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Register a new userU(1)$times$SU(2) Gauge Invariance Made Simple for Density Functional Approximations
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A semi-relativistic density-functional theory that includes spin-orbit couplings and Zeeman fields on equal footing with the electromagnetic potentials, is an appealing framework to develop a unified first-principles computational approach for non-collinear magnetism, spintronics, orbitronics, and topological states. The basic variables of this theory include the paramagnetic current and the spin-current density, besides the particle and the spin density, and the corresponding exchange-correlation (xc) energy functional is invariant under local U(1)$times$SU(2) gauge transformations. The xc-energy functional must be approximated to enable practical applications, but, contrary to the case of the standard density functional theory, finding simple approximations suited to deal with realistic atomistic inhomogeneities has been a long-standing challenge. Here, we propose a way out of this impasse by showing that approximate gauge-invariant functionals can be easily generated from existing approximate functionals of ordinary density-functional theory by applying a simple {it minimal substitution} on the kinetic energy density, which controls the short-range behavior of the exchange hole. Our proposal opens the way to the construction of approximate, yet non-empirical functionals, which do not assume weak inhomogeneity and should therefore have a wide range of applicability in atomic, molecular and condensed matter physics.
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$SO(5) times U(1) times SU(3)$ gauge-Higgs unification model inspired by $SO(11)$ gauge-Higgs grand unification is constructed in the Randall-Sundrum warped space. The 4D Higgs boson is identified with the Aharonov-Bohm phase in the fifth dimension. Fermion multiplets are introduced in the bulk in the spinor, vector and singlet representations of $SO(5)$ such that they are implemented in the spinor and vector representations of $SO(11)$. The mass spectrum of quarks and leptons in three generations is reproduced except for the down quark mass. The small neutrino masses are explained by the gauge-Higgs seesaw mechanism which takes the same form as in the inverse seesaw mechanism in grand unified theories in four dimensions.
Symmetry-protected topological $left(SPTright)$ phases are gapped short-range entangled states with symmetry $G$, which can be systematically described by group cohomology theory. $SU(3)$ and $SU(2)times{U(1)}$ are considered as the basic groups of Quantum Chromodynamics and Weak-Electromagnetic unification, respectively. In two dimension $(2D)$, nonlinear-sigma models with a quantized topological Theta term can be used to describe nontrivial SPT phases. By coupling the system to a probe field and integrating out the group variables, the Theta term becomes the effective action of Chern-Simons theory which can derive the response current density. As a result, the current shows a spin Hall effect, and the quantized number of the spin Hall conductance of SPT phases $SU(3)$ and $SU(2)times{U(1)}$ are same. In addition, relationships between $SU(3)$ and $SU(2)times{U(1)}$ which maps $SU(3)$ to $SU(2)$ with a rotation $U(1)$ will be given.
We study $N=2$ supersymmetric gauge theories on a large family of squashed 4-spheres preserving $SU(2)times U(1)subset SO(4)$ isometry and determine the conditions under which this background is supersymmetric. We then compute the partition function of the theories by using localization technique. The results indicate that for $N=2$ SUSY, including both vector-multiplets and hypermultiplets, the partition function is independent of the arbitrary squashing functions as well as of the other supergravity background fields.
The transition metal dichalcogenide 1$T$-TiSe$_2$ is a quasi-two-dimensional layered material undergoing a commensurate 2 $times$ 2 $times$ 2 charge density wave (CDW) transition with a weak periodic lattice distortion (PLD) below $approx$ 200 K. Scanning tunneling microscopy (STM) combined with intentionally introduced interstitial Ti atoms allows to go beyond the usual spatial resolution of STM and to intimately probe the three-dimensional character of the PLD. Furthermore, the inversion-symmetric, achiral nature of the CDW in the $z$-direction is revealed, contradicting the claimed existence of helical CDW stacking and associated chiral order. This study paves the way to a simultaneous real-space probing of both charge and structural reconstructions in CDW compounds.
The Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix and flavor-changing neutral currents (FCNCs) in the quark sector are examined in the GUT inspired $SO(5) times U(1) times SU(3)$ gauge-Higgs unification in which the 4D Higgs boson is identified with the Aharonov-Bohm phase in the fifth dimension. Gauge invariant brane interactions play an important role for the flavor mixing in the charged-current weak interactions. The CKM matrix is reproduced except that the up quark mass needs to be larger than the observed one. FCNCs are naturally suppressed as a consequence of the gauge invariance, with a factor of order $10^{-6}$. It is also shown that induced flavor-changing Yukawa couplings are extremely small.
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