Symmetric Mass Generation in the 1+1 Dimensional Chiral Fermion 3-4-5-0 Model.

Lattice regularization of chiral fermions has been a long-standing problem in physics. In this Letter, we present the density matrix renormalization group simulation of the 3-4-5-0 model of (1+1)D chiral fermions with an anomaly-free chiral U(1) symmetry, which contains two left-moving and two right-moving fermions carrying U(1) charges 3,4 and 5,0, respectively. Following the Wang-Wen chiral fermion model, we realize the chiral fermions and their mirror partners on the opposite boundaries of a thin strip of (2+1)D lattice model of multilayer Chern insulator, whose finite width implies the quantum system is effectively (1+1)D. By introducing two sets of carefully designed six-fermion local interactions to the mirror sector only, we demonstrate that the mirror fermions can be gapped out by the interaction beyond a critical strength without breaking the chiral U(1) symmetry, via the symmetric mass generation mechanism. We show that the interaction-driven gapping transition is in the Berezinskii-Kosterlitz-Thouless universality class. We determine the evolution of Luttinger parameters before the transition, which confirms that the transition happens exactly at the point when the interaction term becomes marginal. As the mirror sector is gapped after the transition, we check that the fermions in the light chiral fermion sector remain gapless, which provides the desired lattice regularization of chiral fermions.

Boundary Supersymmetry of (1+1)D Fermionic Symmetry-Protected Topological Phases.

We prove that the boundaries of all nontrivial (1+1)-dimensional intrinsically fermionic symmetry-protected-topological phases, protected by finite on-site symmetries (unitary or antiunitary), are supersymmetric quantum mechanical systems. This supersymmetry does not require any fine-tuning of the underlying Hamiltonian, arises entirely as a consequence of the boundary 't Hooft anomaly that classifies the phase, and is related to a "Bose-Fermi" degeneracy different in nature from other well known degeneracies such as Kramers doublets.

Field-theory representation of gauge-gravity symmetry-protected topological invariants, group cohomology, and beyond.

The challenge of identifying symmetry-protected topological states (SPTs) is due to their lack of symmetry-breaking order parameters and intrinsic topological orders. For this reason, it is impossible to formulate SPTs under Ginzburg-Landau theory or probe SPTs via fractionalized bulk excitations and topology-dependent ground state degeneracy. However, the partition functions from path integrals with various symmetry twists are universal SPT invariants, fully characterizing SPTs. In this work, we use gauge fields to represent those symmetry twists in closed spacetimes of any dimensionality and arbitrary topology. This allows us to express the SPT invariants in terms of continuum field theory. We show that SPT invariants of pure gauge actions describe the SPTs predicted by group cohomology, while the mixed gauge-gravity actions describe the beyond-group-cohomology SPTs. We find new examples of mixed gauge-gravity actions for U(1) SPTs in (4+1)D via the gravitational Chern-Simons term. Field theory representations of SPT invariants not only serve as tools for classifying SPTs, but also guide us in designing physical probes for them. In addition, our field theory representations are independently powerful for studying group cohomology within the mathematical context.

Gapped domain walls, gapped boundaries, and topological degeneracy.

Gapped domain walls, as topological line defects between (2+1)D topologically ordered states, are examined. We provide simple criteria to determine the existence of gapped domain walls, which apply to both Abelian and non-Abelian topological orders. Our criteria also determine which (2+1)D topological orders must have gapless edge modes, namely, which (1+1)D global gravitational anomalies ensure gaplessness. Furthermore, we introduce a new mathematical object, the tunneling matrix W, whose entries are the fusion-space dimensions W(ia), to label different types of gapped domain walls. By studying many examples, we find evidence that the tunneling matrices are powerful quantities to classify different types of gapped domain walls. Since a gapped boundary is a gapped domain wall between a bulk topological order and the vacuum, regarded as the trivial topological order, our theory of gapped domain walls inclusively contains the theory of gapped boundaries. In addition, we derive a topological ground state degeneracy formula, applied to arbitrary orientable spatial 2-manifolds with gapped domain walls, including closed 2-manifolds and open 2-manifolds with gapped boundaries.

