Entanglement Entropy From Tensor Network States for Stabilizer Codes

In this paper, we present the construction of tensor network states (TNS) for some of the degenerate ground states of 3D stabilizer codes. We then use the TNS formalism to obtain the entanglement spectrum and entropy of these ground-states for some special cuts. In particular, we work out the examples of the 3D toric code, the X-cube model and the Haah code. The latter two models belong to the category of "fracton" models proposed recently, while the first one belongs to the conventional topological phases. We mention the cases for which the entanglement entropy and spectrum can be calculated exactly: for these, the constructed TNS is the singular value decomposition (SVD) of the ground states with respect to particular entanglement cuts. Apart from the area law, the entanglement entropies also have constant and linear corrections for the fracton models, while the entanglement entropies for the toric code models only have constant corrections. For the cuts we consider, the entanglement spectra of these three models are completely flat. We also conjecture that the negative linear correction to the area law is a signature of extensive ground state degeneracy. Moreover, the transfer matrices of these TNS can be constructed. We show that the transfer matrices are projectors whose eigenvalues are either 1 or 0. The number of nonzero eigenvalues is tightly related to the ground state degeneracy.Comment: 33+9 pages. 16+3 figure

Entanglement entropy of (3+1)D topological orders with excitations

Excitations in (3+1)D topologically ordered phases have very rich structures. (3+1)D topological phases support both point-like and string-like excitations, and in particular the loop (closed string) excitations may admit knotted and linked structures. In this work, we ask the question how different types of topological excitations contribute to the entanglement entropy, or alternatively, can we use the entanglement entropy to detect the structure of excitations, and further obtain the information of the underlying topological orders? We are mainly interested in (3+1)D topological orders that can be realized in Dijkgraaf-Witten gauge theories, which are labeled by a finite group $G$ and its group 4-cocycle $\omega\in\mathcal{H}^4[G;U(1)]$ up to group automorphisms. We find that each topological excitation contributes a universal constant $\ln d_i$ to the entanglement entropy, where $d_i$ is the quantum dimension that depends on both the structure of the excitation and the data $(G,\,\omega)$. The entanglement entropy of the excitations of the linked/unlinked topology can capture different information of the DW theory $(G,\,\omega)$. In particular, the entanglement entropy introduced by Hopf-link loop excitations can distinguish certain group 4-cocycles $\omega$ from the others.Comment: 12 pages, 4 figures; v2: minor changes, published versio

Towards Graph-Aware Diffusion Modeling for Collaborative Filtering

Recovering masked feedback with neural models is a popular paradigm in recommender systems. Seeing the success of diffusion models in solving ill-posed inverse problems, we introduce a conditional diffusion framework for collaborative filtering that iteratively reconstructs a user's hidden preferences guided by its historical interactions. To better align with the intrinsic characteristics of implicit feedback data, we implement forward diffusion by applying synthetic smoothing filters to interaction signals on an item-item graph. The resulting reverse diffusion can be interpreted as a personalized process that gradually refines preference scores. Through graph Fourier transform, we equivalently characterize this model as an anisotropic Gaussian diffusion in the graph spectral domain, establishing both forward and reverse formulations. Our model outperforms state-of-the-art methods by a large margin on one dataset and yields competitive results on the others.Comment: 13 pages, 6 figure

Intrinsically/Purely Gapless-SPT from Non-Invertible Duality Transformations

The Kennedy-Tasaki (KT) transformation was used to construct the gapped symmetry protected topological (SPT) phase from the symmetry breaking phase with open boundary condition, and was generalized in our proceeding work [L. Li et al. arXiv:2301.07899] on a ring by sacrificing the unitarity, and should be understood as a non-invertible duality transformation. In this work, we further apply the KT transformation to systematically construct gapless symmetry protected topological phases. This construction reproduces the known examples of (intrinsically) gapless SPT where the non-trivial topological features come from the gapped sectors by means of decorated defect constructions. We also construct new (intrinsically) purely gapless SPTs where there are no gapped sectors, hence are beyond the decorated defect construction. This construction elucidates the field theory description of the various gapless SPTs, and can also be applied to analytically study the stability of various gapless SPT models on the lattice under certain symmetric perturbations

Structure of the Entanglement Entropy of (3+1)D Gapped Phases of Matter

We study the entanglement entropy of gapped phases of matter in three spatial dimensions. We focus in particular on size-independent contributions to the entropy across entanglement surfaces of arbitrary topologies. We show that for low energy fixed-point theories, the constant part of the entanglement entropy across any surface can be reduced to a linear combination of the entropies across a sphere and a torus. We first derive our results using strong sub-additivity inequalities along with assumptions about the entanglement entropy of fixed-point models, and identify the topological contribution by considering the renormalization group flow; in this way we give an explicit definition of topological entanglement entropy $S_{\mathrm{topo}}$ in (3+1)D, which sharpens previous results. We illustrate our results using several concrete examples and independent calculations, and show adding "twist" terms to the Lagrangian can change $S_{\mathrm{topo}}$ in (3+1)D. For the generalized Walker-Wang models, we find that the ground state degeneracy on a 3-torus is given by $\exp(-3S_{\mathrm{topo}}[T^2])$ in terms of the topological entanglement entropy across a 2-torus. We conjecture that a similar relationship holds for Abelian theories in $(d+1)$ dimensional spacetime, with the ground state degeneracy on the $d$-torus given by $\exp(-dS_{\mathrm{topo}}[T^{d-1}])$.Comment: 34 pages, 16 figure

Symmetry TFTs for Non-Invertible Defects

Given any symmetry acting on a $d$-dimensional quantum field theory, there is an associated $(d+1)$-dimensional topological field theory known as the Symmetry TFT (SymTFT). The SymTFT is useful for decoupling the universal quantities of quantum field theories, such as their generalized global symmetries and 't Hooft anomalies, from their dynamics. In this work, we explore the SymTFT for theories with Kramers-Wannier-like duality symmetry in both $(1+1)$d and $(3+1)$d quantum field theories. After constructing the SymTFT, we use it to reproduce the non-invertible fusion rules of duality defects, and along the way we generalize the concept of duality defects to \textit{higher} duality defects. We also apply the SymTFT to the problem of distinguishing intrinsically versus non-intrinsically non-invertible duality defects in $(1+1)$d.Comment: 119 pages, 46 figures; v2: references added, typos corrected; v3: publication versio