Infrared behavior in systems with a broken continuous symmetry: classical O(N) model vs interacting bosons
IInfrared behavior in systems with a broken continuous symmetry: classical O( N )model vs interacting bosons N. Dupuis
Laboratoire de Physique Théorique de la Matière Condensée, CNRS - UMR 7600,Université Pierre et Marie Curie, 4 Place Jussieu, 75252 Paris Cedex 05, France (Dated: January 14, 2010)In systems with a spontaneously broken continuous symmetry, the perturbative loop expansionis plagued with infrared divergences due to the coupling between transverse and longitudinal fluc-tuations. As a result the longitudinal susceptibility diverges and the self-energy becomes singularat low energy. We study the crossover from the high-energy Gaussian regime, where perturbationtheory remains valid, to the low-energy Goldstone regime characterized by a diverging longitudinalsusceptibility. We consider both the classical linear O( N ) model and interacting bosons at zerotemperature, using a variety of techniques: perturbation theory, hydrodynamic approach (i.e., forbosons, Popov’s theory), large- N limit and non-perturbative renormalization group. We emphasizethe essential role of the Ginzburg momentum scale p G below which the perturbative approach breaksdown. Even though the action of (non-relativistic) bosons includes a first-order time derivative term,we find remarkable similarities in the weak-coupling limit between the classical O( N ) model andinteracting bosons at zero temperature. PACS numbers: 05.30.Jp,05.70.Fh
I. INTRODUCTION
In the context of critical phenomena, it is well knownthat the Gaussian approximation breaks down in thevicinity of a second-order phase transition (below theupper critical dimension). When the Ginzburg criterion | T − T c | /T c (cid:29) t G is violated ( T c denotes the critical tem-perature and | T − T c | /T c ∼ t G defines the Ginzburg tem-perature T G ), the long-distance behavior of the correla-tion functions cannot be described by a Gaussian fluc-tuation theory and more involved techniques, such asthe renormalization group, are required (see e.g. [1]). Atthe critical point ( T = T c ), one can nevertheless distin-guish two regimes in momentum space: a high-energyGaussian regime, where the Gaussian approximation re-mains essentially correct, and a low-energy critical regimewhere the correlation function of the order parameterfield shows a critical behavior characterized by a non-zeroanomalous dimension η . These two regimes are separatedby a characteristic momentum scale p G which defines theGinzburg length ξ G = p − G (see e.g. [2]).In systems with a broken continuous symmetry, thephysics remains non-trivial in the whole low-temperaturephase due to the presence of Goldstone modes, whichimplies that correlations decay algebraically. The cou-pling between transverse and longitudinal order parame-ter fluctuations leads to a divergence of the longitudinalsusceptibility [3–5]. Away from the critical regime (i.e. atsufficiently low temperatures), one can distinguish a high-energy Gaussian regime ( | p | (cid:29) p G ), where the Gaussianapproximation remains correct, and a low-energy Gold-stone regime ( | p | (cid:28) p G ) dominated by the Goldstonemodes and characterized by a divergence of the longitu-dinal susceptibility. Note that the Ginzburg momentumscale p G defined here is the same as the one signaling theonset of the critical regime (in momentum space) when the system is near the phase transition. For instance, forthe ( ϕ ) theory with O( N ) symmetry (classical O( N )model), one finds a transverse susceptibility χ ⊥ ( p ) ∼ / p for p → , while the longitudinal susceptibility χ (cid:107) ( p ) ∼ / | p | − d is also singular in dimensions < d ≤ (the divergence is logarithmic for d = 4 ). At and belowthe lower critical dimension d − c = 2 , transverse fluctua-tions lead to a suppression of long-range order (Mermin-Wagner theorem). There is an analog phenomenon inzero-temperature quantum systems with a broken con-tinuous symmetry. When the Goldstone mode frequency ω = c | p | vanishes linearly with momentum, the longitu-dinal susceptibility χ (cid:107) ( p , ω ) ∼ / ( ω − c p ) (3 − d ) / hasno pole-like structure but a branch-cut for d ≤ , andthe dynamical structure factor exhibits a critical contin-uum above the usual delta peak δ ( ω − c | p | ) due to theGoldstone mode [6–8].Historically, the divergence of the longitudinal sus-ceptibility was encountered (although not recognized assuch) early on in interacting boson systems. The firstattempts to improve the Bogoliubov theory of superflu-idity [9] were made difficult by a singular perturbationtheory plagued by infrared divergences [10–13]. As real-ized later on [14–16], the singular perturbation theory isa direct consequence of the coupling between transverseand longitudinal fluctuations.In this paper, we study the crossover from the high-energy Gaussian regime to the low-energy Goldstoneregime in the ordered phase, both for the classicalO( N ) model and interacting bosons at zero tempera-ture. Even though the action of (non-relativistic) bosonsincludes a first-order time derivative term, which pre-vents a straightforward description in terms of a classicalO(2) model, we find remarkable similarities in the weak-coupling limit between these two models. On the otherhand, the strong-coupling limit of the O( N ) model, i.e. a r X i v : . [ c ond - m a t . qu a n t - g a s ] A p r the critical regime near the phase transition, has no directanalog in zero-temperature interacting boson systems.The classical O( N ) model is studied in Sec. II, whilesuperfluid systems are discussed in Sec. III. First, weshow that the loop expansion about the mean-field so-lution is plagued with infrared divergences and deduce aperturbative estimate of the Ginzburg momentum scale p G (Secs. II A and III A). Then, we use symmetry ar-guments to derive the exact value of the self-energies atvanishing momentum (and frequency) (Secs. II A 3 andIII A 3). In the case of bosons, we obtain Nepomnyashchiiand Nepomnyashchii’s result about the vanishing of theanomalous self-energy [14]. In Secs. II B and III B, weshow that the difficulties of perturbation theory can becircumvented within a hydrodynamic approach (i.e., forbosons, Popov’s theory [17–19]) based on an amplitude-direction representation of the order parameter field.This yields the correlation functions in the hydrody-namic regime defined by a characteristic momentum scale p c (cid:29) p G . The O( N ) model is solved in the large- N limitin Sec. II C. This allows us to obtain the longitudinalcorrelation function in the whole low-temperature phase,including the critical regime in the vicinity of the phasetransition. Finally, we show how the non-perturbativerenormalization group (NPRG) provides a natural frame-work to understand the ordered phase of the O( N ) modeland the superfluid phase of interacting bosons (Secs. II Dand III C). II. THE ( ϕ ) THEORY AT LOWTEMPERATURES
We consider the ( ϕ ) theory defined by the action S [ ϕ ] = (cid:90) d d r (cid:26)
12 ( ∇ ϕ ) + r ϕ + u
4! ( ϕ ) (cid:27) (1)where ϕ is a N -component real field and d the spacedimension. We assume N ≥ and d > . The modelis regularized by a ultraviolet momentum cutoff Λ . Theconnected propagator G ij ( p ) = (cid:104) ϕ i ( p ) ϕ j ( − p ) (cid:105) − (cid:104) ϕ i ( p ) (cid:105)(cid:104) ϕ j ( − p ) (cid:105) (2)is related to the self-energy Σ by Dyson’s equation G − = G − + Σ , where G ,ij ( p ) = δ i,j p + r (3)is the bare propagator. In the low-temperature phase,if we denote by ϕ = (cid:104) ϕ ( r ) (cid:105) the order parameter, theself-energy Σ ij ( p ) = ˆ ϕ ,i ˆ ϕ ,j Σ l ( p ) + ( δ i,j − ˆ ϕ ,i ˆ ϕ ,j )Σ t ( p )= δ i,j [Σ n ( p ) − Σ an ( p )] + 2 ˆ ϕ ,i ˆ ϕ ,j Σ an ( p ) (4)( ˆ ϕ = ϕ / | ϕ | ) can be written in terms of its longitudinal( Σ l ) and transverse ( Σ t ) parts. In the second line of (4), we have introduced the “normal” ( Σ n ) and “anomalous”( Σ an ) self-energies. In the following, we assume that theorder parameter ϕ is along the direction (1 , , · · · , sothat Σ ii ( p ) = (cid:26) Σ n ( p ) + Σ an ( p ) if i = 1 , Σ n ( p ) − Σ an ( p ) if i (cid:54) = 1 . (5)The anomalous self-energy Σ an is related to the spon-taneously broken O( N ) symmetry and vanishes in thehigh-temperature phase. Σ n and Σ an are analogous tothe normal and anomalous self-energies which are usu-ally introduced in the theory of superfluidity [20, 21].For N = 2 , we can introduce the complex field ψ ( r ) = 1 √ ϕ ( r ) + iϕ ( r )] . (6)Making use of the two-component field Ψ( r ) = (cid:18) ψ ( r ) ψ ∗ ( r ) (cid:19) , Ψ † ( r ) = ( ψ ∗ ( r ) , ψ ( r )) , (7)the two-point propagator becomes a × matrix inFourier space, whose inverse is given by (cid:18) p + r + Σ n ( p ) Σ an ( p )Σ an ( p ) p + r + Σ n ( p ) (cid:19) , (8)and bears some similarities with the single-particle prop-agator in a superfluid (Sec. III). A. Gaussian approximation and breakdown ofperturbation theory
Let us begin with a dimensional analysis of the action(1). If we assign the scaling dimension 1 to momenta(i.e. [ p ] = 1 ), the field has engineering dimension [ ϕ ] = d − , [ r ] = 2 and [ u ] = 4 − d . We can then define twocharacteristic length scales, ξ ∼ | r | − / ,ξ G ∼ u / ( d − . (9)In the critical regime of the low-temperature phase ( ξ (cid:29) ξ G ), ξ G is the characteristic length scale associated tothe onset of critical fluctuations, while ξ ≡ ξ J is theJosephson length separating the critical regime from aregime dominated by Goldstone modes [22]. When criti-cal fluctuations are taken into account, one finds that ξ J diverges with a critical exponent ν which differs from themean-field value / . At low temperatures away from thecritical regime ( ξ (cid:28) ξ G ), ξ ≡ ξ c corresponds to a correla-tion length for the gapped amplitude fluctuations whiledirection fluctuations are gapless due to Goldstone’s the-orem. The physical meaning of the Ginzburg length ξ G in this temperature range will become clear below. FIG. 1. One-loop correction Σ (1) to the self-energy. Thedots represent the bare interaction, the zigzag lines the orderparameter ϕ , and the solid lines the connected propagator G (0) .
1. Gaussian approximation
Within the mean-field (or saddle-point) approxima-tion, one finds ϕ = | ϕ | = ( − r /u ) / in the low-temperature phase ( r < ). In the Gaussian approx-imation, one expands the action to quadratic order inthe fluctuations ϕ − ϕ [1]. This yields the (zero-loop)self-energy Σ (0) ii ( p ) = (cid:26) − r if i = 1 , − r if i (cid:54) = 1 , (10)from which we obtain the longitudinal and transversepropagators, G (0) l ( p ) = G (0)11 ( p ) = 1 p + 2 | r | ,G (0) t ( p ) = G (0)22 ( p ) = 1 p . (11)In agreement with Goldstone’s theorem, the transversepropagator is gapless, whereas the longitudinal suscepti-bility G l ( p = 0) = 1 / | r | is finite. We shall see belowthat this last property is an artifact of the Gaussian ap-proximation.
