aa r X i v : . [ h e p - t h ] S e p Accessing Large Global Charge via the ǫ -Expansion Masataka Watanabe Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics, University of Bern,Sidlerstrasse , CH- Bern, Switzerland
Abstract
We compute the lowest operator dimension ∆ ( J ; D ) at large global charge J in the O ( ) Wilson-Fisher model in D = − ǫ dimensions, to leading order in both 1/ J and ǫ . While the effective field theory approach of [ ] could only determine ∆ ( J ; 3 ) as aseries expansion in 1/ J up to an undetermined constant in front of each term, thistime we try to determine the coefficient in front of J in the ǫ -expansion. The finalresult for ∆ ( J ; D ) in the (resummed) ǫ -expansion, valid when J ≫ ǫ ≫
1, turnsout to be ∆ ( J ; D ) = ( D − ) ( D − ) (cid:18) ( D − ) π D (cid:19) D ( D − ) " Γ (cid:0) D (cid:1) π D − ǫ D − ( D − ) × J DD − + O (cid:16) J D − D − (cid:17) where next-to-leading order onwards were not computed here due to technical cum-bersomeness, despite there are no fundamental difficulties. We also compare theresult at ǫ = ∆ ( J ) = × J + · · · to the actual data from the Monte-Carlo simulation in three dimensions [ ], andthe discrepancy of the coefficient 0.293 from the numerics turned out to be 13%.Additionally, we also find a crossover of ∆ ( J ; D ) from ∆ ( J ) ∝ J DD − to ∆ ( J ) ∝ J , ataround J ∼ ǫ , as one decreases J while fixing ǫ (or vice versa), reflecting the factthat there are no interacting fixed-point at ǫ =
0. Based on this behaviour, we proposean interesting double-scaling limit which fixes λ ≡ J ǫ , suitable for probing the regionof the crossover. I will give ∆ ( J ; D ) to next-to-leading order in perturbation theory,either in 1/ λ or in λ , valid when λ ≫ λ ≪
1, respectively. ontents Introduction Large-charge effective action of the O ( ) model in 2 < D < . The field content of the effective field theory at large charge . . . . . . . Sorting effective operators at large charge . . . . . . . . . . . . . . . . . . The lowest operator dimension at large charge in D dimensions . . . . The ǫ -expansion of the lowest operator dimension at large charge . Simplification of the RG flow at large charge . . . . . . . . . . . . . . . . Semi-classical computation . . . . . . . . . . . . . . . . . . . . . . . . . . The double-scaling limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . Three-dimensional Wilson-Fisher fixed point at large charge . . . . . . Conclusions and Outlook Introduction
Not all interesting quantum field theories are solvable, nor even approximately so.However, this does not mean we cannot perform a controlled calculation of someof their physical quantities. It has especially been known that when the systemhas a global symmetry, operator dimensions of or operator product expansion ( ope )coefficients including operators of high charge can be computed to any given orderin perturbation theory in the inverse charge expansion [ , , ]. This method, called the large-charge expansion, simply uses the idea of effec-tive field theory. Effective field theory can be written down by listing all operatorsobeying the symmetry of the system, whereby in the large-charge expansion, thelarge global charge J (or the charge density ρ ) gives a natural scaling of such al-lowed operators. Quite remarkably, this effective theory turns out to be semi-classicaland weakly-coupled as it has the large separation of scales. For example, in D di-mensions, the resulting effective Lagrangian will have its ultraviolet ( uv ) scale at Λ UV ≡ ρ D − and infrared ( ir ) at Λ IR ≡ R geometry for theories without moduli spaceof vacua. This large hierarchy of scales, Λ IR / Λ UV ∝ J − D − , suppresses quantum cor-rections and higher derivative terms, and the effective action at low energies becomesclassically conformally invariant.So powerful is the classical Weyl invariance that it strongly limits the kind ofoperators allowed in the effective action. For the O ( ) Wilson-Fisher ( wf ) fixed pointat large baryon number or the C P N − models at large monopole number in threedimensions, it was argued in [ , ] that there are only two, one at O ( J ) and theother at O ( J ) , allowed effective operators in the effective action at order O ( ) orabove.Now, the peculiarity of such a method using effective Lagrangian is that onecannot determine the coefficient in front of each effective operator. They usually aresome O ( ) numbers which cannot be computed in a controlled fashion unless thereare some other weakly coupled parameters (like the large- N or the ǫ -expansion).This is by no means a limitation of this methodology – Rather, the lesson it offers isthat even when the underlying theory is strongly-coupled, one can at least determinephysical quantities to any given orders in 1/ J expansion, up to some undeterminedconstants which depend on what theory one started with.What is more interesting is that one can start from two different conformal fieldtheories ( cft s) and end up with the same scaling for the physical quantities, upto theory dependent constants which were left undetermined. We could call suchcollection of theories, whose content of the effective action becomes the same (andhence the 1/ J expansion of physical quantities is the same modulo coefficients),as the large-charge universality class. There are countably infinite many knownexamples of different cft s belonging to the same large-charge universality class,which are the C P N − models and the O ( ) wf fixed point in three dimensions. The large-spin expansion of the light-cone bootstrap is parallel to this phenomena [ – ], so we should tryto understand them as large-quantum-number expansion as a whole. aving said that, when one is interested in a particular cft at large global charge,one should compute those theory dependent coefficients in some way or the other.Monte-Carlo simulations are known to work well for such purposes, which (surpris-ingly) showed a remarkable fit down to J =
1, for the O ( ) and the O ( ) wf fixedpoint [ , ].When the theory in question has another weakly-coupled parameter, one canalso utilise it to derive those coefficients. Previous studies have used the 1/ N -expansion for the three dimensional C P N − models [ , ], the ǫ -expansion for thethree-dimensional QED at large monopole number [ ], the expansion in ε ≡ N f / N c − ], or the1/ N -expansion for the three dimensional O ( N ) wf fixed point. The topic of the present paper is to proceed along this direction, to compute an-alytically the lowest operator dimension at leading order in J , for the wf fixed-pointin D = − ǫ . This by no means is a trivial extension of the previous computations:Using the ǫ -expansion at large charge is peculiar compared with other expansionsas it requires partial resuumations of the series, as argued in [ ]. This is becauseEFT (or dimensional analysis) suggests that ∆ ( J ) scales like ∆ ( J ) ∝ J − ǫ − ǫ , so that the ǫ -expansion effectively becomes an expansion in terms of ǫ log J . This means unlesswe resum the series to all orders in the perturbation series, we are meaninglesslyrestricted to the region where ǫ ≫ log J , which of course is not valid when analysingthe D = wf fixed point at large charge.The surprise does not end here – Take exactly ǫ = ∆ ( J ) = J − ǫ − ǫ ,and one gets ∆ ( J ) ∝ J in four dimensions, which is far from true because the theoryis just a free theory in D =
4; One should instead get ∆ ( J ) ∝ J . This suggests thatthere must be a transition of the scaling behaviour as one decreases ǫ , which divertsthe exponent away from 4/3 to 1, and we should be interested in what kind oftransition this is. I will reveal that this transition is a crossover and not a sharp phasetransition of any kind, in the main body of the text. Although this paper seemsto be the first one to show there is such a crossover, but this could be potentiallyinteresting. For example, the fact that the there are no sharp transitions as one divertsaway from free field theory means that the ǫ -expansion can be useful in identifyingthe low-energy field content of the large-charge effective theory in terms of the fieldcontent of the original uv theory.I will also show that this crossover occurs when J = λ / ǫ (where λ is a constant),and that the scaling gradually changes from 1 to − ǫ − ǫ as one increases λ from 0 to ∞ .This suggests a double-scaling limit ǫ → J → ∞ , while λ ≡ ǫ J fixed ( . )where the lowest operator dimension behaves like ∆ ( J ) ∝ J σ ( λ ) , where σ ( ) = σ ( ∞ ) =
