We derive a new exact self-consistent crystalline condensate in the (1+1)-dimensional chiral Gross-Neveu model. This also yields a new exact crystalline solution for the one dimensional Bogoliubov-de Gennes equations and the Eilenberger equation of semiclassical superconductivity. We show that the functional gap equation can be reduced to a solvable nonlinear equation and discuss implications for the temperature-chemical potential phase diagram

Başar, G.

Dunne, G.V.

Carolina Digital Repository

Self-Consistent Crystalline Condensate in Chiral Gross-Neveuand Bogoliubov–de Gennes SystemsGökçe Başar and Gerald V. DunnePhysics Department, University of Connecticut, Storrs, Connecticut 06269, USA(Received 10 March 2008; published 20 May 2008)We derive a new exact self-consistent crystalline condensate in the (1 1)-dimensional chiral Gross-Neveu model. This also yields a new exact crystalline solution for the one dimensional Bogoliubov–deGennes equations and the Eilenberger equation of semiclassical superconductivity. We show that thefunctional gap equation can be reduced to a solvable nonlinear equation and discuss implications for thetemperature-chemical potential phase diagram.DOI: 10.1103/PhysRevLett.100.200404 PACS numbers: 11.10.Kk, 11.30.Rd, 71.10.PmInteracting fermion systems describe a wide range ofphysical phenomena, from particle physics, to solid stateand atomic physics [1–4]. Important paradigms include thePeierls-Frohlich model of conduction [5], the Gorkov–Bogoliubov–de Gennes approach to superconductivity[6], and the Nambu–Jona-Lasinio (NJL) model of symme-try breaking in particle physics [7]. A (1 1)-dimensionalversion of the NJL model, the NJL2 model [also known asthe chiral Gross-Neveu model, GN2] has been widelystudied as it exhibits asymptotic freedom, dynamical massgeneration, and chiral symmetry breaking [8–11]. Surpris-ingly, the temperature-density phase diagram of this sys-tem is not yet fully understood. A gap equation analysisbased on a homogeneous condensate suggests its phasediagram is the same as its discrete-chiral cousin, the origi-nal Gross-Neveu (GN2) model [8], while recent work findsan inhomogeneous Larkin-Ovchinikov-Fulde-Ferrell(LOFF) helical complex condensate (‘‘chiral spiral’’) be-low a critical temperature [12]. In fact, the phase diagramof the discrete-chiral version, the GN2 model, has onlyrecently been solved in the particle physics literature,analytically [13], and on the lattice [14]. There is a crys-talline phase at low temperature and high density, and thisphase is characterized by a periodically inhomogeneous(real) condensate that solves exactly the gap equation. Thisphase is not seen in the old phase diagram which was basedon a uniform condensate [15]. Interestingly, this discrete-chiral GN2 model (with vanishing bare fermion mass) ismathematically equivalent to several models in condensedmatter physics: the real periodic condensate may be iden-tified with a polaron crystal in conducting polymers[1,16,17], with a periodic pair potential in quasi 1D super-conductors [18,19], and with the real order parameter forsuperconductors in a ferromagnetic field [20]. This systemalso is a paradigm of the phenomenon of fermion numberfractionalization [21,22]. Variants of such models alsoapply to ultracold fermionic systems, for which there areinteresting new theoretical and experimental developments[4,23,24].In this Letter, we present an analogous complex crystal-line condensate for the chiral GN system, the NJL2 model.This condensate is an exact inhomogeneous solution to thegap equation, and also provides a new self-consistentsolution to the Bogoliubov–de Gennes (BdG) [6] andEilenberger [25] equations of superconductivity. Our solu-tion may also be relevant for chiral superconductors and forincommensurate charge density waves in quasi 1D systems[26], which have chiral symmetry and an inherently com-plex order parameter.Consider the massless NJL2 model with Lagrangian L   i@6  g22   2    i5 2; (1)which has a continuous chiral symmetry  ! ei5 . Wehave suppressed summation over N flavors, which makesthe semiclassical gap equation analysis exact in the N !1 limit, a limit in which we can consistently discuss chiralsymmetry breaking in 2D. The original GN2 model [8],without the pseudoscalar interaction term   i5 2, has adiscrete chiral symmetry  ! 