Gutzwiller-projected states for the $J_1$-$J_2$ Heisenberg model on the kagome lattice: achievements and pitfalls

TL;DR

This study evaluates the effectiveness of Gutzwiller-projected fermionic wave functions in modeling the ground states of the $J_1$-$J_2$ Heisenberg model on the kagome lattice, revealing phase transitions and limitations of the approach.

Contribution

It demonstrates how the Gutzwiller-projected fermionic wave functions can accurately describe the phase diagram, including magnetic and spin liquid states, and identifies their limitations on small systems.

Findings

Small antiferromagnetic $J_2$ stabilizes the gapless spin liquid.

First-order transition at $J_2/J_1=0.11$ from spin liquid to magnetic order.

Evidence of a first-order transition at $J_2/J_1=-0.065$ in the ferromagnetic regime.

Abstract

We assess the ground-state phase diagram of the $J_1$-$J_2$ Heisenberg model on the kagome lattice by employing Gutzwiller-projected fermionic wave functions. Within this framework, different states can be represented, defined by distinct unprojected fermionic Hamiltonians that include hopping and pairing terms, as well as a coupling to local Zeeman fields to generate magnetic order. For $J_2=0$, the so-called U(1) Dirac state, in which only hopping is present (such as to generate a $\pi$-flux in the hexagons), has been shown to accurately describe the exact ground state [Y. Iqbal, F. Becca, S. Sorella, and D. Poilblanc, Phys. Rev. B 87, 060405 (2013); Y.-C. He, M. P. Zaletel, M. Oshikawa, and F. Pollmann, Phys. Rev. X 7, 031020 (2017)]. Here, we show that its accuracy improves in presence of a small antiferromagnetic super-exchange $J_2$, leading to a finite region where the gapless spin liquid is stable; then, for $J_2/J_1=0.11(1)$, a first-order transition to a magnetic phase with pitch vector ${\bf q}=(0,0)$ is detected, by allowing magnetic order within the fermionic Hamiltonian. Instead, for small ferromagnetic values of $|J_2|/J_1$, the situation is more contradictory. While the U(1) Dirac state remains stable against several perturbations in the fermionic part (i.e., dimerization patterns or chiral terms), its accuracy clearly deteriorates on small systems, most notably on $36$ sites where exact diagonalization is possible. Then, upon increasing the ratio $|J_2|/J_1$, a magnetically ordered state with $\sqrt{3} \times \sqrt{3}$ periodicity eventually overcomes the U(1) Dirac spin liquid. Within the ferromagnetic $J_{2}$ regime, evidence is shown in favor of a first-order transition at $J_2/J_1=-0.065(5)$.