Large-N C P N − 1 sigma model on a finite interval

We analyze the two-dimensional CPN−1 sigma model defined on a finite space interval L, with various boundary conditions, in the large N limit. With the Dirichlet boundary condition at the both ends, we show that the system has a unique phase, which smoothly approaches in the large L limit the standard 2D CPN−1 sigma model in confinement phase, with a constant mass generated for the ni fields. We study the full functional saddle-point equations for finite L, and solve them numerically. The latter reduces to the well-known gap equation in the large L limit. It is found that the solution satisfies actually both the Dirichlet and Neumann conditions.


Introduction
The non linear CP N −1 sigma model in two dimensions has been extensively studied in the past starting from the works [1,2]. It has many properties in common with QCD in four dimensions, such as asymptotic freedom, mass gap, confinement and the existence of a topological sector. Moreover it has the advantage of being solvable in the large-N limit. Recently the CP N −1 sigma model appeared in the study of low energy effective action on certain topological solitons such as the non-Abelian string [3,4,5].
In this paper we study the large-N solution of the bosonic CP N −1 model on a finite interval of length L with various types of boundary conditions. The periodic boundary condition has been previously studied as a thermal compactification in [6,7] and more recently in [8]. One key feature for the solvability of this model is that the periodic compactification preserves translational invariance.
The problem with the Dirichlet boundary condition in a finite interval has been studied recently in [9] by using a translational invariance Ansatz, where two possible phases were found: a confinement phase with mass generation at large L and a Higgs phase for a shorter string L < L crit ∼ 1/Λ.
It is the purpose of the present paper to examine in depth the bosonic CP N −1 model on a finite-width strip, with Dirichlet or Neumann conditions at the space boundaries. The exact functional saddle-point equations will be studied in the large N limit, with no a priori assumption about the translational invariance. It will be seen that while with the periodic boundary condition the translational invariance Ansatz is indeed consistent with the full set of equations of motion, the generalized gap equations cannot be satisfied by a translational invariant (constant) mass-generation Ansatz when the D-D or N-N boundary condition is used. The solution which satisfies the full set of equations is found numerically, for various values of L.
Contrary to a claim made in [9] we find that the system has a unique phase, which smoothly approaches the "confinement phase" in the large L limit. The physical reason for this result may be found in the fact that, for small L, the system reduces in the infrared (the large wavelength) limit effectively to a one-dimensional (quantum mechanics) system where we do not expect to find any phase transitions.

CP N−1 sigma model on a finite space interval
The classical action for the CP N −1 sigma model is given by S = dxdt ((D µ n i ) * D µ n i − λ(n * i n i − r)) , (2.1) where n i with i = 1, . . . , N are N complex scalar fields and the covariant derivative is D µ = ∂ µ − iA µ . Configurations related by a U(1) gauge transformation z i → e iα z i are equivalent: the U(1) gauge field A µ does not have a kinetic term in the classical action. λ is a Lagrange multiplier that enforces the classical condition n * i n i = r , (2.2) where r is the "size" of the CP N −1 manifold, which is the radius of S 2 sphere for N = 2, and can also be expressed in terms of the coupling constant by Since n appears only quadratically in the Lagrangian, it can be integrated out in the partition function thus leaving an effective action for A µ and λ: The condition of stationarity with respect to λ leads to the gap equation which is conveniently written in the Euclidean formulation as where we have set A µ = 0 and λ = m 2 . An expectation value of λ provides a mass for the n i particles (here we assumed a constant vacuum expectation value for λ; see however below). On the infinite line the spectrum is continuous and the gap equation reads leading to the well-known scale-dependent renormalized coupling and to a dynamically generated mass, which in the case of the infinite line can be taken to coincide with the dynamical scale Λ,

