Root Branch Point Research Articles

Monodromy bootstrap for SU(2|2) quantum spectral curves: from Hubbard model to AdS3/CFT2

We propose a procedure to derive quantum spectral curves of AdS/CFT type by requiring that a specially designed analytic continuation around the branch point results in an automorphism of the underlying algebraic structure. In this way we derive four new curves. Two are based on SU(2|2) symmetry, and we show that one of them, under the assumption of square root branch points, describes Hubbard model. Two more are based on SU(2|2) × SU(2|2). In the special subcase of zero central charge, they both reduce to the unique nontrivial curve which furthermore has analytic properties compatible with PSU(1, 1|2) × PSU(1, 1|2) real form. A natural conjecture follows that this is the quantum spectral curve of AdS/CFT integrable system with AdS3 × S3 × T4 background supported by RR-flux. We support the conjecture by verifying its consistency with the massive sector of asymptotic Bethe equations in the large volume regime. For this spectral curve, it is compulsory that branch points are not of the square root type which qualitatively distinguishes it from the previously known cases.

Universal Behavior of the Scattering Matrix Near Thresholds in Photonics.

Scattering thresholds and their associated spectral square root branch points are ubiquitous in photonics. In this Letter, we show that the scattering matrix has a simple universal behavior near scattering thresholds. We use unitarity, reciprocity, and time-reversal symmetry to construct a two-parameter model for a two-port scattering matrix near a threshold. We demonstrate this universal behavior in three different optical systems, namely, a photonic crystal slab, a planar dielectric interface, and a junction between metallic waveguides of different widths.

Short branch cut approximation in two-dimensional hydrodynamics with free surface

A potential motion of ideal incompressible fluid with a free surface and infinite depth is considered in two-dimensional geometry. A time-dependent conformal mapping of the lower complex half-plane of the auxiliary complex variable w into the area filled with fluid is performed with the real line of w mapped into the free fluid’s surface. The fluid dynamics can be fully characterized by the motion of the complex singularities in the analytical continuation of both the conformal mapping and the complex velocity. We consider the short branch cut approximation of the dynamics with the small parameter being the ratio of the length of the branch cut to the distance between its centre and the real line of w . We found that the fluid dynamics in that approximation is reduced to the complex Hopf equation for the complex velocity coupled with the complex transport equation for the conformal mapping. These equations are fully integrable by characteristics producing the infinite family of solutions, including moving square root branch points and poles. These solutions involve practical initial conditions resulting in jets and overturning waves. The solutions are compared with the simulations of the fully nonlinear Eulerian dynamics giving excellent agreement even when the small parameter approaches about one.

Inverse scattering transform for the complex reverse space-time nonlocal modified Korteweg-de Vries equation with nonzero boundary conditions and constant phase shift.

Recently, the complex reverse space-time (RST) nonlocal modified Korteweg-de Vries equation (mKdV) was introduced and shown to be an integrable infinite-dimensional dynamical system. The inverse scattering transform (IST) for zero boundary condition was studied by Ablowitz and Musslimani in 2016. In this paper, the IST for the complex RST nonlocal mKdV equation with nonzero boundary conditions at infinity is presented. The direct and inverse scattering problems are analyzed. The method to carry out the inverse problem employs two Riemann surfaces associated with square root branch points in the eigenfunctions/scattering data, which is more complicated than that for the zero boundary condition. Four cases are considered, special soliton solutions are discussed, and an explicit 1-soliton and two 2-soliton are found for three cases. In another case, there are no solitons.

On the complex singularities of the inverse Langevin function

We study the inverse Langevin function $\mathscr{L}^{-1}(x)$ because of its importance in modelling limited-stretch elasticity where the stress and strain energy become infinite as a certain maximum strain is approached, modelled here by $x\to1$. The only real singularities of the inverse Langevin function $\mathscr{L}^{-1}(x)$ are two simple poles at $x=\pm1$ and we see how to remove their effects either multiplicatively or additively. In addition, we find that $\mathscr{L}^{-1}(x)$ has an infinity of complex singularities. Examination of the Taylor series about the origin of $\mathscr{L}^{-1}(x)$ shows that the four complex singularities nearest the origin are equidistant from the origin and have the same strength; we develop a new algorithm for finding these four complex singularities. Graphical illustration seems to point to these complex singularities being of a square root nature. An exact analysis then proves these are square root branch points.

