Finite size effects in the Gross-Neveu model with isospin chemical potential
The properties of the two-flavored Gross-Neveu model in the (1+1)-dimensional spacetime with compactified space coordinate are investigated in the presence of the isospin chemical potential . The consideration is performed in the limit , i.e. in the case with infinite number of colored quarks. It is shown that at ( is the length of the circumference ) the pion condensation phase is realized for arbitrary small nonzero . At finite values of , the phase portraits of the model in terms of parameters and are obtained both for periodic and antiperiodic boundary conditions of the quark field. It turns out that in the plane there is a strip which lies as a whole inside the pion condensed phase. In this phase the pion condensation gap is an oscillating function vs both (at fixed ) and (at fixed ).
pacs:11.30.Qc, 12.39.-x, 21.65.+f
It is well-known that QCD is a fundamental theory of strong interactions both in the vacuum and in hot and/or dense baryonic matter. However, it can be successfully used only in the region of high energies, temperatures and densities (or chemical potentials), where a weak-coupling expansion is applicable. Away from this region, different nonperturbative methods or effective theories such as chiral effective Lagrangians as well as Nambu – Jona-Lasinio type models (see, e.g., the papers ebert_volkov (); volkov (); hatsuda (); alford () and references therein), are usually employed for the consideration of light meson physics, phase transitions in dense quark matter, etc. In particular, motivated by the fact that in heavy-ion collisions and compact stars the hadronic matter is isotopically asymmetric, different QCD-like effective models were studied at nonzero isospin chemical potential son (); frank (); ek (); andersen (); abuki (). There the charged pion condensation phenomenon, which is generated if is greater than the pion mass , was also considered.
In all above mentioned papers the effective models are i) field theories in usual (3+1)-dimensional spacetime, and ii) they are employed for the description of QCD at rather low energies and densities. At the same time there is another class of theories that can be used as a laboratory for a qualitative consideration of QCD at arbitrary energies. These are the so-called Gross-Neveu (GN) type models, i.e. two-dimensional quantum field theories with four-fermion interactions gn (); VSMT (); ft (). Renormalizability, asymptotic freedom as well as the spontaneous breaking of chiral symmetry (in the vacuum) are the most fundamental features that are inherent both for QCD and all GN type models. In addition, the GN phase portrait in terms of baryon chemical potential vs temperature resembles qualitatively to a great extent the QCD phase diagram wolff (); kgk1 (); barducci (); chodos (); thies (). Due to their relative simplicity in the leading order of a large -expansion ( is a number of colored quarks), it is very convenient to use GN models for considering such a phenomenon of dense QCD as color superconductivity chodos (); zhou () and to elaborate new nonperturbative methods of quantum field theory okopinska (); osipov (); kgk (). Moreover, the influence of the space compactification on chiral symmetry breaking both in the vacuum () kim () and in dense baryon matter () vshivtsev () was studied in terms of GN models (see also the appropriate papers beneventano (); gamayun (); abreu ()).
Before investigating different physical effects relevant to a real (3+1)-dimensional world in the framework of two-dimensional GN models, let us recall that there is a no-go theorem forbidding the spontaneous breaking of continuous symmetries in two dimensions coleman (). However, at present time it is well understood (see, e.g., the discussion in VSMT (); barducci (); chodos (); thies ()) that in the limit this no-go theorem does not apply. This makes it possible to study symmetry breaking effects in terms of GN models as well, but only in the leading order of the -expansion, where most low dimensional theories are exactly solvable. In this sense, for , low dimensional quark models are physically more tractable and appealing than at finite .
In the present paper the pion condensation phenomenon is investigated in the framework of the two-dimensional GN model with two massless quark flavors. In particular, we shall study the influence of the finiteness of the system size on this phenomenon. So our consideration is performed in a spacetime with non-trivial topology, i.e. on the manifold with compactified space coordinate, and the GN model is extended by an isospin chemical potential (for simplicity, we put ). Obviously, the latter issue is motivated by the physics of compact stars, where pion condensation might be realized as a consequence of the isotopic asymmetry of baryon matter. Since all the calculations are carried out on the basis of the leading order of -expansion (i.e. in the case ) we expect that all conclusions concerning the pion condensation phenomenon, caused by a spontaneous breaking of the continuous isospin symmetry, remain qualitatively valid for real QCD.
