Bounds on the density of states and the spectral gap in CFT 2
BBounds on density of states and spectral gap in CFT Shouvik Ganguly, q Sridip Pal ˜ q, ˜ Qq Department of Electrical and Computer Engineering, University of California, San DiegoLa Jolla, CA 92093, USA ˜ q Department of Physics, University of California, San Diego, La Jolla, CA 92093, USA ˜ Q School of Natural Sciences, Institute for Advanced Study Princeton, NJ 08540, USA
E-mail: [email protected] , [email protected] Abstract:
We improve the recently discovered upper and lower bounds on the O (1) correction to the Cardy formula for the density of states integrated over anenergy window (of width 2 δ ), centered at high energy in 2 dimensional conformalfield theory . We prove optimality of the lower bound for δ → − . We prove aconjectured upper bound on the asymptotic gap between two consecutive Virasoroprimaries for a central charge greater than 1 , demonstrating it to be 1 . Furthermore,a systematic method is provided to establish a limit on how tight the bound onthe O (1) correction to the Cardy formula can be made using bandlimited functions.The techniques and the functions used here are of generic importance whenever theTauberian theorems are used to estimate some physical quantities. We unravel acurious connection between the sphere packing problem and the problem of findingthe lower bound. a r X i v : . [ h e p - t h ] J a n ontents Modular invariance is a powerful constraint on the data of 2D conformal field the-ory (CFT). It relates the low temperature data to the high temperature data. Forexample, using the fact that the low temperature behavior of the 2 D CFT partitionfunction is universal and controlled by a single parameter c , the central charge ofthe CFT, we can deduce the universal behavior of the partition function at hightemperature and thereby deduce the asymptotic behavior of the density of states,which controls the high temperature behavior of a 2D CFT [3]. Similar ideas canbe extended to one point functions as well, where the low temperature behavior iscontrolled by the low lying spectra and three point coefficients [4, 5]. Yet anotherremarkable implication of the modular invariance of the partition function is theexistence of infinite Virasoro primaries for CFT with c >
1. Significant progresshas been made in recent years towards exploiting the modular invariance to deduceresults in 2D CFT under the umbrella of modular bootstrap [4–13].Recently, with the use of complex Tauberian theorem, Mukhametzhanov andZhiboedov [18] have explored the regime of validity, as well as corrections, to the The fact that the modular invariance of CFT can predict the asymptotic density of states isexplicitly stated in [1]. One usually takes the inverse Laplace transform of the partition function todeduce such behavior; similar techniques also appeared in [2]. We thank Shouvik Datta for pointingthis out. The usefulness of Tauberian theorems in the context of CFT is pointed out in [14]; subsequently,its importance was emphasized in Appendix C of [5], where the authors used Ingham’s theorem[15]. The fact that going out to the complex plane while using Tauberian theorems would provideextra mileage in controlling the correction terms in various asymptotic quantities of CFT, has beenpointed out in [16]. In particular, the use of [17] turned out to be extremely useful in this context. – 1 –ardy formula with great nuance. In particular, they have investigated the entropy S δ associated with a particular energy window of width δ around a peak value ∆,which is allowed to go to infinity, and found S δ = log (cid:18)(cid:90) ∆+ δ ∆ − δ d ∆ (cid:48) ρ (∆ (cid:48) ) (cid:19) (cid:39) ∆ →∞ π (cid:114) c ∆3 + 14 log (cid:18) cδ (cid:19) + s ( δ, ∆) , (1.1)where ρ (∆) is the density of states, given by a sum of Dirac delta functions peakedat the positions of the operator dimensions. It is shown in [18] that for O (1) energywidth, the O (1) correction s ( δ, ∆) is bounded from above and below: δ = O (1) : s − ( δ, ∆) ≤ s ( δ, ∆) ≤ s + ( δ, ∆) (1.2)In particular, [18] showed that O (1) is the optimal order upto which one can makeuniversal prediction about the density of states without assuming anything further.To paraphrase, one can not theoretically obtain a universal correction, which is fur-ther suppressed compared to the O (1) number . Given this scenario, it is meaningfulto improve upon the O (1) correction and look for the optimal bound. One of thepurposes of the current note is precisely so, to improve the bound and provide asystematic way to estimate how tight the bounds can be made using bandlimitedfunctions. Furthermore, we will show that for a specific value of energy width, it ispossible to achieve the optimal lower bound irrespective of whether one uses ban-dlimited function or not.We prove the conjectured upper bound on the asymptotic gap between Virasoroprimaries, which