Structure and location of branch point singularities for Stokes waves on deep water

The Stokes wave is a finite-amplitude periodic gravity wave propagating with constant velocity in an inviscid fluid. The complex analytical structure of the Stokes wave is analysed using a conformal mapping of a free fluid surface of the Stokes wave onto the real axis with the fluid domain mapped onto the lower complex half-plane. There is one square root branch point per spatial period of the Stokes wave located in the upper complex half-plane at a distance $v_{c}$ from the real axis. The increase of Stokes wave height results in $v_{c}$ approaching zero with the limiting Stokes wave formation at $v_{c}=0$. The limiting Stokes wave has a $2/3$ power-law singularity forming a $2{\rm\pi}/3$ radians angle on the crest which is qualitatively different from the square root singularity valid for arbitrary small but non-zero $v_{c}$, making the limit of zero $v_{c}$ highly non-trivial. That limit is addressed by crossing a branch cut of a square root into the second and subsequently higher sheets of the Riemann surface to find coupled square root singularities at distances $\pm v_{c}$ from the real axis at each sheet. The number of sheets is infinite and the analytical continuation of the Stokes wave into all of these sheets is found together with the series expansion in half-integer powers at singular points within each sheet. It is conjectured that a non-limiting Stokes wave at the leading order consists of an infinite number of nested square root singularities which also implies the existence in the third and higher sheets of additional square root singularities away from the real and imaginary axes. These nested square roots form a $2/3$ power-law singularity of the limiting Stokes wave as $v_{c}$ vanishes.

Diversity and distribution of idioblasts producing calcium oxalate crystals in Dieffenbachia seguine (Araceae)

Although cells that synthesize crystals are known throughout the plant kingdom, their functional significance is still unknown. Mechanical support, mineral balance, waste sequestration, and protection against herbivores have all been proposed as crystal functions. To seek clues to their role(s), I systematically examined all organs except fruit of Dieffenbachia seguine (Araceae) for crystals. Crystals were found in nearly every organ. Raphides (long, slim, pointed crystals) were most common, but druses (crystal aggregates) and prisms were also found. Raphides varied in size by a factor of 10 and also in organization from tightly bundled to loosely organized. Biforines, a type of cell capable of expelling raphides, or biforine-like cells, were found in nearly all organs, but especially in leaves, spathes, and anthers. Different organs had different crystal complements, and characteristic crystals were found at specific locations, such as among pollen, along the undersides of leaf veins, and at root branch points. All crystals appeared to be composed of calcium oxalate, based on acid solubility. Possible roles of the crystals are discussed in light of these findings.

SEMICLASSICAL (QUANTUM FIELD THEORY) AND QUANTUM (STRING) DE SITTER REGIMES: NEW RESULTS

We compute the quantum string entropy S s (m, H) from the microscopic string density of states ρ s (m, H) of mass m in de Sitter space–time. We find for high m (high Hm → c/α') a new phase transition at the critical string temperature T s = (1/2πk B )L cl c2/α', higher than the flat space (Hagedorn) temperature t s (L cl = c/H, the Hubble constant H acts at the transition, producing a smaller string constant α' and thus, a higher tension). T s is the precise quantum dual of the semiclassical (QFT Hawking–Gibbons) de Sitter temperature T sem = ħ c/(2πk B L cl ). By precisely identifying the semiclassical and quantum (string) de Sitter regimes, we find a new formula for the full de Sitter entropy S sem (H), as a function of the usual Bekenstein–Hawking entropy [Formula: see text]. For L cl ≫ ℓ Planck , i.e. for low [Formula: see text] is the leading term, but for high H near c/ℓ Planck , a new phase transition operates and the whole entropy S sem (H) is drastically different from the Bekenstein–Hawking entropy [Formula: see text]. We compute the string quantum emission cross-section σ string by a black hole in de Sitter (or asymptotically de Sitter) space–time (bhdS). For T sem bhdS ℓ T s (early evaporation stage), it shows the QFT Hawking emission with temperature T sem bhdS (semiclassical regime). For T sem bhdS → T s , σ string exhibits a phase transition into a string de Sitter state of size [Formula: see text], [Formula: see text], and string de Sitter temperature T s . Instead of featuring a single pole singularity in the temperature (Carlitz transition), it features a square root branch point (de Vega–Sanchez transition). New bounds on the black hole radius r g emerge in the bhdS string regime: it can become r g = L s /2, or it can reach a more quantum value, r g = 0.365 ℓ s .

