Renormalizable Theory Research Articles

Willis Eugene Lamb Jr

Willis Eugene Lamb Jr

Spin fluctuations probed by NMR in paramagnetic spinelLiV2O4: a self-consistent renormalization theory

Low-frequency spin fluctuation dynamics in paramagnetic spinelLiV2O4, a rare 3d-electron heavy-fermion system, is investigated. A parametrizedself-consistent renormalization (SCR) theory of the dominant AFM spin fluctuations isdeveloped and applied to describe temperature and pressure dependences of thelow-T nuclear spin–latticerelaxation rate 1/T1 in this material. The experimental data for1/T1 availabledown to ∼1 K are well reproduced by the SCR theory, showing the development of AFM spin fluctuationsas the paramagnetic metal approaches a magnetic instability under the applied pressure. Thelow-T upturn of 1/T1T detected below 0.6 K under the highest applied pressure of 4.74 GPa isexplained as the nuclear spin relaxation effect due to the spin freezingof magnetic defects unavoidably present in the measured sample ofLiV2O4.

Magnetic resonant x-ray diffraction study of europium telluride

Here we use magnetic resonant x-ray diffraction to study the magnetic order in a $1.5\text{ }\ensuremath{\mu}\text{m}$ EuTe film grown on (111) ${\text{BaF}}_{2}$ by molecular-beam epitaxy. At $\text{Eu}\text{ }{L}_{\text{II}}$ and ${L}_{\text{III}}$ absorption edges, a resonant enhancement of more than two orders was observed for the $\ensuremath{\sigma}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ensuremath{'}}$ diffracted intensity at half-order reciprocal-lattice points, consistent with the magnetic character of the scattering. We studied the evolution of the $(\frac{1}{2}\frac{1}{2}\frac{1}{2})$ magnetic reflection with temperature. When heating toward the Neel temperature $({T}_{N})$, the integrated intensity decreased monotonously and showed no hysteresis upon cooling again, indicating a second-order phase transition. A power-law fit to the magnetization versus temperature curve yielded ${T}_{N}=9.99(1)\text{ }\text{K}$ and a critical exponent $\ensuremath{\beta}=0.36(1)$, which agrees with the renormalization theory results for three-dimensional Heisenberg magnets. The fits to the sublattice magnetization dependence with temperature, disregarding and considering fourth-order exchange interactions, evidenced the importance of the latter for a correct description of magnetism in EuTe. A value of 0.009 was found for the $(2{j}_{1}+{j}_{2})/{J}_{2}$ ratio between the Heisenberg ${J}_{2}$ and fourth-order ${j}_{1,2}$ exchange constants. The magnetization curve exhibited a round-shaped region just near ${T}_{N}$ accompanied by an increase in the magnetic peak width, which was attributed to critical scattering above ${T}_{N}$. The comparison of the intensity ratio between the $(\frac{1}{2}\frac{1}{2}\frac{1}{2})$ and the $(1\frac{1}{2}1\frac{1}{2}1\frac{1}{2})$ magnetic reflections proved that the ${\text{Eu}}^{2+}$ spins align within the (111) planes, and the azimuthal dependence of the $(\frac{1}{2}\frac{1}{2}\frac{1}{2})$ magnetic peak is consistent with the model of equally populated S domains.

Unusual temperature dependence in the low-temperature specific heat ofU3Ni5Al19

Specific heat has been measured down to 0.053 K on a single crystal of the heavy-fermion antiferromagnet ${\text{U}}_{3}{\text{Ni}}_{5}{\text{Al}}_{19}$ that orders at ${T}_{\text{N}}=23\text{ }\text{K}$. As has been previously reported, these data can be fitted between 0.4 and 4 K by the spin-fluctuation model of Moriya and Takimoto, which describes the contribution of weakly interacting critical spin fluctuations to the specific heat, $C$, where, as $T\ensuremath{\rightarrow}0$, $C/T={\ensuremath{\gamma}}_{0}\ensuremath{-}a\ensuremath{\surd}T$. However, below 0.35 K a noticeable divergence in $C/T\ensuremath{\sim}\text{log}\text{ }T$ dependence, consistent with the existence of strongly interacting fluctuations, is observed. This increase in the divergence of $C/T$ at the lowest temperatures---which is contrary to the self-consistent renormalization theory of Moriya and Takimoto, which predicts $\ensuremath{\surd}T$ dependence for $C/T$ as $T\ensuremath{\rightarrow}0$ and $\text{log}\text{ }T$ dependence at higher temperatures---has been measured as a function of magnetic field to further understand its origin. The field data in the low-temperature regime, where $C/T\ensuremath{\sim}\text{log}\text{ }T$ exhibit scaling with $\ensuremath{\Delta}B/{T}^{1.9}$, further evidence that there exist strongly interacting fluctuations below 0.35 K in ${\text{U}}_{3}{\text{Ni}}_{5}{\text{Al}}_{19}$.

