Fermion states localized on a self-gravitating Skyrmion

We investigate self-gravitating solutions of the Einstein-Skyrme theory coupled to spin-isospin Dirac fermions and consider the dependence of the spectral flow on the effective gravitational coupling constant and on the Yukawa coupling. It is shown that the effects of the backreaction of the fermionic mode may strongly deform the configuration. Depending on the choice of parameters, solutions with positive, negative and zero ADM mass may arise. The occurrence of regular anti-gravitating asymptotically flat solutions with negative ADM mass is caused by the violation of the energy conditions.

A peculiar feature of topological solitons is the occurrence of fermionic zero modes localized on the soliton.In particular, there is a link between the topological charge of the configuration and the number of zero modes [7].Fermionic modes localized on solitons were first discussed in the pioneering work [8].Later, it was shown that such bound states exist on kinks [9][10][11], domain walls [12], monopoles [13], sphalerons [14,15], and Skyrmions [16][17][18].The presence of localized fermion modes gives rise to interesting phenomena like monopole catalysis of proton decay [19,20], emergence of superconducting cosmic strings [21], and appearance of fractional quantum numbers of solitons [9,22].
Here we would like to distinguish between two different types of fermionic zero modes that may arise in the presence of topological solitons, depending on their properties.On the one hand, topological solitons may support exact fermionic zero modes that are localized on them, independent of the Yukawa coupling strength, as is the case, for instance, for monopoles [9].On the other hand, as is the case for Skyrmions, such an exact zero mode is not supported by the topological soliton.Instead, for Skyrmions there is a spectral flow of the eigenvalues.In particular, at some critical value of the Yukawa coupling a fermionic mode emerges from the positive continuum, crosses zero and then slowly approaches the negative continuum as the coupling grows [16][17][18].
A major simplification of most of the above studies is that, until recently, the backreaction of the fermion mode on the soliton was not taken into account.This assumption is justified in the weak coupling limit.How-ever, as the Yukawa coupling increases, the effects of the backreaction can be significant, see e.g.Refs.[23][24][25][26][27][28][29][30][31][32][33].
As compared to boson fields, fermion fields have attracted less attention in General Relativity.Although solutions of the Dirac equation in curved spacetime were constructed many decades ago [34], the consideration of self-gravitating fermions still remains somewhat obscure, because the Dirac field can not be treated on a classical level.Instead, one needs to retain its basic quantum character.Typically this is done by complying with the Pauli exclusion principle.The Dirac field is then treated in terms of normalizable quantum wave functions, and the appropriate occupation number of the relevant fermion mode(s) is imposed.
In particular, one then makes use of the approximation that (i) only single-particle fermion states are considered, (ii) second quantization of the fields is ignored, and (iii) gravity is treated purely classically.Under these assumptions, for instance, the fermion level crossing in the background of the Einstein-Yang-Mills sphaleron was considered in [35].It turned out that self-gravitating spinor fields may give rise to some interesting phenomena in the cosmology of the accelerating Universe [36,37].It was shown that the Einstein-Dirac equations support regular localized solitonic solutions [38], the so-called Dirac stars [39][40][41][42][43].Moreover, it was demonstrated that the backreaction of self-gravitating fermions may significantly affect the metric and, in particular, allow for (traversable) wormholes [44][45][46].
The main objective of the present Letter is to examine subject to the above stated approximation a similar system of a spherically symmetric spin-isospin fermionic mode localized on the self-gravitating Skyrmion, and to study the spectral flow in this system, where for consistency the backreaction of the fermions on the Skyrme field and on the metric is taken into account.
The Model -We consider the (3+1)-dimensional Einstein-Skyrme system, coupled to a Dirac field carry-ing spin and isospin where the gravitational part is the Einstein-Hilbert action, g is the determinant of the metric, R is the curvature scalar, and G is Newton's constant [? ].The Lagrangian of the matter fields L m is given in terms of the Skyrme field U ∈ SU (2) [3,4] minimally coupled to the Dirac isospinor fermions ψ, and L m = L Sk + L sp + L int .We consider the usual Lagrangian of the Skyrme model without a potential term (2) Here f π and a 0 are the parameters of the model with dimensions [f π ] = L −1 and [a 0 ] = L 0 , respectively.
The matrix-valued field U can be decomposed into the scalar component φ 0 and the pion isotriplet φ n via U = φ 0 I + iφ n τ n , where τ n are the Pauli matrices, and the field components φ a = (φ 0 , φ n ) are subject to the sigmamodel constraint, φ a • φ a = 1.
