Abstract The rapid development of neural quantum states (NQS) has established it as a promising framework for studying quantum many-body systems. In this work, by leveraging the cutting-edge transformer-based architectures and developing highly efficient optimization algorithms, we achieve the state-of-the-art results for the doped two-dimensional (2D) Hubbard model, arguably the minimum model for high-Tc superconductivity. Interestingly, we find different attention heads in the NQS ansatz can directly encode correlations at different scales, making it capable of capturing long-range correlations and entanglements in strongly correlated systems. With these advances, we establish the half-filled stripe in the ground state of 2D Hubbard model with the next nearest neighboring hopping, consistent with experimental observations in cuprates. Our work establishes NQS as a powerful tool for solving challenging many-fermions systems.
Abstract The rapid development of neural quantum states (NQS) has established it as a promising framework for studying quantum many-body systems. In this work, by leveraging the cutting-edge transformer-based architectures and developing highly efficient optimization algorithms, we achieve the state-of-the-art results for the doped two-dimensional (2D) Hubbard model, arguably the minimum model for high-Tc superconductivity. Interestingly, we find different attention heads in the NQS ansatz can directly encode correlations at different scales, making it capable of capturing long-range correlations and entanglements in strongly correlated systems. With these advances, we establish the half-filled stripe in the ground state of 2D Hubbard model with the next nearest neighboring hopping, consistent with experimental observations in cuprates. Our work establishes NQS as a powerful tool for solving challenging many-fermions systems.
Whether the ground state of the square lattice Hubbard model exhibits superconductivity remains a major open question, central to understanding high temperature cuprate superconductors and ultra-cold fermions in optical lattices. Numerical studies have found evidence for stripe-ordered states and superconductivity at strong coupling but the phase diagram remains controversial. Here, we show that one can resolve the subtle energetics of metallic, superconducting, and stripe phases using a new class of neural quantum state (NQS) wavefunctions that extend hidden fermion determinant states to Pfaffians. We simulate several hundred electrons using fast Pfaffian algorithms allowing us to measure off-diagonal long range order. At strong coupling and low hole-doping, we find that a non-superconducting filled stripe phase prevails, while superconductivity coexisting with partially-filled stripes is stabilized by a negative next neighbor hopping t-prime, with |t-prime| > 0.1. At larger doping levels, we introduce momentum-space correlation functions to mitigate finite size effects that arise from weakly-bound pairs. These provide evidence for uniform d-wave superconductivity at U = 4, even when t-prime = 0. Our results highlight the potential of NQS approaches, and provide a fresh perspective on superconductivity in the square lattice Hubbard model.
Abstract Simulating the dynamics of many-body quantum systems is a significant challenge, especially in higher dimensions where entanglement grows rapidly. Neural quantum states (NQS) offer a promising tool for representing quantum wavefunctions, but their application to time evolution faces scaling challenges. We introduce the time-dependent neural quantum state (t-NQS), a novel approach incorporating explicit time dependence into the neural network ansatz. This framework optimizes a single, time-independent set of parameters to solve the time-dependent Schrödinger equation across an entire time interval. We detail an autoregressive, attention-based transformer architecture and techniques for extending the model’s applicability. To benchmark and demonstrate our method, we simulate quench dynamics in the 2D transverse field Ising model and the time-dependent preparation of the 2D antiferromagnetic state in a Heisenberg model, demonstrating state of the art performance, scalability, and extrapolation to unseen intervals. These results establish t-NQS as a powerful framework for exploring quantum dynamics in strongly correlated systems.
