A Twisted Kink Crystal in the Chiral Gross-Neveu model

A Twisted Kink Crystal in the Chiral Gross-Neveu model

Gökçe Başar and Gerald V. Dunne Physics Department, University of Connecticut, Storrs CT 06269
Abstract

We present the detailed properties of a self-consistent crystalline chiral condensate in the massless chiral Gross-Neveu model. We show that a suitable ansatz for the Gorkov resolvent reduces the functional gap equation, for the inhomogeneous condensate, to a nonlinear Schrödinger equation, which is exactly soluble. The general crystalline solution includes as special cases all previously known real and complex condensate solutions to the gap equation. Furthermore, the associated Bogoliubov-de Gennes equation is also soluble with this inhomogeneous chiral condensate, and the exact spectral properties are derived. We find an all-orders expansion of the Ginzburg-Landau effective Lagrangian and show how the gap equation is solved order-by-order.

I Introduction

In a recent Letter bd1 (), the authors found a new self-consistent crystalline condensate solution to the gap equation of the massless chiral Gross-Neveu model gross (). For this complex chiral condensate, the amplitude is periodic and the phase winds by a certain angle over each period. Our approach is based on the observation that a carefully motivated ansatz for the associated Gorkov resolvent reduces the gap equation to a simple ordinary differential equation, an explicitly soluble form of the nonlinear Schrödinger equation (NLSE). In general, the gap equation for an inhomogeneous condensate is a highly nontrivial functional differential equation, so the reduction to the NLSE represents a significant simplification. This resolvent approach is complementary to the inverse scattering approach dhn (); shei (), which also dramatically simplifies the gap equation, but which was not developed for periodic inhomogeneities. Our resolvent method is based on an extension, to complex and periodic condensates, of the work of Feinberg and Zee fz (); feinberg (). The general solution to the nonlinear Schrödinger equation contains all previously known self-consistent condensates of the massless Gross-Neveu models [both chiral and non-chiral] as special cases: the single real kink dhn (), the single complex kink shei (), the real kink crystal thies-gn (), the complex chiral spiral schon (), and also yields a new complex kink crystal bd1 (). In the language of condensed matter physics, this crystalline condensate is a new solution of the Eilenberger equation (for the Gorkov resolvent) eilenberger (), and we also present here the complete exact solution of the associated Bogoliubov-de Gennes degennes () equation for this system. The Eilenberger and Bogoliubov-de Gennes equations are fundamental elements of the treatment of a wide class of interacting fermion systems, which are important in many branches of physics, ranging from particle physics, to solid state and atomic physics campbell (); rajagopal (); casalbuoni (); pitaevskii (). Important paradigms include the Peierls-Frohlich model of conduction peierls (), the Gorkov-Bogoliubov-de Gennes approach to superconductivity degennes (), and the Nambu-Jona Lasinio (NJL) model of symmetry breaking in particle physics nambu (). Here we study a -dimensional version of the NJL model, the model [also known as the chiral Gross-Neveu model, ]. This model has been widely studied as it exhibits asymptotic freedom, dynamical mass generation, and chiral symmetry breaking gross (); dhn (); shei (); fz ().

Our primary physical motivation for studying the gap equation of the massless chiral Gross-Neveu model, the model, is to understand the phase diagram of this system. Somewhat surprisingly, the phase diagram of this system is not yet fully understood. A gap equation analysis based on a homogeneous condensate suggests its phase diagram is the same as its discrete-chiral cousin, the original Gross-Neveu () model gross (), while more recent work finds an inhomogeneous Larkin-Ovchinikov-Fulde-Ferrell (LOFF loff ()) helical complex condensate (“chiral spiral”) below a critical temperature schon (). In bd1 (), a Ginzburg-Landau approach was used to show that in a region of the phase diagram the free energy is lower for a complex kink crystal, compared to a uniform condensate or a chiral spiral condensate. In this paper we present the details of the complex crystalline condensate of the system, and also the exact spectral properties of fermions in the presence of such a crystalline condensate. This information will subsequently be used to study the free energy exactly, without resorting to the Ginzburg-Landau approximation.

