Dynamic scaling of the restoration of rotational symmetry in Heisenberg quantum antiferromagnets
Sandvik, Anders W.
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CitationPhillip Weinberg, Anders W. Sandvik. 2017. "Dynamic scaling of the restoration of rotational symmetry in Heisenberg quantum antiferromagnets." Physical Review B, v. 96, Issue 5, 8 p.
We apply imaginary-time evolution with the operator e−τH 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 mzs (the staggered magnetization) in the z spin direction and monitor the expectation value ⟨ms⟩ as a function of imaginary-time τ. Results for different system sizes (lengths) L exhibit an initial essentially size-independent relaxation of ⟨ms⟩ toward its value in the infinite-size spontaneously symmetry-broken state, followed by a strongly size-dependent final decay to zero when the O(3) rotational symmetry of the order paraneter is restored. We develop a generic finite-size scaling theory that shows the relaxation time diverges asymptotically as Lz 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, systematcally 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 as well as 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-dilutes lattice we find z = 3.90(1) or z = 2.056(8)Df , where Df = 91/48 is the fractal dimensionality of the percolating system. This is an improvement on previous estimates of z ≈ 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 ⟨ms⟩ 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 ⟨ms⟩, indicating a mixture of different relaxation time scaling between the low energy states. Nevertheless, the low-energy dynamic here also corresponds to a tower of excitations.