Extremal Lattices Research Articles

Highly Anisotropic to Isotropic Nature and Ultralow Out-of-Plane Lattice Thermal Conductivity of Layered PbClF-Type Materials.

Materials with an extreme lattice thermal conductivity (κl) are indispensable for thermal energy management applications. Layered materials provide an avenue for designing such functional materials due to their intrinsic bonding heterogeneity. Therefore, a microscopic understanding of the crystal structure, bonding, anharmonic lattice dynamics, and phonon transport properties is critically important for layered materials. Alkaline-earth halofluorides exhibit anisotropy from their layered crystal structure, which is strongly determined by axial bond(s), and it is attributed to the large axial ratio (c/a > 2) for CaBrF, CaIF, and SrIF, in which Br/I acts as a rattler, as evidenced from potential energy curves and phonon density of states. The low axial (c/a) ratio leads to relatively isotropic κl values in the BaXF (X = Cl, Br, I) series. MXF (M = Ca, Sr, Ba) compounds exhibit highly anisotropic (a large phonon transport anisotropy ratio of 10.95 for CaIF) to isotropic (a small phonon transport anisotropy ratio of 1.49 for BaBrF) κl values despite their iso-structure. Moreover, ultralow κl (<1 W/m K) values have been predicted for CaBrF, CaIF, and SrIF in the out-of-plane direction due to weak van der Waals (vdWs) bonding. Overall, this comprehensive study on MXF compounds provides insights into designing low κl layered materials with a large axial ratio by fine-tuning out-of-plane bonding from ionic to vdWs bonding.

Fluctuation of the phase boundary in the six-vertex model with domain wall boundary conditions: a Monte Carlo study

We consider the six-vertex model with domain wall boundary conditions in a square lattice of dimension N × N. Our main interest is the study of the fluctuations of the extremal lattice path about the arctic curves. We address the problem through Monte Carlo simulations. At , the fluctuations of the extremal path along any line parallel to the square diagonal were rigorously proven to follow the Tracy-Widom distribution. We provide strong numerical evidence that this is true also for other values of the anisotropy parameter Δ (). We argue that the typical width of the fluctuations of the extremal path about the arctic curves scales as and provide a numerical estimate for the parameters of the scaling random variable.

Pseudo-normalized Hecke eigenform and its application to extremal 2-modular lattices

It is shown that extremal $2$-modular lattices of ranks $32$ and $48$ are generated by their vectors of minimal norm. In the proof, we use certain properties of the difference of normalized Hecke eigenforms. We refer to them as the pseudo-normalized Hecke eigenform, the concept of which is introduced in this paper.

Extremality, left-modularity and semidistributivity

In this article we investigate the relations between three classes of lattices each extending the class of distributive lattices in a different way. In particular, we consider join-semidistributive, join-extremal and left-modular lattices, respectively. Our main motivation is a recent result by Thomas and Williams proving that every semidistributive, extremal lattice is left modular. We prove the converse of this on a slightly more general level. Our main result asserts that every join-semidistributive, left-modular lattice is join extremal. We also relate these properties to the topological notion of lexicographic shellability.

A note on Quebbemann’s extremal lattices of rank 64

By constructing explicit examples, we show that the method of Quebbemann yields many isomorphism classes of extremal lattices of rank 64. Many of these examples have no non-trivial automorphisms.

The irreducible vectors of a lattice:

The main idea behind lattice sieving algorithms is to reduce a sufficiently large number of lattice vectors with each other so that a set of short enough vectors is obtained. It is therefore natural to study vectors which cannot be reduced. In this work we give a concrete definition of an irreducible vector and study the properties of the set of all such vectors. We show that the set of irreducible vectors is a subset of the set of Voronoi relevant vectors and study its properties. For extremal lattices this set may contain as many as 2^n vectors, which leads us to define the notion of a complete system of irreducible vectors, whose size can be upper-bounded by the kissing number. One of our main results shows that modified heuristic sieving algorithms heuristically approximate such a set (modulo sign). We provide experiments in low dimensions which support this theory. Finally we give some applications of this set in the study of lattice problems such as SVP, SIVP and CVPP. The introduced notions, as well as various results derived along the way, may provide further insights into lattice algorithms and motivate new research into understanding these algorithms better.

Self-dual codes over [formula omitted] and s-extremal unimodular lattices

New s-extremal extremal unimodular lattices in dimensions 38, 40, 42 and 44 are constructed from self-dual codes over F5 by Construction A. In the process of constructing these codes, we obtain a self-dual [44,22,14] code over F5. In addition, the code implies a [43,22,13] code over F5. These codes have larger minimum weights than the previously known [44,22] codes and [43,22] codes, respectively.

