PHYSICAL REVIEW D 98, 106007 (2018)
Chiral symmetry breaking and restoration in 2 + 1 dimensions from holography: Magnetic and inverse magnetic catalysis
† ‡ Diego M. Rodrigues,1,* Danning Li,2, Eduardo Folco Capossoli,1,3, and Henrique Boschi-Filho1,§ 1Instituto de Física, Universidade Federal do Rio de Janeiro, 21.941-972, Rio de Janeiro RJ, Brazil 2Department of Physics and Siyuan Laboratory, Jinan University, Guangzhou 510632, China 3Departamento de Física and Mestrado Profissional em Práticas da Educação Básica (MPPEB), Col´egio Pedro II, 20.921-903, Rio de Janeiro RJ, Brazil
(Received 3 August 2018; published 9 November 2018)
We study the chiral symmetry breaking and restoration in (2 þ 1)-dimensional gauge theories from the holographic hardwall and softwall models. We describe the behavior of the chiral condensate in the presence of an external magnetic field for both models at finite temperature. For the hardwall model we find magnetic catalysis (MC) in different setups. For the softwall model we find inverse magnetic catalysis (IMC) and MC in different situations. We also find for the softwall model a crossover transition from IMC to MC at a pseudocritical magnetic field. This study also shows spontaneous symmetry breaking for both models. Interestingly, for B ¼ 0 in the softwall model, we found a nontrivial expectation value for the chiral condensate.
DOI: 10.1103/PhysRevD.98.106007
I. INTRODUCTION in order to reproduce the nonperturbative physics of QCD, one must break the conformal invariance in the Much theoretical progress on the nonperturbative phys- original AdS=CFT correspondence. There are two well- ics of relativistic quantum field theories (RQFT) has been known models which do this symmetry breaking: the achieved recently. In particular, the role played by an hardwall [19–25] and softwall [26–28] models. In par- external magnetic field in these RQFT has been inves- ticular, some researchers have used these models to tigated in many works, especially in connection with QCD discuss the IMC as a perturbation in powers of the [1–12]. In the context of lattice QCD, for the chiral phase magnetic field [9–12], and can only be trusted for weak transition, an inverse magnetic catalysis (IMC) has been 1 2 observed, i.e., the decreasing of the critical temperature fields, that means eB < GeV . Since their approach is (T )with increasing magnetic field (B) for eB ∼ 1 GeV2 perturbative in the magnetic field B, they could not c 1 2 eB ∼ 3 2 predict what would happen for eB > GeV . [13] and more recently for GeV [14]. This is in 2 þ 1 contrast with what would be expected: a magnetic catalysis Concerning free fermions in dimensions in the (MC), meaning the increasing of the critical temperature presence of a magnetic field, we know that, perturbatively, with increasing magnetic field [4]. the phenomenon of magnetic catalysis is due to the effect of A promising approach to investigate the phenomena of the magnetic field itself, which is a strong catalyst of IMC and MC is based on the AdS=CFT correspondence, dynamical chiral symmetry breaking, leading to the gen- or holographic duality [15–18]. Such duality has become eration of a nonzero chiral condensate in the chiral limit → 0 very useful to address strongly coupled gauge theories, (m ), given by [3] including the nonperturbative regime of QCD. However, hψψ¯ i ∝ eB ð Þ 2π : 1 *[email protected] † [email protected] ‡ At finite temperature, also in the perturbative regime, it was [email protected] §[email protected] shown in [29] that the chiral condensate is extremely unstable, meaning that it vanishes as soon as we introduce Published by the American Physical Society under the terms of a heat bath with and without chemical potential. the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to In this work we investigate, nonperturbatively, the chiral the author(s) and the published article’s title, journal citation, phase transition and symmetry restoration in (2 þ 1)- and DOI. Funded by SCOAP3. dimensional holographic gauge theories in an external
