Self-consistent Analytic Solutions in Twisted C P N−1 Model in the Large- N Limit
aa r X i v : . [ h e p - t h ] S e p Prepared for submission to JHEP
Self-consistent Analytic Solutions in Twisted C P N − Model in the Large- N Limit
Muneto Nitta , and Ryosuke Yoshii , Department of Physics, Keio University, 4-1-1 Hiyoshi, Kanagawa 223-8521, Japan Research and Education Center for Natural Sciences, Keio University, 4-1-1 Hiyoshi,Kanagawa 223-8521, Japan Department of Physics, Department of Physics, Chuo University, 1-13-27 Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan
E-mail: [email protected] , [email protected] Abstract:
We construct self-consistent analytic solutions in the C P N − model inthe large- N limit, in which more than one Higgs scalar component take values insidea single or multiple soliton on an infinite space or on a ring, or around boundaries ofa finite interval. ontents C P N − model 11 Nonlinear sigma models in two spacetime dimensions share a lot of non-perturbativeproperties with Yang-Mills theories or QCD in four dimensions, such as asymptoticfreedom, dynamical mass gap, dynamical chiral symmetry breaking and instantons,and therefore the former is regarded as a toy model of the latter [1–10]. In particular,the C P N − model [1–3] corresponding to the SU ( N ) gauge theory has been studiedextensively, such as the supersymmetric C P N − model [11, 12] for which the exactGell-Mann-Low function was obtained [13] and dynamical mass gap was proved bythe mirror symmetry [14]. The low-energy dynamics of a non-Abelian vortex stringin a U ( N ) gauge theory in four dimensions can be described by the C P N − modeldefined on a two-dimensional worldsheet [15–17] (see Refs. [18–21] as a review), givinga more precise correspondence between the C P N − model in two dimensions and the U ( N ) gauge theory in four dimensions [22, 23]. The C P N − model on a compactdirection with twisted boundary conditions has been also studied extensively. Inthis case, an instanton is decomposed into several fractional instantons [26, 27].Recently, bions which are composite states of fractional instanton and anti-instantonshave been studied in the C P N − model for the application to the resurgence theory[28, 29]. The C P N − model at finite density has been also studied [30]. Recently,there is a growing interest on the C P N − model defined on a finite region such asa ring [31, 32], a finite interval [33–36], and a disk [37, 38]. In particular, the case The C P model appears also on a non-Abelian vortex in dense QCD [24] (see Ref. [25] as areivew). – 1 –f a finite interval may correspond to a non-Abelian vortex string stretched betweentwo heavy non-Abelian monopoles [39, 40] or heavy non-Abelian monopole and anti-monopole [41].Since the C P N − model was defined forty years ago, only constant solutionshave been studied for long time except for few cases: the model on a finite intervalstudied recently around whose boundaries a Higgs scalar takes non-zero values [34–36]. Recently, in Ref. [42], a class of self-consistent analytic solutions to gap equationshas been obtained in the C P N − model in the large- N limit in infinite space, byconstructing a map from the gap equations in the C P N − model to those in theGross-Neveu (GN) [43] or Nambu-Jona-Lasino [44] model, or equivalently to theBogoliubov-de Gennes (BdG) equation. The self-consistent analytic solutions thathave been found include inhomogeneous Higgs configurations, such as a soliton inwhich a Higgs scalar field is localized, a lattice of such the soliton and multiplesolitons with arbitrary separations, constructed from a real kink [45, 46], a real kinkcrystal [47, 48], and multiple kink-anti-kink configurations [49–53], respectively in theGN model. The appearance of the nonzero Higgs field, and consequently Nambu-Goldstone modes, are consistent with the Coleman-Mermin-Wagner theorem [54, 