Unrestricted MeanField Analysis for Quantum Materials via Machine Learning
PI: Zhen Bi (Physics)
Tuition for the graduate student may be covered by the PI’s NSFCAREER grant or startup funding.
Mean-field theory– the workhorse for the effects of electron correlation in quantum materials– reduces the problem of interacting many electrons to a noninteracting single-electron problem in which each electron moves in the average field generated by all others. Despite this simplif ication, it remains a cornerstone of materials research: by producing interaction modified band structures, Fermi-surface topologies, and symmetry-breaking orders, mean-field calculations anchor the interpretation of a wide range of condensed matter experiments. In strongly correlated settings– especially in moir´e quantum materials, where interactions are tunable and exotic phases proliferate– mean-field theory is often the first (and sometimes the only) practical tool for surveying an enormous parameter space before resorting to more costly many-body methods.
The traditional workflow typically starts with a random-phase-approximation (RPA) scan to pinpoint the system’s most likely instability, such as spin-density wave, charge order, superconductivity, etc., and then locks that candidate order into a restricted mean-field ansatz. A self-consistent loop then follows: (i) pick an initial set of order parameters ∆(0) i tonian HMF∆(n) i and build the mean-field Hamil; (ii) diagonalize it to obtain the single particle spectrum and eigenstates; (iii) recompute expectation values of the order parameters to generate ∆(n+1) i ∆(n+1) i ≈∆(n) i ; and (iv) iterate until . As the local degrees of freedom (such as spin and electronic orbitals), lattice sites, or momentum patches grows, this fixed-point search must navigate a high-dimensional landscape riddled with metastable minima, becoming hypersensitive to initial guesses and often converging to solutions already biased by the pre-imposed order-parameter manifold. Worse, every new lattice geometry or interaction profile demands a bespoke Hamiltonian generator and hand-tuned convergence heuristics, rendering the overall workflow brittle and labor-intensive.
An improved method via machine learning techniques We recast mean-field theory as a direct free-energy optimisation over an unrestricted order-parameter manifold, thereby sidestepping the self-consistency loop entirely. The mean-field free energy
F{∆i} =−kBT log[Tr[e−HMF[{∆i}]]]
is an explicit, differentiable functional of the order parameters ∆i, and the stationarity condition ∂F ∂∆i = 0ismathematically equivalent to the traditional mean-field equations. Finding the mean-field solution thus boils down to locating the global minimum of F in a parameter space that can swell to tens of thousands of variables once momentum meshes, sub-lattice indices, and spin or orbital flavors are left unrestricted.
This challenge lies squarely in the wheelhouse of modern machine-learning optimizers. Stochastic gradient methods with adaptive moments, quasi-Newton surrogates and Bayesian or evolutionary schemes, all implemented in auto-differentiation frameworks such as PyTorch or JAX and accelerated on GPUs, scale smoothly to these high-dimensional problems. Treating F as the loss function allows us to (i) traverse order-parameter landscapes without preset bias, (ii) sidestep shallow local minima via noise-annealed or population-based strategies, and (iii) reuse a single code base for any material, changing only the input Hamiltonian. In short, machine learning removes long-standing obstacles and potentially offers an unbiased, scalable route to mapping the intricate phase diagrams of correlated quantum materials. This capability is especially valuable for moir´ e quantum materials, where flat electron bands magnify Coulomb interactions and entangle lattice, spin, valley, and layer degrees of freedom in intricate ways. The multitude of competing orders in moir´e systems– superconducting, ferromagnetic, nematic, and correlated insulating– demands an unbiased, high-dimensional optimizer. Machine-learning-driven free-energy minimization offers a fast, systematic way to map and understand these rich physics.
Computational Expertise Needed
1. Machine-learning optimisation with auto-differentiation frameworks Advanced use of PyTorch or JAX to build, train, and tune stochastic–gradient/Adam or evolutionary optimisers for high-dimensional free-energy minimisation.
2. High-performance linear algebra and eigensolvers Implementation and parallelisation of large-scale Hamiltonian diagonalisation on GPUs/CPUs to deliver fast, differentiable evaluations of the mean-field free energy.
The ideal Junior Researcher will pair the expertise outlined above with a working knowledge of band theory and many-body quantum physics.
Specific Short-Term Goals
1. Code development Create a flexible mean-field package capable of treating a wide range of electronic orders in both real and momentum space.
2. Benchmarking Validate the implementation by comparing its results with well-established data in the literature, ensuring accuracy and reliability.
3. First application Deploy the code on an experimentally motivated moir´ e system to demonstrate its practical utility in current research.
Long-term goal Build a portfolio of preliminary results– including validated algorithms, performance benchmarks, and proof-of-concept studies on experimentally relevant moire heterostructures– that will anchor a competitive proposal to the NSF Condensed Matter and Materials Theory (CMMT) program.
Engagement with ICDS Although we have not yet worked with ICDS, we plan to (i) deploy our ML-accelerated mean-field code on Roar-GPU, (ii) take part in ICDS AI workshops, and (iii) mentor an ICDS Junior Researcher through this seed project, using it to foster further collaboration.