Self-Consistent Large-$N$ Analytical Solutions of Inhomogneous Condensates in Quantum ${\mathbb C}P^{N-1}$ Model

We give, for the first time, self-consistent large-$N$ analytical solutions of inhomogeneous condensates in the quantum ${\mathbb C}P^{N-1}$ model in the large-$N$ limit. We find a map from a set of gap equations of the ${\mathbb C}P^{N-1}$ model to those of the Gross-Neveu (GN) model (or the gap equation and the Bogoliubov-de Gennes equation), which enables us to find the self-consistent solutions. We find that the Higgs field of the ${\mathbb C}P^{N-1}$ model is given as a zero mode of solutions of the GN model, and consequently only topologically nontrivial solutions of the GN model yield nontrivial solutions of the ${\mathbb C}P^{N-1}$ model. A stable single soliton is constructed from an anti-kink of the GN model and has a broken (Higgs) phase inside its core,in which ${\mathbb C}P^{N-1}$ modes are localized,with a symmetric (confining) phase outside. We further find a stable periodic soliton lattice constructed from a real kink crystal in the GN model,while the Ablowitz-Kaup-Newell-Segur hierarchy yields multiple solitons at arbitrary separations.


Introduction
Nonlinear sigma models such as the CP N −1 model in 1+1 dimensions [1][2][3] are known to share a number of phenomena common with 3+1 dimensional QCD, e.g. asymptotic freedom, dynamical mass generation, confinement, and instantons [4][5][6][7][8][9][10][11]. The mass gap can be best shown in the large-N analysis in which one solves the gap equations self-consistently, to be consistent with the Coleman-Mermin-Wagner (CMW) theorem forbidding a gapless excitations in 1+1 dimensions [12,13]. The CP N −1 model, or the CP 1 model equivalent to the O(3) sigma model, appears in a wide range of physics from particle physics to condensed matter physics. The relation between the 1+1 dimensional Heisenberg antiferromagnetic spin chain and the O(3) sigma model has been shown in Ref. [14,15]. Recently, the quantum phase transition, so-called deconfined criticality is proposed in the antiferromagnetic system [16,17]. The sigma model with topological term is known to describe the integer quantum Hall effect [18]. The supersymmetric CP N −1 model was also investigated [19,20] for which the all order calculation in coupling constant is possible for Gell-Mann-Low function [11], and dynamical mass gap was proved by the mirror symmetry [21]. The analogy between 3+1 dimensional Yang-Mills theory and 1+1 dimensional sigma model, pointed out in Ref. [4], has been recently revealed in a rather nontrivial way; a non-Abelian vortex string in a U(N) gauge theory with N scalar fields in the fundamental representation carries CP N −1 moduli [22][23][24] (see Refs. [25][26][27][28] as a review), yielding a nontrivial relation between the CP N −1 model on the string worldsheet and the bulk gauge theory [29,30]. The CP N −1 model defined on an interval [31,32] or on a ring [33] was also studied. The CP N −1 model or the O(3) sigma model at finite temperature and/or density was also investigated in which Berezinskii-Kosterlitz-Thouless transition at nonzero density was examined [34]. One of recent developments is a resurgent structure of the CP N −1 model [35,36], in which a molecule of fractional instantons [37,38] called a bion, plays a crucial role. In spite of tremendous studies of the CP N −1 model, there was no study on inhomogeneous configurations (such as solitons) at quantum level,except for a numerical study of the CP N −1 model on an interval [32].
The situation is rather different for an interacting fermionic theory: the Gross-Neveu (GN) [39] or Nambu-Jona-Lasino model [40], exhibiting dynamical symmetry breaking of discrete or continuous chiral symmetry, thereby sharing an important property with QCD [41][42][43]. This model is equivalent at the large-N limit or in the mean field approximation to a set of the Bogoliubov-de Gennes (BdG) equations and the gap equation, appearing in condensed matter systems such as conducting polymers [44][45][46], superconductors, superfluids and ultracold atomic gases [47][48][49]. Selfconsistent analytical solutions such as a real kink [41,44], a twisted (complex) kink [42], a real kink-anti-kink (polaron) [41,50], a real kink-anti-kink-kink [43,51,52] and more general real solutions [53] have been known. Recently, a theoretical progress has been achieved for inhomogeneous condensates in the 1+1 dimensional (chiral) GN model, e.g., the exact self-consistent and inhomogeneous condensates such as a real kink crystal [54] (Larkin-Ovchinnikov(LO) state [55]), a chiral spiral (Fulde-Ferrell(FF) state [56]), and a twisted kink crystal [57] (FF-LO state) have been found by mapping the equations to the nonlinear Schrödinger equation, and such states have been shown to be ground states in a certain region of the phase diagram for finite temperature and density [58]. More generally, multiple twisted kinks with arbitrary phase and positions [59] can be further constructed systematically due to the integrable structure behind the model known as the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy for the nonlinear Schrödinger equation [60][61][62]. Recent developments include time-dependent soliton scatterings [63], multi-component condensates [64,65], a ring geometry [66], and an interval with a Casimir force [67].
