Thermodynamic integration, fermion sign problem, and real-space renormalization

TL;DR

This paper revisits Wilson's real-space renormalization approach for the 2D Ising model, using Monte Carlo and thermodynamic integration to clarify and extend his original construction, and explores its connection to the fermion sign problem.

Contribution

It provides a detailed Monte Carlo implementation of Wilson's real-space renormalization for the Ising model, clarifying historical methods and their relation to the fermion sign problem.

Findings

Successful retracing of Wilson's renormalization computation.

Insights into the connection between real-space renormalization and the fermion sign problem.

Extension of Wilson's approach with multiple Hamiltonian terms.

Abstract

We reconsider real-space renormalization for the two-dimensional Ising model, following the path traced out by Wilson in Sect. VI of his 1975 Reviews of Modern Physics. In that reference, Wilson considerably extended the Kadanoff decimation procedure towards a possibly rigorous construction of a real-space scale-invariant hamiltonian. Wilson's construction has, to the best of our knowledge, never been fully understood and thus neither reproduced nor generalized. In the present work, we use Monte Carlo sampling in combination with thermodynamic integration in order to retrace Wilson's computation for a real-space renormalization with a number of terms in the hamiltonian. We elaborate on the connection of real-space renormalization with the fermion sign problem and discuss to which extent our Monte Carlo procedure actually implements Wilson's program from half a century ago.