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Register a new userDevils Staircase -- Rotation Number of Outer Billiard with Polygonal Invariant Curves
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Zijian Yao
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In this paper, we discuss rotation number on the invariant curve of a one parameter family of outer billiard tables. Given a convex polygon $eta$, we can construct an outer billiard table $T$ by cutting out a fixed area from the interior of $eta$. $T$ is piece-wise hyperbolic and the polygon $eta$ is an invariant curve of $T$ under the billiard map $phi$. We will show that, if $beta $ is a periodic point under the outer billiard map with rational rotation number $tau = p / q$, then the $n$th iteration of the billiard map is not the local identity at $beta$. This proves that the rotation number $tau$ as a function of the area parameter is a devils staircase function.
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Quasi-invariant curves are used in the study of hedgehog dynamics. Denjoy-Yoccoz lemma is the preliminary step for Yoccozs complex renormalization techniques for the study of linearization of analytic circle diffeomorphisms. We give a geometric interpretation of Denjoy-Yoccoz lemma using the hyperbolic metric that gives a direct construction of quasi-invariant curves without renormalization.
Rare earth intermetallic compounds have been fascinating scientists due to rich phenomena induced by the interplay between localized $f$-orbitals and conduction electrons. However, since the energy scale of the crystal-electric-field (CEF) splitting, which defines $f$-orbitals, is very small only in a few meV, the nature of mobile electrons accompanied by CEF-excitations has not been unveiled so far. It thus leaves these systems as frontiers for discovering exotic quasiparticles not yet captured in condensed matter physics. Here, we examined very low-energy electronic structures of CeSb going through the anomalous magnetostructural transitions below the N{{e}}el temperature ($T_{rm{N}}$) $sim$17 K, called devils staircase, by a combination of laser angle-resolved photoemission, Raman and neutron scattering spectroscopies. We found a new type of electron-boson coupling between the mobile electrons and quadrupole CEF-excitations of the 4$f$-orbitals, which renormalizes the Sb 5$p$ band prominently, yielding a remarkable kink at very low-energy ($sim$7 meV). This coupling strength is exceedingly strong and exhibits anomalous step-like enhancement during the devils staircase transition, unveiling a new type of quasiparticle, named multipole polaron, that is a mobile electron largely dressed with a cloud of the quadrupole CEF-polarization.
Solids with competing interactions often undergo complex phase transitions with a variety of long-periodic modulations. Among such transition, devils staircase is the most complex phenomenon, and for it, CeSb is the most famous material, where a number of the distinct phases with long-periodic magnetostructures sequentially appear below the Neel temperature. An evolution of the low-energy electronic structure going through the devils staircase is of special interest, which has, however, been elusive so far despite the 40-years of intense researches. Here we use bulk-sensitive angle-resolved photoemission spectroscopy and reveal the devils staircase transition of the electronic structures. The magnetic reconstruction dramatically alters the band dispersions at each transition. We moreover find that the well-defined band picture largely collapses around the Fermi energy under the long-periodic modulation of the transitional phase, while it recovers at the transition into the lowest-temperature ground state. Our data provide the first direct evidence for a significant reorganization of the electronic structures and spectral functions occurring during the devils staircase.
The devils staircase is a term used to describe surface or an equilibrium phase diagram in which various ordered facets or phases are infinitely closely packed as a function of some model parameter. A classic example is a 1-D Ising model [bak] wherein long-range and short range forces compete, and the periodicity of the gaps between minority species covers all rational values. In many physical cases, crystal growth proceeds by adding surface layers which have the lowest energy, but are then frozen in place. The emerging layered structure is not the thermodynamic ground state, but is uniquely defined by the growth kinetics. It is shown that for such a system, the grown structure tends to the equilibrium ground state via a devils staircase traversing an infinity of intermediate phases. It would be extremely difficult to deduce the simple growth law based on measurement made on such an grown structure.
The temperature ($T$) - magnetic field ($H$) phase diagram for the tetragonal layered compound CeSbSe, is determined from magnetization, specific heat, and electrical resistivity measurements. This system exhibits complex magnetic ordering at $T_{rm{M}}$ $=$ 3 K and the application of a magnetic field results in a cascade of magnetically ordered states for $H$ $lesssim$ 1.8 T which are characterized by fractional integer size steps: i.e., a possible Devils staircase is observed. Electrical transport measurements show a weak temperature dependence and large residual resistivity which suggest a small charge carrier density and strong scattering from the $f$-moments. These features reveal Kondo lattice behavior where the $f$-moments are incompletely screened, resulting in a fine balanced magnetic interaction between different Ce neighbors that is mediated by the RKKY interaction. This produces the nearly degenerate magnetically ordered states that are accessed under an applied magnetic field.
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