Kagome lattice

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Photonic crystal slow light waveguides in a kagome lattice

Photonic crystal slow light waveguides in a kagome lattice

fibre [25] communities, but has attracted only limited interest for PhC slabs [26, 27, 28]. The kagome lattice is a depleted version of the triangular lattice, consist- ing of a sparse super-lattice imposed on the standard triangular lattice. As shown in Fig. 1a, along every second diagonal line of a triangular lattice each sec- ond lattice site is removed to form the kagome lattice. Intuitively, the kagome lattice is better represented by a tight-binding model, compared to the triangu- lar lattice which is better approximated by a “free photon” (or free electron for electronic crystals) ap- proximation. This leads to larger photonic band gaps and flatter bands in the kagome lattice [27, 29]. If we consider a lattice of air holes in a 220 nm-thick silicon slab, the material system used for most slow light PhC work, the kagome lattice supports multiple optical bandgaps for TE polarized light, as shown in Fig. 1b. Furthermore, the sparse super-lattice results in a “compressed’ bandstructure, with each mode oc- cupying a narrower frequency range compared to the triangular lattice. For example, the first bandgap of the kagome lattice starts at a normalized frequency of 0.15, while that of a triangular lattice with the same lattice parameters starts at a normalized frequency of 0.21, see Fig. 1. The compressed bandstructure thus exhibits a reduced slope for all optical modes, result- ing in an increased group index, since the group index is defined as:
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An Improved Finite Temperature Lanczos Method and Its Application to the Spin 1/2 Heisenberg Model on the Kagome Lattice

An Improved Finite Temperature Lanczos Method and Its Application to the Spin 1/2 Heisenberg Model on the Kagome Lattice

In this work, we will present calculations of the specific heat at low T by the exact approach for small clus- ters. The exact approach is very powerful and extensively applied for study at zero T [15]-[17], although the size is severely limited. There are few researches on the exact approach to calculations of the specific heat at fi- nite T . Only some results on L = 18 or L = 24 clusters are found in [12] [13]. Here L is the total number of sites. By these previous works, it is understood that ordinary calculations on L = 27 or larger lattices are quite difficult because the system on the kagome lattice has the complicated structure at low T . We point out the Chebyshev polynomial expansion (CPE) as one method of the exact approaches at finite T [18] [19]. Another method is the finite temperature Lanczos method (FTLM) [20] [21].While the polymial expansion method is well mathematically understood, in the FTLM we should be careful to control numerical errors. Due to this weakness there are a fewer applications of the FTLM, compared with the CPE [18], although both methods give us the same calculation costs.
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Novel S = 1/2 kagome lattice materials : Cs2TiCu3F12 and Rb2TiCu3F12

Novel S = 1/2 kagome lattice materials : Cs2TiCu3F12 and Rb2TiCu3F12

Rb + . These compositions represent new extremes of both A/B′ cation size ratio and B′ absolute cation size within this family. By this size-directed crystal engineering, it was hoped that a perfect kagome lattice might be retained at low temperatures, thus prompting retention of a magnetically-frustrated ground state. However, it is found that Rb 2 TiCu 3 F 12 adopts a highly distorted, triclinic structural variant

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Pair correlations, short range order and dispersive excitations in the quasi-kagome quantum magnet volborthite

Pair correlations, short range order and dispersive excitations in the quasi-kagome quantum magnet volborthite

FIG. 4: (a) Reciprocal space of volborthite: the structural unit cell is indicated by the rectangle, and the extended Bril- louin zone of the kagome lattice by the dotted hexagon. Sym- bols represent Bragg peaks of the orders listed in the text. (b) The experimental S(Q, ω) measured at 0.05 K compared with the powder averaged S(Q, ω) derived from our empirical spin wave model. The dashed line in the left panel indicates the (Q, ω) window of the experiment.

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Greitemann, Jonas
  

(2019):


	Investigation of hidden multipolar spin order in frustrated magnets using interpretable machine learning techniques.


Dissertation, LMU München: Fakultät für Physik

Greitemann, Jonas (2019): Investigation of hidden multipolar spin order in frustrated magnets using interpretable machine learning techniques. Dissertation, LMU München: Fakultät für Physik

This is however where the similarity with the triangular lattice ends. Because adjacent triangles on the kagome lattice only share a single spin, their local planes do not need to co- incide; rather, they may intersect on a line determined by the shared spin. Following a similar constructive approach for building up a ground state as for the triangular case, starting from a single triangle, each consecutive adjacent triangle that is added allows for the choice of one an- gle. However, as triangles start to close loops around the hexagonal plaquettes, their planes are not necessarily compatible. Hence, not all of the aforementioned angles can in fact be chosen freely, but are subject to intricate constraints themselves. The exact degeneracy of the ground state is not known. Still, nonplanar solutions do exist for which the local plane of the triangles is different and becomes independent for triangles at large distances. This is to be contrasted with the triangular lattice, where the constraint implies a coplanar order where all triangles share the same plane, even at (quasi-)long range, thus breaking the O ( 3 ) spin-rotational sym- metry.
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Nature of the spin liquid ground state in a breathing kagome compound studied by NMR and series expansion

