Misc

Convolutional Neural Networks (CNNs) excel at extracting local features hierarchically, but their performance in capturing complex correlations hinges heavily on deep architectures, which are usually computationally demanding and difficult to interpret. To address these issues, we propose a physically-guided shallow model: tensor-augmented CNN (TACNN), which replaces conventional convolution kernels with generic tensors to enhance representational capacity. This choice is motivated by the fact that an order-$N$ tensor naturally encodes an arbitrary quantum superposition state in the Hilbert space of dimension $d^N$, where $d$ is the local physical dimension, thus offering substantially richer expressivity. Furthermore, in our design the convolution output of each layer becomes a multilinear form capable of capturing high-order feature correlations, thereby equipping a shallow multilayer architecture with an expressive power competitive to that of deep CNNs. On the Fashion-MNIST benchmark, TACNN demonstrates clear advantages over conventional CNNs, achieving remarkable accuracies with only a few layers. In particular, a TACNN with only two convolution layers attains a test accuracy of 93.7$\%$, surpassing or matching considerably deeper models such as VGG-16 (93.5$\%$) and GoogLeNet (93.7$\%$). These findings highlight TACNN as a promising framework that strengthens model expressivity while preserving architectural simplicity, paving the way towards more interpretable and efficient deep learning models.

Diagrammatic summation is a common bottleneck in modern applications of projected entangled-pair states (PEPS), especially in computing low-energy excitations of a two dimensional quantum many-body system. To solve this problem, here we extend the generating function approach for tensor network diagrammatic summation, previously proposed in the context of matrix product states. With the excited state expressed as a one-particle excitation, we show that relevant objects in determining the variational parameters, i.e. effective hamiltonian and norm matrices in the variational parameter space can be easily computed in the generating function formalism, which can further be used in evaluating the dynamical structure factor. Our benchmark results for the spin-$1/2$ transverse field Ising model and Heisenberg model on the square lattice provide a desirable accuracy, showing good agreement with known results. Then we study the spin-$1/2$ $J_1-J_2$ model on the same lattice and investigate the dynamical properties of the putative gapless spin liquid phase. We conclude with a discussion on generalizations to multi-particle excitations.

The nature of the quantum spin liquid (QSL) phase in the frustrated $J_1$-$J_2$ Heisenberg model on the triangular lattice remains an open and actively debated problem. In this work, we employ the infinite projected entangled-pair state (PEPS) to systematically investigate the model under different symmetry constraints. Our simulations reveal a direct transition from the $120^\circ$ Néel state to a putative QSL at $J_2/J_1\approx 0.08$, signaled by the collapse of magnetic order. We further show that, through either an appropriate unitary rotation or spontaneous spin long-range order, the stripe antiferromagnetic phase can also be accurately captured within the infinite PEPS framework. A central focus of our study is the role played by the PEPS symmetry in approximating the QSL ground-state sandwiched between the two magnetic phases. We first found that a fully-symmetric topological $\mathbb{Z}_2$ Resonating Valence Bond state, which can be written as a simple PEPS with bond dimension $D=3$, exhibits a reasonably good variational energy. Motivated by this finding, we have further constructed generic $\mathbb{Z}_2$-symmetric PEPS of larger bond dimension (up to $D=7$). We found that, under wavefunction optimization, spinons condense and, simultaneously, topological vison excitations get confined, hence precluding $\mathbb{Z}_2$ topological order. This strongly indicates the gapless (or critical) nature of the QSL phase, which is most naturally consistent with a U(1) Dirac spin liquid scenario.

Studying finite-temperature properties with tensor networks is notoriously difficult, especially at low temperatures, due to the rapid growth of entanglement and the complexity of thermal states. Existing methods like purification and minimally entangled typical thermal states offer partial solutions but struggle with scalability and accuracy in low-temperature regime. To overcome these limitations, we propose a new approach based on generating-function matrix product states (GFMPS). By directly computing a large set of Bloch-type excited states, we construct Gibbs states that moderate the area-law constraint, enabling accurate and efficient approximation of low-temperature thermal behavior. Our benchmark results show magnificent agreement with both exact diagonalization and experimental observations, validating the accuracy of our approach. This method offers a promising new direction for overcoming the longstanding challenges of studying low-temperature properties within the tensor network framework. We also expect that our method will facilitate the numerical simulation of quantum materials in comparison with experimental observations.

Recent advances in combining Clifford circuits with tensor-network (TN) methods have shown that classically simulable disentanglers can suppress substantial portions of the entanglement structure, effectively alleviating the bond-dimension bottleneck in TN simulations. In this work, we develop a variational TN framework based on Grassmann tensor networks, which natively encode fermionic statistics while preserving locality. By incorporating locally defined Clifford circuits within the fermionic formalism, we simulate benchmark models including the tight-binding and $t$-$V$ models. Our results show that Clifford disentangling removes the classically simulable component of entanglement, leading to a reduced bond dimension and improved accuracy in ground-state energy estimates. Interestingly, once the natural Grassmann-evenness requirement of the fermionic formulation is taken into account and Clifford gates with identical entanglement action are grouped together, the original set of 11520 two-qubit Clifford gates reduces to only 12 distinct gates. This strong reduction leads to a more efficient disentangling scheme within the fermionic framework. These findings highlight the potential of Clifford-augmented Grassmann TNs as a scalable and accurate tool for studying strongly correlated fermionic systems, particularly in higher dimensions.

Understanding the dynamical evolution of large-scale moiré systems is crucial for connecting theoretical predictions with experimental observations. Here we develop a machine-learning-based workflow, integrating DeePMD and DeepH frameworks with first-principles calculations, to efficiently investigate time-dependent structural and electronic responses in twisted bilayer transition metal dichalcogenides (TMDs) with experimentally relevant moiré supercells containing over 3000 atoms. Using $\mathrm{WS_2}$ as a representative system, we show that low-temperature lattice vibrations and relaxation deepen the moiré potential wells, narrow the lowest conduction band, and facilitate the formation of strongly localized electronic states. Based on DFT-derived moiré potentials that incorporate these dynamical effects, density-matrix-renormalization-group (DMRG) simulations reveal robust Wigner crystallization and a kagomé-patterned three-electron state, consistent with recent experimental observations. Our workflow provides a practical route for exploring large moiré supercells beyond static configurations and offers new insight into the interplay between lattice dynamics, electronic localization, and emergent correlated states in twisted two-dimensional materials.