Extended Gutzwiller Approximation for Nonlocal Electron-Electron and Electron-Boson Correlations (I): The Theory
Understanding electron-electron and electron-photon correlations is central to uncovering the fundamental mechanisms governing material properties, particularly in systems where strong interactions give rise to emergent phenomena such as superconductivity, magnetism, and polaritonic effects. These correlations play a pivotal role in cavity quantum materials, where hybridized light-matter states enable quantum control over electronic properties. However, capturing both local and nonlocal correlations in these systems presents a significant theoretical challenge. In this work, we extend the Gutzwiller wavefunction method to include nonlocal electron-photon and electron-electron interactions, providing a unified framework to study the intricate interplay between these effects. Our approach accurately captures the long-range correlations induced by photon exchange, enabling the exploration of exotic quantum phases and the effects of cavity coupling on electronic structure. By benchmarking the method across coupling regimes, we reveal the critical role of nonlocal correlations in stabilizing phases, such as superconducting and insulating states, that are inaccessible through local interactions alone. This generalized Gutzwiller framework offers a versatile tool for understanding and designing materials that harness the transformative potential of strong light-matter coupling.