Observables in Inhomogeneous Ground States at Large Global Charge
Simeon Hellerman, Nozomu Kobayashi, Shunsuke Maeda, Masataka Watanabe
IIPMU18-0069
Observables in Inhomogeneous Ground States atLarge Global Charge
Simeon Hellerman , Nozomu Kobayashi , Shunsuke Maeda , andMasataka Watanabe Kavli Institute for the Physics and Mathematics of the Universe (WPI),The University of TokyoInstitutes for Advanced Study, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan Department of Physics, Faculty of Science, The University of Tokyo, Bunkyo-ku, Tokyo 133-0022,Japan
Abstract
As a sequel to previous work, we extend the study of the ground state configu-ration of the D = 3, Wilson-Fisher conformal O (4) model. In this work, we provethat for generic ratios of two charge densities, ρ /ρ , the ground-state configurationis inhomogeneous and that the inhomogeneity expresses itself towards longer spatialperiods. This is the direct extension of the similar statements we previously madefor ρ /ρ (cid:28)
1. We also compute, at fixed set of charges, ρ , ρ , the ground state en-ergy and the two-point function(s) associated with this inhomogeneous configurationon the torus. The ground state energy was found to scale ( ρ + ρ ) / , as dictated bydimensional analysis and similarly to the case of the O (2) model. Unlike the caseof the O (2) model, the ground also strongly violates cluster decomposition in thelarge-volume, fixed-density limit, with a two-point function that is negative definiteat antipodal points of the torus at leading order at large charge. a r X i v : . [ h e p - t h ] A p r ontents O (2) model . . . . . . . . . . . . . . . . . . . . . . 73 The case of the O (4) model with only one nonzero charge . . . . 74 The case of the O (4) model with two nonzero charges . . . . . . 8 O (4) model at arbitrary charge densities 9 ρ , . . . . . . . . . . . . . 123.4 Equation of motion for the conformal sigma model . . . . . . . . . . . . 125 Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 Scales in the equation of motion . . . . . . . . . . . . . . . . . . . 147 Fixing the charge densities . . . . . . . . . . . . . . . . . . . . . . 178 Average energy density . . . . . . . . . . . . . . . . . . . . . . . . 189 Two branches of the solution . . . . . . . . . . . . . . . . . . . . . 19 Introduction
Global symmetries can give conformal field theories interesting and useful simplifi-cations. Asymptotic expansions for dimensions and OPE coe ffi cients of charged localoperators in terms of large global charge are sometimes possible [1–4], to any givenorder, using a small number of unknown coe ffi cients which come from terms in thee ff ective Lagrangian in the large charge sector. These are strikingly parallel to thelarge-spin expansion using the light-cone bootstrap [7, 8], in spite of the fact thatthe method of large global charge includes the use of Lagrangian methods as op-posed to the abstract conformal bootstrap. These methods, of course, work best inthe regime of large charge, and complement the regime of O (1) charges and operatordimensions [9–13], which is e ff ectively accessed using the method of linear program-ming [14–16] to solve the bootstrap equations.In [1–3], the operator dimensions of charged local operators are calculated by radi-ally quantizing the large-charge e ff ective Lagrangian on a spherical spatial slice. Theidea of these papers are as follows: If we consider a system with large global charge J and charge density ρ , on a spatial slice of the scale R geometry , the large-charge e ff ectiveLagrangian has its UV scale at E UV ≡ ρ D − and IR at E IR = 1 /R geometry , and this largehierarchy, E IR /E UV ∝ J − D − , renders the theory weakly coupled. In other words, wecan compute various physical quantities perturbatively in terms of 1 /J , with all thequantum corrections and higher-derivative terms suppressed.Before studying the operator dimensions and various physical quantities, the firstthing necessary here is to know the structure of the large-charge e ff ective Lagrangianand the nature of the ground state with a fixed set of charges.In the limit where thecharge goes to infinity J → ∞ , one may also take the size of the sphere to infinity V → ∞ and consider the system on a infinite flat space with fixed average chargedensity, ρ ≡ J/ V . In this limit, many possibilities may be realized:(a) The ground state may become a homogeneous configuration, related directlyto the thermodynamic limit in infinite volume, at finite chemical potential andzero temperature. In some cases, this is indeed so, and various interesting newphases of matter with spontaneously broken conformal and Lorentz symmetrieshave been derived, which describe such breaking patterns [1–3].(b) The ground state may be homogeneous but the thermodynamic limit in infi-nite volume may not exist. This possibility is realized by some well-known andperfectly well-behaved CFT, including free complex scalar fields, and supercon-formal theories with moduli spaces of supersymmetric vacua in flat space. TheseCFT have discrete spectrum and perfectly well-defined thermodynamics on thesphere, but not in flat space; in the presence of curvature, the ground state atlarge R -charge is homogeneous and satisfies T ∝ ( Ricci scalar ) + ρ , and sothe space of ground states collapses at zero curvature, even in finite volume.Examples include superconformal theories with vacuum manifolds, such as the See also more recent work [5, 6]. D N = 2 superconformal XYZ model at large R -charge and [17], 4 D N = 2 rank1 SCFTs at large R -charge [18, 19].(c-IR) The ground state may be inhomogeneous in finite volume, with the scale of theinhomogeneity set by the scale of the geometry itself. For instance, on a torus,the ground state may break the translational symmetry to ZZ k × ZZ k or ZZ k × U (1),depending on the geometry of the torus and the details of the CFT. This case ispartly studied in [20] and will be expanded in this paper later.