A Model to Study Finite-Size and Magnetic Effects on the Phase Transition of a Fermion Interacting System
Emerson B. S. Corrêa, César A. Linhares, Adolfo P. C. Malbouisson
AA Model to Study Finite-Size and Magnetic Effects on the Phase Transition of aFermion Interacting System
Emerson B. S. Corrˆea ∗ Centro Brasileiro de Pesquisas F´ısicas/MCTI, 22290-180, Rio de Janeiro, RJ, Brazil andFaculdade de F´ısica, Universidade Federal do Sul e Sudeste do Par´a, 68505-080, Marab´a, PA, Brazil
C´esar A. Linhares † Instituto de F´ısica, Universidade do Estado do Rio de Janeiro, 20559-900, Rio de Janeiro, RJ, Brazil
Adolfo P. C. Malbouisson ‡ Centro Brasileiro de Pesquisas F´ısicas/MCTI, 22290-180, Rio de Janeiro, RJ, Brazil
We present a model to study effects from an external magnetic field, chemical potential, and finitesize, on the phase structure of a massive four- and six-fermion interacting system. These effectsare introduced by a method of compactification of coordinates, a generalization of the standardMatsubara prescription. Through the compactification of the z coordinate and of imaginary time,we describe a heated system with the shape of a film of thickness L , at temperature β − undergoingfirst- or second-order phase transition. We have found a strong dependence of the temperaturetransition on the constants couplings λ and η . Besides magnetic catalysis and symmetry breakingfor both kinds of transition, we have found an inverse symmetry breaking phenomenon with respectto first-order phase transition. PACS numbers: 11.30.Qc; 11.10.Wx; 11.10.KkKeywords: phase transitions; four- and six-fermion interaction; finite-size effects
I. INTRODUCTION
Quantum Chromodynamics (QCD) is currently accepted as the theory for describing hadronic matter; its La-grangian is assumed to give all the dynamics of the strong interactions. However, it is not easy to get analyticalresults from it; in particular, when one takes into account temperature and density effects. This is due to the mathe-matically intricate structure of QCD, and for this reason simpler, effective models have called attention over the lastfew decades, for instance, the Nambu–Jona-Lasinio (NJL)[1, 2], Gross–Neveu (GN)[3] and MIT Bag [4] models. Inparticular, the GN model shares some interesting features with QCD, like asymptotic freedom and dynamical symme-try breaking; moreover, it has a peculiar simplicity from the mathematical point of view. Extensions of this effectivemodel have been made in many articles with applications to both condensed matter and hadronic physics [5–19].Interesting effects on the phase transitions in these effective models can be investigated. For example, their behaviourin a uniform magnetic background. This is a very important effect. In ultrarelativistic heavy-ion collisions strongmagnetic fields ( B ≈ G ) can be created perpendicularly to the collision plane. Also, weaker magnetic fields( B ≈ G ) in neutron stars could have an influence in the phase transition of dense hadronic matter [20]. Aninvestigation of the first- and second-order phase transitions in a magnetic background of an extended GN-like model,including four- and six-fermion interactions, has not yet been done, up to our knowledge. We intend to perform thiswork here.Another important factor of influence on the phase transition is size restriction. The methodology to deal with thiseffect was improved in Refs. [21–24] inspired on the imaginary-time formalism, the so-called Matsubara formalism [25].Before presenting size restriction on the model as a quantum field theory on a toroidal topology, let us briefly recallthe imaginary-time formalism. It is constructed in the following way: in a (1 + 3)-dimensional Euclidean manifold,for example, the coordinates will be called ( τ, x, y, z ). To introduce temperature on the system, the Matsubaraprescription tells us to compactify the τ coordinate through the imposition of periodic or antiperiodic boundaryconditions. The imaginary-time formalism defines a theory on a space with topology Γ = S × R , where S isa circumference with length β = 1 /k B T . Here we consider a generalization in which the Matsubara prescription is ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ c ond - m a t . o t h e r] M a y extended to a spatial dimension. Specifically, we will compactify the system in both τ and z directions ( z being thesame direction of the applied magnetic field), with a compactification in the z direction of length