Always-Real-Eigenvalued Non-Hermitian Topological Systems

The effect of non-Hermiticity in band topology has sparked many discussions on non-Hermitian topological physics. It has long been known that non-Hermitian Hamiltonians can exhibit real energy spectra under the condition of parity-time ($PT$) symmetry -- commonly implemented with balanced loss and gain -- but only when non-Hermiticity is relatively weak. Sufficiently strong non-Hermiticity, on the other hand, will destroy the reality of energy spectra, a situation known as spontaneous $PT$-symmetry breaking. Here, based on non-reciprocal coupling, we show a systematic strategy to construct non-Hermitian topological systems exhibiting bulk and boundary energy spectra that are always real, regardless of weak or strong non-Hermiticity. Such nonreciprocal-coupling-based non-Hermiticity can directly drive a topological phase transition and determine the band topology, as demonstrated in a few non-Hermitian systems from 1D to 2D. Our work develops so far the only theory that can guarantee the reality of energy spectra for non-Hermitian Hamiltonians, and offers a new avenue to explore non-Hermitian topological physics

Constrained Likelihood Inference in Instrumental Variable Regression with Invalid Instruments and Its Application to GWAS Summary Data

University of Minnesota Ph.D. dissertation. May 2021. Major: Statistics. Advisors: Xiaotong Shen, Wei Pan. 1 computer file (PDF); viii, 102 pages.There has been increasing interest in instrumental variables regression for causal inference. In genetics, transcriptome-wide association studies (TWAS), also known as PrediXcan, have recently emerged as a widely applied tool to discover causal/target genes by integrating an outcome GWAS dataset with another gene expression/ transcriptome GWAS (called eQTL) dataset; they can not only boost statistical power but also offer biological insights by identifying (putative) causal genes for a GWAS trait, e.g. low-density lipoprotein cholesterol (LDL). Statistically TWAS apply (two-sample) two-stage least squares (2SLS) with multiple correlated SNPs as instrumental variables (IVs) to predict/impute gene expression, in contrast to typical (two-sample) Mendelian randomization (MR) approaches using independent SNPs as IVs, which are expected to be lower-powered. However, some of the SNPs used may not be valid IVs as a result of their (horizontal) pleiotropic/direct effects on the trait not mediated through the gene of interest, leading to false conclusions by TWAS (or MR). We propose a general inferential method for possibly high-dimensional data to account for confounding and invalid IVs while selecting valid IVs simultaneously via two-stage constrained maximum likelihood; we develop a theory for the likelihood method subject to a truncated L1-constraint approximating the L0-constraint for asymptotically valid and efficient statistical inference on causal effects. We demonstrate both theoretically and numerically the superior performance of the proposed method over the standard 2SLS/TWAS and other methods. We apply the methods to identify causal genes for LDL by integrating GWAS summary data with eQTL data.Xue, Haoran. (2021). Constrained Likelihood Inference in Instrumental Variable Regression with Invalid Instruments and Its Application to GWAS Summary Data. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/223167

Unsupervised Learning of Topological Non-Abelian Braiding in Non-Hermitian Bands

The topological classification of energy bands has laid the groundwork for the discovery of various topological phases of matter in recent decades. While this classification has traditionally focused on real-energy bands, recent studies have revealed the intriguing topology of complex-energy, or non-Hermitian bands. For example, the spectral winding of complex-energy bands can from unique topological structures like braids, holding promise for advancing quantum computing. However, discussions of complex-energy braids have been largely limited to the Abelian braid group $\mathbb{B}_2$ for its relative simplicity, while identifying topological non-Abelian braiding is still difficult since it has no universal topological invariant for characterization. Here, we present a machine learning algorithm for the unsupervised identification of non-Abelian braiding of multiple complex-energy bands. The consistency with Artin's well-known topological equivalence conditions in braiding is demonstrated. Inspired by the results from unsupervised learning, we also introduce a winding matrix as a topological invariant in charactering the braiding topology and unveiling the bulk-edge correspondence of non-Abelian braided non-Hermitian bands. Finally, we extend our approach to identify non-Abelian braiding topology in 2D/3D exceptional semimetals and successfully address the unknotting problem in an unsupervised manner

