Multi-term linear fractional nabla difference equations with constant coefficients

We shall consider a linear fractional nabla (backward) fractional difference equation of Riemann–Liouville type with constant coefficients. We apply a transform method to construct solutions. Sufficient conditions in terms of the coefficients are given so that the solutions are absolutely convergent. The method is known for two-term fractional difference equations; the method is new for fractional equations with three or more terms. As a corollary, we exhibit new summation representations of a discrete exponential function, at, t = 0; 1; : : :

Nearly deconfined spinon excitations in the square-lattice spin-1/2 Heisenberg antiferromagnet

We study the spin-excitation spectrum (dynamic structure factor) of the spin-1/2 square-lattice Heisenberg antiferromagnet and an extended model (the J−Q model) including four-spin interactions Q in addition to the Heisenberg exchange J. Using an improved method for stochastic analytic continuation of imaginary-time correlation functions computed with quantum Monte Carlo simulations, we can treat the sharp (δ-function) contribution to the structure factor expected from spin-wave (magnon) excitations, in addition to resolving a continuum above the magnon energy. Spectra for the Heisenberg model are in excellent agreement with recent neutron-scattering experiments on Cu(DCOO)2⋅4D2O, where a broad spectral-weight continuum at wave vector q=(π,0) was interpreted as deconfined spinons, i.e., fractional excitations carrying half of the spin of a magnon. Our results at (π,0) show a similar reduction of the magnon weight and a large continuum, while the continuum is much smaller at q=(π/2,π/2) (as also seen experimentally). We further investigate the reasons for the small magnon weight at (π,0) and the nature of the corresponding excitation by studying the evolution of the spectral functions in the J−Q model. Upon turning on the Q interaction, we observe a rapid reduction of the magnon weight to zero, well before the system undergoes a deconfined quantum phase transition into a nonmagnetic spontaneously dimerized state. Based on these results, we reinterpret the picture of deconfined spinons at (π,0) in the experiments as nearly deconfined spinons—a precursor to deconfined quantum criticality. To further elucidate the picture of a fragile (π,0)-magnon pole in the Heisenberg model and its depletion in the J−Q model, we introduce an effective model of the excitations in which a magnon can split into two spinons that do not separate but fluctuate in and out of the magnon space (in analogy to the resonance between a photon and a particle-hole pair in the exciton-polariton problem). The model can reproduce the reduction of magnon weight and lowered excitation energy at (π,0) in the Heisenberg model, as well as the energy maximum and smaller continuum at (π/2,π/2). It can also account for the rapid loss of the (π,0) magnon with increasing Q and the remarkable persistence of a large magnon pole at q=(π/2,π/2) even at the deconfined critical point. The fragility of the magnons close to (π,0) in the Heisenberg model suggests that various interactions that likely are important in many materials—e.g., longer-range pair exchange, ring exchange, and spin-phonon interactions—may also destroy these magnons and lead to even stronger spinon signatures than in Cu(DCOO)2⋅4D2O.We thank Wenan Guo, Akiko Masaki-Kato, Andrey Mishchenko, Martin Mourigal, Henrik Ronnow, Kai Schmidt, Cenke Xu, and Seiji Yunoki for useful discussions. Experimental data from Ref. [33] were kindly provided by N. B. Christensen and H. M. Ronnow. H. S. was supported by the China Postdoctoral Science Foundation under Grants No. 2016M600034 and No. 2017T100031. St.C. was funded by the NSFC under Grants No. 11574025 and No. U1530401. Y. Q. Q. and Z. Y. M. acknowledge funding from the Ministry of Science and Technology of China through National Key Research and Development Program under Grant No. 2016YFA0300502, from the key research program of the Chinese Academy of Sciences under Grant No. XDPB0803, and from the NSFC under Grants No. 11421092, No. 11574359, and No. 11674370, as well as the National Thousand-Young Talents Program of China. A. W. S. was funded by the NSF under Grants No. DMR-1410126 and No. DMR-1710170, and by the Simons Foundation. In addition H. S., Y. Q. Q., and Sy. C. thank Boston University's Condensed Matter Theory Visitors program for support, and A. W. S. thanks the Beijing Computational Science Research Center and the Institute of Physics, Chinese Academy of Sciences for visitor support. We thank the Center for Quantum Simulation Sciences at the Institute of Physics, Chinese Academy of Sciences, the Tianhe-1A platform at the National Supercomputer Center in Tianjin, Boston University's Shared Computing Cluster, and CALMIP (Toulouse) for their technical support and generous allocation of CPU time. (2016M600034 - China Postdoctoral Science Foundation; 2017T100031 - China Postdoctoral Science Foundation; 11574025 - NSFC; U1530401 - NSFC; 11421092 - NSFC; 11574359 - NSFC; 11674370 - NSFC; 2016YFA0300502 - Ministry of Science and Technology of China; XDPB0803 - Chinese Academy of Sciences; National Thousand-Young Talents Program of China; DMR-1410126 - NSF; DMR-1710170 - NSF; Simons Foundation; Boston University's Condensed Matter Theory Visitors program)Accepted manuscript and published version

Dynamical signature of fractionalization at a deconfined quantum critical point

Deconfined quantum critical points govern continuous quantum phase transitions at which fractionalized (deconfined) degrees of freedom emerge. Here we study dynamical signatures of the fractionalized excitations in a quantum magnet (the easy-plane J-Q model) that realize a deconfined quantum critical point with emergent O(4) symmetry. By means of large-scale quantum Monte Carlo simulations and stochastic analytic continuation of imaginary-time correlation functions, we obtain the dynamic spin-structure factors in the S x and S z channels. In both channels, we observe broad continua that originate from the deconfined excitations. We further identify several distinct spectral features of the deconfined quantum critical point, including the lower edge of the continuum and its form factor on moving through the Brillouin zone. We provide field-theoretical and lattice model calculations that explain the overall shapes of the computed spectra, which highlight the importance of interactions and gauge fluctuations to explain the spectral-weight distribution. We make further comparisons with the conventional Landau O(2) transition in a different quantum magnet, at which no signatures of fractionalization are observed. The distinctive spectral signatures of the deconfined quantum critical point suggest the feasibility of its experimental detection in neutron scattering and nuclear magnetic resonance experiments.First author draf

Revisiting the Role of TGFβ Receptor Internalization for Smad Signaling: It is Not Required in Optogenetic TGFβ Signaling Systems

Endocytosis is an important process by which many signaling receptors reach their intracellular effectors. Accumulating evidence suggests that internalized receptors play critical roles in triggering cellular signaling, including transforming growth factor β (TGFβ) signaling. Despite intensive studies on the TGFβ pathway over the last decades, the necessity of TGFβ receptor endocytosis for downstream TGFβ signaling responses is a subject of debate. In this study, mathematical modeling and synthetic biology approaches are combined to re-evaluate whether TGFβ receptor internalization is indispensable for inducing Smad signaling. It is found that optogenetic systems with plasma membrane-tethered TGFβ receptors can induce fast and sustained Smad2 activation upon light stimulations. Modeling analysis suggests that endocytosis is precluded for the membrane-anchored optogenetic TGFβ receptors. Therefore, this study provides new evidence to support that TGFβ receptor internalization is not required for Smad2 activation