A single hemoglobin gene in Myrica gale retains both symbiotic and non-symbiotic specificity

Here, a hemoglobin gene from the nitrogen-fixing actinorhizal plant Myrica gale was isolated, cloned and sequenced. The gene (MgHb) was a class I hemoglobin with strong sequence homology to non-symbiotic hemoglobin genes. MgHb is highly expressed in symbiotic root nodules, but transcripts and protein were also detected in leaves of M. gale. In Arabidopsis thaliana the MgHb promoter, linked to a beta-glucuronidase coding region, directed expression in the vascular tissue, in shoot meristem and at root branch point--a pattern very similar to the combined expression pattern of the two non-symbiotic A. thaliana hemoglobin promoters AHb1 and AHb2. The results points to a symbiotic as well as a non-symbiotic specificity of MgHb similar to a hemoglobin gene identified in Parasponia andersonii, but different from the situation in Casuarina glauca--a close actinorhizal relative of M. gale.

Self-gravitating phase transitions: Point particles, black holes and strings

We compute the quantum string entropy S s ( m , j ) of the microscopic string states of mass m and spin j in two physically relevant backgrounds: Kerr (rotating) black holes and de Sitter (dS) space–time. We find a new formula for the quantum gravitational entropy S sem ( M , J ) , as a function of the usual Bekenstein–Hawking entropy S sem ( 0 ) ( M , J ) . We compute the quantum string emission by a black hole in de Sitter space–time (bhdS). In all the following cases: (i) strings with the highest spin, and (ii) in dS space–time, (iii) quantum rotating black holes, (iv) quantum dS regime, (v) late bhdS evaporation, we find a new gravitational phase transition with a common distinctive universal feature: a square root branch point singularity in any space–time dimensions. This is the same behavior as for the thermal self-gravitating gas of point particles (de Vega–Sanchez transition), thus describing a new universality class. To cite this article: N.G. Sanchez, C. R. Physique 7 (2006).

Third-order spectral branch points in Krein space related setups: $$\mathcal{P}\mathcal{T}$$ -symmetric matrix toy model, MHD α 2-dynamo and extended Squire equation

The spectra of self-adjoint operators in Krein spaces are known to possess real sectors as well as sectors of pair-wise complex conjugate eigenvalues. Transitions from one spectral sector to the other are a rather generic feature and they usually occur at exceptional points of square root branching type. For certain parameter configurations two or more such exceptional points may happen to coalesce and to form a higher-order branch point. We study the coalescence of two square root branch points semi-analytically for a \(\mathcal{P}\mathcal{T}\)-symmetric 4 × 4 matrix toy model and illustrate its occurrence numerically in the spectrum of the 2 × 2 operator matrix of the magneto-hydrodynamic α 2-dynamo and an extended version of the hydrodynamic Squire equation.

Transcriptional regulation and functional properties of Arabidopsis Pht1;4, a high affinity transporter contributing greatly to phosphate uptake in phosphate deprived plants

Phosphate mobilization into the plant is a complex process requiring numerous transporters for absorption and translocation of this major nutrient. In the genome of Arabidopsis thaliana, nine closely related high affinity phosphate transporters have been identified but their specific roles remain unclear. Here we report the molecular, histological and physiological characterization of Arabidopsis pht1;4 high affinity phosphate transporter mutants. Using GUS-gene trap and in situ hybridization, Pht1;4 was found mainly expressed in inorganic phosphate (Pi) limiting medium in roots, primarily in the epidermis, the cortex and the root cap. In addition to this, expression was also observed at the lateral root branch points on the primary root and in the stele of lateral roots, suggesting a role of Pht1;4 in phosphate absorption and translocation from the growth medium to the different parts of the plant. Pi-starved pht1;4 plantlets exhibited a strong reduction of phosphate uptake capacity (40). This phenotype appears only related to the pht1;4 mutation as there were no obvious changes in the expression of other Pht1 family members in the mutants background. However, after 10 days of growth on phosphate deficient or sufficient medium, the Pi content in the mutants was not significantly different from that of the corresponding wild type controls. Furthermore, the mutants did not display any obvious growth defects or visible phenotypes when grown on a low phosphate containing medium. The work described here offers a first step in the complex genetic dissection of the phosphate transport system in planta.

