Coexistence of Multiple Phases in Magnetized Quark Matter with Vector Repulsion
aa r X i v : . [ h e p - ph ] J un Coexistence of Multiple Phases in Magnetized Quark Matter with Vector Repulsion
Robson Z. Denke
1, 2, ∗ and Marcus Benghi Pinto † Departamento de F´ısica, Universidade Federal de Santa Catarina, 88040-900 Florian´opolis, Santa Catarina, Brazil Departamento de F´ısica, Funda¸c˜ao Universidade Regional de Blumenau, 89012-900 Blumenau, Santa Catarina, Brazil
We explore the phase structure of dense magnetized quark matter when a repulsive vector inter-action, parametrized by G V , is present. Our results show that for a given magnetic field intensity( B ) one may find a value of G V for which quark matter may coexist at three different baryonicdensity values leading to the appearance of two triple points in the phase diagram which have notbeen observed before. Another novel result is that at high pressure and low temperature we observea first order transition which presents a negative slope in the P − T that is reminiscent of the solid-liquid transition line observed within the water phase diagram. These unusual patterns occur for G V and B values which lie within the range presently considered in many investigations related tothe study of magnetars. PACS numbers: 11.10.Wx, 26.60.Kp,21.65.Qr, 25.75.Nq
I. INTRODUCTION
Strongly interacting magnetized matter may be produced in non central relativistic heavy ion collisions [1] and mayalso be present in magnetars [2] reaching up about to ∼ G and ∼ G in each of these two physical situations.As far as heavy ion-collisions are concerned the presence of a strong magnetic field most certainly plays a role despitethe fact that, in principle, the field intensity should decrease very rapidly lasting for about 1-2 fm/c only [1]. Thepossibility that this short time interval may [3] or may not [4] be affected by conductivity remains under dispute.A striking property expected to occur in such extreme conditions is the so called Magnetic Catalysis (MC) phe-nomenon [5], which implies that the order parameter for the chiral transition represented by the quark-antiquarkcondensate rises as the field becomes more intense (see Ref. [6] for a review). It is then natural to ask how the ex-pected quantum chromodynamics (QCD) transitions are affected by the presence of intense magnetic fields, a questionwhich has been addressed in many recent works (see Refs. [7] for an updated discussion). To summarize the mainresults obtained in these investigations let us start by recalling that at finite temperatures and vanishing chemicalpotential both, model approximations [8–11] and lattice QCD (LQCD) evaluations [12, 13] agree that a cross over,which is predicted to occur in the the absence of magnetic fields [14], persists when strong magnetic fields are present.However, a source of disagreement between recent LQCD evaluations [13] and model predictions regards the behav-ior of the pseudocritical temperature ( T pc ), at which the cross over takes place, as a function of the magnetic fieldintensity. The LQCD simulations of Ref. [13], performed with 2 + 1 quark flavors and physical pion mass values,predict that T pc should decrease with B while early model evaluations predict an increase (see Ref. [9] and referencestherein). This problem has been recently addressed by different groups [15] which basically agree that the differentresults stem from the fact that most effective models miss back reaction effects (the indirect interaction of gluons and B ) as well as the QCD asymptotic freedom phenomenon. On the other hand, LQCD results are currently unavailableat high densities and low temperatures so that one has to rely in model approximations [9–11] which predict thatthe first order chiral transition takes place at a coexistence chemical potential value which is lower than the oneobserved for the case of unmagnetized matter leading to the phenomenon called Inverse Magnetic Catalysis (IMC)[16] (see Ref. [17] for a physically intuitive discussion of the IMC phenomenon). Further investigations performedwith the Nambu–Jona-Lasinio model (NJL) have been carried out to locate the critical end point (CP) as well as thecoexistent chemical potentials associated with the first order chiral transition [9, 11]. The results suggest that thesize of first order transition line increases as the field becomes stronger affecting the position of the (second order)CP where the first order transition line terminates. At the same time, the size and location of the coexistence regionin the presence of a magnetic field appears to oscillate around the B = 0 values [18]. Together, all these effects haveinteresting consequences for quantities which depend on the details of the coexistence region such as the surface ten-sion and may have consequences for studies related to the properties of magnetized compact stellar objects [19]. Themotivation for the present investigation stems from an early work by Ebert and collaborators [20] who recognized