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

Abstract

We apply imaginary-time evolution, ${\mathrm{e}}^{-\tau H}$, 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 $m_s^z$ (the staggered magnetization) in the $z$ spin direction and monitor the expectation value $⟨m_s^z⟩$ as a function of the time $\tau$. Different system sizes of lengths $L$ exhibit an initial size-independent relaxation of $⟨m_s^z⟩$ 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 $⟨m_s^z⟩$ 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 $⟨m_s^z⟩$, indicating a mixture of different relaxation time scaling between the low energy states.

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Cited by in corpus (5)

• Solution of the "sign problem" for the half filled Hubbard-Holstein model
• Nonequilibrium dynamics in deconfined quantum critical point revealed by imaginary-time evolution
• Universal short time quantum critical dynamics of finite size systems
• Benchmark study of an auxiliary-field quantum Monte Carlo technique for the Hubbard model with shifted-discrete Hubbard-Stratonovich transformations
• Dual dynamic scaling in deconfined quantum criticality