Ising n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity

We show that the n-fold integrals $\chi^{(n)}$ of the magnetic susceptibility of the Ising model, as well as various other n-fold integrals of the "Ising class", or n-fold integrals from enumerative combinatorics, like lattice Green functions, are actually diagonals of rational functions. As a consequence, the power series expansions of these solutions of linear differential equations "Derived From Geometry" are globally bounded, which means that, after just one rescaling of the expansion variable, they can be cast into series expansions with integer coefficients. Besides, in a more enumerative combinatorics context, we show that generating functions whose coefficients are expressed in terms of nested sums of products of binomial terms can also be shown to be diagonals of rational functions. We give a large set of results illustrating the fact that the unique analytical solution of Calabi-Yau ODEs, and more generally of MUM ODEs, is, almost always, diagonal of rational functions. We revisit Christol's conjecture that globally bounded series of G-operators are necessarily diagonals of rational functions. We provide a large set of examples of globally bounded series, or series with integer coefficients, associated with modular forms, or Hadamard product of modular forms, or associated with Calabi-Yau ODEs, underlying the concept of modularity. We finally address the question of the relations between the notion of integrality (series with integer coefficients, or, more generally, globally bounded series) and the modularity (in particular integrality of the Taylor coefficients of mirror map), introducing new representations of Yukawa couplings.Comment: 100 page

Ising n-fold integrals as diagonals of rational functions and integrality of series expansions

We show that the n-fold integrals $\chi^{(n)}$ of the magnetic susceptibility of the Ising model, as well as various other n-fold integrals of the "Ising class", or n-fold integrals from enumerative combinatorics, like lattice Green functions, correspond to a distinguished class of function generalising algebraic functions: they are actually diagonals of rational functions. As a consequence, the power series expansions of the, analytic at x=0, solutions of these linear differential equations "Derived From Geometry" are globally bounded, which means that, after just one rescaling of the expansion variable, they can be cast into series expansions with integer coefficients. We also give several results showing that the unique analytical solution of Calabi-Yau ODEs, and, more generally, Picard-Fuchs linear ODEs, with solutions of maximal weights, are always diagonal of rational functions. Besides, in a more enumerative combinatorics context, generating functions whose coefficients are expressed in terms of nested sums of products of binomial terms can also be shown to be diagonals of rational functions. We finally address the question of the relations between the notion of integrality (series with integer coefficients, or, more generally, globally bounded series) and the modularity of ODEs.Comment: This paper is the short version of the larger (100 pages) version, available as arXiv:1211.6031 , where all the detailed proofs are given and where a much larger set of examples is displaye

Diagonal Ising susceptibility: elliptic integrals, modular forms and Calabi-Yau equations

We give the exact expressions of the partial susceptibilities $\chi^{(3)}_d$ and $\chi^{(4)}_d$ for the diagonal susceptibility of the Ising model in terms of modular forms and Calabi-Yau ODEs, and more specifically, $_3F_2([1/3,2/3,3/2],\, [1,1];\, z)$ and $_4F_3([1/2,1/2,1/2,1/2],\, [1,1,1]; \, z)$ hypergeometric functions. By solving the connection problems we analytically compute the behavior at all finite singular points for $\chi^{(3)}_d$ and $\chi^{(4)}_d$. We also give new results for $\chi^{(5)}_d$. We see in particular, the emergence of a remarkable order-six operator, which is such that its symmetric square has a rational solution. These new exact results indicate that the linear differential operators occurring in the $n$-fold integrals of the Ising model are not only "Derived from Geometry" (globally nilpotent), but actually correspond to "Special Geometry" (homomorphic to their formal adjoint). This raises the question of seeing if these "special geometry" Ising-operators, are "special" ones, reducing, in fact systematically, to (selected, k-balanced, ...) $_{q+1}F_q$ hypergeometric functions, or correspond to the more general solutions of Calabi-Yau equations.Comment: 35 page

Design of InAs/GaSb superlattice infrared barrier detectors

Design of InAs/GaSb type-II superlattice (T2SL) infrared barrier detectors is theoretically investigated. Each part of the barrier structures is studied in order to achieve optimal device operation at 150 K and 77 K, in the midwave and longwave infrared domain, respectively. Whatever the spectral domain, nBp structure with a p-type absorbing zone and an n-type contact layer is found to be the most favourable detector architecture allowing a reduction of the dark-current associated with generation-recombination processes. The nBp structures are then compared to pin photodiodes. The MWIR nBp detector with 5 μm cut-off wavelength can operate up to 120 K, resulting in an improvement of 20 K on the operating temperature compared to the pin device. The dark-current density of the LWIR nBp device at 77 K is expected to be as low as 3.5 × 10−4 A/cm2 at 50 mV reverse bias, more than one decade lower than the usual T2SL photodiode. This result, for a device having cut-off wavelength at 12 μm, is at the state of the art compared to the well-known MCT ‘rule 07’

