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Federated Learning (FL) has emerged to allow multiple clients to collaboratively train machine learning models on their private data at the network edge. However, training and deploying large-scale models on resource-constrained devices is challenging. Fortunately, Split Federated Learning (SFL) offers a feasible solution by alleviating the computation and/or communication burden on clients. However, existing SFL works often assume sufficient labeled data on clients, which is usually impractical. Besides, data non-IIDness poses another challenge to ensure efficient model training. To our best knowledge, the above two issues have not been simultaneously addressed in SFL. Herein, we propose a novel Semi-supervised SFL system, termed SemiSFL, which incorporates clustering regularization to perform SFL with unlabeled and non-IID client data. Moreover, our theoretical and experimental investigations into model convergence reveal that the inconsistent training processes on labeled and unlabeled data have an influence on the effectiveness of clustering regularization. To mitigate the training inconsistency, we develop an algorithm for dynamically adjusting the global updating frequency, so as to improve training performance. Extensive experiments on benchmark models and datasets show that our system provides a 3.8x speed-up in training time, reduces the communication cost by about 70.3% while reaching the target accuracy, and achieves up to 5.8% improvement in accuracy under non-IID scenarios compared to the state-of-the-art baselines., Comment: 16 pages

The enigma of unconventional superconductivity in doped cuprates presents a formidable challenge in the realm of condensed matter physics. Recent findings of strong near-neighbor attractions in one-dimensional cuprate chains suggest a new avenue for investigating cuprate superconductors. Consequently, we revisited the superconductivity in the extended Hubbard model at the mean-field level. Anticipating a prevalence of antiferromagnetic order due to strong local Coulomb repulsion, our calculations reveal the coexistence of superconducting and antiferromagnetic orders across a wide range of doping at sufficiently low temperatures. The mean-field results capture some key features of cuprate superconductors, including $d$-wave pairing symmetry, a dome-shaped dependence of $T_c$ on doping, and higher superconducting transition temperatures. Additionally, we observed a nearly proportional relationship between $T_c$ and the strength of the nearest-neighbor attraction, reminiscent of experimental findings at the FeSe/SrTiO3 interface. The mean-field results suggest that the extended Hubbard model could be the appropriate framework for investigating cuprate superconductivity and offer insights for more precise calculations within this model in future., Comment: revised,19 pages,4 figures

The Fermi surface topology in the two-dimensional Hubbard model is particularly relevant for the high-temperature superconductors, whereas its theoretical research encounters with the difficulty of the analytical continuation problem. To this end, we proposed the concept of the momentum-dependent compressibility, defined as the variation of the momentum distribution function with respect to the chemical potential. The surface determined by the maximum of the momentum-dependent compressibility is nearly identical to the Fermi surface in the weakly and intermediate coupling regions according to our numerical results. In the correlated region, this surface also exhibits pocket and arc features, just like the Fermi surface in high-temperature superconductors. Therefore, for theoretical studies, this surface can be used as an alternative to determine the underlying Fermi surface. Considering that the momentum-dependent compressibility is closely related to the charge density correlations, our work also shows a connection between the Fermi surface topology and the charge density fluctuations., Comment: 3 pages, 4 figures

The calculation of response functions in correlated electronic systems is one of the most important problems in the condensed matter physics. To obtain a physical response function, preserving both the Ward-Takahashi identity and the fluctuation-dissipation theorem are crucial. Here we propose a self-consistent many body method within the GW framework to calculate the response functions based on the fluctuation-dissipation theorem, which also satisfies the Ward-Takahashi identity. The validity of this methodology is demonstrated on the two-dimensional one-band Hubbard model, where both the Ward-Takahashi identity and fluctuation-dissipation theorem are verified numerically. Moreover, comparing to the accurate spin susceptibility of the determinantal Monte Carlo approach, the results obtained from our method are quite satisfactory and the computational cost are greatly reduced., Comment: 12 pages, 4 figures, supplemental material 15 pages

A modified $GW$ approximation to many - body systems is developed. The approximation has the same computational complexity as the traditional $GW$ approach, but uses a different truncation scheme. This scheme neglects high order connected correlation functions. A covariant (preserving Ward identities due to charge conservation) scheme for two - body correlators is employed, which holds the relation between the charge correlator and charge susceptibility. The method is tested on the two - dimensional one - band Hubbard model. Results are compared with exact diagonalization, the fluctuation - exchange (FLEX) theory and determinantal quantum Monte Carlo (DQMC) approach. The comparison for the (one - body) Green's function demonstrates that it is more precise in strong - coupling regime (especially away from half - filling) than similar - complexity approximations $GW$ or FLEX. The charge correlator is in excellent agreement with the numerically exact result obtained from DQMC., Comment: 21 pages, 7 figures

We developed a systematic non-perturbative method base on Dyson-Schwinger theory and the $\Phi$-derivable theory for Ising model at broken phase. Based on these methods, we obtain critical temperature and spin spin correlation beyond mean field theory. The spectrum of Green function obtained from our methods become gapless at critical point, so the susceptibility become divergent at Tc. The critical temperature of Ising model obtained from this method is fairly good in comparison with other non-cluster methods. It is straightforward to extend this method to more complicate spin models for example with continue symmetry., Comment: 15 pages, 4 figures

Strongly correlated electron systems are generally described by tight binding lattice Hamiltonians with strong local (on site) interactions, the most popular being the Hubbard model. Although the half filled Hubbard model can be simulated by Monte Carlo(MC), physically more interesting cases beyond half filling are plagued by the sign problem. One therefore should resort to other methods. It was demonstrated recently that a systematic truncation of the set of Dyson-Schwinger equations for correlators of the Hubbard, supplemented by a \textquotedblleft covariant" calculation of correlators leads to a convergent series of approximants. The covariance preserves all the Ward identities among correlators describing various condensed matter probes. While first order (classical), second (Hartree-Fock or gaussian) and third (Cubic) covariant approximation were worked out, the fourth (quartic) seems too complicated to be effectively calculable in fermionic systems. It turns out that the complexity of the quartic calculation\ in local interaction models,is manageable computationally. The quartic (Bethe - Salpeter type) approximation is especially important in 1D and 2D models in which the symmetry broken state does not exists (the Mermin - Wagner theorem), although strong fluctuations dominate the physics at strong coupling. Unlike the lower order approximations, it respects the Mermin - Wagner theorem. The scheme is tested and exemplified on the single band 1D and 2D Hubbard model., Comment: 20 pages, 4 figures