2. One-loop correction and the Ginzburg momentum scale
The one-loop correction Σ (1) to the self-energy is showndiagrammatically in Fig. 1. While the first diagram is fi-nite, the second one gives a diverging contribution to Σ in the infrared limit p → when d ≤ . The divergencearises when both internal lines correspond to transversefluctuations, which is possible only for Σ . Thus Σ isfinite at the one-loop level and the normal and anomalousself-energies exhibit the same divergence, Σ (1)n ( p ) (cid:39) Σ (1)an ( p ) (cid:39) − N − u ϕ (cid:90) q q ( p + q ) , (12)where we use the notation (cid:82) q = (cid:82) d d q (2 π ) d . The momentumintegration in (12) gives [23] (cid:90) q q ( p + q ) = (cid:26) A d | p | d − if d < ,A ln(Λ / | p | ) if d = 4 , (13) for | p | (cid:28) Λ , where A d = (cid:40) − − d π − d/ sin( πd/
2) Γ( d/ d − if d < , π if d = 4 . (14)The one-loop correction (12) diverges for p → and theperturbation expansion about the Gaussian approxima-tion breaks down. By comparing the one-loop correc-tion to the zero-loop result, i.e. | Σ (1)n ( p ) | ∼ Σ (0)n ( p ) or | Σ (1)an ( p ) | ∼ Σ (0)an ( p ) , one can nevertheless extract a char-acteristic (Ginzburg) momentum scale, p G ∼ (cid:40) [ A d ( N − u ] / (4 − d ) if d < , Λ exp (cid:16) − A ( N − u (cid:17) if d = 4 , (15)which was obtained previously from dimensional analysis[Eq. (9)]. While the Gaussian or perturbative approachremains valid for | p | (cid:29) p G , the limit | p | (cid:28) p G cannot bestudied perturbatively. We shall see in Sec. II B that thebreakdown of perturbation theory is due to the couplingbetween transverse and longitudinal fluctuations.
3. Exact results for Σ n ( p = 0) and Σ an ( p = 0) Although the one-loop correction Σ (1) ( p ) divergeswhen p → for d ≤ , it is nevertheless possible toobtain the exact value of Σ( p = 0) using the O( N ) sym-metry of the model.Let us consider the effective action Γ[ φ ] = − ln Z [ h ] + (cid:90) d d r h · φ (16)defined as the Legendre transform of the free energy − ln Z [ h ] where h is an external field which couples lin-early to the ϕ field and φ i ( r ) = δ ln Z [ h ] δh i ( r ) = (cid:104) ϕ i ( r ) (cid:105) h . (17)The notation (cid:104)· · ·(cid:105) h means that the average value is com-puted in the presence of the external field h . Γ[ φ ] satisfiesthe equation of state δ Γ[ φ ] δφ i ( r ) = h i ( r ) . (18)At equilibrium and in the absence of external field, theorder parameter ϕ = (cid:104) ϕ ( r ) (cid:105) is obtained from the sta-tionary condition of the effective action, δ Γ[ φ ] δφ i ( r ) (cid:12)(cid:12)(cid:12)(cid:12) φ ( r )= ϕ = 0 . (19) Γ[ φ ] is the generating functional of the one-particle irre-ducible vertices Γ ( n ) i ··· i n ( r , · · · , r n ) = δ ( n ) Γ[ φ ] δφ i ( r ) · · · δφ i n ( r n ) (cid:12)(cid:12)(cid:12)(cid:12) φ ( r )= ϕ . (20)The later fully determine the correlation functions. Inparticular, the two-point vertex Γ (2) is related to thepropagator by Γ (2) = G − = G − + Σ .The O( N ) invariance of the action (1) implies that theeffective action Γ[ φ ] is invariant under a rotation of thefield φ . Let us consider the case N = 3 for simplicity (thefollowing results are easily extended to arbitrary N ). Foran infinitesimal rotation φ → φ + θ n × φ about the axis n ( n = 1 and θ → ), the invariance of the effectiveaction yields (cid:90) d d r (cid:88) ijk δ Γ[ φ ] δφ i ( r ) (cid:15) ijk n j φ k ( r ) = 0 , (21)where (cid:15) ijk is the totally antisymmetric tensor. Takingthe first-order functional derivative δ/δφ l ( r (cid:48) ) and setting φ i ( r ) = δ i, ϕ , we obtain (cid:90) d d r (cid:88) i,j Γ (2) il ( r , r (cid:48) ) (cid:15) ij n j = 0 . (22)With n = (0 , , , this gives Γ (2)22 ( p = 0) = r + Σ ( p = 0) = 0 , (23)where Γ (2) ( p ) denotes Γ (2) ( p , − p ) . Equation (23)is a direct consequence of Goldstone’s theorem. Ifwe now take the second-order functional derivative δ (2) /δφ l ( r (cid:48) ) δφ m ( r (cid:48)(cid:48) ) of (21) and set φ i ( r ) = δ i, ϕ , weobtain the Ward identity (cid:88) i,j (cid:104) Γ (2) im ( r (cid:48) , r (cid:48)(cid:48) ) (cid:15) ijl + Γ (2) il ( r (cid:48)(cid:48) , r (cid:48) ) (cid:15) ijm (cid:105) n j + (cid:90) d d r (cid:88) i,j Γ (3) ilm ( r , r (cid:48) , r (cid:48)(cid:48) ) (cid:15) ij n j ϕ = 0 . (24)Choosing l = 2 , m = 1 and j = 3 , this gives Γ (2)11 ( r (cid:48) , r (cid:48)(cid:48) ) − Γ (2)22 ( r (cid:48)(cid:48) , r (cid:48) ) − ϕ (cid:90) d d r Γ (3)221 ( r , r (cid:48) , r (cid:48)(cid:48) ) = 0 . (25)Integrating over r (cid:48) and r (cid:48)(cid:48) and using (23), we deduce (inFourier space) Γ (3)122 (0 , ,
0) = Γ (2)11 (0 , √ V ϕ , (26)where V is the volume of the system.Let us now consider the exact diagrammatic represen-tation of the self-energy shown in Fig. 2. We know fromperturbation theory that the third diagram in Fig. 2 ispotentially dangerous when the two internal lines corre-spond to transverse fluctuations. We therefore write theself-energy Σ ( p ) as Σ ( p ) = ˜Σ ( p ) − N − √ V u ϕ (cid:88) q G ( q ) G ( p + q ) × Γ (3)122 ( − p , − q , p + q ) , (27) Γ (3) Γ (4) Γ (3) Γ (3) FIG. 2. Exact diagrammatic representation of the self-energyin terms of the three- and four-leg vertices Γ (3) and Γ (4) . Thedots represent the bare interaction, the zigzag lines the orderparameter, and the solid lines the (exact) connected propaga-tor. where ˜Σ ( p ) denotes the part of the self-energy whichis regular in perturbation theory (i.e. the part that doesnot contain pairs of lines corresponding to G G ). Ifwe assume that the transverse propagator G ( q ) is pro-portional to / q for q → (this result will be shownin the following sections), the integral (cid:82) q G ( q ) is in-frared divergent for d ≤ . To obtain a finite self-energy Σ ( p = 0) , one must require that lim q → Γ (3)122 (0 , − q , q ) = Γ (3)122 (0 , ,
0) = 0 . (28)The Ward identity (26) then implies Γ (2)11 ( p = 0) = 0 , sothat we finally obtain Σ n ( p = 0) = − r + 12 [Γ ( p = 0) + Γ ( p = 0)] = − r , Σ an ( p = 0) = 12 [Γ ( p = 0) − Γ ( p = 0)] = 0 . (29)It may appear surprising that the anomalous self-energy,which is related to the spontaneously broken O( N ) sym-metry, vanishes for p = 0 . The equivalent property ininteracting boson systems is a fundamental result of thetheory of superfluidity (Sec. III). B. Amplitude-direction representation
The difficulties of the perturbation theory of Sec. II Acan be avoided if one uses the “good” hydrodynamic vari-ables in the low-temperature phase, namely the ampli-tude and the direction of the ϕ field. We thus write ϕ ( r ) = ρ ( r ) n ( r ) , (30)where n ( r ) = 1 , and obtain the action S [ ρ, n ] = (cid:90) d d r (cid:26)
12 ( ∇ ρ ) + ρ ∇ n ) + r ρ + u ρ (cid:27) . (31)At the mean-field level, the amplitude takes the value ρ = ( − r /u ) / in the low-temperature phase ( r < ). For small amplitude fluctuations ρ (cid:48) = ρ − ρ (which isexpected to be the case at sufficiently low temperatures),we obtain the action S [ ρ (cid:48) , n ] = (cid:90) d d r (cid:26)
12 ( ∇ ρ (cid:48) ) + | r | ρ (cid:48) + ρ ∇ n ) (cid:27) (32)and deduce that the amplitude fluctuations are gapped, (cid:104) ρ (cid:48) ( p ) ρ (cid:48) ( − p ) (cid:105) = 1 p + p c . (33)If we are interested only in momenta | p | (cid:28) p c = (cid:112) | r | ,to first approximation we can ignore the higher-orderterms in ρ (cid:48) that were neglected in (32), since they wouldonly lead to a finite renormalization of the coefficients ofthe action S [ ρ (cid:48) , n ] [23].Equation (32) shows that in the “hydrodynamic”regime | p | (cid:28) p c direction fluctuations are described bya non-linear sigma model. It is convenient to use thestandard parametrization n = ( σ, π ) where σ is the com-ponent of n along the direction of order and π a ( N − -component field ( n = σ + π = 1 ). Integrating over σ ,one obtains S [ ρ (cid:48) , π ] = (cid:90) d d r (cid:26)
