Relativistic quantum theory for charged spinless particles in external vector fields

Guided by a diagonalized form of the classical field-energy we construct a time-dependent canonical pair of Schrödinger fields _t (x) and _t (x) which diagonalizes the field-HamiltonianH _t . These Schrödinger fields in general belong to inequivalent representations of the canonical commutation relations for differentt's.The Heisenberg field is constructed by solving the Heisenberg equation of motion and its time-evolution turns out to be governed by a unitary operator, i.e. the Heisenberg fields at different times are unitarily equivalent.Scattering theory (including eventual incoming and/or outgoing bound-states) is finally constructed.     

From the generalized uncertainty relations on fuzzy <Emphasis Type="Italic">AdS</Emphasis><Subscript>2</Subscript> to the Poincaré geometry

The positive and discrete unitary irreps of SU(1,1) are used to construct fuzzy (Euclidean) AdS ₂. Two different types of uncertainty relation involving the Weyl–Heisenberg and a weaker type are studied. It is shown that there are no generalized coherent states which simultaneously minimize the Weyl–Heisenberg uncertainty relations among three non-commuting embedding coordinates of the fuzzy AdS ₂. However, generalized squeezed states that simultaneously satisfy the three weaker uncertainty relations do exist, and reproduce some properties of the classical AdS ₂. Up to a common scaling factor in terms of the irrep label, the expectation values of the non-commuting coordinates over such states are described in the same manner as the classical AdS ₂, in terms of the Poincaré coordinates. The expectation values on the fuzzy AdS ₂ tend to their corresponding values in the commutative limit.     

Influence of Ag doping on the critical behavior of the heat capacity of manganites

The critical behavior of the heat capacity of Ag doped manganites is studied. Changing regularities of universal critical parameters near the critical point are determined. All investigated samples up to t _min ≈ 10^?3 correspond to 3D Heisenberg universality class of the critical behavior. The universality class of critical behavior of La_{1 – x} Ag _y MnO₃ (y ≤ x) heat capacity is shown to be Ag-concentration independent.     

Specific heat of EuxSr1−xO near the ferromagnetic phase transition

High-resolution measurements of the specific heat,C, near the ferromagnetic phase transition of the diluted ferromagnetic system Eu_xSr_1–xO are reported. Samples with four different concentrations (x=1,x=0.9,x=0.7 andx=0.5) have been studied and the expected phase diagram, i.e. the linear decrease ofT _c with decreasingx, is confirmed. Our specific-heat data of the pure EuO samples yield a critical exponent =–0.12±0.02 in contrast to literature results, but in agreement with the value expected theoretically for this three-dimensional Heisenberg system. The origin of the discrepancies is traced down to differences in data analysis.     

Cu NMR/NQR Studies on Magnetism in Impurity/Hole-Doped Spin-Ladder Compounds

Cu nuclear magnetic resonance (NMR) spectra of impurities (Zn, Ni, and La)-doped spin-1/2 Heisenberg ladder compounds SrCu₂O₃ (Sr123) are broadened with Curie-like temperature (T) dependence. The spectra have been successfully fit by using a quasi-one-dimensional (Q1D) staggered polarization (SP) model. Such a SP has also revealed in Cu NMR measurements of Sr_14–x Ca _x Cu₂₄O₄₁ (Cax) with hole-doped ladders. The origin of possible 3D antiferromagnetic (AF) long-range ordering in (Zn and Ni)-doped Sr123 and Cax around x=12 at low T is considered to be similar. Once unpaired spins S ₀'s are induced and 3D interlayer interaction occurs, the localized spins couple in the whole system.     

Coherent Exchange Cluster calculations for antiferromagnetic disordered systems

The Coherent Exchange Cluster Approach developed in three preceding papers is applied to three-dimensional disordered antiferromagnetic Heisenberg spin systems with Perovskite structure, e.g. to KNi _x Mn_1–x F₃ and KNi _x Zn_1–x F₃. Both the level density of spin wave excitations, and the differential cross sections for inelastic neutron scattering are calculated. The results are compared with recent experiments, and also with results of simpler theories.It is found that the characteristic features of the excitation spectrum can be quite different even for binary alloys, i.e. besides the cases with one or two broadened magnon branches there might also exist systems with a more complicated spectrum.Dedicated to Prof. Dr. W. Döring on the occasion of his 65th birthday.     