Generalized gap equations
The CP N −1 theory on a finite interval of length L, 0 < x < L will now be considered. For this problem the boundary conditions must be specified. One possibility is the Dirichlet boundary condition which, up to a U(N) transformation, is the following constraint on the boundary D-D : For simplicity and for definiteness, we take in this paper the n i field in the same direction in the CP N −1 space at the two boundaries. Another possibility is the Neumann boundary condition N-N : We consider in detail the three possible combinations D-D, N-N and periodic boundary conditions. More possibilities are open for the choice of classical boundary condition but, for reasons to be clarified later, there is no need to list all of them.
The N fields can be separated into a classical component σ ≡ n 1 and the rest, n i (i = 2, . . . , N). Integrating over the N − 1 remaining fields yields the following effective action (2.12) One can take σ real and set the gauge field to zero, and consider the leading contribution at large N only. The total energy can formally be written as where ω 2 n are the eigenvalues of the following operator and f n are the corresponding eigenfunctions. The eigenfunctions f n can be taken to be real and orthonormal The expression for the energy (2.13) is not regularized yet, it still contains quadratic, linear and logarithmic divergences with the UV cutoff. For the moment we shall work with this formally divergent expression for the energy. The eigenfunctions f n 's satisfy also the completeness condition where ∓ refers respectively to the Dirichlet or Neuman conditions. Note that for x, x ′ ∈ [0, L], only the first term δ(x − x ′ ) is relevant; other terms guarantee that the completeness condition is consistent with the boundary condition (the mirror image method).
The two functions σ(x) and λ(x) must be determined by extremizing the energy (2.13). Varying the action with respect to σ(x) one gets In a translationally invariant system (i.e. with constant λ and σ) this equation reduces to λ σ = 0. This implies that either one of the condensates λ or σ must vanish. It follows that there are in general two possible distinct phases which are generally called "confinement phase" (λ = 0, σ = 0) or "Higgs phase" (λ = 0, σ = 0). In a non-translational invariant system there is no net distinction between these two phases since in general both λ(x) and σ(x) are different from zero.
The variation of the spectrum of the operator (2.14) for λ(x) → λ(x) + δλ(x) is given by to first order in δλ(x), as can be seen easily from (2.14). The variation of the energy (2.13) with respect to λ(x) then gives We find it convenient to separate this equation into a constant part (the average in x) plus a non-constant part with zero integral. First write σ 2 as Eq. (2.19) can thus be separated into the one for the constant part and another for the non-constant part So the three equations to be solved for λ(x) and σ(x) are (2.17), (2.21) and (2.22). Only Eq. (2.21) needs to be regularized since it contains a logarithmic divergence with the UV cutoff.
For later use we provide a relation between the expression n fn(x) 2 ωn that appears in the generalized gap equation and the Euclidean propagator. The Euclidean propagator satisfies and, in terms of {f n , ω n }, it is given by We can thus infer the useful relation

Translational invariance Ansatz
Let us first test if the Ansatz of a translationally invariant (constant) λ is consistent with our functional saddle-point equations. Let us consider first the D-D case. If λ = m 2 were constant the eigenfunctions and eigenvalues of the operator (2.14) would be simply given by Here one has an explicit representation of the propagator, Note that each term satisfies the standard Green function equation (2.23) with a single delta function in x on the right hand side. In the case with a constant σ, Eq.(2.21) gives, by using an appropriate regularization which is nothing but the gap equation discussed by Milekhin [9]. Here, taking into account the consistency with Eq.(2.9) for infinite string, the dynamical scale Λ can be introduced as where γ is the Euler-Mascheroni constant. The second term in the last line of Eq. (2.29) comes from the infinite mirror poles: One can verify that Eq. (2.29) gives the local extremum (maximum) of the total energy (2.13) with respect to m 2 , under the translational-invariance Ansatz.
However, the problem is that an additional equation, (2.22), must be satisfied as well.
The left hand side, using σ = 0 (under the assumption of translational invariance), is just which is equal to, by using (2.25), This is clearly not zero: Eq. (2.22) is not satisfied by a constant m.
For N-N boundary conditions and λ = m 2 constant the eigenmodes are In this case, the left hand side of Eq.
which is the same as the D-D case (2.33), except for the sign. The proof is as before by using instead the plus sign in (2.23). Again, Eq. (2.22) cannot be satisfied by a constant λ.
Thus both for DD and NN boundary conditions, the constant generated mass m does not represent a quantum saddle point of our system.
For the periodic boundary condition one can set for convenience the period length to 2L. The complete set of eigenstates and eigenfunctions are exactly given by a D-D set (2.27) plus a N-N set, (2.35), applied in the enlarged space [0, 2L]. In this case, the left hand side of equation (2.22) is the sum of D-D states contribution (2.33) plus the N-N states contribution (2.36) which cancel. As expected, the translational invariance Ansatz is consistent in this case. This system has been studied in [8]. With a periodic condition, the total energy and the gap equation can depend on the Wilson loop exp(i 2L 0 A x dx). With a nonvanishing constant gauge field A x , one finds that the propagator is given by Therefore the energy in the confinement phase (σ = 0), is [8] By minimizing the energy with respect to A x , one finds A x = 0. The saddle point with respect to m 2 is determined by the gap equation In this case no other equations must be satisfied, so this gives the solution of the problem.