This state of affairs should be compared and contrasted with the case of the original Gross-Neveu model gross (), to which we refer as the model, which has a discrete chiral symmetry rather than the continuous chiral symmetry of the model. In the model, the phase diagram has only relatively recently been solved in the particle physics literature, analytically and exactly by a Hartree-Fock analysis thies-gn (), and numerically on the lattice deforcrand (). There is a crystalline phase at low temperature and high density, and this phase is characterized by a periodically inhomogeneous (real) condensate that solves exactly the gap equation. This phase is not seen in the old phase diagram which was based on a uniform condensate wolff (); treml (). Interestingly, some hints of a problem with the homogeneous condensate assumption were found already in an early lattice study karsch (). This discrete-chiral model (with vanishing bare fermion mass) turns out to be mathematically equivalent to several models in condensed matter physics thies-gn (): the real periodic condensate may be identified with a polaron crystal in conducting polymers horovitz (); braz (); campbell (), with a periodic pair potential in quasi 1D superconductors kuper (); mertsching (); buzdin (), and with the real order parameter for superconductors in a ferromagnetic field machida (). The Gross-Neveu models also serve as paradigms of the phenomenon of fermion number fractionalization jackiw (); gw (); niemi (); heeger ().

Here, we consider the massless chiral Gross-Neveu, or , model in dimensions with Lagrangian gross (); shei (); dhn ()

 L=¯ψi∂/ψ+g22[(¯ψψ)2+(¯ψiγ5ψ)2]. (1)

This system has a continuous chiral symmetry under . We have suppressed summation over flavors, which makes the semiclassical gap equation analysis exact in the limit, a limit in which we can consistently discuss chiral symmetry breaking in 2D witten (); affleck (). The original Gross-Neveu model, the model gross (), without the pseudoscalar interaction term , has a discrete chiral symmetry .

There are two equivalent ways to find self-consistent static condensates. First, introduce bosonic scalar and pseudoscalar condensate fields, and , which we combine into a complex condensate field, defined either through its real and imaginary parts, or via its amplitude and phase:

 Δ≡σ−iπ≡Meiχ. (2)

Integrating out the fermion fields we obtain an effective action for the condensate as

 Seff=−12g2∫|Δ|2−iNlndet[i∂/−12(1−γ5)Δ−12(1+γ5)Δ∗] (3)

The corresponding (complex) gap equation is

 Δ(x)=−2iNg2δδΔ(x)∗lndet[i∂/−12(1−γ5)Δ(x)−12(1+γ5)Δ∗(x)] (4)

If the condensate is constant, as is usually assumed, it is straightforward to evaluate the determinant and solve the gap equation wolff (); treml (); casalbuoni (). When the condensate is inhomogeneous this is a much more difficult problem. Dashen, Hasslacher and Neveu dhn () used inverse scattering to find kink-like static but spatially inhomogeneous condensates for the gap equation of the model (where there is no pseudoscalar condensate, so is real). Shei shei () extended this inverse scattering analysis to the chiral Gross-Neveu model, the model, and found a spatially inhomogeneous complex kink. A new approach to the inhomogeneous gap equation, based on the resolvent, was developed by Feinberg and Zee fz () and applied to the kink solutions of both the and models. For the model, Thies used a Hartree-Fock approach to find a periodic extension of the real kink solution, motivated by analogous inhomogeneous condensates in condensed matter systems thies-gn (). In bd1 (), the present authors showed that the complex gap equation (4) can be reduced in an elementary manner to a soluble form of the nonlinear Schrödinger equation. The general solution contains all previously known inhomogeneous condensates (real and complex), and yields a new crystalline extension of Shei’s complex kink.

A second approach to finding a self-consistent condensate is to solve the relativistic Hartree-Fock problem , with single-particle Hamiltonian

 (5)

and subject to the consistency condition

 ⟨¯ψψ⟩−i⟨¯ψiγ5ψ⟩=−Δ/g2 (6)

We choose Dirac matrices , , , to emphasize the natural complex condensate combination in (2). Then the single-particle Hamiltonian is

 H=⎛⎝−iddxΔ(x)Δ∗(x)iddx⎞⎠. (7)

This Hamiltonian is also known as the Bogoliubov-de Gennes (BdG) Hamiltonian, and we will refer to the associated spectral equation

 Hψ=Eψ (8)

as the Bogoliubov-de Gennes (BdG) equation.