32 . ( . ) I thank Domenico Orlando for privately communicating me the result. This type of behaviour at large charge is typical of theories with moduli space of vacua. For more infor-mation, see [ – ]. his double scaling limit naturally coincides the one already noticed in [ – ].Meanwhile, I will also compute the leading order coefficient for the scaling in the ǫ -expansion, when λ ≫
1. This, when we plug in ǫ =
1, gives an analytical estimateon the operator dimension at large charge of the three-dimensional wf theory. Thevalue turns out to be ∆ ( J ; 3 ) = × J + · · · ( . )and surprisingly this is only 13% away from the numerically fitted value of the coef-ficient, . , found in [ ]. Although this accuracy is not good enough to explain theunreasonable effectiveness (in that the fit works even at J =
1) of the large-chargeeffective action, this could be a first step towards that goal.The rest of the paper is organized as follows. In Section , I will review theeffective field theory at large charge of the O ( ) wf fixed point in general dimensions2 < D <
4, and show that there are only two effective operators which contributesat or above O ( J ) . In Section , I will use this information about the effective fieldtheory to track the RG flow directly at large charge and small ǫ , and compute thelowest operator dimension, to leading order in both 1/ J and ǫ . I will find that thereis a crossover in the scaling behaviour of the lowest operator dimension even atleading order, when J becomes J ∼ ǫ , and point out an interesting double-scalinglimit, which fixes J ǫ to a constant. I will also reveal that the competing size of theconformal coupling and the potential term is responsible for this crossover. Finally, Iwill uncontrollably plug ǫ = ∆ ( J ; D ) and compute the lowestoperator dimension of the D = wf fixed point at large charge, which was thencompared with the previous result from the Monte-Carlo simulation. Note:
Three days before this paper was submitted, another paper similar in spiritappeared [ ], which dealt with the similar double scaling limit but with ǫ J fixed.I also learned of a paper in preparation, which should appear on the same day asmine, by Badel, Cuomo, Monin, and Rattazzi which seems to be largely overlappingwhat I did here. Large-charge effective action of the O ( ) modelin 2 < D < . The field content of the effective field theory at large charge
We consider, as in [ ], the O ( ) wf fixed-point, but this time in general dimensions2 < D <
4, which is known to exist as an interacting fixed point [ ]. The existenceof interacting fixed points now makes it possible to apply directly the method oflarge-charge expansion for the wf fixed point in dimensions other than three. Belowwe quickly review the construction of the effective action at large charge for the wf fixed point in general dimension. For further information or references, consult theoriginal paper [ ]. et us start from the complex φ theory in the uv : L UV = − | ∂φ | + g | φ | ( . )As always, giving large charge to the system is equivalent to setting a large dimen-sionful vacuum expectation value ( vev ) to the radial field a ≡ | φ | . This in turn giveslarge mass to the a -field, so that it should be integrated out at large charge. Theremaining field is the angular field χ ∼ χ + π , which is the massless Goldstonemode from the symmetry breaking induced by the aforementioned vev . The vev forthe Goldstone mode is χ ≡ h χ i = ω t , where ω is the chemical potential which fixesthe charge density, and is proportional to the induced mass of the a -field.What is important is that in the deep ir , the effective action should not only beconformally invariant, but also be classically conformally invariant. This is becausethe ir dynamics is free and quantum