2470-0010=2018=98(10)=106007(11) 106007-1 Published by the American Physical Society RODRIGUES, LI, CAPOSSOLI, and BOSCHI-FILHO PHYS. REV. D 98, 106007 (2018) magnetic field from the hardwall and softwall models. We with F2 in (2) given by confirm the picture mentioned before, but we also find 2 2 4 IMC, in the softwall model, for small magnetic fields (in 2 ¼ B z ð Þ F 4 : 9 units of the string tension squared), in agreement with the L lattice results in 3 þ 1 dimensions, while the traditional The general solution for Eqs. (7) and (8) is given by MC behavior happens for large magnetic fields in this 2 4 3 nonperturbative approach. In the hardwall model, we only fðzÞ¼1 þ B z þ bz : ð10Þ find MC. At finite temperature our approach predicts a If we impose the horizon condition fðz ¼ z Þ¼0, we find very fast decrease in the chiral condensate as we increase H 3 the temperature, which means that we do not observe the ð Þ¼1 þ 2 3ð − Þ − z ð Þ fBT z B z z zH 3 : 11 instability found in the perturbative thermal field theory zH computation presented in [29]. This solution was found recently in [31,32]. Equation (11) corresponds to the AdS4 in the presence of a background II. THE CHIRAL CONDENSATE AND CHIRAL magnetic field at finite temperature, T, where zH is the ð ¼ Þ¼0 SYMMETRY BREAKING IN THE horizon position, such that fBT z zH . One can note HOLOGRAPHIC FRAMEWORK that this solution indeed satisfies both differential equa- tions (7) and (8). Note that the solution Eq. (11) implies the A. Background geometry existence of an inner and outer horizon. The outer horizon 0 ð ¼ Þ 0 Via Kaluza-Klein dimensional reduction, the supergrav- satisfies fBT z zH < and is the physically relevant 7 ity theory on AdS4 × S may be consistently truncated to one. The temperature is given by the Hawking formula Einstein-Maxwell theory on AdS4 [30]. The action for this j 0 ð ¼ Þj ¼ fBT z zH ð Þ theory, in Euclidean signature, is given by T 4π : 12 Z 1 pffiffiffi Using (11) and the condition f0 ðz ¼ z Þ < 0, we have S ¼ − d4x gðR − 2Λ − L2F FMNÞ; ð Þ BT H 2κ2 MN 2 4 1 3 3 ð Þ¼ − 2 3 4 ð Þ T zH;B B zH ;zH < 2 : 13 2 4π zH B where κ4 is the four-dimensional coupling constant, which is proportional to the four-dimensional Newton’s constant In what follows and in the rest of this work we set the 2 4 (κ4 ≡ 8πG4); d x ≡ dτdx1dx2dz; R is the Ricci scalar; and AdS radius L ¼ 1. Λ is the negative cosmological constant. The field equations coming from the bulk action (2) B. Holographic setup for chiral symmetry breaking together with the Bianchi identity are [30] Here we describe how to realize the chiral symmetry breaking of SUðNfÞ × SUðNfÞ → SUðNfÞ in the 2 1 2 3 L R diag R ¼ 2L FP F − g F − g ; ð3Þ hardwall and softwall models. For both models we consider MN M NP 4 MN L2 MN the action Z MN 1 3 pffiffiffi −Φ † ∇MF ¼ 0: ð4Þ ¼ − ð M þ S 2 d xdz ge Tr DMX D X VX 2κ4 Our ansatz for the metric and the background magnetic − ð 2 þ 2 ÞÞ ð Þ FL FR ; 14 field to solve (3) are given by where X is a complex scalar field; DM is the covariant 2 2 derivative defined as D X ¼ ∂ X þ iAL X − iXAR , 2 ¼ L ð Þ τ2 þ dz þ 2 þ 2 ð Þ M M M M ds 2 f z d dx1 dx2 ; 5 L;R z fðzÞ with AM being the chiral left- and right-handed gauge fields; FMN the field strength defined as FMN ¼ ∂MAN − ∂ − ½ ¼ 2 † þ λð † Þ2 þ … F ¼ Bdx1 ∧ dx2: ð6Þ NAM i AM;AN ; and VX M4X X X X is the potential for the complex scalar field, where M4 is the Using this ansatz, the field equations (3) are simplified mass of the complex scalar field X. From the AdS4=CFT3 and given by correspondence we have
2 00 0 2 4 2 ¼ ΔðΔ − 3Þ ð Þ z f ðzÞ − 4zf ðzÞþ6fðzÞ − 2B z − 6 ¼ 0; ð7Þ M4 : 15 Since the complex scalar field in four spacetime dimen- 0ð Þ − 3 ð Þ − 2 4 þ 3 ¼ 0 ð Þ zf z f z B z ; 8 sions is supposed to be dual to the chiral condensate
106007-2 CHIRAL SYMMETRY BREAKING AND RESTORATION IN … PHYS. REV. D 98, 106007 (2018)
σ ≡ hψψ¯ i 2 in three spacetime dimensions, whose dimension (z → 0) behavior of χðzÞ, that is, χðzÞ¼mqz þ σz and Δ ¼ 2 2 ¼ −2 is , we have therefore M4 . So, we have the the regularity of χðzÞ at the horizon, zH, meaning ¼ −2 2 þ λ 4 potential VX X X , where the first term is just the χðzHÞ < ∞. mass term and the second is the term needed to realize the spontaneous symmetry-breaking mechanism [33–35]. C. Zero temperature case The field equations coming from (14) are given by Here the background metric is pure AdS, without pffiffiffi pffiffiffi −Φð Þ −Φð Þ singularity, and there is no natural boundary condition at D ½ ge z gMND X − ge z ∂ V ¼ 0: ð16Þ M N X X the horizon. Numerically, one cannot take the horizon Here, we also assume that the expectation value for X takes exactly at infinity. Thus, to analyze the chiral symmetry 1 breaking in the confining region (T ¼ 0), we need to a diagonal form hXi¼ χI2 for the SUð2Þ case [9,11], 2 perform some coordinate transformations in order to bring where I2 is the 2 × 2 identity matrix. For the general case the boundary condition at z → ∞ to a finite position. To of SUðN Þ, we would have hXi¼pffiffiffiffiffiffi1 χI , where the f 2 Nf this end, we are going to split the analysis into two cases: pffiffiffiffiffiffiffiffiffi Nf factor 2N is introduced to maintain the kinetic term (i) T ¼ 0 and B ¼ 0; (ii) T ¼ 0 and B ≠ 0. f ð Þ¼1 of χ canonical [34,35]. In addition, we assume that In the first case (i), f z in Eq. (17),sowehave χð μ Þ ≡ χð Þ x ;z z . With all these assumptions, we can write 2 1 χ00ð Þþ − −Φ0ð Þ χ0ð Þ − ð−2χð Þþ4λχ3ð ÞÞ ¼ 0 (16) as z z z 2 z z : z z 2 0ð Þ 1 00 0 f z 0 ð22Þ χ ðzÞþ − − Φ ðzÞþ χ ðzÞ − ∂χVðχÞ¼0; z fðzÞ z2fðzÞ → 0 ð17Þ In the z coordinate, the two boundaries are z and z → ∞. ¼ 0 where 0 means derivative with respect to z. For our purposes For the hardwall model, k , we can do the following the dilatonic field is ΦðzÞ¼kz2 with k a dimensional coordinate transformation: ¼ 0 constant. For the hardwall model we set k . On the other 1 − t 1 hand, for the softwall model k ≠ 0. Finally, we take z ¼ ; t ¼ : ð23Þ 2 4 t 1 þ z VðχÞ ≡ TrVX ¼ −χ þ λχ , in both models. First, let us consider only the quadratic term in the Under this transformation, the two boundaries in the t ðχÞ ðχÞ¼−χ2 potential V , i.e., V , which implies a linear and coordinate become t ¼ 1ðz → 0Þ and t ¼ 0ðz → ∞Þ, and analytically solvable equation of motion. In this case we the equation of motion (22) becomes can clearly see that the UV behavior (z → 0)of(17) takes the form 2ð2 − Þ 2χð Þ − 4λχ3ð Þ χ00ð Þþ t χ0ð Þþ t t ¼ 0 ð Þ t t 2 2 ; 24 2 2 tð1 − tÞ t ð1 − tÞ χ00ð Þ − χ0ð Þþ χð Þ¼0 ð Þ z z 2 z ; 18 z z which has the following IR (t ¼ 0) and UV (t ¼ 1) asymptotic behaviors, whose solution is given by
