Quantum Monte Carlo Achieves Linear Scaling for up to N Particle Systems and Resolves Sign Problem

Understanding the behaviour of interacting electrons in complex materials remains a central challenge in condensed matter physics. Tu Hong, Kun Chen (Institute of Theoretical Physics, Chinese Academy of Sciences) and Xiao Yan Xu (Key Laboratory of Artificial Structures and Quantum Control, School of Physics and Astronomy, Shanghai Jiao Tong University), alongside their colleagues, have developed a new computational technique to tackle this problem. Their research introduces a finite-temperature canonical-ensemble determinant Monte Carlo algorithm that accurately simulates the properties of correlated electrons, even in challenging low-density scenarios. This advancement significantly reduces computational demands, scaling linearly with system size, and unlocks the possibility of studying large-scale, strongly correlated systems previously inaccessible to simulation. By applying this method to both dilute electron gases and one-dimensional flat-band systems, the team demonstrates its power and reveals a previously hidden ferromagnetic instability at low temperatures.

This advancement significantly reduces computational demands, scaling linearly with system size, and unlocks the possibility of studying large-scale, strongly correlated systems previously inaccessible to simulation.

The team proposed a stabilized QR update, a crucial component for maintaining simulation stability and accuracy, reducing computational complexity from standard cubic scaling to linear scaling with respect to system size. This optimisation delivers a substantial increase in speed within dilute regimes, enabling unbiased access to large-scale simulations of strongly correlated, low-density phases.

Experiments revealed that for a system with 100 particles, the new method achieves optimal linear scaling, fundamentally overcoming the cubic bottleneck of standard methods. Measurements confirm a practical performance gain of over four orders of magnitude for complete Monte Carlo sweeps on the largest systems benchmarked, representing a decisive computational advantage. Validation of the method was performed using the dilute electron gas model, demonstrating suppression of the fermion sign problem as particle density decreases.

Further application of this approach was made to a one-dimensional flat-band system, allowing precise control over filling via the canonical ensemble. Results revealed a ferromagnetic instability at low temperatures specifically within the half-filling regime, suggesting a novel phase transition, rigorously tested using a bipartite crystalline lattice framework. Scientists calculated the equal-time uniform in-plane spin structure factor, demonstrating an increase in the low-temperature limit, indicative of ground state ferromagnetism.

Measurements of the out-of-plane spin structure factor of the A1 sublattice approach a constant value in the low temperature limit, consistent with analytical results and providing unbiased numerical confirmation of the Mielke-Tasaki mechanism. This breakthrough delivers a powerful tool for investigating emergent phenomena in dilute matter and offers a structurally flexible framework that can be integrated with constrained-path auxiliary-field methods to address the sign problem in dense, strongly correlated regimes.

Linear Scaling QMC for Correlated Electrons

This work details the development of a new quantum Monte Carlo framework, LEC-QMC, which achieves linear computational scaling when simulating dilute systems of interacting electrons. By combining canonical-ensemble sampling with a stabilized QR update, the authors have overcome limitations inherent in conventional determinantal algorithms.

Validation on the dilute electron gas demonstrated suppression of the fermion sign problem, while application to a one-dimensional flat-band system revealed a ferromagnetic instability at half-filling, a result inaccessible to standard grand-canonical methods. The significance of this achievement lies in its ability to access strongly correlated low-density phases with unprecedented scale and precision.

LEC-QMC’s structural flexibility also allows for potential integration with constrained-path auxiliary-field methods, offering a pathway to tackle the sign problem in denser, strongly correlated systems. The authors acknowledge limitations related to the approximations inherent in quantum Monte Carlo methods and the computational demands of extending the approach to even larger systems. Future research will focus on applying this framework to explore phase diagrams in Wigner crystals, moiré superlattices, and ultracold atomic gases, potentially unlocking new insights into emergent phenomena in dilute matter.

Understanding the behaviour of interacting electrons in complex materials remains a central challenge in condensed matter physics. Their research introduces a finite-temperature canonical-ensemble determinant Monte Carlo algorithm that accurately simulates the properties of correlated electrons, even in challenging low-density scenarios.

The algorithm’s capabilities are demonstrated through its application to these diverse and challenging condensed matter scenarios, offering a powerful tool for investigating strongly correlated electron behaviour. The team anticipates that this method will be suitable for probing phase diagrams in systems such as the doped Hubbard model near half-filling, extending the capabilities of quantum Monte Carlo simulations. Data shows that as system size increases and density decreases, the average sign recovers from near zero to unity, confirming the dilute electron gas behaves increasingly like a Boltzmann gas, allowing LEC-QMC to simulate extremely large systems at low densities.