Ii The case of spacetime
ii.1 The model and its thermodynamic potential
We consider a two-dimensional model which describes dense quark matter with two massless quark flavors ( and quarks). Its Lagrangian has the form
where the quark field is a flavor doublet ( or ) and color -plet () as well as a two-component Dirac spinor (the summation in (1) over flavor, color, and spinor indices is implied); () are Pauli matrices; the baryon chemical potential in (1) is responsible for the non-zero baryon density of quark matter, whereas the isospin chemical potential is switched on in order to study properties of quark matter at nonzero isospin densities (in this case the densities of and quarks are different). Evidently, the model (1) is a generalization of the two-dimensional Gross-Neveu model gn () with a single massless quark color -plet to the case of two quark flavors and additional chemical potentials. As a result, we have in the case under consideration a more complicated chiral symmetry group. Indeed, at apart from the global color SU() symmetry, the Lagrangian (1) is invariant under transformations from the chiral group. However, at this symmetry is reduced to , where is the third component of the isospin operator (here and above the subscripts mean that the corresponding group acts only on the left, right handed spinors, respectively). Evidently, this symmetry can also be presented as , where is the isospin subgroup and is the axial isospin subgroup. Quarks are transformed under these subgroups as and , respectively. 111 Recall for the following that , .
The linearized version of the Lagrangian (1), which contains composite bosonic fields and , has the following form (in what follows, we use the notation for the quark chemical potential):
From the Lagrangian (2) one gets the equations for the bosonic fields
Obviously, the Lagrangian (2) is equivalent to the Lagrangian (1) when using the equations (3). Furthermore, it is clear from (3) and footnote 1 that the bosonic fields transform under the isospin and axial isospin subgroups in the following manner:
Starting from the theory (2), one obtains in the leading order of the large -expansion (i.e. in the one-fermion loop approximation) the following path integral expression for the effective action of the bosonic and fields:
is a normalization constant. The quark contribution to the effective action, i.e. the term in (5), is given by:
In (6) we have used the notation , where is the unit operator in the -dimensional color space and
is the Dirac operator, which acts in the flavor-, spinor- as well as coordinate spaces only. Using the general formula , one obtains for the effective action the following expression
where the Tr-operation stands for the trace in spinor- (), flavor- () as well as two-dimensional coordinate- () spaces, respectively. Using (8), we obtain the thermodynamic potential (TDP) of the system:
where the - and fields are now -independent quantities, and in the round brackets of (9) just the momentum space representation, , of the Dirac operator appears. Evidently, , where the summation over all four eigenvalues of the 44 matrix is implied and
The TDP is symmetric under the transformations and/or . Hence, it is sufficient to consider only the region . In this case, one can integrate in (11) over with the help of the formula
(which is valid up to an infinite constant independent of quantities , ) and obtain:
(To get the second line in (14) we used the relations and .) In what follows we are going to investigate the -dependence of the global minimum point of the function vs . To simplify the task, let us note that both the quasiparticle energies (12) and hence the TDP (14) depend effectively only on the two combinations and of the bosonic fields, which are invariants with respect to the group, as is easily seen from (4). In this case, without loss of generality, one can put in (14), and study the TDP as a function of only two variables, and . Then the global minimum point of the TDP ,
is the solution of the system of gap equations
where , , and . Evidently, the coordinates and of the global minimum point of the TDP (15) supply us with two order parameters (gaps), which are proportional to the ground state expectation values of the form and , respectively. If the gap is nonzero, then in the ground state of the model the axial isospin symmetry (at ) is spontaneously broken down. Moreover, if the gap , then in the ground state, corresponding to the phase with charged pion condensation, the isospin symmetry is spontaneously broken down.
