Insulator-metal transition on the triangular lattice

Mott insulators with a half-filled band of electrons on the triangular lattice have been recently studied in a variety of organic compounds. All of these compounds undergo transitions to metallic and/or superconducting states under moderate hydrostatic pressure. We describe the Mott insulator using its hypothetical proximity to a $Z_2$ spin liquid of bosonic spinons. This spin liquid has quantum phase transitions to descendant confining states with Néel or valence bond solid order, and the insulator can be on either side of one of these transitions. We present a theory of fermionic charged excitations in these states and describe the route to metallic states with Fermi surfaces. We argue that an excitonic condensate can form near this insulator-metal transition due to the formation of charge neutral pairs of charge $+e$ and charge $-e$ fermions. This condensate breaks the lattice space group symmetry, and we propose its onset as an explanation of a low temperature anomaly in $\kappa \text{−}{\left(\mathrm{ET}\right)}_2{\mathrm{Cu}}_2{\left(\mathrm{C}\mathrm{N}\right)}_3$. We also describe the separate BCS instability of the metallic states to the pairing of like-charge fermions and the onset of superconductivity.

Figure 1

Brillouin zone of the triangular lattice. The first Brillouin zone has the hexagon shape, and the two axes are the dual base with respect to the oblique coordinates we are using in Fig. 8. The symbols corresponds to the momenta of minimal energy of excitations. The solid triangles at $\pm (\frac{2\pi }{3},\frac{2\pi }{3})$ are spinon excitations (see Appendix ); the solid hexagons at $(\pi ,0)$, $(0,\pi )$, and $(\pi ,\pi )$ are doublons $g$; the solid square at the origin is the holon $f$. The hole (bound state of holon and spinon) is located at the same position as the spinon (triangles); the electron (bound state of doublon and spinon) is located at the solid circles: $\pm (-\frac{\pi }{3},\frac{2\pi }{3})$, $\pm (\frac{2\pi }{3},-\frac{\pi }{3})$ and $\pm (\frac{\pi }{3},\frac{\pi }{3})$. The bound states are discussed in Appendix .

Figure 2

Fermi surface in the excitonic metal. Region I is the holelike Fermi surface of the quasiparticle, and region II is the particlelike Fermi surface of the quasiparticle. Because the system is charge neutral, the area of region I is equal to the area of region II. Note that this diagram is drawn in the reduced Brillouin zone. In presence of the pairing, the holon and doublon modes are mixed, and the Brillouin zone shrinks to $1∕4$ of its original size such that the holon and doublon pockets locate at the same momenta.

Figure 3

Phase diagram at fixed masses and interaction $g$. This plot shows the exciton pairing gap $\Delta$ and the area of holelike quasiparticle Fermi surface as a function of the charge gap $U_{\mathrm{ind}}$. It is plotted at $m_h=m_{dy}=3m_{dx}$, and $g∕m_h=0.007$. $U_{\mathrm{ind}}$ and $\Delta$ are measured in unit of the energy scale in the system: $\frac{{\Lambda }^2}{2m_h}$, where $\Lambda$ is the cutoff momentum; the Fermi surface area is in unit of $\pi {\Lambda }^2$. Phases I, II, III, and IV in the diagram are normal insulating phase, excitonic insulator phase, excitonic metal phase, and normal conducting phase, respectively.

Figure 4

Phase diagram at fixed masses. It is plotted at $m_h=m_{dy}=3m_{dx}$. Phases I, II, III, and IV in the diagram are normal insulating phase, excitonic insulator phase, excitonic metal phase, and normal conducting phase, respectively.

Figure 5

Charge order pattern. $q_0$ can be either positive or negative depends on if the relative phase in Eq. (3.19) is $\frac{2\pi }{3}$ or $\frac{5\pi }{3}$.

Figure 6

Schematic illustration of the spectrum of fermionic holon $\left(h\right)$, doublon $\left(d\right)$, and electron $\left(c_{\alpha }\right)$ excitations in various phases which descend from the $Z_2$ spin liquid. The band structures are merely represented by cartoon sketches with the purposes only of indicating the filled states and the location of the chemical potential. More realistically, the Fermi surfaces will form around the respective dispersion minima in Fig. 1.

Figure 7

Interaction.

Figure 8

Configuration of $A_{ij}$ in zero-flux state (Ref. 13). The arrows on the bonds show the direction along which $A_{ij}>0$.

Figure 9

Interaction.

Figure 10

Interaction between spinon and holon due to the hopping term.

Figure 11

One loop correction of the interaction in Fig. 10.

Figure 12

Shape of the interaction function in Eq. (C5).