L ; the topology ofthe system becomes Γ = ( S ) × R − . Then, as argued in [22, 24], L can be interpreted as the separation betweentwo parallel planes orthogonal to the z direction. Thus, after performing these two compactifications, we get a systemin thermal equilibrium in the form of a film. Notice that this is a particular case of a general field theory defined ona toroidal topology[24].In this paper we explore the effects coming from finite size, chemical potential and from a magnetic backgroundon a massive version of an extended fermion interacting model for a system undergoing a first- and a second-orderphase transitions. As in [26], we deal with a one-component fermionic field and we would like to investigate how thetransition temperature is affected by spatial boundaries and the magnetic field. Indeed, this is a generalization, toinclude a magnetic background, of the work developed in [27].The paper is organized as follows. In section II we present the model and fix the conventions on the parameters toensure the system stability and also the Ritus propagator for a fermion in a magnetic background. In section III, wecalculate the two Feynman diagrams used to perform corrections due to the temperature, chemical potential and sizerestriction. In section IV we perform an analysis for for both first- and second-order phase transitions. Indeed, we showthat the catalysis phenomenon occurs for a second-order phase transition, independently of the value of the couplingconstant as well as symmetry breaking at temperatures lower than the critical one. However, for the first-order phasetransition, we find a rather non intuitive phenomenon, namely, inverse symmetry breaking : there are two differentcritical temperatures. Below the smaller critical temperature, t c , we have the usual broken-symmetry phase. Abovethe higher critical temperature, t c , we have also found a broken phase. For intermediate temperatures, t c < t < t c ,we find a symmetric phase. This phenomenon of an inverse symmetry breaking has been already mentioned in theliterature [35, 39–41]. In section V we present our conclusions and notice that no transition exists below a minimalsize of the system in both first- and second-order phase transition and that density (chemical potential) has littleinfluence on the phase structure. We will use natural units c = (cid:126) = k B = 1. II. THE MODEL
We start by considering the Hamiltonian density in Euclidean space for a generalization of the massive Gross–Neveumodel which includes quartic as well as sextic fermionic interactions, H = ¯ ψ ( x )( i /D − m (cid:48) ) ψ ( x ) − λ (cid:2) ¯ ψ ( x ) ψ ( x ) (cid:3) + η (cid:2) ¯ ψ ( x ) ψ ( x ) (cid:3) , (1)where x = ( τ, x, y, z ) and we use the chiral representation for the Dirac matrices. The covariant derivative is given by D µ = ∂ µ + ieA µ , where A µ = (0 , , Bx, B = B ˆ z , B being a constant [28]. The parameter m (cid:48) can be chosen as ± m (the sign is important to ensure the stability of the system). This model is useful to studyboth first- and second-order phase transitions. To describe a first-order transition, we choose m (cid:48) = − m . On theother hand, to study a second-order transition, we fix m (cid:48) = + m .We assume a free-energy density of the Ginzburg–Landau type F = A φ ( x ) + B φ ( x ) + C φ ( x ) , (2)where φ ( x ) denotes the order parameter of the model, namely, φ ( x ) = (cid:113)(cid:10) ¯ ψ ( x ) ψ ( x ) (cid:11) β where, (cid:104)·(cid:105) β means a thermalaverage in the grand-canonical ensemble. We consider A = − m, B = − λ / , C = η /
3, where m = m ( T, µ, L, B )includes temperature, chemical potential, finite size and magnetic field corrections to the m (cid:48) parameter, defined atzero temperature. We do not compute corrections to the coupling constants.The Euclidean Feynman propagator is calculated via the Ritus method, [29, 30], S ( A ) = ω π (cid:88) s = ± ∞ (cid:88) (cid:96) =0 (cid:90) dp τ π dp z π ( / ¯ p − m (cid:48) )¯ p + m , (3)where p = p τ + p z + p + p ; p + p = ω (2 (cid:96) + 1 − s ) , s = ± ω ≡ eB the cyclotronfrequency. The natural numbers (cid:96) label the Landau levels. III. CORRECTIONS TO THE MASS PARAMETER m (cid:48) To obtain corrections due to temperature β , finite size L , chemical potential µ and external field ω , we use just twoFeynman diagrams, namely, the Tadpole e the
Shoestring , depicted in Fig. 1:
Fig. 1 -
Feynman diagrams to be considered: (a) Tadpole; (b) Shoestring.
