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Benchmarking cuprate superconductivity theories: experimental constraints, quantum simulation targets, and a unified framework

Hikaru Wakaura

Published Apr 22, 2026• 0 citations
Publisher

Abstract

Abstract We benchmark five microscopic cuprate superconductivity theories against ten experimental constraints using continuous figures of merit $f_i\in[0,1]$ with explicit acceptance windows. At mean-field level, Model~B (SVILC) scores highest ($F=0.96\pm0.02$), followed by Model~E (spin fluctuation, $0.91\pm0.03$). Failure analysis identifies four ingredients that appear necessary at mean-field level: Mott physics, spin-fluctuation pairing, loop-current order, and three-band CuO$_2$ structure. A unified Hamiltonian incorporating all four yields $\Delta_0=28$~meV and $\mu_{\mathrm{orb}}=0.06\;\mu_{ B }$/Cu on a $24\times24$ cluster. The multi-observable scoring framework provides concrete targets for quantum simulation of cuprate Hamiltonians and enables systematic validation of quantum-computed phase diagrams. Four falsifiable predictions are derived.

DOI
10.21203/rs.3.rs-9480217/v1

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Abstract We benchmark five microscopic cuprate superconductivity theories against ten experimental constraints using continuous figures of merit fi∈[0,1]f_i\in[0,1]fi​∈[0,1] with explicit acceptance windows. At mean-field level, ModelB (SVILC) scores highest (F=0.96±0.02F=0.96\pm0.02F=0.96±0.02), followed by ModelE (spin fluctuation, 0.91±0.030.91\pm0.030.91±0.03). Failure analysis identifies four ingredients that appear necessary at mean-field level: Mott physics, spin-fluctuation pairing, loop-current order, and three-band CuO2_22​ structure. A unified Hamiltonian incorporating all four yields Δ0=28\Delta_0=28Δ0​=28~meV and μorb=0.06  μB\mu_{\mathrm{orb}}=0.06\;\mu_{ B }μorb​=0.06μB​/Cu on a 24×2424\times2424×24 cluster. The multi-observable scoring framework provides concrete targets for quantum simulation of cuprate Hamiltonians and enables systematic validation of quantum-computed phase diagrams. Four falsifiable predictions are derived.
The high-Tc{T}_{c}Tc​ cuprates have a complex phase diagram with many competing phases. We propose a bosonic effective quantum Hamiltonian based on the projected SO(5)\text{SO}(5)SO(5) model with extended interactions, which can be derived from the microscopic models of the cuprates. The global phase diagram of this model is obtained using mean-field theory and the quantum Monte Carlo simulation, which is possible because of the absence of the minus sign problem. We show that this single quantum model can account for most salient features observed in the high-Tc{T}_{c}Tc​ cuprates, with different families of the cuprates attributed to different traces in the global phase diagram. Experimental consequences are discussed and theoretical predictions are presented.
High-temperature superconductivity in copper oxides (cuprates, e.g. La2−xSrxCuO4, YBa2Cu3O6+δ or Bi2Sr2CaCu2O8+δ) remains among the most spectacular phenomena in condensed matter physics. Since its discovery in 1986, an enormous number (∼ 105) of papers on the subject have appeared. So far, there is no single, commonly accepted theory of high-temperature superconductivity. However, it is widely believed that a basic description of this phenomenon can be provided by a single-band Hubbard model or its derivative, the t-J model. The latter model is regarded as a minimal microscopic model, capable of describing the essential aspects of the complex physics of the cuprates. Unfortunately, in general case, neither of those two models can be solved exactly and therefore various approximate methods are used. Among them, the so-called mean-field methods provide a simple, yet fairly reasonable description of the cuprates. In particular, some of the main qualitative features of the phase diagram and the essential features of electronic spectrum are roughly reproduced. A standard mean-field approach to the t-J model, known under the name of renormalized mean-field theory (RMFT) goes beyond the Hartree-Fock approximation. Consequently, its fully consistent treatment requires a novel theoretical approach. This has been our original motivation to develop a general approach to the mean-field models, which is based on the maximum entropy (MaxEnt) principle. The method is presented in detail in Part II of this Thesis, and in Part III it is applied to study RMFT of the t-J model. First, we compare the results obtained within our formalism with those of the frequently used non-variational approach based entirely on the self-consistent equations. Also, various versions of RMFT are compared, and the most satisfactory of them is selected. This optimal version is subsequently used to study different versions of the original t-J Hamiltonian. As a result, upper critical concentration and doping dependence of the selected physical quantities (e.g. the superconducting gap and the Fermi velocity) is determined at low temperatures and in the absence of external magnetic field. We compare our findings both with theoretical results obtained from the Variational Monte Carlo (VMC) methods, as well as with the experimental data for selected cuprates. We show that the version of RMFT approach formulated in this Thesis provides a reasonable qualitative and in some cases semiquantitative rationalization of the principal characteristics of the hole-doped high-temperature superconductors at the optimal doping and in the overdoped regime. Possible extensions of the proposed analysis are mentioned at the end.