Solution of the generalized gap equation
Having proven that a constant λ = m 2 does not satisfy the functional saddle point equations, (2.17), (2.21) and (2.22), for the problem with the Dirichlet or Neumann boundary conditions, we now discuss the solution of these equations, relying also on a numerical analysis. The content of this section constitutes the main new contribution of this paper.

Behavior of the functions λ(x) and σ(x) near the boundaries
Before embarking on the numerical analysis one must first understand the behavior close to the boundaries of the two unknown functions λ(x) and σ(x). For the case of D-D boundary condition, and under the assumption of constant λ, the left hand side of (2.22) is given by (2.33) and thus contains a logarithmic divergence near the boundaries since K 0 (x) ≃ − log x for x ≪ 1. In the propagator formalism (2.34) this leading divergence comes from the closest mirror pole. The logarithmic behavior is due to the fact that at x ≪ 1 one can neglect the potential λ and use the massless propagator. Another crude, but somehow useful, way to understand this logarithm divergence is the following. At fixed x, we split the sum (2.33) into "low-energy" modes and "high-energy" modes. For the low-energy modes we approximate sin 2 nπx L ≃ 0. The high-energy modes instead are summed to zero due to the fast averaging. The result gives the correct log divergent term: We will show below more carefully that this type of divergence near the boundary is really there, at least for the D-D condition, also for a generic, but not too singular, potential λ(x). Let us use the WKB approximation to compute the leading divergence of n fn(x) 2 ωn near the boundaries of the interval. We assume λ(x) to be generic, with λ(x) > 0 and λ(x) = λ(L − x), but not too singular near the boundaries: It will be seen that the potential λ(x) is non-constant and divergent near the boundaries λ(x) → ∞ as x → 0, L. Nevertheless it does satisfy the condition (3.2). This fact also implies that 'the Higgs phase' defined by a configulation in which the potential vanishes identically, λ(x) = 0, cannot be a solution of the generalized gap equation. By using WKB approximation the high-energy modes of the operator (2.14) can be approximated, in the classically allowed region, by and ǫ n and L − ǫ n are the classical turning point The Bohr-Sommerfeld quantization condition is The initial phase θ n is quite important to determine the divergence near the boundaries. It would be equal to π/2 if there were no boundary, but a smooth classical turning point. However, the phase θ n turns out to vanish as can be seen as follows. If λ(x) had the maximum λ M , then for ω n > √ λ M there was no classical turning point and the classically allowed region was [0, L] (ǫ n = 0) and we get θ n = 0 due to the D-D condition. Even in the case where λ diverges at the boundaries, which actually is our case, the same result is obtained because, from the condition (3.2) and the equation (3.5), one gets lim ωn→∞ ω n ǫ n = 0 , (3.8) meaning that the turning point ǫ n goes to zero faster than the typical wave length of the eigenstate, ∼ 1/ω n . This is sufficient for letting us to conclude that for large n the eigenfunctions and eigenvalues are well approximated by 9) and thus which behaves indeed near x = 0 as In other words, the sum of the fluctuations of the n i fields very generally