scholarly journals SU(2) × SU(2) Algebras and the Lorentz Group O(3,3)

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 817
Author(s):  
Martin Walker
Keyword(s):  
Lie Algebra ◽  
Lorentz Group ◽  
Time Reversal ◽  
Standard Form ◽  
Conserved Quantity ◽  
Time Reversal Symmetry ◽  
Planck Constant ◽  
Spatial Rotation ◽  
Units Of Measurement

The Lie algebra of the Lorentz group O(3,3) admits two types of SU(2) × SU(2) subalgebras: a standard form based on spatial rotation generators and a second form based on temporal rotation generators. The units of measurement for the conserved quantity due to invariance under temporal rotations are investigated and found to be the same units of measure as the Planck constant. The breaking of time reversal symmetry is considered and found to affect the chiral properties of a temporal SU(2) × SU(2) algebra. Finally, the symmetry between algebras is explored and pairs of algebras are found to be related by SU(2) × U(1) symmetry, while a group of three algebras are related by SO(4) symmetry.

scholarly journals Odd Willis coupling induced by broken time-reversal symmetry

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Li Quan ◽  
Simon Yves ◽  
Yugui Peng ◽  
Hussein Esfahlani ◽  
Andrea Alù
Keyword(s):  
Time Reversal ◽  
Time Reversal Symmetry ◽  
Acoustic Metamaterials ◽  
Asymmetric Structures ◽  
Sound Control ◽  
Wavefront Shaping ◽  
Scattering Response ◽  
Symmetric Structures ◽  
Signal Routing ◽  
Broad Interest

AbstractWhen sound interacts with geometrically asymmetric structures, it experiences coupling between pressure and particle velocity, known as Willis coupling. While in most instances this phenomenon is perturbative in nature, tailored asymmetries combined with resonances can largely enhance it, enabling exotic acoustic phenomena. In these systems, Willis coupling obeys reciprocity, imposing an even symmetry of the Willis coefficients with respect to time reversal and the impinging wave vector, which translates into stringent constraints on the overall scattering response. In this work, we introduce and experimentally observe a dual form of acoustic Willis coupling, arising in geometrically symmetric structures when time-reversal symmetry is broken, for which the pressure-velocity coupling is purely odd-symmetric. We derive the conditions to maximize this effect, we experimentally verify it in a symmetric subwavelength scatterer biased by angular momentum, and we demonstrate the opportunities for sound scattering enabled by odd Willis coupling. Our study opens directions for acoustic metamaterials, with direct implications for sound control, non-reciprocal scattering, wavefront shaping and signal routing, of broad interest also for nano-optics, photonics, elasto-dynamics, and mechanics.

scholarly journals Symmetry aspects in the macroscopic dynamics of magnetorheological gels and general liquid crystalline magnetic elastomers

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Harald Pleiner ◽  
Helmut R. Brand
Keyword(s):  
Electric Fields ◽  
Time Reversal ◽  
Liquid Crystalline ◽  
Second Law Of Thermodynamics ◽  
Time Reversal Symmetry ◽  
Local Conservation ◽  
Continuous Symmetry ◽  
Polar Order ◽  
Preferred Direction ◽  
Symmetry Properties

Abstract We investigate theoretically the macroscopic dynamics of various types of ordered magnetic fluid, gel, and elastomeric phases. We take a symmetry point of view and emphasize its importance for a macroscopic description. The interactions and couplings among the relevant variables are based on their individual symmetry behavior, irrespective of the detailed nature of the microscopic interactions involved. Concerning the variables we discriminate between conserved variables related to a local conservation law, symmetry variables describing a (spontaneously) broken continuous symmetry (e.g., due to a preferred direction) and slowly relaxing ones that arise from special conditions of the system are considered. Among the relevant symmetries, we consider the behavior under spatial rotations (e.g., discriminating scalars, vectors or tensors), under spatial inversion (discriminating e.g., polar and axial vectors), and under time reversal symmetry (discriminating e.g., velocities from polarizations, or electric fields from magnetic ones). Those symmetries are crucial not only to find the possible cross-couplings correctly but also to get a description of the macroscopic dynamics that is compatible with thermodynamics. In particular, time reversal symmetry is decisive to get the second law of thermodynamics right. We discuss (conventional quadrupolar) nematic order, polar order, active polar order, as well as ferromagnetic order and tetrahedral (octupolar) order. In a second step, we show some of the consequences of the symmetry properties for the various systems that we have worked on within the SPP1681, including magnetic nematic (and cholesteric) elastomers, ferromagnetic nematics (also with tetrahedral order), ferromagnetic elastomers with tetrahedral order, gels and elastomers with polar or active polar order, and finally magnetorheological fluids and gels in a one- and two-fluid description.

scholarly journals Unsplit superconducting and time reversal symmetry breaking transitions in Sr2RuO4 under hydrostatic pressure and disorder

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Vadim Grinenko ◽  
Debarchan Das ◽  
Ritu Gupta ◽  
Bastian Zinkl ◽  
Naoki Kikugawa ◽  
...  
Keyword(s):  
Hydrostatic Pressure ◽  
Symmetry Breaking ◽  
Time Reversal ◽  
Order Parameter ◽  
Muon Spin Rotation ◽  
Horizontal Line ◽  
Time Reversal Symmetry ◽  
Zero Field ◽  
Symmetry Protected ◽  
Line Node

AbstractThere is considerable evidence that the superconducting state of Sr2RuO4 breaks time reversal symmetry. In the experiments showing time reversal symmetry breaking, its onset temperature, TTRSB, is generally found to match the critical temperature, Tc, within resolution. In combination with evidence for even parity, this result has led to consideration of a dxz ± idyz order parameter. The degeneracy of the two components of this order parameter is protected by symmetry, yielding TTRSB = Tc, but it has a hard-to-explain horizontal line node at kz = 0. Therefore, s ± id and d ± ig order parameters are also under consideration. These avoid the horizontal line node, but require tuning to obtain TTRSB ≈ Tc. To obtain evidence distinguishing these two possible scenarios (of symmetry-protected versus accidental degeneracy), we employ zero-field muon spin rotation/relaxation to study pure Sr2RuO4 under hydrostatic pressure, and Sr1.98La0.02RuO4 at zero pressure. Both hydrostatic pressure and La substitution alter Tc without lifting the tetragonal lattice symmetry, so if the degeneracy is symmetry-protected, TTRSB should track changes in Tc, while if it is accidental, these transition temperatures should generally separate. We observe TTRSB to track Tc, supporting the hypothesis of dxz ± idyz order.

Tests of Time-Reversal Symmetry in Compound-Nucleus Reactions

1986 ◽  
Vol 56 (19) ◽  
pp. 2012-2015 ◽  
Author(s):  
D. Boosé ◽  
H. L. Harney ◽  
H. A. Weidenmüller
Keyword(s):  
Compound Nucleus ◽  
Time Reversal ◽  
Time Reversal Symmetry

scholarly journals Time-reversal symmetry breaking superconducting ground state in the doped Mott insulator on the honeycomb lattice

2013 ◽  
Vol 88 (15) ◽  
Author(s):  
Zheng-Cheng Gu ◽  
Hong-Chen Jiang ◽  
D. N. Sheng ◽  
Hong Yao ◽  
Leon Balents ◽  
...  
Keyword(s):  
Symmetry Breaking ◽  
Ground State ◽  
Time Reversal ◽  
Mott Insulator ◽  
Honeycomb Lattice ◽  
Time Reversal Symmetry ◽  
Doped Mott Insulator
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