2 1 2 χðz → 0Þ¼c1z þ c2z ; ð19Þ χ ðtÞ¼pffiffiffiffiffi þ ct þ Oðt Þ; ð25Þ IR 2λ where c1 and c2 are integration constants. Since the χ ð Þ¼− ð − 1Þþð þ σÞð − 1Þ2 þ Oðð − 1Þ3Þ complex scalar field χðzÞ is dual to the operator UV t mq t mq t t ; hψψ¯ i, we can identify these two integration constants ð26Þ as proportional to where c is a constant and mq and σ were defined in z ∝ ð Þ χð → 0Þ ≡ χ ¼ þ σ 2 c1 mq 20 coordinate by z UV mqz z . Now for the softwall model, still in case (i), we have ∝ σ ð Þ 2 c2 ; 21 ΦðzÞ¼−z , where we have set k ¼ −1. In this case equation (22) becomes 2 þ 1 with mq being the fermion mass in dimensions, and σ ¼hψψ¯ i is the chiral condensate. 2 1 χ00ðzÞ − − 2z χ0ðzÞ − ð−2χðzÞþ4λχ3ðzÞÞ ¼ 0; In this work we solve numerically Eq. (17) using the z z2 quartic potential VðχÞ¼−χ2 þ λχ4, with λ ¼ 1. The ð Þ boundary conditions used to solve (17) are the UV 27
106007-3 RODRIGUES, LI, CAPOSSOLI, and BOSCHI-FILHO PHYS. REV. D 98, 106007 (2018) with the leading expansion at z → ∞ given by III. RESULTS FOR THE HARDWALL MODEL
2 In this section we present our results for the hardwall 1 e−z χ ðzÞ¼pffiffiffiffiffi − c ; ð28Þ model concerning the dependence of the chiral conden- IR 2λ z sate, σ, with the external magnetic field, B,forT ¼ 0 and some finite temperatures. We also study the dependence → 0 χð → 0Þ ≡ χ ¼ þ σ 2 σ while at z we still have z UV mqz z . of with temperature, T, for some values of magnetic Note that in this case (the softwall with T ¼ 0 and B ¼ 0) field. These analyses allow us to investigate whether there there is no need to change the coordinates to obtain the is IMC or MC and the role played by the thermal effects. asymptotic behaviors at IR and UV due to the presence of We will also investigate whether the chiral symmetry is the dilaton. spontaneous or explicitly broken at the nonperturbative Finally, in case (ii) for T ¼ 0 and B ≠ 0,wehave(17) level. It is worth remembering that we are working in 31=3 2 þ 1 σ with fðzÞ given by (11) with z ¼ pffiffiffi, where we have used dimensions so that all observable quantities like , H B B, T,andmq pwillffiffiffiffiffi always be measured in units of the the definition of the Hawking temperature (13). Doing the σ coordinate transformation s ¼ z=z , the boundaries in the s string tension ( s). Then, thep temperatureffiffiffiffiffi and mass will H σ coordinate are s ¼ 0 (UV), and s ¼ 1 (IR), and Eq. (17) in be measured in units of s, as, for instance, in this coordinate becomes Refs. [36–38]. On the other hand, the magnetic field and the chiral condensatep willffiffiffiffiffi be measured in units of the pffiffiffi 2 2 string tension squared ð σ Þ . One should note that the 2 2 3k 12s s χ00ðsÞ − − s þ χ0ðsÞ hardwall model can be defined setting k ¼ 0 in Eq. (14) s B ð1 − sÞð1 þ 2s þ 3s2Þ for the dilaton field, given by ΦðzÞ¼kz2 and introducing −2χðsÞþ4λχ3ðsÞ þ ¼ 0 ð Þ a hard cutoff zmax such that the holographic coordinate z 2 2 2 : 29 0 ≤ ≤ – s ð1 − sÞ ð1 þ 2s þ 3s Þ obeys z zmax [19 25]. Note that the temperature of the black hole is related to the horizon position zH, One can check that the near boundary (s → 0) solution of subjected to zH 2 −2χðsÞþ4λχ3ðsÞ χ00ðsÞ − χ0ðsÞþ ¼ 0; ð32Þ 1 − s 6ð1 − sÞ2 whose solution is given by pffiffiffi 1 −1þ 33 0 χ ðsÞ¼pffiffiffiffiffi þ cð1 − sÞ 2 6 þ c × ðsingular solutionÞ; IR 2λ ð33Þ with c and c0 being two integral constants at IR (s ¼ 1). In order to have a nonsingular solution, we set c0 ¼ 0. pffiffiffiffiffi In the following sections we present our results for IMC FIG. 1. The chiral condensate σ in units of σs versus the and MC in the context of the chiral phase transition for the magnetic field B in units of the string tension squared σs in the hardwall and softwall models. confining regime at T ¼ 0. 