Magnetic properties of ferromagnetic diluted Zn x Cd1–x Cr2Se4 spinels are studied by Green's functions,mean field theory and high temperature series expansions theories

The field induced reorientation of the magnetization of ferromagnetic (or antiferromagnetic) structure is treated within the framework of many-body Green's function theory by considering all components of the magnetization. The mean field theory is used to calculate the nearest neighbour and the next-neighbour super-exchange J₁(Cr–Cr) and J₂(Cr–(Zn(Cd)–Se)–Cr), respectively, for the Zn_1–x Cd _x Cr₂Se₄ in the range 0 < x < 1. The intraplanar and the interplanar interactions are deduced. The high temperature series expansions (HTSEs) are derived for the magnetic susceptibility and the two-spin correlation functions for a Heisenberg ferromagnetic model on the B-spinel lattice. The calculations are developed in the framework of the random phase approximation (RPA). The magnetic phase diagram is deduced. A spin glass phase is predicted for intermediate range of concentration. The obtained results are comparable with those obtained by magnetic measurements. The critical exponents associated with the magnetic susceptibility (γ) and the correlation lengths (ν) have been deduced. The obtained values are comparable to those of 3D Heisenberg model.     

String theory, scale relativity and the generalized uncertainty principle

Extensions (modifications) of the Heisenberg uncertainty principle are derived within the framework of the theory of special scale-relativity proposed by Nottale. In particular, generalizations of the stringy uncertainty principle are obtained where the size of the strings is bounded by the Planck scale and the size of the universe. Based on the fractal structures inherent with two dimensional quantum gravity, which has attracted considerable interest recently, we conjecture that the underlying fundamental principle behind string theory should be based on an extension of the scale relativity principle whereboth dynamics as well as scales are incorporated in the same footing.     

The Krichever map,vector bundles over algebraic curves,and Heisenberg algebras

We study the GrassmannianGr _x ⁿ consisting of equivalence classes of rankn algebraic vector bundles over a Riemann surfaceX with an holomorphic trivialization at a fixed pointp. Commutative subalgebras ofgl(n, H ),H being the ring of functions holomorphic on a punctured disc aboutp, define flows on the Grassmannian, giving rise to classes of solutions to multi-component KP hierarchies. These commutative subalgebras correspond to Heisenberg algebras in the Kac-Moody algebra associated togl(n, H ₎. One can obtain, by the Krichever map, points ofGr _x ⁿ (and solutions of mcKP) from coveringsf: YX and other geometric data. Conversely for every point ofGr _x ⁿ and for every choice of Heisenberg algebra we construct, using the cotangent bundle ofGr _x ⁿ , an algebraic curve coveringX and other data, thus inverting the Krichever map. We show the explicit relation between the choice of Heisenberg algebra and the geometry of the covering space.The research was partially supported by US Army grant DAA L03-87-K-0110 and NSF grant DMS 9106938     

On the T Bloch law in magnets with fourth-order exchange interaction

For the ferromagnets EuS and GdMg, in which fourth-order exchange interactions (i.e. biquadratic, three-spin and four-spin interactions) have been identified, the deviation of the spontaneous magnetization with respect to the T =0 value is shown to follow a T² law instead of the famous T3/2 law expected for a Heisenberg ferromagnet. Moreover, the observed T² law holds for temperatures as large as 0.8T_C and the extrapolated magnetization value for does not conform to ferromagnetic saturation. This is because the fourth-order exchange interactions generate a second order-parameter which is assumed to govern the order of the transverse moment components. These moment components have a finite expectation value for at the expense of the Heisenberg order parameter. Like the spontaneous magnetization, the critical field curves B c ( T ) of the metamagnet EuSe and the antiferromagnet EuTe also start decreasing with a T² term for . It is argued that the T² law is a consequence of the fourth-order exchange interactions. This is shown experimentally by a study of the critical field curves [0pt] pertinent to the longitudinal (Heisenberg) order-parameter in the diamagnetically diluted antiferromagnets Eu_xSr_1-xTe. In this solid solution series a particular composition of x c =0.85 exists at which the different fourth-order interaction processes compensate each other in the high temperature average. As a consequence, an Eu_xSr_1-xTe sample with x =0.85 meets the requirements of a Heisenberg antiferromagnet at least if a quantity is considered for which the high-temperature average over all fourth-order interactions is decisive. This seems to be the case for the critical field curve [0pt] which gives the phase boundary to the paramagnetic phase. In fact, a crossover from a T² to a T3/2 law is observed for [0pt] on approaching x_c. This, we believe, shows the frequently observed T² law is caused by the fourth-order interactions. Received 23 July 1998 and Received in final form 12 October 1998     

