SSpatially modulated chiral condensates
Stefano Carignano
INFN, Laboratori Nazionali del Gran Sasso, Via G. Acitelli, 22, I-67100 Assergi (AQ), ItalyE-mail: [email protected]
Abstract.
I discuss some properties of spatially dependent chiral condensates, focusing onone-dimensional modulations. After briefly introducing a generic formalism for studyinginhomogeneous solutions in dense quark matter, I consider a plane-wave and a sinusoidal ansatz,and discuss the relationship between their order parameters and free energies, as well as acomparison of their dispersion relations. I then introduce a real kink crystal, and discuss itsrelationship with the sinusoidal ansatz by performing a Fourier analysis of this kind of solution.
1. Introduction
While the properties of quantum chromodynamics (QCD) at finite density are still poorlyknown, its phase structure is expected to be extremely rich. Matter at low temperatures andasymptotically large baryonic chemical potentials should behave as a color-superconductor, butother pairing mechanisms might become competitive as the density of the system decreases. Inparticular, in the past few years some growing consensus has been building around the idea thatat densities around a few times nuclear matter saturation density and low temperatures, spatiallyinhomogeneous chiral condensates might form, giving rise to crystalline phases. Explicit modelcalculations indeed corroborate this hypothesis by suggesting that an inhomogeneous “island”appears in this region, where the favored structure for the chiral condensate is a spatiallymodulated one (for a recent review, see [1]). A typical phase diagram obtained within thiskind of calculations is shown in Fig. 1.The phenomenon of inhomogeneous chiral symmetry breaking at finite densities stems frompairing of quarks and holes with equal momenta on patches of the Fermi surface. The formedpairs thus carry a nonvanishing total momentum | P pair | ∼ µ , ( µ being the quark chemicalpotential) and the corresponding condensate is spatially non-uniform. Superposition of pairedquarks on different patches of the Fermi surface can then in principle give rise to more involvedcrystalline structures [2].Recent calculations performed within QCD-inspired effective models have shown that thefavored shape for the chiral condensate in the inhomogeneous island is that of a modulationwhich varies in only one spatial direction [3, 4]. In this contribution, I will discuss a fewproperties of some of these one-dimensional solutions within an effective model of QCD.
2. NJL model studies of inhomogeneous chiral condensates
The theoretical framework employed in this work for the characterization of inhomogeneouschiral condensates in dense quark matter is the Nambu–Jona-Lasinio (NJL) model, a populartool often employed in the study of low-energy properties of strong-interaction matter as well a r X i v : . [ h e p - ph ] F e b T ( M e V ) µ (MeV) Figure 1.
Phase diagram obtained within NJL model calculations when allowing forinhomogeneous chiral condensates. The shaded area denotes a region where spatially modulatedchiral condensates are favored over the homogeneous chirally broken and restored solutions.The actual size of the inhomogeneous window depends on the ansatz considered for the spatialdependence of the chiral condensate, as well as on model parameters.as its phase diagram at finite temperature and density. In its simplest form, the NJL-modelLagrangian reads [5] L NJL = ¯ ψ ( iγ µ ∂ µ − m ) ψ + G S (cid:0) ( ¯ ψψ ) + ( ¯ ψiγ τ a ψ ) (cid:1) , (1)where ψ represents a quark field with N f = 2 flavor and N c = 3 color degrees of freedom, γ µ and γ are Dirac matrices, and τ a ( a = 1 , ,
3) are isospin Pauli matrices. The SU (2) L × SU (2) R chiral symmetry of L NJL is explicitly broken by the bare quark mass m , as well as spontaneouslyby non-vanishing condensates of the form φ σ ≡ (cid:104) ¯ ψψ (cid:105) or φ aπ ≡ (cid:104) ¯ ψiγ τ a ψ (cid:105) .In the following I will consider the chiral limit, m = 0, and work within the standard mean-field approximation where fluctuations of the order parameters are neglected. This allows toobtain the model thermodynamic potential per volume V for a given temperature and chemicalpotential, as a function of the scalar and pseudoscalar mean-fields:Ω( T, µ ; φ σ , φ aπ ) = − TV Tr Log (cid:18) S − T (cid:19) + TV (cid:90) V d x E G S ( φ σ + φ π,a ) (2)where the integral is performed in Euclidean space-time, x E = ( τ, x ) with the imaginary time τ = it , and extends over the four-volume V = [0 , T ] × V . The functional trace of the inversequark propagator S − runs over V and internal (color, flavor, and Dirac) degrees of freedom.In order to investigate inhomogeneous chiral-symmetry breaking, within this approach themean-fields are allowed to be spatially dependent, but are still assumed to be static so thatthe temporal part of the functional trace can be evaluated as a standard sum over fermionicMatsubara frequencies.From a technical point of view, the inversion of the quark propagator requires thediagonalization of the effective Hamiltonian H = γ (cid:2) − iγ i ∂ i − G S ( φ σ ( x ) + iγ τ a φ aπ ( x )) (cid:3) . (3)If charged pion condensation is neglected ( φ π = φ π = 0 , φ π ≡ φ π ), after expliciting the Diracstructure it is possible to factorize the Hamiltonian into a direct product of two isospectrallocks in flavor space [6], H = H + ( M ) ⊗ H + ( M ∗ ) , with H + ( M ) = (cid:18) iσ i ∂ i M ( x ) M ∗ ( x ) − iσ i ∂ i (cid:19) , (4)where σ i are Pauli matrices and one customarily defines a constituent mass function M ( x ) = − G S ( φ σ ( x ) + iφ π ( x )) . (5)In presence of spatially modulated condensates, the diagonalization of H becomes extremelyinvolved, as quark momenta are no longer conserved and the Hamiltonian is not diagonal inmomentum space. In practice, it is then useful to restrict the analysis to periodic modulationsof the chiral condensate, an assumption which allows to perform a Fourier decomposition of themass function: M ( x ) = (cid:88) q k ∆ k e i q k · x , (6)with momenta q k forming a reciprocal lattice (RL) in momentum space. Due to the crystalsymmetries, only quark momenta differing by elements of the RL are paired in H , so that onecan effectively diagonalize a Hamiltonian with a discrete momentum structure. The componentsof one of its momentum blocks are given by[ H + ] p m , p n = − (cid:126)σ · p m δ p m , p n (cid:80) q k ∆ k δ p m , p n + q k (cid:80) q k ∆ ∗ k δ p m , p n − q k (cid:126)σ · p m δ p m , p n , (7)where it can be clearly seen that the inhomogeneous chiral condensate couples different quarkmomenta.Any periodic modulation of arbitrary spatial dimension can in principle be implementedwithin this approach by fixing an underlying lattice structure via the definition of the q k and considering an appropriate number of Fourier coefficients. The favored solutions canthen be obtained by minimizing the model thermodynamic potential with respect to thevariational parameters { ∆ k , q k } . In practice, however, the numerical diagonalization of thequark Hamiltonian in momentum space becomes extremely demanding from a numerical pointof view as the mass ansatz becomes more involved. A dramatic simplification of the problemcomes if one considers solutions for which the chiral condensate is modulated only along onespatial direction (which one can take without loss of generality along the z axis), while remainingconstant along the others. In particular, it was shown in [6] that it is usually possible to solvethe eigenvalue problem by diagonalizing a dimensionally-reduced Hamiltonian with p x = p y = 0and subsequently boost the obtained energies along the transverse directions.A systematic NJL model analysis on the favored type of modulation including higher-dimensional ans¨atze has been performed in [3] at zero temperature, and in [4] in proximityof the chiral critical point within a Ginzburg-Landau expansion study. Both agree on findingthat the thermodynamically favored solution is a one-dimensional one, in particular a so-calledreal kink crystal (RKC) which can be expressed in terms of Jacobi elliptic functions. This isin contrast with the situation for the 1+1-dimensional NJL model (NJL ), where the groundstate is given by a plane wave modulation of the chiral condensate often referred to as “chiralspiral” [7]. On the other hand, RKC solutions are found to be favored in the Gross-Neveumodel [8], which has only a discrete chiral symmetry (see also [9] for a comparison of the pairingmechanisms in the two models). From a technical point of view, it is the particular form of theHamiltonian in the NJL model in 3+1 dimensions (Eq. (4)) containing