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

Connor Lenihan, Aaram J. Kim+2 more|arXiv (Cornell University)|Dec 30, 2021

Physics of Superconductivity and Magnetism76 references24 citations

TL;DR

This paper introduces a diagrammatic Monte Carlo (DiagMC) method to detect and characterize second-order phase transitions—specifically the Néel transition—in the doped three-dimensional Hubbard model. By analyzing the asymptotic behavior of high-order diagrammatic series coefficients, the method identifies critical points without finite-size extrapolations, enabling accurate mapping of the phase diagram in the challenging doped regime where traditional methods fail due to the fermion sign problem. The approach reveals a transition to an incommensurate spin density wave state at low temperatures and doping levels above 10%.

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.

Motivation & Objective

• To develop an unbiased method for detecting continuous phase transitions in strongly correlated fermionic systems without relying on finite-size scaling.
• To address the limitations of finite-size techniques, particularly the fermion sign problem in doped regimes of the 3D Hubbard model.
• To map the phase diagram of the doped 3D Hubbard model in regions where conventional quantum Monte Carlo methods fail.
• To validate the method at half-filling and extend it to the doped regime, revealing new physics such as incommensurate spin density wave states.

Proposed method

• The method uses diagrammatic Monte Carlo to stochastically sum all Feynman diagrams to high orders in the perturbative expansion of physical observables.
• It formulates the perturbation series directly in the thermodynamic limit, avoiding finite-size effects and enabling exact summation up to a truncation order.
• Critical points are detected by analyzing the asymptotic scaling of consecutive diagrammatic series coefficients, which diverge at the transition due to non-analyticity.
• The ratio of consecutive coefficients, r_m = a_m / a_{m-1}, is analyzed for convergence to a power-law behavior r_m ∝ m^{−1+γ} U_c, where U_c is the critical interaction.
• An adjusted ratio r̃_m = m^{−1+γ} r_m is used to identify plateaus that signal the critical point U_c with controlled error bars.
• The approach avoids reconstructing physical observables near the transition, instead relying solely on the structure of the series coefficients.

Experimental results

Research questions

• RQ1 Can the Néel transition in the 3D Hubbard model be reliably detected using only the asymptotic behavior of high-order diagrammatic series coefficients?
• RQ2 How does the DiagMC method perform in the doped regime, where finite-size techniques are hindered by the fermion sign problem?
• RQ3 Does the method reveal a transition to an incommensurate spin density wave state at low temperatures and moderate doping?
• RQ4 What is the accuracy and reliability of the DiagMC approach compared to established finite-size quantum Monte Carlo methods at half-filling?
• RQ5 Can the method map the phase diagram in the doped regime with controlled error bars, where no unbiased results previously existed?

Key findings

• The method successfully detects the Néel transition in the 3D Hubbard model at half-filling with accuracy matching state-of-the-art finite-size techniques.
• At low temperatures and doping levels above 10%, the Néel transition is found to be followed by a transition to an incommensurate spin density wave (SDW) state.
• The spin-spin correlation function computed via DiagMC reveals the nature of the underlying correlations, which are amenable to experimental validation using ultracold atoms in optical lattices.
• The approach achieves controlled error bars in the doped regime, where finite-size methods fail due to the exponential scaling of the fermion sign problem.
• The asymptotic scaling of series coefficients provides a robust, unbiased probe of criticality without requiring physical observable reconstruction.
• The technique demonstrates that diagrammatic series divergences at critical points are governed by universal critical exponents, enabling precise determination of U_c with high confidence.