In Section II we review the reduction of the functional gap equation to the nonlinear Schrödinger equation (NLSE), and in Sections III and IV we present the real and complex condensates obtained from solving the NLSE. In Section V we show that the associated Bogoliubov-de Gennes equation can also be solved exactly, and we derive the exact single particle spectrum and density of states. In Section VI we verify the consistency of our solutions by solving the gap equation in the Hartree-Fock approach. An all-orders Ginzburg-Landau expansion of the free energy is presented in Section VII, and we show that the inhomogeneous gap equation is satisfied order-by-order in an interesting and nontrivial way. In a concluding section we review our results and discuss implications for the phase diagram of the chiral Gross-Neveu model.

Ii Reduction of Functional Gap Equation to Nonlinear Schrödinger Equation

In this Section we review the reduction bd1 () of the functional gap equation (4) to the nonlinear Schrödinger equation. The key quantity in our approach is the coincident limit of Gor’kov Green’s function, or the the “diagonal resolvent”:

 R(x;E)≡⟨x|1H−E|x⟩. (9)

The resolvent (9) is clearly a matrix. For a static (but possibly spatially inhomogeneous) condensate, all spectral information is encoded in the resolvent. Indeed, the spectral function characterizing the single-particle spectrum of fermions in the presence of the condensate is

 ρ(E)=1πImTrD,x[R(x;E+iϵ)], (10)

where the trace is a Dirac trace as well as a spatial trace.

Our first, very simple, observation is that the form of the BdG equation (8) places very strong constraints on the possible form of . For any static condensate , must satisfy the following algebraic conditions [these are explained in more detail in Appendix A]:

 R = R† (11) trD(R(x;E)σ3) = 0 (12) detR(x;E) = −14. (13)

Furthermore, must satisfy the first-order differential equation

 ∂∂xR(x;E)σ3 = i[(E−Δ(x)Δ∗(x)−E),R(x;E)σ3] (14)

In superconductivity, (14) is known as the Eilenberger equation eilenberger (); stone (), and in mathematical physics as the Dik’ii equation dickey (). These conditions (11)–(13), and the Eilenberger equation (14), all follow from the simple fact dickey (); waxman (); stone () that for the one-dimensional BdG equation, which involves derivatives with respect to the single variable , the Green’s function can be expressed as a product of two independent solutions to (8):

 R(x;E)=12iW(ψ1ψT2+ψ2ψT1)σ1 (15)

where is the Wronskian of two independent solutions : .

The next step is to note that the gap equation provides further information about the possible form of the resolvent, and this is enough to motivate a specific ansatz form bd1 (). There are two ways of viewing the gap equation (4) in terms of the resolvent. First, for a static condensate we can write the log det term in the effective action (3) as minus the grand canonical potential, in terms of the single-particle spectral function :

 −1β∫∞−∞dEρ(E)ln(1+e−β(E−μ)). (16)

All dependence on resides in the spectral function , via (10). Therefore, inserting this into the gap equation (4), this relates to the diagonal entries of . Further, as a consequence of the condition (12), these diagonal entries are equal. So, the simplest natural solution to the gap equation is for the diagonal entries of to be linear in . A second way to view the gap equation is to evaluate the functional derivative in (4), which for a static condensate leads to:

 Δ(x)=−iNg2trD,E[γ0(1+γ5)R(x;E)] (17)

The Dirac trace then relates the off-diagonal entries of to . Since is hermitean, these off-diagonal entries are complex conjugates of one another.

Summarizing, must be a hermitean matrix with equal diagonal entries, such that [after the spatial and energy trace] the variation of the diagonal terms is proportional to , and with off-diagonal terms linear in , after the energy trace. This suggests taking the resolvent to be of the form

 R(x;E)=N(E)(a(E)+|Δ(x)|2b(E)Δ(x)b(E)Δ∗(x)a(E)+|Δ(x)|2) (18)

where , and are functions of , independent of , and are to be determined. However, this ansatz cannot describe inhomogeneous condensates because the only solution of this form consistent with (13) is a condensate with constant magnitude, independent of . Indeed, taking to be constant (and by a global chiral rotation, real), , the solution to (11)–(14) is simply

 R(x;E)=12√M2−E2(EMME) (19)

as is familiar. This example also illustrates that the hermiticity condition (11) must of course be interpreted with the appropriate prescription for the energy.

