Mott transition and magnetism on the anisotropic triangular lattice

Spin-liquid behavior was recently suggested experimentally in the moderately one-dimensional organic compound $\kappa \text{−}{\mathrm{H}}_3{\left(\text{Cat-EDT-TTF}\right)}_2$. This compound can be modeled by the one-band Hubbard model on the anisotropic triangular lattice with $t^{\prime }/t\simeq 1.5$, where $t^{\prime }$ is the minority hopping. It thus becomes important to extend previous studies, that were performed in the range $0\le t^{\prime }/t\le 1.2$, to find out whether there is a regime where Mott insulating behavior can be found without long-range magnetic order. To this end, we study the above model in the range $1.2\le t^{\prime }/t\le 2$ using cluster dynamical mean-field theory (CDMFT). We argue that it is important to choose a symmetry-preserving cluster rather than a quasi-one-dimensional cluster. We find that, upon increasing $t^{\prime }/t$ beyond $t^{\prime }/t\approx 1.3$, the Mott transition at zero temperature is replaced by a first-order transition separating a metallic state from a collinear magnetic insulating state excluding the possibility to find a quantum spin liquid for the physically relevant value $t^{\prime }/t\simeq 1.5$. The phase diagram obtained in this study can provide a working basis for moderately one-dimensional compounds on the anisotropic triangular lattice.

Figure 1

(a) Illustration of the anisotropic triangular lattice with dashed green lines emphasizing the two mirror planes $v$ and $v^{\prime }$. (b) While the first cluster geometry, called the symmetry-preserving (SP) cluster, displays the same symmetries $v$ and $v^{\prime }$ as the infinite lattice, the second cluster geometry, called the quasi-one-dimensional (Q1d) cluster, does not.

Figure 2

(a) Symmetry preserving (SP) cluster. (b) Quasi-one-dimensional (Q1d) cluster. The four cluster sites are black circles, and the bath sites are green and orange squares. For simplicity, spin indices $\sigma$ for the bath energies ${\epsilon }_{m,\sigma }$ and cluster and spin indices $i,\sigma$ for the bath-cluster hybridization matrix elements ${\theta }_{m,i,\sigma }$ are not explicitly shown. Depending on the symmetry of the phase being explored, some of the variational parameters are taken equal. Note that the bath sites do not have a position in real space and that the Q1d cluster does not share the symmetries of the lattice, contrary to the SP cluster.

Figure 3

(a) Double occupancy $D=\langle {\widehat{n}}_{\uparrow }{\widehat{n}}_{\downarrow }\rangle$ for the SP cluster as a function of $U/t$ for $t^{\prime }/t=1.3$ (sky blue dashed line) and $t^{\prime }/t=1.7$ (dark blue solid line). While the jump at low interaction defines the lower critical ratio $U_{c1}/t$, the change of slope at higher interaction defines the upper critical ratio $U_{c2}/t$. (b) Double occupancy for the Q1d cluster as a function of $U/t$ for $t^{\prime }/t=1.3$ (light green solid line) and $t^{\prime }/t=1.7$ (dark green solid line). Here, the change of slope defines a critical ratio $U_{c2}/t$ analog to the upper critical ratio of the SP cluster. It is investigated further through the low-frequency behavior of the local density of states, Fig. 4. (c) Mott critical ratio $U_{c2}/t$ as a function of $t^{\prime }/t$ for the SP and Q1d clusters, in dark blue and dark green, respectively. For $t^{\prime }/t\le 1$, the results are quantitatively equivalent but differences clearly appear just above $t^{\prime }/t=1$.

Figure 4

Local density of states $A\left(\omega \right)$ for $t^{\prime }/t=1.3$ in the normal state at $U/t=11.84$ and $U/t=11.9$ on the SP cluster. A Lorentzian broadening $\eta =0.035$ was used. A metal-insulator transition occurs between these two values of $U/t$.

Figure 5

(a) Phase diagram for the Hubbard model obtained with CDMFT on the SP cluster. Black curve: Data from a previous study of the low-frustration phase diagram carried out by Kyung et al. (Ref. [28]). Blue curve: Mott transition in the nonmagnetic normal state. The bars indicate the boundaries of the coexistence region. Red triangles: Metal to collinear magnetic state transition. Here, only $U_{c2}^{\mathrm{CMS}}/t$ is displayed. The lower critical interaction $U_{c1}^{\mathrm{CMS}}/t$ cannot always be found due to some numerical instabilities. Red area: The collinear magnetic phase with wave vector $\mathbf{Q}=(0,\pi )$. AF, NMI, CMS, and M denote the Néel state, the nonmagnetic insulator, the collinear magnetic state, and the metallic state, respectively. (b) Same phase diagram with $t^{\prime }$ as energy unit, namely $U/t^{\prime }$ vs $t/t^{\prime }$ for $1.2\le t^{\prime }/t\le 2$. This phase diagram can be more easily compared with the results of Ref. [34].