55]prohibiting Nambu-Goldstone modes in two spacetime dimensions, because the Higgsfield is confined in finite regions in our cases. The integrable structure [56] behindthe GN model [57, 58] and the map give infinite species of self-consistent analyticsolutions. The self-consistent analytic solutions for the case of a finite interval havebeen also obtained [36] from the corresponding solutions in the GN model [59]. TheGN model on a ring was also studied numerically [60] but the map to the C P N − model was not applied to this case. However, all the previous works include onlysolutions in which only one Higgs field component takes a value and so are essentiallysolutions in the C P model.In this paper, we construct self-consistent analytic solutions in which more thanone Higgs scalar component take nonzero values. This is possible when we imposetwisted boundary conditions or equivalently introduce a Wilson line for global flavorsymmetry along spatial direction. We first construct a single soliton and multiplesolitons on an infinite space. Around solitons, the Higgs phase appears with nonzeroHiggs scalar components. We then construct a soliton on a ring. Finally, we con-struct the Higgs and confining phases in a finite interval. For both cases, the Higgsfield diverges around the boundaries. For the former the Higgs fields are nonzero ev-erywhere, while for the latter the Higgs fields are nonzero almost everywhere exceptfor the center of the interval. All of them are genuine solutions in the C P N − modelwith N >
2. These solutions correspond to a non-Abelian vortex string stretched be-tween non-Abelian monopole and anti-monopole whose orientational C P N − modesare not aligned.This paper is organized as follows. In Sec. 2 we define the C P N − model andgive the gap equations in the large- N limit and explain our method. In Sec. 3 we– 2 –ive several examples of self-consistent analytic solutions to the gap equation. Sec. 4is devoted to summary and discussion. In the following, we consider the C P N − model whose Lagrangian is given by L = Z d x X i =1 , , ··· ,N (cid:2) D µ n ∗ i D µ n i − λ (cid:0) | n i | − r (cid:1)(cid:3) (2.1)where n i ( i = 1 , · · · , N ) are N complex scalar fields, D µ = ∂ µ − iA µ , and A µ and λ are auxiliary gauge and scalar fields, respectively. We set A µ = 0 in the following.We impose a twisted boundary condition which generates a twist of the flavor degreesof freedom. By picking up the first m components, we consider a boundary conditionunder which n , · · · , n m are twisted. The twisted boundary condition is equivalentto the presence of a Wilson line for a background SU ( m ) non-dynamical gauge field.Then, the Lagrangian can be rewritten as L = Z d x X i =1 , , ··· ,m (cid:2) ∂ µ n ∗ i ∂ µ n i − λ (cid:0) | n i | − r (cid:1)(cid:3) + Z d x (cid:2) ( D µ Σ) † D µ Σ − λ Σ † Σ (cid:3) , (2.2)where we have defined the m -component vector Σ = ( n , n , · · · , n m ) T and the co-variant derivative D µ = ∂ µ − iA µ with the background non-dynamical gauge potential A µ which is an m × m matrix and A = 0. The stationary condition for λ and theΣ † together with the Lorenz gauge ∂ µ A µ = 0 yields the gap equations[ − ∂ x + λ ( x )] f n ( x ) = ω n f n ( x ) ( n ≥ m ) , (2.3) N − m X n | f n | ω n + Σ † ( x )Σ( x ) − r = 0 , (2.4) − D x Σ( x ) + λ ( x )Σ( x ) = 0 . (2.5)Here the N − m factor appears in the second equation, whence the present calculationis valid in the sense of the N − m expansion. By using the redefinition of the fieldΣ = exp( i R x dxA ) ˜Σ, one obtains[ − ∂ x + λ ] f n = ω n f n ( n ≥ m ) , (2.6) N − m X n | f n | ω n + ˜Σ † ˜Σ − r = 0 , (2.7) − ∂ x ˜Σ + λ ˜Σ = 0 . (2.8)– 3 –n the following we choose the vector potential as A = A aµ T a ⊗ N − m,N − m , which in-duces the phase twist i R L Adx = P i γ i T i . Here T i ’s are SU ( m ) generators. We notethat the “gauge field ” and the mass function λ considered here becomes dynamicalby considering the higher order corrections. However, we restrict ourselves to theleading order, in which these auxiliary fields do not become dynamical.From now on, we focus on the case of m = 2 for simplicity, though the followingargument is straightforwardly applicable to arbitrary m . In the case of m = 2, onecan further rewrite the model as L = Z d x X i =1 , (cid:2) ∂ µ n ∗ i ∂ µ n i − λ (cid:0) | n i | − r (cid:1)(cid:3) + Z d x (cid:0) ∂ µ σ ∗ ∂ µ σ − λ | σ | (cid:1) , (2.9)where σ = ˜ n + i ˜ n or (˜ n , ˜ n ) = ( ℜ σ, ℑ σ ) vice versa. Now the gap equations to solvebecome [ − ∂ x + λ ( x )] f n ( x ) = ω n f n ( x ) ( n ≥ , (2.10) N − X n | f n | ω n + | σ ( x ) | − r = 0 , (2.11) − ∂ x σ ( x ) + λ ( x ) σ ( x ) = 0 . (2.12)In order to ensure the real eigenvalues for ω n , we consider λ to be real such that − ∂ x + λ to be Hermitian. Eq. (2.12) describes a zero mode solution for the Schr¨odingerequation ( − ∂ x + λ ) u = Eu . By rewriting the λ as λ = ∆ + ∂ x ∆ , (2.13)one can solve Eq. (2.12) as σ ( x ) ∝ exp (cid:20)Z x dy ∆( y ) (cid:21) . (2.14)We note that the other linearly independent solution is not normalizable. The re-maining two equations (2.10) and (2.11) can be solved by the mapping given in theprevious work (See Appendix A). The corresponding energy functional is given as E tot = ( N − X n ω n − r Z ∞−∞ dx (∆ + ∂ x ∆) + σ∂ x σ | ∞−∞ . (2.15) In this section we construct self-consistent analytic solutions on an infinite space, aring, and a finite interval in each subsection.– 4 – .1 Infinite system
We first consider the infinite size system. Some solutions for Eqs. (2.10)–(2.12) aregiven by [42] σ = 0 , Ae iφ cosh mx , · · · etc . (3.1)Here we note that a localized Higgs soliton Ae iφ cosh − mx or a localized Higgs solitonlattice which we describe below are not inhibited from the Coleman-Mermin-Wagnertheorem since these solutions do not have a long range order.The C P modes are given by (cid:18) n n (cid:19) = e i R x A ( y ) dy (cid:18) ˜ n ˜ n (cid:19) = e i R x A ( y ) dy (cid:18) ℜ σ ℑ σ (cid:19) . (3.2)As an example, if we choose A = ασ x , we obtain (cid:18) n n (cid:19) = e iαxσ x (cid:18) ℜ σ ℑ σ (cid:19) = (cid:18) cos αx ℜ σ + i sin αx ℑ σ cos αx ℑ σ − i sin αx ℜ σ (cid:19) . (3.3)For the case of A = βσ y , we obtain (cid:18) n n (cid:19) = e iβxσ y (cid:18) ℜ σ ℑ σ (cid:19) = (cid:18) cos βx ℜ σ + sin βx ℑ σ cos βx ℑ σ − sin βx ℜ σ (cid:19) . (3.4)Without loss of generality, we choose A = βσ y and ℜ σ = m, ℑ σ = 0, yielding (cid:18) n n (cid:19) = m (cid:18) cos βx − sin βx (cid:19) . (3.5)In the case of A = γσ z , one obtains (cid:18) n n (cid:19) = e iγxσ z (cid:18) ℜ σ ℑ σ (cid:19) = (cid:18) e iγx ℜ σe − iγx ℑ σ (cid:19) . (3.6)In Fig. 1, we plot the Higgs lattice solution given by (cid:18) n n (cid:19) ∝ (cid:0) −√ ν sn( mx, ν ) + dn( mx, ν ) (cid:1) − / √ ν (cid:18) cos βx − sin βx (cid:19) . (3.7)for A = βσ y which corresponds to ∆ = sn( x, ν ). Here sn and dn are the Jacobi’selliptic functions and ν is the elliptic parameter.In Fig. 2, we plot a doubly localized Higgs soliton solution corresponding to∆ = k tanh[ kx − kδ + R ] − ω b e R [sinh( m + x − kδ + 2 R ) + sinh( m − x + kδ )]cosh( m + x − kδ + 2 R ) + e R cosh( m − x + kδ ) , (3.8)where ω b = √ m − k , R = (1 /
2) ln( m + /m − ), and m ± = m ± k . The left and rightpanels represent the twisted two soliton solution with k = 0 . k = 0 . δ = 0 and m = 1. Bymaking the separation larger, the relative twist of the two peaks becomes larger.– 5 – n Figure 1 . The flavor winding localized Higgs lattice configuration with A = βσ y . Herewe set ν = 0 . β = 3 π/ ν ), and m = 1. The solution has bright soliton latticeconfiguration for the amplitude and the flavor rotates along the spatial axis. n n -5 15 n n -10 10 Figure 2 . The two soliton solution with different elliptic parameters with A = βσ y . Herewe set ν = 0 . ν = 0 .
