Self-consistent large-N analytical solutions of inhomogeneous condensates in quantum ℂPN − 1 model

We give, for the first time, self-consistent large-N analytical solutions of inhomogeneous condensates in the quantum ℂPN − 1 model in the large-N limit. We find a map from a set of gap equations of the ℂPN − 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 ℂPN − 1 model is given as a zero mode of solutions of the GN model, and consequently only topologically non-trivial solutions of the GN model yield nontrivial solutions of the ℂPN − 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 ℂPN − 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][4] are known to share a number of phenomena common with 3+1 dimensional QCD, e.g. asymptotic freedom, dynamical mass generation, confinement, and instantons [5][6][7][8][9][10][11][12]. 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 [13,14]. 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 refs. [15,16]. Recently, the quantum phase transition, so-called deconfined criticality is proposed in the antiferromagnetic system [17][18][19]. The sigma model with topological term is known to describe the integer quantum Hall effect [20]. The supersymmetric CP N −1 model was also investigated [21,22] for which the all order calculation in coupling constant is possible for Gell-Mann-Low function [12], and dynamical mass gap was proved by the mirror symmetry [23]. The analogy between 3+1 dimensional Yang-Mills theory and 1+1 dimensional sigma model, pointed out in ref. [5], 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 [24][25][26][27] (see refs. [28][29][30][31] as a review), yielding a nontrivial relation between the CP N −1 model on the string worldsheet and the bulk gauge theory [32,33]. The CP N −1 model defined on an interval [34][35][36] or on a ring [37,38] 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 JHEP12(2017)145 was examined [39,40]. One of recent developments is a resurgent structure of the CP N −1 model [41][42][43][44][45], in which a molecule of fractional instantons [46][47][48][49] 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 [36].
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 JHEP12(2017)145 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 3) The corresponding gap equations obtained from the static condition with respect to λ and σ are [36] 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 [37,38].
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

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The rather nontrivial step is to rewrite eq. (2.6) as [See appendix A] 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 ∆ [87].
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 and g 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 which has the same form with the gap equation for the GN model. Here we note that corresponding fermionic coupling N g 2 = N/2r is proportional to the 't Hooft coupling in an underlying U(N ) gauge theory N g 2 YM . Since we solve the differentiated one instead of eq. (2.4) itself, we need to fix the integration constant A for σ by substituting eq. (2.8) into eq. (2.4). For the BdG equation (2.10) and gap equation (2.12), various exact selfconsistent solutions are already known. From eq. (2.10) one can immediately find the zero mode solution 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 [88].
We find that the condition 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 [87]. Since all the eigenenergies of this solution are non-negative, the solution is stable. In figure 1 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 [89], 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:

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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 = 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 figure 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.

Higher order self-consistent analytical solutions
In the GN model, the integrable structure enables us to systematically construct all possible exact self-consistent solutions [78,79]. 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 [78,79] does.
Here ω b = √ m 2 − k 2 , R = (1/2) ln(m + /m − ), and m ± = m ± k. In figure 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 figure 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 [74], or in the presence of a chiral chemical potential, equivalent to the constant Zeeman magnetic field on the superconductivity [60]. 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 [34][35][36] is also possible to calculate a Casimir force [90], since the exact solutions in the GN model on an interval have been found recently [86]. Another relation between the CP N −1 model and the GN model in 2 + 1 dimensions has recently been found in ref. [91] 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 [89]. We also leave it as a future problem. Physical consequences of our solitons on a non-Abelian vortex in supersymmetric gauge theories [28][29][30][31] or dense QCD [92] 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 equation (u n = f n , v n = g n ) in 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. [78,79]. 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 equations 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

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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 higher components are calculable with a help of the following formula We can also obtain the nonlinear Schrödinger equations for this system as n+1 j=1 c j M (j) 12 = 0. The AKNS 0 , AKNS 1 , AKNS 2 for instance, yield 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).

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