ii.2 Pion condensation: the case of ,
Since at , the phase structure of different GN models was reasonably well studied both in two dimensions wolff (); barducci (); chodos (); thies () and in three dimensions zhou2 () (in the last case the four-fermion theories are also renormalizable), in this subsection we shall study for simplicity the model (1) only at zero quark chemical potential, i.e. at , but . The corresponding TDP will be denoted as and can be obtained from (15):
where and is the TDP of the system in the vacuum, i.e. at . In the vacuum the TDP is usually called effective potential:
It is easily seen that both the TDP (17) and the effective potential (18) are formally ultraviolet (UV) divergent quantities. So, a few words are needed about the renormalization procedure of the initial model. It is well known that all four-fermion theories of the type (1) are renormalizable in two dimensional spacetime gn (). Moreover, in the leading order of the large expansion only the coupling constant should be renormalized in order to obtain finite (renormalized) expressions for different quantities (see, e.g., kgk1 ()). It means that the bare coupling constant of the model (1) depends on the cutoff parameter , , in such a way that all UV divergences, arising from loop integrations when , are compensated by corresponding terms of . As a consequence, in the limit one must necessarily obtain finite expressions for physical quantities. Of course, different renormalization procedures result in different expressions for the bare coupling constant . However, physical consequences of the theory do not depend on the concrete renormalization scheme. Taking this last remark into account, let us next discuss how to obtain finite renormalized expression for the TDP (17). We see here two ways. On the one hand, one could find an expression for the bare coupling constant such that the UV divergence, arising from the integral in the first line of (17), would be compensated by the term with . Evidently, in this case depends both on the cutoff and . However, we find it more convenient to consider the second way. In this case, one should first of all note that the integral in the second line of (17) is a convergent quantity, and the whole UV divergence is located in the effective potential . Hence, there is a possibility to remove UV divergences using a bare coupling constant which does not depend on . Namely, let us choose
where is the dynamical mass of quarks in the vacuum (for more details, see Appendix A). Then, substituting (19) into (18) and restricting there the range of integration by using the cutoff parameter , it is possible to obtain for the expression (moreover, we omit an inessential infinite constant independent of ):
Since might be considered as a free model parameter, it follows from (19)-(20) that the renormalization procedure of the GN model is accompanied by the dimensional transmutation phenomenon. Indeed, in the initial unrenormalized expressions both for and (see (17) and (18), respectively) the dimensionless coupling constant is present, whereas after renormalization the effective potential (20) is characterized by a dimensional free model parameter .
Due to the relation (20), one can show that the gap equations for the renormalized TDP (17) might have no more than three different solutions. Two of them, and , are present at arbitrary values of , whereas the third one, , appears only at . However, for arbitrary a global minimum point of the TDP lies at the point . This means that in the model (1) the isospin symmetry is always broken down and a charged pion condensate which is equal to the quark mass in the vacuum, is created if .
Since in the vacuum case () chiral symmetry is spontaneously broken down in the model (1), there must exist three massless Nambu–Goldstone bosons which are pions, i.e. . So, we have proved that in the framework of the model (1) the pion condensation phase is realized at , where is the pion mass in the vacuum. Just the same phase structure is predicted by QCD at , son (). In contrast, in the framework of (3+1)-dimensional NJL-type models the pion condensation is not allowed for sufficiently high values of the isospin chemical potential frank (); ek (); andersen (). This fact supports the statement made in the Introduction that the NJL approach is only valid at rather small energies (chemical potentials). Moreover, we have once more demonstrated that in the leading order of the large -expansion the two-dimensional GN models are a quite good theoretical laboratory for qualitative QCD investigations. So we are in a position to believe that the results obtained in the next sections are also inherent to QCD.
Iii The case of spacetime and
In the present section we continue the investigation of the charged pion condensation, this time under the influence of the finite volume occupied by the system. This is obviously a reasonable task, since all physical effects take place in restricted space regions. The consideration of the problem is significantly simplified in the framework of the two-dimensional model (1) at , which is again justified by its similarity to QCD. So we put a system with Lagrangian (1) into a restricted space region of the form (here is the space coordinate). It is well known that in this case the consideration is equivalent to the investigation of the model in a spacetime with nontrivial topology and with quantum fields, satisfying some boundary conditions of the form
where , is the length of the circumference , and the variable means the path along it. Below, we shall use only two values of the parameter : for periodic boundary conditions and for the antiperiodic one.
