Haerter-Shastry kinetic magnetism and metallicity in the triangular Hubbard model

Abstract

The fermionic Hubbard model, when combined with the ingredient of frustration, associated with the breaking of particle-hole symmetry, harbors a rich phase diagram. Aspects of theoretical findings associated with the nature of magnetism and metallicity, in a diverse set of parameter regimes, are now being actively investigated in triangular Hubbard cold atom and solid-state (moiré) based emulators. Building on the theoretical work of Haerter and Shastry [Phys. Rev. Lett. 95,087202 (2005)], we explore the impact of kinetically frustrated magnetism, a phenomenon where antiferromagnetic order emerges without any underlying magnetic interactions, at finite hole density. We numerically study the infinite-$U$ triangular Hubbard model using the density matrix renormalization group algorithm and estimate the extent of stability of the kinetically induced $120^{\circ}$ antiferromagnetic state to hole doping. Beyond the Haerter-Shastry regime, we find an intermediate phase with multimer (involving multiple correlated spins) stripes that eventually gives way to a paramagnet. We also find evidence of gapless charge excitations (metallicity) throughout the phase diagram for finite hole density. We discuss the implications at large, but finite and realistic values of $U/t$, and investigate whether kinetic magnetism and superexchange collaborate or compete.

Figures (11)

• Figure 1: (a) Schematic of a single hole hopping on a triangular plaquette in the classical FM background via two available routes, represented by green and orange arrows. The hole's motion along the two paths interferes destructively. (b) The $2\times 2$ triangular unit cell with periodic boundary conditions (left) and the absolute value of ground state spin-spin correlations on the bonds for three eigensolutions for one hole, two up spins and one down spin (right). (c) Schematic showing available hopping pathways for the $U=0$ and $U=\infty$ models.
• Figure 2: (a) Ground state static spin structure factor $S(\vec{q})$, computed with method 1, of the 72-site (XC-$6$ cylinder of length $12$ restricted to sites in the bulk), triangular Hubbard model for $U/t = \infty$ across a representative sample of hole concentrations $n_{h}$. (b-c) Ground state momentum space fermionic occupation number of one spin species $n_{\sigma}(\vec{q})$ for (b) $U/t = \infty$ and (c) $U/t = 0$. (d) Bulk static spin structure factor at representative momentum space points [$\mathbf{K}$ , $\mathbf{M}_{1}$ and $\mathbf{M}_{2}$ shown in (a)] as a function of $n_h$. (e-f) Real space profile of spin correlations $\langle \vec{S_i}\cdot \vec{S_j}\rangle$ at (e) $n_h=1/4$ and (f) $n_h=1/3$. The red color variation on the bonds represents nearest-neighbor AFM correlations, while the color (red, negative and blue, positive) and size of the dots indicate the expectation value of the spin-spin correlations with respect to a reference site $\chi$.
• Figure 3: (a) Quasiparticle weight, versus hole density $n_h$ computed from the ground state wavefunction for three representative system sizes. (b) Ground state energies for $U/t=\infty$ (denoted by black dots), and $U/t=0$ (red dashed line) versus $n_h$, shown separately for two system sizes. The dotted and black dashed lines represent the $U/t=0$ ground state energy rescaled by $n_h$ and $Z$, respectively. (c) Ground state static charge structure factor $N(\vec{q})$ for $q_{y}=0$ on the 96-site (XC-$4$ of length $24$) ITHM for $n_{h}=0.1$. The linear and quadratic fits close to $q_x\rightarrow 0$ are shown. (d) Metallic weight, $\alpha$, derived numerically from the linear fits of the $N(\vec{q})$, and as defined in the text, versus $n_{h}$ for the same systems as in (a).
• Figure 4: Ground state observables for $n_{h}=0.11$ (within the HS-AFM phase) as a function of $t/U$. (a) Average spin correlation for nearest neighbor bonds of the triangular lattice, (b) average spin correlation for next-to-nearest neighbor bonds, (c) average spin structure factor $S(\vec{q})$ at ordering vector $\mathbf{K}=(4\pi/3,0)$, and (d) average onsite double occupancy.
• Figure 5: Examples of cylindrical lattices used in this study, with periodic boundary conditions applied along the vertical $y$-direction and open boundary conditions in the horizontal $x$- direction. The length 10 XC-6 cylinder does not support the $120^{\circ}$ AFM order, and hence does not accommodate the $\bf K$ and symmetry related points.