999 (right panel). We choose β to be π/
30. We canobserve that when we change the distance between the solitons, one rotates in the flavorspace.
Next we consider the case of the C P N − model on a ring with the circumference L . In this case, the twisted boundary condition can be generated by the Aharonov-Bohm(AB)-like effect for the flavor degrees of freedom. We consider the periodiccondition for all { n i } . By the singular gauge transformation(Σ T , n , · · · , n N ) = ( e i R x dyA ˜Σ T , n , · · · , n N ) , (3.9)the twisted boundary condition Σ( x + L ) = e iβ Σ( x ) (3.10)becomes e i R x + Lx dyA ( y ) ˜Σ( x + L ) = e iβ ˜Σ( x ) , (3.11)– 6 – n n n n n (a) (b)(c) α = π/2 α =-3 π/2α =5 π/2 (a) (b)(c) -3 π/2π/25π/2 (d) Figure 3 . The solution for the twisted boundary condition on the ring. Here we set αL = π/ − π/ π/ d ), we depict the schematics of the phasewinding. whereas the background gauge field is completely eliminated from the self-consistentequations. Here β is 2 × SU (2) generator, we have exp( iασ i x ) = cos αx + iσ i sin αx . Inthe case of A ( y ) = ασ i , the above twisted boundary condition becomes(cos αL + iσ i sin αL ) ˜Σ( x + L ) = e iβ ˜Σ( x ) . (3.12)This shows the periodic structure on α which is similar to the AB oscillation effect.Because of this periodicity, α has 2 π ambiguity. The lowest energy state correspondsto the case of − π ≤ α < π and the solutions with π ≤ | α | corresponds to the higherenergy states. This can be easily shown as follows. If we move to the non-twistedboundary problem with the gauge potential, all of those solutions corresponds to thehomogeneous solution. Thus the energy difference comes only from the gauge fieldpart given by ∝ A and one can show the solution ( a ) has the lowest energy.Apart from the phase winding induced by the boundary twisting, we have anothercondition for ˜Σ ˜Σ( x + L ) † ˜Σ( x + L ) = ˜Σ( x ) † ˜Σ( x ) , (3.13)since e iασ i L is an unitary matrix. This means that the amplitude of the Higgs fieldalso need to be periodic. – 7 –n Fig. 3 we plot a homogeneous solution with the flavor rotation which corre-sponds to σ = me iφ . This solution is not allowed in the infinite system since the gapequation is not satisfied due to the absence of the infrared cutoff played by 1 /L inthe ring case. The absence of the long-range-ordered solution in the infinite systemis consistent with the Coleman-Mermin-Wagner theorem.This solution obviously satisfies the condition (3.13). The Fig. 3 ( a ) shows thesmallest winding solution. The higher winding states are also shown in ( b ) and ( c )for the same boundary condition. The physical meaning of those solutions can beunderstood from the schematic figure ( d ). Because the 2 π periodicity of αL , we haveinfinite branches of the solutions which gives the same twisting of the boundary. Forexample, the solution ( b ) is the solution with the second smallest phase winding inwhich the flavor rotates opposite way compared with the solution ( a ). The solution( b ) can be interpreted as the case of αL = π/ − π which means that the solution ( b )belongs to the neighbor branch to the branch for ( a ). In the same way, the solution( c ) is understood as the case of αL = π/ π . Finally we consider the case of the C P N − model on a finite interval which is rele-vant to describe the C P N − modes on a vortex string connecting heavy monopole and(anti-)monopole in the U ( N ) gauge theory. If we have the flavor twisting between theopposite edges of the string, the solution (3.2), which matches the