12 ( ∇ ρ (cid:48) ) + | r | ρ (cid:48) + ρ ∇ π ) (cid:27) (34)for small transverse fluctuations π [24]. In this limit, wecan treat π i ( r ) as a variable varying between −∞ and ∞ . From (34), we deduce the propagator of the π field, (cid:104) π i ( p ) π j ( − p ) (cid:105) = δ i,j ρ p . (35)Again we note that the terms neglected in (34) wouldonly lead to a finite renormalization of the (bare) stiff-ness ρ of the non-linear sigma model at sufficiently lowtemperature. In fact, equation (34) gives an exact de-scription of the low-energy behavior | p | (cid:28) p c if one re-places ρ by the exact stiffness and p − c = (2 | r | ) − / bythe exact correlation length of the ρ (cid:48) field.We are now in a position to compute the longitudinaland transverse propagators using ϕ l = ρσ = ρ (cid:112) − π (cid:39) ρ + ρ (cid:48) − ρ π , ϕ t = ρ π (cid:39) ρ π . (36)Since the long-distance physics is governed by transversefluctuations, we have retained in (36) the leading contri-butions in π . Making use of (35), one readily obtains G t ( p ) (cid:39) ρ (cid:104) π i ( p ) π i ( − p ) (cid:105) = 1 p . (37)The longitudinal propagator is given by G l ( r ) = (cid:104) ρ (cid:48) ( r ) ρ (cid:48) (0) (cid:105) + 14 ρ (cid:104) π ( r ) π (0) (cid:105) c = (cid:104) ρ (cid:48) ( r ) ρ (cid:48) (0) (cid:105) + N − ρ G t ( r ) , (38) where (cid:104)· · ·(cid:105) c stands for the connected part of (cid:104)· · ·(cid:105) . Thesecond line is obtained using Wick’s theorem. In Fourierspace, this gives G l ( p ) = 1 p + p c + N − ρ (cid:90) q q ( p + q ) , (39)where the momentum integral is given by (13) for | p | (cid:28) Λ and d ≤ . By comparing the two terms in the rhsof (39), we recover the Ginzburg momentum scale (15).For | p | (cid:29) p G , the longitudinal propagator G l ( p ) (cid:39) / ( p + p c ) is dominated by amplitude fluctuations andwe reproduce the result of the Gaussian approximation.On the other hand, for | p | (cid:28) p G , G l ( p ) ∼ / | p | − d is dominated by direction fluctuations and diverges for p → .The divergence of the longitudinal propagator is a di-rect consequence of the coupling between longitudinaland transverse fluctuations [3]. In the long-distance limit,amplitude fluctuations become frozen so that | ϕ | = ρ (cid:39) ρ . This implies that the longitudinal and transversecomponents ϕ l and ϕ t cannot be considered indepen-dently as in the Gaussian approximation (Sec. II A) butsatisfy the constraint ϕ l + ϕ t (cid:39) ρ . To leading order, ϕ l (cid:39) ρ (1 − π ) / and G l ( r ) ∼ G t ( r ) [Eq. (38)], i.e. G l ( p ) ∼ / | p | − d for d ≤ (the divergence is logarith-mic for d = 4 ).Equations (37) and (39) imply that the self-energiesmust satisfy Σ ( p ) = − r − p + C | p | − d , Σ ( p ) = − r + C p , (40)for p → and d < , i.e. Σ n ( p ) = − r + C | p | − d + O ( p ) , Σ an ( p ) = C | p | − d + O ( p ) . (41)For d = 4 , one finds Σ n ( p ) = − r + C ln(Λ / | p | ) + O ( p ) , Σ an ( p ) = C ln(Λ / | p | ) + O ( p ) . (42)For p = 0 , we reproduce the exact results of Sec. II A 3.Equations (41,42) show that Σ n ( p ) and Σ an ( p ) containnon-analytic terms that are dominant for p → . C. Large- N limit In this section, we show that the previous results forthe longitudinal propagator are fully consistent with thelarge- N limit of the ( ϕ ) theory. Furthermore, the large- N limit enables to compute the longitudinal propaga-tor not only at low temperatures but also in the criticalregime near the transition to the high-temperature (dis-ordered) phase.To obtain a meaningful large- N limit, we write thecoefficient of the ( ϕ ) term in Eq. (1) as u /N and takethe limit N → ∞ with u fixed. Following Ref. [23], weexpress the partition function as Z = (cid:90) D [ ϕ , ρ, λ ] e − (cid:82) d d r [ ( ∇ ϕ ) + r ρ + u N ρ + i λ ( ϕ − ρ ) ] . (43)It can be easily verified that by integrating out λ andthen ρ , one recovers the original action S [ ϕ ] . If, instead,one first integrates out ρ , one obtains Z = (cid:90) D [ ϕ , λ ] e − (cid:82) d d r [ ( ∇ ϕ ) + i λ ϕ ] + N u (cid:82) d d r ( iλ − r ) . (44)As in Sec. II B, it is convenient to split the ϕ field into a σ field and a ( N − -component field π . The integrationover the π field gives (cid:90) D [ π ] e − (cid:82) d d r [ ( ∇ π ) + i λ π ] = (det g ) ( N − / , (45)where g − ( r , r (cid:48) ) = − ∇ δ ( r − r (cid:48) ) + iλ ( r ) δ ( r − r (cid:48) ) (46)is the inverse propagator of the π i field in the fluctuating λ field. We thus obtain the action S [ σ, λ ] = 12 (cid:90) d d r (cid:2) ( ∇ σ ) + iλσ (cid:3) − N u (cid:90) d d r ( iλ − r ) + N −
12 Tr ln g − . (47)In the limit N → ∞ , the action becomes proportional to N (this is easily seen by rescaling the σ field, σ → √ N σ )and the saddle-point approximation becomes exact. Foruniform fields σ ( r ) = σ and λ ( r ) = λ , the action is givenby V S [ σ, λ ] = i λσ − N u ( iλ − r ) + N V Tr ln g − (48)(we use N − (cid:39) N for large N ), with g − ( p ) = p + iλ in Fourier space. From (48), we deduce the saddle-pointequations σm = 0 ,σ = 6 Nu ( m − r ) − N (cid:90) p p + m , (49)where we use the notation m = iλ ( iλ is real at thesaddle point). These equations show that the component σ of the ϕ field which was singled out plays the role ofan order parameter.In the low-temperature phase, σ is non-zero and m =0 . The propagator g ( p ) = 1 / p is gapless, thus identi-fying the π i fields as the N − Goldstone modes associ-ated to the spontaneously broken O( N ) symmetry. From Eq. (49), we deduce σ = − Nu ( r − r c ) , (50)where r c = − u (cid:90) p p = − u K d d − d − (51)(with K d = 2 − d π − d/ / Γ( d/ ) is the critical value of r . Since the saddle-point approximation is exact in thelarge- N limit, the effective action Γ[ σ, λ ] is simply givenby the action S [ σ, λ ] [Eq. (47)] [25]. We deduce Γ (2) ( r − r (cid:48) ) = (cid:32) Γ (2) σσ ( r − r (cid:48) ) Γ (2) σλ ( r − r (cid:48) )Γ (2) λσ ( r − r (cid:48) ) Γ (2) λλ ( r − r (cid:48) ) (cid:33) = (cid:18) − ∇ δ ( r − r (cid:48) ) iσδ ( r − r (cid:48) ) iσδ ( r − r (cid:48) ) N Π( r − r (cid:48) ) + Nu δ ( r − r (cid:48) ) (cid:19) , (52)where Π( r − r (cid:48) ) = g ( r − r (cid:48) ) g ( r (cid:48) − r ) (53)and we use the notation Γ (2) σσ ( r − r (cid:48) ) = δ (2) Γ /δσ ( r ) δσ ( r (cid:48) ) ,etc. The two-point vertex Γ (2) is computed for thesaddle-point values of the fields σ and λ . In Fourier space,we obtain Γ (2) ( p ) = (cid:18) p iσiσ N Π( p ) + Nu (cid:19) (54)and the propagator G = Γ (2) − takes the form G ( p ) = 1det Γ (2) ( p ) (cid:18) N Π( p ) + Nu − iσ − iσ p (cid:19) , (55)with det Γ (2) ( p ) = p (cid:20) N p ) + 3 Nu (cid:21) + σ (56)and Π( p ) = (cid:82) q g ( q ) g ( p + q ) . Equation (56), togetherwith the small p behavior of Π( p ) [Eq. (13)], leads us tointroduce three characteristic momentum scales, p G = (cid:18) u A d (cid:19) / (4 − d ) ,p J = (cid:18) σ N A d (cid:19) / ( d − = (cid:20) u A d ( r c − r ) (cid:21) / ( d − ,p c = (cid:18) u σ N (cid:19) / = [2( r c − r )] / . (57)For simplicity, we discuss only the case d < ; equiva-lent results for d = 4 are easily deduced. The Josephsonlength ξ J = p − J – which separates the critical regimefrom the Golstone regime (see below) [22] – diverges atthe transition with the critical exponent ν = 1 / ( d − , FIG. 3. Characteristic momentum scales p G , p J and p c vs T c − T for fixed u [Eqs. (57) with r = ¯ r ( T − T ) ].FIG. 4. Momentum dependence of the longitudinal correla-tion function G σσ ( p ) in the critical and non-critical regimesof the low-temperature phase as obtained from the large- N limit ( < d < ). which also characterizes the divergence of the correlationlength in the high-temperature phase [23]. The momen-tum scales (57) are not independent since p c = p G (cid:18) p J p G (cid:19) d − . (58)If we vary r with u fixed, we find that the three char-acteristic scales (57) are equal when T = T G , where T G is defined by ¯ r ( T c − T G ) = 12 (cid:18) u A d (cid:19) / (4 − d ) (59)(see Fig. 3). We have assumed that r = ¯ r ( T − T ) (with T the mean-field transition temperature) and used r c = ¯ r ( T c − T ) . We recognize in (59) the Ginzburg cri-terion [2] so that we can identify T G with the Ginzburgtemperature separating the critical regime near the tran-sition from the non-critical regime at sufficiently low tem-peratures.In the critical regime ( T c − T (cid:28) T c − T G or p J (cid:28) p G ), using p J (cid:28) p c (cid:28) p G , one finds the longitudinalcorrelation function G σσ ( p ) = p − dJ | p | − d if | p | (cid:28) p J , p if | p | (cid:29) p J , (60) while in the non-critical regime ( T c − T G (cid:28) T c − T or p G (cid:28) p c ), G σσ ( p ) = p c (cid:18) p G | p | (cid:19) − d if | p | (cid:28) p G , p + p c if | p | (cid:29) p G . (61)In the non-critical regime, we recover the results of sec-tion II B. We find two characteristic momentum scales( p G and p c ) and two regimes for the behavior of G σσ ( p ) :i) a Goldstone regime ( | p | (cid:28) p G ) characterized by adiverging longitudinal propagator G σσ ( p ) ∼ / | p | − d ,ii) a Gaussian (perturbative) regime ( | p | (cid:29) p G ) where G σσ ( p ) (cid:39) / ( p + p c ) . The critical regime is char-acterized by two momentum scales ( p J and p G ) andthree regimes for the behavior of G σσ ( p ) : i) a Gold-stone regime ( | p | (cid:28) p J ) with a diverging longitudinalpropagator, ii) a critical regime ( p J (cid:28) | p | (cid:28) p G ) where G σσ ( p ) ∼ / | p | − η with a vanishing anomalous dimen-sion η ( η is O (1 /N ) in the large- N limit [23, 26]), iii)a Gaussian regime ( p G (cid:28) | p | ) where G σσ ( p ) (cid:39) / p .These results are summarized in figure 4. D. The non-perturbative RG
1. The average effective action