5 . There are two equiva-lent ways to find self-consistent static condensates. First,introduce bosonic condensate fields, S    and P  i5 , which we combine into a complex condensatefield:   S iP  Mei. Integrating out the fermionfields we obtain an effective action for the condensate as Seff  12Ng2Zjj2  i ln deti@6 Mei5: (2)The corresponding (complex) gap equation is x  2iNg2x	ln deti@6 Mxei5x: (3)One of the main results of this Letter is that this gapequation can be reduced in an elementary manner to an or-dinary differential equation, which moreover is soluble. Asecond approach to finding a self-consistent condensate isto solve the relativistic Hartree-Fock problem H  E ,subject to the consistency condition h   i  ih  i5 i =g2, with single-particle HamiltonianPRL 100, 200404 (2008) P H Y S I C A L R E V I E W L E T T E R Sweek ending23 MAY 20080031-9007=08=100(20)=200404(4) 200404-1 © 2008 The American Physical Society H  i5ddx 0Mxei5x i ddx x	x i ddx !:(4)This is the Bogoliubov–de Gennes (BdG), or Andreev,Hamiltonian, with x playing the role of the order pa-rameter [6]. We have chosen Dirac matrices 0  1,1  i2, and 5  3, to emphasize the natural com-plex combination   S iP. The key object in ouranalysis is the Gork’ov Green’s function, and, in particular,its coincident-point limit, the ‘‘diagonal resolvent’’: Rx;E  hxj1H  Ejxi: (5)This is a 2 2 matrix, and the spectral function is E 1 Im TrD;xRx;E i, where the trace is both a Dirac andspatial trace. Approximation methods, such as the gradientand semiclassical expansions, of Rx;E have been widelystudied [27]. In one spatial dimension, Rx;E can bewritten in terms of two independent solutions  1;2 to theDirac or BdG equation H  E . That is, Rx;E  1 T2   2 T1 1=2W, with Wronskian W i T12 2. It follows immediately that Rx;E satisfiesthe following first order equation: R05  i5E 0Mei5; R5: (6)In superconductivity, (6) is known as the Eilenberger equa-tion [25], and in mathematical physics as the Dik’ii equa-tion [28]. It is also straightforward to show that R  Ry,detR  1=4, and trR5  0.Our main observation is that the gap Eq. (3) motivates aself-consistent ansatz form of the 2 2 matrix Rx;E, andwhen this is combined with the identity (6), the exact self-consistent condensate and associated resolvent are com-pletely determined. To see this, note that the gap Eq. (3)can be viewed in two ways. First, write the log det term as Z 11dEE1	ln1 e	E (7)All dependence on x resides in the spectral functionE, via TrR, and so the simplest solution to the gapEq. (3) is for diagonal entries of the 2 2 matrix R [recallthey are equal since trR5  0] to be linear in jj2. Onthe other hand, we can also express the gap equation, byperforming the functional derivatives, as x  iNg2trD;E0  1Rx;E: (8)This suggests that the off-diagonal entries of R be propor-tional to  and 	. In fact, consistency between (6) and (8)introduces derivative terms, leading to Rx;E N EaE  jj2 bE i0bE	  i0	 aE  jj2 : (9)The gap equation is satisfied since we find trEN E  0.With this ansatz, the diagonal entries of (6) are identicallysatisfied, while the off-diagonal entries imply that thecondensate  satisfy the equation 00  2jj2 ib 2E0  2a Eb  0: (10)This nonlinear Schrödinger equation (NLSE) is analyti-cally soluble, and all previously known examples of self-consistent condensates in GN2 and NJL2 are special cases.Furthermore, the corresponding Dirac or BdG equationH  E is also exactly soluble and has a spectrumconsisting of a single band in the gap [29]. This providesan elementary explanation of the result from inverse scat-tering [9–11] that self-consistent ground state condensatesare reflectionless (or finite-gap) with a single state (orband). For example, if we specialize to a real condensate,as is relevant for the [discrete-chiral] GN2 model, the[rescaled] self-consistent solution is x psnx;,which satisfies 00  23  1 , where sn is aJacobi elliptic function, with elliptic parameter 0   1 [16–20]. The fermion spectrum has a single band in thegap, centered on E  0, reflecting the charge-conjugationsymmetry of the GN2 system. As ! 