turns out to be 1. This gap is optimal since for the Monster CFT,the gap is precisely 1. This provides a universal bound on how sparse a CFT spec-trum can be asymptotically. In particular, this rules out the possibility of havinga primary spectrum with hadamard gap . We also remark that we have found twodifferent ways to prove the gap, one of which is related to optimal lower bound.We further unravel a curious connection between the sphere packing problem andthe problem of finding the optimal lower bound using bandlimited function. Thisconnection is interesting in its own right and is explored in § δ ≤ δ = 1, the upper bound on the lower bound becomes achievable and thus becomesthe best possible bound using bandlimited functions. It turns out that one can do This approach can be contrasted to the one taken in [19] , where a convergent Rademachersum is written down, nonetheless this has been possible because the much stronger conditions likeholomorphicity is assumed about the CFT. A spectrum { ∆ k } is said to have Hadamard gap if there exists λ > ∆ k +1 ∆ k ≤ λ forall k . – 2 –etter and prove optimality of the lower bound in δ → − limit showing that MonsterCFT staurates the lower bound.Our results, regarding the bound on O (1) correction to the density of states canbe summarized by figure [1], where the green line and dots denote the lower (upper)bound on the upper (lower) bound. The orange lines denote the improved achievablebounds. The brown dots stand for the lower (upper) bound on the upper (lower)bound obtained from implementing the positive definiteness condition on the Fouriertransform of ± ( φ ± − Θ) via Matlab. The bound on bounds represented by the greenline is thus weaker than that represented by the brown dots. In short, the brownshaded region is not achievable by any bandlimited function. In fact, if we use theanlaytic properties of the function involved on top of its bandlimited nature, then itis possible to show that the upper bound on the lower bound becomes 0 . c − ≤ . δ ≤ MZ Improved Lower bound on Upper bound ● ● ● ● ● ● ● ● ● ● ● ● δ [ s + ] MZ Improved upper bound on Lower bound ● ● ● ● ● ● ● ● ● ● δ [ s - ] Figure 1 : Exp[ s ± ] as a function of δ , the half-width of the energy window. Theblue line is the bound obtained in [18]. The orange line denotes the improved boundthat we report here. The green line is the analytical lower (upper) bound on theupper (lower) bound, while the brown dots stand for the lower (upper) bound on theupper (lower) bound obtained from enforcing the positive definiteness condition onthe Fourier transform of ± ( φ ± − Θ) via Matlab. The bound on bounds representedby the green line is thus weaker than that represented by the brown dots. The brownshaded region is not achievable by any bandlimited function. If we use the anlayticproperties of the function involved on top of its bandlimited nature, then it is possibleto show that the upper bound on the lower bound becomes 0 . c − ≤ . δ ≤ δ = 1 − .This indicates one can perhaps do better around that region. In fact, it turns out thatone can do better in the interval [1 − (cid:15), (cid:15) (see figure. [2]). In the δ → − , we analytically show that the lower bound is optimal. This optimality resultfollows from showing Monster CFT saturates this lower bound and thus independent– 3 –f the fact whether we use bandlimited function or not. Improved δ [ s - ] Figure 2 : Exp[ s − ] as a function of δ , the half-width of the energy window around δ = 1. The black curve makes the bounding curve continuous, i.e. the bounding curveis now given by max { orange curve , black curve } .In particular, we show that the upper bound on s ( δ, ∆) is given byexp [ s + ( δ, ∆)] = M Z ( δ ) , δ < . δ (cid:0) δ + π (cid:1) , . < δ ≤ . . , δ > .
785 (1.3)where
M Z ( δ ) is a function introduced in [18] and defined as M Z ( δ ) = π (cid:0) πδ (cid:1) (cid:0) sin (cid:0) πδ (cid:1)(cid:1) − , δ < a ∗ π ∼ . . , δ > a ∗ π ∼ . . (1.4)Here, a ∗ ∼ .
38 satisfies a ∗ = 3 tan( a ∗ / . Eq.(1.3) is an improvement of the upperbound for δ > .
73, as evident from figure [4].The lower bound s − ( δ, ∆) (except for some small interval δ ∈ [1 − (cid:15), s − ( δ, ∆)] = mz ( δ ) , √ π ≤ δ < √ π ∼ . , ( δ − π ) δ , √ π < δ ≤ , . , δ > . (1.5)where mz ( δ ) is a function, introduced in [18] mz ( δ ) = ( δ − π ) δ , √ π ≤ δ < π ∼ . , π ∼ . , δ ≥ π . (1.6)The eq. (1.5) is an improvement of the lower bound for δ > .
94, as evident from fig-ure [4]. In that small interval, the better bound is depicted numerically in figure [2].One can immediately verify the new bounds using the known partition function of2D Ising model and extremal CFTs as shown in figure 3.The rest of the paper details the derivation of the above. In section §
2, wederive the improvement on the bound on the O (1) correction to the Cardy formula.The connection with sphere packing problem has been explored in section ! §