Statistical mechanics of the self-gravitating gas: I. Thermodynamic limit and phase diagrams

We provide a complete picture to the self-gravitating non-relativistic gas at thermal equilibrium using Monte Carlo simulations, analytic mean field methods (MF) and low density expansions. The system is shown to possess an infinite volume limit in the grand canonical (GCE), canonical (CE) and microcanonical (MCE) ensembles when ( N, V)→∞, keeping N/ V 1/3 fixed. We compute the equation of state (we do not assume it as is customary), as well as the energy, free energy, entropy, chemical potential, specific heats, compressibilities and speed of sound; we analyze their properties, signs and singularities. All physical quantities turn out to depend on a single variable η≡ Gm 2N V 1/3T that is kept fixed in the N→∞ and V→∞ limit. The system is in a gaseous phase for η< η T and collapses into a dense object for η> η T in the CE with the pressure becoming large and negative. At η≃ η T the isothermal compressibility diverges. This gravitational phase transition is associated to the Jeans' instability. Our Monte Carlo simulations yield η T ≃1.515. PV/[ NT]= f( η) and all physical magnitudes exhibit a square root branch point at η= η C > η T . The values of η T and η C change by a few percent with the geometry for large N: for spherical symmetry and N=∞ (MF), we find η C =1.561764… while the Monte Carlo simulations for cubic geometry yields η C ≃1.540. In mean field and spherical symmetry c V diverges as ( η C − η) −1/2 for η↑ η C while c P and κ T diverge as ( η 0− η) −1 for η↑η 0=1.51024… . The function f( η) has a second Riemann sheet which is only physically realized in the MCE. In the MCE, the collapse phase transition takes place in this second sheet near η MC=1.26 and the pressure and temperature are larger in the collapsed phase than in the gaseous phase. Both collapse phase transitions (in the CE and in the MCE) are of zeroth order since the Gibbs free energy has a jump at the transitions. The MF equation of state in a sphere, f( η), obeys a first order non-linear differential equation of first kind Abel's type. The MF gives an extremely accurate picture in agreement with the MC simulations both in the CE and MCE. Since we perform the MC simulations on a cubic geometry they describe an isothermal cube while the MF calculations describe an isothermal sphere. The local properties of the gas, scaling behaviour of the particle distribution and its fractal (Haussdorf ) dimension are investigated in the companion paper quoted as paper II in the text: H.J. de Vega, N. Sánchez, astro-ph/0101567, next paper in this issue.

Large-order behaviour of the strong coupling perturbation expansion for anharmonic oscillators

A new formula describing the large-order behaviour of the strong coupling perturbation coefficients for the anharmonic oscillators with the Hamiltonian is suggested. A new method for the accurate calculation of the square root branch points of the energy from the numerical values of the coefficients is also suggested. The branch points and the related minimal values of the coupling constant for which the expansion converges are calculated for the ground state of the quartic, sextic, octic and decadic oscillators.

On the fundamentals of the gradient theory of van der Waals

In this study a simple relation has been derived for the influence parameter in the gradient theory of van der Waals in terms of simple accessible quantities like temperature, the equilibrium densities, and the equilibrium isothermal compressibilities. Application of this relation leads to a substantially better agreement between interfacial tensions computed from the gradient theory and tensions obtained from experiment and simulation. The basis for this novel relation is an expression that connects the influence parameter to the second moment of the direct correlation function of pure fluids at states within the binodal region. The direct correlation functions for this study have been obtained from solving of the Ornstein–Zernike equation and the Percus–Yevick (PY) or hypernetted chain (HNC) closure relations for a Lennard–Jones (LJ) fluid. Special attention was paid to the behavior of solutions in the vicinity of the nonsolution region. It was shown for the PY closure that at the low density side the isothermal compressibility remains finite at the boundary of the nonsolution region. Along the isotherms and isochors the isothermal compressibility terminates at this boundary in so-called square root branch points. The isothermal compressibility diverges on the high density side although the correct location of the spinodal locus could not be found because of numerical inaccuracies. Diverging compressibilities are never encountered as the solution boundary is reached using the HNC closure. In all cases the isothermal compressibility terminates in square root branch points along the isotherms and isochors. In addition, computations show that at this boundary, the second moment of the direct correlation function seems to diverge for both the PY and the HNC closure. Comparison of the tensions obtained from the gradient theory with those obtained from a partial summation of the gradient expansion shows that at low temperatures the former results are ∼50% higher. Comparison of the results obtained from the latter model with experiments and simulations shows good agreement.