Raised solubility in SnTe by GeMnTe2 alloying enables converged valence bands, low thermal conductivity, and high thermoelectric performance

SnTe is a promising alternative of the toxic PbTe, but the concepts of band engineering and thermoelectric performance optimization are restricted by the low solid solubility of effective dopants. Here, we demonstrate a raised Mn solubility limit of around 15% in the SnTe-GeMnTe2 medium-entropy system. Density functional theory calculations show that high doping fraction of Mn gives rise to a near-complete valence band convergence that is responsible for the obviously improved Seebeck coefficient. High-content GeMnTe2 alloying also profoundly creates multiscale microstructure for an extreme low lattice thermal conductivity. These effects yield a high ZTmax of 1.4 at 850 K and a record Vickers hardness of 1.65 GPa in (Sn0.96Sb0.04Te)0.7(Ge0.5Mn0.5Te)0.3. Stem from excellent thermoelectric and mechanical properties, the fabricated thermoelectric module comprised of our p-type SnTe exhibits a conversion efficiency of 6.1% and an output power density of 0.43 W cm–2 under a temperature gradient of 350 °C. This study opens the door for the development of SnTe-based modules, which is promising for the low-grade heat harvest at mid-low temperature.

Extremal hinged lattices do not obey the theory of elasticity

Extremal hinged lattices do not obey the theory of elasticity

Extreme Cosserat elastic hinged lattices with Sarrus links

Hinged lattices are envisaged with Sarrus linkages in the ribs and with rigid nodes where the ribs intersect. Triangulated lattices with Sarrus hinged ribs only deform volumetrically and have a Poisson’s ratio of −1. Lattices containing squares and hexagons can deform axially as well as volumetrically. All the lattices are compliant with respect to tension and compression and rigid with respect to bending. The lattices exhibit extreme Cosserat elasticity corresponding to rotational elasticity.

A novel lattice structure topology optimization method with extreme anisotropic lattice properties

Abstract This research presents a lattice structure topology optimization (LSTO) method that significantly expands the design space by creating a novel candidate lattice that assesses an extremely large range of effective material properties. About the details, topology optimization is employed to design lattices with extreme directional tensile or shear properties subject to different volume fraction limits and the optimized lattices are categorized into groups according to their dominating properties. The novel candidate lattice is developed by combining the optimized elementary lattices, by picking up one from each group, and then parametrized with the elementary lattice relative densities. In this way, the LSTO design space is greatly expanded for the ever increased accessible material property range. Moreover, the effective material constitutive model of the candidate lattice subject to different elementary lattice combinations is pre-established so as to eliminate the tedious in-process repetitive homogenization. Finally, a few numerical examples and experiments are explored to validate the effectiveness of the proposed method. The superiority of the proposed method is proved through comparing with a few existing LSTO methods. The options of concurrent structural topology and lattice optimization are also explored for further enhancement of the mechanical performance.

Non-abelian orbifolds of lattice vertex operator algebras

We construct orbifolds of holomorphic lattice vertex operator algebras for non-abelian finite automorphism groups G. To this end, we construct twisted modules for automorphisms g together with the projective representation of the centralizer of g on the twisted module. This allows us to extract the irreducible modules of the fixed-point VOA VG, and to compute their characters and modular transformation properties. We then construct holomorphic VOAs by adjoining such modules to VG. Applying these methods to extremal lattices in d=48 and d=72, we construct more than fifty new holomorphic VOAs of central charge 48 and 72, many of which have a very small number of light states.

Automorphisms of modular lattices

The methods to classify extremal unimodular lattices with given automorphisms are extended to the situation of modular lattices . A slightly more general notion than the type from the PhD thesis [11] is the det-type. The det-type of an automorphism on L determines the one of all partial dual lattices of L . This easy observation allows to exclude quite a few det-types of automorphisms left open in [11] . Passing to suitable p -maximal lattices, extremal ℓ -modular lattices of composite level ℓ = 14 and ℓ = 15 of dimension 12 and the ones of level ℓ = 6 and dimension 16 are classified.

Wood plastic composites from chopped and pelleted waste plastic bags: effect of concentrations and reprocessing on flexural modulus and strength

The utilization of waste plastic bag, especially as a matrix for wood plastic composites, is expected to reduce the environmental problems throughout the world caused by its use. Wood plastic composites were manufactured via melt mixing of chopped or pelleted waste plastic bags, sawdust and polyethylene grafting maleic anhydrate (MAPE) according to Extreme Lattice Mixture - Design of Experiment (DoE). The effects of concentrations and reprocessing of waste plastic bags on flexural modulus of elasticity and strength of composites were analyzed. In general, flexural modulus of composites from chopped and pelleted waste are increased but still below the value of SNI 8154–2015. Moreover, the flexural strength of the resulted composites from chopped waste also increased than that of the pelleted one. Some of the formulas of composites from chopped waste met the value required by SNI 8154 – 2015. On the contrary, the flexural strengths of composites from pelleted waste are decreased than that of the virgin one. This is probably due to the existence of interaction between pelleted waste and sawdust that prevented the waste plastic to perfectly melted, hence acted as stress concentration sites. Furthermore, the resulted composites were optimized for flexural modulus using response optimizer plot in order to achieve modulus 2000 MPa that required by the standard. However, the optimized flexural modulus of composites from chopped and pelleted waste was in the range of 1500 MPa, which is far below the standard. Therefore, reprocessing of chopped waste into pellets is not necessary due to inferior properties of composites from pelleted waste than that of the chopped one.