Midwave infrared InAs/GaSb superlattice photodiode with a dopant-free p–n junction

Midwave infrared (MWIR) InAs/GaSb superlattice (SL) photodiode with a dopant-free p–n junction was fabricated by molecular beam epitaxy on GaSb substrate. Depending on the thickness ratio between InAs and GaSb layers in the SL period, the residual background carriers of this adjustable material can be either n-type or p-type. Using this flexibility in residual doping of the SL material, the p–n junction of the device is made with different non-intentionally doped (nid) SL structures. The SL photodiode processed shows a cut-off wavelength at 4.65 μm at 77 K, residual carrier concentration equal to 1.75 × 1015 cm−3, dark current density as low as 2.8 × 10−8 A/cm2 at 50 mV reverse bias and R0A product as high as 2 × 106 Ω cm2. The results obtained demonstrate the possibility to fabricate a SL pin photodiode without intentional doping the pn junction

Shape-independent scaling of excitonic confinement in realistic quantum wires

The scaling of exciton binding energy in semiconductor quantum wires is investigated theoretically through a non-variational, fully three-dimensional approach for a wide set of realistic state-of-the-art structures. We find that in the strong confinement limit the same potential-to-kinetic energy ratio holds for quite different wire cross-sections and compositions. As a consequence, a universal (shape- and composition-independent) parameter can be identified that governs the scaling of the binding energy with size. Previous indications that the shape of the wire cross-section may have important effects on exciton binding are discussed in the light of the present results.Comment: To appear in Phys. Rev. Lett. (12 pages + 2 figures in postscript

Radiometric and noise characteristics of InAs-rich T2SL MWIR pin photodiodes

We present a full characterization of the radiometric performances of a type-II InAs/GaSb superlattice pin photodiode operating in the mid-wavelength infrared domain. We first focused our attention on quantum efficiency, responsivity and angular response measurements: quantum efficiency reaches 23% at λ = 2.1 µm for 1 µm thick structure. Noise under illumination measurements are also reported: noise is limited by the Schottky contribution for reverse bias voltage smaller than 1.2 V. The specific detectivity, estimated for 2p field-of-view and 333 K background temperature, was determined equal to 2.29 x 10^10 Jones for -0,8 V bias voltage and 77 K operating temperature

Holonomic functions of several complex variables and singularities of anisotropic Ising n-fold integrals

Lattice statistical mechanics, often provides a natural (holonomic) framework to perform singularity analysis with several complex variables that would, in a general mathematical framework, be too complex, or could not be defined. Considering several Picard-Fuchs systems of two-variables "above" Calabi-Yau ODEs, associated with double hypergeometric series, we show that holonomic functions are actually a good framework for actually finding the singular manifolds. We, then, analyse the singular algebraic varieties of the n-fold integrals $\chi^{(n)}$, corresponding to the decomposition of the magnetic susceptibility of the anisotropic square Ising model. We revisit a set of Nickelian singularities that turns out to be a two-parameter family of elliptic curves. We then find a first set of non-Nickelian singularities for $\chi^{(3)}$ and $\chi^{(4)}$, that also turns out to be rational or ellipic curves. We underline the fact that these singular curves depend on the anisotropy of the Ising model. We address, from a birational viewpoint, the emergence of families of elliptic curves, and of Calabi-Yau manifolds on such problems. We discuss the accumulation of these singular curves for the non-holonomic anisotropic full susceptibility.Comment: 36 page

Thermal ionization of excitons in V-shaped quantum wires

The exciton-to-free-carrier transition in GaAs and In_xGa_{1-x}As V-shaped quantum wires is revealed by means of temperature-dependent magnetoluminescence experiments. The experimental results are in excellent agreement with the diamagnetic shift obtained from a solution of the full two-dimensional Schrödinger equation for electrons and holes including magnetic-field and excitonic effects. In the GaAs wires, the exciton-to-free-carrier transition is found to occur at temperature consistent with the exciton binding energies. In the In_xGa_{1-x}As wires the diamagnetic shift of the luminescence is found to be free-carrier-like, independent of temperature, due to the weakening of the exciton binding energy induced by the internal piezoelectric field