that ∗ Electronic address: [email protected] † Electronic address: [email protected] the MC phenomenon, associated to the filling of Landau levels, could lead to more exotic phase transition patternsas a consequence of the induced magnetic oscillations. To confirm this assumption these authors have considered awide range of the (scalar) coupling values for the two flavor NJL model at vanishing temperatures and, as expected,have observed unusual phase structures as a function of the chemical potential such as an infinite number of masslesschirally symmetric phases, a cascade of massive phases with broken chiral invariance among other features. Morerecently, this seminal study has been extended by a more systematic, and numerically accurate analysis with two andthree flavors in Refs. [21] and [22] respectively. The results confirm that for certain model parametrizations one isable to observe more than one first order phase transition, which is signaled when the thermodynamical potentialdevelops two degenerate minima at different values of the coexistence chemical potential . It is important to remarkthat, in general, weak first order transitions can be easily missed in a numerical evaluation due to the fact thatthe two degenerate minima appear almost at the same location being separated by a tiny potential barrier so thattheir study requires extra care. Physically, this corresponds to a situation where two different (but almost identical)densities coexist at the same chemical potential, temperature and pressure. Here, one of our goals is to extend theinvestigation of these cascades of first order phase transitions, observed in Refs. [20–22], when hot magnetized quarkmatter is subject to the presence of a repulsive vector channel parametrized by G V . This type of interaction providesa saturation mechanism similar to those found in effective nuclear models [24] and is known to be important for anaccurate description of quark matter at high baryonic densities [25]. As we have verified in a previous work [18],the repulsive vector coupling modifies the magnetic effects mainly at lower temperatures and plays an opposite rolecompared to B in the QCD phase diagram. As a matter of fact, the increase of the magnetic field shifts the first ordertransition to lower values of the coexistence chemical potential while a nonzero vector repulsion produces the reverseeffect. This feature has also been recently verified within the three flavor case [26] in the analysis of compact stellarobjects. Here, we shall see that the presence of a vector repulsion allows for further interesting possibilities associatedwith the chiral first order transition due to the fact that this term can stabilize intermediary density magnetic phases.Being carried out at finite temperatures the present investigation also allows for a more complete description of thecoexistence region and makes it possible to better understand the physical nature of the associated phase transitions.As a first novelty we show that within a cascade of the transitions the one which takes place at the highest pressurevalue seems to be reminiscent of the “solid-liquid” type of transition displayed by the phase diagram of water whilethe others are of the usual “liquid-gas” type (commonly observed within QCD effective theories). Motivated by thefact that the Lennard-Jones potential, which describes water, also has a repulsive part we have scanned over the G V values to search for any eventual triple point in the phase diagram of magnetized quark matter since this situationcannot be completely ruled out in the scenario considered here. The reason is that the presence of a magnetic fieldinduces the free energy to develop multiple minima while the repulsive interaction favors stability so that eventuallythree (instead of the usual two) minima could be globally degenerate leading to the coexistence of three phases. As weshall demonstrate the numerical results have confirmed our expectations so that for very particular, but yet realistic,parameter values the phase diagram of strongly interacting magnetized quark matter may indeed display triple points.The work is organized as follows. In the next section we present the free energy in the presence of a repulsivevector channel for the magnetized NJL model within the mean field approximation (MFA) framework. In Sec. III wediscuss how a cascade of the usual type of first order phase transitions, with two degenerate minima, takes place atfinite temperatures. In the same section we extend the analysis to the finite temperature domain in order to drawthe phase diagrams in the T − µ , T − ρ B and P − T planes. Section IV is devoted to the study of unusual first orderphase transitions where the free energy develops more than two degenerate minima allowing for the existence of triplepoints. Our conclusions and final remarks are presented in Sec. V. II. NJL MAGNETIZED FREE ENERGY WITH A REPULSIVE VECTOR INTERACTION