Nature of the spin liquid ground state in a breathing kagome compound studied by NMR and series expansion

which can be accounted for by series expansion for the breathing kagome model. The determination of the ra- tio of the interactions within the breathing kagome lat- tice, J O /J M = 0.55(4), will allow for further theoreti- cal investigations into the effects of the breathing of the S = 1/2 KAFM lattice that are specific to DQVOF. It is hoped that this in turn will shed light on our experimen- tal observation of a gapless spin liquid ground state. The breathing of the kagome lattice appears to be an inter- esting tuning parameter that may be used to control the spin liquid physics and one that warrants further explo- ration. Investigating new anisotropic kagome compounds with different layered structures is also a promising ex- perimental route to furthering our understanding of the rich physics of the S = 1/2 KAFM [35].
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Spin liquid mediated RKKY interaction

Spin liquid mediated RKKY interaction

– the former corresponds to the doubly degenerate ○ and ⬠s and the latter the singular degenerate □s and △s. Example orderings are shown in Fig. 5. It should be noted that the spins are not constrained to the same plane as the kagome lattice because only the relative angle between spins is important to achieve an energy minimum. Stability of the nuclear moment orderings on kagome. To test the stability of the mean field order- ings found above against fluctuations, we investigate how spin-waves modify or destroy these orderings. To do this we select a basis for the directions in spin-space such that the spin on the first site of the Fourier-space unit cell is parallel to the spin z-axis, I Q z 1 ( ) n  . Within this basis, because the regular periodic orderings are co-planar, we can effectively rotate the remaining spins in the unit cell about the y-axis to be parallel with the first spin by defining I Q ′ ( ) = R ( ) ( ) θ I Q
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Extending the family of V4+ S = ½ kagome antiferromagnets

Extending the family of V4+ S = ½ kagome antiferromagnets

Geometrically frustrated magnets are crystalline solids in which the arrangement of spin moments prevents long range magnetic order that typically prevails in magnetic materials upon cooling. Instead, the frustration that can be imposed on the magnetic interactions within a material as a result of its structure leads to a wealth of unusual magnetic behaviours [1]. An archetypal geometrically frustrated structure is the kagome lattice, formed from a two-dimensional corner-sharing triangular network of magnetic ions. When the kagome lattice is dressed with a set of antiferromagnetically interacting S = ½ species an exciting possibility presents itself – the realization of a quantum spin liquid (QSL) [2]. The QSL is a long-sought after magnetic phase of matter that was originally proposed as an alternative to the ordered Néel ground state of an antiferromagnet [3]. It has many intriguing properties, for instance, despite being a system of highly entangled, correlated spins it can evade a conventional symmetry breaking magnetic phase transition even at T = 0 K and instead may exhibit a topological order that could have important consequences for quantum information technology [4]. Furthermore, it can support the existence of unusual magnetic excitations known as spinons, which are relatively well understood in one-dimensional magnets but far less so for magnetic systems of higher dimensionality [5]. In two- dimensions the S = ½ kagome antiferromagnet is the prime candidate to host a QSL state [6].
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Rigidity percolation by next-nearest-neighbor bonds on generic and regular isostatic lattices.

Rigidity percolation by next-nearest-neighbor bonds on generic and regular isostatic lattices.

The reason that there are two distinct transitions in the regular lattices is because sets of braces can form redundancies fairly easily. The coupling of a large number of floppy modes together is not sufficient to completely rigidify the system. In the regular square lattice, this coupling does not even create a single large rigid component, though it does for the regular kagome lattice. Despite the fact that most of the floppy modes become coupled together, many floppy modes remain “isolated”—that is, decoupled from all other modes—and added braces tend to create redundancies rather than remove degrees of freedom. In the bipartite graph representation of the regular square lattice, these floppy modes correspond indeed to isolated vertices, and adding enough braces to the system to couple them all to the giant floppy mode yields a coupon-collector problem [42]. The rigidity transition in the regular kagome lattice proceeds through a similar, but more complicated, process. Thus, in both cases, there is a separate transition to rigidity of the system which occurs much later: when p ∼ ln L/L.
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Surface plasmon dispersion in hexagonal, honeycomb and kagome plasmonic crystals