The strategy of the NPRG is to build a family of the-ories indexed by a momentum scale k such that fluctu-ations are smoothly taken into account as k is loweredfrom the microscopic scale Λ down to 0 [27, 28]. This isachieved by adding to the action (1) the infrared regula-tor ∆ S k [ ϕ ] = 12 (cid:88) p ,i ϕ i ( − p ) R k ( p ) ϕ i ( p ) . (62)The average effective action Γ k [ φ ] = − ln Z k [ J ] + (cid:90) d d r (cid:88) i J i φ i − ∆ S k [ φ ] (63)is defined as a modified Legendre transform of − ln Z k [ J ] which includes the subtraction of ∆ S k [ φ ] . Here J i is anexternal source which couples linearly to the ϕ i field and φ ( r ) = (cid:104) ϕ ( r ) (cid:105) J . The cutoff function R k is chosen suchthat at the microscopic scale Λ it suppresses all fluctua-tions, so that the mean-field approximation Γ Λ [ φ ] = S [ φ ] becomes exact. The effective action of the original model(1) is given by Γ k =0 provided that R k =0 vanishes. For ageneric value of k , the cutoff function R k ( p ) suppressesfluctuations with momentum | p | (cid:46) k but leaves unaf-fected those with | p | (cid:38) k . The variation of the averageeffective action with k is governed by Wetterich’s equa-tion [29] ∂ t Γ k [ φ ] = 12 Tr (cid:26) ˙ R k (cid:16) Γ (2) k [ φ ] + R k (cid:17) − (cid:27) , (64)where t = ln( k/ Λ) and ˙ R k = ∂ t R k . Γ (2) k [ φ ] denotes thesecond-order functional derivative of Γ k [ φ ] . In Fourierspace, the trace involves a sum over momenta as well asthe internal index of the φ field.Because of the regulator term ∆ S k , the vertices Γ ( n ) k,i ··· i n ( p , · · · , p n ) are smooth functions of momentaand can be expanded in powers of p i /k . Thus if weare interested only in the long distance physics, we canuse a derivative expansion of the average effective ac-tion [27, 28]. In the following, we consider the ansatz Γ k [ φ ] = (cid:90) d d r (cid:26) Z k ∇ φ ) + U k ( ρ ) (cid:27) . (65)Because of the O( N ) symmetry, the effective potential U k ( ρ ) must be a function of the O( N ) invariant ρ = φ / .To further simplify the analysis, we expand U k ( ρ ) aboutits minimum ρ ,k , U k ( ρ ) = U k ( ρ ,k ) + λ k ρ − ρ ,k ) . (66)We consider only the ordered phase where ρ ,k > .In a broken symmetry state with order parameter φ =( (cid:112) ρ ,k , , · · · , , the two-point vertex is given by Γ (2) k,ii ( p ) = (cid:26) Z k p + 2 λ k ρ ,k if i = 1 ,Z k p if i (cid:54) = 1 . (67)By inverting Γ (2) k , we obtain the longitudinal and trans-verse parts of the propagator, G k,l ( p ) = 1 Z k p + 2 λ k ρ ,k ,G k,t ( p ) = 1 Z k p . (68)Since these expressions are obtained from a derivativeexpansion of the average effective action, they are validonly in the limit | p | (cid:28) k . In practice however, one canretrieve the momentum dependence of G k =0 ( p ) at finite p by stopping the RG flow at k ∼ | p | , i.e. G k =0 ( p ) (cid:39) G k ∼| p | ( p ) , where G k ∼| p | ( p ) can be approximated by theresult of the derivative expansion. It is possible to ob-tain the full momentum dependence of the correlationfunctions in a more rigorous and precise way, within theso-called Blaizot-Mendez-Weschbor scheme [30–32], butthis requires a much more involved numerical analysis ofthe RG equations.The transverse propagator G k,t ( p ) is gapless [Eq. (68)],in agreement with Goldstone’s theorem, which is a mereconsequence of the O( N ) symmetry of the average effec-tive action (65). On the other hand, the divergence ofthe longitudinal susceptibility obtained in the previoussections suggests that λ k → for k → ( lim k → ρ ,k > in the ordered phase). We shall see that this is indeedthe result obtained from the RG equations.
2. RG flows
It is convenient to work with the dimensionless vari-ables ˜ ρ ,k = Z k k − d ρ ,k , ˜ λ k = Z − k k d − λ k . (69)The flow equations for ˜ ρ ,k , ˜ λ k and Z k are obtained byinserting the ansatz (65,66) into the RG equation (64).The calculation is standard [27, 28] and we only quotethe final result, ∂ t ˜ ρ ,k = (2 − d − η k )˜ ρ ,k −
32 ˜ I k,l − N −
12 ˜ I k,t ,∂ t ˜ λ k = ( d − η k )˜ λ k − ˜ λ k [9 ˜ J k,ll (0) + ( N −
1) ˜ J k,tt (0)] ,η k =2˜ λ k ˜ ρ ,k [ ˜ J (cid:48) k,lt (0) + ˜ J (cid:48) k,tl (0)] , (70)where η k = − ∂ t ln Z k denotes the running anomalousdimension. With the cutoff function R k ( p ) = Z k ( p − k )Θ( p − k ) [33] ( Θ( x ) is the step function), the thresh-old functions appearing in (70) can be calculated analyt-ically (see Appendix A).In Fig. 5 we show ˜ λ k , η k and ˜ ρ ,k vs − t = ln(Λ /k ) for d = 3 and N = 3 . We fix λ k =0 = u / and vary r (i.e. ρ ,k =0 = − r /u ). When the system is in the orderedphase away from the critical regime (red solid lines inFig. 5), i.e. p c (cid:29) p G , we see a crossover for k ∼ p G ( t G = ln( p G / Λ) (cid:39) − ) from the Gaussian regime to theGoldstone regime characterized by ˜ λ k (cid:39) ˜ λ ∗ , η k = 0 and ˜ ρ ,k ∼ k − (i.e. ρ ,k (cid:39) ρ ∗ = lim k → ρ ,k ). Since ˜ λ k (cid:39) ˜ λ ∗ and η k = 0 imply λ k ∼ k , we find that the longitudinalsusceptibility G k,l ( p ) = 1 / λ k ρ ,k ∼ /k diverges when k → . Identifying k with | p | to extract the momentumdependence (as explained above), we recover the singularbehavior G k =0 ,l ( p ) ∼ / | p | in three dimensions. Moregenerally, for an arbitrary dimension, one finds λ k ∼ k (cid:15) ˜ λ ∗ and G k,l ( p ) ∼ /k (cid:15) ≡ / | p | (cid:15) with (cid:15) = 4 − d . Thusin the RG approach, the divergence of the longitudinalsusceptibility is a consequence of the existence of a fixedpoint for the dimensionless coupling constant ˜ λ k .When the system is in the critical regime of the orderedphase (blue dotted lines in Fig. 5), i.e. p G (cid:29) p J , thereis a first crossover from the Gaussian regime to the crit-ical regime for k ∼ p G followed by a second crossoverto the Goldstone regime for k ∼ p J . In the criticalregime p G (cid:29) k (cid:29) p J , ˜ λ k (cid:39) ˜ λ ∗ cr , η k (cid:39) η ∗ and ˜ ρ ,k (cid:39) ˜ ρ ∗ are nearly equal to their values at the critical point be-tween the ordered and disordered phases [34, 35]. Thisgives G k,t ( p ) (cid:39) G k,l ( p ) ∼ /k − η ∗ p , i.e. G k =0 ,t ( p ) (cid:39) G k =0 ,l ( p ) ∼ / | p | − η ∗ if we identify k with | p | .
3. Analytical solution in the low-temperature phase
In the low-temperature phase (away from the criticalregime, i.e. when p c (cid:29) p G ), it is possible to obtain an ˜ λ k p c ≫ p G p G ∼ p c ∼ p J p G ≫ p J η k − t = ln(Λ /k )˜ ρ ,k FIG. 5. (Color online) ˜ λ k , η k and ˜ ρ ,k vs − t = ln(Λ /k ) for d = 3 , N = 3 , Λ = 1 and λ k =0 = 0 . . The (red) solid linecorresponds to ρ ,k =0 = 0 . ( p c (cid:29) p G ) and the (red) dots areobtained from the analytic solution (73). The (green) dashedline corresponds to ρ ,k =0 = 0 . ( p c ∼ p G ∼ p J ) andthe (blue) dotted one to ρ ,k =0 = 0 . ( p G (cid:29) p J ). analytical solution of the flow equations for k (cid:28) p c . Inthis limit, the RG flow is dominated by the Goldstonemodes and the contribution of the longitudinal mode canbe omitted. This amounts to ignoring ˜ J k,ll (0) , ˜ J (cid:48) k,lt (0) and ˜ J (cid:48) k,tl (0) in Eqs. (70), which is justified by noting that ˜ λ k ˜ ρ ,k becomes very large for k (cid:28) p c ( ˜ λ k ˜ ρ ,k ∼ k − d for k → ), where the hydrodynamic scale p c is defined by λ p c ˜ ρ ,p c ∼ . This gives η k = 0 and ∂ t ˜ λ k = − (cid:15) ˜ λ k + 8 v d d ( N − λ k , (71)where v d = [2 d +1 π d/ Γ( d/ − . We have used the ex-pression of the threshold functions given in Appendix A.Equation (71) should be solved with the boundary con-dition ˜ λ k = ˜ λ c for k = Λ (cid:39) p c . For d < , we then find ˜ λ k = (cid:15) ˜ λ c p (cid:15)c (cid:15)k (cid:15) + 8 v d d ( N − λ c ( p (cid:15)c − k (cid:15) ) (cid:39) (cid:15) ˜ λ c p (cid:15)c (cid:15)k (cid:15) + 8 v d d ( N − λ c p (cid:15)c (72)for k (cid:28) p c . The last expression can be rewritten in themore insightful form ˜ λ k = ˜ λ ∗ k/p G ) (cid:15) , (73)where ˜ λ ∗ = lim k → ˜ λ k = (cid:15)d v d ( N − (74)and p G = (cid:20) ( N −
1) 8 v d p (cid:15)c ˜ λ c d(cid:15) (cid:21) /(cid:15) = (cid:20) ( N −
1) 8 v d λ c d(cid:15)Z p c (cid:21) /(cid:15) . (75)Equation (73) is in remarkable agreement with the nu-merical solution of the flow equations (70) (Fig. 5). In theweak-coupling limit p G (cid:28) p c , we can ignore the renor-malization of Z k as well as that of λ k between k = Λ and k = p c , and approximate Z p c (cid:39) and λ c (cid:39) λ k =Λ = u / .We then recover the expression p G (cid:39) (cid:20) ( N −
1) 8 v d u d(cid:15) (cid:21) /(cid:15) (76)of the Ginzburg momentum scale obtained in previoussections. A similar analysis can be made for the case d = 4 . III. INTERACTING BOSONS
We consider interacting bosons at zero temperaturewith the (Euclidean) action S = (cid:90) dx (cid:20) ψ ∗ (cid:18) ∂ τ − µ − ∇ m (cid:19) ψ + g ψ ∗ ψ ) (cid:21) , (77)where ψ ( x ) is a bosonic (complex) field, x = ( r , τ ) , and (cid:82) dx = (cid:82) β dτ (cid:82) d d r . τ ∈ [0 , β ] is an imaginary time, β →∞ the inverse temperature, and µ denotes the chemicalpotential. The interaction is assumed to be local in spaceand the model is regularized by a momentum cutoff Λ .We consider a space dimension d > .Introducing the two-component field Ψ( p ) = (cid:18) ψ ( p ) ψ ∗ ( − p ) (cid:19) , Ψ † ( p ) = (cid:0) ψ ∗ ( p ) , ψ ( − p ) (cid:1) (78)0(with p = ( p , iω ) and ω a Matsubara frequency), the one-particle (connected) propagator becomes a × matrixwhose inverse in Fourier space is given by (cid:18) iω + µ − (cid:15) p − Σ n ( p ) − Σ an ( p ) − Σ ∗ an ( p ) − iω + µ − (cid:15) p − Σ n ( − p ) (cid:19) , (79)where Σ n and Σ an are the normal and anomalous self-energies, respectively, and (cid:15) p = p / m . If we choosethe order parameter (cid:104) ψ ( x ) (cid:105) = √ n to be real (with n the condensate density), then the anomalous self-energy Σ an ( p ) is real.To make contact with the classical ( ϕ ) theory withO( N ) symmetry studied in Sec. II, it is convenient towrite the boson field ψ ( x ) = 1 √ ψ ( x ) + iψ ( x )] (80)in terms of two real fields ψ and ψ and consider the(connected) propagator G ij ( x, x (cid:48) ) = (cid:104) ψ i ( x ) ψ j ( x (cid:48) ) (cid:105) c . Theinverse propagator G − ij ( p ) reads (cid:18) (cid:15) p − µ + Σ ( p ) ω + Σ ( p ) − ω + Σ ( p ) (cid:15) p − µ + Σ ( p ) (cid:19) , (81)where Σ ( p ) = 12 [Σ n ( p ) + Σ n ( − p )] + Σ an ( p ) , Σ ( p ) = 12 [Σ n ( p ) + Σ n ( − p )] − Σ an ( p ) , Σ ( p ) = i n ( p ) − Σ n ( − p )] , Σ ( p ) = − i n ( p ) − Σ n ( − p )] , (82)when Σ an ( p ) is real. A. Perturbation theory and infrared divergences