In the present work, we reveal an unexpected relation between these two completely different theories, the CP N −1 and GN models developed independently. By finding a map from a set of gap equations of the CP N −1 model to those of the GN model, we find self-consistent analytical solutions of stable inhomogeneous condensates in the quantum CP N −1 model, that is, a single soliton, a soliton lattice and multiple solitons at arbitrary separations.

Model and method
We consider the CP N −1 model on an infinite space: where n i (i = 1, · · · , N) are complex scalar fields, D µ = ∂ µ − iA µ , and λ(x) is a Lagrange multiplier. The "radius" r is known to have connection with a coupling constant g YM in the Yang-Mills theory; r = 4π/g 2 YM if we realize this model on a non-Abelian vortex in U(N) gauge theory. Here we note that the model does not have kinetic term for A µ and thus we focus on the case of A µ = 0 throughout this paper. We separate n i fields into a classical field n 1 = σ (real) and n i = τ i (2, · · · , N). Integrating out the τ i fields, we obtain the effective action for σ as In the following we consider and the leading contribution of 1/N expansion and thus we replace N − 1 to N 1 . One can formally write down the total energy functional as The corresponding gap equations obtained from the static condition with respect to λ and σ are [32] N 2 n f 2 n ω n + σ 2 − r = 0, (2.4) respectively, where f n (x) and ω n are orthonormal eigenstates and eigenvalues of the following equation We need to solve Eqs. (2.4)-(2.6) in a self-consistent manner. We here note from Eqs. (2.5) and (2.6) that σ is proportional to a zero mode f 0 . It is well known that assuming a uniform state in infinite system, one finds the confining (unbroken) phase with a constant λ to be a unique solution, to be consistent with the CMW theorem. For the case of a ring, in addition to it, there is a Higgs (broken) phase with a constant σ for a smaller ring [33].
One of the main results of this paper is a map from those equations to the gap equation and eigenvalue equation for the GN model. In order to reduce the number of equations, we introduce the new field ∆ such as (2.7) By using this function, we find a solution to Eq. (2.5): where A is the integral constant. The energy in Eq. (2.3) can be rewritten as The rather nontrivial step is to rewrite Eq.
where g n 's are auxiliary fields and the elimination of g n yields Eq. (2.6). We note that Eq. (2.10) together with Eq. (2.7) describes a supersymmetric quantum mechanics, in which the potential λ is given by the superpotential ∆ [68]. Eq.
The corresponding Hartree-Fock equation becomes Hψ = Eψ, with H = −iγ 5 ∂ x + γ 0 ∆, where γ 5 = −σ 2 and γ 0 = σ 1 with the Pauli matrices σ i . Here ∆ (real) satisfies ψ ψ = −∆/g, which is called a gap equation. It is known that the Z 2 symmetry is spontaneously broken in the GN model, yielding two discrete vacua. With a help of g n = (−∂ x + ∆)f n /ω n , one can show that g n automatically gives a orthonormal set if f n gives a orthonormal set. Eq. (2.10) has the particle-hole symmetry which enables us to obtain the set {−ω n ,f n ,g n } from the set {ω n , f n , g n } byf n = f n andg n = −g n . By taking the derivative of Eq. (2.4) with respect to x and by substituting Eqs. (2.8) and ω n g n = (−∂ x + ∆)f n into that, we obtain where the corresponding auxiliary field is g 0 (x) = 0. The zero mode solutions f 0 in the CP N −1 and GN models are identical. As denoted below Eq. (2.6), the Higgs field σ(x) in the CP N −1 model is proportional to the zero mode, thereby exists only when corresponding ∆ in the GN model is topologically nontrivial with allowing a normalizable zero mode [69] 3 Self-consistent analytical solutions In the GN model, a constant gap ∆ = m is a solution which can be called the Bardeen-Cooper-Schrieffer (BCS) phase, whereas that for m = 0 is called a normal phase. We show that the BCS and normal phases in the GN model correspond to the confining and Higgs phases in the CP N −1 model, respectively. For the constant solution, ω n = (πn/L) 2 + m 2 and the degenerated eigenfunctions are f We find that the condition (3.1) for m = 0 and (3.2) for σ = 0 are equivalent which gives the well known renormalization condition of the coupling constant g 2 = 4π/r. This results in two possibilities {λ = m 2 , σ = 0} (confining phase) and {λ = 0, σ = const} (Higgs phase), but only the former satisfies the gap equation (2.4) and the latter is not allowed in the infinite system [31,33].