106007-4 CHIRAL SYMMETRY BREAKING AND RESTORATION IN … PHYS. REV. D 98, 106007 (2018) FIG. 2. The chiral condensate σ normalized by the fermion pffiffiffiffiffi FIG. 4. The chiral condensate σ normalized by the fermion mass m in units of σ versus the magnetic field B in units of the pffiffiffiffiffi q s mass m in units of σ versus the magnetic field B in units of σ . string tension squared σ in the confining regime at T ¼ 0. q s s s This plot shows MC for the temperature pTffiffiffiffi ¼ 0.200 for three σs values of the fermion mass in the hardwall model. masses pmffiffiffiffiq ¼ 0.001, 0.01, 0.1. Also, in this picture one σs can note that for very weak magnetic fields the chiral σ → 0 condensate is almost independent of the fermion masses, figures show the restoration of the chiral symmetry ( ) implying that, in the chiral limit (m → 0), the chiral for different values of the external magnetic field and of q the fermion mass m . symmetry is spontaneously broken due to the presence of q Comparing Figs. 5 and 6 one can note the MC behavior the magnetic field. As a last comment, for weak fields one since the values of the chiral condensate increases when can see an approximate linear behavior for the chiral the magnetic field is also increased as shown explicitly in condensate consistent with the perturbative result found in Figs. 3 and 4. [3],ascanbeseeninEq.(1). We also show in Fig. 7 the behavior of the critical In Fig. 4 we can see that the increasing of the temper- temperature (T ) against the magnetic field (B)forthe ature still provides MC in the hardwall model. In this case c pTffiffiffiffi hardwall model. Here, we use the definition of Tc as the temperature is now 40 times greater ( σ ¼ 0.200) than s the temperature where σðTÞ¼0,likeinRef.[9].Note in Fig. 3, and we still use the same values for the three that T increases as we increase the magnetic field as fermion masses. c a consequence of the MC observed, for instance, in In Figs. 5 and 6 we present the dependence of the chiral Fig. 3. condensate σ with respect to the temperature T. These σ FIG. 3. The chiralpffiffiffiffiffi condensate normalized by the fermion FIG. 5. The chiral condensate σ normalized by the fermion σ pffiffiffiffiffi mass mq in units of s versus the magnetic field B in units of the mass m versus the temperature T, both in units of σ . This plot σ q s string tension squared s. This plot shows MC for low temper- also shows the restoration of the chiral symmetry ½σðTÞ¼0 for ature pTffiffiffiffi ¼ 0.005 for three values of the fermion mass in the σs the hardwall model for a fixed and weak external magnetic field hardwall model. B ¼ 2.0 and different values of the fermion mass m . σs q 106007-5 RODRIGUES, LI, CAPOSSOLI, and BOSCHI-FILHO PHYS. REV. D 98, 106007 (2018) FIG. 8. The chiral condensate σ versus the magnetic field B, FIG. 6. The chiral condensate σ normalized by the fermion pffiffiffiffiffi both in units of σs. This plot shows IMC for B the confining regime (T ¼ 0) for three fermion masses mq. This result is shown in Fig. 8. Unlike the results for the hardwall model in the previous section, one observes that the chiral condensate in the softwall model presents two different behaviors depending on the value of the magnetic field. These behaviors are separated by a pseudocritical magnetic field, Bc, such that for B pffiffiffiffiffi FIG. 7. The critical temperature, Tc, in units of σs,asa