1. Bogoliubov’s theory
The Bogoliubov approximation is a Gaussian fluctu-ation theory about the saddle point solution ψ ( x ) = √ n = (cid:112) µ/g (i.e. ψ ( x ) = √ n and ψ ( x ) = 0 ). It isequivalent to a zero-loop calculation of the self-energies, Σ (0)n ( p ) = 2 gn , Σ (0)an ( p ) = gn , (83)or, equivalently, Σ (0)11 ( p ) = 3 gn , Σ (0)22 ( p ) = gn , Σ (0)12 ( p ) = 0 . (84)This yields the (connected) propagators G (0)n ( p ) = −(cid:104) ψ ( p ) ψ ∗ ( p ) (cid:105) c = − iω − (cid:15) p − gn ω + E p ,G (0)an ( p ) = −(cid:104) ψ ( p ) ψ ( − p ) (cid:105) c = gn ω + E p , (85) where E p = [ (cid:15) p ( (cid:15) p + 2 gn )] / is the Bogoliubov quasi-particle excitation energy. When | p | is larger than thehealing momentum p c = (2 gmn ) / , the spectrum E p (cid:39) (cid:15) p + gn is particle-like, whereas it becomes sound-likefor | p | (cid:28) p c = √ mc with a velocity c = (cid:112) gn /m .In the weak-coupling limit, n (cid:39) ¯ n ( ¯ n is the meanboson density) and p c can equivalently be defined as p c = (2 gm ¯ n ) / . In the hydrodynamic regime | p | (cid:28) p c , G (0)11 ( p ) = (cid:15) p ω + c p ,G (0)22 ( p ) = 2 gn ω + c p ,G (0)12 ( p ) = − ωω + c p . (86)Note that in the Bogoliubov approximation, the occur-rence of a linear spectrum at low energy (which impliessuperfluidity according to Landau’s criterion), is due to Σ an (0) being nonzero.
2. Infrared divergences and the Ginzburg scale
Let us now consider the one-loop correction Σ (1) to theBogoliubov result Σ (0) . For d ≤ , the second diagram ofFig. 1 gives a divergent contribution when the two inter-nal lines correspond to transverse fluctuations, which ispossible only for Σ . Thus Σ is finite at the one-looplevel and the normal and anomalous self-energies exhibitthe same divergence, Σ (1)n ( p ) (cid:39) Σ (1)an ( p ) (cid:39) − g n (cid:90) q G (0)22 ( q ) G (0)22 ( p + q ) , (87)where we use the notation q = ( q , iω (cid:48) ) and (cid:82) q = (cid:82) ∞−∞ dω (cid:48) π (cid:82) q . For small p , the main contribution to theintegral in (87) comes from momenta | q | (cid:46) p c and fre-quencies | ω (cid:48) | (cid:46) cp c , so that we can use (86) and obtain Σ (1)n ( p ) (cid:39) Σ (1)an ( p ) (cid:39) − g n c (cid:90) Q Q ( Q + P ) , (88)where Q = ( q , ω (cid:48) /c ) and P = ( p , ω/c ) are ( d + 1) -dimensional vectors. The momentum integral in (88)is restricted by | Q | (cid:46) p c and is given by (13), with Λ replaced by p c , d by d + 1 and | p | by ( p + ω /c ) / .We can estimate the characteristic (Ginzburg) momen-tum scale p G below which the Bogoliubov approximationbreaks down from the condition | Σ (1)n ( p ) | ∼ Σ (0)n ( p ) or | Σ (1)an ( p ) | ∼ Σ (0)an ( p ) for | p | = p G and | ω | = cp G , p G ∼ (cid:40) ( A d +1 gmp c ) / (3 − d ) if d < ,p c exp (cid:16) − A gmp c (cid:17) if d = 3 . (89)This result can be rewritten as p G ∼ (cid:40) p c ( A d +1 ˜ g d/ ) / (3 − d ) if d < ,p c exp (cid:16) − A √ g / (cid:17) if d = 3 , (90)1where ˜ g = gm ¯ n − /d ∼ (cid:16) p c ¯ n /d (cid:17) (91)is the dimensionless coupling constant obtained by com-paring the mean interaction energy per particle g ¯ n to thetypical kinetic energy /m ¯ r where ¯ r ∼ ¯ n − /d is the meandistance between particles [36]. A superfluid is weaklycorrelated if ˜ g (cid:28) , i.e. p G (cid:28) p c (cid:28) ¯ n /d (the charac-teristic momentum scale ¯ n /d does however not play anyrole in the weak-coupling limit) [37]. In this case, theBogoliubov theory applies to a large part of the spec-trum where the dispersion is linear (i.e. | p | (cid:46) p c ) andbreaks down only at very small momenta | p | (cid:46) p G (cid:28) p c . In the next sections, we shall see that the weakly-correlated superfluid bears many similarities with the or-dered phase of the classical O( N ) model away from thecritical regime. When the dimensionless coupling ˜ g be-comes of order unity, the three characteristic momen-tum scales p G ∼ p c ∼ ¯ n /d become of the same order.The momentum range [ p G , p c ] where the linear spectrumcan be described by the Bogoliubov theory is then sup-pressed. We expect the strong-coupling regime ˜ g (cid:29) tobe governed by a single characteristic momentum scale,namely ¯ n /d .
3. Vanishing of the anomalous self-energy
The exact values of Σ n ( p = 0) and Σ an ( p = 0) can beobtained using the U(1) symmetry of the action, i.e. theinvariance under the field transformation ψ ( x ) → e iθ ψ ( x ) and ψ ∗ ( x ) → e − iθ ψ ∗ ( x ) [14, 38]. The derivation is similarto that of Sec. II A 3. Let us consider the effective action Γ[ φ ] = − ln Z [ J , J ] + (cid:90) dx [ J φ + J φ ] , (92)where J i is an external source which couples linearly tothe boson field ψ i , and φ i ( x ) = (cid:104) ψ i ( x ) (cid:105) J the superfluidorder parameter. The U(1) symmetry of the action im-plies that Γ[ φ ] is invariant under a uniform rotation of thevector field ( φ ( x ) , φ ( x )) T . For an infinitesimal rotationangle θ , this yields (cid:90) dx (cid:88) i,j δ Γ[ φ ] δφ i ( x ) (cid:15) ij φ j ( x ) = 0 , (93)where (cid:15) ij is the totally antisymmetric tensor. Takingthe functional derivative δ/δφ l ( y ) and setting φ i ( x ) = δ i, √ n leads to Γ (2)2 l ( p = 0) = 0 . (94)For l = 2 , equation (94) yields the Hugenholtz-Pines the-orem [12] Γ (2)22 ( p = 0) = Σ n ( p = 0) − Σ an ( p = 0) − µ = 0 . (95) If we now take the second-order functional derivative δ (2) /δφ l ( y ) δφ m ( z ) of (93) and set φ i ( x ) = δ i, √ n , weobtain the Ward identity (cid:88) i Γ (2) im ( y, z ) (cid:15) il + (cid:88) i Γ (2) il ( z, y ) (cid:15) im − √ n (cid:90) dx Γ (3)2 lm ( x, y, z ) = 0 . (96)Integrating over y and z and setting l = 2 and m = 1 ,we deduce (in Fourier space) Γ (3)122 (0 , ,
0) = 1 √ βV Γ (2)11 (0 , √ n , (97)where we have used (95).The self-energy Σ can be written as Σ ( p ) = ˜Σ ( p ) − g (cid:114) n βV (cid:88) q G ( q ) G ( p + q ) × Γ (3)122 ( − p, − q, p + q ) , (98)where ˜Σ ( p ) denotes the regular part of the self-energy(i.e. the part that does not contain pairs of lines corre-sponding to G G ). If we assume that the transversepropagator G ( q ) ∼ / ( ω + c q ) at low energies (thisresult will be shown in the following sections), the inte-gral (cid:82) q G ( q ) is infrared divergent for d ≤ . To obtaina finite self-energy Σ ( p = 0) , one must require that Γ (3)122 (0 , ,
0) = 0 . The Ward identity (97) then implies Γ (2)11 ( p = 0) = 0 and in turn Σ n ( p = 0) = µ + 12 (cid:104) Γ (2)11 ( p = 0) + Γ (2)22 ( p = 0) (cid:105) = µ Σ an ( p = 0) = 12 (cid:104) Γ (2)11 ( p = 0) − Γ (2)22 ( p = 0) (cid:105) = 0 . (99)The vanishing of the anomalous self-energy Σ an ( p =0) was first proven by Nepomnyashchii and Nepom-nyashchii [14]. To reconcile this result with the existenceof a sound mode with linear dispersion, the self-energies Σ n ( p ) and Σ an ( p ) must necessarily contain non-analyticterms in the limit p → (Sec. III B 4). B. Hydrodynamic approach
It was realized by Popov that the phase-density rep-resentation of the boson field ψ = √ ne iθ leads to a the-ory free of infrared divergences [16–18]. Popov’s theorybears some similarities with the analysis of the ( ϕ ) theory based on the amplitude-direction representation(Sec. II B). In this section, we show how the phase-densityrepresentation can be used to obtain the infrared behav-ior of the propagators G n ( p ) and G an ( p ) without encoun-tering infrared divergences [19]. Our approach is simi-lar to that of Popov (with some technical differences inSec. III B 2).2
1. Perturbative approach
In terms of the density and phase fields, the actionreads S [ n, θ ] = (cid:90) dx (cid:20) in ˙ θ + n m ( ∇ θ ) + ( ∇ n ) mn − µn + g n (cid:21) . (100)At the saddle-point level, n ( x ) = ¯ n = µ/g . Expandingthe action to second order in δn = n − ¯ n , ˙ θ and ∇ θ , weobtain S [ δn, θ ] = (cid:90) dx (cid:20) iδn ˙ θ + ¯ n m ( ∇ θ ) + ( ∇ n ) m ¯ n + g δn ) (cid:21) . (101)The higher-order terms can be taken into account withinperturbation theory and only lead to finite corrections ofthe coefficients of the hydrodynamic action (101) [18].We deduce the correlation functions of the hydrody-namic variables, G nn ( p ) = (cid:104) δn ( p ) δn ( − p ) (cid:105) = ¯ nm p ω + E p ,G nθ ( p ) = (cid:104) δn ( p ) θ ( − p ) (cid:105) = − ωω + E p ,G θθ ( p ) = (cid:104) θ ( p ) θ ( − p ) (cid:105) = p m ¯ n + gω + E p , (102)where E p is the Bogoliubov excitation energy defined inSec. III A 1. In the hydrodynamic regime | p | (cid:28) p c = √ gm ¯ n , G nn ( p ) = ¯ nm p ω + c p ,G nθ ( p ) = − ωω + c p ,G θθ ( p ) = mc ¯ n ω + c p , (103)where c = (cid:112) g ¯ n/m is the Bogoliubov sound mode veloc-ity ( p c = √ mc ).