two blocks differing onlyby the substitution M ↔ M ∗ which makes these real solutions the favored ones [6]. . Comparison of one-dimensional modulations As previously mentioned, in order to perform a model analysis of inhomogeneous condensation,a given ansatz for the spatial dependence of the mean-fields must be chosen.Among all possible one-dimensional solutions, the simplest is a so-called “(dual-)chiral densitywave” (CDW) [10], a plane wave ansatz which can be seen as analogous of the Fulde-Ferrellsolutions introduced in the context of superconductivity and color superconductivity: M CDW ( x ) = ∆ e iqz , (8)with two variational parameters ∆ and q , characterizing the amplitude and the wave number ofthe modulation. This can be seen as a one-dimensional truncated version of the general ansatzof Eq. (6) where only the first Fourier component ∆ is allowed to be nonzero.A very important feature of the CDW is that for this ansatz the explicit spatial dependenceof the condensates in the quark Hamiltonian (Eq. (3)) can be removed by means of a localchiral rotation of the quark fields [11, 12] (see also [1]), thus allowing to evaluate the eigenvaluespectrum analytically. The resulting eigenvalues of the dimensionally reduced Hamiltonian arethe positive and negative square roots of E ± ( p z ) = p z + ∆ + q ± (cid:112) ∆ q + ( qp z ) = (cid:16)(cid:112) p z + ∆ ± q (cid:17) . (9)An alternative ansatz often considered in literature is a simple real sinusoidal modulation ofthe mass function, for which the pseudoscalar part φ π is set to zero: M COS ( x ) = ∆ cos( qz ) . (10)From a technical point of view, this ansatz can be seen as a superposition of two plane wavespropagating in opposite directions, and as such its spatial dependence cannot be rotated awayusing the technique employed for a CDW. In fact, no analytical expression for the eigenvaluespectrum in presence of this modulation is known, so that a numerical diagonalization of themodel Hamiltonian in momentum space must be performed, using, for example, the techniquesdeveloped in [13, 3].After minimization of the thermodynamic potential, one can observe a very close relationshipbetween the order parameters and the free energies of the CDW and COS modulations. Indeed,as can be seen in Fig. 2, throughout almost the entire inhomogeneous window one finds that∆ COS = 2∆
CDW , while q COS and q CDW are similar but not equal and start overlapping onlyclose to the chiral restoration phase transition, a behavior already predicted within a GL analysisin [6]. Another interesting result is that the COS solution is always favored by a factor of 2 infree energy compared to the CDW one, as shown in Fig. 3.In order to understand why the real COS solution is favored over the complex CDW one,it might be also of interest to compare the dispersion relations for the two modulations. Theeigenvalues of the dimensionally-reduced Hamiltonians for the two are plotted in Fig. 4. Thereit can be seen how the CDW spectrum still exhibits branches which are analogous to those ofchirally restored quark matter, whereas for the COS modulation these also acquire a gap. Thisadditional pairing present for the COS modulation allows for a larger energy gain than in theCDW case, since twice as many branches become deformed and allow for the system to furtherlower its energy.The final part of my discussion is devoted to the thermodynamically favored modulation inthe NJL model, which is known to be given by a real-kink crystal. This particular kind of ansatzcan be expressed as [8] (see also [1]) M ( z ) = ∆ √ ν sn(∆ z | ν ) , (11) � � �� � ��� � ��� � ��� � ��� � ��� � ��� � ��� � ��� � ��� � ��� � ��� � ��� � � � � � � � ����� � ��� � ��� �� ��� � ��� � ��� � ��� � ��� � ��� � ��� � ��� � ��� � ��� � ��� � ��� � ��� � ��� � ��� � ��� � � � � � � � ����� � ��� � ��� Figure 2.
Left: comparison of amplitudes at T = 0 for CDW and COS modulation. The solidblue line denotes the thermodynamically favored ∆ for the CDW modulation, the dashed blueline is twice that value, the solid red line is the favored ∆ for the COS modulation. Right:comparison of wave numbers. The blue line is the favored q for the CDW, the red one is the q for the COS. ���������������� � � � ��� � ��� � ��� � ��� � ��� � ��� � ��� � ��� � ��� � � � � �� � � � � � ����� � ��� � ��� �� ��� � ��� Figure 3.