Surface plasmon dispersion in hexagonal, honeycomb and kagome plasmonic crystals

Abstract: We present a systematic experimental study on the optical properties of plasmonic crystals (PlC) with hexagonal symmetry. We compare the dispersion and avoided crossings of surface plasmon modes around the Γ-point of Au-metal hole arrays with a hexagonal, honeycomb and kagome lattice. Symmetry arguments and group theory are used to label the six modes and understand their radiative and dispersive properties. Plasmon-plasmon interaction are accurately described by a coupled mode model, that contains e ff ective scattering amplitudes of surface plasmons on a lattice of air holes under 60 ◦ , 120 ◦ , and 180 ◦ . We determine these rates in the experiment and find that they are dominated by the hole-density and not on the complexity of the unit-cell. Our analysis shows that the observed angle-dependent scattering can be explained by a single-hole model based on electric and magnetic dipoles.
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Revisiting  Orthogonal  Lattice  Attacks  on  Approximate  Common  Divisor  Problems   and  their  Applications

Revisiting Orthogonal Lattice Attacks on Approximate Common Divisor Problems and their Applications

Lattice reduction algorithm is to output a reduced basis consisting of relatively short and nearly orthogonal vectors, which has plenty of cryptographic applica- tions [44]. After the publication of celebrated LLL algorithm [40], a number of lattice reduction algorithms emerged, for example [51, 52, 28, 15, 46, 5]. In prac- tice, the Block-Korkine-Zolotarev (BKZ) algorithm proposed by Schnorr and Euchner [51] has a good performance. In the BKZ algorithm, the running time and output quality depend on an input parameter–blocksize β . Hence, such an algorithm is called BKZ-β. With the increase of β, the output basis becomes much reduced but the cost significantly increases. The BKZ-β proceeds by re- ducing a lattice basis using an SVP oracle in a smaller dimension β. Based on [34], the number of calls to the SVP oracle remains polynomial.
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Off-the-shelf DFT-DISPersion methods: Are they now "on-trend" for organic molecular crystals?

Off-the-shelf DFT-DISPersion methods: Are they now "on-trend" for organic molecular crystals?

COMPASS was parameterized using ab initio calculations and empirical data, which entailed adding new molecular classes to PCFF. In addition, non-bond parameters were re-parameterized whereby the electrostatic and van der Waals terms combine quantum mechanical calculations and fitting to experimental condensed phase properties of liquids and crystals. COMPASS II extends this to include parameters specific to polymers and drug-like molecules, hence its wide adoption in the pharmaceutical industry. However, given the current availability of DFD-DISP options, is it time to re-think this default position of using classical methods to calculate lattice energies? Attempting to answer this question generated the research we present in this paper, in which we assess whether state-of-the-art DFT-DISP methods are comparably predictive to, or better than classical methods We use two different simulation methods to calculate the lattice energies of the organic molecular crystals, namely classical molecular mechanics using a force field, and ab initio density functional theory (DFT) plus dispersion (DFT-DISP). The molecular mechanics was performed using the Forcite module of Materials Studio 62 , with the
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Two heavy-light mesons on a lattice

Two heavy-light mesons on a lattice

a lattice include a contribution from the self-energy of the static source which is unphysical. Thus only energy differ- ences have a physical significance and hence we concentrate especially on the binding energies—the difference of BB en- ergy from twice the B energy. In the special case of R ⫽ 0, we show the actual lattice energy values in Table II to allow us to discuss the extrapolation to large t needed to extract the ground state. Other results are given in Table III for the case of R ⫽ 3 and in Figs. 6–9. We show the results from both 12 3 and 16 3 spatial lattices with the same parameters in order to explore finite size effects. Within errors, we do not see sig- nificant differences in the results between spatial sizes of L ⫽ 12 and 16, which is not unexpected since a study of the B meson using L ⫽ 8 and 12 found 关 16 兴 agreement for the en- ergies of the ground state mesons and a relatively localized Bethe-Saltpeter wave function.
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Scientific and personal recollections of Roberto Petronzio

Scientific and personal recollections of Roberto Petronzio

In spite of the heavy work duty as president of the INFN, he continued to work on lattice QCD. A very remarkable paper of 2007 is QCD with light Wilson quarks on fine lattices: first experiences and physics results [60]. In this paper the universality of the continuum limit and the applicability of renormalized perturbation theory are tested in the SU(2) lattice gauge theory by computing two differ- ent non-perturbatively defined running couplings over a large range of energies [61]. The lattice data (which were generated on the powerful APE computers at Rome II and DESY) are extrapolated to the continuum limit by simulating sequences of lattices with decreasing spacings. The results confirmed the expected universality at all energies to a precision of a few percent. The author found, however, that perturbation theory must be used with care when matching different renormalized couplings at high energies.
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Some Restricted Plane partitions and Associated Lattice Paths

Some Restricted Plane partitions and Associated Lattice Paths

Theorem 2.1 gives us a direct correspondence between a class of plane partitions of an integer ν and a class of partitions with n copies of n of the integer ν. Also, since partitions with n copies of n have a lattice path representation. So, we obtain a lattice path representation for a class of restricted plane partitions. But there is another class of plane partitions for which lattice path representation is still to be found.