(c-UV) The ground state may be inhomogeneous on the scale set by the charge densityitself; this possibility would be realized, for instane, by a striped phase, with theperiodicity set by ρ − times some fixed constant determined by the dynamics.While we know no examples of a relativistic CFT with such a ground state atlarge global charge, it is a logical possibility that may be realized in some not-yet-discovered theories. (However see recent work [21] in which the groundstate with large charge combined with large angular momentum can be shownto break translational invariance spontaneously at the scale of the charge densityitself, in certain limits.)There is actually one other possibility that the ground state is not semiclassical at all,i.e., the fluctuations are not supressed by powers of E IR /µ . This by no means happenswhen all the degrees of freedom are coupled in a generic way. You can nonethelesscome up with several examples where this happens – One is two completly decoupledCFTs, one of which the symemtry acts trivially, and another is just two-dimensionalCFT with a global symmetry, in which the current degress of freedom decouples be-cause of Sugawara decomposition. Note that even in 2D CFTs with currents, whenthe theory has a continuous, rather than discrete, spectrum, the Sugawara decompo-sition doesn’t apply and the theory does not realise this possibility. To the contrary,this kind of theories generically shows a simplification at large charge limit, and hasbeen studied in the context of relativistic strings in the Regge limit [22–28]. We willnot further comment on this non-semiclassical possibility which is irrelevant to thetopic of this paper.Studying and classifying the above possibilities are quite important in checkingthe validity of the e ff ective field theories at large global charge. Especially, whenthe large-charge ground state realizes the case (c-UV), the suppression of quantumfluctuations in powers of E IR /µ is never there in the first place. Reassuringly, though,there are several facts against (c-UV) both in the cases of D = 3, O (2) and O (4) Wilson-fisher fixed points at large charge/set of charges. We will collect evidences against(c-UV) in the first two parts of this paper. In the first part, we are going to, provethat the O (2) model and the O (4) model with vanishing ρ or ρ (charge densities),at large charge realise possibility (a). Also, we prove that for generic charge ratios inthe O (4) model on the torus, it is possible to eliminate the possibility that the groundstate configuration is inhomogeneous in both cycles.In the second part of this paper we will particularly concentrate on the Wilson-Fisher O (4) fixed point in D = 3 for any set of total charges, J , = (cid:82) d ρ , , whichare taken to be large in arbitrary fixed ratio. (We partly answered the same questionin the limit J /J (cid:28) nd the inhomogeneity can be treated as a perturbation.) This case is the simplestnontrivial example with an inhomogeneous ground state at large global charge. (Veryinteresting cases of inhomogeneous ground states combining large charge and angu-lar momentum have also recently been discovered [21].) Extending the result of [2],we find that the ground state solution for large charges in generic charge ratio is inho-mogeneous with a family of classical solutions periodic in one spatial direction; but,that the family of solutions has an energetic preference for longer spatial periods,which is eventually bounded by the longest scale of the geometry of the spatial slice.This means that for any set of large charge densities the ground state configurationvaries very slowly compared to the scale of the charge density, so that the large-chargee ff ective Lagrangian is parametrically reliable, and the possibility (c-IR) is internallyconsistent in this case. Although this analysis supports the consistency of the possi-bility (c-IR) to be realised, it still does not rule out the possibility (c-UV) as we workin the regime of the EFT throughout.In the last part of this note where the issue with inhomogeneity has been settled,we will compute the classical energy and the two-point functions associated with theground state configuration on a torus spatial geometry, (cid:82) t × T using the above result.We show that the classical energy scales like ( ρ + ρ ) / as expected from previousstudies, with subleading correction which goes as ( ρ + ρ ) / . For the ground state ofthe O (2) model, the J term comes from the curvature coupling on the sphere, andvanishes on the torus. For the O (4) model, on the other hand, there is a term scalingas J is nonzero even in flat space: The ( ∂ | ∂χ | ) / | ∂χ | term, which vanishes classicallyfor the ground state solution in the O (2) model, does not vanish classically for theinhomogeneous ground state solutions of the O (4) model with generic charges, andthus makes a contribution of order J even in the absence of curvature.We also compute the two-point function on the torus from the inhomogeneousground state, and see the resulting spatial dependence in term of operator insertions.Interestingly, even at leading order, it exhibits a dramatic di ff erence from that of the O (2) model: The two-point function with insertions at two antipodal points is neg-ative definite and nonzero at leading order in the charge, which can only occur to-gether with a breakdown of cluster decomposition and spontaneous strong spatialinhomogeneity at the infrared scale. We hope these results can be checked againstMonte-Carlo simulations as in [31]. As promised, we are going to derive several facts about (in)homogeneity of the groundstate configuration at large charge. on the torus. This can be done in a simple way bycounting the number of Goldstone bosons and matching with the number of availablelight modes.The following results can and should be generalised from torus spatial slice to thesphere spatial slice. However, because we only deal with the torus time slice in this aper, we are not going to mention the sphere case. Note that the sphere spatial sliceadds a little more complexity to the problem because the symmetry group on it is notAbelian. But this is not in any way an obstacle in expanding these results, as you canuse the method of [32] even in the case of non-Abelian symmetries. A comment on helical symmetries and chemical potentials
