# Competition and duality correspondence between inhomogeneous fermion-antifermion and fermion-fermion condensations in the NJL model

###### Abstract

We investigate the possibility of spatially homogeneous and inhomogeneous chiral fermion-antifermion condensation and superconducting fermion-fermion pairing in the (1+1)-dimensional model by Chodos et al. [ Phys. Rev. D 61, 045011 (2000)] generalized to continuous chiral invariance. The consideration is performed at nonzero values of temperature , electric charge chemical potential and chiral charge chemical potential . It is shown that at , where and are the coupling constants in the fermion-antifermion and fermion-fermion channels, the -phase structure of the model is in a one-to-one correspondence with the phase structure at (called duality correspondence). Under the duality transformation the (inhomogeneous) chiral symmetry breaking (CSB) phase is mapped into the (inhomogeneous) superconducting (SC) phase and vice versa. If , then the phase structure of the model is self-dual. Nevertheless, the degeneracy between the CSB and SC phases is possible in this case only when there is a spatial inhomogeneity of condensates.

## I Introduction

In recent years much attention has been devoted to the investigation of dense quark (or baryonic) matter. The interest is motivated by the possible existence of quark matter inside compact stars or its creation in heavy ion collisions. In many cases, as e.g. in the above-mentioned heavy ion collision experiments the quark matter densities are not too high, so the consideration of its properties is not possible in the framework of perturbative weak coupling QCD. Usually, different effective theories such as the Nambu – Jona-Lasinio (NJL) model, model etc. are more adequate in order to study the QCD and quark matter phase diagram in this case. A variety of spatially nonuniform (inhomogeneous) quark matter phases related to chiral symmetry breaking, color superconductivity, and charged pion condensation phenomenon etc. (see, e.g., 3+1 ; nakano ; Tatsumi:2014cea ; osipov ; nickel ; maedan ; Heinz:2013eu ; pisarski ; miransky ; zfk ; incera ; Anglani:2013gfu and references therein) was predicted in the framework of NJL-like models at rather low values of temperature and baryon density. (A recent interesting review on current model results for inhomogeneous phases in (3+1)-dimensional systems is presented in buballa .)

Moreover, the phenomenon of spatially nonuniform quark pairing was also intensively investigated within different (1+1)-dimensional models which can mimic qualitatively the QCD phase diagram. In this connection, it is necessary to mention Gross-Neveu (GN) type models with four-fermion interactions, symmetrical with respect to the discrete or continuous chiral transformations (in the last case we shall use for such models the notation NJL) and extended by baryon and isospin chemical potentials. In the framework of these models both the inhomogeneous chiral thies ; thies2 ; basar and charged pion condensation phenomena were considered gubina ; gubina2 . (In order to overcome the prohibition on the spontaneous breaking of a continuous symmetry in (1+1) dimensions, the consideration is usually performed in the limit of large , where is the number of quark multiplets.) Inhomogeneous phases in some one-dimensional organic materials and nonrelativistic Fermi gases were recently studied, correspondingly, in caldas and Roscher in terms of (1+1)-dimensional theories with four-fermion interaction.

Among a variety of GN-type models, there is one which describes competition between quark-antiquark (or chiral) and quark-quark (or superconducting) pairing at nonzero temperature and quark number chemical potential chodos . Originally, the model was called for to shed new light on the color superconductivity phenomenon in real dense quark matter. Moreover, in chodos the consideration is performed in the supposition that chiral and superconducting condensates are spatially homogeneous. In this case it was shown there that if , where and are the coupling constants in the chiral and superconducting channels, correspondingly, then at rather high values of quark number chemical potential the superconducting phase is realized in the system.

Since in the true ground state of any system with nonzero density the condensates could be inhomogeneous, the aim of the present paper is to investigate such a possibility. Namely, we shall study the phase structure of the extended model chodos (which is symmetric with respect to continuous chiral group), assuming that both quark-antiquark and quark-quark condensates might have a spatial inhomogeneity in the form of the Fulde-Ferrel single plane wave ansatz ff , for simplicity. Moreover, in addition to the particle (or quark) number chemical potential , we also introduce into consideration the chiral charge chemical potential , which is responsible for a nonzero chiral charge density , i.e. to a nonzero imbalance between densities of left- and right-handed quarks (fermions). In literature, there are some investigations of QCD-like effective theories with and chemical potentials, related to a possible parity breaking phenomena of dense quark gluon plasma (see, e.g., andrianov ; andrianov2 ). Moreover, it was recently established that in heavy ion collision experiments a nonzero chiral charge density can be induced, leading to the so-called chiral magnetic effect ruggieri ; huang . So, we hope that studying the above- mentioned (1+1)-dimensional NJL model with two chemical potentials, and , one can shed new light on the new phenomena of the dense baryonic matter.