1, the period be-comes infinite, and we obtain the famous kink, x tanhx, with a single bound state at E  0. This mid-gap zero mode has many interesting physical consequen-ces in polymer systems, and is the paradigm of the frac-tional fermion number phenomenon [21]. It is also worthnoting that for the [discrete-chiral] GN2 model with abare fermion mass, the NLSE (10) acquires an inhomo-geneous term, in which case the general solution is writ-ten as x    x  x , where  is theWeierstrass zeta function, and this represents a kink-antikink crystal (or bipolaron crystal in the polymer lan-guage [30]), which is a periodic generalization [13] of theDashen-Hasslacher-Neveu (DHN) kink-antikink solution[9]: x  cothb  tanhx  tanhx b.All these are real condensates, and are well known. Theonly previously known complex condensates are (i) thesimple ‘‘chiral spiral’’ or LOFF solution x  Aeiqx; and(ii) Shei’s ‘‘twisted kink’’ solution [10], which we canexpress in complex form x  ei=2coshx sin2 i2coshx sin2: (11) is the angle through which the phase of the condensaterotates as x goes from 1 to 1. The single-particlefermion spectrum has a single bound state within the gap,located at E  cos2, as shown by the dashed line inFig. 1. Consistency with the gap equation requires vanish-ing of the coefficient of 0, which places conditions onparameters of the solution. For example, for Shei’s solutionthe condition is that =2 is equal to the filling fractionof the bound level [10,11].We point out a simple physical interpretation of thiscondition in terms of conserved currents (see also [31]).Both NJL2 and GN2 models have a conserved current j   . Since hjxi  iNtrD;E0Rx, we see that,PRL 100, 200404 (2008) P H Y S I C A L R E V I E W L E T T E R Sweek ending23 MAY 2008200404-2for a static condensate, current conservation follows trivi-ally since hj1xi  0; this is the physical origin of theidentity trD5R  0. The NJL2 model also has a con-served axial current, j5   5 , and the Eilenbergerequation (6) implies @hj5 i  iNtrR0  2Sxh  i5 i  2Pxh   i:(12)Axial current conservation then follows from the gap equa-tion, using precisely the same condition on the coefficientof 0.We have found a nontrivial crystalline solution to (10) x  AiAx =2KiAxK=2expfiAx=2 cs=2  =2 iam=2g: (13)Here,  and  are Weierstrass sigma and zeta functions,A  2sd=4cn=4 and   K, with K theelliptic half-period. For this condensate, both the amplitudeand the phase are x-dependent, as shown in Figs. 2 and 3.The essential parameters of the solution (13) are: (i) theparameter which [via (15) below] characterizes the chiraltwist of x over one period; and (ii) the elliptic parameter which, together with , determines the crystal period.The spinor solutions of the Dirac or BdG equation can alsobe expressed explicitly in terms of elliptic functions, andone can perform the Hartree-Fock analysis, as in the GN2system, to prove self-consistency of the crystalline con-densate [29]. The spectrum is that of a Dirac particlewith a single band in the gap, as shown in Fig. 1.However, unlike the real case where the band lies symmet-rically in the center of the gap, here the band is offset.Indeed, the band edges are given by: E1  1, E2 1 2cn2=4;, E3  1 2cd2=4;, E4  1,as shown in Fig. 1. This spectrum is a band version ofthe Shei spectrum, reducing to the Shei solution in theinfinite period limit. It is also a deformation of the kinkcrystal spectrum of the [discrete-chiral] GN2 system, re-ducing to that case when   2K. We find the exactdiagonal resolvent (9) with N E 141E2  1E E2E E3p ; (14)bE  2E E2  E3, and aE  2E 1E 1 E2  E3=2  1 E2  E3  E2  E32=4.Under a shift through one period L  2K0=A of thecrystal, the BdG Hamiltonian is invariant up to a globalchiral rotation through an angle ’: HxLei5’Hxei5’;’K0=2cs=2 iiK0=2K0:(15)The solutions to H  E acquire a chiral rotationand a Bloch phase under a period shift,  x L eikLei’5 x. The [relativistic] Bloch momentum k isrelated to the spectral function by E  dk=dE.To conclude, we discuss briefly the implications of thisself-consistent solution to the gap equation for the (T, )phase diagram of the NJL2 model. Recall that the corre-sponding real condensate characterizes the inhomogeneouscrystalline phase of the GN2 system [13]. For the NJL2model, the Ginzburg-Landau (GL) approach shows that the‘‘chiral spiral’’ phase identified in [12] has a richer struc-FIG. 3 (color online). Crystalline complex kink (13) plotted asa function of x over one period. The cross section denotesx  Sx  iPx, and indicates a net rotation through thetwist parameter 2’ over one period.χ x DashedM x SolidxFIG. 2. The amplitude Mx [solid line] and phase x[dashed line] of   Mei in (13), over several periods. Theamplitude is periodic while the phase rotates by an angle 2’each period.2K 4K-11EθFIG. 