Effects of Zeeman field on a spin Bose-metal phase

We consider Zeeman field effects on a spin Bose-metal (SBM) phase on a two-leg triangular ladder. This phase was found in a spin-1/2 model with ring exchanges [D. N. Sheng, O. I. Motrunich, and M. P. A. Fisher, Phys. Rev. B 79, 205112 (2009)] and was also proposed to appear in an interacting electronic model with longer-ranged repulsion [H.-H. Lai and O. I. Motrunich, Phys. Rev. B 81, 045105 (2010)]. Using bosonization of a spinon-gauge theory, we study the stability of the SBM phase and its properties under the field. We also explore phases arising from potential instabilities of the SBM; in all cases, we find a gap to spin-1 excitations while spin-nematic correlations are power law. We discuss two-dimensional analogues of these phases where spinons can pair with their own species.

Figure 1

(Color online) Single-particle spectrum in the presence of the Zeeman field, ${\xi }^{\uparrow /\downarrow }\left(k\right)=-2t_1\text{cos}\left(k\right)-2t_2\text{cos}\left(2k\right)\mp \frac{h}{2}-\mu$, shown for parameters $t_2/t_1=1$ and $h/t_1=1/2$. Our $k_F$-s denote right-moving momenta $∊\left(-\pi ,\pi \right)$; with this convention, the half-filling condition reads $k_{F1\uparrow }+k_{F1\downarrow }+k_{F2\uparrow }+k_{F2\downarrow }=-\pi$.

Figure 2

(Color online) Free-electron phase diagram in the $t_2/t_1-h/t_1$ plane. In this paper, we focus solely on the lower region where both spin species have two Fermi seas (bands). For reference, we give the magnetization $M^z\equiv \left(n_{\uparrow }-n_{\downarrow }\right)/\left(n_{\uparrow }+n_{\downarrow }\right)$ at the transition for several band parameters: $M_{\text{crit}}^z=0.32$, 0.46, and 0.54 for $t_2/t_1=1.0$, 1.5, and 2.0.

Figure 3

(Color online) An example of the weak-coupling phase diagram in the electron system under the Zeeman field, using model interactions Eq. (31) with $\kappa =0.5$ and $\gamma =0.3$. We focus on the region where the kinetic energy gives four modes (cf. Figs. 1, 2) and find four phases: metallic phase with four gapless modes evolving out of the C2S2 phase in zero field; phase with three gapless modes where only the $w_{12}^{\uparrow }$ term is relevant and flows to strong coupling; phase with three gapless modes where only the $w_{12}^{\downarrow }$ term is relevant; and phase with two gapless modes where both the $w_{12}^{\uparrow }$ and $w_{12}^{\downarrow }$ are relevant. The $w_{12}^{\uparrow }$ term is relevant in the region with hash lines at roughly $45°$ and the $w_{12}^{\downarrow }$-term is relevant in the region with hash lines at $135°$ with respect to the horizontal axis. Note that the $w_{12}^{\downarrow }$ term always becomes relevant upon approaching the boundary of the two-band structure (Ref. 15).

Figure 4

(Color online) Scaling dimensions $\Delta \left[w_{12}^{\uparrow }\right]$ and $\Delta \left[w_{12}^{\downarrow }\right]$ as a function of $h/t_1$ for fixed $t_2/t_1=1$, calculated in the absence of residual spinon interactions. In this case, the scaling dimensions stay greater than 2 and the SBM phase remains stable under the Zeeman field.

Figure 5

Picture of the valence bond solid order when ${\mathcal{B}}_{\pi }^{\left(1\right)}$ gains an expectation value. Top: 1D chain view. Bottom: the same in 2-leg ladder view. Coexisting with the static order, we also have spin-nematic power-law correlations and power law in either $ϵ$ or $\chi$ channels (these properties are not depicted in any way).

Figure 6

(Color online) Picture of the static period-2 order in spin chirality when ${\chi }_{\pi }$ gains an expectation value. Since ${\mathcal{J}}_{\pi }^{\left(2\right)}\sim {\chi }_{\pi }$, we have static staggered second-neighbor bond currents in the chain view (top figure). In the ladder view (bottom figure), we have oppositely oriented spin currents flowing on the two legs. Coexisting with the static order, we also have spin-nematic power-law correlations and power laws in $ϵ/\chi$ channels (these properties are not depicted in any way).