To find inhomogeneous condensates, we suggested in bd1 () to extend the ansatz (18) to include a first derivative term in the off-diagonal entries:

 R(x;E)=N(E)(a(E)+|Δ(x)|2b(E)Δ(x)−iΔ′(x)b(E)Δ∗(x)+iΔ′∗(x)a(E)+|Δ(x)|2). (20)

This is the simplest extension of (18) that is consistent with the various algebraic constraints and with the Eilenberger equation (14). Indeed, substituting the ansatz (20) into the Eilenberger equation (14), we see that the diagonal entry of this equation is identically satisfied, while the off-diagonal entry implies that must satisfy the following nonlinear Schrödinger equation (NLSE) [and its complex conjugate]:

 Δ′′−2|Δ|2Δ+i(b−2E)Δ′−2(a−Eb)Δ=0. (21)

Two comments are in order. First, it is not immediately obvious that in (20) can satisfy the normalization condition (13) for an inhomogeneous condensate, since

 detR(x;E)=N2{|Δ|4−|Δ′|2+(2a−b2)|Δ|2+ib(Δ′Δ∗−ΔΔ∗′)+a2} (22)

Remarkably, the NLSE (21) implies that is constant:

 ddx(detR(x;E)N2) = (2|Δ|2+2a−b2)(|Δ|2)′−(Δ′′Δ∗′+Δ′Δ∗′′)+ib(Δ′′Δ∗−Δ∗′′Δ) (23) = 0

where we have used the fact that the NLSE (21) implies that , and that . Since is constant, the normalization in (13) can be achieved by suitable choice of . Second, while the ansatz (20) automatically satisfies the dependence of the gap equation in its form coming from (16) [because the trace of is, by construction, linear in ], it doesn’t satisfy the other form of the gap equation (17), until the energy trace is performed. This is because of the terms in the off-diagonal. In Section VI we show that this form of the gap equation is indeed satisfied because the coefficient of the term vanishes due to the energy trace.

Thus, we have reduced the very difficult problem of solving the functional gap equation (4) for a self-consistent condensate to the much simpler problem of solving the NLSE for . In fact, the NLSE (21) is explicitly soluble, as is discussed in the following sections, in which we describe first the real solutions [relevant for the model], and then the complex solutions [relevant for the model].

Iii Real solutions of the NLSE

In this Section we recall the previously known real solutions to the gap equation, and show how they fit in with the NLSE (21) and the resolvent form in (20).

iii.1 Homogeneous condensate

If the condensate is constant, then by a global chiral rotation it can be taken to be real:

 Δ(x)=m (24)

This clearly satisfies the NLSE (21), and we find

 a(E) = 2E2−m2 b(E) = 2E N(E) = 141E√m2−E2 (25)

The spectrum of the associated BdG equation (8) is that of a free fermion with mass , with positive and negative energy continua starting at , the mass scale being set by the amplitude of the condensate.

iii.2 Single real kink condensate

A well known nontrivial solution to the gap equation is the single (real) kink dhn ():

 Δ(x)=mtanh(mx) (26)

This satisfies the NLSE

 Δ′′−2Δ3+2m2Δ=0, (27)

and so we deduce the exact diagonal resolvent to be of the form (20) with

 a(E) = 2E2−m2 b(E) = 2E N(E) = 141E√m2−E2 (28)

The spectrum of the associated BdG equation (8) has positive and negative energy continua starting at , together with a single bound state located at , at the center of the gap. This mid-gap zero mode has many important consequences in a variety of branches of physics jackiw (); niemi ().

iii.3 Real kink crystal condensate

A periodic array of these real kinks also provides a solution to the gap equation. This solution describes a polaron crystal in polymer physics horovitz (); braz (), a periodic pair potential in inhomogeneous superconductors kuper (); mertsching (); machida (), and the crystalline phase of the Gross-Neveu model thies-gn (). Define (the peculiar looking scaling will become clear below)

 Δ(x)=√ν2m1+√νsn(2m1+√νx;ν) (29)

where sn is the Jacobi elliptic function as (); ww (); akhiezer (); lawden () with real elliptic parameter . The sn function has period , where . When (29) reduces to the single kink condensate in (26). The periodic condensate (29) satisfies the NLSE

 (30)

Thus, we deduce the exact diagonal resolvent to be of the form (20) with

 a(E) = 2E2−2m21+ν(1+√ν)2 (31) b(E) = 2E (32) N(E) = 141√m2−E2√E2−m2(1−√ν1+√ν)2 (33)

This periodic condensate is plotted in Figure 1. Note that over the period , the condensate is shaped like a single kink. This reflects the expansion of the Jacobi sn function in terms of an array of periodically displaced tanh functions:

 sn(x;ν)=π2√νK′∞∑n=−∞(−1)ntanh(π2K′(x−2nK)) (34)

where we use the standard notation . In the infinite period limit (), the interval maps to the whole real line, and , and so the kink crystal (29) reduces precisely to the single kink condensate in (26).