function of the magnetic field, B, in units of σs for the hardwall model for different values of the fermion mass. IV. RESULTS FOR THE SOFTWALL MODEL In this section we do the entire analysis of the previous section, but this time for the softwall model [26–28], characterized by a dilatonlike field ΦðzÞ. We assume the dilaton profile to be of the form ΦðzÞ¼kz2, where the k constant k was set as pffiffiffiffi 2 ¼ −1. Further, one can note that ð σsÞ the soft wall linear spectrum persists in 2 þ 1 dimensions, as has been shown in the Appendix of Ref. [31]. There, one uses the dilaton constant k in units of the string tension. In FIG. 9. The chiral condensate σ versus the magnetic field B, σ this model we are also assuming that there is no back- both in units of s. This plot shows IMC for B 106007-6 CHIRAL SYMMETRY BREAKING AND RESTORATION IN … PHYS. REV. D 98, 106007 (2018) FIG. 10. The chiral condensate σ versus the magnetic field B, both in units of σs. This plot shows IMC for B pmffiffiffiffiq condensate behavior in the region of weak magnetic fields. temperatures and three fermion masses σ ¼ 0.001, 0.01, s In Fig. 11 for very low temperature, one can note that the pTffiffiffiffi 0.1. In Fig. 9 the temperature is σ ¼ 0.005, while in s chiral condensate’s behavior is approximately the same for Fig. 10 the temperature is pTffiffiffiffi ¼ 0.200. One should note the three values of the fermion mass, indicating a sponta- σs that these two figures present a crossover between the two neous chiral symmetry breaking. On the other hand, as one phases IMC and MC around a pseudocritical magnetic field increases the temperature as shown in Figs. 12 and 13, its effect on the chiral condensate is more visible for the lowest Bc which depends on the temperature. Note that the fermion masses (pmffiffiffiffiq ≤ 0.01), close to the chiral limit. In comparison between Figs. 9 and 10 also shows implicitly σs a chiral symmetry restoration since σ decreases for an this case we still have an approximate chiral symmetry breaking. Comparing Figs. 12 and 13, one can see that a increasing temperature, for B σ FIG. 11. The chiral condensate σ versus the magnetic field B, FIG. 13. The chiral condensate versus the magnetic field B, both in units of σs for the softwall model with temperature both in units of σs for the softwall model. This plot shows IMC pTffiffiffiffi ¼ 0.200. This plot shows IMC for weak magnetic fields and for weak magnetic fields and low temperature. This plot also σs shows spontaneous breaking of the chiral symmetry. The inde- high temperature and spontaneous breaking of the chiral sym- pendence of the fermion mass for the chiral condensate is more metry. The approximate independence of the fermion mass for the evident for pmffiffiffiffiq ≤ 0.01. chiral condensate is more evident for pmffiffiffiffiq ≤ 0.01. σs σs 106007-7 RODRIGUES, LI, CAPOSSOLI, and BOSCHI-FILHO PHYS. REV. D 98, 106007 (2018) σ σ σ σ FIG. 14. The chiral condensatepffiffiffiffiffi , in units of s, versus the FIG. 16. The chiral condensatepffiffiffiffiffi , in units of s, versus the temperature T, in units of σs for the softwall model. This plot temperature T, in units of σs for the softwall model. This plot shows the chiral symmetry restoration ½σðTÞ¼0 for different shows the chiral symmetry restoration ½σðTÞ¼0 for different values of the fermion mass without external magnetic field. This values of the fermion mass with external magnetic field B ¼ 20. σs picture also shows an approximate independence of the chiral This picture also shows an approximate independence of the