Gene-mating dynamic evolution theory II: global stability of N-gender-mating polyploid systems.

Extending the previous 2-gender dioecious diploid gene-mating evolution model, we attempt to answer "whether the Hardy-Weinberg global stability and the exact analytic dynamical solutions can be found in the generalized N-gender N-polyploid gene-mating system with arbitrary number of alleles?" For a 2-gender gene-mating evolution model, a pair of male and female determines the trait of their offspring. Each of the pair contributes one inherited character, the allele, to combine into the genotype of their offspring. Hence, for an N-gender N-polypoid gene-mating model, each of N different genders contributes one allele to combine into the genotype of their offspring. We exactly solve the analytic solution of N-gender-mating $(n+1)$-alleles governing highly nonlinear coupled differential equations in the genotype frequency parameter space for any positive integer N and $n$. For an analogy, the 2-gender to N-gender gene-mating equation generalization is analogs to the 2-body collision to the N-body collision Boltzmann equations with continuous distribution functions of discretized variables instead of continuous variables. We find their globally stable solution as a continuous manifold and find no chaos. Our solution implies that the Laws of Nature, under our assumptions, provide no obstruction and no chaos to support an N-gender gene-mating stable system.

Gene-mating dynamic evolution theory: fundamental assumptions, exactly solvable models and analytic solutions.

Fundamental properties of macroscopic gene-mating dynamic evolutionary systems are investigated. A model is studied to describe a large class of systems within population genetics. We focus on a single locus, any number of alleles in a two-gender dioecious population. Our governing equations are time-dependent continuous differential equations labeled by a set of parameters, where each parameter stands for a population percentage carrying certain common genotypes. The full parameter space consists of all allowed parameters of these genotype frequencies. Our equations are uniquely derived from four fundamental assumptions within any population: (1) a closed system; (2) average-and-random mating process (mean-field behavior); (3) Mendelian inheritance; and (4) exponential growth and exponential death. Even though our equations are nonlinear with time-evolutionary dynamics, we have obtained an exact analytic time-dependent solution and an exactly solvable model. Our findings are summarized from phenomenological and mathematical viewpoints. From the phenomenological viewpoint, any initial parameter of genotype frequencies of a closed system will eventually approach a stable fixed point. Under time evolution, we show (1) the monotonic behavior of genotype frequencies, (2) any genotype or allele that appears in the population will never become extinct, (3) the Hardy-Weinberg law and (4) the global stability without chaos in the parameter space. To demonstrate the experimental evidence for our theory, as an example, we show a mapping from the data of blood type genotype frequencies of world ethnic groups to our stable fixed-point solutions. From the mathematical viewpoint, our highly symmetric governing equations result in continuous global stable equilibrium solutions: these solutions altogether consist of a continuous curved manifold as a subspace of the whole parameter space of genotype frequencies. This fixed-point manifold is a global stable attractor known as the Hardy-Weinberg manifold, attracting any initial point in any Euclidean fiber bounded within the genotype frequency space to the fixed point where this fiber is attached. The stable base manifold and its attached fibers form a fiber bundle, which fills in the whole genotype frequency space completely. We can define the genetic distance of two populations as their geodesic distance on the equilibrium manifold. In addition, the modification of our theory under the process of natural selection and mutation is addressed.

Dynamics of growing surfaces by linear equations in 2+1 dimensions.

An extensive study on the (2+1) -dimensional super-rough growth processes, described by a special class of linear growth equations, is undertaken. This special class of growth equations is of theoretical interests since they are exactly solvable and thus provide a window for understanding the intriguing anomalous scaling behaviors of super-rough interfaces. We first work out the exact solutions of the interfacial heights and the equal-time height difference correlation functions. Through our rigorous analysis, the detailed asymptotics of the correlation function in various time regimes are derived. Our obtained analytical results not only affirm the applicability of anomalous dynamic scaling ansatz but also offer a solid example for understanding a distinct universal feature of super-rough interfaces: the local roughness exponent is always equal to 1. Furthermore, we also perform some numerical simulations for illustration. Finally, we discuss what are the essential ingredients for constructing super-rough growth equations.