Generative models, particularly normalizing flows, have shown exceptional performance in learning probability distributions across various domains of physics, including statistical mechanics, collider physics, and lattice field theory. In the context of lattice field theory, normalizing flows have been successfully applied to accurately learn the Boltzmann distribution, enabling a range of tasks such as direct estimation of thermodynamic observables and sampling independent and identically distributed (i.i.d.) configurations. In this work, we present a proof-of-concept demonstration that normalizing flows can be used to learn the Boltzmann distribution for the Hubbard model. This model is widely employed to study the electronic structure of graphene and other carbon nanomaterials. State-of-the-art numerical simulations of the Hubbard model, such as those based on Hybrid Monte Carlo (HMC) methods, often suffer from ergodicity issues, potentially leading to biased estimates of physical observables. Our numerical experiments demonstrate that leveraging i.i.d. sampling from the normalizing flow effectively addresses these issues.
Generative models, particularly normalizing flows, have shown exceptional performance in learning probability distributions across various domains of physics, including statistical mechanics, collider physics, and lattice field theory. In the context of lattice field theory, normalizing flows have been successfully applied to accurately learn the Boltzmann distribution, enabling a range of tasks such as direct estimation of thermodynamic observables and sampling independent and identically distributed (i.i.d.) configurations. In this work, we present a proof-of-concept demonstration that normalizing flows can be used to learn the Boltzmann distribution for the Hubbard model. This model is widely employed to study the electronic structure of graphene and other carbon nanomaterials. State-of-the-art numerical simulations of the Hubbard model, such as those based on Hybrid Monte Carlo (HMC) methods, often suffer from ergodicity issues, potentially leading to biased estimates of physical observables. Our numerical experiments demonstrate that leveraging i.i.d. sampling from the normalizing flow effectively addresses these issues.
We present the first proof of principle that normalizing flows can accurately learn the Boltzmann distribution of the fermionic Hubbard model—a key framework for describing the electronic structure of graphene and related materials. State-of-the-art methods like hybrid Monte Carlo often suffer from ergodicity issues near the time-continuum limit, leading to biased estimates. Leveraging symmetry-aware architectures as well as independent and identically distributed sampling, our approach resolves these issues and achieves significant speedups over traditional methods.
Understanding strongly correlated electron systems remains one of the central challenges in condensed matter physics. Finding ab initio descriptions of non-perturbative systems is of great importance for modeling carbon-based materials such as graphene or fullerenes. In this regime, computational methods become indispensable tools for performing simulations, making predictions, and ultimately comparing to experiments. The Hubbard model offers a minimal description of the dynamics of electrons in such materials. As exact methods are limited to small system sizes by the exponentially growing Hilbert space, larger-scale first-principle studies rely on numerical approaches. Alongside powerful Hamiltonian-based methods, the auxiliary-field description of the model offers a natural setting for Lagrangian-based Monte Carlo sampling. However, substantial challenges, such as lack of ergodicity and the numerical sign problem, arise in these approaches, leading to long autocorrelation times, biased results, and large statistical uncertainties. In recent years, generative models, especially normalizing flows, have been applied extensively to various theories to address shortcomings of traditional Monte Carlo-based algorithms, such as critical slowing down or topological freezing. While these approaches have shown promise in alleviating these issues, a poorly understood and unfavorable scaling of these methods has stalled their progress, and whether they can reach state-of-the-art system sizes in lattice quantum chromodynamics remains to be seen. This motivates the application of these methods to condensed matter systems, where physically relevant system sizes are reachable at substantially smaller scales. In this work, we study for the first time the application of generative models to the finite-temperature auxiliary-field Hubbard model. We begin by employing an equivariant normalizing flow that respects multiple symmetries inherent to the Boltzmann distribution, demonstrating a substantial improvement in sampling efficiency over Hybrid Monte Carlo and standard non-equivariant flows. We further develop Symmetry-Enforcing Stochastic Modulation (SESaMo), a more scalable algorithm capable of modeling both exact and broken symmetries in normalizing flows, and show that it achieves significantly higher effective sample sizes than traditional equivariant techniques at large system sizes. Equipped with SESaMo, we compute physical observables at half-filling in excellent agreement with exact diagonalization at substantially lower computational cost than Hybrid Monte Carlo. We then turn to the doped Hubbard model, where the theory exhibits a numerical sign problem. We introduce an annealing scheme to simulate the Hubbard model at finite chemical potential in the spin basis. Making use of the less severe sign problem in this description, the annealing scheme enables ergodic sampling of the multi-modal distribution, yielding more accurate results with substantially reduced uncertainties compared to state-of-the-art optimized Hybrid Monte Carlo. Lastly, we employ a masked autoencoder for distribution estimation to learn the Boltzmann distribution of the discrete auxiliary-field Hubbard model. Preliminary results show a significantly higher effective sample size compared to normalizing flows with continuous degrees of freedom, while also exhibiting a milder sign problem than the continuous spin basis.
