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

## TL;DR

This paper investigates the relaxation dynamics of quantum antiferromagnets using imaginary-time evolution and finite-size scaling, accurately determining the dynamic exponent and revealing differences between regular and diluted lattices.

## Contribution

The study introduces a finite-size scaling method to extract the dynamic exponent from quantum Monte Carlo data, improving previous estimates and analyzing symmetry restoration in different lattice geometries.

## Key findings

- For 2D square lattice, the dynamic exponent z=2.001(5), matching theoretical expectations.
- For site-diluted lattice at percolation threshold, z=3.90(1), refining earlier estimates.
- Scaling behavior differs between regular and diluted lattices, indicating different relaxation mechanisms.

## Abstract

We apply imaginary-time evolution, ${\rm e}^{-\tau H}$, to study relaxation dynamics of gapless quantum antiferromagnets described by the spin-rotation invariant Heisenberg Hamiltonian ($H$). Using quantum Monte Carlo simulations, we propagate an initial state with maximal order parameter $m^z_s$ (the staggered magnetization) in the $z$ spin direction and monitor the expectation value $\langle m^z_s\rangle $ as a function of the time $\tau$. Different system sizes of lengths $L$ exhibit an initial size-independent relaxation of $\langle m^z_s\rangle$ toward its value the spontaneously symmetry-broken state, followed by a size-dependent final decay to zero. We develop a generic finite-size scaling theory which shows that 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 way of extracting the dynamic exponent from the numerical finite-size data. We apply the method to spin-$1/2$ Heisenberg antiferromagnets on two different lattice geometries; the two-dimensional (2D) square lattice as well as a site-diluted 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-dilutes lattice we find $z=3.90(1)$. This is an improvement on previous estimates of $z\approx 3.7$. The scaling results also show a fundamental difference between the two cases: In the 2D system the data can be collapsed onto a common scaling function even when $\langle m^z_s\rangle$ is relatively large, reflecting the Anderson tower of quantum rotor states with a common dynamic exponent $z=2$. For the diluted lattice, the scaling works only for small $\langle m^z_s\rangle$, indicating a mixture of different relaxation time scaling between the low energy states.