Bad metallic transport in geometrically frustrated models

We study the transport properties for a family of geometrically frustrated models on the triangular lattice with an interaction scale far exceeding the single-particle bandwidth. Starting from the interaction-only limit, which can be solved exactly, we analyze the transport and thermodynamic behavior as a function of filling and temperature at the leading nontrivial order in the single-particle hopping. Over a broad range of intermediate temperatures, we find evidence of a DC resistivity scaling linearly with temperature and with typical values far exceeding the quantum of resistance $h/e^2$. At a sequence of commensurate fillings, the bad metallic regime eventually crosses over into interaction induced insulating phases in the limit of low temperatures. We discuss the relevance of our results to experiments in cold-atom and moiré heterostructure based platforms.

Figure 1

(a) An illustration of the relevant interaction terms in the modified Hamiltonian $H_{int,c}+\delta H_{int,c}$ for a particular configuration. Note that $\delta H_{int,c}$ modifies the nearest-neighbor interaction term (orange line). (b) A schematic phase diagram as a function of temperature ($T$) and filling ($0<n<1$). The nearest-neighbor (Ising) interaction drives a transition to a $\sqrt{3}\times \sqrt{3}$ charge density wave (or generalized Wigner crystal) at fillings, $n=1/3,2/3$ below $T\sim O\left(V\right)$. For $V^{\prime }\ne 0$, additional correlated insulating states can appear at a sequence of commensurate fractions, $n=p/q$, at lower temperatures.

Figure 2

(a) Compressibility as a function of temperature for $V^{\prime }=0.01V$ for a range of chemical potentials. In the limit of low temperatures, four different incompressible states appear as domes over an extended range of $\mu$. The cooling curves corresponding to a fixed average density are shown as colored lines. (b) The rescaled compressibility calculated from the MC simulations shows distinct regimes of a $1/T$ behavior at high and low temperatures connected by a smooth crossover. (Error bars correspond to a $1\sigma$ standard deviation extracted from a bootstrap resampling; see Appendix pp1 for details.) The solid and dashed lines correspond to asymptotic expansions for $\chi$ at high and low temperatures, respectively.

Figure 3

Typical configurations generated in our Monte Carlo simulations at different temperatures. Red dots represent occupied sites and blue lines denote the unfrustrated bonds (see text for discussion). (a) At half-filling and for $T\gg V$, the charge configuration is uncorrelated without any restriction on the number of frustrated bonds in a triangle. (b) A snapshot of a regime with liquidlike, long-range correlations at $T=0.1V$. The configurations are in a one-to-one correspondence with rhombic tillings of the plane; the defects in the tilling are exponentially suppressed at low temperatures. (c) Charge ordered state at $n=1/3$ at $T=0.09V$. (d) Stripe phase at half-filling determined by the strength of further neighbor interactions at $T=0.01V$ with $V^{\prime }=5V$.

Figure 4

Thermodynamic susceptibilities at fixed temperatures as a function of filling. The susceptibilities have been rescaled by appropriate factors of $T$ such that their high temperature limit is $T$ independent. The dashed line in each figure corresponds to the result expected from the high-temperature expansion.

Figure 5

(a) and (b) Typical configurations associated with an electron hop from ${\mathit{r}}^{\prime }$ to $\mathit{r}$ and the corresponding energetics. Orange lines denote bonds that contribute to ${\varepsilon }_{{\mathit{r}}^{\prime }}^i$. Dashed (dotted) lines represent second (third) nearest-neighbor interactions, respectively. The local energies at leading order [i.e., ignoring contributions at $O\left(V^{\prime }\right)]$ are summarized in the inset. The change in the local energy associated with the particular hopping process contributes to the spectral weight in the optical conductivity at a frequency $\sim 2V$. (c) Optical conductivity calculated using Eq. (15) at $T=V$ as a function of increasing $V^{\prime }/V$. (d) A magnified view of the central peak in (c).

Figure 6

(a) The central peak of the optical conductivity for fixed $V^{\prime }=0.01V, t=0.1V^{\prime }$, and $n=1/2$ as a function of decreasing temperature. The DC resistivity is extracted from the peak height, which is strongly temperature dependent; see inset. The peak width is mildly temperature dependent. (b) The DC resistivity as a function of temperature for a few representative fillings. The low-temperature behavior depends sensitively on the degree of commensuration.

Figure 7

(a) ${\rho }_{DC}$ as a function of temperature for varying $V^{\prime }$ at half-filling [see inset of (d) for $V^{\prime }$ legend). The transition temperature to the insulating stripe phase is controlled by $V^{\prime }$. (b) ${\rho }_{DC}$ as a function of temperature for varying $V^{\prime }$ at $n=25/144$. There are no insulating transitions down to the lowest temperatures considered. Scaling collapse of the resistivity at (c) half-filling and (d) $n=25/144$.

Figure 8

Effects of finite size effects for different thermodynamic quantities as a function of filling and temperature.

Figure 9

Thermodynamic and transport coefficients that contribute to thermoelectric coupling. See text for definition of individual quantities.

Figure 10

Inverse eigenvalues of the diffusion matrix. Numerical errors prevent the accurate determination of these diffusion constants for dilute fillings and for low temperatures deep in the correlated regime.

Figure 11

Comparison between the exact solution by Wannier and MC calculation for a system size of $30\times 30$.