Normalizing flows have recently demonstrated the ability to learn the Boltzmann distribution of the Hubbard model, opening new avenues for generative modeling in condensed matter physics. In this work, we investigate the steps required to extend such simulations to larger lattice sizes and lower temperatures, with a focus on enhancing stability and efficiency. Additionally, we present the scaling behavior of stochastic normalizing flows and non-equilibrium Markov chain Monte Carlo methods for this fermionic system.
Normalizing flows have recently demonstrated the ability to learn the Boltzmann distribution of the Hubbard model, opening new avenues for generative modeling in condensed matter physics. In this work, we investigate the steps required to extend such simulations to larger lattice sizes and lower temperatures, with a focus on enhancing stability and efficiency. Additionally, we present the scaling behavior of stochastic normalizing flows and non-equilibrium Markov chain Monte Carlo methods for this fermionic system.
The Hubbard model arises naturally when electron-electron interactions are added to the tight-binding descriptions of many condensed matter systems. For instance, the two-dimensional Hubbard model on the honeycomb lattice is central to the ab initio description of the electronic structure of carbon nanomaterials, such as graphene. Such low-dimensional Hubbard models are advantageously studied with Markov chain Monte Carlo methods, such as hybrid Monte Carlo (HMC). HMC is the standard algorithm of the lattice gauge theory community, as it is well suited to theories of dynamical fermions. As HMC performs continuous, global updates of the lattice degrees of freedom, it provides superior scaling with system size relative to local updating methods. A potential drawback of HMC is its susceptibility to ergodicity problems due to so-called exceptional configurations, for which the fermion operator cannot be inverted. Recently, ergodicity problems were found in some formulations of HMC simulations of the Hubbard model. Here, we address this issue directly and clarify under what conditions ergodicity is maintained or violated in HMC simulations of the Hubbard model. We study different lattice formulations of the fermion operator and provide explicit, representative calculations for small systems, often comparing to exact results. We show that a fermion operator can be found which is both computationally convenient and free of ergodicity problems.
Learning many-body quantum systems is of fundamental importance in quantum information processing, however, it is a challenging task which typically requires estimating quantum distributions of dimensionality exponentially scaling to the system size. As generative models have shown a great scalability to learn high-dimensional distributions and found wide applications in the domain of image and text, they can be a powerful tool to facilitate us to accomplish the challenging quantum tasks. In this work, we propose using normalizing flow (NF) models with fast sampling to learn discrete quantum distributions for quantum state tomography. Particularly, three NF models, including denoising flow, argmax flow, and tree flow, are first adapted to the task of explicit quantum probability density estimation. We then perform extensive experiments on a large scale of quantum systems, and our numerical results demonstrate that these discrete NFs admit an excellent sampling efficiency in the sense that they are insensitive to the system size to learn the high-dimensional quantum distributions, without compromising the learning performance. Finally, in comparison to the other generative models, such as autoregressive, the NFs avoid the problem of slow sequential sampling.