corrections at scale Λ only comes in positivepowers in Λ / Λ UV , where Λ UV ∝ ω , which is large compared to the ir scale. This,combined with the fact that the χ -field has dimension 0, constrains the form of op-erators appearing in the effective action a lot. Especially, at leading order in the J -scaling, the effective action should be simply L = b χ | ∂χ | D + · · · , ( . )where b χ is an undetermined constant, as in the case of the three-dimensional wf fixed point. As was emphasized in the introduction, this constant should be analyt-ically computable once we have a weak-coupling parameter, which is exactly whatwe will do in later sections. . Sorting effective operators at large charge
Rules for sorting operators
Because of the fact that the effective action must be classically Weyl-invariant, wehave the following rules for allowed operators.• The term must have Weyl weight D .• The term must be O ( ) invariant ( i.e., it must respect the shift symmetry of χ ).• The term must be charge-conjugation invariant, i.e., invariant under χ ↔ − χ .• Only | ∂χ | can appear in the denominator, because it is the mass for the a -field.The semi-classical leading order action tells us the scaling of operators in termsof the charge density ρ , which we now fix and take large (in units of the size of theunderlying geometry). The rules to keep track of in counting the ρ -scaling are thefollowing.• ∂χ ∝ ρ D − • ∂ · · · ∂χ ∝ ρ − D ( D − ) • The leading order equation of motion, ∂ µ (cid:0) | ∂χ | D − ∂ µ χ (cid:1) =
0, can be used. he second rule comes from the ρ -scaling of the fluctuation of χ , as the vev for χ vanishes upon acting on more than one derivatives. The last rule is because wheneversuch a combination appears, it can be replaced by something of the lower ρ -scaling.One thing one should potentially be careful about is the meaning of operatorlisting in fractional dimensions. In this paper, I employ the hypothesis that oneonly allows fractional powers of | ∂χ | , as one does not seem to be able to generatefractional powers of anything else, from the original Lagrangian. One might evenbe able to check this statement in the ǫ -expansion. Also, assuming that the analyticcontinuation in D has nice properties in the limit D → ∞ , excluding behavioursincluding trigonometric functions in D , one sees that fractional powers of anythingother than | ∂φ | must be excluded. In other words, such a hypothesis for fractional D is sufficient to reproduce the scaling rules for any integer D . Effective operators at large charge in general dimensions
We are going to list operators that are bigger than or equal to O ( J ) at large charge.First notice that we can only schematically allow for operators of the form ∂ n (cid:2) ( ∂χ ) m (cid:3) | ∂χ | n + m − D , ( . )aside from terms including the curvature. The ρ -scaling of the operator of this formis ∆ ≡ ( − ℓ ) D − n ( D − ) , ( . )where ℓ indicates how many ∂ · · · ∂χ there are in the numerator, and 1 ℓ min ( n , m ) when n ≥ ℓ can only be 0 when n =
0, trivially).Because we are looking for operators that does not vanish in the large chargelimit, we impose ∆ >
0, or n ( − ℓ ) D . )This brings down the number of operators to consider to finite, and we can examinethem one by one. In Table we show the table for the values of allowed ( n , ℓ ) andthe resulting ρ -scaling (we assume D ρ -scaling. Order ρ DD − The only operator at this order is | ∂χ | D ( . )which is the leading order contribution. Order ρ D − D − The only operator at this order (on the non-warped geometry) is
Ric | ∂χ | D − ( . )which, to be precise, must be Weyl-completed by | ∂χ | D − ( ∂ | ∂χ | ) , but this has ascaling that is lower than O ( ρ ) and we discard it. n , ℓ ) ρ -scaling D = D = D = (
0, 0 ) DD − (
1, 1 ) D − ( D − )
13 14 0 (
2, 1 ) D − ( D − ) − − : We show the ρ -scaling of operators of the form ( . ) which do not vanish in thelarge-charge limit, assuming D Order ρ D − ( D − ) There are two operators at this order, but they do not appear in theeffective Lagrangian because they are odd under the charge conjugation symmetry.
