Counting rule of Nambu–Goldstone modes for internal and spacetime symmetries: Bogoliubov theory approach

Takahashi, Daisuke; Nitta, Muneto

doi:10.1016/j.aop.2014.12.009

Cited by 50 publications

(110 citation statements)

References 75 publications

(223 reference statements)

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“…, which has the physical meaning of the zero-mode solution originated from translational symmetry breaking 32 . (See also Sec.…”

Section: Main Results and Numerical Evidence A Kelvin Modesmentioning

confidence: 99%

“…Differentiating the GP equation by θ, x 0 , and y 0 , we obtain the following SSB-originated zero mode solutions 32 for the Bogoliubov equation:…”

Section: A Fundamental Equations and Zero Modesmentioning

confidence: 99%

“…Both Kelvin modes 31,32 and ripple modes 18,32 have quadratic dispersion relations, ǫ ∼ (log R)k 2 and ǫ ∼ √ Lk 2 , in finite-size systems with R and L denoting system lengths perpendicular to a vortex and DW, respectively. These facts are consistent with the general argument that an NGM with quadratic dispersion corresponds to two broken symmetries [26][27][28][29] .…”

Section: Introductionmentioning

confidence: 99%

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Nambu-Goldstone modes propagating along topological defects: Kelvin and ripple modes from small to large systems

2015

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Nambu-Goldstone modes associated with (topological) defects such as vortices and domain walls in (super)fluids are known to possess quadratic/non-integer dispersion relations in finite/infinite-size systems. Here, we report interpolating formulas connecting the dispersion relations in finite-and infinite-size systems for Kelvin modes along a quantum vortex and ripplons on a domain wall in superfluids. Our method can provide not only the dispersion relations but also the explicit forms of quasiparticle wavefunctions (u, v). We find a complete agreement between the analytical formulas and numerical simulations. All these formulas are derived in a fully analytical way, and hence not empirical ones. We also discuss common structures in the derivation of these formulas and speculate on the general procedure.

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“…, which has the physical meaning of the zero-mode solution originated from translational symmetry breaking 32 . (See also Sec.…”

Section: Main Results and Numerical Evidence A Kelvin Modesmentioning

confidence: 99%

“…Differentiating the GP equation by θ, x 0 , and y 0 , we obtain the following SSB-originated zero mode solutions 32 for the Bogoliubov equation:…”

Section: A Fundamental Equations and Zero Modesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Nambu-Goldstone modes propagating along topological defects: Kelvin and ripple modes from small to large systems

2015

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“…(This simplicity contrasts with the bosonic problem, reducing to the symplectic group S p(2N, R) and its Lie algebra, where classification of standard forms is complicated [90][91][92][93]. See also [94].)…”

Section: A Bdg Equation and Bogoliubov Transformationmentioning

confidence: 99%

Bogoliubov–de Gennes soliton dynamics in unconventional Fermi superfluids

Takahashi

2016

Phys. Rev. B

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Exact self-consistent soliton dynamics based on the Bogoliubov-de Gennes (BdG) formalism in unconventional Fermi superfluids/superconductors possessing an S U(d)-symmetric two-body interaction is presented. The derivation is based on the ansatz having the similar form as the Gelfand-Levitan-Marchenko equation in the inverse scattering theory. Our solutions can be regarded as a multicomponent generalization of the solutions recently derived by Dunne and Thies [Phys. Rev. Lett. 111, 121602 (2013)]. We also propose superpositions of occupation states, which make it possible to realize various filling rates even in one-flavor systems, and include Dirac and Majorana fermions. The soliton solutions in the d = 2 systems, which describe the mixture of singlet s-wave and triplet p-wave superfluids, exhibit a variety of phenomena such as rotating polar phases by soliton spins, SU(2)-DHN breathers, Majorana triplet states, s-p mixed dynamics, and so on. These solutions are illustrated by animations, where order parameters are visualized by spherical harmonic functions. The full formulation of the BdG theory is also supported, and the double-counting problem of BdG eigenstates and N-flavor generalization are discussed.

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“…At the quantum level, the U(1) 2 symmetry is recovered and there are two Tomonaga-Luttinger liquids as far as the so-called TomonagaLuttinger parameter in the isospin sector is greater than 1 (See appendix B). Finally, fate of type-II NG modes localized on domain walls [41,42] and skyrmion lines [43] in the presence of quantum effects is one interesting future problem to explore. …”

Section: Jhep09(2014)098mentioning

confidence: 99%

Quantum exact non-abelian vortices in non-relativistic theories

Nitta

Uchino

Vinci

2014

J. High Energ. Phys.

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Non-Abelian vortices arise when a non-Abelian global symmetry is exact in the ground state but spontaneously broken in the vicinity of their cores. In this case, there appear (non-Abelian) Nambu-Goldstone (NG) modes confined and propagating along the vortex. In relativistic theories, the Coleman-Mermin-Wagner theorem forbids the existence of a spontaneous symmetry breaking, or a long-range order, in 1+1 dimensions: quantum corrections restore the symmetry along the vortex and the NG modes acquire a mass gap. We show that in non-relativistic theories NG modes with quadratic dispersion relation confined on a vortex can remain gapless at quantum level. We provide a concrete and experimentally realizable example of a three-component Bose-Einstein condensate with U(1) × U(2) symmetry. We first show, at the classical level, the existence of S 3 S 1 S 2 (S 1 fibered over S 2 ) NG modes associated to the breaking U(2) → U(1) on vortices, where S 1 and S 2 correspond to type I and II NG modes, respectively. We then show, by using a Bethe ansatz technique, that the U(1) symmetry is restored, while the SU(2) symmery remains broken non-pertubatively at quantum level. Accordingly, the U(1) NG mode turns into a c = 1 conformal field theory, the Tomonaga-Luttinger liquid, while the S 2 NG mode remains gapless, describing a ferromagnetic liquid. This allows the vortex to be genuinely non-Abelian at quantum level.

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Counting rule of Nambu–Goldstone modes for internal and spacetime symmetries: Bogoliubov theory approach

Cited by 50 publications

References 75 publications

Nambu-Goldstone modes propagating along topological defects: Kelvin and ripple modes from small to large systems

Nambu-Goldstone modes propagating along topological defects: Kelvin and ripple modes from small to large systems

Bogoliubov–de Gennes soliton dynamics in unconventional Fermi superfluids

Quantum exact non-abelian vortices in non-relativistic theories

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