It is worth noting that this periodic kink crystal (29) can be written in an equivalent, but different looking, form, by use of a Landen transformation as (); ww (); akhiezer (); lawden () of the Jacobi functions. That is, by rescaling the elliptic parameter together with the argument , we can write

 Δ(x)=m~νsn(mx;~ν)cn(mx;~ν)dn(mx;~ν);~ν=4√ν(1+√ν)2. (35)

This is the form in which this periodic kink solution is presented in the work of Thies et al thies-gn () on the crystalline phase of the Gross-Neveu model, while the form (29) was used in the condensed matter literature in kuper (); horovitz (); braz (); mertsching (); machida (); buzdin ().

The spectrum of the associated BdG equation (8) has positive and negative energy continua starting at , together with a single bound band in the middle of the gap, with band edges at . This band lies symmetrically in the center of the gap. The spectrum is plotted in Figure 2 as a function of the elliptic parameter . Notice that there is just one bound band in the energy gap, and when [the infinite period limit], the bound band at the center of the gap contracts smoothly to the single bound zero mode of the kink condensate.

Iv Complex solutions of the NLSE

iv.1 Single plane wave condensate

The simplest complex solution to the NLSE is a single plane wave:

 Δ=meiqx (36)

This satisfies the NLSE (21) with

 a(E) = 2(E−q2)2−m2 b(E) = 2E−2q N(E) = 141(E−q/2)√m2−(E−q/2)2 (37)

This plane wave condensate behaves just like a constant one, but with the energy shifted by , as can be seen by making a local chiral rotation:

 Δ→e−iqxΔ,ψ→e−iqx/2γ5ψ. (38)

It is clear from the BdG equation (8) that such a transformation has the effect of shifting the entire energy spectrum by . This illustrates an important point: for complex solutions of the NLSE, one can always multiply by an arbitrary plane wave phase factor , and this simply corresponds to shifting the entire energy spectrum. In general, a local chiral rotation through angle leads to a local chemical potential wilczek (); aitchison (), or local electrostatic potential, , and therefore a local electric field . For the single plane wave condensate in (36), is linear in , and so there is no associated electric field.

iv.2 Single complex kink condensate

Shei shei () found a solution to the gap equation for the model, in which both the scalar and pseudoscalar condensates have a kink-like form:

 σ(x) = m[cos2(θ/2)+sin2(θ/2)tanh(msin(θ/2)x)] π(x) = −m2sin(θ)[1−tanh(msin(θ/2)x)] (39)

where is a parameter.

These kinks are plotted in Figure 3. This complex kink (39) has also been extensively studied in the resolvent approach by Feinberg and Zee fz (). In our analysis it is more natural to combine these into the complex condensate , [as in (2)]:

 Δ(x)=mcosh(msin(θ/2)x−iθ/2)cosh(msin(θ/2)x)eiθ/2. (40)

This complex form is plotted in Figure 4. This illustrates the role of the parameter as the net rotation angle of the kink as goes from to :

 Δ(x=+∞)=e−iθΔ(x=−∞) (41)

Observe that when , the complex kink (40) is in fact real, and reduces to the familiar real kink solution in (26); this real kink changes its sign [i.e., rotates through ] in passing from to . Another useful representation of this kink is in terms of the magnitude and phase :

 M2(x) = m2[1−sin2(θ/2)sech2(msin(θ/2)x)] (42) χ(x) = arctan(sinθcosθ+e2mxsin(θ/2)) (43)

The complex kink condensate (40) satisfies the NLSE:

 Δ′′−2|Δ|2Δ−2imcos(θ/2)Δ′+2m2Δ=0 (44)