Many discussions of finite-density ground states in the condensed matter literature,as well as some recent work on large quantum-number expansions in CFT [2], makeuse of chemical potentials in order to describe the large charge ground state. This isnatural in the thermodynamic limit though slightly less so in finite volume, wherethe legendre transform between chemical potentials and densities must be replacedby a Fourier transform of the quantum partition function [4].For this and other reasons, in the present work, we describe the ground state interms of a classical solution with a helical symmetry, i.e., a symmetry under a com-bined time translation and global symmetry translation [1] [17] [25] [26]. In classi-cal mechanics the two notions are precisely equivalent after a change of variables:The overall lowest-energy classical solution of the system with chemical potentialis always static on general grounds of Hamiltonian mechanics; therefore after a time-dependent global symmetry transformation that removes the chemical potential termfrom the Hamiltonian, the lowest-energy ground state with a given charge must havea helical symmetry. Quantum corrections to the classical picture of the ground statecan be added systematically in an asymptotic expansion in the inverse charge [4] [17].The equivalence of these two descriptions of the ground state also emphasizes animportant point that is sometimes ignored: A chemical potential always preservesthe same symmetries – Lorentz and global symmetries – of the system that are pre-served by the Hamiltonian without chemical potential: A constant chemical potentialcan be removed by a change of variables. The change in the ground state of the sys-tem induced by a chemical potential (at zero temperature) should always be viewedas spontaneous rather than explicit breaking. The description in terms of a helicalsolution merely emphasizes this fact which is otherwise obscured by the nontrivialtransformation law after the change of variables.While the helical frequency is a spontaneous rather than explicit breaking, theonly light goldstone modes are those corresponding to symmetries commuting withthe generator ˆ g defining the helical time-dependence exp { i ˆ gωt } , since the helicalground state is only time-independent up to a symmetry transformation by ˆ g . Thesymmetries commuting with the helical frequency are precisely those which wouldcommute with the chemical potential after the change of variables, and it is only thesethat generate light goldstone modes. The generators not commuting with the helicalfrequency are "massive goldstone bosons" [32] whose masses are above the cuto ff butstill precisely fixed by the symmetry algebra (See the supersymmetric version in the W = Φ model for [1], and the non-supersymmetric, O (2 N ) version [2] for examplesof massive goldstone fermions and bosons, respectively, in the sense of [32] in the con-text of the large charge expansion.). For purposes of counting light goldstone modeswe can ignore these and count only symmetry generators that are spontaneously bro- en by the solution at fixed time, and commute with the helical frequency. We will not refer further to the chemical potential in the present paper; we haveincluded these comments only to allow the reader to translate without di ffi culty be-tween the two points of view. The case of the O (2) model The discussion so far has led us to a simple rule for the counting of light modes: Thelight modes that can be understood as goldstone bosons, correspond one to one withsymmetry generators that are spontaneously broken by the configuration at fixed time t = 0, and which commute with the generator ˆ g describing the helical frequency. Asan example we will now apply this rule to the case of the large-charge ground stateof the O (2) model. We will see the rule gives a simple explanation for the spatialhomogeneity of the ground state, that is independent of the details of the Wilsonianaction at the conformal fixed point.In the case of the O (2) model the nature of the large-charge ground state is bynow well-understood: In addition to the helical symmetry, the ground state is alsospatially homogeneous, and the goldstone-counting rule shows this must be the case,because the O (2) model does not have enough massless fields to realize more than onegoldstone boson.The proof of the homogeneity goes as follows. Assume otherwise, then the inho-mogeneity is at some particular scale set by the charge density itself. Then in the IR,the e ff ective action should contain one or more translational Goldstone bosons andone axion from the spontaneous broken O (2) (combined with broken time translationsymmetry). Now, remember we started out from a theory of one complex scalar, withtwo real degrees of freedom. The renomalization group flow takes this UV theory toan IR theory which inevitably includes fewer than two real light degrees of freedom,and hence contradicts with the above statement. So by contradiction, we know thatthe ground state configuration must be homogeneous for the O (2) WF fixed point atlarge charge.One could have considered a logical possibility that the charged ground state maybe inhomogeneous, and indeed O (2) or U (1) symmetric CFT with a larger number ofdegrees of freedom, may spontaneously break translational symmetries, because theyhave enough degrees of freedom from the start, to do so. In the O (2) case, the homo-geneity of the ground state is related to the small number of light fields available, anddoes not follow automatically from the symmetries of the conformal fixed point. The case of the O (4) model with only one nonzero charge Now we consider the next-simplest case, already analyzed in [2], and apply the goldstone-counting argument to reproduce some results of that paper regarding the spatial ho-mogeneity of the ground state for various choices of ratios of charge density. In [2] itwas shown that a particular choice of (conjugacy classes of) large-charge limit, have This formulation of the goldstone-counting rule, while not manifestly equivalent to the way of countingin [32], is more convenient for our purposes and does work out to the same answer, as illustrated in [2]. spatially homogeneous ground state, namely those in which the charge density ofthe SO (4) (cid:39) SU (2) × SU (2) symmetry of the CFT is chosen to lie entirely inside oneof the two SU (2) factors. That is, letting ρ , ρ be the Cartan eigenvalues of the totalcharge matrix (divided by a factor of the spatial volume), one can show that a classicalground state of the system with those charges, is homogeneous.Here without loss of generality we set ρ = 0, and let us use the notation q as in(3.1). In terms of q , this condition translates to, by looking at (3.5) and (3.6), q ∂ t q =0. Just assume q = 0 for the moment, and then the subgroup of O (4) which preservesthis condition is SU (2) × U (1) (overall phase rotation and SU (2) rotation of q ).Because of how q is defined, q takes the following form, q = (cid:32) q q (cid:33) = (cid:32) e iω t sin( p ( x )) e iω t cos( p ( x )) (cid:33) . (2.1)So in order to have q = 0, you inevitably have p ( x ) = π/