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On Stone sublattices of the lattice of totally local Fitting classes

On Stone sublattices of the lattice of totally local Fitting classes

Being the extremal case, totally local classes have a series of specific properties. In particular we note that for every non-negative integer n the lattices of all n-multiply local formations, of all n-multiply local hered- itary formations, of all n-multiply local normally hereditary formations etc. are modular but all of them are not distributive even in the class of all soluble groups S (see [13, Chapter 2] and [14, Chapter 4]). Moreover as it was mentioned in [15] (see also [13, 14]) for every two non-negative integers n and m the systems of all laws of the lattices of all n-multiply local and all m-multiply local formations coincide. On the other hand the lattice of all soluble totally local formations is distributive [13] but we know nothing about it in the general case (see [14, Question 4.2.14] and [16, Question 14.80]).
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An Attack Bound for Small Multiplicative Inverse of ) mod e) mode with a Composed Prime Sum pq Using Sublattice Based Techniques

An Attack Bound for Small Multiplicative Inverse of ) mod e) mode with a Composed Prime Sum pq Using Sublattice Based Techniques

a polynomial time when d is less than N 0.25 for N = pq and q < p < 2q. Using lattice reduction approach based on the Coppersmith techniques [7] for finding small solutions of modular bivariate integer polynomial equations, D. Boneh and G. Durfee [4] improved the wiener result from N 0.25 to N 0.292 in 2000 and J. Bl¨ omer and A. May [5] has given an RSA attack for d less than N 0.29 in 2001, that requires lattices of dimension smaller than the approach by Boneh and Durfee. In 2006, E. Jochemsz and A. May [10], described a strategy for finding small modular and integer roots of multivariate polynomial using lattice-based Coppersmith tech- niques and by implementing this strategy they gave a new attack on an RSA variant called common prime RSA. In our paper [2], first we described an attack on RSA when ϕ(N ) has small multiplicative inverse k of modulo e, the public encryption exponent by using lattice and sublattice based techniques. Let N = pq, q < p < 2q, p − q = N β and e = N α > p + q. As (e, ϕ(N )) = 1, there exist unique r, s such that (p − 1)r ≡
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Lattice Structures on Z+ Induced by Convolutions

Lattice Structures on Z+ Induced by Convolutions

a i ). A partially ordered set (X , ≤) is called a meet semi lattice if a ∧ b (=glb{a, b}) exists for all a and b ∈ X . (X , ≤) is called a join semi lattice if a ∨ b (=lub{a, b}) exists for all a and b ∈ X . A poset (X , ≤) is called a lattice if it is both a meet and join semi lattice. Equivalently, lattice can also be defined as an algebraic system (X , ∧, ∨), where ∧ and ∨ are binary operations which are associative, commutative and idempotent and satisfying the absorption laws, namely a ∧ (a ∨ b) = a = a ∨ (a ∧ b) for all a, b ∈ X ; in this case the partial order ≤ on X is such that a ∧ b and a ∨ b are respectively the glb and lub of {a, b}. The algebraic operations ∧ and ∨ and the partial order ≤ are related by
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Soft valuation on a generalized soft lattice

Soft valuation on a generalized soft lattice

Abstract. Lattice theory play an important role in mathematics as well as in other disciplines such as computer science, engineering, cryptography, etc. In this paper, we introduce the concept of generalized soft lattice (gs lattice) and investigate some of its fundamental properties. Further we define soft valuation on a generalized soft lattice (gs lattice) and study its major properties. In the last section we discuss the notion of soft distance function and express it in terms of soft valuation. Here we discuss the notions of soft pseudo metric lattice and soft metric lattice.
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Concept Lattice: A rough set approach

Concept Lattice: A rough set approach

Concept lattice is an efficient tool for knowledge representation and knowledge discovery and is applied to many fields successfully. However, in many real life applications, the problem under investigation cannot be described by formal concepts. Such concepts are called the non-definable concepts. The hierarchical structure of formal concept (called concept lattice) represents a structural information which obtained automatically from the input data table. We deal with the problem in which how further additional information be supplied to utilize the basic object attribute data table. In this paper , we provide rough concept lattice to incorporate the rough set into the concept lattice by using equivalence relation. Some results are established to illustrate the paper.
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