3. Thesection § § § The basic ingredients for estimating the asymptotic growth of the density of statesare two functions φ ± such that the following holds: φ − (∆ (cid:48) ) < Θ (∆ (cid:48) ∈ [∆ − δ, ∆ + δ ]) < φ + (∆ (cid:48) ) . (2.1)We refer the readers to section 4 of [18] for details of the procedure leading to abound when ∆ goes to infinity. A nice and detailed exposition of the technique incontext of asymptotics on ( h, ¯ h ) plane can be found in [20]. The notion of asymptoteis far more rich on ( h, ¯ h ) plane, hence the one can see the usefulness of Tauberiantechniques in a detailed and transparent manner in such a scenario. The key pointis to bound the number of states in an order one window by two increasing positivefunctions from above and below such that they differ by an order one multiplicativenumber. This forces the growth of number of states in that window to follow the– 5 – D Ising, δ =
200 400 600 800 1000 Δ - - - ( Δ )
2D Ising, δ =
200 400 600 800 1000 Δ - - - ( Δ ) Monster, δ = Δ - - - ( Δ ) Monster, δ = Δ - - ( Δ ) Figure 3 : Verification of the bound on s ( δ, ∆), order one correction to entropy using2D Ising model (the top row) and non-chiral Monster CFT (the bottom row). Wehave plotted for different δ , the half width of the interval under consideration. Onecan see for high enough ∆, the bound is satisfied, indicating the asymptotic natureof the bound. The partition function for chiral Monster CFT can be reinterpretedas a S modular invariant particle function of a non chiral CFT with c = 12. Thedense cyan curve in between the red and the black line is obtained from the actualpartition function.same growth as in the bounds. Subsequently, the multiplicative order one numbersappearing in the bounds contain finer information about the spectrum. In particular,one can derive: c − ρ (∆) ≤ δ (cid:90) ∆+ δ ∆ − δ d ∆ (cid:48) ρ (∆ (cid:48) ) ≤ c + ρ (∆) , (2.2)where ρ (∆) reproduces the contribution from the vacuum at high temperature andis given by ρ (∆) = π (cid:114) c I (cid:16) π (cid:113) c (cid:0) ∆ − c (cid:1)(cid:17)(cid:112) ∆ − c Θ (cid:16) ∆ − c (cid:17) + δ (cid:16) ∆ − c (cid:17) . (2.3)– 6 –he above is in fact the leading result for the density of states at high energy andcaptures the growth of states at high energy. Furthermore, c ± is defined as c ± = 12 (cid:90) ∞−∞ dx φ ± (∆ + δx ) . (2.4)As mentioned, c ± are order one numbers containing finer details about the spectrum.At this point, a technical but very important remark is in order: the eq. (2.2) holdsif the Fourier transform of φ ± has a support on an interval which lies entirely within[ − π, π ]. This ensures that one can ignore the contribution coming from heavystates towards the density of states when doing the high temperature expansionof the partition function. With this constraint in mind, we consider the followingfunctions: φ + (∆ (cid:48) ) = sin (cid:16) Λ + (∆ (cid:48) − ∆)6 (cid:17) Λ + (∆ (cid:48) − ∆)6 (cid:18) (cid:48) − ∆) δ (cid:19) , (2.5) φ − (∆ (cid:48) ) = sin (cid:16) Λ − (∆ (cid:48) − ∆)6 (cid:17) Λ − (∆ (cid:48) − ∆)6 (cid:18) − (∆ (cid:48) − ∆) δ (cid:19) . (2.6)In order to ensure that the indicator function on the interval [∆ − δ, ∆ + δ ] is boundedabove by φ + , we need to have δ Λ + ≤ . . (2.7)The number in the eq. (2.7) is obtained by requiring that φ + (∆ ± δ ) >
1. Thefunctions φ ± have Fourier transforms with bounded supports [ − Λ ± , Λ ± ] , respectively.Thus, in order for this support to lie within [ − π, π ] , we also require that Λ ± < π .The bound is then obtained by minimizing (or maximizing) c ± = 12 δ (cid:90) d x φ ± (∆ + x ) = 3 π (cid:0) δ Λ ± ± (cid:1) δ Λ ± (2.8)for a given δ by varying Λ ± subject to the constraint given by the eq. (2.7), as wellas Λ ± < π . From the eq. (2.2), one can conclude [18] that c − ≤ exp [ s ( δ, ∆)] ≤ c + . (2.9)– 7 –ince for a fixed δ , c + is a monotonically decreasing function of Λ + , we deduce that c + should be minimized byΛ + = min (cid:26) π, . δ (cid:27) = (cid:40) π, δ < . , . δ , δ > . . (2.10)This explains the number 0 .
785 appearing in the bounds in the eq. (1.3). The finalbound can be obtained by combining these results with the result of [18]. A similaranalysis can be performed on c − . These procedures yield the eq. (1.3) for the upperbound, while the lower bound is given byexp [ s − ( δ, ∆)] = mz ( δ ) , √ π ≤ δ < √ π ∼ . , ( δ − π ) δ , √ π < δ < √ π ∼ . , (cid:113) π ∼ . , δ > √ π ∼ . . (2.11). δ [ s + ] MZImproved δ [ s - ] MZImproved
Figure 4 : Exp[ s ± ] : The orange line denotes the improved lower (upper) boundwhile the blue line is from [18].The lower bound can be further improved for δ > − πδ , πδ ] .φ Sphere − (∆ (cid:48) ) := 11 − (cid:0) ∆ (cid:48) − ∆ δ (cid:1) sin (cid:16) π (∆ (cid:48) − ∆) δ (cid:17) π (∆ (cid:48) − ∆) δ . (2.12)This yields c − = 0 .
5, which is an improvement over the above; see figure 5.
