Electron correlation effects and spin-liquid state in the Herbertsmithite Kagome lattice

TL;DR

This work tackles electron correlation and the potential quantum spin-liquid state in the kagome-lattice material $Zn_xCu_{4-x}O_6$ (Herbertsmithite) using real-space variational Monte Carlo with RVB-based wave functions. By comparing three wave-function ansatzes ($JDFT$, $JSD$, and $JAGP$) and two Jastrow forms, the authors quantify static and dynamic correlations across doping levels $x=0,\frac{1}{3},\frac{2}{3},1$, highlighting the critical role of full wave-function optimization in capturing strong correlations. The results show correlation energy grows with Zn doping and is amplified by long-range Jastrow tails, supporting a spin-liquid state stabilized by long-range correlations rather than local pairing; the RVB pairing energy in the CuO$_2$ plane is small and short-ranged, with finite-size effects evident. Overall, the study demonstrates that comprehensive optimization within the JAGP framework is essential for accurately describing strongly correlated kagome systems and provides evidence for a spin-liquid state in Herbertsmithite driven by long-range electronic correlations.

Abstract

We employ real-space variational quantum Monte Carlo methods with resonating valence bond many-body wave functions to investigate electron correlation effects in the Kagome system $Zn_xCu_{4-x}O_6$. Using three trial wave functions of the Slater-Jastrow type, where (i) only the Jastrow correlation factor is optimized and the orbitals obtained by density functional theory, (ii) both the Jastrow factor and the Slater determinant are optimized, and (iii) additionally the Slater determinant is substituted by an antisymmetrized-geminal power wave function, we analyze static and dynamic correlation energies across concentrations $x=0, \frac{1}{3}, \frac{2}{3}, 1$. Our results show that the correlation energy increases with the concentration of $Zn_x$. Optimizing the Slater determinant significantly enhances the correlation energy by approximately $\sim -73(4) mHa$ per electron. Eventually, the emergence of a quantum spin liquid state driven by a long-range correlation energy is discussed.

Figures (5)

• Figure 1: (Top panel) Crystal structure of Herbertsmithite $ZnCu_3(OH)_6Cl_2$. Zinc atoms are black, copper blue, and oxygen atoms are red. Chlorine and hydrogen atoms are not shown for clarity. There are three sites between $CuO_2$ planes along the $c$ axis which we considered to be occupied by $Cu$ or $Zn$. (Bottom panel) Kagome lattice structure of two-dimensional $CuO_2$ plane.
• Figure 2: The difference between VMC optimization energy and DFT ($\Delta E$) as a function of the number of energy minimisation steps for $Zn_xCu_{4-x}O_6$ system obtained by three different wave functions of $J_{u_i}^{cof}DFT$, $J_{u_i}DFT$, $J_{u_i}SD$ where $i=C, F$ represent two different forms for two-body Jastrow term as explained in the text. Only the Jastrow coefficients and $a$, and $b$ parameters of two-body $u$ term were optimised in $J_{u_i}^{cof}DFT$ wave function. The Jastrow terms are fully optimised, both exponents and coefficients, in $J_{u_i}DFT$. Jastrow and Slater determinant are optimised simultaneously in $J_{u_i}SD$ wave function. The results are obtained for $Zn_x$ concentrations of $x=0., 1/3, 2/3, 1.$.
• Figure 3: The difference between VMC and DFT energies as a function of $Zn_x$ concentration in $Zn_xCu_{4-x}O_6$ obtained by $J_{u_i}DFT$ (left panel) and $J_{u_i}SD$ (right panel) wave functions where $i=F,C$.
• Figure 4: The VMC correlation energies as a function of $Zn_x$ concentration in $Zn_xCu_{4-x}O_6$ system. (left panel) VMC energies were calculated using two forms of 2b-Jastrow $u_C$ and $u_F$ where in both wave functions the DFT-Slater determinant was not optimised. (right panel) The 2b-Jastrow term is the same as the left panel but the single-particle Slater determinant and the Jastrow were optimised simultaneously.
• Figure 5: The difference between VMC energy minimization and DFT ($\Delta E$) as a function of the number of optimization steps for the two-dimensional $CuO_2$ layer. Two system sizes with the number of electrons per simulation cell of $N_{el}=150$ (top panel) and $N_{el}=300$ (bottom panel) were used. The VMC energies were obtained with three trial wave functions of $JDFT$, $JSD$, and $JAGP$ as explained in the text.