Finite-size scaling of the single-particle gap using quadratic fits. (b) Expectation value of the interaction term, corresponding to the derivative of the free energy with respect to $U$. Here, $\lambda =0,\alpha =1$.

Figure 5

Real-space charge and spin correlations relative to the central site for (a), (b) the Hubbard model $\left(\alpha =0\right)$, and (c), (d) a long-range interaction $\left(\alpha =1.23\right)$. Here, $\lambda =0,U/t=3.5,L=15$.

Figure 6

QSHI-AFMI quantum phase transition. (a) Finite-size scaling of the magnetization $m_{xy}$ using quadratic fits. (b) Scaling intersection using the critical exponents of the 3D XY model [30, 47]. (c) Scaling collapse using $U_{\mathrm{c}}/t=8.4$. Here, $\lambda /t=0.2,\alpha =1$.

Figure 7

(a) Finite-size extrapolated single-particle gap. (b) Expectation value of the interaction term, corresponding to the derivative of the free energy with respect to $U$. Here, $\lambda /t=0.2,\alpha =1$.