Order ρ D − ( D − ) There are superficially two operators at this order, but they are theform of the total derivative modulo operators smaller than O ( ρ ) , so there are nooperators at this order. Order ρ Especially, there are no operators at this order, when we assume 2 D D = | ∂χ | D − . . The lowest operator dimension at large charge in D di-mensions Using the above result for the effective operators, the lowest operator dimension atlarge charge goes as follows: ∆ ( J ) = c leading J DD − + c Ricci J D − D − + γ ( D ) + · · · , ( . )where γ ( D ) is the universal one-loop Casimir energy at order O ( J ) . In three dimen-sions, this is known to take a value γ ( D = ) = − , ]. The ǫ -expansion of the lowest operator dimen-sion at large charge . Simplification of the RG flow at large charge
If one is only interested in the leading order result in the ǫ -expansion, the computa-tion of the lowest operator dimension at large charge is most easily done by directly racking the flow of the renormalization group. The argument can be thought of asthe precise version of what was given in Sec. of [ ]. An important remark is thatall statements below should be only understood at leading order in the ǫ expansion.Let us start from the renormalised Lagrangian as follows: L = Z φ | ∂φ | + m Z m | φ | + µ ǫ gZ g | φ | , ( . )and we use dimensional regularization and minimal subtraction in regularizing andrenormalising the divergences. As one starts out from uv , the value for the couplingconstant g quickly reaches its attractive fixed point, g ⋆ , which is already known inthe ǫ -expansion as g ⋆ = π ǫ . ( . )One can also consider fine-tuning the coupling so that we already are on the fixedpoint in the uv .Now, turning on the vev for the field a ≡ | φ | changes the renormalization groupflow when µ ∼ m a , after which the RG evolution of µ ǫ g completely freezes. Here, m a is the mass of the a -field induced by the vev , which is ( m a ) ∝ µ ǫ g | a | Thisprocess is usually not under control, and one cannot usually see what value µ ǫ g takes. However, in the ǫ expansion, because such a shift of g ⋆ due to the vev startsonly at order O ( g ⋆ ) = O ( ǫ ) , one can just plug in the value of g ⋆ into µ ǫ g . As onestill do not know what value of µ ∼ O ( m a ) one should plug in, we will just plug in µ = K × m a = K × µ ǫ /2 g ⋆ | a | , where K is some O ( ) constant. Solving for µ , we get µ ǫ = ( K √ g ⋆ ) ǫ − ǫ /2 a ǫ − ǫ /2 , ( . )and therefore we generate the potential that goes as1 V ( a ) = ( K √ g ⋆ ) ǫ − ǫ /2 × g ⋆ × a + − ǫ /2 = g ⋆ | φ | + ǫ − ǫ /2 × ( + O ( ǫ )) ( . ) . Semi-classical computation
The semi-classical Lagrangian
At large charge, the Lagrangian including this potential term can be treated semi-classically, which at leading order in O ( ǫ ) reads, on the unit sphere, L = | ∂φ | + ( D − ) | φ | + g ⋆ | φ | DD − ( . )where the second term is the conformal coupling of the scalar field. It is also veryimportant to not expand | φ | DD − in terms of ǫ . It would give terms like ( ǫ log a ) n butfrom effective field theory analysis, we know they must finally resum to a DD − in theend. In other words, we conduct computation assuming ǫ is not small, except that weuse the O ( ǫ ) result for the coefficient in front of | φ | DD − , as this value is contaminated t O ( ǫ ) by ǫ log K or the running of the coupling at around µ ∼ m a , in the presenceof the vev .Now we solve the equation of motion at fixed vev for the a -field, assuming thehelical configuration for the lowest energy state, φ = a × e i ω t . The equation of motionthen gives ω = g ⋆ D ( D − ) a D − + ( D − ) . )The charge J and the energy on the unit sphere ∆ goes as follows, in