2, so that the configuration ishomogeneous.Now consider the case where ∂ t q = 0. The subgroup of O (4) which preserves thiscondition is the same SU (2) × U (1) as before. The solution to the equation of motionspontaneously breaks this SU (2) × U (1) into a smaller subgroup. If we assume theground state configuration is homogeneous, one of such solutions is ∂ t q = (cid:32) (cid:33) (2.2)but the subgroup of SU (2) × U (1) which respects this particular ground state is only U (1) phase rotation of ∂ t q .So, schematically, we get the following breaking pattern; O (4) helical frequency −−−−−−−−−−−−−−−→ U (2) = U (1) × SU (2) spontaneous breaking −−−−−−−−−−−−−−−−−−−→ U (1) (2.3)The dimension of the coset is dim ( U (2) /U (1)) = 3, so we have as many as 3 Gold-stone bosons. Note that this counting is precisely what is given in [2]. If you wereto break the ground state homogeneity, you add one or more translational Goldstonebosons to these, making the total number of them 4 or more. But you only have threelight real degrees of freedom in the IR; thus by contradiction, the ground state con-figuration must always be homogeneous in the one-charge case. The case of the O (4) model with two nonzero charges In this case, we know from [2] that the configuration is inhomogeneous, so let us firstassume the configuration is inhomogeneous only in one direction of the torus. As youstill have the freedom to phase rotate q and q , respectively the pattern of breakingby the helical frequency becomes O (4) helical frequency −−−−−−−−−−−−−−−→ U (1) × U (1) (2.4) he inhomoegenity will not let us choose a ground state configuration which is spe-cial, like ∂ t q = (cid:32) (cid:33) . Rather, the ground state configuration can only lie at a very genericpoint, hence preserves no subgroup of U (1) × U (1). The spontaneous breaking patternof it, along with translational symmetries, is therefore, U (1) × U (1) × { translation } −−−−−−−−−−−−−−−−−−−→ { translation } (2.5)so that you have three Goldstone boson in the system by looking at the coset dimen-sion.Now, if you were to break one more translational symmetry, there would be fourlight Goldstone bosons in the system; however the theory has only had three light realfields with which to realize such excitations, which would be a contradiction, regard-less of the form of the Wilsonian action for the light fields. Therefore the translationalsymmetry of the torus can be only broken in one direction for the case of the O (4)model at generic set of large charges. O (4) model at arbitrary charge densities We move on to prove that the possibility (c-IR) is self-consistent by extending theresult in [20] to the case of any sets of charges, J , . In this paper, we denote by J , the two independent positive real eigenvalues of the matrix defined by the SO (4)Noether charge. Now, rather than requiring the limit where one of the global charge ismuch less than the other as was done in [20], we study the ground state of the critical O (4) model, fixing the spatially averaged global charge densities ρ , ≡ J , / V in anarbitrary ratio J /J . As in [20] we integrate out the mode which becomes heavy atlarge charge and work within a weakly-coupled conformal sigma model with targetspace S . The Lagrangian density for this conformal e ff ective theory is singular in thevacuum but is not meant to be used there; it is only meant to be used to study statesof energy O (1) or less, above the ground state of large charge density.In [20], we reproduced the no-go result of [2] in the context of the conformal sigmamodel: For generic charge densities or chemical potentials for the two independent O (4) charges, there is no spatially homogeneous classical solution. The only homoge-neous ground states occur when the two chemical potentials are equal, or equivalentlywhen the real antisymmetric 4 × O (4) charge of the state has avanishing determinant, which in our conventions and those of [2], means ρ = 0 or ρ = 0.For any classical theory, the lowest energy solution carrying given global chargesalways leaves unbroken a helical symmetry, by which we mean a symmetry undercombined time translation and global symmetry rotation. Regarding the spatial con-figurations are necessarily inhomogeneous in the O (4) model with generic O (4) charges,by virtue of [2], as we have stated earlier. Hence in this note, we search for the lowest-energy inhomogeneous helical solution. or inhomogeneous ground states, the most important qualitative question iswhether the inhomogeneity is on the ultraviolet scale, set by the charge density it-self, or on the infrared scale, set by the boundary conditions or overall geometry andtopology of the spatial slice. It is only in the case where the inhomogeneity is on theinfrared scale, that the large-charge e ff ective theory can be used in a straightforwardmanner.In the space of CFT with global symmetries, this question does not have a simplis-tic universal answer: The answer appears to depend on the theory and on the choiceof global symmetry quantum numbers. For angular momentum in a single plane, forexample, the lowest state is generally a small number of quanta each carrying a largeangular momentum [7, 8] and thus the inhomogeneity is on the UV rather than IRscale. On the other hand, for theories in D ≥ O (4) charges ρ , taken large in fixed ratio, it was shown in [20] that the loweststate has inhomogeneity on the infrared scale, for ρ /ρ (or ρ /ρ ) su ffi ciently small.In the present paper, we show that this result holds for any value of the ratio ρ /ρ ,and thus the ground state properties at large charge can be analyzed consistently inthe e ff ective conformal sigma model for any charge ratio.The simplest candidate helical solutions leave the translational symmetry unbro-ken in one spatial direction and break it in the other direction down to a discretesubgroup whose period is (cid:96) . Two natural questions, which are closely related arisebecause of this fact. One is which value of (cid:96) has the lowest energy. The other one isthe range of (cid:96) where the e ff ective field theory is reliable. We will now try to answerthis question using e ff ective field theory, and are going to show that where (cid:96) (cid:29) (cid:112) ρ − ,the lowest energy is achieved where (cid:96) is largest, that is the size of the underlyinggeometry itself.Note that it does not logically exclude the possibility of a striped phase, where thescale is set by (cid:112) ρ − itself so that the EFT breaks down. Although this possibility couldnot be realized in the case of the O (2) model due to Goldstone counting, in the caseof the O (4) model we could not rule out this possibility on the basis of Goldstone-counting alone, due to the fact that any value of (cid:96) would realize the same