2. Exact hydrodynamic description
In this section, we show that equations (103) are exactin the low-energy limit | p | , | ω | /c (cid:28) p c provided that c is the exact sound mode velocity and ¯ n the actual meandensity (which may differ from µ/g ). Let us consider theeffective action Γ[ n, θ ] defined as the Legendre transformof the free energy − ln Z [ J n , J θ ] ( J n and J θ are externalsources linearly coupled to n and θ ) [39]. At zero temper-ature, Γ[ n, θ ] inherits Galilean invariance from the action(100). In a Galilean transformation (in imaginary time), r (cid:48) = r + i v τ and τ (cid:48) = τ , the fields transform as n (cid:48) ( x (cid:48) ) = n ( x ) ,θ (cid:48) ( x (cid:48) ) = θ ( x ) − i m v τ − m v · r , (104) where x (cid:48) = ( r (cid:48) , τ (cid:48) ) . n ( x ) , ∇ n ( x ) and i∂ τ θ + m ( ∇ θ ) areGalilean invariant (but ∂ τ n ( x ) is not). ∇ θ is also in-variant but is odd under time-reversal symmetry. Thus,to second order in derivatives, the most general effec-tive action compatible with Galilean invariance and time-reversal symmetry reads Γ[ n, θ ] = (cid:90) dx (cid:26) Y ( n )8 m ( ∇ n ) + U ( n )+ (cid:88) p =1 c p ( n ) (cid:104) i∂ τ θ + 12 m ( ∇ θ ) (cid:105) p (cid:27) , (105)up to an additive (field-independent) term. Y ( n ) , U ( n ) and c p ( n ) are arbitrary functions of n .To determine c p ( n ) , we now consider the system in thepresence of a fictitious vector potential ( A , A ) , S [ n, θ ; A µ ] = (cid:90) dx (cid:20) in ( ∂ τ θ − A ) + n m ( ∇ θ − A ) + ( ∇ n ) mn − µn + g n (cid:21) . (106)The action is invariant under the local U(1) transfor-mation θ → θ + α and A µ → A µ + ∂ µ α where α ( x ) is an arbitrary phase. By requiring that Γ[ n, θ ; A µ ] =Γ[ n, θ + α ; A µ + ∂ µ α ] shares the same invariance, we de-duce Γ[ n, θ ; A µ ] = (cid:90) dx (cid:26) Y ( n )8 m ( ∇ n ) + U ( n )+ (cid:88) p =1 c p ( n ) (cid:104) i∂ τ θ − iA + 12 m ( ∇ θ − A ) (cid:105) p (cid:27) . (107)Noting that n ( x ) = δ ln Z [ J n , J θ ; A µ ] iδA ( x ) = − δ Γ[ n, θ ; A µ ] iδA ( x ) , (108)we must have c ( n ) = n and c p ( n ) = 0 for p ≥ . Weconclude that Γ[ n, θ ] = (cid:90) dx (cid:26) Y ( n )8 m ( ∇ n ) + U ( n )+ n (cid:20) i∂ τ θ + ( ∇ θ ) m (cid:21)(cid:27) (109)to second order in derivatives.From (109), we obtain the two-point vertex in constantfields n ( x ) = ¯ n and θ ( x ) = const (with ¯ n the actual bosondensity), Γ (2) ( p ) = (cid:32) Γ (2) nn ( p ) Γ (2) nθ ( p )Γ (2) θn ( p ) Γ (2) θθ ( p ) (cid:33) = (cid:18) Y (¯ n )4 m p + U (cid:48)(cid:48) (¯ n ) ω − ω ¯ nm p (cid:19) . (110)3By inverting Γ (2) ( p ) , we recover the propagators(103) in the small momentum limit | p | (cid:28) p c =[4 mU (cid:48)(cid:48) (¯ n ) /Y (¯ n )] / but with a sound mode velocity c given by c = (cid:114) ¯ nU (cid:48)(cid:48) (¯ n ) m . (111)Noting that the compressibility κ = ¯ n − d ¯ n/dµ can alsobe expressed as [40] κ = 1¯ n U (cid:48)(cid:48) (¯ n ) , (112)we conclude that the Bogoliubov sound mode velocity c is equal to the macroscopic sound velocity ( m ¯ nκ ) − / .Moreover, since the superfluid density n s is defined by Γ (2)22 ( p ,
0) = n s m p for p → [8], we find that at zerotemperature n s = ¯ n is given by the fluid density [13].
3. Normal and anomalous propagators
To compute the propagator of the ψ field, we write ψ ( x ) = (cid:112) n + δn ( x ) e iθ ( x ) , (113)where n = |(cid:104) ψ ( x ) (cid:105)| = |(cid:104) (cid:112) n ( x ) e iθ ( x ) (cid:105)| is the conden-sate density. For a weakly interacting superfluid, n (cid:39) ¯ n ,and we expect the fluctuations δn to be small. Let us as-sume that the superfluid order parameter (cid:104) ψ ( x ) (cid:105) = √ n is real. Transverse and longitudinal fluctuations are thenexpressed as δψ = √ n θ + · · · δψ = δn √ n − (cid:114) n θ + · · · (114)where the ellipses stand for subleading contributions tothe low-energy behavior of the correlation functions. Forthe transverse propagator, we obtain G ( p ) (cid:39) n G θθ ( p ) = 2 n mc ¯ n ω + c p (115)to leading order in the hydrodynamic regime, while G ( p ) (cid:39) G nθ ( p ) = − ωω + c p . (116)The longitudinal propagator is given by G ( x ) = 12 n G nn ( x ) + n (cid:104) θ ( x ) θ (0) (cid:105) c = 12 n G nn ( x ) + n G θθ ( x ) , (117)where the second line is obtained using Wick’s theorem(which is justified since the Goldstone (phase) mode is effectively non-interacting in the hydrodynamic limit).In Fourier space, G ( p ) = ¯ n mn p ω + c p + n G θθ (cid:63) G θθ ( p ) , (118)where G θθ (cid:63) G θθ ( p ) = (cid:90) q G θθ ( q ) G θθ ( p + q ) (119)with the dominant contribution to the integral comingfrom momenta | q | (cid:46) p c and frequencies | ω (cid:48) | /c (cid:46) p c . Us-ing (13), we find G θθ (cid:63) G θθ ( p )= A d +1 c (cid:0) m ¯ n (cid:1) (cid:16) p + ω c (cid:17) ( d − / if d < , A c (cid:0) m ¯ n (cid:1) ln (cid:18) p c p + ω c (cid:19) if d = 3 . (120)By comparing the two terms in the rhs of (118) with | p | = p G and | ω | = cp G , we recover the Ginzburg scale(89). For | p | , | ω | /c (cid:29) p G , the last term in the rhs of(118) can be neglected and we reproduce the result ofthe Bogoliubov theory (noting that ¯ n (cid:39) n ), while for | p | , | ω | /c (cid:28) p G , G ( p ) ∼ / ( ω + c p ) (3 − d ) / is domi-nated by phase fluctuations. The longitudinal suscepti-bility G ( p , iω = 0) ∼ / | p | − d for p → in contrast tothe Bogoliubov approximation G ( p , iω = 0) = 1 / mc .From these results, we deduce the hydrodynamic be-havior of the normal propagator, G n ( p ) = −
12 [ G ( p ) − iG ( p ) + G ( p )]= − n mc ¯ n ω + c p − iωω + c p − G ( p ) , (121)as well as that of the anomalous propagator, G an ( p ) = −
12 [ G ( p ) − G ( p )]= n mc ¯ n ω + c p − G ( p ) , (122)where G ( p ) is given by (118). The leading order termsin (121) and (122) agree with the results of Gavoret andNozières [13] and are exact (see next section). The con-tribution of the diverging longitudinal correlation func-tion was first identified by Nepomnyashchii and Nepom-nyashchii [15], and later in Refs. [19, 41–44].4
4. Normal and anomalous self-energies
To compute the self-energies Σ n ( p ) and Σ an ( p ) , we usethe relations Σ n ( p ) = G − ( p ) − G n ( − p ) G n ( p ) G n ( − p ) − G an ( p ) , Σ an ( p ) = G an ( p ) G n ( p ) G n ( − p ) − G an ( p ) , (123)with G n ( p ) G n ( − p ) − G an ( p ) = G ( p ) G ( p ) + G ( p ) = G ( p ) (cid:20) n G θθ (cid:63) G θθ ( p ) + ¯ n n mc (cid:21) . (124)Setting G n ( p ) (cid:39) − G ( p ) ,G an ( p ) (cid:39) G ( p ) , (125)in the numerator of Eqs. (123), we obtain Σ an ( p ) = Σ n ( p ) − G − ( p )= ¯ n A d +1 c − d n m ( ω + c p ) (3 − d ) / if d < , ¯ n A cn m (cid:104) ln (cid:16) c p c ω + c p (cid:17)(cid:105) − if d = 3 , (126)in the infrared limit | p | , | ω | /c (cid:28) p G , where G − ( p ) = iω − (cid:15) p + µ . Equations (126) agree with the exact results(99) and show that Σ n ( p ) and Σ an ( p ) are dominated bynon-analytic terms for p → . This non-analyticity re-flects the singular behavior of the longitudinal correlationfunction G ( p ) (cid:39) an ( p ) (127)in the low-energy limit.It should be noted that the singularity of the self-energies is crucial to reconcile the existence of a soundmode with a linear dispersion and the vanishing of theanomalous self-energy Σ an ( p = 0) [14]. In the low-energylimit, Σ an ( p ) = ∆Σ( p ) + ˜Σ an ( p ) , Σ n ( p ) − G − ( p ) = ∆Σ( p ) + ˜Σ n ( p ) , (128)where ∆Σ( p ) denotes the singular part (126) while ˜Σ n ( p ) and ˜Σ an ( p ) are regular contributions of order p , ω . Us-ing ∆Σ( p ) (cid:29) ˜Σ n ( p ) − G − ( p ) , ˜Σ an ( p ) for p → , by in-verting (79) we obtain G n ( p ) (cid:39) −
12[ ˜Σ n ( p ) − ˜Σ an ( p )] ,G an ( p ) (cid:39)