Strongly-correlated electrons in transition-metal oxides give rise to intriguing emergent phenomena, including high-temperature superconductivity in cuprates. While simplified one-band Hubbard models capture some aspects, explicitly describing the interplay of copper and oxygen orbitals -- as in the three-band Emery model -- is essential to capture the full phenomenology of cuprates. Quantum simulators based on ultracold atoms offer a promising route to study such systems in a controlled setting, but realizing realistic multi-orbital Hubbard models remains challenging. Here we propose an optical superlattice architecture that implements the three-band Emery model with ultracold fermions. By combining lattice beams with controllable interference, we engineer orbital degrees of freedom that reproduce key features of the cuprate band structure, while enabling independent control of orbital-dependent interactions and charge-transfer energy. We show that single-particle quantum walks can benchmark the resulting tight-binding model. Using determinant quantum Monte Carlo, we further investigate thermodynamic properties in the undoped regime and find a finite-temperature metal-insulator crossover accompanied by the onset of antiferromagnetic correlations accessible in current experiments. Finally, we apply a Hamiltonian learning protocol enabling to infer effective single-band Hubbard models from experimental realizations of Emery models. Our results provide a practical pathway to simulate multi-orbital Hubbard physics with quantum gas microscopes.
Achieving materials-specific predictions for large, realistic molecules and solids with strong electron correlations remains a long-standing challenge in quantum chemistry. This dissertation develops scalable ab initio quantum many-body methods for crystalline materials, demonstrating that rigorous optimization of theoretical approximations and numerical algorithms enables predictive computation even in strongly correlated systems. The work tackles three interconnected challenges, namely describing material-specific properties of strongly correlated superconductors, achieving linear scaling with k-points for large-scale periodic calculations, and establishing reliable methods for electron-phonon interactions. The following chapters address each challenge in turn. Chapter 2 demonstrates that ab initio density matrix embedding theory, combined with symmetry-breaking strategies for superconducting order parameters, reproduces experimental trends in cuprate superconductivity, including pressure and layer effects on transition temperatures. By directly solving the electronic Schrödinger equation from material structures, these calculations capture multi-orbital covalency effects that simplified model Hamiltonians neglect, revealing their essential role in the spin fluctuations driving pairing. This predictive capability establishes a foundation for efficient computational screening of superconducting materials. Extending these calculations to larger k-point meshes and broader phase diagram explorations, however, demands more efficient computational algorithms. To overcome this computational bottleneck, Chapter 3 introduces interpolative separable density fitting framework. This development enables thermodynamic limit calculations with up to 1000 k-points, as validated on diamond, carbon dioxide, nickel monoxide, and cuprate systems. Integration with density matrix embedding theory and local natural orbital methods further yields converged ground-state energies from accurate correlated wavefunction methods such as CCSD(T) for crystalline materials in the thermodynamic limit. With computational efficiency established, the remaining challenge lies in extending the theoretical framework beyond purely electronic degrees of freedom. Chapter 4 addresses this need by establishing exponential ansatz wavefunctions as competitive approaches for electron-phonon systems. Unlike previous treatments based on perturbation theory, systematic benchmarking of coupled cluster theory and variational Lang-Firsov methods on the Holstein model demonstrates, for the first time, that these approaches accurately describe polaron formation across coupling regimes when combined with appropriate reference state optimization. These results provide a foundation for incorporating phonon effects into ab initio studies of superconducting materials. In addition to these methodological advances, GPU acceleration and MPI parallelization enhance computational throughput, enabling efficient large-scale calculations. Together, these developments constitute a computational framework for predictive ab initio quantum many-body calculations in crystalline materials, opening new avenues for understanding and designing strongly correlated quantum materials.
Toward the Detection of Spin-Vortex-Induced Loop Currents in a Single Bilayer Bi2_22​Sr2_22​CaCu2_22​O8+δ_{8+\delta }8+δ​ Thin Film and Their Possible Use as Qubits: Model Calculations for Three Nano-island Architecture
Abstract A theory for cuprate superconductivity predicts the existence of nano-sized loop currents called “spin-vortex-induced loop currents (SVILCs).” In this work, we first calculate magnetic fields produced by them in a single bilayer Bi 2_22​ 2 Sr 2_22​ 2 CaCu 2_22​ 2 O 8+δ_{8+\delta }8+δ​ 8 + δ (Bi-2212) thin film for the purpose of detecting the SVILCs. The estimated magnitude of the magnetic field at the point 10a (a is the lattice constant of the CuO 2_22​ 2 plane) above the surface could be in the order of 100 mT; thus, they may be detectable by currently available detection methods. Next, we investigate the use of them as qubits (the “SVILC qubits”) in an architecture composed of three nano-islands of the thin film and consider the use of the detection of the magnetic field generated by the SVILCs as the qubit readout. We show there are a number of energy levels suitable for qubit states that can be manipulated by external current feeding, and the magnetic field generated by the SVILCs is large enough to be used for the readout.
A possibility of the electronic origin of the high-temperature superconductivity in cuprates is probed with the quantum Monte Carlo method by revisiting the three-band Hubbard model comprising Cu ${3d}_{{x}^{2}{\ensuremath{-}y}^{2}}$ and O ${2p}_{\ensuremath{\sigma}}$ orbitals. The ${d}_{{x}^{2}{\ensuremath{-}y}^{2}}$ pairing correlation is found to turn into an increasing function of the repulsion Ud{U}_{d}Ud​ within the ddd orbitals or the ddd- ppp level offset $\ensuremath{\Delta}\ensuremath{\varepsilon}$, where the normalized correlation grows with the system size. We have detected this in both the charge-transfer and Mott-Hubbard regimes upon entering the strong-correlation region ( Ud{U}_{d}Ud​ or $\ensuremath{\Delta}\ensuremath{\varepsilon}>$ bare bandwidth).