twisted bound-ary conditions, becomes a solution. The other important point for this boundarycondition is that the Higgs field Σ unavoidably diverges at the edges.One of self-consistent analytical solutions of this problem can be given byΣ ∝ exp (cid:20)Z dx L sn(4K x/L, ν ) (cid:21) e iγσ i x/L (cid:18) cos α sin α (cid:19) , (3.14)where we considered the twisted boundary condition on Σ such asΣ(0) = (cos γ + iσ i sin γ ) Σ( L ) . (3.15)Here γ is a constant parameter which characterizes the twisting of the boundarycondition, ν is the elliptic parameter, and K( ν ) is the complete elliptic integral ofthe first kind. We also have another parameter α which could be set to α = 0 by afield redefinition since it corresponds to the overall U (1) factor of the n + in . Thus,we obtain Σ ∝ exp (cid:20)Z dx L sn(4K x/L, ν ) (cid:21) e iγσ i x/L (cid:18) (cid:19) . (3.16)This vanishes in the infinite size limit ( L → ∞ , ν → λ becomes constant and thus this solution corresponds to the confiningphase in the large size limit. We plot the confining phase solutions in Fig. 4.– 8 – n n n n (a) (b) Figure 4 . The winding solution for the Dirichlet boundary condition corresponding tothe confining phase. The left panel is an untwisted solution constructed in Ref. [36] whilethe right panel corresponds to the case with a twist of γ = π/
3. For the both cases we set ν = 0. The Higgs fields vanish at the center. n n n n γ Figure 5 . The winding solution for the Dirichlet boundary condition corresponding to theHiggs phase. The left panel is an untwisted solution constructed in Ref. [36] while the rightpanel corresponds to the case with a twist of γ = π/
3. For the both cases we set ν = 0.The Higgs field does not vanish anywhere in contrast to the case of the confining phase. There is another solution which corresponds to the Higgs phase in the infinitesize limit where the Higgs field becomes the plane-wave like solution Σ( x ) = const · exp( iσ i ˜ γx ) · (1 , T :Σ ∝ exp (cid:20) − Z dx x/L + K , ν ) L sn(2K x/L + K , ν ) (cid:21) e iγσ i x/L (cid:18) (cid:19) . (3.17)We plot the Higgs phase solutions in Fig. 5. This solution is inhibited in the infinitesystem since the gap equation is no longer satisfied in the limit. In other words,this solution is possible only in a finite system, to be consistent with the Coleman-Mermin-Wagner theorem. For both solutions, the energy is scaled as 1 /L [36].– 9 – Summary and discussion
In the paper, we have constructed self-consistent analytic solutions of the C P N − model in the large- N limit with the twisted boundary condition or equivalently withthe background SU ( m ) ( m < N ) gauge field in the flavor space. The resultingsolutions describe the various Higgs configurations with the SU ( m ) flavor rotation.In the present analysis, we have assumed that the flavor rotation is uniform sincea nonuniform rotation costs more energy. However, nonuniform backgrounds couldappear in a specific setup, which remains as a future problem.In this paper, we have used the mapping from the Gross-Neveu model to the C P N − model. It might be possible to generalize this mapping to the case of the chi-ral Gross-Neveu model, where the complex kink solution[61], complex kink crystalsolution [62], and the complex kink with arbitrary separation [63] have been ob-tained. In Ref. [64], a confining soliton in the Higgs phase was obtained, in whicha confinement phase is localized in the soliton core. This solution can be twisted aswell.Our solutions could also be applicable to the condensed matter