Lower bound and its optimality in δ → − : We can see that there is a disconti-nuity in the orange curve, as in figure 5. It turns out that for δ <
1, an improvement– 8 – .0 1.2 1.4 1.6 1.8 2.0 δ [ s - ] ImprovedFurther Improved
Figure 5 : The orange line represents the improvement on the lower bound by usingthe function φ Sphere − appearing in the sphere packing problem.is possible which makes the bounding curve continuous (see figure. [2]). The im-provement is possible by considering the following function for σ ∈ (0 ,
1) and letting (cid:15) → + and σ → − at at the end of the day: φ − (∆ (cid:48) ) = σ − (cid:0) ∆ (cid:48) − ∆ (cid:15) (cid:1) sin (cid:16) π (∆ (cid:48) − ∆) (cid:15) (cid:17) π (∆ (cid:48) − ∆) (cid:15) + (1 − σ ) cos (cid:16) π (∆ (cid:48) − ∆) (cid:15) (cid:17) − (cid:0) ∆ (cid:48) − ∆ (cid:15) (cid:1) (2.13)We consider this function in (cid:15) → + limit, which enforces the constraint on the sup-port of its Fourier transform i.e. we have Λ → (2 π ) − . Now, f ( x ) ≡ φ − (∆ + x ) isnegative for | x | > δ ( σ ), a function of σ . We can find out δ ( σ ) numerically from thefunction f by noting that δ ( σ ) is the least positive root of the equation f ( x ) = 0.This, in particular, shows that as σ → − , we have δ ( σ ) → − , the new bound c − indeed approaches 0 . . δ → − . To achievethis, we will consider Monster CFT and a sequence of energy window centered at∆ n = n and of half-width δ → − . Clearly, the window contains all the states with∆ = n and nothing else. From the estimation of [21, 22], we know that the numberof states with ∆ = n is asymptotically given by ρ ( n ). Thus we have12 δ (cid:90) n + δn − δ d∆ (cid:48) ρ (∆ (cid:48) ) (cid:39) . ρ ( n ) , for δ → − (2.14)One can check this numerically as well using Monster CFT as depicted below infigure [6]. One needs to go to really high enough ∆ to verify this. Further verificationcan be obtained by looking at the subsequence ∆ n = n and verifying against thebound, as depicted in figure [7]. – 9 – onster, δ = Δ - - - - - - ( Δ ) Monster, δ = Δ - - - - - - ( .9794, Δ ) Figure 6 : In cyan, we have s ( δ, ∆) for Monster CFT as a function of ∆, for δ =0 . δ = 0 . k . Figure 7 : In black, we have s ( δ, ∆ n = n ) for Monster CFT as a function of n for δ = 0 . δ <
1. In fact it saturates to . . . Weemphasize that the optimality is proven for δ → − only. As we approach δ = 1 frombelow, the red line approaches log[0 .
5] and the black dots asymptotes to log[0 . The purpose of this section is to elucidate a curious connection between the sphere– 10 –acking problem and the problem of finding the lower bound. We will be showingthat using the bandlimited functions, one can do better than 0 . δ ≤
1. Thekeyrole is played by the analytical property of the function φ − , coupled with theoremsand techniques appearing in the sphere packing literature. As a starter, we observethat the function in the eq. (2.12) also appears in the context of one dimensionalsphere packing problem [23]. In fact, there is an uncanny similarity between thefunctions required in the two problems, especially if we look at the requirements onthe function producing the lower bound . In the sphere packing problem, one has aFourier transform pair f, ˆ f satisfying f ( x ) ≤ | x | > , (3.1)ˆ f ( k ) ≥ . (3.2)In our case, we have x ↔ ∆ (cid:48) and k ↔ t and we require that ˆ f ( k ) has bounded sup-port. In both scenarios, the goal is to maximize ˆ f (0). In the case of sphere packing,we also normalize f (0) to one. For more details on the relevance of sphere packingto CFT and exciting new results on modular bootstrap, we refer the reader to therecent article [24].In one dimension [23], where the sphere packing problem is trivial, the relevantfunction as given in the eq. (2.12) has bounded support in the Fourier domain and ispositive . This appears to hint that if we want to further improve our bound usingbandlimited functions, we might need a function whose Fourier transform becomesnegative within the band. We have already shown in the previous section that 0 . δ → − limit. The connection with sphere packing problem helpsus to a prove a weaker statement stating that 0 . − π, π ] for δ = 1. We further show that for δ <
1, one can not do better than 0 . . SP thanks John McGreevy for pointing to [24], where sphere packing plays a pivotal role. For higher dimensions too, bandlimited functions are used (see, for example, Proposition 6.1 in[23]); nonetheless, they do not provide the tightest bound for the higher dimensional sphere packingproblem. For n = 1, the function appearing in the said proposition is related to the one that wehave used. For other values of n, we obtain bounds strictly less than 1 /
2. We thank Tom Hartmanfor pointing this out. – 11 – is an admissible function i.e. there exists a κ > and M > such that | f ( x ) | ≤ M (1 + | x | ) − − κ (3.3) Furthermore assume that ˜ f ( t ) is in L and has a bounded support, which is a subsetof [ − π, π ] . Furthermore, we have f ( x ) ≤ for | x | ≥ . One can then show that ˜ f (0) ≤ f (0) . (3.4)Now we note that the function φ − (∆ (cid:48) ), hence, f ( x ) = φ (∆ + x ) appearing in theTauberian analysis is in L . This follows from the fact that ˜ f ( t ) is in L ∞ , implyingthat it is in L because it has a bounded support. The finite value of ˜ f (0) implies that (cid:82) d x f ( x ) is finite. Now since f ( x ) ≤ Θ([ − , (cid:82) d x | f ( x ) | is finite.This implies that f is in L . In particular, we would impose the following technicalassumption on f that it is an admissible function. This automatically guaranteesthat f is in L . Thus we are justified to apply the theorem in our context and deduce˜ f (0) ≤ f (0) ≤ . (3.5)Thus the maximum possible value of c − = ˜ f (0) is 0 .