As a consequence, to obtain the thermodynamic potential of the initial system placed in the restricted domain and at , one must simply replace the integration in (17) by an infinite series, using the rule:
Moreover, instead of it is necessary to use the effective potential of the model in the vacuum (see Appendix A). As a result, the TDP (17) will be replaced by the corresponding expression for the spacetime of the form , i.e.
where , and the function is defined in (46). In what follows, it will be convenient to use the dimensionless quantities
where is the dynamical quark mass in the vacuum. Moreover, since the phase structure of the model in the two particular cases , and , was already considered in section II.2 and in the Appendix A, we will now investigate the phase structure only at .
iii.1 The case of periodic boundary conditions
where is the Euler’s constant poisson (). We consider the TDP (25) as a function of two variables, . Moreover, are free parameters there. Since the information about the phase structure of the model in the case of the periodic boundary conditions is contained in the global minimum point of the function (25) vs , it is first of all necessary to study the gap equations and then to investigate the behavior of the global minimum point vs parameters . In particular, it is possible to show that for each fixed point of the plane (with and ) the global minimum point of the TDP (25) might be located at two different points only, i) and ii) , where the nonzero gap is the solution of the equation . The point i) corresponds to the symmetric phase of the model (without charged pion condensation). On the other hand, if the global minimum of the function (25) is situated at the point ii), then in the ground state of the model the isospin symmetry is spontaneously broken down, and the pion condensation takes place. Let us denote by the critical curve which separates the region of the plane with symmetric phase from the points , corresponding to the pion condensed phase of the model. Since in each point of the curve there is a phase transition of the second order from the symmetric phase to the pion condensed one and vice versa, the gap must vanish on this curve. So the critical curve is defined by the following equation
To represent the curve in the plane , it is convenient to divide this plane into an infinite set of regions :
In accordance with the division (27), the critical curve can also be presented as a set of pieces, . Obviously, each piece of the whole critical curve lies inside the corresponding -th region and obeys the following equation ()
which is valid only at . Here is the logarithmic derivative of the Euler’s function poisson (). Before drawing the critical curve , we would like to point out one its peculiarity. Using the well-known property of the function, , as well as the periodicity of , the equation (29) can be reduced to the following one:
where and . Since the absolute minimum of the function from (30) corresponds to the point , each branch of the critical curve lies to the right of the vertical line (in the plane ), where , i.e. . All the branches of the critical curve as well as the phase portrait of the initial model in terms of are presented in Fig. 1 (left picture). Clearly, there is a strip which lies, as a whole, inside the region, corresponding to the pion condensed phase.
In the right picture of Fig. 1 the behavior of the pion condensation gap vs is depicted at . It is easily seen that this quantity oscillates as a function of . However, the amplitude of this oscillations is a rapidly decreasing function of when . Similar oscillations of different physical quantities such as gaps, critical curves, particle densities etc vs were also observed in some NJL-type models with one compactified space coordinate, but in a qualitatively alternative case with nonzero baryonic chemical potential vshivtsev1 (). Moreover, oscillating phenomena as functions of curvature are inherent to NJL models in the Einstein universe, i.e. in the curved spacetime of the form tyukov (). In Fig. 2 the behavior of the gap vs is depicted at (left picture) and (right picture). Concerning this type of oscillations of the gap , it is necessary to note first of all that its period is equal to . Moreover, it is clear from Fig. 2, and this fact is supported by numerical calculations, that the amplitude of the oscillations of the quantity vs is a very slowly decreasing function of . Finally, it is evident that the smaller , the smaller the amplitude of this -oscillations of the gap .
iii.2 The case of antiperiodic boundary conditions
The critical curve which divides the parameter plane into a region with symmetric phase and the region, corresponding to a pion condensed phase, is now defined by the following equation
As in the case with periodic boundary conditions, for solving the equation (32) it is convenient to represent the parameter -plane as the union of regions, , where
Accordingly, in this case the critical curve is composed of different pieces, i.e. , where is the part of , arranged in the corresponding region . Obviously, the equation for is just the equation (32) with omitted absolute value symbols. However, the equations for () look like
where and for , whereas for . Note that the equation (36) coincides with (30) except for the different dependence of the variable . In Fig. 3 (left picture) the first several branches of the whole critical curve , which divides the plane into a region with pion condensed phase (the number 2 in the figure) and a region corresponding to a symmetric phase (the number 1 in the figure), are represented. Note that the strip of the plane belongs to the region 2 with the pion condensation phase.