Inability of the hypernetted chain integral equation to exhibit a spinodal line

The hypernetted chain (HNC) integral equation is solved for a variety of systems including the simple fluid of hard spheres with attractive Yukawa tail, the symmetrical and highly asymmetrical mixtures of charged particles, and the mixtures of hard spheres with positively nonadditive diameters. All these systems exhibit a liquid–gas or fluid–fluid phase separation. We are concerned with the behavior of the HNC solution near the two-phase region in the phase diagram. In all cases, the HNC equation does not have solutions inside a certain region whose boundary line is not a spinodal line. As the boundary is approached, the isothermal compressibility deduced from the compressibility route does not diverge, but tends to finite values. The isotherms and isochores terminate at the boundary line in square root branch points. This behavior is directly correlated to the existence of multiple HNC solutions. Physical and nonphysical solutions coincide on the boundary line. This universal HNC behavior is compared with that of the mean spherical approximation and Percus–Yevick integral equations.

Multi‐valued solutions and branch point singularities for nonlinear hyperbolic or elliptic systems

AbstractMulti‐valued solutions are constructed for 2 × 2 first‐order systems using a generalization of the hodograph transformation. The solution is found as a complex analytic function on a complex Riemann surface for which the branch points move as part of the solution. The branch point singularities are envelopes for the characteristics and thus move at the characteristic speeds. We perform an analysis of stability of these singularities with respect to perturbations of the initial data. The generic singularity types are folds, cusps, and nondegenerate umbilic points with non‐zero 3‐jet. An isolated singularity is generically a square root branch point corresponding to a fold. Two types of collisions between singularities are generic: At a “tangential” collision between two singularities moving at the same characteristic speed, a cube root branch point is formed, corresponding to a cusp. A “non‐tangential” collision, between two square root branch points moving at different characteristic speeds, remains a square root branch point at the collision and corresponds to a nondegenerate umbilic point. These results are also valid for a diagonalizable n‐th order system for which there are exactly two speeds. © 1993 John Wiley &amp; Sons, Inc.

Critical and off-critical conformal analysis of the Ising quantum chain in an imaginary field

The critical curve with central charge c=-22/5 of the Ising quantum chain in an imaginary longitudinal field is calculated by finite-size extrapolation of the Yang-Lee square root branch point position. Various power laws are found. The field dependence of the off-criticality mass scale and the four lowest off-criticality scaling functions are calculated. The latter turn out to be universal along the critical line and agree very well with the results of the truncated Hilbert space method for the c=-22/5 continuum theory. The author obtains the universal bulk scaling coefficient in qualitative agreement with the thermodynamic Bethe ansatz result. The scattering phaseshift calculated by Luscher's method (1988) reproduces the Cardy-Mussardo minimal solution (1989) for the S-matrix. If a boundary defect is introduced, the only non-trivial primary field decouples, so that the spectrum consists only of the tower of the unit operator. In the antiferromagnetic regime with a non-staggered imaginary field the author finds boundary-dependent lines where the gap vanishes.

Threshold singularities of two-cluster–two-cluster scattering amplitudes for dilation analytic potentials

Two-cluster–two-cluster scattering amplitudes are studied for N-body quantum systems with potentials that are both dilation analytic and exponentially decaying. It is proved that under quite broad assumptions these amplitudes can be meromorphically continued in the energy, with square root or logarithmic branch points at the two-cluster thresholds.

Existence and analyticity of many-body scattering amplitudes at low energies

Two-cluster–two-cluster scattering amplitudes for N-body quantum systems are studied. Our attention is restricted to energies below the lowest three-cluster threshold. For potentials falling off like r−1−δ it is proved that in this energy range these amplitudes exist, are continuous, and that the asymptotic completeness holds. Moreover, if the potentials fall off exponentially it is proved that these amplitudes can be meromorphically continued in the energy, with square root or logarithmic branch points at the two-cluster thresholds.