The solution ∆ = −m tanh mx is known as a topological kink solution interpolating two discrete vacua of the GN model, which has a zero mode localized near the kink. In the case of kink solution, the eigenvalue is the same with the constant solution ω n = (πn/L) 2 + m 2 while the degenerated eigenfunctions are f n = √ 2 cos πnx/L. We also have a normalizable zero mode f 0 (x) ∝ 1/ cosh mx, g 0 (x) = 0. Thus Eq. (2.12) yields which has a bright solitonic profile. Again, it is indeed proportional to the zero mode solution. In this case, the mass gap function becomes which has a gray soliton configuration and is called the Pöschl-Teller potential [68]. Since all the eigenenergies of this solution are non-negative, the solution is stable. In Fig. 1, we plot the configuration of σ(x) and the mass gap function λ(x). The energy of the soliton can be calculated by the energy E s for the soliton configuration in Eqs. (3.6) and (3.7) subtracted by E 0 for the confining phase (σ 0 = 0 and λ 0 = m 2 ), for both of which the third term in Eq. (2.9) vanishes from the equation of motion (2.5) and ω n 's are the same. We thus obtain Since σ has a localized profile function, soliton core is in the Higgs (broken) phase where the CP N −1 modes are localized, while the bulk is in the confining (symmetric) phase, in contrast to a uniform system allowing only the confining phase in infinite system to be consistent with the CMW theorem. It is known that the correlation function behaves at large distance as x −1/N in 1+1 dimension [70], which inhibits the long-range order for finite N. Here we have obtained the Higgs phase localized with length ∼ 1/m, thus the robustness of our solitonic solution is expected if N is sufficiently large as ln(1/m) ≪ N 2 . The above solutions can be obtained from a soliton lattice obtained from a real kink crystal in the GN model: where sn, cn, and dn (appearing later) are the Jacobi functions and ν is elliptic parameter. Here the periodicity of the above solution is given by ℓ = 4K(ν)/m, where K(ν) is a complete elliptic integral of the first kind. This solution together with Eq. (2.8) gives a soliton lattice: (3.10) In Fig. 2, we plot the mass gap function λ and σ for ν = 10 −2 and ν = 1 − 10 −2 .
The auxiliary field ∆ are also plotted. The Higgs field σ in this solution has a bright soliton lattice profile. By taking ν = 1 limit for ∆ = msn(mx + K(ν), ν), λ becomes constant and σ = 0 in the whole system. This limit corresponds to the constant solution discussed above. On the other hand, ∆ = msn(mx + 2K(ν), ν)

Higher order self-consistent analytical solutions
In the GN model, the integrable structure enables us to systematically construct all possible exact self-consistent solutions [61,62]. The above solutions belong to the lowest order (n = 1) of the AKNS hierarchy (denoted by AKNS n for n = 1, 2, · · · ) for the nonlinear Schrödinger equation [61,62] , and m ± = m ± k. In Fig.  3, we plot the configurations of σ, λ, and ∆ for various parameter choices. The symmetric case δ = 0 (a) looks like a double copy of a single soliton in Fig. 1. For larger δ the middle kink is closer to the right anti-kink than the left anti-kink in ∆ as (b), and then the amplitude of the Higgs field σ localized in the right soliton of λ decreases with increasing δ. On the other hand, the parameter k controls the soliton-soliton distance [(a), (c), and (d)]. The two solitons merge for larger k and eventually becomes one soliton in k → 1. This is possible because the three kink solution belongs to the same topological sector with the single kink solution in the GN model. In general, AKNS 2k+1 (k = 1, 2, · · · ) yields solutions of k solitons with arbitrary positions exhibiting the similar behaviors.