symmetry-breaking pattern, and so the Hamiltonian could a priori favor either long or shortdistance scales for (cid:96) . The EFT analysis shows only that a solution with period (cid:96) that islong on the infrared scale, energetically prefers as long a spatial period (cid:96) as possible,so that there is no internal instability in the EFT towards inhomogeneity on shortscales. A convenient parametrization for the O (4) charge is as follows. The charge densitiestake value in the adjoint of O (4), the group of antisymmetric 4 × hich has real eigenvalues that occur in pairs with equal magnitude and oppositesign. We denote two (out of four) positive eigenvalues of the charge density matrix tobe J and J ; in the infinite volume limit V → ∞ , we may take these to infinity, fixingthe spatially averaged charge densities ρ , ≡ J , / V .For helical solutions, this specific choice of parametrization is equivalent to choos-ing a complex basis for the fundamental of U (2) ⊂ SO (4) and parametrizing thecharge generator by the two matrix elements on the diagonal. Note that the o ff -diagonal matrix elements always vanish because the lowest-energy classical solutionis always helical. This is the same convention used in the earlier work, [2].We are now going to put the system on (cid:82) × (cid:82) and seek for the ground state config-uration. This should be regarded as the infinite volume limit of the geometry (cid:82) × S or (cid:82) × T . We will comment on the ground state configurations in finite volume caselater on. Now that the convenient parametrization of the charge densities has been made, wedescribe the O (4) model by four real scalars X , , , , which is then organised into Q ≡ (cid:32) X + iX X + iX (cid:33) , a complex SU (2) doublet. The interacting IR fixed point of the modelis given by starting from the UV Lagrangian with the kinetic term for Q plus a quarticpotential proportional to | Q | , with a fine-tuned mass term so that it actually flows toa non-trivial fixed point. We parametrise Q as amplitudes and angles, which is givenby Q = A × q, q = (cid:32) q q (cid:33) , (3.1)where | q | + | q | = 1. We give a large VEV to the A -field, and the resulting leadingaction in the IR includes a term that is proportional to A , as explained in [1]. The IRLagrangian we get, henceforth, is L IR = 12 ( ∂A ) + γ A ∂q † ∂q − h A . (3.2)This is under a RG normalization condition that the two-derivative kinetic term of A is canonical.Note that we have omitted other terms like Ricci coupling and higher derivativeterms, because we are only using the leading large-charge-density term. We alreadyknow that there are no homogeneous ground state configurations, so the suppressionof these terms, however, should be proven a posteriori . Those terms are only sup-pressed when the scale of the inhomogeneity, L , is much larger than the UV scale,( ρ + ρ ) − – otherwise the large-charge EFT is not within its range of validity, and thehigher-derivative operators and quantum corrections are out of control, and there isno simplification of the dynamics at large charge densities. In the following we firstassume L to be much smaller than the UV scale and derive the ground state configu-ration, and then justify this assumption later, a posteriori . nder this assumption, we first integrate the A field out, which has a mass scaledefined by the charge density itself, which is above the Wilsonian scale we are talkingabout. By virtue of the EOM for A , we have, as an equilibrium value of A , δ L IR δA = 0 ⇐⇒ A = (cid:115) ∂q † ∂qγ − h (3.3)Now by using this and the original IR Lagrangian, (3.2), we get L = b q L / = b q ( ∂q † ∂q ) / , (3.4)which is the conformal sigma model whose target space is S and where | q | = 1 and b q = (cid:112) γ h − / ffi cient from the original large-charge e ff ectiveaction [1]. ρ , We now put the theory on (cid:82) , so we inevitably have to use the concept of the “fixedaverage charge density” instead of that of total charge, which is ill-defined. We there-fore impose the following conditions onto Noether currents, − ib q (cid:90) dx i (cid:112) L (cid:104) q † ∂ t q − c . c . (cid:105)(cid:44) V = ρ + ρ (3.5) − ib q (cid:90) dx i (cid:112) L (cid:104) q † σ ∂ t q − c . c . (cid:105)(cid:44) V = ρ − ρ , (3.6)where V indicates the total volume of the space. Also, let us set ρ < ρ for simplicity,but we will comment on the ρ < ρ case later on.The energy density can also be derived from the Lagrangian, which is H = b q (cid:113) ˙ q † ˙ q − ∂ i q † ∂ i q × (cid:16) q † ˙ q + ∂ i q † ∂ i q (cid:17) . (3.7)We will therefore look for the minimizer of above under constraints (3.5) and (3.6). To achieve the ground state solution of (3.4), we set an ansatz that the solution is atleast homogeneous in one of the spatial directions, y , and only varies spatially in the x direction.We also use the helical nature of the ground state solution and the invarianceunder the combination of t → − t and the complex conjugation. Basically these ansatzsets the solution of the form q = (cid:32) q q (cid:33) = (cid:32) e iω t sin( p ( x )) e iω t cos( p ( x )) (cid:33) , (3.8) here we are free to set ω > ω and p ( x ) takes value in (cid:82) . Under this parametriza-tion, (3.4) becomes L = b q (cid:104) − p (cid:48) ( x ) + V ( p ) (cid:105) / , (3.9)where V ( p ) = ω + (cid:16) ω − ω (cid:17) sin ( p ) . (3.10)For general helical solutions depending on no more than one spatial direction,we can simplify the equations by reducing them to first order. Local conservation ofmomentum in the x -direction implies 0 = ∂ µ T xµ . For a helical solution independent ofthe y -direction, the stress tensor is independent altogether of y and t , so the pressure T xx in the x -direction is simply a constant: ∂ µ T xx = 0 . Define a constant κ , of mass dimension +1 as the cube root of the pressure, with thecoe ffi cient of proportionality b − q to simplify the formulae: κ ≡ b − q T xx . Now we will use the general formula for the stress tensor in a theory with aLagrangian that is first-order in derivatives acting on fields of vanishing conformalweight: T µν ≡ δ µν L − (cid:88) A Θ A,µ δ L δ Θ A,ν (3.11)where Θ A runs over all the fields in the system, in this case the three Goldstonesparametrizing the S target space. For a helical solution, with ˙ χ , = ω , , the La-grangian density is L = b q L , L ≡ V − p (cid:48) ,V ≡ ω + (cid:16) ω − ω (cid:17) sin ( p ) . modulo terms of second order in χ (cid:48) , , which do not contribute to the stress tensor ina helical solution because χ (cid:48) , vanishes in the helical solution