The self-energy becomes m ( β, L, µ, ω ) = m (cid:48) + Σ ( a ) ( β, L, µ, ω ) + Σ ( b ) ( β, L, µ, ω ) , (4)where Σ represents the corresponding diagram to be calculated. A. Tadpole diagram contribution
Following the method developed in [22], we start fromΣ ( a ) ( β, L, µ, ω ) = λ tr (cid:34) ω π (cid:88) s = ± ∞ (cid:88) (cid:96) =0 (cid:90) dp τ π dp z π ( / ¯ p − m (cid:48) )¯ p + m (cid:35) . To insert temperature, chemical potential and finite size, we use the method described in Ref.[22]. In this way, weintroduce two compactifications: (cid:90) dp τ π → β + ∞ (cid:88) n = −∞ ; p τ → πβ (cid:18) n + 12 − i µβ π (cid:19) ; (cid:90) dp z π → L + ∞ (cid:88) n = −∞ ; p z → πL (cid:18) n + 12 (cid:19) . Therefore, we obtain Σ ( a ) ( β, L, µ, ω ) = λ (cid:34) ω π (cid:88) s = ± ∞ (cid:88) (cid:96) =0 + ∞ (cid:88) n ,n = −∞ − m (cid:48) βL π m × (cid:32) a ( n − b ) + a ( n − b ) + c (cid:96),s (cid:33)(cid:35) , (5)where we have defined the dimensionless quantities a = ( m β ) − = t ; a = ( m L ) − = ξ ; c (cid:96),s = [ δ (2 (cid:96) + 1 − s ) + 1] / π ; γ = µ/m ; δ = ω/m ; b = iβµ/ π − / iγ/t π − / b = − / . In terms of these dimensionless quantities, we can rewrite the expression in Eq. (5) asΣ ( a ) ( t, ξ, γ, δ ) = λ (cid:26) − δ π (cid:0) m m (cid:48) √ a a (cid:1) Z c (cid:96),s [1 , { a j } , { b j } ] (cid:27) , (6)where we have introduced the Epstein–Hurwitz inhomogeneous zeta function, Z c (cid:96),s [ ν, { a j } , { b j } ] = + ∞ (cid:88) n ,n = −∞ (cid:104) a ( n − b ) + a ( n − b ) + c (cid:96),s (cid:105) − ν , with j = 1 ,
2. Its analytical extension to the whole complex set of values on the ν − plane is given by[31] Z c (cid:96),s [ ν, { a j } , { b j } ] = π ( c (cid:96),s ) − ν √ a a Γ( ν − ν ) + 4 π ν √ a a ν ) (cid:101) Z c (cid:96),s ( ν ) , (7)where (cid:101) Z c (cid:96),s ( ν ) = (cid:88) j =1 ∞ (cid:88) n j =1 cos(2 πn j b j ) (cid:18) n j c (cid:96),s √ a j (cid:19) ν − K ν − (cid:18) πc (cid:96),s n j √ a j (cid:19) +2 ∞ (cid:88) n ,n =1 cos(2 πn b ) cos(2 πn b ) c (cid:96),s (cid:115) n a + n a ν − × K ν − πc (cid:96),s (cid:115) n a + n a . (8)Since Γ( ν − / Γ( ν ) = 1 / ( ν − ν = 1. We apply the modifiedminimal subtraction prescription to eliminate this term, so that Eq. (6) can be expressed asΣ ( a ) ( t, ξ, γ, δ ) = − (cid:18) m (cid:48) λδπ (cid:19) R (1 , t, ξ, γ, δ ) , (9)where the dimensionless constant coupling was defined as λ = λ m and the function R (1 , t, ξ, γ, δ ) is given by R ( ν, t, ξ, γ, δ ) = (cid:88) s = ± ∞ (cid:88) (cid:96) =0 + ∞ (cid:88) n =1 ( − n cosh( n γ/t ) (cid:32) πn t (cid:112) δ (2 (cid:96) + 1 − s ) + 1 (cid:33) ν − × K ν − (cid:16) n t (cid:112) δ (2 (cid:96) + 1 − s ) + 1 (cid:17) + + ∞ (cid:88) n =1 ( − n (cid:32) πn ξ (cid:112) δ (2 (cid:96) + 1 − s ) + 1 (cid:33) ν − K ν − (cid:18) n ξ (cid:112) δ (2 (cid:96) + 1 − s ) + 1 (cid:19) +2 + ∞ (cid:88) n ,n =1 ( − n + n cosh( n γ/t ) π (cid:112) δ (2 (cid:96) + 1 − s ) + 1 (cid:115) n t + n ξ ν − × K ν − (cid:112) δ (2 (cid:96) + 1 − s ) + 1 (cid:115) n t + n ξ . (10) B. Shoestring diagram contribution
