Supersymmetric construction of self-consistent condensates in the large $N$ Gross-Neveu model: Solitons on finite-gap potentials

In the present work, the set of stationary solutions of the Gross-Neveu model in the ’t Hooft limit is extended. Such an extension is obtained by striving for a hidden supersymmetry associated with disconnected sets of stationary solutions. How the supersymmetry arises from the Darboux-Miura transformations between Lax pairs of the stationary modified Korteweg–de Vries and the stationary Korteweg–de Vries hierarchies is shown, associating the correspondent superpotentials with self-consistent condensates for the Gross-Neveu model.

Figure 1

Examples of stationary condensates of the GN model: solitonic defects on a homogeneous background. For each graphic are shown the shape of the condensate $\mathrm{\Delta }(x)$, in a blue continuous line, and its spectrum $\sigma (H^D)$, in green thick lines the continuous spectrum, in dashed red line the band gaps, and in cyan dots the bound states and the band edges. Note that $-\mathrm{\Delta }(x)$ is a condensate solution of the GN model too, with an identical spectrum. (a) Trivial zero solution, (b) homogeneous background with two kink-antikink defects, and (c) kink on a homogeneous background with two kink-antikink defects.

Figure 2

Examples of stationary condensates of the GN model: solitonic defects on two- and three-gap scalar Dirac potentials. The same symbology as in Fig. 1 is used but with a brown dashed thin line added to show the shape of the finite-gap background: (a) a two-gap condensate with two modulation shape solitonic defects that support each two bound states in the external gaps, one in each forbidden band; (b) a three-gap background with three solitonic defects, two of them as modulations and one kink-antikink defect that supports two bound states in the central band gap; and (c) a kink version of the previous condensate, where its spectrum additionally contains a zero energy state.

Figure 3

Kink defects look like a domain wall that pushes away two different phases of the condensate, while soliton defects with energies very close to zero allow one to have two transitions between these two phases, which is due to the spontaneous appearance of a kink and an antikink. The width of the kink-antikink increases while the energy of the defect approaches zero, while in the zero limit one or both domain walls can disappear in the spatial infinities.

Figure 4

Occupation of fermion matter. The occupation of the fermionic states by each flavor is characterized by the spectrum of the condensate $\mathrm{\Delta }$. Such a spectrum defines the poles and the branch cuts in the argument of the trace in Eq. (7.1). Here the meaning of the integration paths for ground state configurations is shown: (a) Examples of paths for ground states; the first path leaves holes in Dirac sea (antiparticles) so its charge $Q$ will be “negative.” The second case corresponds to a full Dirac sea, where there are no particles nor antiparticles, and thus, the total charge is zero. In the third case, there are only particles, so it has a “positive” charge. Since the paths cross the axis only once, these examples correspond to ground state type paths. (b) The path $C_r$ for the $r$th fermion flavor. The path $C_r$ in (b) is identical to the path $C_r^{\prime }$ in (c). $C_r^{\prime }$ results from adding a closed integration path in the upper plane to $C_r$, and such a closed integration path covers the axis $\mathrm{Im}[z]=0$ with direction $\mathrm{Re}[z]=\infty \to \mathrm{Re}[z]=-\infty$ and whose integral, by means of the residue theorem, is equal to zero. (d) The superposition of the integration path of all flavors construct the path $C$. To simplify, the scattering zones, of particles and holes, are chosen empty. In other words, the lower allowed band is occupied by all the flavors, i.e., the path $C$ cover $N$ times the respective branch cut, and the upper allowed band is empty. On the other hand, some flavors occupy the bound states, and then the poles are turned by $C$ a number $n<N$ of times. Another assumption for simplicity is to choose all the states in the bands occupied by the same number of fermions.