Fermi-liquid-like phase driven by next-nearest-neighbor couplings in a lightly doped kagome-lattice $t$-$J$ model

TL;DR

This study tackles how next-nearest-neighbor couplings $t_2$ and $J_2$ influence charge order and metallic behavior in a lightly doped kagome $t$-$J$ model. Using unbiased DMRG on kagome YC6 and YC8 cylinders with $t_1/J_1=3$, the authors map a phase diagram showing a CDW phase at small $t_2,t_1$ and a robust Fermi-liquid-like phase stabilized by increasing $t_2$ and $J_2$, characterized by power-law decays of $G(r)$ and $D(r)$ and a central charge $c oughly 2$. They find no evidence of hole pairing in the FL-like phase, with $P_{aa}(r)$ decaying as a power with $ ext{K}_{ m sc}\, oughly 2.3$ and $n_{AA}(f k)$ featuring a small $oldsymbol{ m oldsymbol{\Gamma}}$-point pocket, while spin correlations acquire a three-sublattice structure. The FL-like state persists across a range of dopings on YC6 and YC8, suggesting possible stabilization in two dimensions, though superconductivity remains elusive within the studied parameter space. These results highlight the delicate balance between charge order, spin correlations, and doped carriers in frustrated kagome systems and guide future exploration of interaction terms that could promote pairing.

Abstract

Due to the interplay between charge fluctuation and geometry frustration, the doped kagome-lattice Mott insulator is a fascinating platform to realize exotic quantum states. Through the state-of-the-art density matrix renormalization group calculation, we explore the quantum phases of the lightly doped kagome-lattice $t$-$J$ model in the presence of the next-nearest-neighbor electron hopping $t_2$ and spin interaction $J_2$. On the $L_y = 3$ cylinder ($L_y$ is the number of unit cells along the circumference direction), we establish a quantum phase diagram with tuning $t_2 > 0$ and $J_2 > 0$, showing an emergent Fermi-liquid-like phase driven by increased $t_2$ and $J_2$, which sits at the neighbor of the previously identified charge density wave (CDW) phase. Compared with the CDW phase, the charge order is significantly suppressed in the Fermi-liquid-like phase, and most correlation functions are greatly enhanced with power-law decay. In particular, we find the absence of hole pairing and a strong three-sublattice magnetic correlation. On the wider $L_y = 4$ cylinder, this Fermi-liquid-like phase persists at low doping levels, strongly suggesting that this state might be stable in the two-dimensional kagome system.

Figures (15)

• Figure 1: Schematic figure of kagome-lattice $t$-$J$ model and quantum phase diagram of the YC6 system by tuning $t_2$ and $J_2$. (a) The studied $t$-$J$ model on the YC6 kagome cylinder, where the electrons and doped holes live at the vertices (solid circles). The model has both the nearest-neighbor and next-nearest-neighbor hoppings ($t_1$ and $t_2$), as well as corresponding spin exchange interactions ($J_1$ and $J_2$). The periodic boundary conditions and open boundary conditions are imposed, respectively, along the directions specified by the lattice vectors, ${\bf e}_2$ and ${\bf e}_1$. Each unit cell (denoted by the small triangle in the shaded region) has three sites ($A$, $B$, and $C$) and three bonds ($a$, $b$, and $c$). $L_x$ and $L_y$ denote the numbers of unit cells along the ${\bf e}_1$ and ${\bf e}_2$ directions, respectively. (b) Quantum phase diagram of the kagome-lattice $t$-$J$ model obtained in the parameter region $0 \leq t_2/t_1 \leq 0.7$ and $0 \leq J_2 /J_1 \leq 0.7$, at the given doping ratio $\delta = 1/18$ and $t_1/J_1 = 3$. Besides the charge density wave (CDW) phase identified previously kagome-tJ-Jiang-2017kagome-tJlike-PCheng-2021, we find a Fermi-liquid-like phase. The phase boundaries are determined by examining charge density profile and correlation functions. The green region indicates an intermediate region, in which most of physical quantities are similar to those in the Fermi-liquid-like phase but some quantities change slowly with tuning couplings.
• Figure 2: Charge density profile on the YC6 cylinder. The averaged charge density of the unit cell in each column $n_x$ is defined as $n_x = \frac{1}{3L_y} \sum_{y=1}^{L_y} \sum_{i=1}^{3} \langle \hat{n}_{x,y,i} \rangle$. (a) $L_x = 32$ cylinder with $t_2/t_1 = 0.2$, $J_2/J_1=0.04$, and $\delta = 1/18$ in the CDW phase. (b)-(d) show the results in the Fermi-liquid-like phase with $t_2/t_1 = 0.5$, $J_2/J_1=0.25$, and $\delta = 1/18$, $1/27$, and $1/36$ respectively. The blue lines in (a), (c), (d) are the fitting curves to the function $n_x = n_0 + A_\text{CDW} \cos(Qx + \phi)$, where $A_\text{CDW} = A_0 [x^{-K_c/2} + (L_x + 1 - x)^{-K_c/2}]$ and $Q$ are the CDW amplitude and wave vector, respectively. $\phi$ is a phase shift.
• Figure 3: Density correlation function $D(r)$ and single-particle Green’s function $G(r)$ on the YC6 cylinder. (a) $D(r)$ for different $t_2/t_1$ values along the line $(t_2/t_1)^2 = J_2/J_1$ with $\delta = 1/18$. $\xi_{cdw}$ and $K_{cdw}$ are the fitting exponents in exponential decay function and power-law decay function, respectively. (b) $D(r)$ at $t_2/t_1=0.5$ and $J_2/J_1=0.25$, for different bond dimensions in the range of $D = 6000-18000$. The dashed line denotes the power-law fitting of the extrapolated $D \rightarrow \infty$ results. (c) $D(r)$ at $t_2/t_1 = 0.5$ and $J_2/J_1=0.25$ for different doping levels and system sizes. (d)-(f) Similar plots for the single-particle Green’s function $G(r)$.
• Figure 4: Double-logarithmic plot of SC pairing correlation $P_{aa}$ on the YC6 cylinder. (a) $P_{aa}$ for different $t_2/t_1$ values along the line $(t_2/t_1)^2 = J_2/J_1$ with $\delta = 1/18$. (b) $P_{aa}$ for different doping levels and system sizes at $t_2/t_1 = 0.5$, $J_2/J_1=0.25$ and $\delta = 1/18$. (c) and (d) show the results for different bond dimensions at $\delta = 1/18$ for $t_2/t_1 = 0.5$, $J_2/J_1=0.25$, and $t_2/t_1 = 0.7$, $J_2/J_1=0.49$, respectively. (e) and (f) compare pairing correlation $P_{aa}$ with the the square of single-particle Green’s function $(G(r)/2)^2$ at $\delta = 1/18$ for $t_2/t_1 = 0.5$, $J_2/J_1=0.25$, and $t_2/t_1 = 0.7$, $J_2/J_1=0.49$, respectively.
• Figure 5: Spin correlation $S(r)$ on the YC6 cylinder. Semi-logarithmic plot of $S(r)$: (a) for different $t_2/t_1$ values along the line $(t_2/t_1)^2 = J_2/J_1$ at $\delta = 1/18$, and (b) for different doping levels and system sizes at $t_2/t_1 = 0.5$ and $J_2/J_1 = 0.25$. (c) Semi-logarithmic plot and (d) double-logarithmic plot of $S(r)$ for various bond dimensions in the range $D = 8000-18000$ at $\delta = 1/18$ for $t_2/t_1 = 0.5$ and $J_2/J_1=0.25$.