The paper is organized as follows. In Sec. II the duality property of the model is established. It means that there is a correspondence between properties (phase structure) of the model at and . After obtaining the thermodynamic potential (TDP), we will first investigate it in the next Sec. III under the supposition that both superconducting and chiral condensates are spatially homogeneous. In this section a rather rich -phase structure of the model is established at . In addition, we will show here that there is an invariance of the TDP with respect to a duality transformation (when , and superconductivity chiral symmetry breaking). As a result, the -phase structure of the model at is a dual mapping of the phase portrait at . In Sec. IV the phase structure of the model is investigated in the assumption that both condensates might be spatially inhomogeneous. Then at the chiral density wave phase is realized for arbitrary values of and . On the other hand, at there is an inhomogeneous superconducting phase in the whole plane. Note, that there is a dual correspondence between these phases. Finally, Sec. V presents a summary and some concluding remarks. The discussion of some technical problems are relegated to four Appendixes.

## Ii The model and its thermodynamic potential

### ii.1 The duality property of the model

Our investigation is based on a (1+1)-dimensional NJL-type model with massless fermions belonging to a fundamental multiplet of the flavor group. Its Lagrangian describes the interaction in the fermion–antifermion and scalar fermion-fermion channels,

(1) | |||||

where is a fermion number chemical potential (conjugated to a fermion, or electric charge, number density) and is an axial chemical potential conjugated to a nonzero density of chiral charge , which represents an imbalance in densities of the right- and left-handed fermions ruggieri . As it is noted above, all fermion fields () form a fundamental multiplet of the group. Moreover, each field is a two-component Dirac spinor (the symbol denotes the transposition operation). The quantities (), , and in (1) are matrices in the two-dimensional spinor space,

(2) |

It follows from (2) that . Clearly,
the Lagrangian is invariant under transformations from the internal
group, which is introduced here in order to make it possible to perform all the
calculations in the framework of the nonperturbative large- expansion
method. Physically more interesting is that the model (1) is invariant under
transformations from the group, where is the
fermion number conservation group,
(), and is the group of continuous chiral transformations,
(). ^{1}^{1}1Earlier in
chodos a similar model symmetric under discrete chiral
transformation was investigated. However, only the possibility for the
spatially homogeneous chiral and difermion condensates was considered there. In
our paper, the invariance of the model considered by Chodos et al. chodos
is generalized to the case of continuous chiral symmetry in order to study the
inhomogeneous chiral condensates in the form of chiral spirals (or chiral
density waves). The linearized version of Lagrangian (1) that contains
auxiliary scalar bosonic fields , , ,
has the following form:

(3) | |||||

(Here and in what follows, summation over repeated indices is implied.) Clearly, the Lagrangians (1) and (3) are equivalent, as can be seen by using the Euler-Lagrange equations of motion for scalar bosonic fields which take the form

(4) |

One can easily see from (4) that the (neutral) fields and
are real quantities, i.e. ,
(the superscript symbol denotes the
Hermitian conjugation), but the (charged) difermion scalar fields
and are Hermitian conjugated complex quantities, so
and vice versa. Clearly, all the fields
(4) are singlets with respect to the group. ^{2}^{2}2Note that
the field is a flavor singlet, since the representations of
this group are real. If the scalar difermion field has a nonzero
ground state expectation value, i.e. , the Abelian
fermion number symmetry of the model is spontaneously broken down.
However, if then the continuous chiral symmetry of the
model is spontaneously broken.

Before studying the thermodynamics of the model, we want first of all to consider its duality property. To this end, it is very useful to form an infinite set composed of all Lagrangians (3) when the free model parameters and take arbitrary admissible values, i.e. at arbitrary fixed values of coupling constants and chemical potentials . Then, let us perform in (3) the so-called Pauli-Gursey transformation of spinor fields pauli , accompanied with corresponding simultaneous transformations of auxiliary scalar fields (4),

(5) |

Taking into account that all spinor fields anticommute with each other, it is easy to see that under the action of the transformations (5) each element (auxiliary Lagrangian) of the set is transformed into another element of the set according to the following rule

(6) |

i.e. the set is invariant under the field transformations (5). Owing to the relation (6) there is a connection between properties of the model when free model parameters and vary in different regions. Due to this reason, we will call the relation (6) the duality property of the model.

