Emus live on the Gross-Neveu-Yukawa archipelago

Abstract: It is expected that the Gross-Neveu-Yukawa (GNY) chiral Ising transition of $N$ Majorana (or $N_f=N/4$ four-component Dirac) fermions coupled with scalar field in (2+1)D will be the first fermionic quantum critical point that various methods such as conformal bootstrap [1], perturbative renormalization group [2] and quantum Monte Carlo (QMC) simulations [3], would yield the converged critical exponents -- serving the same textbook role as the Ising and $O(N)$ models in the statistical and quantum phase transition. However, such expectation has not been fully realized from the lattice QMC simulations due to the obstacles introduced by the UV finite size effect. In this work, by means of the elective-momentum ultra-size (EMUS) QMC method [4], we compute the critical exponents of the GNY $N=8$ chiral Ising transition on a 2D $\pi$-flux fermion lattice model between Dirac semimetal and quantum spin Hall insulator phases [3, 5]. With the matching of fermionic and bosonic momentum transfer and collective update in momentum space, our QMC results provide the fully consistent exponents with those obtained from the bootstrap and perturbative approaches. In this way, the Emus now live happily on the $N=8$ island and could explore the Gross-Neveu-Yukawa archipelago [1] with ease.

• The paper presents EMUS-QMC to resolve finite-size challenges in determining the critical exponents, yielding 1/ν = 1.07(12), ηφ = 0.72(6), and ηψ = 0.04(2).
• It introduces a momentum patching technique on a 2D π-flux fermion lattice that efficiently captures the infrared critical behavior of Dirac fermions.
• The findings bridge lattice QMC, conformal bootstrap, and renormalization group approaches, enhancing our understanding of fermionic quantum phase transitions.

Overview of the Chiral Gross-Neveu-Yukawa Transition Research

This paper addresses the chiral Gross-Neveu-Yukawa (GNY) transition, which represents a pivotal point of study in the domain of quantum phase transitions involving Dirac fermions. The authors utilize an innovative computational approach termed Elective-Momentum Ultra-Size Quantum Monte Carlo (EMUS-QMC) to investigate the critical exponents of the $O(N/2)^2 \rtimes \mathbb{Z}_2$ GNY model for $N=8$. This research aims to reconcile results from various theoretical frameworks, including lattice QMC, conformal bootstrap, and perturbative renormalization group methods, critically enhancing our understanding of such fermionic quantum critical points.

Insights and Methodology

The study design centers on a 2D $\pi$-flux fermion lattice model characterizing transitions between Dirac semimetals (DSM) and quantum spin Hall (QSH) insulators. Previous endeavors to resolve the critical exponents across different methodologies were hindered by significant finite-size effects intrinsic to lattice QMC simulations. These effects have been notably reduced in the study through the adoption of the EMUS-QMC approach, which strategically bypasses computational intensity by focusing on linear dispersion regions crucial to the infrared critical behavior.

In this sophisticated QMC method, simulations implement a "momentum patching" technique that primarily focuses on critical segments of the Brillouin zone, thereby efficiently modeling the system in momentum space rather than the complete lattice. This adjustment, combined with careful subdivision, matches momenta to ensure an accurate representation of critical phenomena while mitigating the ultraviolet (UV) size effects. The authors provide robust numerical exponents that match other theoretical predictions:

• Inverse correlation length: $1/\nu = 1.07(12)$
• Bosonic anomalous dimension: $\eta_\phi = 0.72(6)$
• Fermionic anomalous dimension: $\eta_\psi = 0.04(2)$

These results now align with estimates derived from both the bootstrap and the epsilon-expansion approaches.

Implications and Future Directions

The advanced computational strategy exhibited in this research addresses prior discrepancies in calculating critical exponents inherent in complex quantum systems involving fermions. These findings stimulate further exploration into other regions within the GNY archipelago by extending these techniques to different system symmetries and scales.

The accuracy of these results has potential practical implications for understanding phase transition phenomena in contemporary materials, such as graphene and other moiré structure-related systems, where fermionic interactions are central to observed quantum behaviors. Moreover, EMUS-QMC can be adapted to explore a broader array of quantum transitions, including those in superconductors and topologically ordered systems.

The research signifies a substantial stride in integrating computational efficiencies with theoretical approaches to quantum criticality, promising advancements in characterizing and predicting the behavior of fermionic systems under extreme conditions. Future developments could yield further optimizations in computational techniques, enabling more exhaustive studies across larger, more complex systems, extending to real-world condensed matter applications.