The QCD interaction between quarks can be effectively described by the well-known NJL theory [27] which repro-duces, at lower energies, the main features of chiral symmetry breaking. In the usual two flavor version the samecoupling constant, G S , sets the interaction strength in both the scalar and pseudo-scalar channels. However, in finitedensity investigations the model produces more realistic results if an additional repulsive vector channel, parametrizedby G V , is introduced. In this case, the corresponding Lagrangian density [25, 28] can be written as L = ¯ ψ ( iγ µ ∂ µ − m ) ψ + G S [( ¯ ψψ ) + ( ¯ ψiγ ~τ ψ ) ] − G V ( ¯ ψγ µ ψ ) , (2.1) This fact has also been recently observed to arise within another effective four fermion theory described by the 2 + 1 d Gross-Neveumodel [23]. where m = m u ≃ m d is the bare quark mass. In order to derive the thermodynamical potential within the MFA thequadratic interaction terms appearing in the above Lagrangian are linearized by the introduction of the mean fieldsexpressed in terms of the scalar condensate, φ = h ¯ ψψ i , and the quark number density, ρ = h ψ + ψ i ( ¯ ψψ ) ≃ φ ¯ ψψ − φ and ( ¯ ψγ ψ ) ≃ ρψ + ψ − ρ , (2.2)where quadratic terms in the fluctuations have been neglected while the pseudo-scalar term does not contribute atthis level. Then, in the case of symmetric quark matter ( µ = µ u = µ d ) the theory is described by L = ¯ ψ ( iγ µ ∂ µ − M + ˜ µγ ) ψ − ( M − m ) G S + ( µ − ˜ µ ) G V , (2.3)where the effective quark mass, M , and the effective chemical potential, ˜ µ , are determined upon applying the cor-responding minimization conditions, δ Ω /δM = 0 and δ Ω /δ ˜ µ = 0. Integrating over the fermionic fields yields thethermodynamical potential Ω = ( M − m ) G S − ( µ − ˜ µ ) G V + i Z d p (2 π ) ln[ − p + M ] . (2.4)One can then include the effects of a static magnetic field and a thermal bath to this dense quark matter system byapplying the following replacements [29] to Eq. (2.4): p → i ( ω ν − iµ ) ,p → p z + (2 n + 1 − s ) | q f | B , with s = ± , n = 0 , , ... , Z + ∞−∞ d p (2 π ) → i T | q f | B π ∞ X ν = −∞ ∞ X n =0 Z + ∞−∞ dp z π , where ω ν = (2 ν + 1) πT ( ν = 0 , ± , ± ... ) represents the Matsubara frequencies for fermions. The Landau levels arelabelled by n while the absolute values of quark electric charges | q f | are | q u | = 2 e/ | q d | = e/ e = 1 / √ . Then, following Ref. [30] we can write the thermodynamical potential asΩ = ( M − m ) G S − ( µ − ˜ µ ) G V + Ω vac + Ω mag + Ω med , (2.5)where the vacuum contribution reads Ω vac = − N c N f Z d p (2 π ) p p + M , (2.6)and, as usual, can be regularized by a non-covariant sharp cut-off, Λ, yieldingΩ vac = N c N f π (cid:26) M ln (cid:20) (Λ + ǫ Λ ) M (cid:21) − ǫ Λ Λ[Λ + ǫ Λ2 ] (cid:27) , (2.7)where ǫ Λ represents the energy √ Λ + M at the cutoff momentum value Λ. We remark that in Refs. [31, 32] theauthors choose a smooth cut off to avoid unphysical oscillations, which appear when the pairing interaction is included Our results are expressed in Gaussian natural units where 1 MeV = 1 . × G . because the sharp cut off limits the allowed momenta. In the present work, no superconducting phase (that wouldrequire the pairing gap equation to be solved) is used and hence we do not face the problem of unphysical solutions.The magnetic contribution of the thermodynamical potential is given byΩ mag = − d X f = u N c ( | q f | B ) π (cid:26) ζ ′ [ − , x f ] −
12 ( x f − x f ) ln x f + x f (cid:27) , (2.8)whre we have used the definition x f = M / (2 | q f | B ) and the derivative of the Riemann-Hurwitz zeta function ζ ′ ( − , x f ) = dζ ( z, x f ) /dz | z = − (see the appendix of Ref. [30] for detailed steps). Finally, the term Ω med repre-sents the in-medium contributionΩ med = − N c π d X f = u ∞ X k =0 α k ( | q f | B ) Z ∞−∞ dp z π n T ln [1 + e − ( E p +˜ µ ) /T ] + T ln [1 + e − ( E p − ˜ µ ) /T ] o , (2.9)where α k = 2 − δ k and E p = p p z + 2 k | q f | B + M . A similar expression for the magnetized thermodynamicalpotential at G V = 0 was originally obtained in Ref. [33] where Schwinger’s proper time approach has been used.Solving δ Ω /δM = 0 and δ Ω /δ ˜ µ = 0 we get the following coupled self consistent equations M = m − G S φ , (2.10)and ˜ µ = µ − G V ρ , (2.11)Note also that, in principle, one should have two coupled gap equations for the two distinct flavors: M u = m u − G S ( h ¯ uu i + h ¯ dd i ) and M d = m d − G S ( h ¯ dd i + h ¯ uu i ) where h ¯ uu i and h ¯ dd i represent the quark condensates which differ,due to the different electric charges. However, in the two flavor case, the different condensates contribute to M u and M d in a symmetric way and since m u = m d = m one has M u = M d = M . The quantities φ = φ vac + φ mag + φ med and ρ appearing in Eqs. (2.10) and (2.11) are given by φ vac = − M N c N f π (cid:26) Λ ǫ Λ − M (cid:20) (Λ + ǫ Λ ) M (cid:21)(cid:27) , (2.12) φ mag = − M N c π d X f = u | q f | B (cid:20) ln Γ( x f ) −
12 ln (2 π ) + x f −
12 (2 x f −