The recent development of Neural Quantum States (NQS) has established them as one of the most accurate methods for studying strongly correlated many-fermion systems, outperforming existing many-body approaches for large systems. However, NQS calculations are currently highly resource intensive. In this work, we introduce an improved Pareto frontier of efficiency and accuracy for NQS in the simulation of strongly correlated lattice fermions. This frontier is defined by two complementary backflow-related architectures: the Sparse Convolutional Ansatz for Lattice Electrons (SCALE), which achieves high computational efficiency, and the Accurate Convolutional ansatz for lattice Electrons (ACE), which delivers high variational accuracy, based on benchmark results on the Hubbard model and the t – J model on large lattice sizes. SCALE utilizes a tailored convolutional design to enable an efficient local update based on the low-rank update of determinants. This structure reduces the computational scaling from O ( N 4 ) to O ( N 3 ) in backflow methods, resulting in a practical speedup exceeding 40 × in our test cases while retaining accurate variational accuracy. As an application, we study the 1 / 8 -doped pure Hubbard model up to size 32 × 32 and find no significant energy difference between the horizontal and vertical unit-filled stripe states, in contrast to the half-filled stripe state when next-nearest-neighbor hoppings are included. ACE, conversely, employs a deep convolutional stack to maximize expressive power, achieving high accuracy on large systems. We perform extensive benchmark calculations on the Hubbard and t – J models with the two architectures. On the challenging repulsive Hubbard model, SCALE achieves variational energies competitive with leading methods at a fraction of the computational cost. Meanwhile, ACE achieves lower variational energies than previously reported results while requiring only one-sixth of the run-time for system size 16 × 4 . These NQS approaches provide , , and tools for exploring the rich physics of strongly correlated fermionic systems.
Understanding the physics underlying high-temperature superconductivity in cuprates and, more recently, infinite-layer nickelates has remained a central challenge in condensed-matter physics. We establish neural quantum states (NQS), specifically Hidden Fermion Determinant States (HFDS), as a scalable variational approach to the three-band Emery model of the copper- and nickel-oxide layers in these materials. After benchmarking HFDS against matrix product states on width-two geometries, we study ground states of fully two-dimensional (2D) systems of up to 10×10 unit cells (300 sites). We characterize the momentum-space distribution of dopants and find a pronounced electron-hole dichotomy similar to cuprate experiments. We further downfold the three-band model to effective single-band descriptions by constructing interacting Wannier functions. We consider a wide range of parameters -- from the charge-transfer regime relevant to cuprates to the Hubbard-Mott regime of nickelates, as well as systematic scans of the charge-transfer gap that has been demonstrated to impact the critical superconducting temperatures. Across all regimes, the effective model significantly deviates from the usual Fermi-Hubbard model: The typical ratio U/t is enhanced, some parameters experience a significant doping dependence, and sizable terms beyond the conventional Hubbard model are present, most notably a density-assisted hopping tn. Notably, in all effective models, tn has the largest contribution to particle-hole asymmetry, rather than next-nearest-neighbor hopping contributions. The effective parameters sensitively depend on the charge-transfer energy, doping, and interaction ratios. Our results establish HFDS as an efficient tool for studying the 2D Emery model and demonstrate that single-band descriptions can require interaction terms generated by the underlying multi-band models.
Accurate simulations of the Hubbard model are crucial to understanding strongly correlated phenomena, where small energy differences between competing orders demand high numerical precision. In this work, Neural Quantum States are used to probe the strongly coupled and underdoped regime of the square-lattice Hubbard model. We systematically compare the Hidden Fermion Determinant State and the Jastrow-Backflow ansatz, parametrized by a Vision Transformer, finding that in practice, their accuracy is similar. We also test different symmetrization strategies, finding that output averaging yields the lowest energies, though it becomes costly for larger system sizes. On cylindrical systems, we consistently observe filled stripes. On the torus, our calculations display features consistent with a doped Mott insulator, including antiferromagnetic correlations and suppressed density fluctuations. Our results demonstrate both the promise and current challenges of neural quantum states for correlated fermions.
Abstract Numerically simulating large, spinful, fermionic systems is of great interest in condensed matter physics. However, the exponential growth of the Hilbert space dimension with system size renders exact quantum state parameterizations impractical. Owing to their representative power, neural networks often allow to overcome this exponential scaling. Here, we investigate the ability of neural quantum states (NQS) to represent the bosonic and fermionict − Jmodel – the high interaction limit of the Hubbard model – on various 1D and 2D lattices. Using autoregressive, tensorized recurrent neural networks (RNNs), we study ground state representations upon hole doping the half-filled system. Additionally, we propose a method to calculate quasiparticle dispersions, applicable to any network architecture or lattice geometry, and allowing to infer the low-energy physics from NQS. By analyzing the strengths and weaknesses of the RNN ansatz we shed light on the challenges and promises of NQS for simulating bosonic and fermionic systems.