Self-Consistent Multiple Complex-Kink Solutions in Bogoliubov–de Gennes and Chiral Gross-Neveu Systems

Abstract

We exhaust all exact self-consistent solutions of complex-valued fermionic condensates in the 1+1 dimensional Bogoliubov-de Gennes and chiral Gross-Neveu systems under uniform boundary conditions. We obtain n complex (twisted) kinks, or grey solitons, with 2n parameters corresponding to their positions and phase shifts. Each soliton can be placed at an arbitrary position while the self-consistency requires its phase shift to be quantized by π/N for N flavors.PACS numbers: 11.10. Kk, 03.75.Ss, Introduction.-The search for inhomogeneous selfconsistent fermionic condensates including states such as the Fulde-Ferrell (FF) [1] and Larkin-Ovchinnikov (LO) [2] states having phase and amplitude modulations, respectively, in superconductors has attracted considerable attentions for more than half a century since theoretical predictions were made about their existence. While amplitude modulations are well studied in conducting polymers [3][4][5][6][7], the FFLO state is theoretically shown to be a ground state of superconductors under a magnetic field [8]. Recently, the FFLO state has also been discussed in the context of an ultracold atomic Fermi gas [9,10]. In general, inhomogeneous self-consistent fermionic condensates with a gap function and fermionic excitations can be treated simultaneously using the Bogoliubov-de Gennes (BdG) and gap equations [11]. The gap functions are real and complex for conducting polymers [12] and superconductors, respectively. In the quantum field theory, these systems correspond to the Gross-Neveu (GN) model [13] and the Nambu-Jona-Lasinio (or chiral GN) model [14], which were proposed as models of dynamical chiral symmetry breaking in 1+1 or 2+1 dimensions. Therefore, BdG and (chiral) GN systems have been studied and developed together from the viewpoint of both condensed matter physics and high energy physics (see Ref.[15] for a review). For instance, fermion number fractionization is one of the topics that has been studied from this viewpoint [16,17]. Recently, it has been shown that the solutions in 1+1 dimensions can be promoted to 3+1 dimensions [18,19], thereby leading to extensive study of the modulated phases of these systems in terms of quantum chromodynamics (QCD) [20].Inhomogeneous self-consistent solutions are often studied numerically because analytic solutions are generally difficult to obtain. However, several analytic solutions are available in the case of the real-valued condensates in 1+1 dimensions, which describe the conducting polymers and the real GN model. Under uniform boundary conditions at spatial infinities, a real kink was constructed by Dashen et. al. [21] by using the inverse scattering method, and later, it was reconstructed in polyacetylene [22] in the continuum limit of the

Bogoliubov–de Gennes soliton dynamics in unconventional Fermi superfluids

“…[39], we use the latter convention in the present work. By this correspondence, the self-consistent condition (2.25) becomes equivalent to Ref.…”

“…When this convention is used, the subscripts ↑, ↓ are no longer necessary to label the eigenstates, so we can simply write (u j↑ , v j↓ ) = (u j , v j ) without confusion. Finally, we give a remark on the difference of convention between the present work and our previous work [39]. For the one-dimensional spin-1/2 s-wave system, both (ii) and (iii) are applicable, so we have two choices.…”

“…By this correspondence, the self-consistent condition (2.25) becomes equivalent to Ref. [39]. [Rewrite Eq.…”

“…While the time evolution of bosonic mean fields, e.g., Bose-Einstein condensates (BECs), is modeled by a partial differential equation (PDE) such as the Gross-Pitaevskii or nonlinear Schrödinger (NLS) equation, fermionic ones need to solve the BdG and gap equations self-consistently, and order parameters do not solely satisfy a simple PDE. The inverse scattering theory (IST) of the Zakharov-Shabat (ZS) operator [67,68] can powerfully solve the initial-value problem of the NLS equation [69][70][71], but it is applicable only for a stationary problem for the fermionic BdG systems (e.g., [17,18,39]). These circumstances may be phrased as "mathematics underlying in bosonic soliton dynamics ≃ that in fermionic soliton statics".…”

Integrable Gross-Neveu models with fermion-fermion and fermion-antifermion pairing

“…While this does not seem to play any role in the calculation of thermodynamic quantities, integrability permits to solve static and even time dependent soliton problems in the massless GN and NJL models explicitly. Thus, scattering problems involving any number of kinks, kink-antikink baryons, compound bound states and breathers have been solved analytically by time-dependent Hartree-Fock methods (TDHF) recently [19][20][21][22]. Nothing comparable has been achieved for massive GN models, or any variant with different interaction terms, for that matter, so that integrability is undoubtedly crucial here.…”