Ground State Degeneracy of Infinite-Component Chern-Simons-Maxwell Theories

Infinite-component Chern-Simons-Maxwell theories with a periodic $K$ matrix provide abundant examples of gapped and gapless, foliated and non-foliated fracton orders. In this paper, we study the ground state degeneracy of these theories. We show that the ground state degeneracy exhibit various patterns as a function of the linear system size -- the size of the $K$ matrix. It can grow exponentially or polynomially, cycle over finitely many values, or fluctuate erratically inside an exponential envelope. We relate these different patterns of the ground state degeneracy with the roots of the ``determinant polynomial'', a Laurent polynomial, associated to the periodic $K$ matrix. These roots also determine whether the theory is gapped or gapless. Based on the ground state degeneracy, we formulate a necessary condition for a gapped theory to be a foliated fracton order.Comment: 15 pages, 1 figure; the authors are ordered alphabeticall

Ground state degeneracy of the Ising cage-net model

The Ising cage-net model, first proposed in Phys. Rev. X 9, 021010 (2019), is a representative type I fracton model with nontrivial non-abelian features. In this paper, we calculate the ground state degeneracy of this model and find that, even though it follows a similar coupled layer structure as the X-cube model, the Ising cage-net model cannot be "foliated" in the same sense as X-cube as defined in Phys. Rev. X 8, 031051 (2018). A more generalized notion of "foliation'' is hence needed to understand the renormalization group transformation of the Ising cage-net model. The calculation is done using an operator algebra approach that we develop in this paper, and we demonstrate its validity through a series of examples

Fractonic order in infinite-component Chern-Simons gauge theories

2+1D multi-component U(1) gauge theories with a Chern-Simons (CS) term provide a simple and complete characterization of 2+1D Abelian topological orders. In this paper, we extend the theory by taking the number of component gauge fields to infinity and find that they can describe interesting types of 3+1D "fractonic" order. "Fractonic" describes the peculiar phenomena that point excitations in certain strongly interacting systems either cannot move at all or are only allowed to move in a lower dimensional sub-manifold. In the simplest cases of infinite-component CS gauge theory, different components do not couple to each other and the theory describes a decoupled stack of 2+1D fractional Quantum Hall systems with quasi-particles moving only in 2D planes -- hence a fractonic system. We find that when the component gauge fields do couple through the CS term, more varieties of fractonic orders are possible. For example, they may describe foliated fractonic systems for which increasing the system size requires insertion of nontrivial 2+1D topological states. Moreover, we find examples which lie beyond the foliation framework, characterized by 2D excitations of infinite order and braiding statistics that are not strictly local

Fractonic Orders from Lattice Models and Field Theories

Fracton models are characterized by exotic features such as point-like excitations with restricted mobility, sub-extensive ground state degeneracy and UV/IR mixing. They have been studied previously using exactly solvable lattice models, higher rank gauge theories, etc. In an effort to classify fracton models into phases (i.e., fractonic orders), the so-called foliation structure has been introduced and shown to exist in many previously known models. A natural question then arises concerning the feasibility of the foliation paradigm in general. In this thesis, I study fracton models beyond the foliation paradigm and give simple diagnostics for the absence of a foliation structure. New notions of fractonic orders therefore need to be conceived, and I present such a conception which is a generalization of the foliation RG. In Chapters 2 - 4, I introduce new fracton models obtained from infinite-component Chern-Simons (CS∞) theories. By calculating observables such as ground state degeneracy and planon braiding statistics, I prove that most CS∞ theories are not foliated. A CS∞ theory can also be gapless with certain choices of parameters, and I show that such a theory is a stable gapless fracton model. Furthermore, I discuss topological features of a large subclass of gapless CS∞ theories and present fully continuous effective field theories for this subclass. In Chapters 5 - 6, I discuss a new notion of fractonic orders by studying the example of the Ising cage-net model. I begin by calculating the ground state degeneracy of the model, which shows that the model is not foliated. The calculation uses an operator algebra approach which relies only on intrinsic physical properties of the model rather than microscopic details, and I establish the framework of this approach conceptually and via examples. I then argue why this intrinsic approach, despite being a tool for calculation initially, may be a useful characterization of a fractonic order. Finally, I present a generalized foliation RG scheme, apply it to the Ising cage-net model, and discuss its limitations.</p

A Study on Using Location-Information-Based Flow Field Reconstruction to Model the Characteristics of a Discharging Valve in a Hydrodynamic Retarder

In modeling the characteristics of a discharging valve in a hydrodynamic retarder, it is commonly required to determine the value of the flow area to calculate the force on the spool. However, the flow area often relies heavily on empirical or simulation data, which leads to increased uncertainty and computational cost, especially with the variation in the spool displacement. To overcome these shortcomings, Res-SE-U-Nets (networks that combine residual connections, squeeze-and-excitation blocks, and U-Net) are used to reconstruct the velocity field, and they have shown exceptional performance in image-to-image mapping tasks. The dataset of computational fluid dynamics (CFD) results for the velocity field is collected and verified using particle image velocimetry (PIV). The results show that Res-SE-U-Nets can capture the location information of the flow field using a training set of only 120 data points. By utilizing location information in velocity field reconstruction, the flow area can be directly obtained under different spool displacements and pressures to calculate the spool force. The valve characteristics calculated with this method show an error of less than 2% when compared with CFD results, which confirms the validity and effectiveness of this method. The proposed method, which utilizes location information extracted from flow field prediction results, is capable of calculating valve characteristics. This approach also demonstrates the feasibility of using Res-SE-U-Nets for flow field reconstruction

The effect of text-based math task on dynamic stability control during stair descent (ID: BM-D-20-00079R3)

© 2020 Elsevier Ltd Stair descent imposes a significant challenge for dynamic stability among young adults. The effect of a concurrent text-based math task on dynamic stability control remains unclear during stair descent when considering the influence of gait velocity. Twenty-six participants performed three successful stair descent trials under TEXTING or NO-TEXTING conditions at their preferred speed. Synchronous kinematics and kinetics were collected by an eight-camera Vicon infrared motion capture system and two force platforms. Repeated measures analysis of covariance and Wilcoxon signed rank test were used to analyze the differences between the two different task conditions with gait velocity as a covariate. The outcomes indicated that under TEXTING condition, sagittal margin of stability increased at right-foot-landing; step cadence, double-support percentage, sagittal and frontal joint moment decreased; and sagittal and frontal joint angles were also modified. It is concluded that concurrent TEXTING impaired sagittal and frontal stability control during stair descent despite slowing down the step cadence. Knee and ankle joint adjustment strategies were mainly adopted in response to stability control in the sagittal plane with the interference of TEXTING, whereas the hip joint adjustment strategy was adopted in the frontal plane. In conclusion, texting behaviors on mobile phones should be minimized during stair descent