1) ln ( x f ) (cid:21) , (2.13) φ med = M N c π d X f = u ∞ X k =0 α k ( | q f | B ) Z ∞−∞ dp z π E p [ n p (˜ µ, T ) + ¯ n p (˜ µ, T )] , (2.14)and ρ = N c π d X f = u ∞ X k =0 α k ( | q f | B ) Z ∞−∞ dp z π [ n p (˜ µ, T ) − ¯ n p (˜ µ, T )] , (2.15)where n p (˜ µ, T ) = 11 + e ( E p − ˜ µ ) /T and ¯ n p (˜ µ, T ) = 11 + e ( E p +˜ µ ) /T , (2.16)represent, respectively, the Fermi occupation number for quarks and antiquarks.At T = 0 the relevant in-medium terms appearing in Ω, M and ˜ µ can be written asΩ med = − N c π d X f = u k f,max X k =0 α k | q f | B ( ˜ µk F ( k, B ) − s f ( k, B ) ln (cid:20) ˜ µ + k F ( k, B ) s f ( k, B ) (cid:21)) , (2.17) φ med = d X f = u k f,max X k =0 α k M N c ( | q f | B )2 π ln (cid:20) ˜ µ + k F ( k, B ) s f ( k, B ) (cid:21) , (2.18)and ρ = d X f = u k f,max X k =0 α k | q f | BN c π k F ( k, B ) , (2.19)where k F ( k, B ) represents the Fermi momentum, k F = p ˜ µ − s f ( k, B ) , and s f ( k, B ) = p M + 2 | q f | kB . Theupper Landau level (or the nearest integer) is defined by k f,max = (cid:22) ˜ µ − M | q f | B (cid:23) . (2.20)To obtain numerical results we must now fix the model parameters and, as usual, the cut-off value together the otherparameters G S , m are chosen to reproduce the phenomenological values [28] for the pion mass ( m π ≃
140 MeV),the pion decay constant ( f π ≃
93 MeV) and the quark condensate ( h ¯ ψψ i / ≃
250 MeV). Here, we consider the setΛ = 590 MeV, G S Λ = 2 .
435 and m = 6 . G V poses and additional problem since this quantity should be fixed using the ρ meson mass which, ingeneral, happens to be higher than the maximum energy scale set by Λ. At present, the vector term coupling G V cannot be determined from experiments and lattice QCD simulations (LQCD) but eventually, the combination ofneutron star observations and the energy scan of the phase-transition signals at FAIR/NICA may provide us somehints on its precise numerical value. Therefore, at the present stage the vector coupling G V is usually taken to bea free parameter whose accepted values lie within the range 0 . G S − . G S [34–36]. It is worth to point out thatthe explicit use of G V within the NJL model can be avoided provided that the evaluations be performed beyond thelarge- N c (or MFA) level. In this case two loop (exchange like) terms bring finite N c corrections to the pressure suchas ( G S /N c ) ρ so that the same type of physics can be observed with one less parameter [37]. III. CASCADE OF FIRST ORDER CHIRAL TRANSITIONS WITH TWO COEXISTING DENSITIES
Within the usual first order transition scenario the free energy develops a pair of degenerate global minima definingtwo distinct values for the effective quark masses as well as for the quark number density. Usually, when B = 0, onlyone first order chiral transition takes place so that, for a given temperature, a unique value for the coexistence chemicalpotential exists. However, as already emphasized, the presence of a magnetic field causes an oscillatory behavior whichmay induce more exotic patterns like the appearance of a cascade of first order transitions such as the ones studiedin Refs. [20–22] where only the case of vanishing temperatures, in the absence of repulsion, has been considered.One of the main outcomes of these applications is that, for some phenomenologically acceptable parametrizations,several transitions are needed to move from the vacuum phase to the (approximately) chirally symmetric phase. Morerecently, Allen, Pagura and Scoccola [38] have observed a similar situation when considering a a generalized versionof the two flavor NJL model, which includes a G V term, but again at vanishing temperatures. In this section we willextend the investigation performed at G V = 0 to the finite temperature case to gain extra insight on the effects of B and G V by exploring the phase diagram in planes such as T − µ , T − ρ B , and P − T . This exercise will allow us notonly to review some of the main aspects related to the appearance of a cascade of first order phase transitions as theones studied in Refs. [20–22, 38] but also to explore more realistic finite temperature situations.Let us first recall that, within this model, the order parameter associated with the chiral transition is ( h ¯ uu i + h ¯ dd i ) / B and G V over M since this quantity also determines the behavior of the associated EoS. The left panelof Fig. 1, which was originally obtained in Ref. [18], displays the effective quark mass, at T = 0, as a function of µ for G V = 0 . G S and different values of the magnetic field. The figure indicates that, in the vacuum, the value of M tends to increase with B which is in accordance with the magnetic catalysis effect [5]. Also, due to the filling ofLandau levels, one observes the typical de Haas-van Alphen oscillations which are more pronounced for small valuesof B . Only the segments of the curves where dM/dµ < G V = 0 . G S ) we observe that for eB = 5 m π and eB = 8 m π some of these solutions arestable leading to intermediate transitions. For example, at eB = 5 m π one sees a first transition at µ = 388 .