Free energy associated with different modulations of the chiral condensate in theregion where inhomogeneous phases are thermodynamically favored, as a function of quarkchemical potential ( T = 0). Solid blue line: CDW, Red line: COS. The dashed blue line,corresponding to twice the CDW free energy, is basically overlapping the COS one throughoutthe whole inhomogeneous window. The purple dash-dotted line denotes the RKC free energy,which is slightly lower than the COS one for lower µ but rapidly becomes degenerate with it.where sn is a Jacobi elliptic function and the parameter ν , called elliptic modulus and rangingfrom 0 to 1, describes the shape and periodicity of the modulation.By inspecting the behavior of the mass function as chemical potential increases, one can seethat at the onset of the inhomogeneous phase this solution has the limit M ( z ) → ∆ tanh(∆ z ),which can be seen as a single soliton interpolating between the two homogeneous chirally brokensolutions. As µ increases, solitons start stacking up against each other and the order parameterprogressively assumes a sinusoidal shape [14, 15]. This kind of solution and its behavior canbe encompassed within the framework introduced in the previous section by evaluating theFourier coefficients associated with it, as function of chemical potential. The resulting valuesare presented in Table 1. There one can see, as expected, that as the chemical potential increasesthe higher Fourier coefficients rapidly drop to zero, and the RKC solution assumes the form ofa simple cosine. � � ��� � ��� � ��� � ��� � ��� � ��� � ��� � ��� � � � �� � ��� � ��� � ��� � ��� � ��� � ��� � ��� � � � � � � � � � � ����� � � � ��� � ��� � ��� � ��� � ��� � ��� � ��� � ��� � � � �� � ��� � ��� � ��� � ��� � ��� � ��� � ��� � � � � � � � � � � ����� Figure 4.
Dispersion relations for the CDW (left) and COS (right) modulations. Both havebeen evaluated for values of the order parameters corresponding to the respective minima at T = 0 , µ = 325 MeV (recall at the minimum q COS (cid:39) q CDW and ∆
COS (cid:39) CDW ). Thechemical potential is plotted as reference as an horizontal blue line.
Table 1.
Values of the higher order nonzero harmonics for a RKC modulation at differentvalues of chemical potential close to the onset of the inhomogeneous phase ( T = 0). Results arenormalized to the value of the first harmonic ( n = 1) in order to show their relative weights.Being a real modulation, one also finds ∆ − n = ∆ n . Even coefficients are found to be zero. µ ∆ / ∆ ∆ / ∆ ∆ / ∆ ∆ / ∆
4. Conclusions
In this contribution I discussed some properties of one-dimensional modulations of the chiralcondensate in dense quark matter within NJL model calculations. In particular, I focused ontwo possible one-dimensional solutions, namely a chiral density wave and a single cosine, which,in the framework of a Fourier expansion of the order parameter, can be seen as prototypes forcomplex and real ans¨atze, respectively. I showed that the thermodynamically favored values forthe wave numbers of the chiral density wave and that of the cosine are very similar throughoutthe whole inhomogeneous window, while the amplitude of the cosine is twice that of a planewave. Furthermore, by inspecting the quasiparticle dispersion relations I argued why a realsinusoidal modulation is favored over a complex plane wave one. In particular, the fact that forthe COS modulation twice as many branches as for the CDW become gapped could explain thefact that the energy gain for the former is twice the one of the latter.Finally, I considered the solution which is found to be the most favored throughout the wholeinhomogeneous window, a real kink crystal, and investigated its relationship with a simplerreal sinusoidal modulation by performing a Fourier decomposition of this type of ansatz. TheRKC effectively reduces to a simple cosine rapidly after the onset of the inhomogeneous phase.This in turn implies that for most intents and purposes the RKC solution can be reasonablyapproximated by considering a single cosine, with the exception of the immediate vicinity of thephase transition associated with the onset of inhomogeneous condensation. In particular, it isimportant to remember that the nature of such phase transition depends strongly on the type ofodulation considered. Among those discussed here, only the RKC can smoothly reduce in thethermodynamic limit to an homogeneous chirally broken solution. Indeed, in order to reproducean homogeneous solution which has an infinite period, an arbitrarily large set of Fourier modesneeds to be considered. As such, only the RKC solution can realize a second-order transitionto the homogeneous chirally broken phase, whereas for the other simpler ans¨atze a first-ordertransition must occur.
5. Acknowledgments
I would like to thank M. Buballa for helpful discussions.
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