Functional techniques in quantum field theory and two-dimensional models

Paulo F. Bedaque, Hersh Kumar, Suryansh Rajawat +1 more

We present a variational approach to quantum field theory based on wavefunctions parameterized by neural networks. While variational methods have a celebrated history across many fields, their application to quantum field theory has been limited by well-known challenges. We show that neural-network wavefunctions, combined with modern machine-learning techniques, enable competitive variational calculations in nontrivial field theories. As a demonstration, we reproduce the essential features of the two-dimensional nonlinear $σ$-model: asymptotic freedom, dynamical mass generation and the model's step-scaling function.

Nima Moshayedi

Khavtgain Namsrai

Chuxin Chen, Qiang Luo, N. E. Bickers

A variety of self-consistent-field techniques have recently been used to calculate correlation functions of the two-dimensional Hubbard model. Such techniques extend Hartree–Fock theory by including a large class of fluctuation processes. The computational requirements of the self-consistent-field approach decrease in lower dimensional systems. For this reason, a thorough test of techniques may be performed by examining the Anderson impurity model, the zero-dimensional limit of the Hubbard model. Using the fluctuation exchange and pseudopotential parquet approximations applied previously in two dimensions, we calculate static and dynamic correlation functions for the Anderson model and compare with quantum Monte Carlo results.

Barry M. McCoy, Craig A. Tracy, Tai Tsun Wu

For both $T>{T}_{c}$ and $T<{T}_{c}$, we report exact expressions for the $n$-spin correlation function of the two-dimensional Ising model suitable for studying large separations between spins. In particular, the scaling limit of these correlation functions can be shown to exist and yields $n$-point Schwinger functions of some relativistic quantum field theories.

Allen C. Hirshfeld, Hajo Leschke, Tsuneo Suzuki