Holon Wigner Crystal in a Lightly Doped Kagome Quantum Spin Liquid

We address the problem of a lightly doped spin liquid through a large-scale density-matrix renormalization group study of the $t\text{−}J$ model on a kagome lattice with a small but nonzero concentration $\delta$ of doped holes. It is now widely accepted that the undoped ($\delta =0$) spin-$1/2$ Heisenberg antiferromagnet has a spin-liquid ground state. Theoretical arguments have been presented that light doping of such a spin liquid could give rise to a high temperature superconductor or an exotic topological Fermi liquid metal. Instead, we infer that the doped holes form an insulating charge-density wave state with one doped hole per unit cell, i.e., a Wigner crystal. Spin correlations remain short ranged, as in the spin-liquid parent state, from which we infer that the state is a crystal of spinless holons, rather than of holes. Our results may be relevant to kagome lattice herbertsmithite upon doping.

Figure 1

The $t\text{−}J$ model on a kagome cylinder, where the electrons live at the vertices (solid circles). Periodic (PBC) and open (OBC) boundary conditions are imposed, respectively, along the directions specified by the lattice basis vectors, $e_2$ and $e_1$. Each basis (denoted by small triangle in the shaded region) has three sites ($A$, $B$, and $C$) and three bonds ($a$,$b$, and $c$). $t$ and $J$ are hopping integral and spin exchange interactions between NN sites. $L_x$ and $L_y$ are the number of unit cells in the $e_1$ and $e_2$ directions.

Figure 2

The spin-spin correlation functions $F(r)$ along the ${\mathbf{e}}_1$ direction for (a) YC-6 and (b) YC-8 cylinders at different hole doping concentrations $\delta .r$ is the distance between two sites where the reference $A$ site is in the middle of the system. Inset: The spin-spin correlation length ${\xi }_s$ as a function of $\delta$, by fitting $F(r)$ to an exponential function $F(r)\sim e^{-r/{\xi }_s}$. Here $L_x=16–24$.

Figure 3

The superconducting pair-field correlation functions ${\mathrm{\Phi }}_{aa}(r)$ along the ${\mathbf{e}}_1$ direction between ($a─a$) bonds for (a) YC-6 and (b) YC-8 cylinders at different hole doping concentrations $\delta .r$ is the distance between two bonds with the reference bond in the middle of the system. (c) The superconducting correlation length ${\xi }_{sc}$ as a function of $\delta$, by fitting $\mathrm{\Phi }(r)$ to an exponential function $\mathrm{\Phi }(r)\sim e^{-r/{\xi }_{sc}}$. Here $L_x=16–24$.

Figure 4

The charge density profile $n_h(x,y)$ at different hole doping concentrations $\delta$ for YC-6 in (a)–(b), YC-8 in (c)–(d), and YC-10 cylinders in (e).

Figure 5

The charge density profile $n_h(x)$ (includes both $A$ and $B$ sites) at different hole doping concentration $\delta$ for YC-6 in (a)–(c) and YC-8 cylinders in (d)–(e). (f) CDW order parameter $A_{\mathrm{CDW}}$ by fitting to $n_h(x)$ using function $n_h(x)=A_{\mathrm{CDW}}\cos (Q_{\mathrm{CDW}}x+\theta )+\cdots$, where $Q_{\mathrm{CDW}}$ is the ordering wave vector.