Particle Jetting Induced by the Impulsive Loadings

Particle rings/shells/cylinders dispersed by the radial impulsive loadings ranging from strong blast waves to moderate shock waves form a dual coherent jetting structure consisting of particle jets which have different dimensions. In both circumstances, the primary jets are found to initiate from the inner surface of particle layers and propagate through the thickness of particle layers, which are superimposed by a large number of much smaller secondary jets initiating from the outer surface of particle layers upon the reflection of the shock wave. This chapter first presents a summary of the experimental observations of the hierarchical particle jetting mainly via the cinematographic techniques, focusing on the characteristics of the primary particle jet structure. Due to the distinct behaviors of particles subjected to the strong blast and moderate shock waves, specifically solid-like and fluid-like responses, respectively, the explosive and shock-induced particle jetting should be attributed to distinct mechanisms. A dual particle jetting model from the perspective of continuum is proposed to account for the explosive-induced particle jetting. By contrast the shock-induced particle jetting arises from the localized particle shear flows around the inner surface of particle layers which result from the heterogeneous network of force chains

Non-Hermitian Dirac Cones

Non-Hermitian systems, which contain gain or loss, commonly host exceptional point degeneracies rather than the diabolic points found in Hermitian systems. We present a class of non-Hermitian lattice models with symmetry-stabilized diabolic points, such as Dirac or Weyl points. They exhibit non-Hermiticity-induced phenomena previously existing in the Hermitian regime, including topological phase transitions, Landau levels induced by pseudo-magnetic fields, and Fermi arc surface states. These behaviors are controllable via gain and loss, with promising applications in tunable active topological devices

Observation of Protected Photonic Edge States Induced By Real-Space Topological Lattice Defects

Topological defects (TDs) in crystal lattices are elementary lattice imperfections that cannot be removed by local perturbations, due to their real space topology. We show that adding TDs into a valley photonic crystal generates a lattice disclination that acts like a domain wall and hosts topological edge states. The disclination functions as a freeform waveguide connecting a pair of TDs of opposite topological charge. This interplay between the real-space topology of lattice defects and band topology provides a novel scheme to implement large-scale photonic structures with complex arrangements of robust topological waveguides and resonators

Scattering Dynamics and Boundary States of a Non-Hermitian Dirac Equation

We study a non-Hermitian variant of the (2+1)-dimensional Dirac wave equation, which hosts a real energy spectrum with pairwise-orthogonal eigenstates. In the spatially uniform case, the Hamiltonian's non-Hermitian symmetries allow its eigenstates to be mapped to a pair of Hermitian Dirac subsystems. When a wave is transmitted across an interface between two spatially uniform domains with different model parameters, an anomalous form of Klein tunneling can occur, whereby reflection is suppressed while the transmitted flux is substantially higher or lower than the incident flux. The interface can even function as a simultaneous laser and coherent perfect absorber. Remarkably, the violation of flux conservation occurs entirely at the interface, as no wave amplification or damping takes place in the bulk. Moreover, at energies within the Dirac mass gaps, the interface can support exponentially localized boundary states with real energies. These features of the continuum model can also be reproduced in non-Hermitian lattice models

Inference of nonlinear causal effects with GWAS summary data

Large-scale genome-wide association studies (GWAS) have offered an exciting opportunity to discover putative causal genes or risk factors associated with diseases by using SNPs as instrumental variables (IVs). However, conventional approaches assume linear causal relations partly for simplicity and partly for the only availability of GWAS summary data. In this work, we propose a novel model {for transcriptome-wide association studies (TWAS)} to incorporate nonlinear relationships across IVs, an exposure, and an outcome, which is robust against violations of the valid IV assumptions and permits the use of GWAS summary data. We decouple the estimation of a marginal causal effect and a nonlinear transformation, where the former is estimated via sliced inverse regression and a sparse instrumental variable regression, and the latter is estimated by a ratio-adjusted inverse regression. On this ground, we propose an inferential procedure. An application of the proposed method to the ADNI gene expression data and the IGAP GWAS summary data identifies 18 causal genes associated with Alzheimer's disease, including APOE and TOMM40, in addition to 7 other genes missed by two-stage least squares considering only linear relationships. Our findings suggest that nonlinear modeling is required to unleash the power of IV regression for identifying potentially nonlinear gene-trait associations. Accompanying this paper is our Python library nl-causal(https://github.com/nl-causal/nonlinear-causal) that implements the proposed method.Comment: 36 pages, 8 figure