The Dirac Lagrangian is where m is a bare mass of the fermions, γ µ are the Dirac matrices in the standard representation in a curved spacetime, / D = γ µ Dµ , and the isospinor covariant derivative on a curved spacetime is defined as (see, e.g., Ref. [47]) Dµ ψ = (∂ µ − Γ µ )ψ.Here Γ µ are the spin connection matrices [47,48].In the numerical calculations we restrict ourselves to the case of fermions with zero bare mass, m = 0.
Finally, the Skyrmion-fermion chiral interaction Lagrangian is where h is the Yukawa coupling constant and γ5 is the corresponding Dirac matrix in curved spacetime.
It is convenient to introduce the dimensionless radial coordinate r = a 0 f π r, the effective gravitational coupling α 2 = 4πGf 2 π , and to rescale the Dirac field, the Yukawa coupling constant and the bare fermion mass as ψ → ψ/ a 0 f 3 π , h → h/(a 0 f π ), and m → m/(a 0 f π ), respectively.
To construct spherically symmetric solutions of the model (1) we implement the above set of assumptions.We treat the gravitational field purely classically and employ Schwarzschild-like coordinates with a spherically symmetric metric, following closely the usual considerations of self-gravitating Skyrmions (see, e.g., Refs.[49][50][51][52]), static spherically symmetric Skyrmion of topological degree one, we then make use of the usual hedgehog parametrization U = cos (F (r)) I + ı sin (F (r)) (σ a n a ) , where n a is the unit radial vector.
The isospin carrying Dirac field is treated by a normalized quantum wave function, and its fermionic nature is imposed at the level of the occupation number, in accordance with Pauli's exclusion principle.Thus we consider here a normalized single particle state to describe the spectral flow of the system consisting of a Skyrmion with topological number one, that is interacting with a single isospinor fermion.The spherically symmetric Ansatz for the isospinor fermion field localized on the Skyrmion can be written in terms of two 2 × 2 matrices χ and η [9,53] as ψ = e −ıωt χ η with Here u(r) and v(r) are two real functions of the radial coordinate only, and ω is the eigenvalue of the Dirac operator.
Varying the total reduced action of the spherically symmetric self-gravitating Skyrmion coupled to the isospin fermion with respect to the functions F, u, v, N, σ, we get the set of five coupled mixed order ordinary differential equations supplemented by the normalization condition of the single localized fermion mode dV ψ † ψ = 1.This system is solved numerically together with the constraint imposed by the normalization condition.The boundary conditions are found from the asymptotic expansion of the solutions on the boundaries of the domain of integration together with the assumption of regularity  4. The point 2 corresponds to a configuration with ADM mass M = 0 (cf.Fig. 3).The inset shows the behavior of the curves for α 2 = 0.14 and 0.15 in the neighbourhood of ω = 0. and asymptotic flatness.The Skyrmion profile function F (r) corresponds to the configuration of topological degree one.
Numerical results -In the decoupled limit (h → 0) the dependence of the regular self-gravitating Skyrmion on the effective gravitational coupling α 2 = 4πGf 2 π is well known.There are two branches of solutions which are characterized by their limiting behavior as α tends to zero [49][50][51][52].The first branch originates from the flat spacetime Skyrmion (see the solid purple curve labeled as "Pure Skyrmion" in the inset of Fig. 1).It extends up to a maximal value α 2 max ≈ 0.0404, where it bifurcates with the second, upper mass branch (the dotted purple curve in the inset of Fig. 1).The second (backward) branch extends down to the limit α → 0 which is approached as f π → 0. Thus the sigma-model term in the Skyrme Lagrangian ( 2) is vanishing and the configuration tends to the scaled Bartnik-McKinnon (BM) solution.
In the presence of the fermions, the limit α = 0 with G = 0 corresponds to the fermionic mode localized on the Skyrmion in Minkowski spacetime.This mode emerges from the positive continuum at some critical value of the Yukawa coupling h cr ≈ −0.40.Further increase of the modulus of the Yukawa coupling decreases the scaled eigenvalue ω/|h| of the Dirac operator.For some critical value of the coupling h the curve ω(h) then crosses zero, as seen in Fig. 2.There is a single fermionic level which monotonically flows from positive to negative values as the coupling decreases.