An analogous calculation leads to the 2-loop contributionΣ ( b ) ( t, ξ, γ, δ ) = (cid:18) m ηδ π (cid:19) [ R (1 , t, ξ, γ, δ )] , (11)where η = η m .The total correction to the mass parameter, at first order in the coupling constants, due to temperature, finite size,chemical potential, and magnetic field, for the system is, then, m ( t, ξ, γ, δ ) = m (cid:48) − (cid:18) m (cid:48) λδπ (cid:19) R (1 , t, ξ, γ, δ ) + (cid:18) m ηδ π (cid:19) [ R (1 , t, ξ, γ, δ )] . (12)In the next section, we solve numerically Eq. (12) and describe the phase structure of the model. IV. PHASE TRANSITION
At criticality, we obtain different conditions to the first- and second-order phase transitions. For this reason, eachcase is presented separately. The modified Ginzburg–Landau free energy density is rewritten as F m ≡ f ( ϕ ) = − m ( t, ξ, γ, δ ) m ϕ − λ ϕ + η ϕ , (13)where we take the rescaled order parameter φ : ϕ = φ/m / . A. First-Order Phase Transition
To describe a first-order phase transition, we choose in Eqs. (12) and (13), η > , λ >
0. The Ginzburg–Landaufree-energy density has three minima, one of them at the origin, ϕ = 0, the other two symmetrically located withrespect to the origin. Also, we have two local maxima, symmetrically located with respect to ϕ = 0. The criticalitycondition for a first-order phase transition, is obtained by noticing that when we are at criticality, the three minima theGinzburg–Landau free-energy density, lie on the same horizontal line, which can be taken as the horizontal axis[22, 32].Mathematically, this can be represented by the conditions f = df /dϕ = 0. Then using Eq. (13) it is easily shown thatthe corrected mass, at criticality, is given by m ( t c , ξ, γ, δ ) = − (cid:18) λm η (cid:19) . (14)Eq. (12) is to be replaced into Eq. (14) and we have to solve a transcendental equation for the critical temperature, t c , in terms of the other parameters. In Fig. 2, we see the magnetic catalysis effect, i.e., the magnetic field contributesto decrease the critical temperature of the system (all figures in color in the online version). Ξ t c Fig. 2 -
Critical temperature as a function of the inverse length of the system. For the three curves, we have fixed γ = 0 . , λ = 2 . and η = 1 . . Also, we have used δ = 1 . (dashed curve); δ = 3 . (dot-dashed curve); δ = 4 . (continuous curve). We also see that as the field increases, ξ max decreases. In the other words, as the magnetic field becomes stronger,the minimal thickness of the system becomes larger, to ensure that the transition will happen.From Fig. 3, we can see the small influence that the chemical potential has on the model. Furthermore, the minimalsize of the system, ξ max , seems to be independent of the chemical potential. Ξ t c Fig. 3 -
Influence of the chemical potential on the model. In the two curves, we have fixed λ = 2 . ; η = 1 . and δ = 3 . . We have used γ = 0 . (dashed curve); γ = 0 . (full curve). From Eqs. (12) and (10), we have seen that the product
P ≡ cosh ( γn /t ) × K [( n /t ) (cid:112) δ (2 (cid:96) + 1 − s ) + 1] comes up.Since at small temperatures we can use the asymptotic formula K ν ( z ) = (cid:112) π/ − z )] / √ z , valid when | z | → ∞ , | arg z | < π , Re ν > − / γ ≤ (cid:112) δ (2 (cid:96) + 1 − s ) + 1, we get that the product P goes to zero at the limit t c →
0. Therefore, Eq.(12) is independent of γ at t c ≈ m ( t c , ξ, γ, δ ) /m = − λ/ η : thesmaller one, t c (from Fig. 4, in the bulk form, i.e., ξ ≈ L →
0, its value is t c ≈ .
