We provide a detailed description of a general procedure by which a nano/micro-mechanical resonator can be calibrated using its thermal motion. A brief introduction to the equations of motion for such a resonator is presented, followed by a detailed derivation of the corresponding power spectral density (PSD) function. The effective masses for a number of different resonator geometries are determined using both finite element method (FEM) modeling and analytical calculations.

We critique a Padé analytic continuation method whereby a rational polynomial function is fit to a set of input points by means of a single matrix inversion. This procedure is accomplished to an extremely high accuracy using a novel symbolic computation algorithm. As an example of this method in action, it is applied to the problem of determining the spectral function of a single-particle thermal Green's function known only at a finite number of Matsubara frequencies with two example self energies drawn from the Tmatrix theory of the Hubbard model. We present a systematic analysis of the effects of error in the input points on the analytic continuation, and this leads us to propose a procedure to test quantitatively the reliability of the resulting continuation, thus eliminating the black magic label frequently attached to this procedure.

In a system with an even number of SU(2) spins, there is an overcomplete set of states--consisting of all possible pairings of the spins into valence bonds--that spans the S=0 Hilbert subspace. Operator expectation values in this basis are related to the properties of the closed loops that are formed by the overlap of valence bond states. We construct a generating function for spin correlation functions of arbitrary order and show that all nonvanishing contributions arise from configurations that are topologically irreducible. We derive explicit formulas for the correlation functions at second, fourth, and sixth order. We then extend the valence bond basis to include triplet bonds and discuss how to compute properties that are related to operators acting outside the singlet sector. These results are relevant to analytical calculations and to numerical valence bond simulations using quantum Monte Carlo, variational wavefunctions, or exact diagonalization.Comment: 22 pages, 14 figure

We use quantum Monte Carlo (stochastic series expansion) and finite-size scaling to study the quantum critical points of two S = 1/2 Heisenberg antiferromagnets in two dimensions: a bilayer and a Kondo-lattice-like system (incomplete bilayer), each with intra-and inter-plane couplings J and J ⊥ . We discuss the ground-state finite-size scaling properties of three different quantities-the Binder moment ratio, the spin stiffness, and the long-wavelength magnetic susceptibility-which we use to extract the critical value of the coupling ratio g = J ⊥ /J. The individual estimates of gc are consistent provided that subleading finite-size corrections are properly taken into account. In the case of the complete bilayer, the Binder ratio leads to the most precise estimate of the critical coupling, although the subleading finite-size corrections to the stiffness are considerably smaller. For the incomplete bilayer, the subleading corrections to the stiffness are extremely small, and this quantity then gives the best estimate of the critical point. Contrary to predictions, we do not find a universal prefactor of the ∼ 1/L spin stiffness scaling at the critical point, whereas the Binder ratio is consistent with a universal value. Our results for the critical coupling ratios are gc = 2.52181(3) (full bilayer) and gc = 1.38882(2) (incomplete bilayer), which represent improvements of two orders of magnitude relative to the previous best estimates. For the correlation length exponent we obtain ν = 0.7106(9), consistent with the expected 3D Heisenberg universality class.

A quantum phase transition is typically induced by tuning an external parameter that appears as a coupling constant in the Hamiltonian. Another route is to vary the global symmetry of the system, generalizing, e.g., SU(2) to SU(N ). In that case, however, the discrete nature of the control parameter prevents one from identifying and characterizing the transition. We show how this limitation can be overcome for the SU(N ) Heisenberg model with the help of a singlet projector algorithm that can treat N continuously. On the square lattice, we find a direct, continuous phase transition between Néel-ordered and crystalline bond-ordered phases at Nc = 4.57(5) with critical exponents z = 1 and β/ν = 0.81(3).