Two-band electronic metal and neighboring spin Bose-metal on a zigzag strip with longer-ranged repulsion

We consider an electronic model for realizing the spin Bose-metal (SBM) phase on a two-leg triangular strip—a spin liquid phase found by Sheng et al. [Phys. Rev. B 79, 205112 (2009)] in a spin-1/2 model with ring exchanges. The SBM can be viewed as a “C1S2” Mott insulator of electrons where the overall charge transporting mode is gapped out. We start from a two-band “C2S2” metal and consider extended repulsion motivated by recent ab initio derivation of electronic model for $\kappa \text{-ET}$ spin liquid material [K. Nakamura et al., J. Phys. Soc. Jpn. 78, 083710 (2009)]. Using weak coupling renormalization group analysis, we find that the extended interactions allow much wider C2S2 metallic phase than in the Hubbard model with on-site repulsion only. An eight-fermion umklapp term plays a crucial role in producing a Mott insulator but cannot be treated in weak coupling. We use bosonization to extend the analysis to intermediate coupling and study phases obtained out of the C2S2 metal upon increasing overall repulsion strength, finding that the SBM phase is a natural outcome for extended interactions.

Figure 1

(Color online) Electron band for $t_2>0.5t_1$ has two occupied Fermi sea segments. This is free fermion C2S2 metal.

Figure 2

(Color online) Example of RG flows in the C2S2 phase. The model potential is Eq. (32) with $\kappa =1/2$, $\gamma =2/5$; the band parameter is $t_2/t_1=0.9$. We choose logarithm of the couplings to be the $y$ axis and RG “time” $\ell$ to be the $x$-axis. We see that $w_{12}^{\rho /\sigma }$ flow toward 0 rapidly (irrelevant couplings); ${\lambda }_{ab}^{\rho }$ saturate very fast (strictly marginal couplings); while ${\lambda }_{ab}^{\sigma }$ flow to 0 slowly (marginally irrelevant). More generally, if we fix these $\kappa$ and $\gamma$ values, for $t_2/t_1<0.99$ the flows are similar to those shown here and the phase is C2S2.

Figure 3

(Color online) Example of RG flows of selected couplings in the C1S0 phase. The model is the same as in Fig. 2, but with $t_2/t_1=1.05$. We see that the selected couplings diverge after some time. For example, once the ${\lambda }_{11}^{\sigma }$ and ${\lambda }_{22}^{\sigma }$ become negative while $w_{12}^{\rho }$ and $w_{12}^{\sigma }$ remain positive, the RG Eqs. (23, 24, 25, 26, 27, 28, 29, 30) drive the ${\lambda }_{11}^{\sigma }$ and ${\lambda }_{22}^{\sigma }$ to $-\infty$ and in turn $w_{12}^{\rho }$ and $w_{12}^{\sigma }$ to $+\infty$, and then all couplings diverge. More generally, if we fix $\gamma =2/5$, for $t_2/t_1>0.99$ the flows are similar to those shown here and we call this C1S0 phase. Varying $\gamma$, we obtain the phase diagram Fig. 4.

Figure 4

(Color online) Stabilization of the C2S2 metal by extended interactions. The model potential is Eq. (32) with $\kappa =0.5$. The noninteracting problem has one band for $t_2/t_1<0.5$ and two bands for $t_2/t_1>0.5$, cf. Figure 1, and we focus on the latter region. The limit $\gamma \to \infty$ corresponds to the Hubbard model with on-site repulsion only, and the C2S2 phase is stable only over a narrow window $t_2/t_1∊\left[0.5\cdots 0.57\right]$ (Refs. 18, 23). The C2S2 region becomes progressively wider as we increase the interaction range $1/\gamma$.

Figure 5

(Color online) Same as Fig. 4 but for the potential Eq. (32) truncated at the fourth neighbor.

Figure 6

(Color online) Projection of phases obtained out of the C2S2 of Fig. 4 as we increase overall repulsion strength $V$, which we imagine to be the $z$ axis perpendicular to the page (Fig. 7 gives one cut at $\gamma =0.4$ with such $V$ axis shown explicitly). The results are obtained in the intermediate coupling procedure as explained in the text. White region is C1S0 at weak coupling, cf. Fig. 4, and is not considered here.

Figure 7

(Color online) Intermediate coupling analysis of the model with potential Eq. (32) for $\kappa =0.5$ and $\gamma =0.4$. Here the horizontal range is equal to the extent of the C2S2 phase in the weak coupling analysis from Fig. 4. We start in the C2S2 at small $U$. The boundary where the charge-spin coupling term $W$ becomes relevant first is indicated with blue triangles and the system goes into the C1S0; the next stage where the C1S0 in turn becomes unstable and the system goes into the C0S0 is marked with green circles. The boundary where the umklapp term $H_8$ becomes relevant first is indicated with red squares and the system goes into the C1S2, which is the SBM phase of Ref. 14; upon further increase in the interaction strength the C1S2 eventually becomes unstable and goes to the C0S0 at locations marked with black diamonds. Note that the discontinuity shown with dotted vertical line is not meaningful and is due to our crude analysis performed separately out of the C1S0 and C1S2; in either case, the final C0S0 is likely the same phase. Also note that the C1 mode content is distinct in the $\text{C}1\left[\rho +\right]\text{S}0$ (conducting) and $\text{C}1\left[\rho -\right]\text{S}2$ (insulating) cases and any transition between them is first order. The C2S2 to C1S2 transition is Kosterlitz-Thouless-like.

Figure 8

(Color online) Schematic merging of the weak and intermediate coupling results in the model regimes like in Fig. 7 in the whole range with $t_2/t_1>0.5$. In weak coupling, the C2S2 phase is unstable beyond the shaded region in Fig. 4. However, due to the crudeness of our intermediate coupling procedure, Fig. 7 shows monotonic growth of the C2S2 phase with $t_2/t_1$ past this instability. This discrepancy arises because our intermediate coupling procedure completely ignores the spin sector. More realistically, we expect the C2S2 phase to peak somewhere in the middle of the range shown in Fig. 7 and be bounded by the C1S0 for larger $t_2/t_1$. Similar considerations apply to the C1S2 phase, which is bounded by the C0S0.

Figure 9

(Color online) Same as Fig. 6 but for the model with interactions [Eq. (32)] truncated at the fourth neighbor and starting out of the C2S2 of Fig. 5.