Emergent Fermi-liquid-like phase by melting a holon Wigner crystal in a doped Mott insulator on the kagome lattice

TL;DR

The paper investigates how doping a kagome-lattice quantum spin liquid in the $t$-$J$ model evolves as holes are introduced, using large-scale DMRG on YC6 and YC8 cylinders across $δ$ from $0.027$ to $0.36$. It identifies a transition from charge-density-wave holon crystals at low $δ$ to an emergent Fermi-liquid-like phase around $δ≈0.15$, evidenced by uniform charge density, algebraic correlations, and a finite central charge $c$ in a gapless state. In a narrow near-$1/3$ doping window, a state with exponential $G(r)$ but enhanced pairing relative to $G^2$ appears on YC6, suggesting a superconductivity precursor, though this behavior is not robust on YC8, indicating lattice-size dependence. The results illuminate a nontrivial melting pathway from holon-Wigner crystallization to a metallic state and point to a possible doping range for superconductivity in doped kagome Mott insulators, with implications for understanding exotic metallic states in frustrated quantum magnets.

Abstract

The doped quantum spin liquid on the kagome lattice provides a fascinating platform to explore exotic quantum states, such as the reported holon Wigner crystal at low doping. By extending the doping range to $δ= 0.027$ - $0.36$, we study the kagome-lattice $t$-$J$ model using the state-of-the-art density matrix renormalization group calculation. 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 increasing doping level. In addition to the charge density wave (CDW) states at lower doping, we find an emergent Fermi-liquid-like phase by melting the holon Wigner crystal at $δ\approx 0.15$, which is characterized by suppression of charge density oscillation and power-law decay of various correlation functions. On the wider $L_y = 4$ cylinder, the bond-dimension extrapolated correlation functions also support such a Fermi-liquid-like state, suggesting its stability with increasing system size. In a narrow doping range near $δ= 1/3$ on the $L_y = 3$ cylinder, we find a state with an exponential decay of single-particle correlation but the other correlation functions preserving the features in the Fermi-liquid-like phase, which may be a precursor of a superconducting state. Nevertheless, this peculiar state near $δ= 1/3$ disappears on the $L_y = 4$ cylinder, implying a possible lattice size dependence. Our results reveal a quantum melting from a holon Wigner crystal to a Fermi-liquid-like state with increasing hole density, and suggest a doping regime to explore superconductivity for future study.

Figures (9)

• Figure 1: Schematic figure of kagome-lattice $t$-$J$ model and quantum phase diagram of the YC6 system with increasing doping ratio $\delta = 0.027 - 0.36$. (a) The $t$-$J$ model on the kagome Y-cylinder (YC), where the periodic and open boundary conditions are imposed, respectively, along the directions specified by the lattice vectors ${\bf e}_2$ and ${\bf e}_1$. The electrons and doped holes reside at lattice vertices (solid circles). 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 $L_y$ are the numbers of unit cells along the ${\bf e}_1$ and ${\bf e}_2$ directions, respectively. Note that the column and row indices (x, y) will be used later to specify the coordinates of individual lattice sites. In the example illustrated in the figure, sites $A$ and $B$ correspond to coordinates $(3, 3)$ and $(4, 3)$, respectively. (b) Quantum phase diagram of the kagome-lattice $t$-$J$ model ($t/J = 3$) in the YC6 ($L_y = 3$) system. Besides the charge density wave (CDW) states observed previously kagome-tJ-Jiang-2017kagome-tJlike-PCheng-2021, we identify a new Fermi-liquid-like phase. In a narrow regime near $1/3$ doping, the state shows an exponential decay of single-particle correlation but the other correlation functions remain the features in the Fermi-liquid-like phase.
• Figure 2: Charge density profiles in the YC6 systems with $L_x=32$. $n(x,y)$ denotes the electron density at each lattice site, where $x$ ($y$) is the lattice index along the ${\bf e}_1$ (${\bf e}_2$) direction with the NN distance as the unit of length. The column-averaged electron density is denoted as $n_x$. (a) The hole density distribution $n_h(x,y) = 1 - n(x,y)$ for $\delta \approx 0.11$ in the CDW phase. (b)-(d) $n_x$ in the Fermi-liquid-like phase at $\delta \approx 0.16$, $0.20$, and $0.24$, respectively.
• Figure 3: SC pairing correlations and single-particle correlation $G(r)$ in the YC6 systems. (a) $|P_{aa}|$ for different doping levels. $\xi_{\rm sc}$ and $K_{\rm sc}$ are the fitted exponents in exponential and power-law decay, respectively. (b) $|P_{aa}|$ for $\delta \approx 0.24$ with different bond dimensions in the range of $D = 8000 - 28000$. The dashed line denotes the power-law fitting of the extrapolated $D \rightarrow \infty$ results. (c) Pairing correlation functions between different bonds $|P_{\alpha \beta}(r)|$ at $\delta \approx 0.24$. (d) and (e) are the similar plots for the single-particle Green’s function $G(r)$. (f) Comparison of pairing correlation $|P_{aa}|$ with the square of single-particle correlation $|G(r)/2|^2$ at $\delta \approx 0.24$.
• Figure 4: Density correlation $D(r)$, spin correlation $S(r)$, and entanglement entropy $S_E$ in the YC6 systems with $L_x=32$. (a) $|D(r)|$ for different doping levels. (b) $|D(r)|$ for $\delta \approx 0.24$ at different bond dimensions $D = 8000-28000$. (c) and (d) are the similar plots for the spin correlation $S(r)$. (e) Entanglement entropy for $\delta \approx 0.24$ at different bond dimensions. The $D = \infty$ results are extrapolated from the finite bond-dimension data. The label $l$ denotes the unit cell index along the ${\bf e}_1$ direction. (f) Fitting of the central charge $c$ from the formula $S_E(l) = (c/6)\ln \left[(L_x/\pi) \sin \left(l \pi/L_x \right) \right] + g$ for the $D = 28000$ and $D = \infty$ data, giving a finite central charge.
• Figure 5: Single-particle Green's function $G(r)$ and spin correlation $S(r)$ in the YC8 systems. (a) Double-logarithmic plot and (b) semi-logarithmic plot of $G(r)$ for $\delta\approx0.24$ on the YC8 cylinder, with the bond dimensions $D = 12000-20000$. (c) and (d) Similar plots of spin correlation $S(r)$. (e) and (f) Comparisons of $G(r)$ and $S(r)$ between YC6 and YC8 cylinders at $\delta \approx 0.24$. (g) and (h) Comparisons of $G(r)$ and $S(r)$ at $\delta \approx 0.11$ and $\delta \approx 0.24$ on the YC8 cylinder.