Extended Watson integrals for the cubic lattices

Glasser, M. L.; Zucker, I. J.

doi:10.1073/pnas.74.5.1800

Cited by 96 publications

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“…A continuum of states is, of course, indicative of conductive behaviour. Many similar integrals -which can be exactly solvedappear in the literature 6,10 for the lattices with a cubic unit cell but not with all the sums and products of H appearing together. In particular, the last summations of cos 2x in the P -function do not appear in the canonical form in Eq.…”

Section: Is Given Bymentioning

confidence: 99%

“…There currently is hardly any exactly known result for this lattice, even for non-interacting models. However it was conjectured 6 that lattices with cubic cells should have a lattice Green's functions expressable in a "canonical" integral form, which, in general, can be exactly evaluated; a rationale for this proposition was later given 7 . Glasser's canonical form for the Green's function is…”

Section: Hyperkagome Latticementioning

confidence: 99%

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Lattice Green's functions for kagome, diced, and hyperkagome lattices

Varma¹,

Monien²

2013

Phys. Rev. E

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Lattice Green's functions (LGF) and density of states (DOS) for non-interacting models on 3 related lattices are presented. The DOS and LGF at the origin for the kagome and diced lattices are rederived. Furthermore, from the form obtained for the DOS of the hyperkagome lattice, it seems highly suggestive that a closed form expression can be obtained if a particular complicated yet highly symmetrical integral can be solved. The results have been checked numerically. The continuum of energy states in the hyperkagome lattice and the appearance of a zero in the kagome lattice may be explicitly seen from the formulae obtained.Lattice Green's functions (LGF) of systems are ubiquitous 1,2 in solid state physics appearing in problems of lattice vibrations, of spin wave theory of magnetic systems, of localized oscillation modes at lattice defects, and in combinatorial problems in lattices 3 . The LGF G(t, r) in the complex energy variable t = s − iǫ between the (arbitrary) lattice origin and the point r is defined as the solution to the inhomogenous difference equationcorresponding to the linear, Hermitian, time-independent operator H( r). This operator is generally taken to be the discrete Laplacian for lattice problems. Its action on an n-variable function φ( r ≡ {r 1 , r 2 · · · r n }) is given bywhere N denotes the set of nearest neighbours of r and z ≡ |N | is the number of such elements. In subsequent analysis, we will neglect the second term in Eq. (2) because it merely adds a constant shift to the energy variable t. With that Eq. (1) defining the LGF is expressed asThe above equation may be readily solved by taking its Fourier transform into k-space, noting that G(t, r+ ∆) ↔ e ( k. ∆)G (t, k), whereG denotes the Fourier transformed variable. For a simple cubic lattice in d-dimensions with unit lattice distances, this can be written, after inverse Fourier transforming, aswhereThe general form for the LGF for other lattices like body-centred cubic, face-centred cubic remain the same 1 . In many cases such integrals may be exactly solved in terms of the elliptic integrals of the first and second kind. Once a solution for the LGF is obtained, we may obtain information about the eigenvalues and eigenfunctions of the operator H( r). And any solution of the inhomogenous equation may be constructed using G(t, r). An important quantity of the lattice problem is the density of states (DOS) ρ(s), which will be a focus of this paper, given aswhere ℑ refers to the imaginary part.In this brief report, we present results for three related lattices at r = 0. In the first section, we derive the LGF and DOS for the kagome lattice, which is shown in Fig. 1(a). In the second section we perform similar calculations for the diced lattice, which is shown in Fig.

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Section: Is Given Bymentioning

confidence: 99%

Section: Hyperkagome Latticementioning

confidence: 99%

Lattice Green's functions for kagome, diced, and hyperkagome lattices

Varma¹,

Monien²

2013

Phys. Rev. E

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“…We have found that these coefficients have been quoted sometimes incorrectly in the literature: The first term, I 3 (3) = A, has been expressed analytically in terms of functions [31,32] but in apparent disagreement with the correct numerical value [27,32],…”

Section: Appendix A: Derivation Of the Effective-range Expansion In Lmentioning

confidence: 99%

“…A mathematical complication here is that each term in the expansion of the principal-valued integral has to be evaluated numerically. Previously, the triple integral I 3 (z) at z = 3 + 0 was analyzed by Watson [29], and for |z| 3 was studied [27,[30][31][32] in connection to random walks on lattices and to lattice dynamics in condensed matter. The expansion about z = 3 + 0 was found to be [27,30] …”

Section: Appendix A: Derivation Of the Effective-range Expansion In Lmentioning

confidence: 99%

Effective field theory of nucleon-nucleon scattering on large discrete lattices

Seki

Kolck

2006

Phys. Rev. C

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“…The critical value of a is given by (This integral is actually Green's integral for the cubic lattice, and has the closed form F(1/24)F(5/24)F(7/24)F(ll/24)(6)I/2/32zr3; see [24].) In the case discussed in [52] and [40] an optimal solution was proposed informally for the case a .5; our solution agrees exactly.…”

Section: Theorem 34 (Duality) V(p) >-_ V(pe) >= V(p*) If the Dual mentioning

confidence: 99%

Partially-Finite Programming in $L_1 $ and the Existence of Maximum Entropy Estimates

Borwein¹,

Lewis²

1993

SIAM J. Optim.

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Abstract. Best entropy estimation is a technique that has been widely applied in many areas of science. It consists of estimating an unknown density from some of its moments by maximizing some measure of the entropy of the estimate. This problem can be modelled as a partially-finite convex program, with an integrable function as the variable. A complete duality and existence theory is developed for this problem and for an associated extended problem which allows singular, measure-theoretic solutions. This theory explains the appearance of singular components observed in the literature when the Burg entropy is used. It also provides a unified treatment of existence conditions when the Burg, Boltzmann-Shannon, or some other entropy is used as the objective. Some examples are discussed.

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Extended Watson integrals for the cubic lattices

Abstract: The = (4/7r2) (2 + a)2 )K(k+)K(k-) [7] (2 + a2 + a2Z)Kk)Kk) [7 [3] [3a]in which k.2 =1 ±2za(2 + a2)1/2(a4 + 2a2 + z)'/2

Cited by 96 publications

References 1 publication

Lattice Green's functions for kagome, diced, and hyperkagome lattices

Lattice Green's functions for kagome, diced, and hyperkagome lattices

Effective field theory of nucleon-nucleon scattering on large discrete lattices

Partially-Finite Programming in $L_1 $ and the Existence of Maximum Entropy Estimates

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