### ii.2 The thermodynamic potential at

We begin an investigation of a phase structure of the four-fermion model (1) using the equivalent semibosonized Lagrangian (3). In the leading order of the large- approximation, the effective action of the considered model is expressed by means of the path integral over fermion fields:

where

(7) |

The fermion contribution to the effective action, i.e. the term in (7), is given by

(8) |

The ground state expectation values , , etc. of the composite bosonic fields are determined by the saddle point equations,

(9) |

In vacuum, i.e. in the state corresponding to an empty space with zero particle density and zero values of the chemical potentials and , the above-mentioned quantities , etc. do not depend on space coordinates. However, in a dense medium, when and/or , the ground state expectation values of bosonic fields (4) might have a nontrivial dependence on the spatial coordinate . In particular, in this paper we will use the following ansatz:

(10) |

where and are real constant quantities. (It means that we
suppose for and the chiral spiral (or chiral
density wave) ansatz, and the Fulde-Ferrel ff single plane wave ansatz
for difermion condensates.) In fact, they are coordinates of the global minimum
point of the thermodynamic potential .
^{3}^{3}3Here and in what follows we will use a conventional notation ”global”
minimum in the sense that among all our numerically found local minima the
thermodynamical potential takes in their case the lowest value. This does not
exclude the possibility that there exist other inhomogeneous condensates,
different from (10), which lead to ground states with even lower values of
the TDP. In the leading order of the large -expansion it is defined by the
following expression:

which gives

(11) | |||||

where .
In principle, one way to evaluate the path integral in (11) is to
extend the technique of the paper basar , where a more simple
model with single quark-antiquark
channel of interaction was investigated, to the case under
consideration, i.e. to the GN model (1) with additional
superconducting interaction of quarks. The rigorous method of
basar is based on finding the resolvent function corresponding
to the Hamiltonian of the system. However, technically it is very
difficult to use this approach in the framework of the model (1). So,
in order to simplify the problem we first perform in (11)
Weinberg (or chiral) transformation of spinor fields weinberg ,
and . Since Weinberg transformation of
fermion fields does not change the path integral measure in (11),
^{4}^{4}4Strictly speaking, performing Weinberg transformation of
fermion fields in (11), one can obtain in the path integral
measure a factor, which however does not depend on the dynamical
variables , , , and . Hence, we ignore this
unessential factor in the following calculations. Note that only in the
case when there is an interaction between spinor and gauge fields
there might appear a nontrivial, i.e. dependent on dynamical
variables, path integral measure, generated by Weinberg
transformation of spinors. This unobvious fact follows from the
investigations by Fujikawa fujikawa . we see
that the system is reduced by the Weinberg transformation from a
spatially modulated to a uniform one; i.e. we obtain the following
expression for the thermodynamic potential:

(12) | |||||

where

(13) |

The path integration in the expression (12) is evaluated in Appendix
A ^{5}^{5}5In Appendix A we consider for simplicity the case
; however the procedure is easily generalized to the case with .
(see also kzz for similar integrals), so we have for the TDP

(14) |

where are presented in (67) and superscript “un“ denotes the unrenormalized quantity. Note, the TDP (14) describes thermodynamics of the model at zero temperature . In the following we will study the behavior of the global minimum point of this TDP as a function of dynamical variables vs the external parameters and in two qualitatively different cases: (i) the case of homogeneous condensates, i.e. when in (10) and (14) both and are supposed from the very beginning, without any proof, to be zero, and (ii) the case of spatially inhomogeneous condensates, i.e. when the quantities and are defined dynamically by the gap equations of the TDP (14). Moreover, the influence of temperature on the phase structure is also taken into account.

## Iii The homogeneous case of the ansatz (10) for condensates: and

### iii.1 Dual invariance of the TDP

In the present section we suppose that all the condensates are spatially homogeneous, i.e. we put in the ansatz (10) and in the TDP (14) and . So, the TDP is considered a priori as a function of only two variables, and ( and are treated as external parameters). Note that the subject and results of the section are largely preparatory for considering the main purpose of the paper, i.e. to clarify (see the next section) a genuine ground state structure of the model in the framework of the inhomogeneous ansatz (10) for condensates.

Taking into account the expressions (67) for , we obtain the unrenormalized TDP in this case:

(15) |

where

(16) | |||||

Expanding the right-hand side of (16) in powers of , one can obtain an equivalent expression for . Namely,

(17) | |||||

We would like to stress once more that there is an identical equality between the expressions (16) and (17).