More generally, there is a family of solutions depending continuously on two parameters, the Yukawa coupling constant h and the eigenvalue of the Dirac oper- ator ω, for each particular value of the effective gravitational coupling α.Since the appearance of a single zero crossing fermionic level is related to the underlying topology of the Skyrme field, we may expect that, as the self-gravitating configuration evolves towards the topologically trivial BM solution, this mode undergoes a certain transition.
For any non-zero value of the gravitational coupling, the spherically symmetric fermionic mode localized on the Skyrmion is no longer linked to the positive continuum, as seen in Fig. 2. Instead, it arises at some particular value of the Yukawa coupling h max (α) < h cr with a scaled eigenvalue ω/|h| smaller than the threshold value.Physically, this situation reflects the energy balance of the system of a self-gravitating Skyrmion interacting with the isospinor fermion: the added gravitational interaction must be compensated by the force of the Yukawa interaction.Notably, the spectral flow of the fermionic Hamiltonian bifurcates at this point: as h decreases below h max , two branches arise as displayed in Fig. 2.
We can understand qualitatively this pattern by analogy with the appearance of two branches of solutions for self-gravitating Skyrmions.The evolution of the configuration along one branch is related to the decrease of the Newton constant G, whereas the second branch may be considered as being obtained by decreasing the pion decay constant f π .In both cases the effective gravitational coupling α is the same and the configuration remains in equilibrium for some particular value of the Yukawa coupling.By analogy with the case of the usual selfgravitating Skyrmions, we will refer to these branches of the spectral flow as the "Skyrmion branch" and the "BM branch", respectively.To distinguish these branches visually, we plot them using solid (for the Skyrmion branch) and dotted (for the BM branch) lines in Figs.1-3.
As α remains relatively weak, α 2 0.14, the Skyrmion The field components of the self-gravitating Skyrmion-fermion system are shown as functions of the compactified radial coordinate x = r 1+r for α 2 = 0.1 for the points 1-6 in Fig. 2. The solution with negative ADM mass is labeled as 1, while the solution with zero ADM mass is labeled as 2.
branch of the spectral flow still crosses zero, and the BM branch slowly approaches zero from above, as seen in Fig. 2. Further increase of the gravitational coupling excludes the zero eigenvalue of the Dirac operator, and ω/|h| remains negative along both branches.On the BM branch, it then tends to zero from below, as the Yukawa coupling becomes stronger.Transitional branches between nodeless branches and those with one node are the branches for the values of α ≈ 0.14 when the spectral flow has two zeros, as seen in the inset of Fig. 2.
A rather intriguing observation is that the ADM mass of the coupled configuration becomes negative as h decreases along the Skyrmion branch, as seen in Fig. 3. On the other hand, the ADM mass remains always positive along the BM branch.
In order to get a better understanding, let us now consider the pattern of evolution of the components of the coupled system.Fig. 4 presents the profile functions of the self-gravitating Skyrmion coupled to the localized fermionic mode for α 2 = 0.1 and a set of values of the Yukawa coupling.Note that, as h decreases below the critical value, labeled as the point 5 in Fig. 2, the size of the configuration on the Skyrmion branch increases.As the scaled eigenvalue ω/h crosses zero (the point 4) and becomes positive, the minimal values of the metric functions N (r) and σ(r) are increasing.
A further increase of the strength of the Skyrmionfermion coupling for a fixed value of α yields a very unusual picture: the metric function N (r) increases above unity in the interior of the Skyrmion, while it becomes less than unity in the outer region, as seen in Fig. 4 for the solution labeled by the point 3 in Fig. 2.
Notably, the ADM mass of the configuration becomes zero at some particular value of the Yukawa coupling, as seen in Fig. 3.At this critical point the metric component g 00 is nearly unity almost everywhere in space and the first derivative of the metric function N at spatial infinity is vanishing, as displayed in Fig. 4.
As the Yukawa coupling becomes even stronger (the point 1 in Fig. 2), the metric function N (r) is greater than unity, except at the boundaries, and the metric function σ(r) becomes greater than unity in the inner region of the configuration.In contrast, the solution on the BM branch, labeled as the point 6 in Fig. 2, behaves as expected.The configuration becomes increasingly localized by the stronger gravitational attraction.