itself.So the stress tensor is T xx = L − p (cid:48) ( x ) δ L δp (cid:48) = b q (cid:113) − p (cid:48) ( x ) + V ( p ( x )) (cid:16) p (cid:48) ( x ) + V ( p ( x )) (cid:17) , (3.12)which we know is a constant and have already set to be equal to b q κ . We now havethe EOM, which is − κ − b − q T xx (cid:16) p (cid:48) ( x ) − V ( p ( x )) (cid:17) (cid:32) p (cid:48) ( x ) + V ( p ( x ))2 (cid:33) , (3.13) here κ > T xx . The meat of this is that the equation ofmotion has now been reduced to a first-order equation (3.13) with one undeterminedconstant of motion, κ . In principle we could invert (3.13) algebraically to solve for( p (cid:48) ) in terms of V ( p ) for a given κ , using Cardano’s formula for the general solutionto a cubic equation, and then solve the first-order autonomous ODE for p as a functionof x . However most of the complication involved in such a solution is unnecessary,because we are working only within the regime where the fields are varying on scales L long compared to the ultraviolet scale set by ω , , so we need only solve the EOMunder the condition p (cid:48) (cid:28) ω , . (3.14)Indeed, there are other terms in the e ff ective Lagrangian that we have omitted, whichwould become important if we were to work outside this regime. But now we willnow organize the first-order EOM in such a way as to exploit the condition (3.14) inorder to solve it. Remark
Before proceeding, let us make a few comments about how to choose the right solutionwhen solving (3.13) for ( p (cid:48) ) . By imposing ( p (cid:48) ) > κ >
0, we can have multiplesolutions depending on the value of κ . Here, however, we only restrict attention tothe case where 0 (cid:54) ( p (cid:48) ) (cid:54) V ( p ) /
2. This is equivalent to imposing a condition that p ( x ) must have a point where its derivative is vanishing. This is natural when weeventually want to put the system on S and compute the dimensions of operatorsusing the Neumann boundary condition on the poles. When we put the system ona torus, T , there can possibly be solutions on di ff erent “winding numbers”, that arecharacterised by p ( x/(cid:96) ) = p (0) + nπ . The winding number here is not a topologicalcharge, because the map from T to S can only be trivial homotopically. This meansthat on the torus we will also have to compute their energies separately, to know thetrue ground state configuration. We will, however, just assume the lowest solutionis achieved when n = 0 even on the torus – at least we know that this is the casein the homogeneous case, and the continuity requires the statement is also true in acertain subset of J /J near zero. Also, this is physically related to the existence of softmodes discussed in [2], and adding soft modes intuitively should increase the energyof the system, not otherwise. Again for these reasons, we will hereafter only considersolutions which has a point at which its derivative vanishes. Scales in the equation of motion
The dimensionful quantities in the EOM are p (cid:48) , ω and ω , and we are going to con-sider the regime (3.14). The two frequencies ω , are independent a priori, but theirrelationship will be fixed in terms of the spatial period of the solution.First thing to notice is that the spatial period of the solution goes to infinity at ω = ω , becuase the EOM just gives p (cid:48) ( x ) = 0. This tells us straight away that the di ff erencein frequencies, ω − ω must scale di ff erently than either individual frequency: whilewe can hold ω or ω fixed while taking p (cid:48) →
0, we see that ω − ω must vanish n the limit p (cid:48) →
0. Let us now rewrite the first order EOM (3.13) to emphasize thedistinction in scalings. We define ω − ≡ ω − ω and expand κ as κ [0] + ∆ κ , where κ [0] is the value of κ for a homogeneous solution with a given ω , and ∆ κ ≡ κ − κ [0] is thedi ff erence, which must scale as a positive power of ω − . We could further expand ∆ κ as a series κ [1] + κ [2] + · · · where κ [ p ] is the term of order ω p − ; this will not be necessary,however, as we will only be interested in first-order quantities. So using the formula(3.10), which we recap here, V ( p ) = ω + (cid:16) ω − ω (cid:17) sin ( p ) , (3.15)we see that V = O ( ω )+ O ( ω ω − ). So V is of order O ( ω )+ O ( ω ω − ). So now, the LHSof (3.13) is identically independent of spacetime; therefore the x -dependent parts ofthe RHS will have to cancel order by order in ω − /ω .The two types of x -dependent terms on the RHS are p (cid:48) ω and ω − ω sin ( p ). Inorder for them to cancel, if one is treating p as O (1), one needs to scale ω − as p (cid:48) /ω .To make this more concrete, we want to define the length scale (cid:96) as the inverseof the maximum x -derivative of the p -field. This isn’t quite the right thing, becausewe want to get the right order of magnitude for the length scale not only when theamplitude of the oscillations is of O (1) but also when it is small, whereas assigning (cid:96) to be p (cid:48) max would go to infinity in the limit when the amplitude of the oscillationsis small but the period is fixed. To repair this deficiency, we multiply by sin p , anddefine: (cid:96) ≡ sin( p ) p (cid:48) max . (3.16)We will express (cid:96) in terms of the actual period of the solution later.This defines the general length scale characterizing the solution, and we expect theactual spatial period of the solution to be of order (cid:96) ; we will confirm this expectationwhen we find the ground-state classical solution exactly.In terms of (cid:96) , then, we see ω − must scale as ω − = O (cid:16) (cid:96) ω (cid:17) . We require that there be a point at which p (cid:48) vanishes, and at that point we have to beable to satisfy the EOM anyway. So this fixes ∆ κ at first order completely in terms of ω , ω − and p : ∆ κ = κ [1] + O (cid:16) (cid:96) − ω − (cid:17) ,κ [1] = ω − sin ( p ) , which we can also write as κ [1] = ω η , η ≡ ω − ω sin ( p ) . ow, at first order in ω − , the EOM reads: p (cid:48) = 2 ω ω − (cid:104) sin ( p ) − sin ( p ) (cid:105) , (3.17)which we can also write as p (cid:48) = 2 ω (cid:104) η − (cid:15) sin ( p ) (cid:105) , (cid:15) ≡ ω − ω . (3.18)The quantities (cid:15) and η are related simply by η = sin ( p ) (cid:15) , so the ratio η/(cid:15) is always less than 1, and goes to zero in the linearized limit, wherethe amplitude of oscillation is small.The maximum value of p (cid:48) occurs when p = 0, where it takes the value( p (cid:48) ) max = (cid:112) ω ω − | sin( p ) | = √ (cid:15) ω | sin( p ) | = (cid:112) η ω , (3.19)and so the general scale (cid:96) of