Synthesis and Magnetic Properties of Ni3AlCx

The itinerant-electron ferromagnet Ni 3 Al and its carbon-intercalated compounds Ni 3 AlC x ( x = 0.01, 0.02, 0.04, 0.05, 0.08, and 0.1) were synthesized and investigated by magnetization measurements between 2 and 300 K with the magnetic field H up to 5 T and 27 Al NMR measurements at H = 7.4847 T. The Curie temperature is suppressed and disappears only by the carbon intercalation with x = 0.02. For the samples with x ≤0.02, the T 4/3 -temperature dependence of M s 2 ( M s : spontaneous magnetization) and χ 0 -1 (χ 0 : initial susceptibility) are newly observed, being consistent with the predication of the self-consistent renormalization theory of spin fluctuations for weak itinerant ferromagnets. The coupling constant between the 27 Al Knight shift and the bulk susceptibility of Ni 3 AlC 0.1 is three times larger than that of Ni 3 Al, suggesting the drastic change of the electronic state by the carbon intercalation.

Structural Relations between Nested Harmonic Sums

We describe the structural relations between nested harmonic sums emerging in the description of physical single scale quantities up to the 3–loop level in renormalizable gauge field theories. These are weight w=6 harmonic sums. We identify universal basic functions which allow to describe a large class of physical quantities and derive their complex analysis. For the 3–loop QCD Wilson coefficients 35 basic functions are required, whereas a subset of 15 describes the 3–loop anomalous dimensions.

Four loop vacuum bubbles for two-dimensional four-fermi theory renormalization

We review recent four loop calculations in the two dimensional S U ( N ) Gross-Neveu model involving four loop massive vacuum bubble diagrams relevant to the determination of the MS ¯ mass anomalous dimension.

Noncommutative Geometry

Noncommutative geometry applies ideas from geometry to mathematical structures determined by noncommuting variables. Within mathematics, it is a highly interdisciplinary subject drawing ideas and methods from many areas of mathematics and physics. Natural questions involving noncommuting variables arise in abundance in many parts of mathematics and quantum mathematical physics. On the basis of ideas and methods from algebraic topology and Riemannian geometry, as well as from the theory of operator algebras and from homological algebra, an extensive machinery has been developed which permits the formulation and investigation of the geometric properties of noncommutative structures. This includes K-theory, cyclic homology and the theory of spectral triples. Areas of intense research in recent years are related to topics such as index theory, quantum groups and Hopf algebras, the Novikov- and Baum-Connes conjectures as well as to the study of specific questions in other fields such as number theory, modular forms, topological dynamical systems, renormalization theory, theoretical high-energy physics and string theory. Many results elucidate important properties of fascinating specific classes of examples that arise in many applications. The talks covered substantial new results and insights in several of the different areas in Noncommutative Geometry. The workshop was attended by 53 participants including 6 young researchers supported by the European Union.

Scalar radius of the pion in the Kroll-Lee-Zumino renormalizable theory

The Kroll-Lee-Zumino renormalizable Abelian quantum field theory of pions and a massive $\ensuremath{\rho}$-meson is used to calculate the scalar radius of the pion at next-to-leading (one-loop) order in perturbation theory. Because of renormalizability, this determination involves no free parameters. The result is $⟨{r}_{\ensuremath{\pi}}^{2}{⟩}_{s}=0.40\text{ }\text{ }{\mathrm{fm}}^{2}$. This value gives for ${\overline{\ensuremath{\ell}}}_{4}$, the low energy constant of chiral perturbation theory, ${\overline{\ensuremath{\ell}}}_{4}=3.4$, and ${F}_{\ensuremath{\pi}}/F=1.05$, where $F$ is the pion decay constant in the chiral limit. Given the level of accuracy in the masses and the $\ensuremath{\rho}\ensuremath{\pi}\ensuremath{\pi}$ coupling, the only sizable uncertainty in this result is due to the (uncalculated) next-to-next-to-leading order contribution.

Ghost condensate busting

Applying the Thomas–Fermi approximation to renormalizable field theories, we constructghost condensation models that are free of the instabilities associated with violations of thenull-energy condition.

