Theta-vacuum and large N limit in CP^{N-1} sigma models
aa r X i v : . [ h e p - t h ] D ec Theta-vacuum and large N limitin C P N − σ models M. Aguado ∗ , M. Asorey † Max-Planck-Institut f¨ur QuantenoptikHans-Kopfermann-Str. 1, D-85748 Garching (Germany) Departamento de F´ısica Te´orica, Facultad de CienciasUniversidad de Zaragoza, E-50009 Zaragoza (Spain)
Abstract
The θ dependence of the vacuum energy density in C P N − models isre-analysed in the semiclassical approach, the 1 /N expansion and argu-ments based on the nodal structure of vacuum wavefunctionals. The 1 /N expansion is shown not to be in contradiction with instanton physics atfinite (spacetime) volume V . The interplay of large volume V and large N parameter gives rise to two regimes with different θ dependence, onebehaving as a dilute instanton gas and the other dominated by the tra-ditional large N picture, where instantons reappear as resonances of theone-loop effective action, even in the absence of regular instantonic solu-tions. The realms of the two regimes are given in terms of the mass gap m by m V ≪ N and m V ≫ N , respectively. The small volume regime m V ≪ N is relevant for physical effects associated to the physics of theboundary, like the leading rˆole of edge states in the quantum Hall effect,which, however, do not play any rˆole in the thermodynamic limit at large N . Depending on the order in which the limits N → ∞ and V → ∞ aretaken, two different theories are obtained; this is the hallmark of a phasetransition at 1 /N = 0. A lack of nonperturbative analytical methods haunts the study of the infraredbehaviour of confining field theories such as QCD. The main tools used for thispurpose rely on approximations (e.g., semiclassical, large number of colours),and rigorous results are attainable in few corners of parameter space. Thebordering region between topology and field physics is especially troubling, sincedifferent methods arise sometimes from apparently incompatible hypotheses andphysical pictures.Two-dimensional C P N − sigma models [1, 2, 3] are regarded as a convenienttesting ground to prepare the assault on four-dimensional gauge theories, be-cause both kinds of theories share a number of important properties: conformal ∗ [email protected] † [email protected] θ − π − π π π E θ − π − π π π Figure 1: Structure of the vacuum energy density in the traditional large N pic-ture (above) and in the semiclassical picture dominated by instantons (below).invariance at the classical level, asymptotic freedom, dynamical mass genera-tion, confinement, existence of a topological term θ and instantons for all valuesof the number of colours N .A relevant problem in these theories with topological properties is the θ dependence of the vacuum energy density, the quantity that determines thephase structure of the theory (for a recent review, see [4]). In particular, thefate of the discrete parity symmetry P upon quantisation at the values θ = 0and θ = π (the only values for which it is classically conserved) is an issue.Perhaps the simplest model in which the subtlety of the θ -dependence ofthe vacuum energy E is manifest is the quantum rotor [5], i.e., the quantummechanical problem of the dynamics of a charged particle on the circumferenceS enclosing a magnetic flux θ . In the absence of perturbations, the vacuumenergy is quadratic in θ ; periodicity of the physics in θ → θ + 2 π imposesthat the ground level is twofold degenerate for θ = π (i.e., half a flux quantumacross the region bounded by S ) and there parity is spontaneously broken.This, as we will see, mimicks the traditional picture of the large N expansion in C P N − models. However, even slight perturbations compatible with reflectionsymmetry lift the degeneracy of the rotor, by a level repulsion mechanism,making the curve E ( θ ) smooth at θ = π . A convenient approximate methodis that of the dilute instanton gas, where the vacuum is understood in termsof tunnelling processes among classical vacua. In the dilute approximation, thevacuum energy (a pure nonperturbative effect) is a smooth periodic function of θ proportional to (1 − cos θ ). This corresponds to the semiclassical approximationin C P N − models, where instantons play an all-important rˆole.These two regimes have the following paradigmatic expressions for E ( θ ),2llustrated in figure 1: E ( θ ) ∝ min (cid:8) ( θ + 2 πk ) ; k ∈ Z (cid:9) (large N ), (1) E ( θ ) ∝ − cos θ (semiclassical). (2)We now consider the situation in C P N − models. Exact solutions are knownfor the quantum C P model (equivalent to the O(3) model) both at θ = 0 and θ = π . In the first case [6], the solution exhibits a mass gap, the spectrumconsists of an SU(2) triplet, and parity is conserved. This agrees with theHaldane map [7], which transforms this model into a chain of integer classicalspins. Vafa and Witten [8] argued that there is no first order phase transitionwith spontaneous parity breaking at θ = 0 for QCD, their argument beingapplicable straightforwardly to all C P N − models (see [9] for a proof of theVafa-Witten theorem using the topological charge as an order parameter).The exact solution of the quantum C P model at θ = π [10] also conservesparity but shows no mass gap (this result was anticipated in [11]). The criticalbehaviour of the model is described by an SU(2) WZNW model at level k = 1.This also agrees with the Haldane map, which transforms this model into achain of half-odd spins. By the Lieb-Schulz-Mattis theorem [12], the absenceof mass gap implies that P is conserved. On the other hand, the absence of afirst order phase transition with spontaneous P breakdown at θ = π has beenargued to hold for all C P N − models [13], by analyzing the nodal structure ofthe vacuum in the Hamiltonian formalism [14] in analogy with QCD [15].For the intermediate region 0 < θ < π , analytical techniques are lacking,and we must rely on approximations and numerical simulations. We will dis-cuss two important approximations, which have been argued to be mutuallyincompatible: the semiclassical method and the 1 /N expansion.The semiclassical approach [16] is based, as in the case of the rotor, on thepicture of the quantum vacuum of 4 d gauge theories and 2 d C P N − models builtfrom tunnelling processes among classical vacua. These nonperturbative pro-cesses are dominated by instantons and antiinstantons, (anti)selfdual solutionsof the classical Euclidean equations of motion. A dilute gas approximation givesa θ dependence of the vacuum energy density of the form E ( θ ) ∝ m (1 − cos θ ) , (3)where m is the mass gap. This dependence cannot be seen in perturbationtheory due to the nonanalytic dependence of the mass gap on the coupling.However, a vacuum based on a dilute gas of instantons and antiinstantonsis not satisfactory, since the statistical ensemble is dominated by the infrareddivergent contribution of arbitrarily large instantons, whose density n as a func-tion of size ρ is n ( ρ )d ρ ∝ (Λ ρ ) N d ρρ (4)for the C P N − model, with Λ a typical scale of the theory. A statistical me-chanical treatment of interacting instanton fluids has been developed [17, 18],bringing about the instanton liquid picture of the QCD vacuum [19]. This maybe very relevant for the behaviour of these theories at finite temperature andhigh density. We note in passing that the dilute gas approximation breaks downas well in the ultraviolet for the C P model, as pointed out by L¨uscher [20] build-ing on his work with Berg [21] on the geometric definition of a topological charge3ensity on the lattice. Technically, the topological susceptibility in this modeldoes not scale according to the perturbative renormalisation group due to smalldistance fluctuations. This is reflected in the singularity of (4) as ρ → N = 2; it may also be understood a consequence of the slow vanishing rate ofthe density of Lee-Yang zeros as θ → C P model, numerical simulations suggest a similar pathology [22].The 1 /N expansion [23, 24] stands as an alternative to the semiclassicalmethod. This technique is based on the simplification of both 4 d SU( N ) gaugetheories and 2 d C P N − models when N is taken to infinity keeping certainparameter combinations constant.The 1 /N expansion of C P N − models, as developed in [3] and [25], agrees at θ = 0 with the known spectrum, given by a massive particle in the adjoint rep-resentation of SU( N ). The mass m T is generated dynamically, and the particleturns out to be a composite state of two fundamental fields, bound together bya Coulomb potential. For θ = 0, this analysis predicts a quadratic θ dependence E ( θ ) = 32 π m θ N (5)of the vacuum energy density around θ = 0. This dependence can be madeto agree with the fundamental requirement that physics be periodic in θ withperiod 2 π only if there is a first order cusp at odd multiples of θ = π , i.e., afirst order phase transition accompanied by spontaneous parity breakdown, asshown in the upper part of figure 1.Instanton effects, being nonperturbative, are not visible in the perturbativeexpression (5). This led Witten [25] to argue that the 1 /N expansion is notsensitive to instantons — equivalently, that instantons play no significant rˆolein the quantum C P N − models (or in 4 d gauge theories) to the extent that the1 /N expansion is a good approximation thereof. Jevicki [26], however, arguedthat instantons resurface in the 1 /N expansion as poles of the integrand ofthe partition function Z (eqn. 8), and that Z can be computed both by thesaddle point method and by using a functional Cauchy theorem summing theresidues of all these poles (representing resonances). Then the large N limit andinstanton effects would not be a priori incompatible with each other.The quadratic dependence (5) agrees with the holographic picture providedby the Maldacena conjecture [27], and moreover with lattice measurements ofthe topological susceptibility of C P N − models (see [4] for a review). TheWitten-Veneziano formula [28, 29], derived in this approximation, gives a phe-nomenologically correct value of the η ′ mass in terms of the topological suscep-tibility at θ = 0. However, the appearance of a first order cusp at θ = π is incontradiction with the results arising from the nodal analysis of the vacuum [13],and with the intuition that level repulsion generically destroys level crossings.In this work we show how this discrepancy stems from the fact that thelarge N limit and the thermodynamical limit do not commute. The traditionalformulation of the 1 /N expansion starts directly at infinite spacetime volume V = LT = ∞ . As we shall see, a procedure in which the thermodynamiclimit is taken after the N → ∞ limit provides results compatible both withinstanton physics and with the rigorous results at θ = π , and different from thereverse order of limits. A finite volume analysis is in order. This agrees withSchwab’s [30, 31] and M¨unster’s [32, 33] approach in the case of the sphere;4e moreover outline the application of Jevicki’s residue method. We find thecase of the torus much more tractable and amenable to explicit computationafter integration over the dual torus parametrising the different holomorphicbundle structures within each topological charge sector. In particular, Jevicki’sapproach requires computation of the residues of meromorphic functions insteadof functionals, and his programme can be carried out in the simplest casesexhibiting the reappearance of instantons as resonances (poles) in the one-loopeffective action. Remarkably, in spite of the absence of regular unit chargeinstantons on the torus [34], the contribution of this sector is nonzero in theresonance approach: this we interpret as