the solution, which we defined in (3.16), is (cid:96) = 1 √ ω ω − = 1 √ (cid:15) ω . (3.20)This low-energy EOM (3.17) for the helical solution is exactly the equation of mo-tion for the angle θ of the pendulum with length L in a gravitational field g under theidentification of p = θ / ω ω − = g / L We know the exact solution to this type ofdi ff erential equations so now we can simply solve the equation (3.17) and using it, wecan compute the energy of the large charge ground state.Now we have the analytic solution to (3.17), which issin( p )sin( p ) = sn (cid:18) x(cid:96) ; sin( p ) (cid:19) (3.21)where (cid:96) was already given in (3.20), and sn( x ; k ) is the Jacobian elliptic function withmodulus k . The quarter period of the solution, L , is given by L = (cid:96) F (cid:18) π p ) (cid:19) (3.22)where F ( p ; k ) ≡ (cid:90) p d ˆ p (cid:113) − k sin ( ˆ p ) = sn − ( sin( p ); k ) (3.23) ixing the charge densities We have now solved for the spatial dependence p ( x ) in terms of the amplitude p and the frequencies ω , . However we do not really want (cid:96) to be an output and wedo not really want the spatial frequencies to be inputs. We would like to invert therelationship between the frequencies ω , and the densities ρ , so that the latter arethe independent variable.The expressions for the average charge densities given by (3.5) and (3.6), are ρ = 8 b q V (cid:90) dx i ω (cid:113) − p (cid:48) ( x ) + V ( p ( x )) sin ( p ( x )) (3.24) ρ = 8 b q V (cid:90) dx i ω (cid:113) − p (cid:48) ( x ) + V ( p ( x )) cos ( p ( x )) , (3.25)We can compute these integrals analytically at leading-order in the low-energyexpansion (cid:96) (cid:29) /ω . Using the equation of motion (3.20), we can substitute (cid:113) − p (cid:48) ( x ) + V ( p ( x ))) = 1 (cid:112) η p (cid:48) ( x )(1 + η ) (cid:113) − (cid:15)η sin ( p ( x )) + O ( (cid:15) ) . (3.26)Now as the previous subsection essentially states that the low-energy expansionwe are interested in is in terms of (cid:15), η (cid:28) η/(cid:15) = sin ( p ) = O (1), we only haveto keep track of the leading order contribution in terms of this expansion rule. (3.26),therefore means that at first order in (cid:15) and η , we can now change the variable in theintegrand from x to p itself. Doing this, and using η/(cid:15) = sin ( p ), we have ρ = 8 b q ω (1+ (cid:15) )(1 + η )3 L (cid:112) η p (cid:90) dp sin ( p ) (cid:114) − sin ( p )sin ( p ) (3.27)= 8 b q ω (1+ (cid:15) + η )3 × ∆ (cid:18) π p ) (cid:19) (3.28) ρ = 8 b q ω (1 + η )3 (cid:20) − ∆ (cid:18) π p ) (cid:19) (cid:21) (3.29)where ∆ ( p ; k ) ≡ F ( p ; k ) − E ( p ; k ) F ( p ; k ) F ( x ; k ) ≡ (cid:90) p d ˆ p (cid:113) − k sin ( ˆ p ) E ( p ; k ) ≡ (cid:90) p d ˆ p (cid:113) − k sin ( ˆ p ) hese expressions are a bit complicated, and it is instructive to see how they be-have in the linearized limit p (cid:28)
1, using the fact ∆ (cid:16) π ; sin( p ) (cid:17) ∼ sin ( p )2 ∼ p at lead-ing order in p . We can also immediately reproduce the statement that limit p (cid:28) J (cid:28) J is the same thing. Average energy density
The average energy density for the ground state configuration can be calculated from(3.7) as E = b q V (cid:90) d x (cid:113) − p (cid:48) ( x ) + V ( p ( x ))( p (cid:48) ( x ) + 2 V ( p ( x ))) (3.30)= b q κ ρ ω + ρ ω ) . (3.31)Now by using ( ω ) = 3( ρ + ρ )8 b q (cid:32) − η − (cid:15)ρ ρ + ρ (cid:33) , (3.32)we get E = 3 √ (cid:112) b q ( ρ + ρ ) / × (cid:32) η (cid:15)ρ ρ + ρ ) (cid:33) (3.33)or you could also write this way using η = (cid:15) sin ( p ), E = 3 √ (cid:112) b q ( ρ + ρ ) / × (cid:34) η (cid:32) ρ sin ( p )( ρ + ρ ) (cid:33)(cid:35) (3.34)Now we also have the constraint due to fixed charges, which is ρ ρ + ρ = ∆ (cid:18) π p ) (cid:19) (cid:18) (cid:15) × ∆ (cid:18) π p ) (cid:19)(cid:19) (3.35)or equivalently, ∆ (cid:18) π p ) (cid:19) = ρ ρ + ρ (cid:32) − (cid:15)ρ ρ + ρ (cid:33) , (3.36)which determines sin( p ) = (cid:112) η/(cid:15) . We also have, from the constraint on the charges, ρ = 4 b q sin ( p )3 (cid:96) (cid:20) − ∆ (cid:18) π p ) (cid:19) (cid:21) × (cid:32) η (cid:33) (3.37)hence η = ρ + ρ ) (cid:96) b q sin ( p ) − − = 4 b q sin ( p )3( ρ + ρ ) (cid:96) (cid:16) O (1 /(cid:96) ) (cid:17) (3.38)The energy density, in terms of the length scale (cid:96) is, therefore, E = 3 √ (cid:112) b q ( ρ + ρ ) / × (cid:18) A(cid:96) (cid:19) , (3.39) here A ≡ b q ρ + ρ ) (cid:32) sin ( p ) + ρ ρ + ρ (cid:33) > . (3.40)(This reproduces the result in [20] for ρ / ( ρ + ρ ) → L = (cid:96)F (cid:16) π ; sin( p ) (cid:17) → ∞ , which is the same result wehave already got for p (cid:28) Two branches of the solution
We have worked in the regime where ρ < ρ , but what happens if we make ρ > ρ ?Does sin( p ) go very close to 1 and as a consequence? The answer is no: one immediatereason is that the expression for the energy must be symmetric under the exchangeof ρ and ρ . We could also say that the exchanging the role of them when ρ > ρ isenergetically favourble. Hence, when ρ exceeds the value of ρ , there is a first orderphase transition (strictly only a crossover but gets infinitely sharp at large- J limit)and sin( p ) decrease as ρ /ρ gets bigger, eventually reaching the same homogeneoussolution, p = 0. We compute various quantities for the D = 3 Wilson-Fisher conformal O (4) model on T × (cid:82) in this section. Here we set (cid:96) < (cid:96) to be the spatial period of the torus, thespatial slice. Imposing the boundary condition on the toric geometry, then, we have4 (cid:96)F (cid:16) π ; sin( p ) (cid:17) = 4 L = (cid:96) or equivalently, (cid:96) = (cid:96) F ( π ;sin( p ) ) Let us compute the total energy of the system when ρ = ρ = ρ/ E = 3 √ (cid:112) b q ρ / × (cid:18) A(cid:96) (cid:19) , (4.1)where A = b q ρ (cid:18) sin ( p ) + 12 (cid:19) (4.2)and ∆ (cid:18) π p ) (cid:19) = 12 ⇐⇒ sin ( p ) = 0 . . . . . (4.3)Using the value for sin ( p ), we also have F (cid:18) π p ) (cid:19) = 2 . . . . . (4.4) o sum up, the total energy of the system becomes E = (cid:96) (cid:96) E = 3 √ (cid:96) (cid:96) ρ / (cid:112) b q × (cid:32) . × b q ρ ( (cid:96) ) (cid:33) , (4.5)or using J ≡ ρ(cid:96) (cid:96) , E = 3 √ J / (cid:112) b q (cid:96) (cid:96) × (cid:32) . × b q (cid:96) J(cid:96) (cid:33) . (4.6)This shows that the classical ground state energy of the O (4) model at large charge J = J scales as ( J + J ) / . There is also the subleading term scaling O ( J / ) here andthis is also uncorrected by the quantum corrections on the torus. This is because theonly e ff ective operator scaling as O ( J / ) is Ric | q | , which vanishes on the torus. Let us calculate the two point function (cid:104) q ∗ (0 , q ( x , x ) (cid:105) . At leading order there arejust classical contributions, which amount to (cid:104) q ∗ (0) q ( x ) (cid:105) ∝ (cid:90) dy dy q ∗ ( y , y ) q ( x + y , x + y ) (4.7)because of the translational symmetry. Then we have (cid:104) q ∗ (0) q ( x ) (cid:105) = (cid:90) dy dy q ∗ ( y , y ) q ( x + y , x + y ) (4.8) ∝ (cid:90) dy sin( p ( y )) sin( p ( x + y )) (4.9)= σ L (cid:90) dy sn (cid:18) y (cid:96) ; σ (cid:19) sn (cid:18) x + y (cid:96) ; σ (cid:19) (4.10)where ∆ ( π/ σ ) = ρ / ( ρ + ρ ). Likewise, we have (cid:104) q ∗ (0) q ( x ) (cid:105) ∝ L (cid:90) dy dn (cid:18) y (cid:96) ; σ (cid:19) dn (cid:18) x + y (cid:96) ; σ (cid:19) (4.11)Note that we have kept all the σ dependence to check the result in the homogeneouslimit, σ →