terms of a . J = α ( D ) s g ⋆ D ( D − ) a ( D − ) D − × s + ( D − ) D g ⋆ a D − ( . ) ∆ = α ( D ) g ⋆ ( D − ) D − a DD − × + ( D − ) ( D − ) g ⋆ a D − ! , ( . )where α ( D ) ≡ π D /2 Γ ( D /2 ) is the area of the unit ( D − ) -sphere. This is sufficient toinfer the relation between ∆ and J . Because it is analytically hard to compute thisdirectly, we will take g ⋆ a large or small and compute ∆ ( J ) in the form of the Taylorexpansion in terms of it. The scaling structure from such an analysis will be ∆ ( J ) = c leading J DD − + c Ricci J D − D − + · · · ( . )when g ⋆ a ≫ D = ∆ ( J ) = J + c Ricci J + · · · ( . )when g ⋆ a ≪ The effect of the conformal coupling and the crossover
Because there are two competing small parameters in the system, g ⋆ ∼ ǫ and 1/ a , therelative size of those scales becomes important. We examine below the cases where ǫ a D − scales as ǫ a D − = λ a p , seperately when 0 < p < D − , p =
0, and p < (a) p >
0: Semi-classical regime at J ≫ ǫ In this case, because we take a large, we can approximate ( . ) and ( . ) as follows: J p > = α ( D ) s g ⋆ D ( D − ) a ( D − ) D − ( . ) ∆ p < = α ( D ) g ⋆ ( D − ) D − a DD − , ( . )The relationship between ∆ and J therefore becomes ∆ ( J ) = c ( D ) × J DD − ( . ) ( D ) = ( D − ) ( D − ) (cid:18) ( D − ) D π (cid:19) D ( D − ) Γ (cid:18) D (cid:19) D − g D − ( D − ) ⋆ , ( . )and the condition ǫ a D − = λ a p can be rewritten as J ∝ ǫ (cid:16) − p − (cid:17) ≫ ǫ ( p > ) ( . ) (b) p =
0: Crossover region at J ∼ ǫ One should in theory be able to express ∆ in terms of J , but it would be too cumbersome. We can understand this region J ∝ ǫ as the crossover region from region (a), J ≫ ǫ , to region (c), J ≪ ǫ . The analysis ofthis regime will be done in perturbation theory in J ǫ ≪ ( J ǫ ) − ≪ (c) p <
0: Free theory regime at J ≪ ǫ In this region, what dominates takesover, and we can approximate ( . ) and ( . ) as follows: J p < = α ( D )( D − ) a ( . ) ∆ p < = α ( D )( D − ) a , ( . )so that ∆ ( J ) = D − J , ( . )whose coefficient of course is the mass dimension for the scalar field in D dimen-sions. The condition ǫ a D − = λ a p translates to J ∝ ǫ + | p | ≪ ǫ ( p < ) ( . )To summarise, ∆ ( J ) behaves in the following way, depending on how large J is: ∆ ( J ) = J DD − semi-classical behaviour at J ≫ ǫ J σ ( J ) crossover at J ∼ ǫ J free theory behaviour at J ≪ ǫ ( . ) . The double-scaling limit
One can use various double-scaling limit for this system at large J and small ǫ ,depending on what regime one is interested in. For example, when one is interestedin the regime (a), one can use the double-scaling limit with ǫ J + | α | fixed. This willensure that the limit taken leads to the operator scaling ∆ ∝ J − ǫ − ǫ + · · · ( when J + | α | is fixed ) ( . ) n the other hand, one can use the double-scaling limit like ǫ J −| α | fixed, to reach theregion (c). This will in turn ensure that the limit taken leads to the operator scaling ∆ ∝ J + · · · ( when J −| α | is fixed ) ( . )Somewhat more interesting is the double-scaling limit which fixes λ ≡ ǫ J , whichaccesses the crossover region (b), in the regime of weak-coupling. The operatorscaling will take the form ∆ ( J ) ∝ J σ ( λ ) + · · · , ( . )where σ ( λ ) is an increasing function in λ , with σ ( ) = σ ( ∞ ) = DD − . Perturbative expansion in the double-scaling limit
We treat cases where λ ≫ λ ≪ g separately, and see what are the correctionto the leading formula in 1/ J and in ǫ in the double-scaling limit. When J = λǫ and λ ≫ By using the Taylor expansion, we get ∆ ( J ; D ) = c ( D = ) × J + √ √ · ǫ J ! + (cid:18) √ · ǫ J (cid:19) + · · · ( . ) When J = λǫ and λ ≫ By using the Taylor expansion, we get ∆ ( J ; D ) = J + ǫ J − ( ǫ J ) + · · · ! ( . ) . Three-dimensional Wilson-Fisher fixed point at large charge