Boundaries of Siegel disks: Numerical studies of their dynamics and regularity

Siegel disks are domains around fixed points of holomorphic maps in which the maps are locally linearizable (i.e., become a rotation under an appropriate change of coordinates which is analytic in a neighborhood of the origin). The dynamical behavior of the iterates of the map on the boundary of the Siegel disk exhibits strong scaling properties which have been intensively studied in the physical and mathematical literature. In the cases we study, the boundary of the Siegel disk is a Jordan curve containing a critical point of the map (we consider critical maps of different orders), and there exists a natural parametrization which transforms the dynamics on the boundary into a rotation. We compute numerically this parameterization and use methods of harmonic analysis to compute the global Hölder regularity of the parametrization for different maps and rotation numbers. We obtain that the regularity of the boundaries and the scaling exponents are universal numbers in the sense of renormalization theory (i.e., they do not depend on the map when the map ranges in an open set), and only depend on the order of the critical point of the map in the boundary of the Siegel disk and the tail of the continued function expansion of the rotation number. We also discuss some possible relations between the regularity of the parametrization of the boundaries and the corresponding scaling exponents.

Field-theoretical Renormalization-Group approach to critical dynamics of crosslinked polymer blends

We consider a crosslinked polymer blend that may undergo a microphase separation. When the temperature is changed from an initial value towards a final one very close to the spinodal point, the mixture is out equilibrium. The aim is the study of dynamics at a given time t, before the system reaches its final equilibrium state. The dynamics is investigated through the structure factor, S(q, t), which is a function of the wave vector q, temperature T, time t, and reticulation dose D. To determine the phase behavior of this dynamic structure factor, we start from a generalized Langevin equation (model C) solved by the time composition fluctuation. Beside the standard de Gennes Hamiltonian, this equation incorporates a Gaussian local noise, zeta. First, by averaging over zeta, we get an effective Hamiltonian. Second, we renormalize this dynamic field theory and write a Renormalization-Group equation for the dynamic structure factor. Third, solving this equation yields the behavior of S(q, t), in space of relevant parameters. As result, S(q, t) depends on three kinds of lengths, which are the wavelength q(-1), a time length scale R(t) approximately t(1/z), and the mesh size xi*. The scale R(t) is interpreted as the size of growing microdomains at time t. When R(t) becomes of the order of xi*, the dynamics is stopped. The final time, t*, then scales as t* approximately xi*z, with the dynamic exponent z = 6-eta. Here, eta is the usual Ising critical exponent. Since the final size of microdomains xi* is very small (few nanometers), the dynamics is of short time. Finally, all these results we obtained from renormalization theory are compared to those we stated in some recent work using a scaling argument.

On the vacuum states for non-commutative gauge theory

Candidates for renormalizable gauge theory models on Moyal spaces constructed recently have non-trivial vacua. We show that these models support vacuum states that are invariant under both global rotations and symplectic isomorphisms which form a global symmetry group for the action. We compute the explicit expression in position space for these vacuum configurations in two and four dimensions.

A lattice Monte Carlo simulation of the FePt alloy using a first-principles renormalized four-body interaction

Although a lattice Monte Carlo method provides an effective, simple, and fast way to study thermodynamic properties of substitutional alloys, it cannot treat by itself the off-lattice effects, such as thermal vibrations and local distortions. Therefore, even if the interaction among atoms at lattice points is calculated accurately by means of first-principles calculations, the lattice Monte Carlo simulation overestimates the order-disorder phase transition temperature. In this paper, we treat this problem in the investigation of the FePt alloy, which has recently attracted considerable interest in its magnetic properties. We apply a simple version of the potential renormalization theory to determine the interaction among atoms, including partly the off-lattice effects by means of first-principles calculations. Then, we use the interaction to perform a lattice Monte Carlo simulation of the FePt alloy on a fcc lattice. From the results, we find that the transition temperature obtained after the present renormalization procedure becomes closer to the experimental value.

AN EQUIVALENCE OF TWO MASS GENERATION MECHANISMS FOR GAUGE FIELDS

Two mass generation mechanisms for gauge theories are studied. It is proved that in the Abelian case the topological mass generation mechanism introduced in Refs. 4, 12 and 15 is equivalent to the mass generation mechanism defined in Refs. 5 and 20 with the help of "localization" of a nonlocal gauge invariant action. In the non-Abelian case the former mechanism is known to generate a unitary renormalizable quantum field theory, describing a massive vector field.