Summary
We have found the map from the GN model to the CP N −1 model, which enables us to construct, for the first time, the exact self-consistent stable inhomogeneous solutions of the CP N −1 model; a single soliton, a soliton lattice and multiple solitons with arbitrary separations. The Higgs (broken) phase appears inside the soliton cores where the Higgs field σ has bright solitonic profiles and the CP N −1 moduli are confined.
It is an open question whether there is a map to the chiral GN model with continuous chiral symmetry, which allows a variety of complex solutions. In the (chiral) GN model, the inhomogeneous phase is stabilized at the low temperature and high density [58], or in the presence of a chiral chemical potential, equivalent to the constant Zeeman magnetic field on the superconductivity [47]. Such analogies in the CP N −1 model may imply a possibility of a crystaline phase. While our periodic soliton lattice can be put on a ring, an extension to an interval [31,32] is also possible to calculate a Casimir force [71], since the exact solutions in the GN model on an interval have been found recently [67]. Another relation between the the CP N −1 model and the GN model in 2 + 1 dimensions has recently been found in Ref. [72] in which the large-N free energy densities for the both theories are found to be remarkably similar. Though it would be important to see whether the similar structure also appears in the 1 + 1 dimensions, we leave it as a future problem. The connection between our formalism and the bosonization scheme in 1 + 1 dimensions should be also important. The former gives the coincidence of the self-consistent equations in CP N −1 model and the GN model, whereas the latter yields the sine-Gordon model as the bosonized model of the GN model [70]. We also leave it as a future problem. Physical consequences of our solitons on a non-Abelian vortex in supersymmetric gauge theories [25][26][27][28] or dense QCD [73] will be an important problem to be explored.

A Alternative mapping
In this Appendix, we show an alternative map from the Gross-Neveu model to the CP N −1 model. In our formalism, f n 's are chosen as upper components of BdG For the same ∆, one can also definẽ These functions satisfy In this Appendix, we briefly summarize the self-consistent treatment of Gross-Neveu model studied in Refs. [61,62]. The Lagrangian of the chiral Gross-Neveu model with N flavor is given by where g > 0. By introducing the auxiliary fields ∆ 1 = −g 2 ψ ψ and ∆ 2 = −g 2 ψ iγ 5 ψ , and by taking the large N approximation (or mean field approximation) one can obtain the following effective Lagrangian Thus we obtain the following total energy with the Bogoliubov-de Gennes (BdG) Hamiltonian The consistency condition of the auxiliary field ∆ 1 and ∆ 2 are called the gap equa- which must be solved in a consistent manner with the BdG equation Hψ = Eψ. Here the left hand sides of the gap equations can be, respectively, rewritten as N ψ 1 ψ 1 and N ψ 1 iγ 5 ψ 1 , since the N flavors gives the same contributions, e.g., ψ 1 ψ 1 = ψ 2 ψ 2 = · · · = ψ N ψ N . Thus we can rewrite the gap equations as In the following, we use the chiral representation γ 0 = σ 1 , γ 1 = −iσ 2 , and γ 5 = σ 3 . For the BdG Hamiltonian, the Gor'kov resolvent R(x; E) = 1/ x|(H − E)|x satisfies the Dikii-Eilenberger equation where ∆ = ∆ 1 − i∆ 2 . We note that the BdG equation can be written as ∂ x ψ = Qψ. The Gor'kov resolvent must satisfies the conditions det R = − 1 4 , TrRσ 3 = 0, and R † = R.
The Dikii-Eilenberger equation and the BdG equation can be rewritten as with the constraint ∂ t Q = 0. The first equation is the integrable condition (zero curvature condition) of this system; ∂ x ∂ t ψ = ∂ t ∂ x ψ. Since we find the connection between BdG system to the AKNS system, by using the machinery of the integrable system, one can systematically expand the resolvent Rσ 3 which yields AKNS n as The fermionic solutions are also calculable as where ψ = (ψ 1 , ψ 2 ) T and C is the normalization constant. The square-root of those function must be taken such as v/u = iV 21 /(iV 11 − ω).