the effect of rough configurations nearthe forbidden regular instanton.At finite volume there are two regimes: one dominated by instantons for lowmass theories, m V ≪ N , and another regime where they are are strongly sup-pressed, m V ≫ N . The second regime is the relevant one for C P N − theoriesin the thermodynamic limit, but the other regime is relevant for effects wherethe finite volume or space topology play a leading rˆole, like in the appearanceof edge states in the quantum Hall effect.The structure of the article is as follows. In section 2, the traditional large N picture of C P N − models is reviewed. The θ dependence of C P N − modelsformulated on the sphere is considered in section 3. The corresponding analysisfor the case of the torus is performed in section 4. The consequences of thisanalysis are discussed in section 5. /N expansion The traditional large N picture of C P N − models was developed in [3] and [25].We shall now give a brief account of it before the analysis in finite volume.The large N method is based in a saddle point approximation of the par-tition function, defined on the infinite 2 d Euclidean plane, after integration ofthe fundamental Ψ, Ψ † fields (taking values in C N , i.e., in representatives ofprojective classes in C P N − ). We introduce the dummy U(1) gauge field A ν = − i (cid:0) Ψ † ∂ µ Ψ − ( ∂ µ Ψ) † Ψ (cid:1) , (6)and a scalar field α ( x ) imposing the constraint Ψ † Ψ = 1 at each point as aLagrange multiplier. Starting from the full partition function at θ = 0, Z = Z D Ψ D Ψ † D A µ δ (cid:2) Ψ † Ψ − (cid:3) exp (cid:26) − N g Z R d x | D µ Ψ | (cid:27) = Z D Ψ D Ψ † D A µ D α × exp (cid:26) − N g Z R d x | D µ Ψ | − N g Z R d x α ( x ) (cid:0) Ψ † Ψ − (cid:1)(cid:27) , (7)we perform the Gaussian integration over Ψ, Ψ † to obtain Z = Z D A µ D α e − NS eff [ A µ , α ] , (8)5here the effective action is S eff [ A µ , α ] = Tr ln (cid:0) − D µ − α ( x ) (cid:1) − g Z R d x α ( x ) . (9)The saddle point equations δS eff δα ( x ) = 1 − D µ + α ( x, x ) − g = 0 , (10) δS eff δA µ ( x ) = 2 i D µ − D µ + α ( x, x ) = 0 , (11)can be solved within a renormalisation scheme to yield a saddle configuration A µ = 0 , α = m ≡ µ exp (cid:26) − πg R ( µ ) (cid:27) , (12)where µ is a mass scale and g R the corresponding renormalised coupling.Perturbation theory around the saddle configuration reveals a dynamicalsystem of an N -plet of charged scalars Ψ, with a short range interaction dueto the field α , and electromagnetic interaction due to the field A µ . The latterdevelops an effective kinetic term and couples to the scalars with effective electriccharge e eff = p πm / N . Thus there is a confining Coulomb interactionbetween scalars, and the spectrum at θ = 0 consists of (ΨΨ † ) bound states,with mass gap 2 m T , living in the adjoint representation of SU( N ).For θ = 0, the topological term in the action is − iθQ = − i θ π Z d x F , (13) Q being the magnetic flux associated with the U(1) field A µ and its field strength F µν , or equivalently, the topological charge of the field Ψ, which is an integerfor smooth finite-action configurations. This term plays the rˆole of an externalelectric field in electrodynamics. Therefore, in this picture of the C P N − models,its contribution to the vacuum energy density is E ( θ ) = 12 e (cid:18) θ π (cid:19) = 32 π m θ N , (14)yielding a topological susceptibility χ t = (cid:18) d E ( θ )d θ (cid:19) θ =0 = 3 m πN . (15)This quadratic dependence is perturbative, i.e., it can be seen in terms of Feyn-man diagrams. Instanton effects, nonperturbative in nature, were argued in [25]to be exponentially suppressed in the 1 /N expansion, and therefore irrelevantfor the physics of the C P N − models. However, we have seen that the levelcrossing and first order phase transition at θ = π implied by (14) and the re-quirement of 2 π -periodicity in θ are in contradiction with the nodal argumentsof [13]. We will next go over to a compact space with the purpose of showingthat this incompatibility stems from the infinite volume starting point of thetraditional 1 /N analysis. 6 /N expansion on S N and the volume, and the effects of taking these toinfinity in different orders, requires the C P N − models to be first formulated ina compact (Euclidean) space.Schwab [30, 31] and M¨unster [32, 33] studied the 1 /N expansion of C P N − models on S , and observed that the k = 1 contribution to the partition functionis dominated, for large N , by a saddle point given by a rotationally invariantinstanton (in the sense that global U( N ) transformations can be compensatedby O(3) rotations in Euclidean space). The saddle point equations1 − D µ D µ + α ( x, x ) = 12 g , D ν − D µ D µ + α ( x, x ) = 0 , (16)admit for N > | k | , in a uniform topological charge density background, solutionswith constant α ( x ). Indeed, the second equation holds due to parity. The firstequation states rotation invariance of the propagator G ( x, x ) of a particle withmass √ α . It is easy to show that G ( x, y ) depends only on the geodesic distancebetween x and y , and therefore the first equation has solutions with constant α . Thus, we begin [35, 36] with a spherical spacetime of radius R and volume V = 4 πR , and the action of the C P N − model on a background of topologicalcharge k , S k = − N g Z Ψ † ∆ k Ψ + N g Z m (cid:0) Ψ † Ψ − (cid:1) , (17)where integration implies the measure d x √ g , and ∆ k is the covariant Laplacianin the background chosen. The magnetic flux for the composite U(1) field isquantised, Φ B = 4 πR B = 2 πk, k ∈ Z . (18)We rewrite the constant saddle point value of the α field as m , variable still tobe integrated upon.Integrating out Ψ, Ψ † yields the functional determinant of the operator − ∆ k + m , which is computed in the ζ function renormalisation scheme atenergy scale µ . Discarding unessential factors, Z k = Z d m e − NS eff k , (19)with effective action S eff k = ln det (cid:18) − µ ∆ k + m µ (cid:19) − πR g m ≡ ln det A − πR g m . (20)7he eigenvalues of A are λ n = 1 µ R (cid:20)(cid:18) n + | k | (cid:19) (cid:18) n + | k | (cid:19) − k m R (cid:21) ≡ e λ n µ R , n = 0 , , , . . . , (21)with degeneracy d n = 2 n + | k | + 1.We use the ζ -function definition of the determinant, equivalent to a renor-malisation at scale µ :ln det ζ A = ∞ X n =0 d n ln λ n = ∞ X n =0 d n ln e λ n − ∞ X n =0 d n ! ln( µ R ) → − ζ ′ e A (0) − ζ e A (0) ln( µ R ) , (22) ζ e A being the ζ function associated with the operator e A ≡ µ R A , i.e. theanalytic continuation of ζ e A ( s ) ≡ ∞ X n =0 d n e λ sn , (Re s >