Reducible expansions and related sharp crossovers in Feigenbaum’s renormalization field

We discuss reducible aspects of Mao and Hu's multiple scaling expansion [J. Stat. Phys. 46, 111 (1987); Int. J. Mod. Phys. B 2, 65 (1988)] in the framework of renormalization theory. After establishing a suitable form of reduced expansion, we present numerical evidence showing sharp crossovers from Feigenbaum's constant (delta) to Mao and Hu's constant (delta (')) in the first-order reduced expansion. We find that the crossover is caused by the universal scaling relation existing in constant coefficients of Mao and Hu's expansion. Special attention is paid to constant coefficients corresponding to scaling terms including delta ('). We show numerically that they converge to zero in universal ways with convergence ratios larger than delta. Here, the convergence direction is transversal to the unstable eigendirection of the linearized renormalization operator. From this observation, we propose a concise form of expansion for Feigenbaum's universal function g(r)(x).

Self-consistent renormalization theory of spin fluctuations in paramagnetic spinelLiV2O4

A phenomenological description for the dynamical spin susceptibility $\ensuremath{\chi}(\mathbf{q},\ensuremath{\omega};T)$ observed in inelastic neutron scattering measurements on powder samples of $\mathrm{Li}{\mathrm{V}}_{2}{\mathrm{O}}_{4}$ is developed in terms of the parametrized self-consistent renormalization (SCR) theory of spin fluctuations. Compatible with previous studies at $T\ensuremath{\rightarrow}0$, a peculiar distribution in $\mathbf{q}$ space of strongly enhanced and slow spin fluctuations at $q\ensuremath{\sim}{Q}_{c}\ensuremath{\simeq}0.6\phantom{\rule{0.3em}{0ex}}{\mathrm{\AA{}}}^{\ensuremath{-}1}$ in $\mathrm{Li}{\mathrm{V}}_{2}{\mathrm{O}}_{4}$ is involved to derive the mode-mode coupling term entering the basic equation of the SCR theory. The equation is solved self-consistently with the parameter values found from a fit of theoretical results to experimental data. For low temperatures, $T\ensuremath{\lesssim}30\phantom{\rule{0.3em}{0ex}}\mathrm{K}$, where the SCR theory is more reliable, the observed temperature variations of the static spin susceptibility $\ensuremath{\chi}({Q}_{c};T)$ and the relaxation rate ${\ensuremath{\Gamma}}_{Q}(T)$ at $q\ensuremath{\sim}{Q}_{c}$ are well reproduced by those suggested by the theory. For $T\ensuremath{\gtrsim}30\phantom{\rule{0.3em}{0ex}}\mathrm{K}$, the present SCR is capable of predicting only main trends in $T$ dependences of $\ensuremath{\chi}({Q}_{c};T)$ and ${\ensuremath{\Gamma}}_{Q}(T)$. The discussion is focused on a marked evolution (from $q\ensuremath{\sim}{Q}_{c}$ at $T\ensuremath{\rightarrow}0$ toward low $q$ values at higher temperatures) of the dominant low-$\ensuremath{\omega}$ integrated neutron scattering intensity $I(q;T)$.

Recursion and Growth Estimates in Renormalizable Quantum Field Theory

In this paper we show that there is a Lipatov bound for the radius of convergence for superficially divergent one-particle irreducible Green functions in a renormalizable quantum field theory if there is such a bound for the superficially convergent ones. In the nonnegative case the radius of convergence turns out to be min{ρ,1/b 1}, where ρ is the bound on the convergent ones, the instanton radius, and b 1 the first coefficient of the β-function, while in general it is bounded by the above.

Parametric Representation of “Covariant” Noncommutative QFT Models

We extend the parametric representation of renormalizable non commutative quantum field theories to a class of theories which we call critical, because their power counting is definitely more difficult to obtain. This class of theories is important since it includes gauge theories, which should be relevant for the quantum Hall effect.

Noncommutative induced gauge theories on Moyal spaces

Noncommutative field theories on Moyal spaces can be conveniently handled within a framework of noncommutative geometry. Several renormalisable matter field theories that are now identified are briefly reviewed. The construction of renormalisable gauge theories on these noncommutative Moyal spaces, which remains so far a challenging problem, is then closely examined. The computation in 4-D of the one-loop effective gauge theory generated from the integration over a scalar field appearing in a renormalisable theory minimally coupled to an external gauge potential is presented. The gauge invariant effective action is found to involve, beyond the expected noncommutative version of the pure Yang-Mills action, additional terms that may be interpreted as the gauge theory counterpart of the harmonic term, which for the noncommutative φ4-theory on Moyal space ensures renormalisability. A class of possible candidates for renormalisable gauge theory actions defined on Moyal space is presented and discussed.