55 MeVwhen the mass jumps from 409 MeV to 313 MeV. This is followed by another transition from M = 312 . ææææææææ ææææææææææ
300 350 400 450 5000100200300400500 Μ @ MeV D M @ M e V D eB = = m Π eB = m Π eB = m Π eB = m Π eB = m Π ææ ææ ææææææ ææææ ææ ææææ ææ ææææ ææ ææ
300 350 400 450 5000100200300400500 Μ @ MeV D M @ M e V D G V = V = S G V = S G V = S G V = S G V = S FIG. 1: Cascade of first order transitions. Left panel: The effective quark mass, M , at T = 0, as a function of µ for differentvalues of eB at G V = 0 . G S . Right panel: The effective quark mass, M , at T = 0, as a function of µ for different values of G V at eB = 5 m π . In both cases the thick lines represent stable solutions to the gap equation. M = 190 MeV at µ = 390 MeV. To complete the “cascade” of (three) first order phase transitions one observes afinal transition from M = 150 MeV to 59 MeV at µ = 402 .
65 MeV. The numerical results illustrated by the figureclearly display the effective mass oscillatory behavior showing that at relatively weak magnetic fields we observe manyoscillations of quark mass values (due to the many Landau levels available). When the field becomes stronger, thequantity of oscillations in the quark effective mass is reduced since there are less Landau levels available. The leftpanel also shows that, for a fixed value of G V , the transition to the chiral phase (lowest mass value) occurs at lowerchemical potential coexistence values as B increases in accordance with the IMC phenomenon. The right panel showsthat, for a fixed B , the transition to the chiral phase occurs at higher chemical potential coexistence values as G V increases (for a complete discussion on these issues the reader is referred to Ref. [18]). For our present purposes it isimportant to note how the insertion of a repulsive vector coupling G V between quarks brings stability ( dM/dµ ≤ FIG. 2: Quark matter phase diagrams in the T − µ plane (left panel) and in the P − T plane (right panel) for eB = 5 . m π and G V = 0 . G S . In both figures the continuous thin line represents the case with G V = 0 for comparison. When G V = 0 . G S a cascade of three first order transitions appear at very low temperatures. Each first order transition line terminates at theindicated critical point. Figure 2 shows the T − µ and P − T phase diagrams for the case of a field strength eB = 5 . m π and a vectorcoupling magnitude of G V = 0 . G S , which is the value recently suggested by Sugano et al. [36], compared to the G V = 0 case. At temperatures close to zero, one observes a splitting of the G V = 0 first order transition line intothree lines occurring at different µ values as one could expect from the discussion related to figure 1. As alreadyemphasized this exotic scenario may also appear in the G V = 0 case provided that one uses a different parametrizationfor G S as discussed in Refs. [20–22]. Here, we have instead adopted a rather canonical value for G S so that only onetransition occurs if G V = 0. This standard choice is well suited to achieve our goal since the role played by the vectorchannel itself can be further highlighted. The left panel of this figure shows that, as expected [39], G V weakens thefirst order transition and shifts the coexistence chemical potential to high values. Note also that for G V = 0 the thirdtransition (thick continuous line) is quickly washed out for temperatures higher than ≈ .
75 MeV. The right panelshows the phase diagram in the physically more intuitive P − T plane. It is interesting to note that the transitionterminating at CP3, which is associated with the Landau level jump k d = 1 →
2, has a negative ∆ P/ ∆ T slope(related the Clausius-Clayperon equation) while the ones terminating at CP CP
2, associated with k d = 0 → G V = 0. Therefore, apart from the usual “liquid-gas”type of transition (positive slope) it appears that the combined presence of G V and B may also induce a transition ofthe “solid-liquid” type observed in the water phase diagram (negative slope) which, as far as we know, has not beenobserved before within QCD motivated models.Next, in the left panel of Fig. 3 we show the phase diagram in the T − ρ B plane which could not be analyzed inthe previous applications at T = 0 [20–22, 38]. The figure shows that for the chosen G V and B a values a shrinkageof the coexistence regions also takes place. At T = 0 the figure shows a coexistence between the following pairs ofdensities: ρ B = 0 and ρ B = 0 . ρ (at µ = 388 .
