Lie Algebras in Particle Physics

A symmetry broken phase of a system with internal degrees of freedom often features a complex order parameter, which generates a rich variety of topological excitations and imposes topological constraints on their interaction (topological influence); yet the very complexity of the order parameter makes it difficult to treat topological excitations and topological influence systematically. To overcome this problem, we develop a general method to calculate homotopy groups and derive decomposition formulas which express homotopy groups of the order parameter manifold G/H in terms of those of the symmetry G of a system and those of the remaining symmetry H of the state. By applying these formulas to general monopoles and three-dimensional skyrmions, we show that their textures are obtained through substitution of the corresponding su(2)-subalgebra for the su(2)-spin. We also show that a discrete symmetry of H is necessary for the presence of topological influence and find topological influence on a skyrmion characterized by a non-Abelian permutation group of three elements in the ground state of an SU(3)-Heisenberg model.

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“…The Lie algebra g of a compact Lie group G has a convenient basis called the Cartan canonical form [45]:…”

Section: Cartan Canonical Form and Lattices Of A Compact Lie Groupmentioning

confidence: 99%

Influence of topological constraints and topological excitations: Decomposition formulas for calculating homotopy groups of symmetry-broken phases

Higashikawa

Ueda

2017

Phys. Rev. B

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“…The number of possible arrangements that could lead to different maximal singular values is simply given by the number of standard Young tableaux consisting of d 1 × d 2 boxes, arranged in a single block. This number is given by the so-called hook-length formula [14] …”

Section: C: Connection To the Theory Of Young Tableauxmentioning

confidence: 99%

“…Nevertheless, the complexity can be reduced, as we have to consider only those permutations which lead to different maximal singular values. In Appendix C [11]) we discuss this simplification which leads to the theory of Young tableaux [14]. It turns out that for a decom-…”

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confidence: 99%

Characterizing Genuine Multilevel Entanglement

et al. 2018

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Entanglement of high-dimensional quantum systems has become increasingly important for quantum communication and experimental tests of nonlocality. However, many effects of highdimensional entanglement can be simulated by using multiple copies of low-dimensional systems. We present a general theory to characterize those high-dimensional quantum states for which the correlations cannot simply be simulated by low-dimensional systems. Our approach leads to general criteria for detecting multilevel entanglement in multiparticle quantum states, which can be used to verify these phenomena experimentally.

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“…model (2.1) depicts a non-linearity which is removed in the SU (3) space by reformulating the model in terms of Gell-Mann matrices 8 and neglecting an Abelian term without affecting the quantities of central interest 6 (see next paragraph). It should also be noted that when the SU (3) symmetry breaks down and the transverse drive is switched off (D = ω = 0) the model Eq.…”

Section: Modelmentioning

confidence: 99%

“…When three diabatic levels cross and form a triangular geometry, this hides the dynamical symmetry of the SU (3) group (spanned in the Lie space by Gell-Mann matrices 8 ) and may be exploited as a quantum interferometer 6,7 or used as the building block for qutrits (the unit of ternary quantum computing 7 ). If in addition the inter-level distance between level positions is maintained constant throughout the course of variation of a control parameter (time, chemical potential, flux, magnetic field, pressure, temperature etc), the relevant model leads to the so called SU (3) LZSM interferometry 6,7 .…”

Section: Introductionmentioning

confidence: 99%

SU(3) Landau-Zener interferometry with a transverse periodic drive

2017

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Quantum triangles can work as interferometers. Depending on their geometric size and interactions between paths, "beats" and/or "steps" patterns are observed. We show that when inter-level distances between level positions in quantum triangles periodically change with time, formation of beats and/or steps no longer depends only on the geometric size of the triangles but also on the characteristic frequency of the transverse signal. For large-size triangles, we observe the coexistence of beats and steps when the frequency of the signal matches that of non-adiabatic oscillations and for large frequencies, a maximum of four steps instead of two as in the case with constant interactions is observed. Small-size triangles also revealed counter-intuitive interesting dynamics for large frequencies of the field: unexpected two-step patterns are observed. When the frequency is large and tuned such that it matches the uniaxial anisotropy, three-step patterns are observed. We have equally observed that when the transverse signal possesses a static part, steps maximize to six. These effects are semi-classically explained in terms of Fresnel integrals and quantum mechanically in terms of quantized fields with a photon-induced tunneling process. Our expressions for populations are in excellent agreement with the gross temporal profiles of exact numerical solutions. We compare the semi-classical and quantum dynamics in the triangle and establish the conditions for their equivalence.

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Lie Algebras in Particle Physics

Cited by 149 publications

References 0 publications

Influence of topological constraints and topological excitations: Decomposition formulas for calculating homotopy groups of symmetry-broken phases

Influence of topological constraints and topological excitations: Decomposition formulas for calculating homotopy groups of symmetry-broken phases

Characterizing Genuine Multilevel Entanglement

SU(3) Landau-Zener interferometry with a transverse periodic drive

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