Protection of Weyl Semimetals in Optical Lattices by Gauge-Color-Translation Symmetry
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In this paper, we develop the quantum theory of particles that has discrete Poincar\'{e} symmetry on the one-dimensional Bravais lattice. We review the recently discovered discrete Lorentz symmetry, which is the unique Lorentz symmetry that coexists with the discrete space translational symmetry on a Bravais lattice. The discrete Lorentz transformations and spacetime translations form the discrete Poincar\'{e} group, which are represented by unitary operators in a quantum theory. We find the conditions for the existence of representation, which are expressed as the congruence relation between quasi-momentum and quasi-energy. We then build the Lorentz-invariant many-body theory of indistinguishable particles by expressing both the unitary operators and Floquet Hamiltonians in terms of the field operators. Some typical Hamiltonians include the long-range hopping which fluctuates as the distance between sites increases. We calculate the Green's functions of the lattice theory. The spacetime points where the Green's function is nonzero display a lattice structure. During the propagation, the particles stay localized on a single or a few sites to preserve the Lorentz symmetry.
Bravais lattice
Translational symmetry
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A classification of the possible symmetric principal bundles with a compact gauge group, a compact symmetry group and a base manifold which is regularly foliated by the orbits of the symmetry group is derived. A generalization of Wang's theorem (classifying the invariant connections) is proven and local expressions for the gauge potential of an invariant connection are given.
BRST quantization
Gauge group
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Mirror symmetry
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We describe a set of periodic lattices in (1+1)-dimensional Minkowski space, where each lattice has an associated symmetry group consisting of inhomogeneous Lorentz transformations that map the lattice onto itself. Our results show how ideas of crystal structure in Euclidean space generalize to Minkowski space and provide an example that illustrates basic concepts of spacetime symmetry.
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Lorentz group
Minkowski's theorem
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Mirror symmetry
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We present a study on the existence of synthetic Weyl points with various planar symmetries in phononic crystals. We design a set of phononic crystals to display Weyl points at high-symmetry points with different symmetries in both square and honeycomb lattices. We demonstrate that, in a square lattice, double Weyl points are protected by C4 rotation symmetry, while in a honeycomb lattice, they are protected by C3 rotation symmetry. Additionally, we investigated the effects of symmetry on double Weyl points. The results indicate that double Weyl points would split into two single Weyl points along high-symmetry lines if we break the corresponding symmetries. The distributions of Weyl points in various symmetries are presented systematically. Finally, gapless surface states and the robust one-way acoustic transport in a square lattice are demonstrated in a double Weyl system.
Square lattice
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Mirror symmetry
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Symmetry plays fundamental role in physics and the nature of symmetry changes in non-Hermitian physics. Here the symmetry-protected scattering in non-Hermitian linear systems is investigated by employing the discrete symmetries that classify the random matrices. The even-parity symmetries impose strict constraints on the scattering coefficients: the time-reversal ( C and K ) symmetries protect the symmetric transmission or reflection; the pseudo-Hermiticity ( Q symmetry) or the inversion ( P ) symmetry protects the symmetric transmission and reflection. For the inversion-combined time-reversal symmetries, the symmetric features on the transmission and reflection interchange. The odd-parity symmetries including the particle-hole symmetry, chiral symmetry, and sublattice symmetry cannot ensure the scattering to be symmetric. These guiding principles are valid for both Hermitian and non-Hermitian linear systems. Our findings provide fundamental insights into symmetry and scattering ranging from condensed matter physics to quantum physics and optics.
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