1) (23)for all complex s = 1.From the small s expansion of (23), we obtain the effective action for topo-logical sector k : S eff k = − ζ ′ e A (0) − ζ e A (0) ln( µ R ) − πR g R m = 2 (cid:18) k − m R (cid:19) + (cid:18) m R − (cid:19) ln( µ R ) − πR g R m + 2 r k − m R ln Γ (cid:18) | k | +12 + q k + − m R (cid:19) Γ (cid:18) | k | +12 − q k + − m R (cid:19) − ζ ′ H − | k | + 12 + r k − m R ! − ζ ′ H − | k | + 12 − r k − m R ! . (24)Here we have used the Hurwitz zeta function ζ H ( s ; v ), defined by analyticalcontinuation to all s = 1 of ζ H ( s ; v ) = ∞ X n =0 ( n + v ) − s = 1Γ( s ) Z ∞ d t t s − e − vt − e − t , Re s > , (25)and its derivative ζ ′ H ( s ; v ) with respect to s . Function (24) is defined for allcomplex values of m R , bar isolated singularities.8
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PSfrag replacements m R S eff k Figure 2: Effective action in the k = 1 sector on the sphere, for real m R . In order to compute Z k , integration over m is still to be performed, throughimaginary values in order to ensure convergence.Let us study the behaviour of the effective action for real m R (see figure2). To begin with, the integrand of Z k has N (2 n + | k | + 1)-fold poles at m R = p n = − (cid:18) n + | k | (cid:19) (cid:18) n + | k | (cid:19) + k , n = 0 , , , . . . , (26)reproducing the eigenvalues and degeneracies of − ∆ k + m . In this sense, in theway predicted by Jevicki, the partition function can be computed by deformingthe integration curve so as to surround the poles, and summing the residues ateach of them. However, the problem of computing and summing the residues istoo difficult to be tackled analytically, although the previous formulae can beused in a numerical approach (more progress can be made analytically in thecase of the torus, as will be seen in next section).Alternatively, we can use the saddle point method. The zeros of the deriva-tive,d S eff k d( m R ) = ln( µ R ) − π g R − ψ | k | + 12 + r k + 14 − m R ! − ψ | k | + 12 − r k + 14 − m R ! , (27)alternate with the poles in the real m R axis, as seen in figure 2 (here ψ ( z ) =Γ ′ ( z ) / Γ( z ) is the digamma function). There is a unique saddle point s to theright of the first pole p = − | k | , which we assume to be dominant.The partition function of sector k in the saddle point approximation is Z ( s ) k = 1 R e − NS eff k ( s ) s πN | S eff k ′′ ( s ) | , (28)9p to quadratic order. Explicit results can be obtained for large m R , in whichregion the effective action can be expanded as S eff k = − (cid:18) m R − (cid:19) ln m µ − πR g R m + m R + (cid:18) k − (cid:19) m R + (cid:18) k − (cid:19) m R + O (cid:0) m − R − (cid:1) , (29)and the saddle point is found to be m k R = m R + 13 − (cid:18) k − (cid:19) m R − (cid:18) k − (cid:19) m R + O ( m − R − ) , (30)where m = µ exp (cid:8) − π/g R (cid:9) is the infinite volume saddle point in (12).The total partition function after summing all topological sectors is Z ( s ) ( θ ) = X k ∈ Z Z ( s ) k e − ikθ = r πN m T R exp (cid:26) πN g R − N m R + N m R (cid:27) × X k ∈ Z exp (cid:26) − N k m R − ikθ (cid:27) (cid:0) O (cid:0) m − R − (cid:1)(cid:1) ≡ r πN m T R exp (cid:26) πN g R − N m R + N m R (cid:27) × ϑ (cid:18) θ π (cid:12)(cid:12)(cid:12)(cid:12) iN πm R (cid:19) (cid:0) O (cid:0) m − R − (cid:1)(cid:1) , (31)where the last equation uses Jacobi’s ϑ function: ϑ ( z | τ ) = X n ∈ Z e iπτn e i πnz . (32)Two asymptotic regimes for (31) can be analysed. For N ≫ m R , thesum therein can be truncated, keeping just the k = − , , θ dependence, E ( θ ) − E (0) = − πR ln Z ( s ) ( θ ) Z ( s ) (0) ≈ πR exp (cid:26) − N m R (cid:27) (1 − cos θ ) . (33)But if m R ≫ N , using the Poisson resummation formula for the θ functionin (31) and keeping the dominant term in the dual sum, we have Z ( s ) ( θ ) ≈ √ πm N exp (cid:26) πN g R − N m R + N m R − m R N e θ (cid:27) , (34)10 θ being the angle in ( − π, + π ] differing from θ by an integer. The correspondingvacuum energy density coincides with the traditional large N prediction: E ( s )0 ( θ ) − E ( s )0 (0) ≈ m πN e θ , (35)which is periodic in θ and undergoes first order phase transitions with levelcrossing at θ = (2 ℓ + 1) π , ℓ ∈ Z .Before commenting on these two different limiting procedures, let us performthe same analysis on the torus [35, 36]. /N expansion on T L and spacetime volume V = L .Functional integration over the fields of the C P N − model on the torus in-volves an additional variable, the complex coordinate u ∈ ˆT in the dual torusparametrising the different holomorphic bundle structures associated with thecomplex line bundle E k (T , C ) [37].In this case, the effective action S eff u [ A µ , α ] resulting from integration of theΨ, Ψ † fields does not have saddle points. Specifically, the saddle point equations1 − D µ + α ( x, x ) = 12 g , D µ − D µ + α ( x, x ) = 0 , (36)do not have solutions with constant α and topological charge density. T The arguments used for the sphere can, nevertheless, be adapted to the torus,generalising the saddle point method. By integrating over u (i.e., averaging over u ), the irregularities of the saddle point configurations are swept off [35, 36].Upon integration over u , a reduced effective action S red obtains,exp {− S red [ A µ , α, N ] } = Z ˆT d u e − NS eff u [ A µ , α ] , (37)which can be argued to be dominated by constant topological density in thelarge N limit. The generalised saddle point equations δδA µ ∂∂N S red [ A µ , α, N ] = 0 ,δδα ∂∂N S red [ A µ , α, N ] = 0 , (38)hold in this case because they remain finite as N → ∞ . Their solutions we call quantum saddle points . 