55 MeV); ρ B = 0 . ρ and ρ B = 1 . ρ (at µ = 390 MeV); and ρ B = 2 ρ and ρ B = 2 . ρ (at µ = 402 .
65 MeV) whereas for the G V = 0 case the dilute phase occurs at vanishing density andthe dense phase occurs at ρ B = 3 ρ . As usual the baryonic density is defined as ρ B = ρ/ ρ = 0 .
17 fm − .According to Refs [19, 40] one can then expect that the surface tension between the coexistence phases will be lowerwhen G V = 0. The right panel of Fig. 3 shows a three dimensional plot displaying the P − T − ρ B phase diagramwith the Andrews isotherms that define the equation of state. This figure indicates that the cascade of three firstorder transition observed so far always refer to the coexistence of two phases occurring at distinct pressures. FIG. 3: Left Panel: Coexistence phase diagram in the T − ρ B plane for eB = 5 . m π and G V = 0 . G S . The case G V = 0(thin continuous line) is shown for comparison. Right panel: The phase diagram in the P − T − ρ B space. In both panels it ispossible to distinguish the three independent first order phase transitions defined by the coexistence of two distinct densitiesat the same pressure. Up to this point we have seen that, due to the LL filling procedure, chiral symmetry restoration may take place via The choice of this particular value will become clear in the sequel. successive first order transitions between different magnetized phases. From the free energy perspective this successionhappens because, depending on the values of the couplings, the presence of a magnetic field may induce the appearanceof more than two minima so that by varying a control parameter, such as µ , one may observe multiple transitions.This scenario can happen either when G V = 0 and G S is relatively weak (leading to effective quark masses such as M ≈
200 MeV) [20–22] or when G V = 0 but G S leads to more standard M values as we have shown. At this point itis crucial to note that, with the values of G V and B considered so far, we have only observed the occurrence of twodegenerate global minima signaling the usual type of first order phase phase transition for a given value of µ while anyeventual extra minima remain local. Then, at another chemical potential value a global minimum turns into a localone and vice versa leading us to observe a cascade of transitions where only two densities coexist for a given value of µ . The results found in this section suggest that perhaps there are certain values of G V at which three degenerateglobal minima will emerge signaling the coexistence of three different phases at the same T, µ and B values as we shalldiscuss next. IV. FIRST ORDER CHIRAL TRANSITIONS WITH THREE COEXISTING DENSITIES
The previous discussion shows that the B and G V values considered so far produce a cascade of first order transitionswhere, at a given coexistence µ value, only a pair of degenerate (global) minima coexist with other (local) minima.However, for other parameter values, it may be energetically preferable that one of these local minima becomes aglobal one so that the ground state is triply degenerated. With this motivation let us now scan G V and eB aroundthe values G V = 0 . G S and eB = 5 m π . Fig. 4 shows the thermodynamic potential evaluated at eB = 5 . m π and G V = 0 . G S at various temperatures. To facilitate the understanding of the unusual phase diagrams which appearin the sequel let us discuss the results shown in this figure with some detail. Starting with the T = 0 case oneobserves (left panel) three coincident minima occurring at the same null pressure coexistence chemical potential value µ T P = 388 .
05 MeV where the subscript stands for “triple point 1”.
FIG. 4: Thermodynamic potential at G V = 0 . G S and eB = 5 . m π for some selected temperatures. Left panel: at T = 0the potential is triply degenerate at the coexistence chemical potential value µ TP = 388 .
05 MeV. At T = 10 MeV thepotential is doubly degenerate at two different coexistence chemical potential values, µ Ia = 387 MeV and µ Ib = 387 .
65 MeV.At T = 21 . µ TP = 384 .
82 MeV while at T = 30 MeV it is doubly degenerate at µ II = 381 .
08 MeV. Right panel: At T = 10 MeV this triple coexistence at the same chemical potential no longer survives and one observes thedissociation into two separate first order transitions occurring at two distinct coexistence chemical potential values µ Ia = 387 MeV and µ Ib = 387 .
65 MeV, respectively which signals a “cascade” of two subsequent first order phasetransitions. At a higher temperature, near T T P = 21 . µ T P = 384 .
82 MeV. Above this temperature, we observe the usualbehavior of the chiral with only one first order transition which evolves to the critical point as T increases. The rightpanel of Fig. 4 shows a final transition line starting at T = 0 and µ III = 404 .