produces a divergent behavior for the first two terms of Eq. (2.22) near the boundaries, quite independently of the form of λ(x), as long as the latter satisfies the condition (3.2). The only way to solve Eq. (2.22) is then to cancel this divergence with the classical field σ But this is exactly the expected behavior, as it corresponds to the classical constraint (2.2), or to the D-D boundary condition, (2.10), as can be seen once a renormalized coupling (2.8) with energy scale µ = 1/x is substituted. Approaching the boundaries, x ∼ ǫ, L − ǫ, is thus similar to going into the UV, hence the classical value for σ(x) = n 1 (x) there (see more on this point below).
The divergence of the σ field near the boundaries (3.12) on the other hand implies, through equation (2.17), that the potential λ also diverges. Its leading behavior at x ≃ 0 is given by which indeed satisfies the condition, (3.2).
Thus the behavior of the functions λ(x) and σ(x) near the boundaries has been determined self-consistently to be given by (3.12) and (3.13) (and similarly for x ∼ L, with Physically, such a behavior can be understood as follows. At any fixed x, far from the boundaries, the UV divergences due to the higher modes are cancelled by a constant (and divergent) r ǫ = 4π g 2 ǫ , exactly as in the standard CP N −1 sigma model without the space boundaries. This is so because the UV divergences are local effects in x: far from the boundaries the effects of the latter are negligible. Near the boundaries where the quantum fluctuations of the n i fields are suppressed, however, this cancellation cannot occur, r ǫ being a constant. This is where the classical field σ = n 1 comes to rescue. The generalized gap equation (2.19) is now satisfied by balancing the "ultraviolet" divergent behavior of σ(x) at x ∼ ǫ, with r ǫ .
Even though this way of interpreting (3.12) and (3.13) is straightforward in the case of the D-D condition, actually the same results hold for the system with the N-N boundary condition as well. They just tell that at the boundaries, where the quantum fluctuations of the n i fields are suppressed and cannot produce the logarithmic divergences due to the lack of the two-dimensional spacetime, the defining condition (2.26) of the CP N −1 theory must be satisfied by the classical field 1 , independently of the type of the boundary condition. The numerical solutions found below indeed hold true both for the case of the D-D and the N-N boundary conditions. This conclusion can be further confirmed by studying the wave functions f n (x) analytically near the boundaries, assuming λ(x) of the form of (3.13), as is done in Appendix A.
The resulting behavior is (3.14) Thus not only do they satisfy the Dirichlet boundary condition but also the Neumann boundary condition. As already anticipated, this is the indication one gets from the numerical solution found by the recursive procedure to be described in the next subsection. The results found for λ(x) and σ(x) turn out to be the same, irrespectively of the boundary condition used to solve the wave equation (2.14).
Finally we note that the case of the periodic boundary condition is qualitatively different. In that case, the points x = 0, L are no special points, the quantum fluctuations are effective there, as well as at any other point, and our results (3.12) and (3.13) do not apply. Such system has been recently studied in [8].