Obviously, the function
(15) is symmetric with respect to the transformations and/or
. Moreover, it is invariant under the transformations
and/or . ^{6}^{6}6Indeed, if simultaneously with
and/or transformations we perform in the integral
(15) the following change of variables, and/or , then one can easily see that the expression (16) remains intact.
Hence, without loss of generality, we restrict ourselves by the constraints:
, , , and . However, there is one
more discrete transformation of the TDP (15), which leaves it invariant.
It follows from a comparison between (16) and (17). Indeed, if in
(16) for the transformations
and are performed simultaneously, then the expression
(17) will be obtained, which is equal to the original expression
(16) for . So the TDP (15) is invariant with respect
to the following duality transformation :

(18) |

Taking into account that the TDP (15) is symmetric with respect to and/or , it is possible to conclude that the dual invariance of the TDP (15) is a particular realization of the dual property (6) of the initial model. Suppose now that at some fixed particular values of the model parameters, i.e. at , and , , the global minimum point of the TDP lies at the point . Then it follows from the dual invariance (18) of the TDP that the permutation of the coupling constant and chemical potential values (i.e. at , and , ) moves the global minimum point of the TDP to the point . In particular, if in the original model with , and , the global minimum point of the TDP lies at the point (as a result, in this case the continuous chiral symmetry is spontaneously broken down), then in the model with , and , the global minimum point of the TDP lies at the point and the symmetry is spontaneously broken. The duality correspondence between these two particular cases of the original model (1) was discussed in thies1 . (Even earlier, a special case with of the duality between chiral symmetry breaking and superconductivity phenomena was considered in the framework of the simplest two-dimensional Gross-Neveu model Ojima:1977cg ; Vasiliev:1995qp .) Hence, a knowledge of a phase structure of the model (1) at is sufficient to construct, by applying the duality transformation (18), the phase structure at ; i.e. in the model under consideration there is a duality correspondence between chiral symmetry breaking and superconducting phases.

To investigate the TDP (15) it is necessary to renormalize it.

### iii.2 The vacuum case: ,

First of all we will consider the renormalization procedure and the phase structure of the model in the vacuum case, i.e. when . Putting and in (15), we have in this case the following expression for the unrenormalized effective potential (in vacuum TDP is usually called an effective potential):

(19) |

Integrating in (19) over (see Appendix B in gubina2 for similar integrals) and cutting the integration region, , one obtains the regularized effective potential :

(20) |

Since this expression diverges at , it is necessary to renormalize it, assuming that and have appropriate dependencies. It is easy to establish that if

(21) |

where and are some finite and cutoff independent parameters with dimensionality of mass, then integrating in (20) over and ignoring there an unessential term - one can obtain in the limit a finite and renormalization invariant expression for the effective potential,

(22) |

Now two remarks are in order. First, since and can be considered as free model parameters, it is clear that the renormalization procedure of the NJL model (1) is accompanied by the dimensional transmutation phenomenon. Indeed, there are two dimensionless bare coupling constants in the initial unrenormalized expression (19) for , whereas after renormalization the effective potential (22) is characterized by two dimensional, and , free model parameters. Moreover, and are renormalization invariant quantities, i.e. they do not depend on the normalization points. (The physical sense of and will be discussed below.) Second, the transposition of the bare coupling constants before renormalization is equivalent, as it is clear from (21), to the transposition after renormalization procedure. Hence, the vacuum effective potential (22) of the model is invariant with respect to the duality transformation (18) which now, i.e. in vacuum, looks like , .

Note also that the effective potential written in the form (22) has a singularity at , which is really fictitious. Indeed, the expression (22) may be presented in an equivalent form that is more convenient for both numerical and analytical investigations:

(23) |

where

(24) |

The expression (23) is now a smooth function at . As it is clear from (23), instead of two massive and parameters the renormalized model can be characterized by one massive and one dimensionless parameter and , respectively. (In this case only the partial dimensional transmutation phenomenon takes place.) Just this set of parameters, i.e. and , was used in early investigations of the initial model (1) at chodos . In spite of the fact that the dual invariance (18) of the effective potential in the form (23), i.e. its symmetry with respect to simultaneous transformations and , is not so evident as in the form (22), in the following we will treat the model properties in terms of the parameters and as well.