From this NLSE, we deduce the exact diagonal resolvent to be of the form (20) with

 a(E) = 2E2−2mcos(θ/2)E−m2 b(E) = 2E−mcos(θ/2) N(E) = 14(E−mcos(θ/2))√m2−E2 (45)

The spectrum of the associated BdG equation (8) has positive and negative energy continua starting at , together with a single bound state located at . When , where this complex kink reduces to the standard real kink, the bound state is once again a zero mode. But for other values of the single bound state lies asymmetrically inside the gap, as plotted in Figure 6 . As goes from to , one state moves from the positive to the negative energy continuum. As is clear from the previous subsection, we can always multiply the complex kink solution (40) by a plane-wave factor , which has the net effect of displacing the fermion spectrum by , with the corresponding simple modifications to the resolvent functions , and in (45).

At this stage we have shown that Shei’s complex kink condensate (40) solves the NLSE, and we have found the corresponding exact diagonal resolvent (20) with , and given in (45). This agrees with the spectral properties derived from inverse scattering shei (). Shei further showed shei () that this complex condensate solves the gap equation provided a further restriction is applied to the winding parameter . This condition states that is equal to the filling fraction , in the large flavor limit, of the single bound state in the gap by flavors, with fixed as shei (); fz ():

 θ2π=nN (46)

In Section VI we show that in our approach this same condition arises from demanding that the coefficient of the term in (17) vanishes after the energy trace, a necessary requirement to satisfy the gap equation.

iv.3 Complex kink crystal condensate

A new complex condensate was presented in bd1 (). This new solution is a periodic array of Shei’s complex kink (40). Physically, it is associated with a crystalline phase of the system bd1 (), just as the real kink crystal condensate (29) is associated with a crystalline phase of the system thies-gn (). Up to a plane wave factor [as in Section IV.1], this complex crystalline condensate is the general solution to the NLSE (21), and all other solutions [both real and complex] can be obtained from it by suitable choices of parameters. This solution can be written 111Note that we have made a Jacobi imaginary transformation , which involves interchanging the real and imaginary periods, relative to the notation in bd1 (), in order to be consistent with the elliptic parameter conventions of thies-gn (). in terms of Weierstrass elliptic functions [or, alternatively but equivalently, in terms of Jacobi theta functions]

 Δ(x) = −Aσ(Ax+iK′−iθ/2)σ(Ax+iK′)σ(iθ/2)exp[iAx(−iζ(iθ/2)+ins(iθ/2))+iθη3/2] (47)

The parameter sets the scale of the condensate and its length scale:

 A=A(m,θ,ν)≡−2imsc(iθ/4)nd(iθ/4) (48)

where sc=sn/cn and nd=1/dn are Jacobi elliptic functions as (); ww (). The functions and are the Weierstrass sigma and zeta functions as (); ww (); akhiezer (); lawden (), some relevant properties of which are given in Appendix B. We have chosen real and imaginary half-periods: , and . Both periods are therefore controlled by the single [real] elliptic parameter . Also, is purely imaginary. The parameter is related to the angle through which the condensate rotates in one period :

 Δ(x+L)=e2iφΔ(x) (49)

where the angle is a function of and

 φ=K(−iζ(iθ/2)+ins(iθ/2)−ηθ2K) (50)

Here we used the quasi-periodicity property (127) of the function. Note that and are real, and when , we have . This crystalline complex kink is plotted in Figure 7 showing the winding of the kink over a period.

It is also useful to visualize the condensate (47) in terms of its amplitude and phase: . The modulus squared is a bounded periodic function, with period :

 M2≡|Δ(x)|2=A2(P(Ax+iK′)−P(iθ/2)) (51)

Here we used the quasi-periodicity property (127) of the function, together with the product identity (137) relating the and functions. The phase can be expressed as

 χ(x)=A(−iζ(iθ/2)+ins(iθ/2))x+i2ln(σ(Ax+iK′+iθ/2)σ(Ax+iK′−iθ/2))+η3θ2 (52)

The amplitude and phase are plotted in Figure 8. Note that the amplitude is periodic while the phase changes by over each period.

