ar X iv : h ep - l at / 0 40 80 41 v 1 2 5 A ug 2 00 4 Finite Size Scaling , Fisher Zeroes and N = 4 Super Yang - Mills

We investigate critical slowing down in the local updating continuous-time Quantum Monte Carlo method by relating the finite size scaling of Fisher Zeroes to the dynamically generated gap, through the scaling of their respective critical exponents. As we comment, the nonlinear sigma model representation derived through the hamiltonian of our lattice spin model can also be used to give a effective treatment of planar anomalous dimensions in N =4 SYM. We present scaling arguments from our FSS analysis to discuss quantum corrections and recent 2-loop results, and further comment on the prospects of extending this approach for calculating higher twist parton distributions. We investigate the critical behavior of Valence-Bond-Solid (VBS) states in quantum spin chains by means of Quantum Monte Carlo (QMC) simulation using the continuous-time loop cluster algorithm [1]. Following Haldane's conjecture a variety of exotic magnetic phenomena can be attributed to the formation of ground states with an energy-gap in integer spin systems at low temperatures. These gapped states are predicted to have transitions with a massless excitation between phases, and can be effectively expressed in terms of a VBS state (spin-singlet) picture. The Lieb-Schultz-Mattis argument to this conjecture has led to the recent introduction of a string exact order parameter to characterise these VBS transitions. We investigate the Finite Size Scaling (FSS) properties of a generalised twist order parameter for a periodic mixed-spin chain [2] with a unit cell of the form (1, 1, 1 2 , 1 2). Also determining the FSS properties of the complex-temperature (Fisher) zeroes of the partition function [3], evaluated in a new scheme through knowledge of the QMC transfer matrix. This enables us to separate pseudocritical and critical point scaling behavior relating to the correlation length exponent and also to locate the VBS state transition points through an independent and expeditious means. Leading corrections to the FSS of the zeroes are sa sb sb sa Jab Jbb Jab Jaa Figure 1. Periodic mixed spin cell, [4]. also known exactly for comparable models such as the 2D Ising model with Brascamp-Kunz boundary value conditions [5]. We compare the FSS of these indicators to evaluate the critical exponents for the VBS transitions, comparing with nonlin-ear σ-model predictions.