Quantum Monte Carlo in the Interaction Representation --- Application to a Spin-Peierls Model

A quantum Monte Carlo algorithm is constructed starting from the standard perturbation expansion in the interaction representation. The resulting configuration space is strongly related to that of the Stochastic Series Expansion (SSE) method, which is based on a direct power series expansion of exp(-beta*H). Sampling procedures previously developed for the SSE method can therefore be used also in the interaction representation formulation. The new method is first tested on the S=1/2 Heisenberg chain. Then, as an application to a model of great current interest, a Heisenberg chain including phonon degrees of freedom is studied. Einstein phonons are coupled to the spins via a linear modulation of the nearest-neighbor exchange. The simulation algorithm is implemented in the phonon occupation number basis, without Hilbert space truncations, and is exact. Results are presented for the magnetic properties of the system in a wide temperature regime, including the T-->0 limit where the chain undergoes a spin-Peierls transition. Some aspects of the phonon dynamics are also discussed. The results suggest that the effects of dynamic phonons in spin-Peierls compounds such as GeCuO3 and NaV2O5 must be included in order to obtain a correct quantitative description of their magnetic properties, both above and below the dimerization temperature.

From Parallel Sequence Representations to Calligraphic Control: A Conspiracy of Neural Circuits

Calligraphic writing presents a rich set of challenges to the human movement control system. These challenges include: initial learning, and recall from memory, of prescribed stroke sequences; critical timing of stroke onsets and durations; fine control of grip and contact forces; and letter-form invariance under voluntary size scaling, which entails fine control of stroke direction and amplitude during recruitment and derecruitment of musculoskeletal degrees of freedom. Experimental and computational studies in behavioral neuroscience have made rapid progress toward explaining the learning, planning and contTOl exercised in tasks that share features with calligraphic writing and drawing. This article summarizes computational neuroscience models and related neurobiological data that reveal critical operations spanning from parallel sequence representations to fine force control. Part one addresses stroke sequencing. It treats competitive queuing (CQ) models of sequence representation, performance, learning, and recall. Part two addresses letter size scaling and motor equivalence. It treats cursive handwriting models together with models in which sensory-motor tmnsformations are performed by circuits that learn inverse differential kinematic mappings. Part three addresses fine-grained control of timing and transient forces, by treating circuit models that learn to solve inverse dynamics problems.

A Network for Learning Kinematics with Application to Human Reaching Models

A model for self-organization of the coordinate transformations required for spatial reaching is presented. During a motor babbling phase, a mapping from spatial coordinate directions to joint motion directions is learned. After learning, the model is able to produce straight-line spatial velocity trajectories with characteristic bell-shaped spatial velocity profiles, as observed in human reaches. Simulation results are presented for transverse plane reaching using a two degree-of-freedom arm.