chiral mq condensate with the fermion mass, in particular for pffiffiffiffi ≤ 0.01. pmffiffiffiffiq ≤ 0 01 σs condensate with the fermion mass, in particular for . . σs pTffiffiffiffi 2 þ 1 small increase of the temperature from σ ¼ 0.150 to concerning the chiral symmetry breaking in dimen- s ¼ 0 pTffiffiffiffi sions without magnetic field (B )weredoneinthe σ ¼ 0.200 affects the lighter fermions’ masses more. s context of QED3, within the large N approximation Figures 14–16 show the behavior of the chiral con- [39,40]. There it was shown that it is energetically densate with respect to the temperature for different values preferable for this theory to dynamically generate a mass of the fermion mass and different values of the magnetic for fermions. In a subsequent analysis by the same authors field. All three figures show the restoration of the chiral it was shown, in the lowest order in the large N symmetry as one increases the temperature, so that the approximation, that the dynamical chiral symmetry break- chiral condensate goes to zero. In particular, in Fig. 14, ing happens only when the number of fermions N is less one notes that even at zero external magnetic field this 2 than a critical number Nc ¼ 32=π [41]. After that, in model provides a nonzero value for the chiral condensate [42], it was shown that higher order corrections do not at low temperatures. This is in contrast with the pertur- modify the nature of the chiral symmetry breaking. bative result, Eq. (1), so one could associate this behavior We also plot in Figs. 17 and 18 the behavior of the with a nonperturbative effect. However previous studies critical temperature (Tc) against the magnetic field (B). In this case for the softwall model Tc is defined such that σðTÞ decreases fastest [a maximum of −dσðTÞ=dT], as in Ref. [35]. Note that for weak fields, Fig. 17, Tc decreases σ σ FIG. 15. The chiral condensatepffiffiffiffiffi , in units of s, versus the temperature T, in units of σs for the softwall model. This plot shows the chiral symmetry restoration ½σðTÞ¼0 for different B values of the fermion mass with external magnetic field σ ¼ 2.0. s pffiffiffiffiffi This picture also shows an approximate independence of the FIG. 17. Critical temperature, Tc, in units of σs, as a function of chiral condensate with the fermion mass, in particular for the magnetic field, B, in units of σs, for the softwall model in the pmffiffiffiffiq ≤ 0.01. IMC phase (low B regime), for different values of the fermion mass. σs 106007-8 CHIRAL SYMMETRY BREAKING AND RESTORATION IN … PHYS. REV. D 98, 106007 (2018) concluded, for zero and nonzero B, that the breaking of the chiral symmetry is a spontaneous one, since the dependence on the chiral condensate σ with the magnetic field B for different fermion masses mq is approximately the same, as can be seen in the plots of σ versus B, represented by Figs. 11–13. Remarkably, the softwall model provides a crossover between the IMC/MC phases as shown in the plots of σ versus B in Figs. 8–10, characterized by a pseudocritical magnetic field Bc for each fermion mass for a given temperature. Therefore, in this sense, for B>Bc we recover approximately the linear behavior σ ∼ B as in the perturbative result given by (1).On pffiffiffiffiffi the other hand, for B 106007-9 RODRIGUES, LI, CAPOSSOLI, and BOSCHI-FILHO PHYS. REV. D 98, 106007 (2018) [1] K. G. Klimenko, Three-dimensional Gross-Neveu model in [21] H. Boschi-Filho and N. R. F. Braga, QCD/string holo- an external magnetic field, Teor. Mat. 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