Numerical solutions for λ(x) and σ(x)
The method we employ to solve the set of equations (2.17), (2.21), and (2.22) numerically is as follows. We first introduce a finite cutoff n max on the number of modes. Accordingly, equation (2.21) can be regularized as (see Eq. (2.8) where, by using (3.9), µ(n max ) = n max π L . (3.18) The algorithm we use here to find the solution, somewhat reminiscent of Hartree's equations in atomic physics, is a recursive procedure in which at each step k one has a certain function λ k (x), with some initial λ 0 (x) satisfying (3.13). Given λ k (x), the Schrödinger equation (2.14) is solved to give {ω n , f n (x)} for the desired number of modes; (2.22) and (3.17) are then used to find the corresponding value of σ 2 k : Note that the right hand side of this equation depends only on λ k (x).
The next iteration for λ is found by use of Eq. (2.17): This procedure can be repeated until a convergence to a consistent set of λ(x) and σ(x) is achieved 3 . We set L fixed and then study various values of Λ. We take the starting point 2 A careful change of the regularization gives the factor 2 in front of µ(n max ) so that it is consistent with Eq.(2.9). Omitting higher modes larger than n max introduces a translation between the two UV cut-off parameters ǫ and µ(n max ) as 1/ǫ = µe γ . Therefore, Eq.(2.30) giving Λ is translated as of the iteration to be λ 0 (x) = 0 for the first value of Λ. For the successive values of Λ, the self-consistent solution λ(x) obtained with the previous value of Λ will be taken as the starting point λ 0 (x) of recursion.
The method as described above works well for LΛ up to 1 after which it loses convergence. The delicate point in the algorithm is the step (3.20) which is exceedingly sensitive to small variations of σ k (x) in the denominator. A technical improvement we implement is the following. We substitute (3.20) with a "smoother" recursive formula with some conveniently chosen K. Having a larger value of K makes the convergence slower but at the same time it allows to reach higher values of LΛ. For example by choosing K = 15 (as is done to produce the plots for large LΛ given below) one can reach LΛ up to 4.
Another technical issue is that σ k (x), as computed in (3.19), cannot be inserted as it is in the second derivative of the expression for λ k+1 (x). In passing from (3.19) to (3.21) one must perform a numerical fit to approximate σ k (x) with a sum of analytical functions. In the plots given below σ k (x) and its analytical fit are superimposed: the two curves are indistinguishable, within the resolution of the graphs.

Results
In Figure 1     Another visualization of the results is given in Figure 3 where the value of the fields at The results for larger values of LΛ are shown in Figure 4, for Λ = 1 fixed and L = 1, 2, 3, 4. From the figures we see the expected pattern emerging: by going to larger L at fixed Λ one expects to recover the confined phase of the standard CP N −1 sigma model, Eq. (2.9). We indeed see that λ(x) → Λ in the middle of the interval whereas the condensate σ approaches zero there at the same time. The effects of the boundary remain concentrated near the two extremes and do not propagate beyond 1/Λ.  The values of the fields at the middle point of the interval are given in Figure 5 where in the bottom line are the corresponding logarithmic plots to show better the rate of convergence to the confinement phase. A linear fit to the logarithmic plots at large L, is a fairly clear signal that the boundary effects are exponentially suppressed at fixed x, for large values of L.
As a further check of our conclusion, Figure 6 shows that the field σ(x) becomes indeed well approximated by a hyperbolic cosine function in the central region for large L. As seen from Eq. (2.17) this is what is to be expected for σ(x), upon dynamical generation of the mass for n i , λ = Λ 2 .

Absence of the "Higgs" phase
The preceding analysis clearly shows that the solution of the set of the generalized gap equations is unique: it smoothly approaches in the large L limit the well-known physics of the CP (N − 1) model (at large N) in "confinement phase" with (See Fig. 1 -Fig. 5 ). It is however really a simple matter to prove directly the absence of the "Higgs phase", characterized by the condensates for any value of L. The proof is basically identical to the proof already given (Subsection 2.2) of the fact that translationally invariant condensates, λ and σ constant, are not consistent with the generalized gap equations: rather, it is a special case of it. Indeed, if the n field mass is not generated, λ ≡ 0, the approximation used for higher levels (n ≫ 1) (see Eq. (3.9), Eq. (3.10)) is actually valid for all levels. Thus one finds that the first term of the gap equation (2.22) is given exactly by, which is certainly not equal to 0 identically. This proves that the Higgs phase is not possible in the presence of the boundaries, except for the case with the periodic boundary conditions.