0. Also incidentally, because of the charge conservation, (cid:104) q ∗ (0) q ( x ) (cid:105) mustvanish.These integrals can be performed analytically too, but this gives rather involvedexpressions involving compositions of elliptic integrals and their inverses, and deriva-tives of those. It is simpler for practical purposes to express the observables in termsof numerically evaluated integrals; as an illustration, we compute these integrals nu-merically when ρ = ρ . We also set (cid:96) = 1 with no loss of generality, since the theory q ( ) q ( x ) >< q ( ) q ( x ) > x - - Two - point functions Figure 1: 2 π (cid:104) q ∗ (0) q ( x ) (cid:105) and 2 π (cid:104) q ∗ (0) q ( x ) (cid:105) in terms of x when J = J . Note that theycoincide with each other at x = 0. is conformal. Note that the constraint ∆ ( π/ σ ) = ρ / ( ρ + ρ ) = 1 / σ = 0 . . . . . Now by using this, we have the graph of the two-point function below(Fig. 1). Meanwhile in the limit where ρ → p ( x ) = 0 and we justget (cid:104) q ∗ (0) q ( x ) (cid:105) = 0 and (cid:104) q ∗ (0) q ( x ) (cid:105) = 1. We have seen that the large-charge ground state of the system with a generic chargeratio J /J , is inhomogeneous, and that the inhomogeneity is energetically favored atthe longest possible distance scales. In finite volume, with a toroidal spatial slice,this pattern of global and translational symmetry breaking, produces a distinctivesignature in the correlation functions in the large-charge ground state. Although ourfocus is on finite-volume observables, it is worth trying to understand the meaning ofthe infrared-enhanced inhomogeneity in the infinite-volume limit.Since the inhomogeneity is clearly relevant at the longest distance scales, theseIR-inhomogeneous solutions, may be understood as realizing a disordered phase ofsome kind, in which certain local operators fail to cluster. This is distinct from astriped phase, in which the inhomogeneity would have a characteristic scale , whichis inevitably fixed by the average charge density itself, in the case of a conformal fieldtheory such as the critical O (4) model.The phenomenon of violation of cluster decomposition is quite common underrenormalization group flow. What is unfamiliar here is the direct visibility of cluster-nondecomposition in large-charge perturbation theory, despite the strong coupling ofthe underlying model, where the perturbation parameter is 1 / ( µ | x | ) , with | x | being the istance between operators and µ being the chemical potential. The non-clustering ofthe two-point function comes from averaging over classical solutions that break thetranslational invariance spontaneously at the IR scale. Other examples of perturba-tively calculable breakdown of cluster decomposition under renormalization groupflow are known, and are related by dualities to cases where defect operators fail tocluster due to strong coupling e ff ects: For instance, see [33], [34], [35]. In this paper we began with a general argument, by counting the number of Gold-stones and matching with the number of avaliable light modes, that for the O (2)model at large global charge, the ground state configuration can only be homoge-neous regardless of the detailed form of the Wilsonian action at the fixed point. Wewent on to extend the argument to give similar constraints for the ground state of the O (4) model, especially that there can be inhomogeneity in only one direction in thecase of generic charge densities.We have also followed up on the argument due to [2] that for the O (4) model witha generic set of fixed large charges, the lowest energy classical solution is inhomoge-neous: Making the argument more concrete, we have constructed the inhomogeneousground state solution explicitly. We also see that there are two branches of the config-uration, namely when J < J and when vice versa; there is a continuous but nondif-ferentiable dependence of observables on the charge ratio, at the point J /J = 1. Thissolution confirms the analysis of [20] at charge densities close to the homogeneouscase, J /J (cid:28) J + J ) / at leading order just like the O (2) model. We computed the two-point func-tions on the torus too. These results can be checked with Monte-Carlo simulations,and it would be interesting if one could compute those quantities numerically andverify our results. However be warned that the result can only be checked by runninga simulation at quite a low temperature, due to the presence of the soft modes of [2].The appearance of perturbatively calculable disorder is interesting, and shouldmake it possible to be far more explicit about the long-distance behavior of the O (4)model (and other O (2 N ) models) at finite chemical potential. In principle one couldcalculate explicitly which sets of operators obey cluster decomposition and which donot. However such a calculation would take a more thorough study of the classicalsolutions than we have performed so far. For instance, one would have to be morecareful about the possibility of ground states preserving nontrivial combinations oftranslational and internal global symmetries while breaking each separately; we haveignored those possible breaking patterns in this paper.In future work we hope to study the large-charge EFT on the sphere, in order tocompute ground-state conformal dimensions via radial quantization.There the sym-metry braking pattern will be much more interesting because of the non-Abelian na-ture of the symmetry group on the sphere spatial slice. Also, the accuracy of the arge- J expansion can be improved by computing subleading corrections from higher-derivative operators in the EFT and quantum e ff ects. Acknowledgements
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