Dynamic scaling of the restoration of rotational symmetry in Heisenberg quantum antiferromagnets

Abstract

We apply imaginary-time evolution with the operator ${e}^{\ensuremath{-}\ensuremath{\tau}H}$ to study relaxation dynamics of gapless quantum antiferromagnets described by the spin-rotation-invariant Heisenberg Hamiltonian $H$. Using quantum Monte Carlo simulations to obtain unbiased results, we propagate an initial state with maximal order parameter ${m}_{s}^{z}$ (the staggered magnetization) in the $z$ spin direction and monitor the expectation value $\ensuremath{\langle}{m}_{s}\ensuremath{\rangle}$ as a function of imaginary time $\ensuremath{\tau}$. Results for different system sizes (lengths) $L$ exhibit an initial essentially size independent relaxation of $\ensuremath{\langle}{m}_{s}\ensuremath{\rangle}$ toward its value in the infinite-size spontaneously symmetry broken state, followed by a strongly size dependent final decay to zero when the $\mathrm{O}(3)$ rotational symmetry of the order parameter is restored. We develop a generic finite-size scaling theory that shows the relaxation time diverges asymptotically as ${L}^{z}$, where $z$ is the dynamic exponent of the low-energy excitations. We use the scaling theory to develop a practical way of extracting the dynamic exponent from the numerical finite-size data, systematically eliminating scaling corrections. We apply the method to spin-$1/2$ Heisenberg antiferromagnets on two different lattice geometries: the standard two-dimensional (2D) square lattice and a site-diluted 2D square lattice at the percolation threshold. In the 2D case we obtain $z=2.001(5)$, which is consistent with the known value $z=2$, while for the site-diluted lattice we find $z=3.90(1)$ or $z=2.056(8){D}_{f}$, where ${D}_{f}=91/48$ is the fractal dimensionality of the percolating system. This is an improvement on previous estimates of $z\ensuremath{\approx}3.7$. The scaling results also show a fundamental difference between the two cases; for the 2D square lattice, the data can be collapsed onto a common scaling function even when $\ensuremath{\langle}{m}_{s}\ensuremath{\rangle}$ is relatively large, reflecting the Anderson tower of quantum rotor states with a common dynamic exponent $z=2$. For the diluted 2D square lattice, the scaling works well only for small $\ensuremath{\langle}{m}_{s}\ensuremath{\rangle}$, indicating a mixture of different relaxation-time scalings between the low-energy states. Nevertheless, the low-energy dynamic here also corresponds to a tower of excitations.