70 MeV and terminating at the CPlocated at T CP = 4 .
75 MeV and µ CP = 401 .
60 MeV.Notice that in the previous section we have deliberately shown results for the cases G V = 0 . eB = 5 MeVas well as for G V = 0 . eB = 5 . G V = 0 . eB = 5 . µ forthe T, G V and B values considered in Fig. 4. With this aim we present Fig. 5 where one may observe the transitionsdiscussed in connection with the former figure from a different perspective. The squares locate the first triple point,the triangles locate the second triple point while the ordinary type of first order transitions is denoted by the dots. FIG. 5: Sequence of plots showing first order chiral transitions for eB = 5 . m π and G V = 0 . G S . The squares and trianglesindicate triply degenerate transitions while the dots indicate doubly degenerate ones. From the sequence shown, it becomes evident how the triple point
T P
1, related to three different mass values(marked by the squares), determines the same chemical potential µ T P = 388 .
05 MeV when T = 0. Note that thethe three mass values, satisfying the gap equation, differ by an approximately equal amount (close to 100 MeV).Still at T = 0 but at a higher chemical potential value one observes the occurrence of another (ordinary) transitionsignaled by two different mass values (marked by the dots) at µ III = 404 .
70 MeV. Then, rising the temperature to T = 10 MeV one sees that the triple coexistence, observed at T = 0, decouples into two ordinary first order transitionsoccurring at two different, but yet very similar, chemical potentials given by µ Ia = 387 MeV and µ Ib = 387 .
65 MeV.At this temperature the first order line starting at µ III and T = 0 has already vanished. Note also that the low massvalue occurring at µ Ia and the high mass value occurring µ Ib are almost identical (the difference amounts to about5 MeV). Near T = 21 . T P
2) at µ T P marked by the triangles. In this case the high and the intermediate mass values becomealmost identical but differ substantially from the low value (by almost 200 MeV). Above this temperature, as the T = 30 MeV panel suggests, only the ordinary scenario takes place until the first order transition line ends at a criticalpoint, CP
1, located at T CP = 57 .
65 MeV and µ CP = 361 .
90 MeV.0We are now in position to map all these transitions into phase diagrams such as the ones shown in Fig. 6 in orderto display the phase boundaries in the T − µ and P − T planes. FIG. 6: Phase diagrams in the T − µ plane (Left Panel) and in the P − T plane (Right Panel) for B = 5 . m π and G V = 0 . G S .At T = 0, both diagrams show a triple point (TP1) splits into two common dual-phase coexistence lines as the zoomed areasshow. At T TP = 21 . P/ ∆ T slope. Analyzing these figures we verify that the usual phase diagrams are drastically modified from the ordinary case at G V = 0 since we now have a first-order coexistence line originating from a triple point at ( T = 0 , µ T P = 388 .
05 MeV)and then splitting into two new branches of ordinary first order transitions, as shown in the zoomed region, to finallymerge again at a second triple point ( T T P = 21 . , µ T P = 384 .
82 MeV). Then, for higher temperatures thetransition always follows the usual pattern until the line ends a critical point (CP1). The figure also shows anordinary first order phase transition line which starting at ( T = 0 , µ III = 404 .
70 MeV) and terminating at CP2( T CP = 4 .
75 MeV , µ CP = 401 .
60 MeV). The P − T allows to further understand these two different first order linesby observing the dP/dT slope in each case. The diagram shows that the line terminating at CP1 has a positive slopejust like in the well known “liquid-gas” type of transition while the line terminating at CP2 presents a negative slopewhich is reminiscent of the “solid-liquid” transition occuring in the water phase diagram which does not appear tohave been reported previously (at least within the context of strongly interacting systems). The left panel of Fig. 7displays the coexistence phase diagram in the T − ρ B plane while the right panel of the same figure shows a threedimensional plot displaying the P − T − ρ B phase diagram with the Andrews isotherms.It is well known that at zero temperature this type of model, as well as the quark meson model, typically predictphase coexistence between the vacuum and the dense quark phase, similar to the “liquid-gas” transition. However,we now observe that when a vector repulsion and a magnetic field are present it is possible that three phases coexistfor particular parameter values. As Fig. 7 shows, for eB = 5 . , m π and G V = 0 . G S , one observes the coexistencebetween phases with densities ρ B /ρ = 0 , . , and 1 .
