Bulk-surface partial differential equations(BS-PDEs)are prevalent in manyapplications such as cellular,developmental and plant biology as well as in engineeringand material sciences.Novel numerical methods for BS-PDEs...Bulk-surface partial differential equations(BS-PDEs)are prevalent in manyapplications such as cellular,developmental and plant biology as well as in engineeringand material sciences.Novel numerical methods for BS-PDEs in three space dimensions(3D)are sparse.In this work,we present a bulk-surface virtual elementmethod(BS-VEM)for bulk-surface reaction-diffusion systems,a form of semilinearparabolic BS-PDEs in 3D.Unlike previous studies in two space dimensions(2D),the3D bulk is approximated with general polyhedra,whose outer faces constitute a flatpolygonal approximation of the surface.For this reason,the method is restricted tothe lowest order case where the geometric error is not dominant.The BS-VEM guaranteesall the advantages of polyhedral methods such as easy mesh generation andfast matrix assembly on general geometries.Such advantages are much more relevantthan in 2D.Despite allowing for general polyhedra,general nonlinear reaction kineticsand general surface curvature,the method only relies on nodal values without needingadditional evaluations usually associated with the quadrature of general reactionkinetics.This latter is particularly costly in 3D.The BS-VEM as implemented in thisstudy retains optimal convergence of second order in space.展开更多
Topological phase of matter is now a mainstream of research in condensed matter physics, of which the classification, synthesis, and detection of topological states have brought excitements over the recent decade whil...Topological phase of matter is now a mainstream of research in condensed matter physics, of which the classification, synthesis, and detection of topological states have brought excitements over the recent decade while remain incomplete with ongoing challenges in both theory and experiment. Here we propose to establish a universal non-equilibrium characterization of the equilibrium topological quantum phases classified by integers, and further propose the high-precision dynamical schemes to detect such states. The framework of the dynamical classification theory consists of basic theorems. First, we uncover that classifying a d-dimensional(dD) gapped topological phase of generic multibands can reduce to a(d-1)D invariant defined on so-called band inversion surfaces(BISs), rendering a bulk-surface duality which simplifies the topological characterization. Further, we show in quenching across phase boundary the(pseudo) spin dynamics to exhibit unique topological patterns on BISs, which are attributed to the post-quench bulk topology and manifest a dynamical bulk-surface correspondence. For this the topological phase is classified by a dynamical topological invariant measured from an emergent dynamical spintexture field on the BISs. Applications to quenching experiments on feasible models are proposed and studied, demonstrating the new experimental strategies to detect topological phases with high feasibility. This work opens a broad new direction to classify and detect topological phases by non-equilibrium quantum dynamics.展开更多
基金Regione Puglia(Italy)through the research programme REFIN-Research for Innovation(protocol code 901D2CAA,project No.UNISAL026)MF acknowledges support from the Italian National Institute of High Mathematics(INdAM)through the INdAM-GNCS Project no.CUP E55F22000270001+3 种基金the Global Challenges Research Fund through the Engineering and Physical Sciences Research Council grant number EP/T00410X/1:UK-Africa Postgraduate Advanced Study Institute in Mathematical Sciences,the Health Foundation(1902431)the NIHR(NIHR133761)and by the Discovery Grant awarded by Canadian Natural Sciences and Engineering Research Council(2023-2028)AM acknowledges support from the Royal Society Wolfson Research Merit Award funded generously by the Wolfson Foundation(2016-2021)AM is a Distinguished Visiting Scholar to the Department of Mathematics,University of Johannesburg,South Africa,and the University of Pretoria in South Africa.IS and MF are members of the INdAM-GNCS activity group.The work of IS is supported by the PRIN 2020 research project(no.2020F3NCPX)”Mathematics for Industry 4.0”,and from the”National Centre for High Performance Computing,Big Data and Quantum Computing”funded by European Union-NextGenerationEU,PNRR project code CN00000013,CUP F83C22000740001.
文摘Bulk-surface partial differential equations(BS-PDEs)are prevalent in manyapplications such as cellular,developmental and plant biology as well as in engineeringand material sciences.Novel numerical methods for BS-PDEs in three space dimensions(3D)are sparse.In this work,we present a bulk-surface virtual elementmethod(BS-VEM)for bulk-surface reaction-diffusion systems,a form of semilinearparabolic BS-PDEs in 3D.Unlike previous studies in two space dimensions(2D),the3D bulk is approximated with general polyhedra,whose outer faces constitute a flatpolygonal approximation of the surface.For this reason,the method is restricted tothe lowest order case where the geometric error is not dominant.The BS-VEM guaranteesall the advantages of polyhedral methods such as easy mesh generation andfast matrix assembly on general geometries.Such advantages are much more relevantthan in 2D.Despite allowing for general polyhedra,general nonlinear reaction kineticsand general surface curvature,the method only relies on nodal values without needingadditional evaluations usually associated with the quadrature of general reactionkinetics.This latter is particularly costly in 3D.The BS-VEM as implemented in thisstudy retains optimal convergence of second order in space.
基金supported by the National Key Research and Development Program of China (2016YFA0301604)National Natural Science Foundation of China (11574008 and 11761161003)the Thousand-Young-Talent Program of China
文摘Topological phase of matter is now a mainstream of research in condensed matter physics, of which the classification, synthesis, and detection of topological states have brought excitements over the recent decade while remain incomplete with ongoing challenges in both theory and experiment. Here we propose to establish a universal non-equilibrium characterization of the equilibrium topological quantum phases classified by integers, and further propose the high-precision dynamical schemes to detect such states. The framework of the dynamical classification theory consists of basic theorems. First, we uncover that classifying a d-dimensional(dD) gapped topological phase of generic multibands can reduce to a(d-1)D invariant defined on so-called band inversion surfaces(BISs), rendering a bulk-surface duality which simplifies the topological characterization. Further, we show in quenching across phase boundary the(pseudo) spin dynamics to exhibit unique topological patterns on BISs, which are attributed to the post-quench bulk topology and manifest a dynamical bulk-surface correspondence. For this the topological phase is classified by a dynamical topological invariant measured from an emergent dynamical spintexture field on the BISs. Applications to quenching experiments on feasible models are proposed and studied, demonstrating the new experimental strategies to detect topological phases with high feasibility. This work opens a broad new direction to classify and detect topological phases by non-equilibrium quantum dynamics.