Discussion
In this paper we have studied in detail the two-dimensional CP N −1 model with a finite space interval. In other words the system is defined on a finite-width worldstrip, in which the fields propagate. We find that the system has a unique phase which smoothly approaches in the large L limit the standard 2D CP N −1 system in confinement phase, with (constant) mass generation for the n i fields, which are then confined by a 2D Coulomb force. This kind of model is of great interest in physics. For instance, it appears as the low-energy effective theory describing the quantum excitations of monopole-vortex soliton complex [10]. In such a context, the CP N −1 model describes the nonAbelian orientational zeromodes of the nonAbelian vortex (string), whereas its boundaries represent the monopoles arising from a higher-scale gauge-symmetry breaking, carrying the same orientational CP N −1 moduli. NonAbelian monopoles, which are not plagued by the well-known pathologies, emerge in such a context, and the infrared properties of such a soliton complex is an important question, defining the dynamical properties of the nonAbelian monopoles.
As the CP N −1 sigma model in two dimensions is asymptotically free, i.e., it is a theory which is scale invariant in the UV, one might wonder what the meaning of the "finite interval" L is. The point is that the behavior of the system in the UV -where the effects of the space boundaries should be negligible -contains in itself by dimensional transmutation the renormalization-invariant mass scale Λ (equivalent to specifying the value of the coupling constant at the ultraviolet cutoff, Λ U V ), whether or not the system dynamically flows actually into the infrared regime at 1/Λ where the interactions become strong. The space interval L can be defined in reference to 1/Λ. But this implies that for L ≪ 1/Λ quantum fluctuations would not become strongly coupled, their wavelengths constrained to be smaller than 1/Λ. In other words, the system effectively reduces to a finite dimensional object propagating in time, i.e., quantum mechanics, rather than a proper 2D quantum field theory. This would explain our finding that the CP N −1 model with a space boundary has a unique phase.
With the periodic boundary condition, the story could be different. The translational invariance is unbroken, and it could be a property of the vacuum: the constant mass λ(x) = m 2 (or σ) generation of the order of Λ is certainly consistent with the functional saddle-point equations, as we have shown. Even L < 1/Λ the system thus maintains 2D field-theoretic properties, and it could be [8] that, at large N, the system shows a phase transition at a critical length L crit = O(1/Λ), from confinement (m ∼ Λ, σ = 0) to Higgs (σ ∼ Λ, m = 0) phase.

A Eigenfunctions near the boundaries
λ(x) satisfying the generalized gap equation was found to behave as and Λ 0 is some constant. In terms of the new coordinate y,λ(y) ≡ λ(x) has an essential singularity at y = 0. It is natural to assume that λ(x) can be expanded around the boundary at x = 0 (y = 0) as where v n (y) are functions with no essential singularities. Especially, we set for v 0 (y) v 0 (y) = y 2 + ∞ n=2 a n y n . Note that the knowledge of an exact form of v 0 (y) and poles of v 1 (y) is needed to determine the divergent contribution to the following integral and to renormalize the total energy. An incomplete knowledge of the subdominant divergences in λ(x) and σ(x) for the moment prevents us from removing these divergences from the energy unambiguously. We expect that these divergences are independent of the length L; the resulting finite contribution may be interpreted as the mass of the endpoints (the monopole mass).
Given the potential λ(x), the behavior of an eigenfunction f (x) near the boundaries can be inferred directly from where we dropped the level number, as the leading behavior near x = 0 or x = L turns out to be independent of the energy level. We assume an expansion of the form Similarly, by considering terms of the order of exp −n+2−ρ y , one finds a recursive relation for h n (y) which has a unique solution, given {h 0 (y), h 1 (y), · · · , h n−1 (y), ω}.
Our boundary condition requires that the functions f n (x) describing the quantum fluctuations of the n i fields choose which is the result reported in (3.14). On the other hand, the classical field σ, corresponding to the zero (ω = 0) mode, was found to opt for (see Eq. (3.12))