76) and the larger one, t c (again from Fig. 4 and in the bulk form, t c ≈ . inverse symmetry breaking . Ξ t c Fig. 4 -
Inverse symmetry breaking phenomenon. There are two regions in which a phase transition occurs for thesame values of ( γ, δ, λ, η ) . We have fixed γ = 0 . , λ = 4 . and η = 2 . . We have used δ = 1 . . At temperatures lower than t c , we have that the inner region defined by t c is the region of symmetry breaking. Inthe region between the two curves of the Fig. 4, i.e., with temperatures in the range t c < t < t c , there is a disorderedphase. But this disordered phase does not persist. Indeed, at temperatures t > t c , we get again the ordered phase.Thus, to temperatures above t c , we also have a symmetry breaking region. As a consequence of the presence of thesetwo critical temperatures, we have two minimal thicknesses of the system that support the transition.In Figs. 5 and 6, we show the behaviour of the Ginzburg–Landau free energy of the system in bulk form. Noticethat in Fig. 5, we have the usual behaviour of a first-order phase transition, i.e., we pass from the symmetric phaseto the broken symmetry phase as the temperature decreases. On the other hand, we have the opposite behaviour inFig. 6. (cid:45) (cid:45) (cid:45) (cid:106)(cid:45) (cid:72) (cid:106) (cid:76) Fig. 5 -
Free-energy density in the bulk region (inner part of Fig. ). We observe a standard symmetry breaking. Wehave fixed γ = 0 . , λ = 4 . and η = 2 . . We have used δ = 1 . . In the full curve, we have t = 1 . . Criticality isachieved at t c ≈ . (dashed curve). Below t c , we have the broken symmetry region, as usual (dotted curve, where t = 0 . ). (cid:45) (cid:45) (cid:45) (cid:106)(cid:45) (cid:45) (cid:72) (cid:106) (cid:76) Fig. 6 -
Free-energy density in the bulk (but now we are in external part of Fig. ). We have an inverse symmetrybreaking. We have fixed γ = 0 . , λ = 4 . and η = 2 . . We have used δ = 1 . . In the full curve, we have t = 1 . .Criticality is achieved at t c ≈ . (dashed curve). Even above t c , we have a broken symmetry region (dotted curve,we fixed t = 1 . ). B. Second-Order Phase Transition
To describe a second-order phase transition, we have to fix η = 0 and choose λ < m ( t c , ξ, γ, δ ) = 0, as in the Ginzburg-Landau model. In Fig. 7 we observe that the catalysiseffect occurs again. ξ t c Fig. 7 -
Critical temperature as a function of the inverse length. In the three curves, we have fixed γ = 0 . , λ = 4 . . We have used δ = 2 . (dashed curve); δ = 3 . (dot-dashed curve); δ = 4 . (full curve). In the same way as in the case of the first-order phase transition, we notice a minimal size of the system belowwhich the second-order phase transition disappears.In Fig. 8, we observe a strong dependence of the coupling constant on the critical temperature. Ξ t c Fig. 8 -
Critical temperature as a function of the inverse length for high values of the coupling constant. We havetaken γ = 0 . , δ = 2 . . We have used λ = 2 . (full curve); λ = 4 . (dot-dashed curve). In Fig. 9, we show the critical temperature behaviour of the model taking into account the chemical potential. Ξ t c Fig. 9 -
Critical temperature as a function of the inverse length for two values of the chemical potential. We havetaken λ = δ = 2 . . We have fixed γ = 0 . (dashed curve); γ = 0 . (continuous curve). In Fig. 10, we show the Ginzburg–Landau free-energy density for the bulk region of Fig. 7. (cid:45) (cid:45) (cid:106) (cid:72) (cid:106) (cid:76)
Fig. 10 -
Free energy density for the second-order phase transition. We take δ = 2 . , λ = 4 . and γ = 0 . . For thefull curve, we have t = 1 . . The critical temperature occurs at t c ≈ . (dashed curve). In the dot-dashed curve, wehave t = 0 . . To get the critical temperature we have to numerically solve the transcendental equation m ( t c , , γ, δ ) = 0, i.e., inthe bulk region. We obtain that at temperatures above t c ≈ .
