Quantum Phase Transition in Coupled Spin Ladders

The ground state of an array of coupled, spin-half, antiferromagnetic ladders is studied using spin-wave theory, exact diagonalization (up to 36 sites) and quantum Monte Carlo techniques (up to 256 sites). Our results clearly indicate the occurrence of a zero-temperature phase transition between a N\'eel ordered and a non-magnetic phase at a finite value of the inter-ladder coupling ($\alpha_c\simeq0.3$). This transition is marked by remarkable changes in the structure of the excitation spectrum.Comment: 4 pages, 6 postscript figures, to appear in Physical Review

A Closed-Form Approximation of Likelihood Functions for Discretely Sampled Diffusions: the Exponent Expansion

In this paper we discuss a closed-form approximation of the likelihood functions of an arbitrary diffusion process. The approximation is based on an exponential ansatz of the transition probability for a finite time step $\Delta t$, and a series expansion of the deviation of its logarithm from that of a Gaussian distribution. Through this procedure, dubbed {\em exponent expansion}, the transition probability is obtained as a power series in $\Delta t$. This becomes asymptotically exact if an increasing number of terms is included, and provides remarkably accurate results even when truncated to the first few (say 3) terms. The coefficients of such expansion can be determined straightforwardly through a recursion, and involve simple one-dimensional integrals. We present several examples of financial interest, and we compare our results with the state-of-the-art approximation of discretely sampled diffusions [A\"it-Sahalia, {\it Journal of Finance} {\bf 54}, 1361 (1999)]. We find that the exponent expansion provides a similar accuracy in most of the cases, but a better behavior in the low-volatility regime. Furthermore the implementation of the present approach turns out to be simpler. Within the functional integration framework the exponent expansion allows one to obtain remarkably good approximations of the pricing kernels of financial derivatives. This is illustrated with the application to simple path-dependent interest rate derivatives. Finally we discuss how these results can also be used to increase the efficiency of numerical (both deterministic and stochastic) approaches to derivative pricing.Comment: 28 pages, 7 figure

Finite-size spin-wave theory of a collinear antiferromagnet

The ground-state and low-energy properties of the two-dimensional $J_1{-}J_2$ Heisenberg model in the collinear phase are investigated using finite-size spin-wave theory [Q. F. Zhong and S. Sorella, {\em Europhys. Lett.} {\bf 21}, 629 (1993)], and Lanczos exact diagonalizations. For spin one-half -- where the effects of quantization are the strongest -- the spin-wave expansion turns out to be quantitatively accurate for $J_2/J_1\gtrsim 0.8$. In this regime, both the magnetic structure factor and the spin susceptibility are very close to the spin-wave predictions. The spin-wave estimate of the order parameter in the collinear phase, $m^\dagger\simeq 0.3$, is in remarkable agreement with recent neutron scattering measurements on ${\rm Li_2VOSiO_4}$.Comment: 10 pages, 3 figure

Algorithmic differentiation and the calculation of forces by quantum Monte Carlo

We describe an efficient algorithm to compute forces in quantum Monte Carlo using adjoint algorithmic differentiation. This allows us to apply the space warp coordinate transformation in differential form, and compute all the 3M force components of a system with M atoms with a computational effort comparable with the one to obtain the total energy. Few examples illustrating the method for an electronic system containing several water molecules are presented. With the present technique, the calculation of finite-temperature thermodynamic properties of materials with quantum Monte Carlo will be feasible in the near future.Comment: 32 pages, 4 figure, to appear in The Journal of Chemical Physic

Effect of local charge fluctuations on spin physics in the Neel state of La2_2CuO4_4

We explore the effect of local charge fluctuations on the spin response of a Mott insulator by deriving an effective spin model, and studying it using Schwinger boson mean field theory. Applying this to La$_2$CuO$_4$, we show that an accurate fit to the magnon dispersion relation, measured by Coldea {\em et al.} [Phys. Rev. Lett. {\bf 86}, 5377 (2001)] is obtained with Hubbard model parameters $U \approx 2.34 eV$, and $t \approx 360 meV$. These parameters lead to estimates of the staggered magnetization ($m_s \approx 0.25$), spin wave velocity ($c\approx 800 meV$-\AA), and spin stiffness ($\rho_s \approx 24 meV$). In particular the staggered moment as well as the effective local moment are renormalized to smaller values compared to the Heisenberg model due to local charge fluctuations in the Hubbard model. The dynamical structure factor shows considerable weight in the continuum along the zone boundary as well as secondary peaks that may be observed in high resolution neutron scattering experiments.Comment: Manuscript considerably revised following referee comments. Also added a brief discussion of sum rules. 8 pages, 6 eps figure

Quantum Effects and Broken Symmetries in Frustrated Antiferromagnets

We investigate the interplay between frustration and zero-point quantum fluctuations in the ground state of the triangular and $J_1{-}J_2$ Heisenberg antiferromagnets, using finite-size spin-wave theory, exact diagonalization, and quantum Monte Carlo methods. In the triangular Heisenberg antiferromagnet, by performing a systematic size-scaling analysis, we have obtained strong evidences for a gapless spectrum and a finite value of the thermodynamic order parameter, thus confirming the existence of long-range N\'eel order.The good agreement between the finite-size spin-wave results and the exact and quantum Monte Carlo data also supports the reliability of the spin-wave expansion to describe both the ground state and the low-energy spin excitations of the triangular Heisenberg antiferromagnet. In the $J_1{-}J_2$ Heisenberg model, our results indicate the opening of a finite gap in the thermodynamic excitation spectrum at $J_2/J_1 \simeq 0.4$, marking the melting of the antiferromagnetic N\'eel order and the onset of a non-magnetic ground state. In order to characterize the nature of the latter quantum-disordered phase we have computed the susceptibilities for the most important crystal symmetry breaking operators. In the ordered phase the effectiveness of the spin-wave theory in reproducing the low-energy excitation spectrum suggests that the uniform spin susceptibility of the model is very close to the linear spin-wave prediction.Comment: Review article, 44 pages, 18 figures. See also PRL 87, 097201 (2001