11o compute the effective action in the sectors of nonzero topological charge,the ζ function method is used, renormalising at energy scale µ : S eff k = ln det (cid:18) − µ ∆ k + m µ (cid:19) − m L g ≡ − ζ ′B (0) − ζ B (0) ln µ L π | k | − m L g . (39)The spectrum of the Laplacian on the torus, in a background with uniformlydistributed topological charge 2 πk = 0, is independent of the holonomies u andconsists of Landau levels − ω ( n + 1 / n = 0 , , . . . , with ω = | B | = π | k | L ,where L is the linear size of the torus. These levels have | k | -fold degeneracy[38]. The zeta function for operator B = (cid:0) − L ∆ k + m L (cid:1) / (4 π | k | ) is ζ B ( s ) = ∞ X n =0 | k | (cid:18) n + m L π | k | + 12 (cid:19) − s = | k | ζ H (cid:18) s ; m L π | k | + 12 (cid:19) = −| k | m L π | k | + s | k | ln (cid:26) √ π Γ (cid:18) m L π | k | + 12 (cid:19)(cid:27) + O ( s ) , (40)yielding the effective action (see [39]) S eff k = − m L π (cid:26) πg R + ln 4 π | k | µ L (cid:27) − | k | ln (cid:26) √ π Γ (cid:18) m L π | k | + 12 (cid:19)(cid:27) = − m L π ln 4 π | k | m L − | k | ln (cid:26) √ π Γ (cid:18) m L π | k | + 12 (cid:19)(cid:27) (41)where m is the (constant) saddle point value of the α field, and g R = g R ( µ )is the renormalised coupling at scale µ . In the last equation, m = µ exp (cid:8) − π/g R ( µ ) (cid:9) stands for the large N dynamically generated mass at infinite vol-ume.Expression (41) can be checked to coincide with the dominant term in alarge volume, constant B expansion of the corresponding effective action (24)for the sphere: S S ef ,k V →∞ −→ S T ef ,k + O ( V ) ( B = const) . (42)The contribution of topological sector k to the partition function on thetorus now depends only on m , all other fields having been integrated out: Z k = Z d m e − NS ef k . (43)As in the previous sections, the integration is performed through imaginaryvalues of m to guarantee convergence.In order to make the functional dependences in some the following expres-sions clear, it is useful to define dimensionless variables y = m L π | k | , y = m L π | k | . (44)12 PSfrag replacements y e S ( y )Figure 3: Effective action e S ( y ) for the torus, with y ∈ R .Then the k -sector partition function is Z k = 4 π | k | L Z d y Γ (cid:0) y + (cid:1) √ π ! N | k | e − N | k | y ln y = 4 π | k | L Z d y e − N | k | e S ( y ) , (45)where the function e S ( y ) = S eff k | k | = y ln y − ln Γ ( y + 1 / √ π (46)is defined for all complex values of y , except for a series of poles of the integrandof Z k , as can be seen in figure 3.For small values of y , that is, when | k | ≪ m L , the exponent simplifies: e S ( y ) = − y ln yy + y + 124 y − y + O (cid:0) y − (cid:1) , (47)meaning that the effective action has an expansion in powers of the topologicalnumber k where the first nontrivial term is quadratic: S eff k = − m L π ln m m + m L π + πk m L + O (cid:18) k m L (cid:19) . (48)In the opposite limit, when | k | ≫ m L , that is, for large y , e S ( y ) = ln 22 + (cid:8) ln y − ψ (1 / (cid:9) y − π y − ψ ′′ (1 / y − π y + O ( y ) . (49)Written in terms of the effective action, we see that the leading term is linearin the absolute value of the topological number: S eff k = ln 22 | k | + m L π (cid:26) ln m L π | k | − ψ (1 / (cid:27) + O (cid:18) m L | k | (cid:19) . (50)Notice the change of asymptotic behaviour of the effective action S eff k in thedifferent topological sectors. According to equations (48) and (50), for smallvalues of the topological charge, | k | < m V , the effective action is quadratic in k , whereas for large topological charges its leading term is linear in | k | [9]. Thischange of asymptotic behaviour has important physical consequences.13 b s b s b s b s r p ≡ I r p ≡ I ( ¯ II ) r p ≡ I ( ¯ II ) r p ≡ I ( ¯ II ) Figure 4: Structure of poles p n of the effective action (squares) and saddlepoints s n of the integrand of the partition function (circles) on the torus, in thecomplex plane of the variable y = m L / (4 π | k | ). Note that zeros and polestend to coalesce as n grows. Extrema of e S ( y ) allow us to perform a saddle point approximation by deformingthe integration contour so that it passes through the dominant extremum. Theexponent in the Z k integral has an overall N | k | factor, therefore the saddle pointapproximation is a large N | k | expansion, and results in terms of y are generalfor all k = 0 sectors.Poles of the integrand of Z k give us a chance of testing Jevicki’s proposal,since functional integration has been reduced to integration along a path in thecomplex plane. The integration contour must be deformed so that it surroundseach pole. In spite of the fact that various fields have been integrated out inthe effective action we are working with, we shall see that instantons reappearin these poles.The structure of saddle points of e S k and poles of the integrand of Z k isrepresented in figure 4. Let us consider the poles. There is an infinite series of N | k | -fold poles at values m n of m such that y = m n L π | k | ≡ p n = − (cid:18) n + 12 (cid:19) , n = 0 , , , . . . . (51)Their multiplicity is equal to the complex dimension of the moduli space ofcharge k instantons in the C P N − model (see [40]).14he partition function can be written as a sum over residues `a la Jevicki, Z k = 8 π | k | L ∞ X n =0 Res y → p n Γ (cid:0) y + (cid:1) √ π ! N | k | e − N | k | y ln y = 8 π | k | L (cid:16) y π (cid:17) N | k | ∞ X n =0 e nN | k | ln y Res ε → (cid:8) y − ε Γ( − n + ε ) (cid:9) N | k | . (52)There is no difficulty in computing and summing the residues for the first cases, N | k | = 1 and N | k | = 2: Z N | k | =1 = √ πm T L exp (cid:26) − m L π (cid:27) (53)and Z N | k | =2 = m K (cid:18) m L π | k | (cid:19) (54)(where K is a modified Bessel function), but the partition function for highervalues of N | k | turns out to be more difficult to compute. From the expansionof (52), we can write it as Z k = 8 π | k | L (cid:18) y √ π (cid:19) N | k | ∞ X n =0 (cid:18) ( − y ) n n ! (cid:19) N | k | T N | k |− , n ( y ) . (55)The function T R, n ( y ) is given by T R, n ( y ) = ∞ X r,s,t =0 δ r + s + t, R a r b s c t,n , (56)with coefficients defined by the expansions y − ε = ∞ X r =0 a r ε r ,πε sin πε ∼ ∞ X s =0 b s ε s ,n !Γ( n + 1 − ε ) ∼ ∞ X t =0 c t,n ε s . (57)Expressions for a r , b s are readily found, a r = ( − ln y ) r r ! ,b s = s − − π s | B s | s ! , s even , , s odd , (58)where B s are Bernoulli numbers. As for c t,n , it can be written as a sum overYoung tableaux of order t , c t,n = X Y . T . ( t ) ( − t − P tj =1 ν j Q ti =1 (cid:2) ψ ( i − ( n + 1) (cid:3) ν i Q tℓ =1 ℓ ! ν ℓ ν ℓ ! , (59)15here ν j , j = 1 , . . . , t, are the numbers of rows with j elements, such that P tj =1 jν j = t , and ψ is the digamma function.As an example, for the k = 1 sector in the C P model we need T ,n = ψ ( n + 1) − ln y , (60)from which equation (54) obtains. For the k = 2 sector in the same model,expression (59) is already too cumbersome to compute (56) explicitly: T ,n = −
16 (ln y ) − π y + (cid:20)
12 (ln y ) + π (cid:21) ψ ( n + 1) − (ln y ) (cid:18) ψ ( n + 1) − ψ ′ ( n + 1) (cid:19) + 16 ψ ( n + 1) − ψ ( n + 1) ψ ′ ( n + 1) + 16 ψ ′′ ( n + 1) . (61)However, it is not necessary to perform the summation in (55) to realise thatthe pole structure has a natural interpretation in terms of instantons. From theoriginal classical action S [Ψ , Ψ † , A µ , α ] = N g Z T d x | D µ Ψ | + N g Z T d x α ( x ) (cid:0) Ψ † Ψ − (cid:1) , (62)the classical equations of motion (cid:0) − D µ + α (cid:1) Ψ = 0 , α = Ψ † D µ Ψ (63)ensure that, for classical solutions, the value of the action is given by the integralof − α : S cl = N g Z T d x | D µ Ψ | = N g Z T d x ( − α ) . (64)The n th pole of the integrand of Z k corresponds to a value of α = m such that m L π | k | = − (cid:18) n + 12 (cid:19) = ⇒ S cl = N g m L = (1 + 2 n ) N g π | k | , (65)i.e., it exactly matches the classical action of a multiinstanton configurationcomposed of an instanton and n instanton-antiinstanton pairs (figure 4). Thisis in contrast with the case of the sphere, where the structure of poles of theintegrand of Z k does not correspond to charge k multiinstanton configurations.Notice that, although there are no unit charge instantons on the torus [34],there is a nontrivial contribution of the k = ± .2.2 The saddle point method Now let us consider the extrema of e S ( y ). These are the zeros of its derivative,d e S ( y )d y = ln y − ψ ( y + 1 / , (66)and constitute a sequence y = s n , n = 0 , , . . . Saddle points and poles alter-nate, s > p > s > p > · · · , as seen in figures 3 and 4. The saddle pointsapproach the poles for large n , lim n →∞ s n /p n = 1.If the dominant saddle point s lies in the region y ≫
1, we can find itslocation as an expansion in powers of y − starting from (47): s = y + 124 y − y + O (cid:0) y − (cid:1) . (67)Equivalently, the infinite volume value m of the saddle point receives finitevolume corrections, m s = m (cid:18) π | k | m L + O (cid:0) y − (cid:1)(cid:19) . (68)We evaluate the partition function in sector k , up to quadratic order, Z ( s ) k ≈ π | k | L s πN | k | e − N | k | e S ( s ) d e S d y ! − / s , (69)from which Z ( s ) k = 4 π | k | L s πy N | k | exp (cid:26) − N | k | y − N | k | y + 116 y + 29 N | k | y + O (cid:0) y − (cid:1)(cid:27) = 4 πm T √ N L exp ( − N m L π − π N k m L + (cid:18) π | k | m L (cid:19) + O (cid:0) y − (cid:1)) . (70)The partition function in the trivial topological sector can be computed inthe large m L limit by substituting an integral for the sum in the definition ofthe ζ function, Z = Z d m exp (cid:26) N m L π (cid:18) ln m m − (cid:19)(cid:27) , (71)and coincides with the B → Z k . There are no poles in thissector, which is compatible with the absence of instantons of zero topologicalcharge, and its only saddle point yields an approximation Z ( s )0 ≈ πm T √ N L exp (cid:26) − N m L π (cid:27) , (72)agreeing with the result (70) for the nontrivial sectors.17he full partition function in terms of the vacuum angle θ , in the saddlepoint approximation, is the sum Z ( s ) ( θ ) = X k ∈ Z Z ( s ) k e − ikθ ≈ πm T √ N L exp (cid:26) − N m L π (cid:27) X k ∈ Z exp (cid:26) − π N k m L − ikθ (cid:27) = 4 πm T √ N L exp (cid:26) − N m L π (cid:27) ϑ (cid:18) θ π (cid:12)(cid:12)(cid:12)(cid:12) iN m L (cid:19) , (73)Still in the region y ≫
1, i.e. m L ≫ π | k | , let us analyse this partitionfunction in two different regimes, depending on the relation between the number N and m L .For N ≫ m L , the contribution of high topological sectors (large | k | ) to(73) can be neglected. Keeping just sectors k = 0 , ±