9, we have a symmetric phase. On the other hand, attemperatures below t c ≈ .
9, we have symmetry breaking.
V. COMMENTS AND CONCLUSIONS
In this paper we present a model to describe in a unified way first- and second-order phase transitions for a fermionicself-interacting system. Using results from quantum field theory on a toroidal topology and a one-component massiveextended GN model, we have done an investigation about chemical potential, finite-size and magnetic effects for bothtransitions. In the first-order phase transition, as well as in the second-order one, we see that the system must have aminimal size, L min , corresponding to a reduced inverse size, ξ max . There is no transition below this minimal thickness.This is a reasonable result, since our system corresponds to a heated film of thickness L along the z -direction. Noticethat the limiting situation L = 0 is not well defined in the formalism. We can see that the effect of the magnetic fieldis to lower the critical temperature, the so-called magnetic catalysis. We notice that the minimal size is independentof the chemical potential and that the strength of the couplings have a significant influence on the phase transition ofthe system. Namely, for higher values of the couplings, we obtain larger minimal sizes. In particular, for the first-ordertransition, higher values of λ and η lead to two minimal sizes and two regions under transition. This phenomenonin called inverse symmetry breaking which is already discussed in the literature, for instance, in [27, 34–41]. FromFig. 4, we can see that there exists an ordered phase at greater thicknesses and/or low temperatures (internal part ofFig. 4, as usual). Also, there is a transition for small thicknesses and/or high temperatures (external part of Fig. 4.In the region between the curves in Fig. 4, there is a disordered phase. In nature, the inverse symmetry breakingphenomenon has been observed in some composed systems, like the Rochelle Salt . This salt has an orthorhombicstructure (ordered phase) at temperatures below t ≈ − ◦ C and above t ≈ ◦ C . For temperatures in the range t < t < t , the Rochelle Salt has a monoclinic unit cell (disordered phase). In Refs. [42, 43] we find other systemsthat suffer the same effect;, even water, under special pressure conditions, can present inverse symmetry breaking. Wehave quantitatively studied this behaviour in the model we have considered, for the system in bulk form, in Figs. 5and 6. We have found that both transitions suffer little influence of the chemical potential. In fact, we have shownexplicitly that near t c ≈
0, there is no dependence of the model on the chemical potential, as shown in Figs. 3 and 9.In future developments, we would like to investigate whether these results hold for other backgrounds.
