Trivial Lattice Research Articles

Topologically valley-polarized edge states in elastic phononic plates yielded by lattice defects

Lattice defects (including disclinations and dislocations) which commonly exist in various materials, have shown a fantastic ability to produce excellent mechanical and physical properties of matters. In this paper, lattice defects are introduced into valley-polarized elastic phononic plates. Deformations of lattices yield interfaces expressing as topologically protected waveguides, due to the valley-polarized phase transition of phononic crystals (PnCs) across interfaces. The topological interface states immunize against the finite sizes and the moderate disturbances of plates, essentially differing from the trivial lattice defect states. The discovery of topologically valley-polarized edge states yielded by lattice defects unveils a new horizon in topological mechanics and physics. It provides a novel platform to implement elastic devices with robust topological waveguides.

Effect of truncation on photonic corner states in a Kagome lattice

We theoretically present a simple nanophotonics platform, where the photonic corner states can be obtained in high frequencies. For the breathing Kagome lattice with three bearded truncations, we show that a pair of bonding and anti-bonding corner states can be observed in the gapped bulk bands, even for a topologically trivial lattice. In particular, this anti-bonding corner state is totally embedded in the bulk modes for topologically non-trivial structures, forming a bound state in continuum. We further verify that these corner states are protected by the symmetry and can be excited by merely one point source, even in a loss system. The newly found corner modes may expand our understanding of the high-order corner modes in a more general framework, involving both topology and symmetry.

Quantum anomalous Hall phase stabilized via realistic interactions on a kagome lattice

We study the quantum phases of spinless fermion at one-third filling on a Kagome lattice featuring a quadratic band touching Fermi point. In the presence of weak first and second nearest-neighbor repulsive interactions ($V_1$ and $V_2$), we demonstrate an interaction driven quantum anomalous Hall effect by employing exact diagonalization and density-matrix renormalization group methods. The time-reversal symmetry is broken spontaneously by forming loop currents that exhibit long-range correlation. Quantized Hall conductance corresponding to Chern number of $\pm1$ is obtained by measuring the pumped charge through inserting flux in a cylinder geometry. We find that the energy gap, which topologically protects the emerging ground states, can be enhanced remarkably by a moderate $V_2 < V_1$ via calculating the spectrum and charge excitation gaps, which highlights the experimentally feasible scheme of realizing interaction driven topological phase by spatially decaying interactions on topologically trivial lattice models.

Distributive lattices determined by weighted double skeletons

Related to his S-glued sum construction, the skeleton S(L) of a finite lattice L was introduced by C. Herrmann in 1973. Our theorem asserts that if D is a finite distributive lattice and its second skeleton, S(S(D)), is the trivial lattice, then D is characterized by its weighted double skeleton, introduced by the second author in 2006. The assumption on the second skeleton is essential.

A single-element extension of antimatroids

An antimatroid is a family of sets which is accessible, closed under union, and includes an empty set. A number of examples of antimatroids arise from various kinds of shellings and searches on combinatorial objects, such as, edge/node shelling of trees, poset shelling, node-search on graphs, etc. (Discrete Math. 78 (1989) 223; Geom. Dedicata 19 (1985) 247; Greedoids, Springer, Berlin, 1980) [1–3]. We introduce a one-element extension of antimatroids, called a lifting, and the converse operation, called a reduction. It is shown that a family of sets is an antimatroid if and only if it is constructed by applying lifting repeatedly to a trivial lattice. Furthermore, we introduce two specific types of liftings, 1-lifting and 2-lifting, and show that a family of sets is an antimatroid of poset shelling if and only if it is constructed from a trivial lattice by repeating 1-lifting. Similarly, an antimatroid of edge-shelling of a tree is shown to be constructed by repeating 2-lifting, and vice versa.

Quark propagator in the Landau gauge

The Landau gauge quark propagator in momentum space is investigated using the O(a)-improved Sheikholeslami--Wohlert (SW) quark action with a tree-level mean-field improved coefficient c_sw. We have studied the unimproved definition of the quark propagator, as well as two different tree-level O(a)-improved propagators. The ultraviolet behavior of the free lattice propagator is studied for each of these in order to establish which of them provides the most reliable description of the quark propagator up to the medium momentum regime. A general method of tree-level correction is introduced. This exploits asymptotic freedom and removes much of the trivial lattice artifacts at medium to high momenta. We obtain results for the quark propagator which are qualitatively similar to those typically used in quark models. A simple extrapolation of the infrared quark mass M(p^2=0) to the chiral limit gives (298+/-8+/-30) MeV, which is consistent with phenomenological expectations.

Reducible classes of finite lattices

In this paper we study a notion of reducibility in finite lattices. An element x of a (finite) lattice L satisfying certain properties is deletable if L-x is a lattice satisfying the same properties. A class of lattices is reducible if each lattice of this class admits (at least) one deletable element (equivalently if one can go from any lattice in this class to the trivial lattice by a sequence of lattices of the class obtained by deleting one element in each step). First we characterize the deletable elements in a pseudocomplemented lattice what allows to prove that the class of pseudocomplemented lattices is reducible. Then we characterize the deletable elements in semimodular, modular and distributive lattices what allows to prove that the classes of semimodular and locally distributive lattices are reducible. In conclusion the notion of reducibility for a class of lattices is compared with some other notions like the notion of order variety.

LO Phonon Raman Lineshape in Highly Excited GaAs

AbstractIt is shown that the LO phonon Raman line in bulk GaAs excited with pulsed IR radiation of high intensity is shifted, broadened, and distorted. The TO line is not affected. These phenomena cannot be explained by trivial lattice heating but are quantitatively accounted for by invoking mixing of the LO phonons with photo‐excited carrier collective modes.

Lattice homomorphisms induced by group homomorphisms

Introduction. By a lattice homorphism of a group G onto a group G' we mean a single-valued mapping sb of the lattice L(G) of subgroups of G onto the lattice L(G') of subgroups of G', which preserves all unions and intersections, that is, which is subject to the conditions 1. (U,S,)(k U= (S,(k), 2. (nfS,)ck w= s, for every (finite or infinite) set of subgroups S, of G. We call proper any lattice homomorphism which is neither a lattice isomorphism (1-1), nor a trivial lattice homomorphism (So = 1 for every S in L(G)). The general problem which now presents itself is the characterization of those (subgroup) lattice mappings which are proper lattice homomorphisms, and of those groups which admit proper lattice homomorphisms (cf. Whitman [1],2 Zappa [1] and [2], and Suzuki [1]). A specialization of this problem arises when we consider a lattice mapping which is induced by a (group) homomorphism. It is easily seen that such a mapping preserves unions. Under what conditions will it be a (proper) lattice homomorphism? The main concern of this note is the characterization of those homomorphisms which induce proper lattice homomorphisms, and of those groups which admit such homomorphisms. A further specialization occurs in connection with a method suggested by G. Zappa for constructing lattice mappings which preserve intersections. Let H be a proper subgroup of a group G. Then the mapping ,6 of L(G) onto L(H) defined by