Direct Calculation of the Spin Stiffness in the $J_1$--$J_2$ Heisenberg Antiferromagnet

TL;DR

This paper computes the spin stiffness of the $J_1$--$J_2$ Heisenberg antiferromagnet using exact diagonalization, revealing a vanishing stiffness in the frustrated regime indicating a possible non-magnetic phase.

Contribution

It provides the first precise calculations of spin stiffness across the frustration parameter, highlighting the transition region where magnetic order disappears.

Findings

Spin stiffness matches known results in the unfrustrated case.

Stiffness vanishes for $0.4 \,\lesssim\, J_2/J_1 \lesssim 0.6$.

Finite-size scaling is unreliable in the intermediate frustration region.

Abstract

We calculate the spin stiffness $\rho_s$ for the frustrated spin-$\frac{1}{2}$ Heisenberg antiferromagnet on a square lattice by exact diagonalizations on finite clusters of up to $36$ sites followed by extrapolations to the thermodynamic limit. For the non-frustrated case, we find that $\rho_s = (0.183\pm 0.003)J_1$, in excellent agreement with the best results obtained by other means. Turning on frustration, the extrapolated stiffness vanishes for $0.4 \lesssim J_2/J_1 \lesssim 0.6$. In this intermediate region, the finite-size scaling works poorly -- an additional sign that their is neither N\'eel nor collinear magnetic order. Using a hydrodynamic relation, and previous results for the transverse susceptibility, we also estimate the spin-wave velocity in the N\'eel-ordered region.