Fig. 1 displays the dependence of the ADM mass of the configurations on the effective gravitational coupling α 2 .As before, two branches bifurcate at some maximal value α max , which increases as the absolute value of the Yukawa coupling h becomes larger.However, unlike the self-gravitating Skyrmions in the decoupling limit h = 0, we can not extend the BM branches of the solutions with a localized fermionic mode all the way down to the limit α → 0, as seen in Fig. 1.For every BM branch, there occurs some limiting value of α min (h) for which this branch terminates in our numerical calculations.Our calculations indicate that there might be a critical value of the Yukawa coupling h ≈ −0.79 for which the ADM mass attains its maximum value M/α 2 ≈ 11.88.This would correspond to the minimal value of the gravitational coupling α 2 min ≈ 0.02.Correspondingly, the increase or decrease in the strength of the Yukawa interaction would increase the minimal value α min (h), while the ADM mass of the corresponding limiting systems would decrease, as seen in Fig. 1.Here the dashed-dotted line then possibly corresponds to the sequence of limiting configurations on the BM branch which would terminate at α min as h varies.In other words, the presence of the fermions may possibly prohibit the BM limit.However, further numerical work will be necessary to confirm or refute such a limit.
On the other hand, the domain of existence of the solutions is restricted by the bifurcation points, from which the Skyrmion and BM branches originate.These points are connected by the critical line (the dashed line in Fig. 1) which restricts the domain of existence of the solutions from above.This line starts at α 2 ≈ 0.07 and h ≈ −0.437 and extends up to the last calculated point at α 2 ≈ 1.22 and h = −2.4.In turn, in the range −0.4 h −0.437, BM branches seems to be already absent, and only Skyrmion branches are found, which degenerate at h ≈ −0.4 and α = 0 into one flat spacetime configuration, cf.Fig. 2. Curiously, for the fixed value of the Yukawa coupling h = −1.565there is a certain family of configurations on the Skyrmion branch, whose mass remains approximately constant as α varies up to the bifurcation point (as seen in Fig. 1).
The unexpected behavior of the solutions of the self-FIG.5.The function ǫ + p and the energy density ǫ for the solutions labeled as 1-5 in Fig. 2.
gravitating Skyrmion-fermion system on the Skyrmion branch is related to the violation of the null and weak energy conditions, which state that the stress-energy tensor T µν satisfies, respectively, the inequalities T µν k µ k ν ≥ 0, and T µν V µ V ν ≥ 0 for any light-like vector k µ , and for any time-like vector V µ (for a review, see e.g.Ref. [54]).
The null/weak energy conditions for the self-gravitating coupled spherically symmetric Skyrmion-fermion system become ǫ + p ≡ T 0 0 − T 1 1 ≥ 0, and the weak energy condition also implies ǫ ≡ T 0 0 ≥ 0, where ǫ and p are the energy density and radial pressure, respectively.
In Fig. 5 we display the combination ǫ + p and the energy density ǫ for the configurations labeled by the points 1−5 on the Skyrmion branch for α 2 = 0.1 in Fig. 2. Clearly, the null/weak energy conditions become violated already for the solution labeled as the point 3. Notice also that for all solutions 1−5 the pressure p is always positive and the violation of the null/weak energy conditions is caused only by the negativeness of the energy density.
Conclusion -Static, spherically symmetric Skyrmions minimally coupled to gravity are the pioneering example of solitonic and hairy black hole solutions in General Relativity [49-52, 55, 56].The index theorem secures the existence of a bound fermionic mode localized on the self-gravitating Skyrmion.However, this mode is not an exact zero mode, independent of the Skyrmion-fermion coupling.Instead the Skyrmion-fermion system exhibits spectral flow.In this work, we have shown that the localization of the backreacting fermionic mode may have dramatic consequences, in particular, the energy conditions may be violated and regular self-gravitating asymptotically flat solutions with negative and zero ADM mass may emerge.

2 FIG. 1 .
FIG.1.The ADM mass of the self-gravitating Skyrmionfermion system is shown as a function of the effective gravitational coupling α 2 for several values of h.The inset shows the behavior of the mass for small α 2 in the neighbourhood of the maximum mass obtained.

2 FIG. 2 .
FIG.2.The normalized eigenvalue of the localized fermionic states is shown as a function of the Skyrmion-fermion coupling h for several values of α 2 .The points 1-6 along the curve α 2 = 0.1 correspond to the values of the parameters ω and h for which profiles of the matter and metric fields are shown in Fig.4.The point 2 corresponds to a configuration with ADM mass M = 0 (cf.Fig.3).The inset shows the behavior of the curves for α 2 = 0.14 and 0.15 in the neighbourhood of ω = 0.

20 FIG. 3 .
FIG.3.The ADM mass of the self-gravitating Skyrmionfermion system is shown as a function of the Yukawa coupling h for several values of α 2 .