Neural-Quantum-States Impurity Solver for Quantum Embedding Problems

Such systems exhibit a variety of electronic phases, including metallic, insulating, and superconducting states [1], and represent a versatile design space for technological applications in electronics, quantum computing, and sensing. Designing new correlated materials with targeted properties therefore depends on the ability to solve the many-body electronic Hamiltonian, a computationally demanding task [2]. Quantum embedding (QE) methods provide a robust framework for overcoming these challenges [3-5]. The common strategy underlying these methods is to describe each fragment of interacting orbitals through an effective model, where its complex environment is replaced by a simpler, entangled quantum bath, designed to approximate its influence [3, 6]. The link between this effective model and the original system is established through self-consistency conditions, and different embedding schemes are defined by their choice of which physical property to match [3]. Dynamic mean-field theory (DMFT) [7-14], for instance, uses frequency-dependent one-body Green's functions, whereas density matrix embedding theory (DMET) [5, 15-19] typically uses one-and two-body density matrices. The recently developed gGA [20-26] is a powerful variational method that generalizes the standard Gutzwiller Approximation (GA) [27-34] by systematically extending its variational space with auxiliary "ghost" fermionic degrees of freedom. This approach yields results in remarkable agreement with DMFT but at a much lower computational cost, as it requires calculating only the ground state of a finite-size impurity model, whereas DMFT requires the full spectra from an impurity model that corresponds to an infinite bath. Successful applications of gGA include accurate modelling of the Anderson lattice systems [21], excitonic phenomena [20], non-equilibrium systems [35] and altermagnetic systems [36], along with extensions that achieve charge self-consistency with density functional theory (DFT) [22], demonstrating its versatility and practical utility in real-material contexts.