12[ ˜Σ n ( p ) − ˜Σ an ( p )] . (129) Since both ˜Σ n ( p ) and ˜Σ an ( p ) can be expanded to order p , ω , we conclude that equations (129) predict the exis-tence of a sound mode with linear dispersion. Of course,Eqs. (129) are nothing but our previous equations (115)and (125).In deriving the low-energy expression (126) of the self-energies, we have assumed that the hydrodynamic de-scription holds up to the momentum scale p c and ignoredthe contribution of the non-hydrodynamic modes. InPopov’s original approach [19], one introduces a momen-tum cutoff p satisfying p G (cid:28) p (cid:28) p c . Since p (cid:29) p G ,modes with momenta | p | ≥ p can be taken into accountwithin standard perturbation theory (see Sec. III A). Onthe other hand, low-momentum modes | p | ≤ p (cid:28) p c are naturally treated in the hydrodynamic approach dis-cussed in this section. The final results are independentof p . The only difference with our results (126) is that p c in the expression of Σ an ( p ) for d = 3 is replaced by asmaller momentum scale [45]. C. The non-perturbative RG
The NPRG approach to zero-temperature interactingbosons has been discussed in detail in Refs. [8, 43, 44, 46–51]. Our aim in this section is to briefly summarize themain results [52] while emphasizing the common pointswith the classical O( N ) model studied in Sec. II D.To implement the NPRG, we add to the action an in-frared regulator term ∆ S k [ ψ ∗ , ψ ] = (cid:88) p ψ ∗ ( p ) R k ( p ) ψ ( p ) , (130)which suppresses fluctuations with momen-tum/frequency below a characteristic scale k butleaves high momentum/frequency modes unaffected.The average effective action is defined as Γ k [ φ ∗ , φ ] = − ln Z k [ J ∗ , J ] + (cid:88) p [ J ∗ ( p ) φ ( p ) + c . c . ] − ∆ S k [ φ ∗ , φ ] , (131)where φ ( x ) = (cid:104) ψ ( x ) (cid:105) J is the superfluid order parameter. J denotes a complex external source that couples linearlyto the boson field. Γ k satisfies the RG equation (64).As in Sec. II D, we choose the cutoff function R k suchthat all fluctuations are suppressed for k = Λ (so that Γ Λ [ φ ∗ , φ ] = S [ φ ∗ , φ ] ) and R k =0 ( p ) = 0 . In practice, wetake [8] R k ( p ) = Z A,k m (cid:18) p + ω c (cid:19) r (cid:18) p k + ω k c (cid:19) , (132)where r ( Y ) = ( e Y − − . The k -dependent variable Z A,k is defined below. A natural choice for the veloc-ity c would be the actual ( k -dependent) velocity of theGoldstone mode. In the weak coupling limit, however,the Goldstone mode velocity renormalizes only weaklyand is well approximated by the k -independent value c = (cid:112) g ¯ n/m .5
1. Derivative expansion and infrared behavior
The infrared regulator ensures that the vertices areregular functions of p for | p | (cid:28) k and | ω | /c (cid:28) k evenwhen they become singular functions of ( p , iω ) at k = 0 ( c ≡ c k (cid:39) c k =0 is the velocity of the Goldstone mode).In the low-energy limit | p | , | ω | /c (cid:28) k , we can thereforeuse a derivative expansion of the average effective action.We consider the ansatz Γ k [ φ ∗ , φ ] = (cid:90) dx (cid:104) φ ∗ (cid:16) Z C,k ∂ τ − V A,k ∂ τ − Z A,k m ∇ (cid:17) φ + λ k n − n ,k ) (cid:105) (133)( n = | φ | ), which is similar to the one used in the clas-sical O( N ) model. n ,k denotes the condensate densityin the equilibrium state. Note that we have introduced asecond-order time derivative term. Although not presentin the initial average effective action Γ Λ , we shall see thatthis term plays a crucial role when d ≤ [46, 48]. Aspointed out in Sec. II D, the derivative expansion givesaccess only to the low-energy limit | p | , | ω | /c (cid:28) k of thecorrelation functions. It is however possible to extractthe p dependence of the correlation functions by stop-ping the flow at k ∼ ( p + ω /c ) / [8].In a broken symmetry state with order parameter φ = √ n , φ = 0 , the two-point vertex is given by Γ (2) k, ( p ) = V A,k ω + Z A,k (cid:15) p + 2 λ k n ,k , Γ (2) k, ( p ) = V A,k ω + Z A,k (cid:15) p , Γ (2) k, ( p ) = Z C,k ω. (134)Using (82), we then find Σ k, n ( p ) = G − ( p ) + 12 (cid:104) Γ (2) k, ( p ) + Γ (2) k, ( p ) (cid:105) − i Γ (2) k, ( p )= µ + V A,k ω + (1 − Z C,k ) iω − (1 − Z A,k ) (cid:15) p + λ k n ,k (135)and Σ k, an ( p ) = 12 (cid:104) Γ (2) k, ( p ) − Γ (2) k, ( p ) (cid:105) = λ k n ,k . (136)At the initial stage of the flow, Z A, Λ = Z C, Λ = 1 , V A, Λ =0 , λ Λ = g and n , Λ = µ/g , which reproduces the resultsof the Bogoliubov approximation.Since the anomalous self-energy Σ k =0 , an ( p ) ∼ ( ω + c p ) (3 − d ) / is singular for | p | , | ω | /c (cid:28) p G and d ≤ ,we expect Σ k, an ( p = 0) ∼ k − d for k (cid:28) p G (given theequivalence between k and ( p + ω /c ) / ), i.e. λ k ∼ k − d . (137)The hypothesis (137) is sufficient, when combined toGalilean and gauge invariances, to obtain the exact in-frared behavior of the propagator. Furthermore we shall see that it is internally consistent. In the domain of va-lidity of the derivative expansion, | p | , | ω | /c (cid:28) k (cid:28) k − d for k → , one obtains from (134) G k, ( p ) = 12 λ k n ,k ,G k, ( p ) = 1 V A,k ω + c k p ,G k, ( p ) = − Z C,k λ k n ,k V A,k ωω + c k p , (138)where c k = (cid:32) Z A,k / mV A,k + Z C,k / λ k n ,k (cid:33) / (139)is the velocity of the Goldstone mode. From (137) and(138), we recover the divergence of the longitudinal sus-ceptibility if we identify k with ( p + ω /c ) / .The parameters Z A,k , Z C,k and V A,k can be related tothermodynamic quantities using Ward identities [8, 13,44, 53], n s,k = Z A,k n ,k = ¯ n k ,V A,k = − n ,k ∂ U k ∂µ (cid:12)(cid:12)(cid:12)(cid:12) n ,k ,Z C,k = − ∂ U k ∂n∂µ (cid:12)(cid:12)(cid:12)(cid:12) n ,k = λ k dn ,k dµ , (140)where ¯ n k is the mean boson density and n s,k the su-perfluid density. Here we consider the effective poten-tial U k as a function of the two independent variables n and µ . The first of equations (140) states that in aGalilean invariant superfluid at zero temperature, thesuperfluid density is given by the full density of thefluid [13]. Equations (140) also imply that the Goldstonemode velocity c k coincides with the macroscopic soundvelocity [8, 13, 44], i.e. d ¯ n k dµ = ¯ n k mc k . (141)Since thermodynamic quantities, including the conden-sate “compressibility” dn ,k /dµ should remain finite inthe k → limit, we deduce from (140) that Z C,k ∼ λ k ∼ k − d vanishes in the infrared limit, and lim k → c k = lim k → (cid:18) Z A,k mV A,k (cid:19) / . (142)Both Z A,k = ¯ n k /n ,k and the macroscopic sound velocity c k being finite at k = 0 , V A,k (which vanishes in theBogoliubov approximation) takes a non-zero value when k → .The suppression of Z C,k , together with a finite valueof V A,k =0 shows that the effective action (133) exhibits a“relativistic” invariance in the infrared limit and therefore6becomes equivalent to that of the classical O(2) model indimensions d + 1 [54]. In the ordered phase, the couplingconstant of this model vanishes as λ k ∼ k − ( d +1) (seeSec. II D), which is nothing but our starting assumption(137). For k → , the existence of a linear spectrumis due to the relativistic form of the average effectiveaction (rather than a non-zero value of λ k n ,k as in theBogoliubov approximation). To neglect the term Z C,k ∂ τ in the average effective action (133) (and therefore obtaina relativistic symmetry), it is necessary that λ k (cid:29) k [8],a condition which is related to the singularity of the self-energies in the limit p → . Thus we recover the factthat singular self-energies are crucial to obtain a linearspectrum in spite of the vanishing of the anomalous self-energy.To obtain the limit k = 0 of the propagators (atfixed p ), one should in principle stop the flow when k ∼ ( p + ω /c ) / . Since thermodynamic quantitiesare not expected to flow in the infrared limit, they canbe approximated by their k = 0 values. As for the lon-gitudinal correlation function, its value is obtained fromthe replacement λ k → C ( ω + c p ) (3 − d ) / (with C a con-stant). From (138) and (140), we then deduce the exactinfrared behavior of the normal and anomalous propaga-tors (at k = 0 ), G n ( p ) = − n mc ¯ n ω + c p − mc ¯ n dn dµ iωω + c p − G ( p ) ,G an ( p ) = n mc ¯ n ω + c p − G ( p ) , (143)where G ( p ) = 12 n C ( ω + c p ) (3 − d ) / . (144)The hydrodynamic approach of Sec. III B correctly pre-dicts the leading terms of (143) but approximates dn /dµ by ¯ n/mc . On the other hand, it gives an explicit expres-sion of the coefficient C in the longitudinal correlationfunction (144).
