Lattice Monte Carlo meets the lattice functional renormalization group: A quantitative comparison

Lattice Monte Carlo simulations and the functional renormalization group (RG) are powerful approaches that allow for quantitative studies of nonperturbative phenomena such as bound-state formation, spontaneous symmetry breaking, and phase transitions. While results from both methods have recently shown remarkable agreement for many observables, e.g., in quantum chromodynamics, an analysis of deviations in certain quantities turns out to be challenging. This is because calculations with the two methods are based on different approximations, regularizations, and scale fixing procedures. In the present work, we present a framework for a more direct comparison by formulating the functional RG approach on a finite spacetime lattice. This removes all ambiguities of regularization, finite-size, and scale fixing procedures in concrete studies. By investigating the emergence of spontaneous symmetry breaking and phase transitions in a <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>Z</a:mi> <a:mo stretchy="false">(</a:mo> <a:mn>2</a:mn> <a:mo stretchy="false">)</a:mo> </a:math> scalar theory in <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>d</e:mi> <e:mo>=</e:mo> <e:mn>1</e:mn> </e:math> , 2, and 3 spacetime dimensions, we demonstrate at the example of the local potential approximation how this framework can be used to evaluate and compare the systematic errors of both approaches.