5. Suppose, we have a function f such that f ( x ) ≤ | x | > δ . If δ ≤ f belongs to the class of the function, con-sidered in the theorem. Thus we have shown that for δ ≤
1, ˜ f (0) ≤ c − ≤ . δ = 1.Before moving on to the discussion of the bound on bounds, we pause to remarkthat the following class of functions parameterized by α can not be used to improvethe bound from above: φ ( α )+ (∆ (cid:48) ) = sin (cid:16) Λ + δα (cid:17) Λ + δα − α sin (cid:16) Λ + (∆ (cid:48) − ∆) α (cid:17) Λ + (∆ (cid:48) − ∆) α α , α ≥ . (3.6)Within this class of functions, α = 4 gives the tightest bound as found in [18]. In this section, we provide a systematic algorithm to estimate how tight the boundscan be made using bandlimited functions φ ± . This provides us with a quantitativeestimate of the limitation of the procedure which produces these bounds on the O (1)correction to the Cardy formula. If one drops the requirement that the function bebandlimited, one might hope to do better. For the rest of this section, we will restrict– 12 –urselves to bandlimited functions only.We recall that the functions φ ± are chosen in such a way that they satisfy φ − (∆ (cid:48) ) < Θ (∆ (cid:48) ∈ [∆ − δ, ∆ + δ ]) < φ + (∆ (cid:48) ) . (4.1)This inequality gives a trivial bound on c ± : c − ≤ ≤ c + . (4.2)In what follows, we make this inequality tighter. In this context, the followingcharacterization of the Fourier transform of a positive function in terms of a positivedefinite function turns out to be extremely useful. Before delving into the proof, letus define the notion of positive definiteness of a function. Unless otherwise specified,here we will be dealing with functions from the real line to the complex plane. Afunction f ( t ) is said to be positive definite if for every positive integer n and for everyset of distinct points t , . . . , t n chosen from the real line, the n × n matrix A definedby A ij = f ( t i − t j ) (4.3)is positive definite. A function g (∆) is said to be positive if g (∆) > .Now, let us explore how this characterization can improve the eq. (4.2). Withoutloss of generality, we set ∆ = 0 henceforth, and define g ± (∆ (cid:48) ) = ± [ φ ± (∆ (cid:48) ) − Θ (∆ (cid:48) ∈ [ − δ, δ ])] . (4.4)At this point we use the fact that φ ± is a bandlimited function, i.e., it has a boundedsupport [ − Λ ± , Λ ± ] , and that Λ ± < π . This requirement stems from the procedurefollowed in [18]. Thus we arrive at the following: (cid:101) g ± (0) = ± δ ( c ± − , (4.5) (cid:101) g ± ( t ) = ∓ δ (cid:18) sin( tδ ) tδ (cid:19) for | t | ≥ π. (4.6)The eq. (4.2) states that ˜ g (0) / δ >
0. In order to improve this, we construct 2 × t > π : G (2) ± = (cid:20) (cid:101) g ± (0) (cid:101) g ± ( t ) (cid:101) g ± ( t ) (cid:101) g ± (0) (cid:21) . (4.7) The proof is given in a box separately at the end of this subsection for those who are interested. – 13 –or a fixed δ , we consider the first positive peak of ˜ g ± outside t > π . If this occursat t = t ( δ ), we choose t = t ( δ ). Subsequently, the positive definiteness of the matrix G (2) ± boils down to the inequality (cid:101) g ± (0) > (cid:101) g ± ( t ( δ )) , (4.8)where t ( δ ) is the first positive peak of (cid:101) g ± outside t > π . For example, we can showthat (see the green lines in Fig. 8): MZ Improved Lower bound on Upper bound δ [ s + ] MZ Improved upper bound on Lower bound δ [ s - ] Figure 8 : Exp[ s ± ] : The green line is the analytical lower and upper bound on upperand lower bound i.e. c ± respectively. The green shaded region is not achievable byany bandlimited function. c + > . , δ < . , . , . > δ > . , . , . > δ > . , (4.9) c − < (cid:40) . , δ < . , . , . > δ > . . (4.10) Positive function ⇔ Positive definite Function:
Fourier transformWe will show that the Fourier transform of an even and positive function is apositive definite function. Consider a function g (∆) and let us define the Fouriertransform as (cid:101) g ( t ) = (cid:90) ∞−∞ dt g (∆) e − ı ∆ t = 2 (cid:90) ∞ dt cos(∆ t ) g (∆) . (4.11)– 14 –ow, we construct the matrix G ij = g ( t i − t j ) = 2 (cid:90) ∞ dt cos [∆( t i − t j )] g (∆) . (4.12)In order to show that G is a positive definite matrix, i.e., (cid:80) ij v i v j G ij > v i ∈ R such that (cid:80) i v i (cid:54) = 0, we think of an auxiliary 2 dimensional spacewith n vectors (cid:126)v ( i ) , (for clarity, we remark that i labels the vector itself, not itscomponent) such that we have (cid:126)v ( i ) ≡ ( | v i | cos(∆ t i ) , | v i | sin(∆ t i )) . (4.13)Thus, we have (cid:88) ij v i v j G ij = 2 (cid:90) ∞ dt (cid:32)(cid:88) ij v i v j cos [∆( t i − t j )] (cid:33) g (∆) (4.14)= 2 (cid:90) ∞ dt (cid:16) (cid:126)V · (cid:126)V (cid:17) g (∆) > t , . . . , t n are distinct. Here, (cid:126)V is given by (cid:126)V = (cid:88) i sign( v i ) (cid:126)v ( i ) . (4.16)This completes the proof that the Fourier transform of an even positive functionis a positive definite function. First of all, it is easy to see that c ± , and hence theinequality, is insensitive to the midpoint of the interval, i.e., ∆, so we set it to 0and this makes the functions φ ± and Θ even. In particular, we will be applyingthis theorem to φ + (∆ (cid:48) ) − Θ (∆ (cid:48) ∈ [∆ − δ, ∆ + δ ]) and Θ (∆ (cid:48) ∈ [∆ − δ, ∆ + δ ]) − φ − (∆ (cid:48) ). We make one more remark before exploring the consequences of this.The above result is true for any function, not necessarily even. The converse isalso true due to Bochner’s Theorem , but in what follows, we do not require theconverse statement.