Acknowledgments
We thank financial support from CNPq (Brazilian agency). EBSC also thanks theCAPES/PRODOUTORAL/UNIFESSPA program. [1] Y. Nambu and G. Jona-Lasinio, Phys. Rev. , 345 (1961).[2] Y. Nambu and G. Jona-Lasinio, Phys. Rev. , 246 (1961).[3] D.J. Gross and A. Neveu, Phys. Rev. D , 2599 (1974).[5] A. Barducci, R. Casalbuoni, M. Modugno, G. Pettini, and R. Gatto, Phys. Rev. D , 3042 (1995).[6] C.-T. Hou and C.C.C. Mudry, Phys. Rev. Lett. , 186809 (2007).[7] R. Jackiw and S.-Y. Pi, Phys. Rev. Lett. , 266402 (2007).[8] G.W. Semenoff, Phys. Rev. Lett. , 2449 (1984).[9] V. Juriˇci´c, I.F. Herbut, and G.W. Semenoff, Phys. Rev. B , 081405(R) (2009).[10] B.R. Zhou, Commun. Theor. Phys. , 425 (1999).[11] A. Brzoska and M. Thies, Phys. Rev. D , 125001 (2002).[12] J.-L. Kneur, M.B. Pinto, R.O. Ramos, and E. Staudt, Phys. Rev. D , 045020 (2007).[13] C. Boehmer, U. Fritsch, S. Kraus, and M. Thies, Phys. Rev. D , 065043 (2008).[14] H. Kohyama, Phys. Rev. D , 045016 (2008).[15] J. Feinberg, Int. J. Mod. Phys. , 898 (2002).[16] A.P.C. Malbouisson, J.M.C. Malbouisson, A.E. Santana, and J.C. Silva, Phys. Lett. B , 373 (2004). [17] F.C. Khanna, A.P.C. Malbouisson, J.M.C. Malbouisson, and A.E. Santana, Phys. Rev. D , 085015 (2012).[18] L.M. Abreu, A.P.C. Malbouisson, and J.M.C. Malbouisson, Phys. Rev. D , 025001 (2011).[19] L.M. Abreu, A.P.C. Malbouisson, and J.M.C. Malbouisson, Phys. Rev. D , 065036 (2011).[20] D. Kharzeev, K. Landsteiner, A. Schmitt, and H.U. Yee (eds.) Strong Interacting Matter in Magnetic Fields , LecturesNotes in Physics , Springer-Verlag, Berlin-Heidelbeg (2013).[21] N.D. Birrell and L.H. Ford, Phys. Rev. D , 330 (1980).[22] F.C. Khanna, A.P.C. Malbouisson, J.M.C. Malbouisson, and A.E. Santana, Thermal Quantum Field Theory: AlgebraicAspects and Applications , World Scientific, Singapore (2009).[23] F.C. Khanna, A.P.C. Malbouisson, J.M.C. Malbouisson, and A.E. Santana, Ann. Phys. , 2364 (2011).[24] F.C. Khanna, A.P.C. Malbouisson, J.M.C. Malbouisson, and A.E. Santana, Phys. Rep. , 135 (2014).[25] T. Matsubara, Prog. Theor. Phys. , 351 (1955).[26] F.C. Khanna, A.P.C. Malbouisson, J.M.C. Malbouisson, and A.E. Santana, EPL , 11002 (2012).[27] L.M. Abreu, F.C. Khanna, A.P.C. Malbouisson, J.M.C. Malbouisson, and A.E. Santana, Phys. Lett. A , 2597 (2014).[28] I.D. Lawrie, Phys. Rev. Lett. , 131 (1997).[29] V.I. Ritus, Ann. Phys. , 555 (1972); Soviet Phys. JETP , 788 (1978).[30] G. Murgu´ıa, A. Raya, ´A. S´anchez, and E. Reyes, Am. J. Phys. , 700 (2010).[31] E. Elizalde, Ten Physical Applications of Spectral Zeta-Functions, Springer-Verlag, Berlin, (1995).[32] M. Le Bellac, Quantum and Statistical Field Theory , Oxford University Press, New York (1991).[33] E.B.S. Corrˆea, C.A. Linhares, and A.P.C. Malbouisson, Phys. Lett. A 377, 1984 (2013).[34] S. Weinberg, Phys. Rev. D , 3357 (1974).[35] G. Bimonte and G.Lozano, Phys. Lett. B , 248 (1996).[36] M.B. Pinto and R.O. Ramos, Phys. Rev. D , 125016 (2000).[37] S.I. Hong and J.B.Kogut, Phys. Rev. D , 085014 (2001).[38] H.C.G. Caldas, Phys. Rev. D , 105015 (2002).[39] M.B. Pinto, R.O. Ramos, and J.E. Parreira, Phys. Rev. D , 123519 (2005).[40] T.H. Phat, L.V. Hoa, N.T. Anh, and N.V. Long, Phys. Rev. D , 125027 (2007).[41] M. Sakamoto and K. Takenaga, Phys. Rev. D80