Let us now plug ǫ = J dependence of the lowest operator dimension. We can, as in [ ], ignore the conformalcoupling, which is equivalent to analysing the semi-classical region, J ≫ ǫ = c ( D ) will not be truncated to O ( ǫ ) but considered to be exact, which is the expectation from the EFT at largecharge.Using the expression for c ( D = ) and plugging in g ⋆ = π ǫ (cid:12)(cid:12)(cid:12) ǫ = = π , wehave c ( D ) = . ( . )This gives the lowest operator dimension at leading order at large charge as ∆ ( J ) = × J + · · · ( . )whose value is different from the result of the Monte-Carlo simulation found in [ ],which is ∆ ( J ; D ) = J , by 13%. I thank Zohar Komargodski for making me notice that presenting this as a double-scaling limit is inter-esting. omparison with the C P model Because the ir fixed point of the C P model is the O ( ) wf fixed point, we can usethe result from the large- N expansion of the C P N − model [ ] and again just plugin N =
1. The result yields ∆ ( J ) ∝ × J , ( . )which gives a slightly better number compared to ours. Conclusions and Outlook
I have computed the lowest operator dimension at large charge J in D = − ǫ dimensional wf theory to leading order in both 1/ J and ǫ , with corrections that goas powers of ( J ǫ ) − ≪ J ǫ ≪
1. Especially, by extrapolating to D = D = wf model to leading order in the 1/ J -expansion: ∆ ( J ) = × J + · · · . ( . )I also compared the result with the previous Monte-Carlo result and found a dis-crepancy of 13%.I also found an interesting crossover in the scaling behaviour of ∆ ( J ) , and foundthat it scales ∆ ( J ) ∝ J DD − when J ≫ ǫ and ∆ ( J ) ∝ J when J ≪ ǫ . I alsopointed out that the crossover can be accessed in the weak-coupling limit by takinga double-scaling limit, where 1/ ǫ , J → ∞ while ǫ J fixed.There are a number of important future directions. Non-trivial check of the method of large-charge expansion
The lowest operator dimension expanded in 1/ J in the large-charge universalityclass of the O ( ) wf fixed point has a theory-independent part at order O ( J ) .Going to higher-orders in ǫ to reproduce this number gives a consistency checkof the method of the large charge expansion. Computing and comparing thesubleading coefficients to the numerical data will be of great importance too. Checking dualities at large charge
If two theories are dual to each other, the lowest operator dimension at charge J should match to all orders in 1/ J -expansion, including the coefficients. Com-puting the coefficients will give more precise check of the duality than justlooking at the scaling behaviour. Contributions from massive modes
Massive modes (in this example, the a -field), contributes as O ( exp ( − m a )) ,where m a ∝ J α . This can be explicitly seen by actually computing the freeenergy as a function of the chemical potential µ for the charge density – onecan directly see the exponential contribution, coming from the one-loop en-ergy shift of the massive a -field. One should also compare the result with thedouble-scaling limit (which fixes the mass of the BPS dyon) in rank-1 SCFTsintroduced in [ ]. This will be reported in the forthcoming paper. cknowledgements The author is grateful to Andrew Gasbarro, Simeon Hellerman, Nozomu Kobayashi,Zohar Komargodski, Keita Nii, Domenico Orlando and Susanne Reffert for valuablediscussions. The author also thank the Simons Center for Geometry and Physics forhospitality during the conference “Quantum-Mechanical Systems at Large QuantumNumber” while this paper was being completed.
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