Gauge Theory Models Cuprate Superconductivity and Spin Liquid Behavior

High-temperature superconductivity, initially observed in cuprate ceramics, continues to present a significant challenge to condensed matter physics. These materials exhibit superconductivity—the complete absence of electrical resistance—at temperatures considerably higher than conventional superconductors, although substantial cooling remains necessary for practical operation. Understanding the underlying mechanism driving this phenomenon remains a central goal, with the pseudogap phase playing a crucial role in the emergence of this state of matter.

The behaviour of hole-doped cuprates, where electrons are removed creating ‘holes’ that act as charge carriers, is particularly intriguing, demonstrating the highest superconducting critical temperatures at ambient pressure. A key characteristic of these materials is the proximity of the pseudogap phase to an insulating antiferromagnetic state, suggesting a potential link between magnetism and superconductivity.

Two theoretical frameworks attempt to explain the pseudogap and its relation to superconductivity. The first proposes that the pseudogap arises from fluctuating superconducting order, utilising concepts from the XY model—a statistical mechanics model describing interacting spins—to describe this fluctuating phase. The second framework focuses on fluctuating spin density waves as the origin of the pseudogap, offering an alternative explanation.

Recent research explores a gauge theory approach, employing an SU(2) gauge field to describe the interactions within the pseudogap phase, offering a novel perspective on the interplay of charge and spin. This theory introduces two distinct fermionic sectors, one consisting of conventional electron-like quasiparticles and the other a ‘critical spin liquid’ composed of electrically neutral spinons, providing a framework to understand the complex behaviour of these materials. These sectors interact via a Yukawa coupling to a spinless Higgs boson.

This research presents a theoretical framework for understanding the pseudogap phase and the emergence of superconductivity in underdoped cuprates, employing a methodology that bridges concepts from gauge theory, condensed matter physics, and numerical simulation. The core of the approach lies in constructing a square lattice SU(2) gauge theory, adapting a mathematical structure typically used to describe fundamental forces to model the complex interactions within the material. Researchers derive this gauge theory from the square lattice Hubbard model—a standard starting point for understanding strongly correlated electron systems—by augmenting it with ancilla qubits, thereby enhancing the accuracy of theoretical calculations.

The model postulates two distinct fermionic matter sectors, with one sector comprising conventional, electrically charged quasiparticles forming hole pockets, regions where electrons are missing from the material’s electronic structure. The other sector consists of electrically neutral spinons, particles carrying spin but no electric charge, arranged in a critical spin liquid state, where spins are highly entangled and do not order even at very low temperatures. These two sectors interact via a Yukawa coupling to a spinless Higgs boson, which itself carries electric charge and transforms under the SU(2) gauge group.

To simplify calculations, researchers employ a Born-Oppenheimer-type approximation, a common technique in quantum chemistry, treating the Higgs bosons classically. This allows them to focus on the quantum mechanical behaviour of the fermions while still capturing the essential influence of the bosons. The resulting effective boson energy functional is then subjected to Monte Carlo simulations, a numerical method that uses random sampling to approximate the solution, revealing a Kosterlitz-Thouless transition, a specific type of phase transition driven by topological defects known as vortices, consistent with the onset of d-wave superconductivity.

Crucially, the model’s predictions align with experimental observations, demonstrating the formation of a halo of charge order surrounding each vortex, mirroring observations from scanning tunneling microscopy experiments. Furthermore, the model accurately reproduces the “Fermi arcs” observed in angle-resolved photoemission spectroscopy (ARPES) data, and predicts oscillations under a magnetic field, consistent with recent evidence from the Yamaji effect.

The behaviour of high-temperature cuprate superconductors centres on the complex interplay of multiple electronic orders, rather than a single superconducting mechanism. These investigations suggest that cuprates frequently exist near quantum critical points, where quantum fluctuations drive transitions between different phases even at zero temperature.

Computational simulations, employing a Born-Oppenheimer approximation, reveal a Kosterlitz-Thouless transition involving vortices, which drives the onset of d-wave superconductivity. Importantly, each vortex exhibits a surrounding halo of charge order, a feature consistent with observations from scanning tunnelling microscopy. Above the superconducting transition temperature, the model predicts the existence of ‘Fermi arcs’ in the electron spectral function, aligning with experimental data obtained through angle-resolved photoemission spectroscopy (ARPES). Furthermore, the theoretical framework predicts oscillations in the area under a magnetic field, aligning with recent experimental evidence obtained through the Yamaji effect.

This research highlights the importance of considering multiple, intertwined orders when attempting to understand high-temperature superconductivity, suggesting that a comprehensive understanding requires a holistic approach. The theoretical framework and computational results presented offer a compelling explanation for several key experimental observations, including the pseudogap phase, Fermi arcs, and the Yamaji effect. Recent publications from 2023-2025 demonstrate an active and evolving field, with ongoing investigations into the role of polaronic correlations and the development of advanced computational techniques, such as those utilising ancilla qubits, to model these complex systems.

Future work should focus on extending this model to incorporate the effects of disorder and dimensionality, potentially revealing a more complete understanding of the superconducting mechanism. Investigating the role of different spin liquid states and exploring the possibility of incorporating additional bosonic degrees of freedom also represent promising avenues for further research.