1 (color online). Band spectrum of the BdG Hamiltonian(4) for the complex crystal condensate (13), as a function of thetwist parameter . The dashed line denotes the infinite periodlimit [in this case ! 0], which is Shei’s solution (11). At  2K, we recover the symmetric spectrum of the real kink crystal,relevant to the GN2 system.PRL 100, 200404 (2008) P H Y S I C A L R E V I E W L E T T E R Sweek ending23 MAY 2008200404-3ture, characterized by a crystalline complex condensate ofthe form (13). Near the tricritical point of the masslessNJL2 system, the GL effective Lagrangian is LGL  c0  c2jj2  c3Im0	  c4jj4  j0j2 c5Im00  3jj20	  c6f2jj6 8jj2j0j2  2Re02	2  j00j2g: (16)Here, the coefficients cn are known functions of T and [32]. (This GL approach has been used previously in [32]to describe the phase diagram of the massive and masslessNJL2 models, in the vicinity of the tricritical point. For themassive NJL2 model no (complex) exact solution to thegap equation is known, so [32] is the current state-of-the-art for the massive model.) In the GN2 model, which has areal condensate, there is a tricritical point at T  0:3183,  0:6082, given by the point c2  c4  0. In the [chi-ral] NJL2 model, which has a complex condensate, there isa tricritical point at T  0:5669,   0; given by c2 c3  0. To search for possible crystalline phases near thetricritical point, we keep terms up to c4 and study theeffective equation of motion c400  ic30  c2  2c4jj2  0: (17)Note that this has precisely the same form as the NLSE(10) found from the gap equation and the Eilenbergerequation. Thus, we can use our solution (13) as a varia-tional ansatz and compute the free energy. We have foundthat the resulting free energy is lower than that of theLOFF-form ‘‘chiral spiral’’ variational ansatz x Aeiqx, for T < Tc. Noting that our ansatz reduces to thechiral spiral in the perturbative limit ( just as the GN2 self-consistent crystal condensate reduces to the LOFF formx  A sinqx in this limit [13]), we are led to the phasediagram shown in Fig. 4. For a detailed discussion of thephase diagram, based on the full free energy (beyond theGL approximation), using our exact spectral data, see [29].We thank the DOE for support through Grant No. DE-FG02-92ER40716.[1] D. K. Campbell and A. R. Bishop, Nucl. Phys. B 200, 297(1982).[2] K. Rajagopal and F. Wilczek, arXiv:hep-ph/0011333.[3] R. Casalbuoni and G. Nardulli, Rev. Mod. Phys. 76, 263(2004).[4] S. Giorgini, L. P. Pitaevskii, and S. Stringari,arXiv:0706.3360 [Rev. Mod. Phys. (to be published)].[5] R. Peierls, The Quantum Theory of Solids (Oxford,New York, 1955).[6] P. G. de Gennes, Superconductivity of Metals and Alloys(Addison-Wesley, Redwood City, CA, 1989).[7] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345(1961).[8] D. J. Gross and A. Neveu, Phys. Rev. D 10, 3235 (1974).[9] R. F. Dashen, B. Hasslacher, and A. Neveu, Phys. Rev. D12, 2443 (1975).[10] S. S. Shei, Phys. Rev. D 14, 535 (1976).[11] J. Feinberg and A. Zee, Phys. Rev. D 56, 5050 (1997);arXiv:hep-th/0109045.[12] V. 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Stone, and A. J. Leggett, Phys.Rev. B 58, 9365 (1998); S. Kos and M. Stone, Phys.Rev. B 59, 9545 (1999).[28] L. A. Dickey, Soliton Equations and Hamiltonian Systems(World Scientific, Singapore, 1991).[29] G. Basar and G. V. Dunne (to be published).[30] S. A. Brazovskii, N. N. Kirova, and S. I. Matveenko, Sov.Phys. JETP 59, 434 (1984).[31] F. Karbstein and M. Thies, Phys. Rev. D 76, 085009(2007).[32] C. Boehmer, M. Thies, and K. Urlichs, Phys. Rev. D 75,105017 (2007).TcTµFIG. 4 (color online). Phase diagram of the chiral NJL2 model,from a Ginzburg-Landau analysis based on a crystalline con-densate of the form in (13). Along the T axis (below Tc) thecondensate is of the form of Shei’s twisted kink (11), while onthe T  Tc line, the condensate reduces to the form of the LOFFchiral spiral [12]. The condensate is depicted as in Fig. 3.PRL 100, 200404 (2008) P H Y S I C A L R E V I E W L E T T E R Sweek ending23 MAY 2008200404-4

Self-consistent crystalline condensate in chiral Gross-Neveu and bogoliubov-de gennes systems

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Self-consistent crystalline condensate in chiral Gross-Neveu and bogoliubov-de gennes systems

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