physics, for ex-ample, to the magnetic order [65–68].In the case of a single component at a finite interval with the Dirichlet bound-ary condition, the Casimir force depending on the size of the system was discussedbefore [35, 36], where the Casimir force gives either attractive or repulsive pressureto the system size. For the case of the twisted boundary conditions studied in thispaper, one can further discuss a Casimir force acting on the flavor internal spaceof C P N − . In this case, the force gives either attraction or repulsion between the C P N − modes on the boundaries. In the context of a non-Abelian string stretchedbetween a monopole and (anti-)monopole, this force attains ferromagnetic or anti-ferromagnetic properties, respectively, on the monopole and (anti-)monopole.The twisted boundary condition in the temporal direction has also been investi-gated. The large-N volume independence and the absence of the Affleck transitionhave been shown in the setup [69]. Our formalism and inhomogeneous solutions maypossibly be used also in this case.During completion of this paper, we were informed that the authors of Ref. [35]were writing a paper on similar configurations. Acknowledgement
We thank Sven Bjarke Gudnason, Kenichi Konishi and Keisuke Ohashi for corre-spondence. The support of the Ministry of Education, Culture, Sports, Science(MEXT)-Supported Program for the Strategic Research Foundation at Private Uni-versities ‘Topological Science’ (Grant No. S1511006) is gratefully acknowledged. The– 10 –ork of M. N. is supported in part by the Japan Society for the Promotion of Sci-ence (JSPS) Grant-in-Aid for Scientific Research (KAKENHI Grant No. 16H03984and 18H01217) and by a Grant-in-Aid for Scientific Research on Innovative Areas“Topological Materials Science” (KAKENHI Grant No. 15H05855) from the MEXTof Japan.
A Mapping between the Gross-Neveu model and C P N − model In this appendix, we review the mapping between the GN model and the C P N − model [42]. We consider the gap equations (2.10)–(2.12) which have to be solvedself-consistently. By using ∆ defined in Eq. (2.13), the Klein-Gordon-like equation(2.10) can be rewritten as the following Dirac-like equation (cid:18) ∂ x + ∆ − ∂ x + ∆ 0 (cid:19) (cid:18) f n g n (cid:19) = ω n (cid:18) f n g n (cid:19) . (A.1)By eliminating g n , one obtains Eq. (2.10). The same procedure for Eq. (2.12) yields (cid:18) ∂ x + ∆ − ∂ x + ∆ 0 (cid:19) (cid:18) στ (cid:19) = 0 . (A.2)This equation is nothing but an equation for zero modes. Thus, the solution is givenby Eq. (2.14). Differentiating Eq. (2.11) by x and substituting the solution (2.14)into that, one obtains ∆ = N − r X n f n g n . (A.3)This equation self-consistently determines ∆ together with Eq. (A.1).Now the three gap equations reduces to the two equations (A.1) and (A.3). Thelatter two equations coincide to the gap equations appearing in the GN model; theso-called Bogoliubov-de Gennes equation (A.1) and the gap equation (A.3). Bydetermining ∆ from those equations, one can calculate σ by Eq. (2.14). It shouldbe noted that the normalization of σ must be fixed from Eq. (2.11), since Eq. (A.3)is obtained from the differentiation of Eq. (2.11) and thus the information for thenormalization is lacking. For example, the constant solution ∆ = m exists for Eqs.(A.1) and (A.3) in the infinite system, but this solution cannot satisfy Eq. (2.11). References [1] H. Eichenherr, “SU(N) Invariant Nonlinear Sigma Models,” Nucl. Phys. B , 215(1978). – 11 –
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