D.S. Sivia

Applications of Maximum Entropy and Bayesian Methods in Computational Many-Body Physics

Computer simulations of many-body systems can present difficult data analysis problems. An example is the determination of dynamical properties from quantum Monte Carlo (QMC) simulations. Data on imaginary time Matsubara Greens’ functions implicitly contain information about real time behavior. However, the spectral representation which relates imaginary time to real time is similar to a Laplace transform, and the numerical inversion of such transforms is notoriously unstable in the presence of statistical noise. Another example is the determination of the densities of states and the thermodynamic functions from imprecise knowledge of a finite number of moments. Data analysis problems in computer simulations are similar in many respects to those found in experimental research. They are often inverse problems, they can be ill-posed, the data may be incomplete, and they may be subject to statistical and systematic errors. Fortunately, the recent conceptual and algorithmic advances in maximum entropy (Max-Ent) and Bayesian data analysis methods1 are equally applicable to computer simulations. We illustrate such applications by calculating the dynamical properties of the Anderson model for dilute magnetic alloys and the densities of states of the 2-D Heisenberg model.

Applications of Maximum Entropy and Bayesian Methods in Neutron Scattering

We report on the use of Maximum Entropy (MaxEnt) and Bayesian methods applied to problems in neutron scattering at Los Alamos over the past year. Although the first applications were straight-forward deconvolutions, the work has been extended to make routine use of multi-channel entropy to additionally determine (broad) unknown backgrounds. A more exotic example of the use of MaxEnt involves the study of aggregation in a biological sample using Fourier-like data from small angle neutron scattering. We have also been considering the question of how to optimise instrumental hardware, leading to the derivation of better “figures-of-merit” for spectrometers and moderators, which may result in a far-reaching revision of ideas on the design of neutron scattering facilities.

Applications of Maxent to Quantum Monte Carlo

We consider the application of maximum entropy methods to the analysis of data produced by computer simulations. The focus is the calculation of the dynamical properties of quantum many-body systems by Monte Carlo methods, which is termed the “Analytic Continuation Problem.” For the Anderson model of dilute magnetic impurities in metals, we obtain spectral functions and transport coefficients which obey “Kondo Universality.”

Dynamics of the Anderson Model for Dilute Magnetic Alloys: A Quantum Monte Carlo and Maximum Entropy Study

The single-impurity Anderson model1 was invented nearly thirty years ago to describe dilute magnetic impurities in metallic hosts, including the formation and properties of itinerant local moments in alloys. The model is a prerequisite for the understanding of mixed valent and heavy-fermion phenomena.2 A limit of the model is the famous Kondo model3 which describes resistivity minima and saturation. The model is simple to state and its properties are readily measureable in the laboratory.4 However, the solution of the model is a difficult many-body problem. In recent years considerable progress in understanding the static properties of the model has been achieved using non-perturbative methods such as the Bethe ansatz,5 the renormalization group,6 and quantum Monte Carlo (QMC).7 Nevertheless, the understanding of the dynamical properties has remained elusive. These include the spectral density of the impurity state, the transport coeffecients, and the dynamical magnetic susceptibility. The spectral density has been calculated reliably only for large orbital degeneracy8 or for expansion parameters below the range of most physical interest.9 It has been obtained only at zero temperature by the renormalization group method,10 and it is extremely difficult to calculate reliably from QMC.11-15