The effect of spin dilution on magnetism of the linear chain system β-Cu2?xZnxV2O7

We have measured the magnetic susceptibility (χ) and heat capacity (C _p) of β-Cu_2−x Zn _x V₂O₇ (x = 0, 0.05, 0.1, 0.15, 0.2, 0.3, 2) in the temperature range 2–300 K. A one-dimensional alternating exchange Heisenberg antiferromagnetism (HAF) is observed in all compositions with chains of infinite length. The intra-chain exchange remains uniform and decreases marginally with dilution of the magnetic state. A cooperative ordering is seen in the magnetic chains for all Zn concentrations (x ≤ 0.3). The temperature of occurrence of this transition decreases with increasing Zn concentration. Though the conventional spin-wave theory has been used here to describe the properties of the ordered phase, the presence of some contributions like the lattice heat capacity in C _p and the Curie-Weiss term in susceptibility introduces some uncertainties in the estimation of the proportions contributed by the spin system. Therefore, the nature of the ordered phase could not be ascertained unambiguously.     

It?’s Lemma with Quantum Calculus (q-Calculus): Some Implications

q-derivatives are part of so called quantum calculus. In this paper we investigate how such derivatives can possibly be used in It?’s lemma. This leads us to consider how such derivatives can be used in a social science setting. We conclude that in a It? Lemma setting we cannot use a macroscopic version of the Heisenberg uncertainty principle with q-derivatives.     

Domain wall dynamics of the spin-$\frac{1}{2}$ Ising-like antiferromagnetic chain in presence of Dzyaloshinskii-Moriya interactions

We consider a one-dimensional Ising-like S = \frac12\frac{1}{2} Heisenberg antiferromagnetic Hamiltonian and study the dynamics of domain wall excitations in presence of both uniform and staggered Dzyaloshinskii-Moriya interactions. We obtain dispersion relations and dynamical spin correlation functions, S ^{x x} (q, ω) using the basis of domain wall pair states. It is shown that the line shapes of S ^{x x} (q, ω) are highly asymmetric over the whole Brillouin zone such that the spectral weights mainly concentrate in the low energy side. It is observed that presence of staggered Dzyaloshinskii-Moriya interaction explains the experimental results on the Ising-like antiferromagnetic compounds CsCoCl₃ and CsCoBr₃ very well.     

Compositional effect of bcc Co100−x Fe x electrodes on magnetoresistance in AlO x -based magnetic tunnel junctions

The effect of the composition of ferromagnetic bcc Co_100−x Fe _x electrodes on tunneling magnetoresistance (TMR) of Co_100−x Fe _x /AlO _x /Co_100−x Fe _x /IrMn magnetic tunnel junctions was studied. The epitaxial growth of the bottom Co_100−x Fe _x electrode leads to a high-quality electrode and interface, which significantly enhances the TMR ratio and the desired effect for study. Other factors that could also affect TMR, such as interface roughness, tunneling barrier properties, and exchange-bias properties, were kept the same within the uncertainty of the experiment in order to minimize their effects. The observed TMR dependence on composition is attributed to the variation of the s-like electron densities of state of the bcc Co_100−x Fe _x electrodes with different compositions.     