In the right picture of Fig. 3 as well as in Fig. 4 the oscillating behavior of the pion condensation gap vs and, correspondingly, vs is depicted. The properties of these oscillations are the same as in the periodic case. Namely, at fixed the gap is a quickly damping oscillating function of when , whereas at fixed the gap oscillates at with a very weak damping.
Iv Summary and discussion
In the present paper we have studied the phase structure of a two-dimensional GN model at nonzero isospin chemical potential and in the spacetime with nontrivial topology, when the space coordinate is compactified into a circumference of a finite length . The consideration is performed in the leading order of the large- expansion technique. 222 It should be noted that the problem of the IR-behavior of the correlation function of quantum fluctuations in two-dimensional QFT models was extensively discussed in literature with relation to the Coleman-Mermin-Wagner theorem. One may mention the papers on the 2-dimensional GN model VSMT (); barducci (); chodos (); thies (), where this problem has been investigated and it was demonstrated that for the limit of infinite this theorem is not valid, and hence spontaneous symmetry breaking may take place.
It turns out that in the case with the pion condensed phase is realized in the model at arbitrary nonzero values of . In this phase the corresponding order parameter, the pion condensate , does not depend on the isospin chemical potential and is equal to , i.e. to the dynamical quark mass in the vacuum. The same phase structure at occurs in the chirally symmetric QCD, where pions are massless particles, so one more common property is found which is shared both by the GN model and QCD. As a result, the assurance that the finite size () effects of the GN model are also inherent to compactified QCD at is raised.
If is finite, then the phase portraits of the model in terms of and are found for the case of periodic (see Fig.1) and antiperiodic (see Fig. 3) boundary conditions. Among the most interesting properties of these phase diagrams is the fact that the strip lies as a whole inside the pion condensed phase. We have shown also that the pion condensed gap is an oscillating function vs both (at fixed ) and (at fixed ). The same is true for other thermodynamic quantities of the model such as pressure, particle densities etc, and is inherent also to the (3+1)-dimensional NJL models with curved spacetimes tyukov () or spacetimes with non-trivial topology vshivtsev1 ().
One more interesting issue should also be mentioned. Although in the present paper the spatially uniform pion condensation was assumed for simplicity, however, for a sufficiently high values of , the pion superfluidity with inhomogeneous condensate might be realized in isotopically asymmetric and spatially infinite quark matter systems jin (). A detailed investigation of this possibility in the case of finite space volume is outside the scope of this paper and should be left for further studies in the framework of different QCD-like models including the GN model.
Two of the authors (V.Ch.Zh. and A.V.T.) are grateful to Prof. M. Mueller-Preussker for his kind hospitality during their stay in the particle theory group at the Institute of Physics of Humboldt-University, where part of this work has been done, and also to DAAD for financial support. This work was also supported in part by DFG-grant 436 RUS 113/477.
Appendix A Effective potential in the vacuum ()
Note that in the vacuum and in the spacetime, the expression for the effective potential of the initial model (1) can be found starting from the TDP (11) at , where without loss of generality it is possible to put ():
Here the second equality is obtained with the help of formula (13). Since in (37) the last integral is an UV divergent one, we regularize it by cutting off the integration region, i.e. supposing that . The effective potential , as a whole, must be a finite quantity at . So the UV divergence of the integral term in (37) must be compensated by the term with the bare coupling constant , which, of course, has to be a -dependent quantity. To find an appropriate expression for , let us recall that in the vacuum the chiral symmetry is necessarily broken down in the framework of the model (1), and quarks acquire a nonzero dynamical mass which is a nontrivial solution of the gap equation . Taking into account this circumstance, one can immediately obtain from the gap equation the expression (19) for the bare coupling . Substituting it in (37), it is possible to get for the effective potential (20) of the initial model in the vacuum and in the spacetime. (More details about the above renormalization procedure for are presented, e.g., in kgk (); vshivtsev ()).
Now let us find the effective potential of the model (1), when the spacetime has a nontrivial topology of the form and quark fields obey the most general boundary conditions (21). In this case one can start from the equation (37), in which it is necessary to perform the euclidian rotation () and then use the transformations according to the rule (22). As a result, we have