1, we obtain Z ( s ) ( θ ) ≈ πm T √ N L exp (cid:26) − N m L π (cid:27) (cid:18) (cid:26) − π Nm L (cid:27) cos θ (cid:19) , (74)giving rise to a vacuum energy density E ( s )0 ( θ ) − E ( s )0 (0) = − L ln Z ( s ) ( θ ) Z ( s ) (0) ≈ L exp (cid:26) − π Nm L (cid:27) (1 − cos θ ) , (75)i.e., the typical E ( θ ) dependence of a dilute instanton gas. This is a 2 π -periodicfunction, smooth for all values of θ including θ = ± π . Hence, there is no firstorder phase transition.This regime would be compatible with a definition of the 1 /N expansionin which both the large N limit and the 1 /N corrections are studied in finitevolume.However, if we move to the region m L ≫ N , it proves convenient to usethe Poisson formula in (73) to get Z ( s ) ( θ ) = 4 √ πm N exp (cid:26) − N m L π (cid:27) ϑ (cid:20) θ π (cid:21) (cid:18) (cid:12)(cid:12)(cid:12)(cid:12) i m L N (cid:19) , = 4 √ πm N exp (cid:26) − N m L π (cid:27) X q ∈ Z exp ( − πm L N (cid:18) q + θ π (cid:19) ) , (76)where the ϑ function with characteristics is given by ϑ (cid:20) ab (cid:21) ( z | τ ) = X n ∈ Z exp (cid:8) iπτ ( n + a ) + i π ( n + a )( z + b ) (cid:9) . (77)Now, it is the dual sum in q that is dominated by the low | q | terms. Definingagain e θ as the angle in [ − π, + π ] differing from θ by an integer, Z ( s ) ( θ ) ≈ √ πm N exp (cid:26) − N m L π − m L πN e θ (cid:27) . (78)18his reproduces the traditional large N picture, where instanton effects aresuppressed, and the vacuum energy density depends quadratically on θ withinthe interval [ − π, + π ]: E ( s )0 ( θ ) − E ( s )0 (0) ≈ m πN e θ . (79)Periodicity in θ is guaranteed because E ( θ ) depends on the periodic variable e θ , but this function is not smooth at odd multiples of π , where e θ is doublydefined, levels cross and a first order phase transition occurs.This regime is compatible with a definition of the 1 /N expansion in which thethermodynamic limit is performed first, and N is taken to infinity afterwards.These results agree with the analysis of C P N − models on the sphere. Sincethe physical pictures pertaining to the regimes N ≫ m L and m L ≫ N aredifferent, the limits N → ∞ and V → ∞ do not commute, and the orders inwhich these limits are taken determine different theories. This behaviour pointstowards the existence of a phase transition in the N → ∞ theory. After the careful analysis of the large N method on the sphere and the torus, weconclude that the apparent incompatibility between instanton physics and the1 /N expansion has its cause in the formulation of the latter in infinite volumeand is a subtle effect of the noncommutativity of the large N and thermodynamiclimits.To clarify this, consider the essential dependence of the vacuum energy den-sity on the angle θ , the volume V , and the number of colours N , at fixed saddlepoint mass m T E ( θ ) = − V ln ϑ (cid:16) θ π (cid:12)(cid:12)(cid:12) iN m V (cid:17) ϑ (cid:16) (cid:12)(cid:12)(cid:12) iN m V (cid:17) , (80)which is valid in the cases of the sphere and the torus provided we define thefree energy as a function of θ by subtraction of the contribution at θ = 0 foreach N .The various limits of (80) are best discussed in terms of a dimensionlessvariable x ≡ N/ (6 m V ) and the function N m E ( θ ) ≡ f ( θ, x ) = − x ln ϑ (cid:0) θ π | ix (cid:1) ϑ (0 | ix ) = − x ln ϑ (cid:20) θ/ (2 π )0 (cid:21) (cid:0) (cid:12)(cid:12) ix (cid:1) ϑ (cid:0) (cid:12)(cid:12) ix (cid:1) . (81)The last equation is obtained by applying the Poisson resummation formula,that is, the modular transformation of the theta functions. For x strictly posi-tive, f is a well defined real analytic function of θ ∈ R . The nonanalyticities forcomplex θ are branch cuts located at the zeros of the Jacobi ϑ function, thatis, for θ = (1 + ix ) π and its translations by integer multiples of 2 π and of i πx .These zeros never occur for real values of θ .The two limiting regimes we have been discussing are given in terms of thedimensionless quantities as: 19 Semiclassical: send x to infinity (meaning N ≫ m V ). In this case ϑ (cid:18) θ π (cid:12)(cid:12)(cid:12)(cid:12) ix (cid:19) = 1 + 2 e − πx cos θ + O (e − πx ) , (82)and we recover the nonperturbative result f ( θ, x ) = x e − πx (1 − cos θ ) + O (e − πx ) , (83)equivalent to E ( θ ) = 1 V exp (cid:26) − πN m V (cid:27) (1 − cos θ ) + O (cid:18) m N exp (cid:26) − πN m V (cid:27)(cid:19) , (84)equivalent to (2) (and vanishing as x → ∞ together with all of its deriva-tives). • Traditional large N : send x to zero (meaning N ≪ m V ). This is aproblematic limit for the modular parameter τ = ix , which leaves theupper half plane. We consider separately the regions θ ∈ ( − π, π ) and θ = π (the rest of the function is obtained by periodicity): f (cid:0) θ ∈ ( − π, π ) , x (cid:1) = θ π + O (cid:18) exp (cid:26) − π (1 − θ/π ) x (cid:27)(cid:19) ,f ( θ = π, x ) = π π − x ln 2 + O (e − π/x ) , (85)reproducing (1), including the first order cusp at θ = π : E (cid:0) θ ∈ ( − π, π ) (cid:1) = 3 m πN θ + O (cid:18) m N exp (cid:26) − πm V (1 − θ/π ) N (cid:27)(cid:19) , E ( θ = π ) = 3 m πN π − ln 2 V + O (cid:18) m N exp (cid:26) − πm VN (cid:27)(cid:19) , (86)The rˆole of ix as a modular parameter suggests an analogy with finite tem-perature models, where the number of colours corresponds to the inverse tem-perature β . In this sense, we expect that the physics at 1 /N = 0 correspondsto zero temperature phenomena. The fact that the thermodynamic and large N limits do not commute (reflected in the behaviour of x ) would suggest thepresence of a phase transition exactly at zero temperature, i.e., 1 /N = 0.Returning to physical quantities, we have shown that finite volume effectsin the θ dependence of the vacuum energy density for C P N − models on S and T , when all topological sectors are taken into account, give rise to twoasymptotic regimes, one dominated by instanton effects (when N ≫ m V ) andthe other by the conventional large N picture (when N ≪ m V ). These aresmoothly connected by an interpolating region.It should be realised that the basic hypotheses of the method of large N donot hold when N ≪ m V , for which precisely the traditional large N resultsobtain. The saddle point technique needs N to be the largest dimensionlessparameter of the theory, in particular larger than m V . Two very differenttheories are defined by interchanging the noncommuting limits N → ∞ and V → . In principle, the only procedure consistent with the saddle point method istaking the large N limit first, and then going over to the thermodynamic limit.However, for small values of θ , where the large N approximation is expectedto hold [41], both procedures seem to make sense. Lattice measurements of the θ = 0 topological susceptibility agree with the traditional large N picture, cor-responding to performing first the V → ∞ limit, and then taking N to infinity.However, these simulations have been only carried out for values of the parame-ters such that m V & N (see Table 1 and references [46, 50, 51]), agreeing withour analysis in the region not validated by the saddle point method. It wouldbe interesting to rerun these simulations for smaller volumes, still close to thethermodynamic limit with a stable mass gap, but where an instanton-dominated θ dependence of the vacuum energy density could emerge and eventually takeover. Notice that the singularity of the topological susceptibility pointed out bylattice simulations in the C P model is also found in the semiclassical scenario.The existence of such a singularity can also be understood by the presence of afamily of Lee-Yang zeros of the analytic continuation of the partition functionin the complex θ -plane, converging to θ = 0 in the thermodynamic limit [9].Reference m L m L N [42] Blatter et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. m for different C P N − models atdifferent volumes m L . The models are in the thermodynamic regime m L >N in all cases. The rˆole of instantons is only manifest in the cases C P and C P [22]As regards the neighbourhood of θ = π , the nodal analysis of [13] appears21ncompatible with a first order phase transition at θ = π . This is the behaviourof the system for lower values of N for any volume, i.e. C P and C P models.For larger values of N , for instance N >
4, this behaviour is only observed forsmall volumes, i.e. volumes which verify m V ≪ N . For larger volumes theeffect is swept off by the infrared fluctuations and the system undergoes a phasetransition at θ = π with spontaneous CP symmetry breaking.The behaviour for intermediate values of θ has so far proved elusive to nu-merical techniques, due to the inaccuracies inherent to lattice simulations inthis region. Let us however remark that a novel technique [52] based on analyt-ically continuing the θ dependence to imaginary values has been introduced toovercome this problem. This technique has been applied to the C P model [53],with conclusions agreeing with the usual large N expansion. The consistency ofthis technique for any value of N is supported by the absence of singularities inthe analytic extension of the partition function to the whole θ -plane [9].Summing up, the large N method is compatible with instanton effects. Be-sides, the results on θ dependence obtained with this tool agree with the knownbehaviour at θ = 0, i.e., the vacuum energy density is differentiable there andthe Vafa-Witten theorem holds. It is also compatible with the numerical deter-mination of the topological susceptibility at θ = 0. The analysis of the polesof the partition function on the torus supports the method of Jevicki, wherebyinstantonic effects appear in the large N limit in the form of resonances. Firstorder phase transitions with spontaneous parity breaking at θ = π appear inthe formulation of the models directly at infinite volume, and we have exposedthe analytic roots of this fact. The large N method is thus in agreement withall exact results on θ dependence, and provides a valuable bridge between theangles θ = 0 and θ = π .Let us remark that the behaviour of the theory in finite volume plays afundamental rˆole in condensed matter settings, where sigma models can beused as effective theories for the quantum Hall effect [54, 55, 56, 57]. In thiscontext, the Hall conductivity is identified with the coupling of the topologicalterm, and the stability of Hall plateaux is linked to the renormalisation grouprunning of the couplings (including θ ). The large N limit of C P N − modelswas studied in connection with this phenomenon in a series of papers (see, e.g.,[58, 59, 60, 61, 62]), in which the different regimes we have discussed were alsoidentified; in this case the traditional large N limit at infinite volume is blindto edge effects, which of course are crucial for the physics of the Hall effect.In particular, edge currents are a finite size effect and this suggests that the m V ≪ N regime of C P N − sigma models is the relevant regime for theirdescription.Finally, we remark that the difference between the two regimes is due to theasymptotic behaviour of the effective action in the different topological sectors.According to equations (48) and (50), for small values of the topological charge, | k | < m V , the effective action is quadratic in k , whereas for large topologicalcharges its leading term is linear in | k | [9]. The two regimes also differ atfinite temperature. Since the spacetime volume is V = LT , the change ofasymptotic dependence of the effective action on the topological charge can beassociated with a finite temperature crossover from the low temperature regime β = 1 /T > m L/ | q | to the high temperature regime β < m L/ | q | and cannot berelated to any phase transition [63]. One might expect a similar phenomenon inQCD, although in that case there is a finite temperature phase transition [64].22 cknowledgements We thank D. Garc´ıa-Alvarez for discussions on the early stages of this paper.M. Aguado thanks Departamento de F´ısica Te´orica of the University of Zaragozafor hospitality in visits where this project was developed. M. Asorey was par-tially supported by the Spanish CICYT grant FPA2009-09638 and DGIID-DGA(grant 2009-E24/2).
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