75 which is a rather interesting result since the vanishing densityrepresents the vacuum, the intermediate density value is very close to that of ordinary nuclear matter while the thirdis close to the values this model predicts for the dense quark phase. Eventually, by scanning over parameter values onemay force the intermediate density to take place at ρ B /ρ = 1 so that this could be identified as nuclear matter. As thetemperature increases, the three-phase coexistence vanishes and one observes a decoupling into two separate branches Ia and Ib , which are related to two first order transitions taking place at distinct pressure values. This bifurcationin two coexistence branches creates a stability island between the regions Ia and Ib . Again, at T T P = 21 . T P II ends at an ordinary critical point CP III inthe left panel of Fig. 7. At T = 0 the coxistence densities for this region are ρ B /ρ = 2 . ρ B /ρ = 2 .
6. Then,when the temperature reaches the critical value T CP = 4 .
75 MeV this coexistence region, which corresponds to a“solid-liquid” type of transition, terminates at CP B and G V values which give rise to multiple densities coexisting in the same transition chemical potential µ . Let us1 FIG. 7: Phase diagram in the T − ρ B plane (Left Panel) and EOS isotherms in the P − T − ρ B space (Right Panel) for the B = 5 . m π and G V = 0 . G S case. Filled squares and triangles indicate the triple points and filled circles indicate the criticalend points in accordance with the previous figures. now call G cV the critical vector coupling value at which three degenerate global minima appear for a given B value.Then, when mapping the parameter space of ( B, G cV , µ ) values which result in a three-phase coexistence at T = 0 weencounter the oscillatory behavior shown in Fig. 8. FIG. 8: Magnetic field dependence of the critical values, G cV , related to the transitions ( k u , k d ) → (0 ,
0) and ( k u , k d ) → (0 , T = 0. The dot labels the values used in our analyzes and the open circles mark the excluded G V = 0 case. Each critical vector coupling G cV can be associated with the emergence of a new intermediary (stable) value of theorder parameter. Depending on the field strength, more than one intermediary Landau level configuration ( k u , k d )can appear and each one has its own G cV value. So, it must exist an infinite set of critical values G cV associated withthe infinite number of intermediary states ( k u , k d ) in the limit of B → G V → ∞ . The figure only shows thefirst two possibilities characterized by the transitions ( k u , k d ) → (0 ,
0) and ( k u , k d ) → (0 , G V = 0 − . G S and eB = 2 − m π ). Note that for high fields eB > ∼ m π the emergence of triple points is suppressedsince only the LLL is occupied. We have tested coupling values above and below the critical line observing that when G V < G cV only the traditional case with one first order transition line emerges. Then, when G V = G cV the multiplecoexistence becomes possible and a new first order transition lines starts from the original one at T = 0 as shown2in Figs. 6 and 7. Finally, when G V > G cV the chiral symmetry restoration takes place via more than one first ordertransition line in a cascade mode as discussed in Section III. We close this section by remarking that we have alsoobserved the coexistence of four phases but this is resctrited to the T = 0 case only. V. CONCLUSIONS
We have investigated how the presence of a magnetic field and a repulsive vector interaction may influence thephase diagram of strongly interacting matter generating unusual transition patterns associated to first order chiraltransitions. In the first part of the present study we have considered parameter values which produce a cascade offirst order transitions similar to the ones analyzed in Refs. [20–22] at vanishing temperatures. We have taken a stepforward by incorporating the vector interaction as well as by pushing the evaluations to the finite temperature domainin order to better understand the physical nature of such transitions. Mapping the transition into the P − T planehas allowed us to observe that the transition which takes place at high pressure has a negative dP/dT slope which isreminiscent of the “solid-liquid” transition observed in the water phase diagram while the remaining (low pressure)transitions have a positive slope and therefore resemble the “liquid-gas” transition which is usually observed withineffective quark models. Having in mind that, due to the inherent oscillations caused by the filling of Landau levels,the magnetic field induces the free energy to develop multiple minima while the repulsive vector interaction favorsstability we have scanned over B and G V values to check for the possibility of observing three degenerate globalminima which would then lead to the existence of a triple point. Our expectation was confirmed by the numericalinvestigation and we were able to find, for a given value of B , a certain value of G V so that three (instead of the usualtwo) phases coexist which, as far as we know, has not been observed before. In contrast to the “solid-liquid” type oftransition, which may be find when canonical parametrizations are used, the existence of triple points is only possiblewhen very specific values of G V are chosen. Finally, we point out remark that the coexistence of multiple phases hasalso been recently observed by Manso and Ramos [41] within the 2 + 1 d Gross-Neveu model despite the fact that arepulsive term has not been considered. Instead the authors have considered a tilted magnetic field observing that theparallel component tends to stabilize the free energy (just as G V in our case) so that three degenerate minima werealso observed. Together with our observations this result leads us to conclude that the phase diagram of magnetizedmatter may display the coexistence of multiple phases if the dynamics contains terms which bring stability to the freeenergy. Acknowledgments
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