Normal Coordinates and Primitive Elements in the Hopf Algebra of Renormalization

We introduce normal coordinates on the infinite dimensional group $G$ introduced by Connes and Kreimer in their analysis of the Hopf algebra of rooted trees. We study the primitive elements of the algebra and show that they are generated by a simple application of the inverse Poincar\'e lemma, given a closed left invariant 1-form on $G$. For the special case of the ladder primitives, we find a second description that relates them to the Hopf algebra of functionals on power series with the usual product. Either approach shows that the ladder primitives are given by the Schur polynomials. The relevance of the lower central series of the dual Lie algebra in the process of renormalization is also discussed, leading to a natural concept of $k$-primitiveness, which is shown to be equivalent to the one already in the literature.Comment: Latex, 24 pages. Submitted to Commun. Math. Phy

Baby-Step Giant-Step Algorithms for the Symmetric Group

We study discrete logarithms in the setting of group actions. Suppose that $G$ is a group that acts on a set $S$. When $r,s \in S$, a solution $g \in G$ to $r^g = s$ can be thought of as a kind of logarithm. In this paper, we study the case where $G = S_n$, and develop analogs to the Shanks baby-step / giant-step procedure for ordinary discrete logarithms. Specifically, we compute two sets $A, B \subseteq S_n$ such that every permutation of $S_n$ can be written as a product $ab$ of elements $a \in A$ and $b \in B$. Our deterministic procedure is optimal up to constant factors, in the sense that $A$ and $B$ can be computed in optimal asymptotic complexity, and $|A|$ and $|B|$ are a small constant from $\sqrt{n!}$ in size. We also analyze randomized "collision" algorithms for the same problem

Glassy relaxation without freezing in a random dipolar-coupled Ising magnet

We have measured the magnetic susceptibility, χ’+iχ’’, of the dilute dipolar-coupled Ising magnet LiHo_(0.045)Y_(0.955)F_4 over six decades of frequency from 0.02 Hz to 20 kHz. The system behaves as an ideal relaxational glass with Arrhenius behavior in temperature of the peak in χ’’. Scaling data from T=100 mK to T=300 mK by the peak in χ’’ shows an enhanced low-frequency response at high temperatures, in contrast to expectations for spin-glasses and random-field magnets

Continuous and Discontinuous Quantum Phase Transitions in a Model Two-Dimensional Magnet

The Shastry-Sutherland model, which consists of a set of spin 1/2 dimers on a 2-dimensional square lattice, is simple and soluble, but captures a central theme of condensed matter physics by sitting precariously on the quantum edge between isolated, gapped excitations and collective, ordered ground states. We compress the model Shastry-Sutherland material, SrCu2(BO3)2, in a diamond anvil cell at cryogenic temperatures to continuously tune the coupling energies and induce changes in state. High-resolution x-ray measurements exploit what emerges as a remarkably strong spin-lattice coupling to both monitor the magnetic behavior and the absence or presence of structural discontinuities. In the low-pressure spin-singlet regime, the onset of magnetism results in an expansion of the lattice with decreasing temperature, which permits a determination of the pressure dependent energy gap and the almost isotropic spin-lattice coupling energies. The singlet-triplet gap energy is suppressed continuously with increasing pressure, vanishing completely by 2 GPa. This continuous quantum phase transition is followed by a structural distortion at higher pressure.Comment: 16 pages, 4 figures. Accepted for publication in PNA

Ferromagnetism, glassiness, and metastability in a dilute dipolar-coupled magnet

We have measured the ac magnetic susceptibility of the model dilute dipolar-coupled Ising system LiHo_xY_(1−x)F_4. The x=0.46 material displays an ordinary ferromagnetic transition, while the x=0.045 and 0.167 samples are two very different magnetic glasses. Thermal relaxation times are more than five times longer for x=0.167 than for x=0.045. In addition, the more concentrated glass shows history dependence and metastability upon field cooling

Critical Behavior of the Conductivity of Si:P at the Metal-Insulator Transition under Uniaxial Stress

We report new measurements of the electrical conductivity sigma of the canonical three-dimensional metal-insulator system Si:P under uniaxial stress S. The zero-temperature extrapolation of sigma(S,T -> 0) ~\S - S_c\^mu shows an unprecidentedly sharp onset of finite conductivity at S_c with an exponent mu = 1. The value of mu differs significantly from that of earlier stress-tuning results. Our data show dynamical sigma(S,T) scaling on both metallic and insulating sides, viz. sigma(S,T) = sigma_c(T) F(\S - S_cT^y) where sigma_c(T) is the conductivity at the critical stress S_c. We find y = 1/znu = 0.34 where nu is the correlation-length exponent and z the dynamic critical exponent.Comment: 5 pages, 4 figure

From classical to quantum glass

We study the effects of a transverse magnetic field on the dynamics of the randomly diluted, dipolar coupled, Ising magnet LiHo_(0.167)Y_(0.833)F_4. The transverse field mixes the eigenfunctions of the ground-state Ising doublet with the otherwise inaccessible excited-state levels. We observe a rapid decrease in the characteristic relaxation times, large changes in the spectral form of the relaxation, and a depression of the spin-glass transition temperature with the introduction of quantum fluctuations

Anisotropy of Transverse Sound in the Heavy-Fermion Superconductor UPt_3

We report the first measurements of the attenuation of ultrasound in the basal plane of superconducting UPt_3. Transverse sound propagating along the b axis shows a marked anisotropy in its temperature dependence when the polarization is rotated in and out of the basal plane. For polarization in the basal plane the attenuation varies linearly with temperature down to 35 mK and the slope scales as the square of the frequency. Our results appear to indicate the presence of an additional attenuation mechanism when compared with recent theories of anisotropic superconductors in the dirty limit