48 dimensional even unimodular nearly extremal lattices.

In this paper we introduce the notion of an even unimodular nearly extremal lattice in the case of 48 dimension. We prove two basic properties of such lattices $L$. First we prove that any nearly extremal lattice $L$ is generated by the vectors of norm 4 and norm 6 in $L$. Next we prove that the Siegel theta series of degree 2 associated with an even unimodular nearly extremal lattice is determined by the Fourier coefficients which are connected with the vectors of norm 4.

Low dimensional strongly perfect lattices IV: The dual strongly perfect lattices of dimension 16

We classify the dual strongly perfect lattices in dimension 16. There are four pairs of such lattices, the famous Barnes-Wall lattice Λ16, the extremal 5-modular lattice N16, the odd Barnes-Wall lattice O16 and its dual, and one pair of new lattices Γ16 and its dual. The latter pair belongs to a new infinite series of dual strongly perfect lattices, the sandwiched Barnes-Wall lattices, described by the authors in a previous paper. An updated table of all known strongly perfect lattices up to dimension 26 is available in the catalogue of lattices [15].

Rowmotion in slow motion

Rowmotion is a simple cyclic action on the distributive lattice of order ideals of a poset: it sends the order ideal x to the order ideal generated by the minimal elements not in x. It can also be computed in ‘slow motion’ as a sequence of local moves. We use the setting of trim lattices to generalize both definitions of rowmotion, proving many structural results along the way. We introduce a flag simplicial complex (similar to the canonical join complex of a semidistributive lattice) and relate our results to recent work of Barnard by proving that extremal semidistributive lattices are trim. As a corollary, we prove that if A is a representation finite algebra and mod A has no cycles, then the torsion classes of A ordered by inclusion form a trim lattice.

Fractional quantum Hall states with gapped boundaries in an extreme lattice limit

We present a detailed microscopic investigation of fractional quantum Hall states with gapped boundaries in a coupled bilayer lattice model featuring holes whose counterpropagating chiral edge states are hybridized and gapped out. We focus on a lattice limit for cold-atom experiments, in which each hole just consists of a single removed site. Although the holes distort the original band structure and lead to ingap remnants of the continuum edge modes, we find that the lowest nearly flat band representing a higher-genus system may naturally form by controlling the local hopping terms that gap out the boundaries. Remarkably, local interactions in this new flat band lead to various Abelian and non-Abelian fractional quantum Hall states with gapped boundaries residing on emergent higher-genus surfaces, which we identify by extracting the nontrivial topological ground-state degeneracies and the fractional statistics of quasiparticles. These results demonstrate the feasibility of realizing novel fractional quantum Hall states with gapped boundaries even in the extreme lattice limit, thus enabling a possible new route towards universal topological quantum computation.

Convex Geometries are Extremal for the Generalized Sauer-Shelah Bound

The Sauer-Shelah lemma provides an exact upper bound on the size of set families with bounded Vapnik-Chervonekis dimension. When applied to lattices represented as closure systems, this lemma outlines a class of extremal lattices obtaining this bound. Here we show that the Sauer-Shelah bound can be easily generalized to arbitrary antichains, and extremal objects for this generalized bound are exactly convex geometries. We also show that the problem of classification of antichains admitting such extremal objects is NP-complete.

On the Secrecy Gain of Extremal Even l-modular Lattices

ABSTRACTThe secrecy gain is a lattice invariant that appears in the context of wiretap lattice coding. It has been studied for unimodular lattices, for 2 −, 3 −, and 5 −modular lattices. This paper studies the secrecy gain for extremal even l-modular lattices, for l ∈ {2, 3, 5, 6, 7, 11, 14, 15, 23}. We compute the highest secrecy gains as a function of the lattice dimension and the lattice level l. We show in particular that l = 2, 3, 6, 7, 11 are best for the respective ranges of dimensions {80, 76, 72}, {68, 64, 60, 56, 52, 48}, {44, 40, 36}, {34, 32, 30, 28, 26, 24, 22}, {18, 16, 14, 12, 10, 8}. This suggests that within a range of dimensions where different levels exist, the highest value of l tends to give the best secrecy gain. A lower bound computation on the maximal secrecy gain further shows that extremal lattices provide secrecy gains which are very close to this lower bound, thus confirming the good behavior of this class of lattices with respect to the secrecy gain.