The complex crystalline condensate in (47) satisfies the NLSE:

 Δ′′−2|Δ|2Δ−i(2Ains(iθ/2))Δ′−A2(3P(iθ/2)−ns2(iθ/2))Δ=0 (53)

Comparing this equation with the NLSE (21) we can extract the functions , and appearing in (20), thereby determining the exact diagonal resolvent. To express these functions in a compact form, we define some properties of the associated fermionic spectrum for the BdG equation (8). This spectrum has positive and negative energy continua starting at , together with a single bound band in the gap, as depicted in Figure 9. In contrast to the case for the real kink crystal in Section III, here the bound band is not centered in the middle of the gap, but is displaced from the center. The parameter characterizes this asymmetry in the spectrum. The band edges are functions of both the winding angle and the elliptic parameter :

 E1 = −m E2 = m(−1+2nc2(iθ/4;ν)) E3 = m(−1+2nd2(iθ/4;ν)) E4 = +m (54)

In the infinite period limit [], the band contracts to a single bound state, with , and this is precisely the bound state of the single complex kink, as shown in Figure 6. At finite period, but when , we find , and the band is centered symmetrically about ; this is precisely the band spectrum of the real kink array in Section III. Thus, we can roughly think of the parameter as setting the offset location of the band inside the gap, while the parameter , together with , plays the role of determining the period of the crystal, and hence the width of the band. In terms of the band edges, the resolvent functions , and that appear in the resolvent (20) take the following simple form 222In general, when we allow for an overall shift of the energy spectrum by including an extra plane-wave phase in the condensate , these functions can be written as , , and .:

 a(E) = 2E2−(E2+E3)E−(E2−E3)24−m2 b(E) = 2E−(E2+E3) N(E) = 14√(m2−E2)(E−E2)(E−E3) (55)

Here the normalization is fixed by the property .

iv.3.1 Solution of NLSE by separation into amplitude and phase

We now comment on two ways to derive this solution to the NLSE (21). The first is to separate the NLSE into two equations, one for the amplitude and one for the phase. Writing , and considering , we immediately find that the phase is related to the amplitude by:

 χ′(x)=−12(b−2E)+C1M2(x) (56)

where is a constant. Note that this is indeed true for all the complex condensate solutions discussed above. For the crystalline complex solution in (51, 52) the relation (56) can be verified using the Weierstrass function properties listed in Appendix B, in particular equation (140).

Next, considering , we find the following nonlinear equation for :

 ((M2)′)2=4M6+2(4(a−Eb)−12(b−2E)2)M4+C2M2−4C21 (57)

where is another constant. From the form of this equation, comparing with the equation (135) for the Weierstrass function as (); ww (); akhiezer (); lawden (), we recognize the solution to be of the form

 M2=A2[P(Ax+iK′)−P(C3)] (58)

The shift by ensures that is bounded for on the real axis. This explains the result in (51), identifying . The constants and are related to as

 C1 = −i2A3P′(iθ/2) C2 = 2A4P′′(iθ/2) (59)

Given this solution for the amplitude, the phase follows from (56), using the integration formula as ():

 1A∫P′(C3)P(Ax+iK)−P(C3)dx=ln(σ(Ax+iK−C3)σ(Ax+iK+C3))+2(Ax+iK)ζ(C3) (60)

Notice that the phase can always be shifted by , which amounts to multiplying the solution by a plane wave factor, as discussed in Section IV.1. In our solution (47), the plane wave factor has been chosen so that the associated fermion spectrum has the form shown in Figure 9. An additional plane wave factor in displaces this entire spectrum by a constant.

iv.3.2 Solution of NLSE as periodic array of complex kinks

Another way to derive the general periodic complex solution to the NLSE is to make an educated guess for a periodic array of Shei’s complex kink solution (40), using known properties of the Weierstrass functions. Shei’s complex kink condensate(40) can be written as

 (61)

This can be made quasi-periodic along the real axis by generalizing the hyperbolic sine function to its doubly-periodic form, which is the Weierstrass sigma function. Thus, we are led to try the form

 Δ(x)=Aeiθη3/2σ(Ax+iK′−iθ/2)σ(Ax+iK′)σ(iθ/2)eiλx (62)

where we have included a possible plane wave factor , to be determined, and the normalization factors have been chosen for convenience.