2. RG flows
The conclusions of the preceding section can be ob-tained more rigorously from the RG equation satisfiedby the average effective action. The dimensionless vari-ables ˜ n ,k = k − d Z C,k n ,k , ˜ λ k = k d (cid:15) − k Z − A,k Z − C,k λ k , ˜ V A,k = (cid:15) k Z A,k Z − C,k V A,k , (145) satisfy the RG equations ∂ t ˜ n ,k = − ( d + η C,k )˜ n ,k + 32 ˜ I k,ll + 12 ˜ I k,tt ,∂ t ˜ λ k = ( d − η A,k + η C,k )˜ λ k − ˜ λ k (cid:2) J k ; ll,ll (0) − J k ; lt,lt (0) + ˜ J k ; tt,tt (0) (cid:3) ,η A,k = 2˜ λ k ˜ n ,k ∂∂y (cid:2) ˜ J k ; ll,tt ( p ) + ˜ J k ; tt,ll ( p )+ 2 ˜ J k ; lt,lt ( p ) (cid:3) p =0 ,η C,k = − λ k ˜ n ,k ∂∂ ˜ ω (cid:2) ˜ J k ; tt,lt ( p ) − ˜ J k ; lt,tt ( p ) − J k ; ll,lt ( p ) + 3 ˜ J k ; lt,ll ( p ) (cid:3) p =0 ,∂ t ˜ V A,k = (2 − η A,k + 2 η C,k ) ˜ V A,k − λ k ˜ n ,k ∂∂ ˜ ω (cid:2) ˜ J k ; ll,tt ( p ) + ˜ J k ; tt,ll ( p )+ 2 ˜ J k ; lt,lt ( p ) (cid:3) p =0 , (146)where η A,k = − ∂ t ln Z A,k , η C,k = − ∂ t ln Z C,k , y = p /k and ˜ ω = ωZ C,k /Z A,k (cid:15) k . The definition of the thresholdfunctions ˜ I and ˜ J can be found in Ref. [8].The flow of λ k , Z C,k and V A,k is shown in Fig. 6for a two-dimensional system in the weak-couplinglimit. We clearly see that the Bogoliubov approxima-tion breaks down at a characteristic momentum scale p G ∼ (cid:112) ( gm ) ¯ n . In the Goldstone regime k (cid:28) p G , wefind that both λ k and Z C,k vanish linearly with k in agree-ment with the conclusions of Sec. III C 1. Furthermore, V A,k takes a finite value in the limit k → in agreementwith the limiting value (142) of the Goldstone mode ve-locity. Figure 7 shows the behavior of the condensatedensity n ,k , the superfluid density n s,k = Z A,k n ,k andthe velocity c k . Since Z A,k =0 (cid:39) . , the mean bo-son density ¯ n k = n s,k is nearly equal to the condensatedensity n ,k . Apart from a slight variation at the be-ginning of the flow, n ,k , n s,k = Z A,k n ,k and c k do notchange with k . In particular, they are not sensitive tothe Ginzburg scale p G . This result is quite remarkablefor the Goldstone mode velocity c k , whose expression(139) involves the parameters λ k , Z C,k and V A,k , whichall strongly vary when k ∼ p G . These findings are a niceillustration of the fact that the divergence of the longitu-dinal susceptibility does not affect local gauge invariantquantities [8, 44].
3. Analytical results in the infrared limit
In the Goldstone regime k (cid:28) p G , the physics is dom-inated by the Goldstone (phase) mode and longitudinalfluctuations can be ignored. If we take the regulator (132)with r ( Y ) = − YY Θ( Y ) , the threshold functions ˜ I and ˜ J -10 -5 0 5 1000.51 ln( p G /k ) λ k /gZ C,k V A,k /V A,k =0 -3 -2 -1 -7 -6 -5 -4 -3 -2 p G mg FIG. 6. (Color online) λ k , Z C,k and V A,k vs ln( p G /k ) where p G = (cid:112) ( gm ) ¯ n/ π for ¯ n = 0 . , mg = 0 . and d = 2 [ ln( p G /p c ) (cid:39) − . ]. The inset shows p G vs mg obtainedfrom the criterion V A,p G = V A,k =0 / [the Green solid line isa fit to p G ∼ (2 mg ) / ]. -10 -5 0 5 100.50.60.70.80.91 ln( p G /k ) n ,k /n ,k =Λ n s,k /n s,k =Λ c k /c k =Λ FIG. 7. (Color online) Condensate density n ,k , superfluiddensity n s,k and Goldstone mode velocity c k vs ln( p G /k ) . Theparameters are the same as in Fig. 6. -10 -5 0 5 10010203040 ln( p G /k ) ˜ λ ′ k FIG. 8. (Color online) ˜ λ (cid:48) k vs ln( p G /k ) [Eq. (150)]. The pa-rameters are the same as in Fig. 6. can be computed exactly and one obtains [8] ∂ t ˜ n ,k = − ( d + η C,k )˜ n ,k ,∂ t ˜ λ k = ( d − η C,k )˜ λ k + 8 v d +1 d + 1 ˜ λ k ˜ V / A,k ,η C,k = − v d +1 d + 1 ˜ λ k ˜ V / A,k ,∂ t ˜ V A,k = (2 + 2 η C,k ) ˜ V A,k , (147) while η A,k (cid:39) . The first and last of these equationscan be rewritten as n ,k = n ,k =0 and V A,k = V A,k =0 .From (147), we deduce ∂ t ˜ λ k = (1 − (cid:15) )˜ λ k ,∂ t η C,k = − (cid:15)η C,k − η C,k , (148)where (cid:15) = 3 − d . For d < , this yields ˜ λ k ∼ k ( − (cid:15) ) and lim k → η C,k = − (cid:15), (149)i.e. λ k , Z C,k ∼ k (cid:15) in agreement with the numericalresults of Sec. III C 2 and the analysis of Sec. III C 1.The anisotropy between time and space in the Goldstoneregime k (cid:28) p G (where the average effective action takesa relativistic form) can be eliminated by an appropri-ate rescaling of frequencies of fields. This leads to anisotropic relativistic model with dimensionless conden-sate density and coupling constant defined by [8] ˜ n (cid:48) ,k = (cid:113) ˜ V A,k ˜ n ,k , ˜ λ (cid:48) k = ˜ λ k (cid:113) ˜ V A,k . (150) ˜ λ (cid:48) k satisfies the RG equation ∂ t ˜ λ (cid:48) k = − (cid:15) ˜ λ (cid:48) k + 8 v d +1 d + 1 ˜ λ (cid:48) k , (151)which is nothing but the RG equation of the couplingconstant of the classical O(2) model in dimensions d + 1 [Eq. (71)]. The corresponding fixed point value can bededuced from (74) [55]. In the infrared limit, we find λ k = k − d ( Z A,k (cid:15) k ) / V / A,k ˜ λ (cid:48) k ∼ k (cid:15) ˜ λ (cid:48) k (152)if we approximate Z A,k (cid:39) Z A,k =0 and V A,k (cid:39) V A,k =0 .The vanishing of λ k ∼ k (cid:15) and the divergence of the longi-tudinal susceptibility is therefore the consequence of theexistence of a fixed point ˜ λ (cid:48) k ∗ for the coupling constantof the effective ( d + 1 )-dimensional O(2) model whichdescribes the Goldstone regime k (cid:28) p G . To describethe entire hydrodynamic regime k (cid:28) p c , we should inprinciple relax the assumption V A,k (cid:39) V A,k =0 , since V A,k strongly varies for k ∼ p G , which makes the analyticalsolution of the RG equations much more difficult. InRef. [51], it was shown that Eq. (151) is nevertheless ingood agreement with the numerical solution of the flowequations in the entire hydrodynamic regime. We canthen use (76) to obtain the Ginzburg momentum scale p G (cid:39) (cid:20) v d +1 g ( d + 1) (cid:15) (cid:21) /(cid:15) (153)in the weak-coupling limit, which agrees with the resultsof Secs. III A and III B.8 IV. CONCLUSION
In conclusion, we have studied the classical linearO( N ) model and zero-temperature interacting bosons us-ing a variety of techniques: perturbation theory, hydro-dynamic approach, large- N limit and NPRG. We haveshown that in the weak-coupling limit these two systemscan be described along similar lines. They are character-ized by two momentum scales, the hydrodynamic scale(or healing scale for bosons) p c and the Ginzburg scale p G . For momenta | p | (cid:28) p c , we can use a hydrodynamicdescription in terms of amplitude and direction of thevector field ϕ in the O( N ) model, or density and phasein interacting boson systems. The hydrodynamic de-scription allows us to derive the order parameter correla-tion function without encountering infrared divergences.In the Goldstone regime | p | (cid:28) p G , amplitude (density)fluctuations play no role any more and both the trans-verse and longitudinal correlation functions are fully de-termined by direction (phase) fluctuations. In this mo-mentum range, the coupling between transverse and lon-gitudinal fluctuations leads to a divergence of the lon-gitudinal susceptibility and singular self-energies. A di-rect computation of the order parameter correlation func-tion (without relying on the hydrodynamic description)is possible, but one then has to solve the problem of in-frared divergences which appear in perturbation theorywhen | p | (cid:46) p G and signal the breakdown of the Gaussianapproximation. The NPRG provides a natural frame-work for such a calculation. In the case of bosons, itshows that in the Goldstone regime | p | , | ω | /c (cid:28) p G , thesystem is described by an effective action with relativis-tic invariance similar to that of the ( d + 1) -dimensionalclassical O(2) model.These strong similarities between the classical linearO( N ) model and zero-temperature interacting bosonsdisappear in the strong-coupling limit. For the O( N )model, this limit corresponds to the critical regime nearthe phase transition, which has no direct analog in zero-temperature interacting boson systems. The only ap-proach that one can hope to extend to strongly-correlatedbosons is the NPRG. Recent progress in that direc-tion, based on the Bose-Hubbard model, is reported inRef. [56]. ACKNOWLEDGMENTS
We would like to thank B. Svistunov for useful corre-spondence.
Appendix A: Threshold functions
The threshold functions appearing in the NPRG equa-tions for the O( N ) model (Sec. II D) are defined by I α = − (cid:90) q ˙ R ( q ) G α ( q ) ,J αβ ( p ) = − (cid:90) q ˙ R ( q ) G α ( q ) G β ( p + q ) ,J (cid:48) αβ ( p ) = ∂ p J αβ ( p ) . (A1)where α, β ∈ { l, t } . To alleviate the notations, we dropthe k index. In dimensionless form, ˜ I α = 2 v d (cid:90) ∞ dy y d/ ( ηr + 2 yr (cid:48) ) ˜ G α , ˜ J αβ (0) = 2 v d (cid:90) ∞ dy y d/ ( ηr + 2 yr (cid:48) ) ˜ G α ˜ G β , ˜ J (cid:48) αβ (0) = 4 v d d (cid:90) ∞ dy y d/ (cid:8) [ ηr + ( η + 4) yr (cid:48) + 2 y r (cid:48)(cid:48) ] ˜ G α − r + yr (cid:48) )( ηr + 2 yr (cid:48) ) ˜ G α (cid:9) (1 + r + yr (cid:48) ) ˜ G β , (A2)where ˜ G l = 1 y (1 + r ) + 2˜ λ ˜ ρ , ˜ G t = 1 y (1 + r ) , (A3)and we have written the cutoff function as R k ( p ) = Z k p r ( y ) with y = p /k and r ( y ) a k independentfunction. For the theta cutoff function introduce inSec. II D 2, r = − yy Θ(1 − y ) , and the threshold functionscan be computed analytically ˜ I α = − v d d (cid:18) − ηd + 2 (cid:19) ˜ A α , ˜ J αβ (0) = − v d d (cid:18) − ηd + 2 (cid:19) ˜ A α ˜ A β , ˜ J (cid:48) αβ (0) = 4 v d d ˜ A l , (A4)where ˜ A l = 11 + 2˜ λ ˜ ρ , ˜ A t = 1 . (A5) [1] S. K. Ma, Modern Theory of Critical Phenomena (Ad-vanced Books Classics, New edition, 2000). [2] P. M. Chaikin and T. C. Lubensky,
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