Comment on ``Quantum Phase Transition of the Randomly Diluted Heisenberg Antiferromagnet on a Square Lattice''

In Phys. Rev. Lett. 84, 4204 (2000) (cond-mat/9905379), Kato et al. presented quantum Monte Carlo results indicating that the critical concentration of random non-magnetic sites in the two-dimensional antiferromagnetic Heisenberg model equals the classical percolation density; pc=0.407254. The data also suggested a surprising dependence of the critical exponents on the spin S of the magnetic sites, with a gradual approach to the classical percolation exponents as S goes to infinity. I here argue that the exponents in fact are S-independent and equal to those of classical percolation. The apparent S-dependent behavior found by Kato et al. is due to temperature effects in the simulations as well as a quantum effect that masks the true asymptotic scaling behavior for small lattices.Comment: Comment on Phys. Rev. Lett. 84, 4204 (2000), by K. Kato et al.; 1 page, 1 figur

Finite-size scaling and boundary effects in two-dimensional valence-bond-solids

Various lattice geometries and boundaries are used to investigate valence-bond-solid (VBS) ordering in the ground state of an S=1/2 square-lattice quantum spin model---the J-Q model, in which 4- or 6-spin interactions Q are added to the Heisenberg exchange J. Ground state results for finite systems (with up to thousands of spins) are obtained using a projector QMC method. Great care has to be taken when extrapolating the order parameter to infinite size, in particular in cylinder geometry. Even though strong 2D VBS order exists and is established clearly with increasing system size on L*L lattices (or Lx* Ly lattices with a fixed Lx/Ly), only short-range VBS correlations are observed on long cylinders (when Lx -> infinity at fixed Ly). The correlation length increases with Ly, until long-range order sets in at a "critical" Ly. This width is large even when the 2D order is strong, e.g, for a system where the order parameter is 70% of the largest possible value, Ly=8 is required for ordering. Correlation functions for small L*L lattices can also be misleading. For a 20%-ordered system results for L up to 20 appear to extrapolate to a vanishing order parameter, while for larger L the behavior crosses over and extrapolates to a non-zero value (with exponentially small finite size corrections). The VBS order also exhibits interesting edge effects related to emergent U(1) symmetry, which, if not considered properly, can lead to wrong conclusions for the thermodynamic limit. The finite-size behavior for small L*L lattices and long cylinders is similar to that predicted for a Z2 spin liquid. The results raise concerns about recent works claiming Z2 spin liquid ground states in frustrated 2D systems, in particular, the Heisenberg model with nearest and next-nearest-neighbor couplings. VBS state in this system cannot be ruled out.Comment: 26 pages, 28 figures. v2: final, published versio

Optimal taxation and normalisations

There still seems to be some confusion about the consequences of normalisations in the optimal taxation litterature. We claim that: 1) Normalisations do not matter for the real solution of optimal taxation problem. 2) Normalisations do matter for good characterisations of the solutions to optimal taxation problems. Whereas the first point is uncontroversial, the second one is less well understood. By postponing the normalisation of consumer prices, we detail how normalisations of consumer prices affect the characterisation of optimal commodity taxes, derive the preferred characterisation, and show how it depends on the normalisation.Optimal taxation; normalisation; marginal cost of public funds.

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

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