Given this form of , since , we find

 Δ′=A{ζ(Ax+iK′−iθ/2)−ζ(Ax+iK′)+iλ/A}Δ (63)

and since , we find

 Δ′′=A2{[−P(Ax+iK′−iθ/2)+P(Ax+iK′)]+[ζ(Ax+iK′−iθ/2)−ζ(Ax+iK′)+iλ/A]2}Δ (64)

Furthermore, using the quasiperiodicity properties of the Weierstrass functions (127) and the Weierstrass function product formula (137), it follows that

 |Δ(x)|2 = A2σ(Ax+iK′−iθ/2)σ(Ax+iK′+iθ/2)σ2(Ax+iK′)σ(iθ/2)σ(−iθ/2) (65) = A2[P(Ax+iK′)−P(iθ/2)]

Therefore,

 Δ′′−2|Δ|2Δ = A2{−[P(Ax+iK′−iθ/2)+P(Ax+iK′)] (66) +[ζ(Ax+iK′−iθ/2)−ζ(Ax+iK′)+iλ/A]2 +2P(iθ/2)}Δ

where note that the relative sign between the first functions has been flipped by the subtraction of .

Now we use the remarkable identity (139) that relates the function to squares of the function, to find that

 Δ′′−2Δ2Δ=A2[3P(iθ/2)+(λ/A+iζ(iθ/2))2]Δ+2iA[λ/A+iζ(iθ/2)]Δ′ (67)

Thus, of the form in (62) does indeed satisfy the NLSE (21), and one simply needs to match the constants algebraically in order to express , and in terms of . This determines in (62) up to an additive constant, which can be fixed by matching the fermion spectrum to the form in (54).

iv.4 Reduction of general solution to special cases

To conclude this section describing our new complex crystalline solution (47), we note that it incorporates all previously known solutions as special cases.

iv.4.1 Reduction to real kink crystal condensate

When , the scale factor , and the condensate reduces to the real kink crystal in (29):

 Δ → 2m√ν1+√νsn(2mx1+√ν;ν) (68) = m~νsn(mx;~ν)cn(mx;~ν)dn(mx;~ν),~ν=4√ν(1+√ν)2

In this limit, the amplitude vanishes at , and one sees that the kink ”winds” through angle by passing through zero. For other values of , the complex kink crystal winds around zero, but without the amplitude actually vanishing. For , the bound band becomes symmetric about , because the band edges (54) reduce to:

 E2 → −m(1−√ν1+√ν) E3 → m(1−√ν1+√ν) (69)

Accordingly, the resolvent functions , and in (55) reduce smoothly when to the corresponding expressions for the real kink crystal in (33).

iv.4.2 Reduction to single complex kink condensate

For general , when , the complex crystalline condensate in (47) reduces to Shei’s complex kink solution (40). To see this, first observe that in this limit, , and diverges, while . Thus the period diverges. Also, the band edges contract as , so the band shrinks to a single bound state, whose dependence matches that of the bound state for Shei’s complex kink condensate. Furthermore, as , the Weierstrass functions simplify:

 σ(x) → sinh(x)e−x2/6 ζ(x) → −x3+coth(x) P(x) → 13+1sinh2(x) (70)

These relations show that on the interval , in the limit , the complex crystalline condensate (47) reduces to Shei’s single complex kink condensate (40) up to an unimportant constant phase factor.

iv.4.3 Reduction to single plane wave condensate

In the opposite limit, as , the period remains finite, because , and . As , the Weierstrass functions simplify:

 σ(x) → sin(x)ex2/6 ζ(x) → x3+cot(x) P(x) → −13+1sin2(x) (71)

Thus, the complex crystalline condensate (47) reduces to a single plane wave (chiral spiral) form:

 Δ→msech2(θ/4)e−2imtanh2(θ/4)x (72)

V Solutions to the Bogoliubov/de Gennes equation

v.1 Spinor solutions

Remarkably, not only is it possible to solve the NLSE (21) exactly, we can also solve exactly the associated BdG equation smirnov (). In the previous sections we described the spectrum; here we present the explicit spinor solutions and express the spectral information in a more compact and useful form. We write the BdG equation (8) as

 ⎛⎝−iddxΔ(x)Δ∗(x)iddx⎞⎠ψ=Eψ (73)

The solutions for the real condensates in Section III are well known dhn (); feinberg (). For the complex plane wave, , the solutions are simply chiral rotations of free spinors, and are discussed for example in ohwa (). For Shei’s complex kink condensate (40), the spinor solutions are given in shei (); thies-gl (). For the complex kink crystal (47), the two independent spinor solutions can be written as

 ψ−