Matlab implementation
We implement the above argument using more than two points and making surethat | t i − t j | ≥ π . For a fixed δ , we use a random number generator to sample thepoints t i with the mentioned constraint. We do this multiple times and each time,we test the positive definiteness of the matrix G by providing as an input the valueof ± ( c ± − ± ( c ± −
1) is chosen to be from the first peak t ( δ ) tillsome value larger than the achievable bound given in (1.3) and (2.11). This in turn– 15 –ields a lower bound (or upper bound) for c ± for each trial . Subsequently, we pickout the best possible bound among all the trials. For example, we provide a table [1]showing the outputs from a typical run for improving the bound on the upper bound.The tables [1] and [2] improve the lower (upper) bound for c ± and this is shown inthe figure [1], where the brown dots are the stronger bounds over the green lines anddisallow a larger region. δ Number of iterations c + ) Lower Bound0.4 10000 300 2.2 1.70420.5 1000 300 2.02 1.69050.5 10000 200 2.02 1.70020.5 10000 300 2.02 1.71790.6 1000 200 2.02 1.60860.6 10000 200 2.02 1.59170.7 10000 200 2.02 1.42460.7 10000 250 2.02 1.42700.8 10000 200 1.757 1.36920.8 10000 200 2.757 1.36980.9 10000 200 2.757 1.37981 20000 200 1.757 1.37591.1 10000 200 2.757 1.33311.20 10000 150 2.757 1.25971.25 10000 150 2.757 1.25811.3 10000 170 2.757 1.25311.4 10000 150 2.757 1.25811.5 10000 150 1.757 1.25991.5 10000 150 2.757 1.25971.5 10000 150 2.757 1.25971.6 10000 150 1.757 1.23131.7 10000 150 1.757 1.1933 Table 1 : Typical output from a run yielding lower bounds for the upper bound c + .The Max( c + ) column contains a number that is greater than or equal to what canalready be achieved. In this section, we switch gear and explore the asymptotic spectral gap. In [18], ithas recently been shown that the asymptotic gap between Virasoro primaries arebounded above by 2 (cid:113) π (cid:39) . We assume that the mesh size for c ± − – 16 – Iteration Number c − ) Upper Bound0.6 1000 200 0.173 0.57380.6 10000 200 0.173 0.55350.7 10000 200 0.362 0.56040.7 10000 250 0.362 0.55590.8 10000 200 0.44 0.55670.9 10000 200 0.46 0.58531 10000 200 0.48 0.69601.1 10000 200 0.49 0.71121.2 10000 150 0.49 0.71611.2 10000 180 0.49 0.71611.3 10000 170 0.49 0.71111.4 10000 150 0.49 0.72431.5 10000 150 0.49 0.77881.6 20000 150 0.49 0.78951.7 20000 150 0.49 0.7861 Table 2 : Typical output from a run providing upper bound for the lower bound c − .The Min( c − ) column contains a number that is smaller than or equal to what canalready be achieved.should be 1. The example of Monster CFT tells us that the gap can not be belowthan 1, hence 1 should be the optimal number. In this section, we show that theprevious bound 2 (cid:113) π can be improved and made arbitrarily closer to the optimalvalue 1. The idea of the gap comes from binning the states and putting a positivelower bound on that. If the width of the bin is very small, one finds that the lowerbound on that number of states become negative. This indicates that if the binwidth is very small, we might land up with no states in the bin. Thus we need tofind a minimum bin width which would still allow us to prove a positive lower bound.Ideally, to prove this one should find out a function f (which will eventually play therole of φ − in this game, to be precise f (∆ (cid:48) ) = φ − (∆ + ∆ (cid:48) )) such that following holds: f (∆ (cid:48) ) ≤ Θ (cid:16) ∆ (cid:48) ∈ (cid:104) − (cid:15) , (cid:15) (cid:105)(cid:17) (5.1)and ˜ f ( t ) = 0 for | t | ≥ π(cid:15) , (cid:15) > f (0) > (cid:90) ∆+ δ ∆ − δ d ∆ (cid:48) ρ (∆ (cid:48) ) > f (0) = 0 ? One needs to go back to the original derivationand reconsider it carefully. Hence instead of the eq. (2.2), we consider a more basicinequality[18]:exp [ β (∆ − δ )] (cid:90) d ∆ (cid:48) ρ (∆ (cid:48) ) e − β ∆ (cid:48) φ − (∆ (cid:48) ) − Z H (cid:18) π ββ + Λ − (cid:19) e − β c (cid:90) Λ − − Λ − dt | ˆ φ ( t ) |≤ (cid:90) ∆+ δ ∆ − δ d ∆ (cid:48) ρ (∆ (cid:48) ) (5.5)where Λ − = π(cid:15) and Z H ( β ) is the contribution from the heavy states and defined as Z H ( β ) = (cid:88) ∆ > ∆ H > c e − β ( ∆ − c ) . (5.6)Now we make the following choice for φ − : φ − (∆ (cid:48) ) = cos (cid:16) π (∆ (cid:48) − ∆) (cid:15) (cid:17) − (cid:0) ∆ (cid:48) − ∆ (cid:15) (cid:1) , f (∆ (cid:48) ) = cos (cid:0) π ∆ (cid:48) (cid:15) (cid:1) − (cid:0) ∆ (cid:48) (cid:15) (cid:1) (5.7)This function f has following properties: f (∆ (cid:48) ) ≤ Θ (cid:16) ∆ (cid:48) ∈ (cid:104) − (cid:15) , (cid:15) (cid:105)(cid:17) (5.8)˜ f ( t ) = 0 for | t | ≥ π(cid:15) (5.9)˜ f (0) = 0 ⇒ c − = 0 (5.10)Since c − = 0, one can not readily evaluate the integral appearing in (5.5) by saddlepoint method and deduce exp [ β (∆ − δ )] (cid:82) d ∆ (cid:48) ρ (∆ (cid:48) ) e − β ∆ (cid:48) φ − (∆ (cid:48) ) = c − ρ (∆), so welook for subleading corrections to the saddle point approximation. We find that theleading behavior is given by, after setting β = π (cid:112) c ,exp [ β (∆ − δ )] (cid:90) d ∆ (cid:48) ρ (∆ (cid:48) ) e − β ∆ (cid:48) φ − (∆ (cid:48) ) = Cρ (∆) , (5.11)– 18 –here C turns out to be C = (cid:90) ∞ dx (cid:32) cos (cid:0) π x(cid:15) (cid:1) − x (cid:15) (cid:33) exp (cid:34) − x π (cid:112) c ∆ (cid:35) . (5.12)We remark that C > ρ (∆) − ( − (cid:15) ). The analysis forthis second term is exactly same as done in [18]. For sufficiently large ∆, it can benumerically verified that ρ (∆) − ( − (cid:15) ) is subleading compared to Cρ (∆) as longas (cid:15) > ρ (∆) = ∆ →∞ (cid:16) c (cid:17) exp (cid:34) π (cid:114) c ∆3 (cid:35) . (5.13)One can analytically show that ρ (∆) − ( − (cid:15) ) is subleading to Cρ (∆) for large∆. One way to show this is to have an estimate for C . We start with the observationthat the integrand is positive in (cid:0) , (cid:15) (cid:1) and negative in (cid:0) (cid:15) , ∞ (cid:1) . Furthermore, we have (cid:90) ∞ d ∆ (cid:48) f (∆ (cid:48) ) = 0 (5.14)Using the above facts, one can always choose 0 < (cid:15) < (cid:15) and (cid:15) < (cid:15) < ∞ such that (cid:90) (cid:15) d ∆ (cid:48) f (∆ (cid:48) ) = − (cid:90) ∞ (cid:15) d ∆ (cid:48) f (∆ (cid:48) ) (5.15) (cid:90) (cid:15) (cid:15) d ∆ (cid:48) f (∆ (cid:48) ) = 0 (5.16)This is basically guaranteed by the continuity. We choose (cid:15) such that 0 < (cid:15) < (cid:15) and consider the function F ( y ) = (cid:82) y(cid:15) dx f ( x ). Now F ( y ) is a continuous function. Itis positive when y = (cid:15) and negative when y → ∞ . Thus by continuity, there exists (cid:15) < (cid:15) < ∞ such that the eq. (5.15) holds. The shaded region in the figure. 9 is thearea under the function f restricted to the interval [ (cid:15) , (cid:15) ] so that the eq. (5.15) issatisfied.Now we note that (cid:90) (cid:15) (cid:15) dx f ( x ) exp (cid:34) − x π (cid:112) c ∆ (cid:35) ≥ unction for Spectral Gap - - ( x ) Figure 9 : The function (cid:18) cos ( π x(cid:15) ) − x (cid:15) (cid:19) , the shaded region is the area under the functionrestricted to the interval [ (cid:15) , (cid:15) ]. Here (cid:15) = 0 . , (cid:15) = 1 . , (cid:15) = 0 . (cid:90) (cid:15) dx f ( x ) exp (cid:34) − x π (cid:112) c ∆ (cid:35) ≥ exp (cid:34) − (cid:15) π (cid:112) c ∆ (cid:35) (cid:90) (cid:15) dx f ( x ) (5.18) (cid:90) ∞ (cid:15) dx f ( x ) exp (cid:34) − x π (cid:112) c ∆ (cid:35) ≥ exp (cid:34) − (cid:15) π (cid:112) c ∆ (cid:35) (cid:90) ∞ (cid:15) dx f ( x ) (5.19)where in the second inequality, we have used negativity of f ( x ) for x > (cid:15) . Combiningthe last four equations i.e (5.15),(5.17),(5.18),(5.19) we can write C ≥ Ω (cid:32) exp (cid:34) − (cid:15) π (cid:112) c ∆ (cid:35) − exp (cid:34) − (cid:15) π (cid:112) c ∆ (cid:35)(cid:33) (cid:39) ∆ →∞ ( (cid:15) − (cid:15) ) Ω2 π (cid:112) c ∆ > (cid:82) (cid:15) dx f ( x ) > (cid:15) >
1, we can neglect the second piece i.e. contributions from the heavystates due to its subleading nature. In fact, one can do much better and show that We thank Alexander Zhiboedov for pointing this out in an email exchange. – 20 – falls like ∆ − / by noting the following: C = (cid:15)π (cid:34) − π (cid:112) c ∆ / (cid:35) Erfi (cid:113) π (cid:112) c ∆ / − (cid:15)π e − √ π √ c ∆3 / Im Erf (cid:112) π (cid:16) π + i √ π √ c ∆ / (cid:17) √ (cid:113) √ c ∆ / (cid:39) ∆ →∞ (cid:15) (cid:18) c (cid:19) / ∆ − / . (5.21)To summarize, we have proved that for sufficiently large ∆, (cid:90) ∆+ (cid:15) ∆ − (cid:15) d ∆ (cid:48) ρ (∆ (cid:48) ) ≥ Cρ (∆) > (cid:15) , where (cid:15) >
1. Now one can choose (cid:15) to be arbitrarily close to 1, which proves that the optimal bound is exactly 1. Theanalysis can be carried over to the case for Virasoro primaries, as pointed out in [18].This implies that the asymptotic gap between two consecutive Virasoro primaries isbounded above by 1, thereby proves the conjecture made in [18].