Biquadratic exchange and critical behaviour in the diluted antiferromagnet EuxSr1−xTe

Magnetization, specific heat, magnetocaloric effect, and neutron diffraction measurements have been performed in order to better understand the magnetic interactions and phase transitions in the diluted antiferromagnetic system Eu_xSr_1–xTe. Due to the low Néel temperature ofT _N =9.8 K and the associated small critical field ofB _C (T=0)=7.5 T, EuTe provides the opportunity to change the angle between the antiferromagnetic moments continuously between 180° and 0° by applying a magnetic field and allows thus to probe the type of the magnetic interaction. It is found that in addition to the common bilinear Heisenberg exchange mechanism there is evidence for a small biquadratic exchange term, which has important consequences on the critical behaviour of Eu_xSr_1–xTe. In the case of EuTe the biquadratic exchange supports the antiferromagnetic order, such that virtually no anisotropy exists and this restricts the anti-Herromagnetic (AF) to spin-flop (SF) transition toB ₀=0. On dilution with strontium, the biquadratic interaction decreases strongly and changes its sign at x0.83 thus favouring the perpendicular moment orientation. This acts like an anisotropy on the dominating antiferromagnetic bilinear interaction and shifts the AF-SF phase line to finite fields which reach values of 0.3 T forT0. No bicritical behaviour is observed, but instead, the SF phase is the state of lowest energy in a small temperature interval belowT _N . The critical exponent (x) of the specific heat decreases in a non-linear way with dilution starting from a large positive value for EuTe. (x) is close to the Heisenberg value –0.12 nearx0.83 where the biquadratic interaction is zero. The same applies for the exponent of the sublattice magnetization.     

Magnetic phase diagram of diluted spinel Zn1−xCuxCr2Se4 system

By using mean field theory, we have evaluated the nearest-neighbour and the next-neighbour super-exchange J₁(x) and J₂(x), respectively, for Zn_1−xCu_xCr₂Se₄ in the range 0?x?1. The intraplanar and the interplanar interactions are deduced. High-temperature series expansions are derived for the magnetic susceptibility and two-spin correlation functions for a Heisenberg ferromagnetic model on the B-spinel lattice. The calculations are developed in the framework of the random phase approximation. The magnetic phase diagram is deduced. A spin glass phase is predicted for intermediate range of concentration. The results are comparable with those obtained by magnetic measurements. The critical exponents associated with the magnetic susceptibility (γ) and the correlation lengths (ν) have been deduced. The values are comparable to those of the 3D Heisenberg model, and are insensitive to the dilution x.     

Influence of quantum fluctuations on the Néel temperature of a dilute two-dimensional anisotropic antiferromagnet

A quantum Monte Carlo procedure is used to calculate the energy, sublattice magnetization, Néel temperature, and the slopes of the S=[1/T _N(x=0)]dT _N(x)/dx curves as functions of the hole concentration and the exchange anisotropy Δ=1−J _x,y/J _z in the Heisenberg model with anisotropic negative interactions between nearest neighbors in a square lattice with dilution among the lattice sites. The slope diverges in the limit Δ→0: S∼ln(6.5/Δ). Fiz. Tverd. Tela (St. Petersburg) 39, 898–900 (May 1997)     

Thermal entanglement in two-qutrit spin-1 anisotropic Heisenberg model with inhomogeneous magnetic field

The thermal entanglement of a two-qutrit spin-1 anisotropic Heisenberg XXZ chain in an inhomogeneous magnetic field is studied in detail. The effects of the external magnetic field (B), a parameter b which controls the inhomogeneity of B, and the bilinear interaction parameters J_{x=Jy≠Jz on the thermal variation of the negativity are studied in detail. It is found that negativity N decreases when the values of magnetic field, inhomogeneity b and temperature are increasing. In addition, N remains at higher temperatures for higher values of Jz and lower values of B and b.}     

The Curie temperature of dilute ferromagnetic alloys near the percolation threshold

Upper limit for the Curie temperature T_c(x) of the dilute Heisenberg ferromagnet near percolation threshold is derived. Qualitative arguments are put forward that this limit gives a correct dependence of T_c on the concentration of magnetic atoms x.     

Uncertainty relation for successive measurements

It is noted that the Heisenberg uncertainty relations set a lower bound on the product of variances of two observablesA, B when they are separately measured on two distinct, but identically prepared ensembles. A new uncertainty relation is derived for the product of the variances of the two observablesA, B when they are measured sequentially on a single ensemble of systems. It is shown that the two uncertainty relations differ significantly wheneverA andB are not compatible.