Non-perturbative Analytic and Quantum Monte Carlo Study of Strongly Correlated Quantum Systems

When Interactions Define New Physics -- Many of the most fascinating phenomena in quantum materials arise precisely where conventional approximations become least reliable. In weakly interacting systems, one can often begin with independent electrons and incorporate interactions perturbatively. In strongly correlated quantum systems, however, interactions are not merely corrections to single-particle physics -- they can reorganize the many-body state and generate fundamentally new phenomena such as magnetism, unconventional superconductivity, Mott physics, and other forms of collective quantum order. Understanding these systems therefore requires non-perturbative approaches: methods capable of treating strong interactions without assuming that they are small.

Our group's research in this direction uniquely combines two key interconnected non-perturbative tools, exact analytic methods in mathematical physics and sign-problem-free quantum Monte Carlo (QMC) simulations, to study itinerant ferromagnetism and correlated flat band physics as central strongly correlated problems. Exact mathematical results can establish rigorously that a many-body phenomenon must occur and reveal the fundamental mechanism responsible for it. Quantum Monte Carlo can then access large interacting systems at finite temperature and determine thermodynamic behavior, phase transitions, and competing orders beyond what can normally be obtained analytically. Together, they provide a route from rigorous microscopic understanding to quantitative many-body physics.

Nagaoka ferromagnetism and the 15-puzzle

This program led to an unexpected connection we discovered between a classic mathematical puzzle and quantum magnetism. In Exact Results on Itinerant Ferromagnetism and the 15-Puzzle Problem (Physical Review B, 2018), we showed that the connectivity problem underlying Nagaoka ferromagnetism can be mapped onto the permutation structure of the classical 15-puzzle. The result provided general lattice-connectivity conditions for kinetic ferromagnetism and connected rigorous many-body physics with combinatorics. It was subsequently discussed in Hal Tasaki's graduate textbook Physics and Mathematics of Quantum Many-Body Systems and featured by Quanta Magazine in “A Child’s Puzzle Has Helped Unlock the Secrets of Magnetism.”

The experimental context has changed substantially since this work was published. Nagaoka and related kinetic ferromagnetism have now been observed or directly probed in quantum-dot arrays, moiré quantum materials, and ultracold-atom Hubbard simulators, turning a historically idealized many-body mechanism into an experimentally accessible phenomenon.

Analogy of Nagaoka ferromagnetism with the 15 puzzle game in a 4 by 4 square lattice.

Flat bands and an exact ferromagnetic percolation representation

The physics of strong correlations can sometimes be understood by identifying the right emergent geometric representation of the many-body Hilbert space. In “Ferromagnetic Percolation Transition in a Multiorbital Flat Band Assisted by Hund’s Coupling,” (Physical Review B 104, 064442 (2021)), we connected an interacting flat band ferromagnetic transition with a problem of correlated geometric percolation. The central theoretical advance is that Hund's coupling makes possible a rigorous percolation representation that is unavailable from the Hubbard interaction alone in honeycomb and triangular lattices. This turns the strongly interacting quantum many-body problem into a classical statistical problem of cluster formation with spin labels, where the only remaining degrees of freedom are the sizes and connectivity of these spin-polarized clusters. This provides a controlled route from microscopic kinetics and interactions to macroscopic magnetism: when clusters remain small the system behaves paramagnetically, while the emergence of a system-spanning cluster forces all spins to align, producing ferromagnetism.

Exact itinerant ferromagnetism in multiorbital systems

In Exact Results for Itinerant Ferromagnetism in Multi-Orbital Systems on Square and Cubic Lattices (Physical Review Letters, 2014), we established rigorous ferromagnetic ground states in strongly interacting multiorbital systems. The work showed that orbital degrees of freedom are not merely additional complexity: their directional structure can fundamentally reorganize the many-body problem and make robust itinerant ferromagnetism possible.

This PRL work provided the analytic foundation for a broader research program on orbital-driven itinerant magnetism, in which the geometry of orbital hopping and the resulting connectivity of the many-body Hilbert space play a central role. In particular, it identified precise conditions under which kinetic frustration and interaction effects cooperate to stabilize fully polarized ground states, establishing a controlled starting point for studying itinerant ferromagnetism beyond single-band systems.

This program was subsequently extended to t_{2g}-orbital systems (2014) and then to finite temperature through sign-problem-free quantum Monte Carlo study (2015). These developments were built on the mathematical structural revealed in this work.

Unbiased finite-temperature SSE quantum Monte Carlo study of itinerant magnetism

In Sign-Problem-Free Quantum Monte Carlo Study on Thermodynamic Properties and Magnetic Phase Transitions in Orbital-Active Itinerant Ferromagnets (Physical Review X, 2015), we developed the first sign-problem-free quantum Monte Carlo study of ferromagnetic phase transition at a wide range of filling in a strongly correlated itinerant multi-orbital electron system, enabling non-perturbative characterization of the finite-temperature phase transition and its associated thermodynamic and critical behavior across the itinerant ferromagnetic transition.

A central methodological advance was the first application of stochastic series expansion (SSE) quantum Monte Carlo to an itinerant electron problem, extending a technique traditionally used for quantum spin systems to mobile fermions with orbital degrees of freedom. This development enabled, for the first time, a controlled numerical investigation of how ferromagnetism emerges from itinerant electrons in a multiorbital system. The work has recently received renewed attention in connection with interaction-driven spin and valley ferromagnetism in correlated graphene systems.

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The journey into the realm of strongly correlated quantum materials is never ending. As we deepen our understanding through innovative combination of analytic and numeric methods, we can expect to see groundbreaking discoveries that will shape the future of quantum matter and materials.

Selected works

Ferromagnetic Percolation Transition in a Multiorbital Flat Band Assisted by Hund's Coupling

Eric Bobrow, Junjia Zhang, Yi Li

Physical Review B 104, 064442 (2021)

Exact Results on Itinerant Ferromagnetism and the 15-Puzzle Problem

Eric Bobrow, Keaton Stubis, Yi Li

Physical Review B 98, 180101(R) (2018)

Majorana Positivity and the Fermion Sign Problem of Quantum Monte Carlo Simulations

Zhong Chao Wei, Congjun Wu, Yi Li, Shiwei Zhang, Tao Xiang

Physical Review Letters 116, 250601 (2016)

Sign-Problem-Free Quantum Monte Carlo Study on Thermodynamic Properties and Magnetic Phase Transitions in Orbital-Active Itinerant Ferromagnets

Shenglong Xu, Yi Li, Congjun Wu

Physical Review X 5, 021032 (2015)

Exact Results for Itinerant Ferromagnetism in Multi-Orbital Systems on Square and Cubic Lattices

Yi Li, Elliott H. Lieb, Congjun Wu

Physical Review Letters 112, 217201 (2014)