So, if , i.e., as is easily seen from (24) and (21), at or , the global minimum of the effective potential (23) lies at the point . This means that if interaction in the fermion-antifermion channel is greater than that in the difermion one, then the chiral symmetry of the model is spontaneously broken down and fermions acquire dynamically a nonzero Dirac mass, which is equal just to the free model parameter . Further, in order to establish the phase structure of the model (or, equivalently, to find the global minimum point of the function ) at , i.e. at , we do not need a straightforward analytical (or numerical) study of the function (23) on the extremum. In this case it is enough to take into account the dual invariance (18) of the TDP (15) (at it is reduced to a symmetry of the effective potential with respect to simultaneous permutations , ) and conclude (see also the discussion just after (18)) that at the effective potential (23) has a global minimum at the point , where . Since in this case only the difermion condensate, which is equal to , is nonzero, the fermion number symmetry is spontaneously broken and the superconducting phase is realized in the model. Hence, the parameter is a Majorana mass of fermions, which appears dynamically in superconducting phase of the model.

### iii.3 The case , and

Taking into account the expression (83) (see Appendix B), in this case the unrenormalized TDP (15) can be presented in the following form

(25) |

where quasiparticle and quasiantiparticle energies , and , , respectively, are presented in (72). It is shown in Appendix B (see the text below formula (80)) how one can find the asymptotic expansion of the integrand in (25) at . As a consequence of this prescription we have obtained the asymptotic expansions (81) and, as a result, the following expansion:

(26) |

It means that the integral in (25) is an ultraviolet (UV) divergent, so we need to renormalize the TDP . Using the momentum cutoff regularization scheme, we obtain

(27) | |||||

(28) | |||||

where is given in (20). Note that the leading terms of the asymptotic expansion (26) do not depend on and . So the quantity

(29) |

has the same asymptotic expansion (26) at . Hence, the integral term in (28) is a convergent one, and all UV divergences are located in the first term . The UV divergences are eliminated, if the dependencies (21) of the bare coupling constants and are supposed. In this case we have from (28) at the following expression for the renormalized TDP:

(30) |

where is the TDP (effective potential) (22)-(23) of the model at and . Let us denote by the global minimum point (GMP) of the TDP (30). Then, investigating the behavior of this point vs and it is possible to construct the -phase portrait (diagram) of the model. A numerical algorithm for finding the quasi(anti)particle energies , , , and is elaborated in Appendix B. Based on this, it can be shown numerically that GMP of the TDP can never be of the form . Hence, at arbitrary fixed values of and , i.e. at arbitrary values of (24), it is enough to study the projections and of the TDP (30) to the and axes, correspondingly. Taking into account the relations (85) and (86) for the sum at or , it is possible to obtain the following expressions for these quantities,

(31) | |||||

(32) |

(Details of the derivation of these expressions are given in Appendix C.) Now, to find the GMP of the whole TDP (30) and, as a consequence, to obtain the phase structure of the model, it is sufficient to compare the minimal values of the functions (31) and (32). Recall that, up to an unessential constant, each of the functions and is just a well-known TDP of the usual massless Gross-Neveu model at zero temperature and nonzero chemical potential. It was investigated, e.g., in Klimenko:1986uq . So, one can conclude that at () the GMP of the function (of the function )) lies at the point (at the point ). Whereas at (at ) the GMP is at the point (). Moreover, the corresponding minimal values are the following:

(33) |

Comparing the least values (33) of the TDPs (31) and (32) for different values of the chemical potentials and , it is possible to obtain the -phase portrait of the model, which consists of only three phases, the chiral symmetry breaking phase, the superconducting phase and, finally, symmetrical phase. Moreover, it is evident that in the CSB phase the GMP of the TDP (30) has the form , and in the SC phase it lies at the point , whereas in the symmetrical phase the least value of the TDP (30) is reached at the point . Note that the phase structure of the model depends essentially on the relation between and . Indeed, let us first suppose that . In this case the typical -phase portrait of the model is presented in Fig. 1. It is evident that the region of the figure corresponds to the symmetrical phase of the model. Moreover, in the region (in the region ) of the figure the CSB phase (the SC phase) is arranged. The competition between CSB and SC phases takes place in the region . Namely, the critical curve of Fig. 1 is defined by the equation , i.e. by the equation

(34) |

The curve divides this region into two subregions. To the left of the CSB phase is arranged, whereas to the right of we have the SC phase. Furthermore, it is clear from (34) and (33) that it is possible to obtain an exact analytical expression for ,

(35) |

In a similar way it is possible to construct a -phase portrait of the model when (the typical -phase portrait is presented in Fig. 2). The critical curve of the figure is given by the relation

(36) |

Finally, if , then the typical -phase portrait of the model is given in Fig. 3.

Suppose that the values of and , for which the phase portrait of Fig. 2 is drawn, are obtained by rearrangement of the corresponding , values for which Fig. 1 is depicted (and vice versa). For example, let us assume that , <