Alternate proof:
This proof starts from considering the following function for σ ∈ (0 , φ − (∆ (cid:48) ) = σ − (cid:0) ∆ (cid:48) − ∆ (cid:15) (cid:1) sin (cid:16) π (∆ (cid:48) − ∆) (cid:15) (cid:17) π (∆ (cid:48) − ∆) (cid:15) + (1 − σ ) cos (cid:16) π (∆ (cid:48) − ∆) (cid:15) (cid:17) − (cid:0) ∆ (cid:48) − ∆ (cid:15) (cid:1) (5.23)It can be verified that for (cid:15) → + and σ → + , c + approaches to 0 from thepositive side and the max { x : φ − (∆ + x ) > } approaches to 0 . x < . (cid:15) = 1, c − , as defined in eq. (2.4) approaches to 0 . σ → − , starting from c − = 0 for σ = 0, as shown in figure [10]. This observation regarding the behavior of c − around δ = 1 is related to what is depicted in figure 2. In this work, we have improved the existing upper and lower bound on the O (1)correction to the density of states in 2D CFT at high energy. Since one can not the-oretically deduce a universal correction suppressed compared to O (1) correction, anyimprovement in the bound on O (1) correction is meaningful. Furthermore, we haveshown the optimality of the lower bound in the limit δ → − . We have also proven– 21 – .5 0.6 0.7 0.8 0.9 1.0 δ c - Figure 10 : Exp[ s ± ] ≡ c − as a function of δ , the half-width of the energy window,obtained from the function defined in eq. (5.23).the conjectured upper bound on the gap between Virasoro primaries. In particular,we have shown that there always exists a Virasoro primary in the energy window ofwidth greater than 1 at large ∆. This is a quantitative universal measure of/boundon how sparse a CFT spectrum can be asymptotically. In particular, this rules outthe possibility of having CFTs with hadamard like gap. We have also unraveled acurious connection between the sphere packing problem and the problem of find-ing the lower bound using bandlimited functions. This puts an upper bound on thelower bound for δ ≤ δ = 1, the bestbound is achievable and in fact the optimal one owing to saturation by Monster CFT.We have provided a systematic way to estimate how tight the bound can be madeusing bandlimited functions. Since there is still a gap between the achievable boundand the bound on the bound, there is scope for further improvement (except in thelimit δ → − , where we have shown optimality). Ideally, one would like to close thisgap, which might be possible either by sampling more points and leveraging the pos-itive definiteness condition on a bigger matrix, or by choosing some suitable functionwhich would make the achievable bound closer to the bound on the bound. Anotherpossible way to obtain the bound on bound is to use a known 2D CFT partitionfunctions, for example 2D Ising model and explicitly evaluate s ( δ, ∆). It would beinteresting to see how the bound on bound obtained in this paper compares to theone which can be obtained from the 2D Ising model. For example, one can verifythat the bound on bound obtained here is stronger than that could be obtained from2D Ising model for δ = 1. In fact, this happens to be case for different other values We thank Alexander Zhiboedov for raising this question of how our bound compares to s (1 . , ∆)for the 2D Ising model, as found in [18]. – 22 –f δ as well. It would be interesting to explore this further analytically.The utility of the technique developed here lies beyond the O (1) correction tothe Cardy formula. We expect the technique to be useful whenever one wants toleverage the complex Tauberian theorems, for example in [20, 26, 27]. As emphasizedin [18], the importance of Tauberian theorems lies beyond the discussion of 2D CFTpartition functions, especially in investigating Eigenstate Thermalization Hypothesis[28–31] in 2D CFTs[32–43]. It worths mentioning that the use of extremal functionalsprovided us with sharper inequalities in CFT [24, 44–46]. One can hope to blend theTauberian techniques with techniques involving extremal functionals to investigatethe landscape of 2D CFT more. We end with a cautious remark that if we relaxthe condition of using bandlimited functions, the bound on bounds would not beapplicable and it might be possible to obtain nicer achievable bounds on the O (1)correction to the Cardy formula. Nonetheless, we emphasize that the lower boundfor δ → − is indeed optimal, thus not restricting to bandlimited functions wouldnot provide us with anything more. Acknowledgements
The authors thank Denny’s for staying open throughout the night. The authors ac-knowledge helpful comments and suggestions from Tom Hartman, Baur Mukhamet-zhanov, and especially, Alexander Zhiboedov. The authors thank Shouvik Datta forencouragement. SG wishes to acknowledge Pinar Sen for some fruitful and illumi-nating discussions on what makes the Fourier transform of a function positive, whichhelped him arrive at the answer by thinking about autocorrelation functions andpower spectral densities. SP acknowledges a debt of gratitude towards Ken Intrili-gator and John McGreevy for fruitful discussions and encouragement. SP thanksShouvik Datta and Diptarka Das for introducing him to the rich literature of CFTin 2017. This work was in part supported by the US Department of Energy (DOE)under cooperative research agreement DE-SC0009919 and Simons Foundation award
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