Evaluating second-order phase transitions with Diagrammatic Monte Carlo: N\'{e}el Transition in the doped three-dimensional Hubbard model

TL;DR

This paper demonstrates how Diagrammatic Monte Carlo can detect second-order phase transitions directly in the thermodynamic limit, providing accurate phase diagrams of the doped 3D Hubbard model without finite-size limitations.

Contribution

It introduces a novel application of Diagrammatic Monte Carlo to identify phase transitions via series divergence, especially in doped regimes where traditional methods face challenges.

Findings

Successfully mapped the phase diagram of the doped 3D Hubbard model.

Detected the Ne9el transition and incommensurate spin density wave at low temperatures.

Achieved high accuracy comparable or superior to finite-size techniques.

Abstract

Diagrammatic Monte Carlo -- the technique for numerically exact summation of all Feynman diagrams to high orders -- offers a unique unbiased probe of continuous phase transitions. Being formulated directly in the thermodynamic limit, the diagrammatic series is bound to diverge and is not resummable at the transition due to the non-analyticity of physical observables. This enables the detection of the transition with controlled error bars from an analysis of the series coefficients alone, avoiding the challenge of evaluating physical observables near the transition. We demonstrate this technique by the example of the N\'eel transition in the $3d$ Hubbard model. At half-filling and higher temperatures, the method matches the accuracy of state-of-the-art finite-size techniques, but surpasses it at low temperatures and allows us to map the phase diagram in the doped regime, where finite-size techniques struggle from the fermion sign problem. At low temperatures and sufficient doping, the transition to an incommensurate spin density wave state is observed.