Conformal bootstrap at large charge Daniel L. Jafferis Harvard University 20 Years Later: The Many Faces of AdS/CFT Princeton Nov 3, 2017 DLJ, Baur Mukhametzhanov, Sasha Zhiboedov
Exploring heavy operators Macroscopic limits and effective field theory Conformal blocks and crossing in the large charge limit Results and future directions 2
What do we know about general CFTs? Beyond weak coupling, abstract self-consistency relations like crossing and unitarity are our best methods of characterizing the space of all CFTs. For low dimension operators, the numerical bootstrap has been used to great effect. There are also powerful analytic techniques at fixed twist. 3
Generic heavy operators Correspond to high energy states on the cylinder, which must have exponentially large degeneracy. The asymptotic density of states is governed by thermal physics. They are dual to black hole microstates. The leading correlation functions are given by the ETH, which encodes semi-classical expectation values. [Lashkari Dymarsky Liu] It would be very interesting to understand finer details. One expects a relation to hydrodynamics. 4
UV constraints on bulk EFT In addition to giving complementary information to low dimension bootstrap constraints, finding constraints from the high dimension spectrum is equivalent to finding UV constraints on EFTs in AdS. This is important in going beyond generalized free field theory + perturbations, which automatically satisfies crossing among low dimension operators. It s hard because the number of operators appearing in the OPE channels scales as e S. 5
Large charge limit A simpler situation the lightest heavy operator with a given large charge. May expect a superfluid like state on the cylinder, with a finite gap above the vacuum with fixed charge. [Hellerman Orlando Reffert Watanabe] Today I ll present the first steps to analyzing this situation using the bootstrap. 6
Goldstone EFT The sector with a fixed large charge spontaneously breaks the global symmetry, as well as dilations (time translations on the cylinder). However, a linear combination must be preserved, since the state is an eigenstate. Nonrelativistic Goldstone theory of the broken symmetry should describe the fluctuations. The coupling is the ratio of the UV scale set by the charge density and the IR scale set by the radius. [Hellerman Orlando Reffert Watanabe, Monin Pirtskhalava Rattazzi Seibold] 7
Macroscopic limits In general, it is interesting to consider the flat space limit obtained by bringing the light operators together while scaling the dimension of the heavy state to infinity, such that correlation functions remain finite. In the EFT case, this results in finite energy and charge densities. The EFT is a standard Goldstone theory. For free scalars, and more generally, when there is a moduli space of vacua, the scaling is different, and the charge and energy densities vanish. [Hellerman Maeda Watanabe] 8
Gravity dual The gravity dual of the EFT is superconductor in AdS order Q light charged particles, held apart by electrostatic repulsion and contained in the gravity trap of AdS. The extremal Reissner-Nordstrom black hole has been pointed out to have the same scaling of dimension versus charge. It would have an exponentially dense spectrum. Which has lower energy depends on the charge/mass ratio of the particle. The existence of a gap is closely related to the weak gravity conjecture. 9
Explore using the bootstrap In abstract CFTs, how general is the large charge EFT? To this end, one should study n+2 point functions of n light operators and 2 heavy operators. Examine the constraints of crossing in such a four point function. The hope is that universal features will emerge in controlled 1/Q expansions. Under some assumptions, matching of the two HL channels, together with some input from a LL scaling regime, the Goldstone EFT will emerge as the minimal solution. 10
EFT results Expanding the EFT action around the saddle, one finds the dimension of the lightest charged operator. At leading order, the spectrum looks like Goldstone Fock space excitations. The sound speed at half the speed of light is dictated by conformal invariance. 11
Correlation functions The correlators of light charged operators in the heavy state are given in terms of their overlap with the Goldstone field The three and four point functions are then Here D is the Green s function for the free Goldstone. 12
Large Q limit In the large Q limit, the general expression for the s- and u- blocks in terms of power series and Gegenbauer polynomials, where coefficients are given by the Casimir equation: simplify. Descendants suppressed by powers of Q We want to approximate the spectrum by the lightest large Q operators and finite energies on top of it. 13
Crossing at large Q It is easy to see that at leading order, crossing is satisfied by a single operator. In the OPE limit, one always has But at large Q, descendants are suppressed, so This is just continuity of and. Would be spoiled if there were multiple operators. 14
The bad news There is no overlapping region of convergence of the large charge limits of s- and u-channel conformal block expansions. This is because the large charge limit of each block is dominated by the primary operator in the z frame, and descendants are suppressed. Moreover, the excitations over the large charge vacuum only dominate the 4 point function within the z unit disk. The t-channel is dominated by unknown heavy operators in the domain of validity of the above expansions. [Pappadopulo Rychkov Espin Rattazzi] still valid for exact answer 15
The good news Crossing can be satisfied by a finite number of Regge trajectories at the first nontrivial order! At leading order, recall that it was satisfied by a single operator. A finite number of Regge trajectories is analogous to having a finite number of fields in the EFT. Under this assumption, smoothly matching the s- and u- channels at their common boundary of convergence, and input from the existence of the macroscopic limit, leads to a solvable bootstrap problem. 16
Checking crossing in EFT This has to hold in the effective field theory. What remains is to expand HHLL into conformal blocks. The Goldstone Green s function can be expanded Therefore one has It actually is analytic away from z=1. This requires an interplay between the correction to vacuum dimension and the single Goldstone Regge trajectory. 17
Domains of validity 18
Large charge bootstrap conditions The four point function is smooth across the unit circle, away from z=1. Matching all derivatives gives infinitely many conditions. More carefully, integrate the discontinuity in the n th derivative with a Legendre. This gives an unknown but controllable piece from the macroscopic limit region, appearing in infinitely many equations. 19
Input from the macroscopic limit The existence of the macroscopic limit implies that is finite. This is the scaling which results in a finite charge density. The prefactor is due to the mapping between plane and cylinder. Thus f can grow to be at most, and thus is homogeneous of degree minus 1 for small 20
Unique solution with a single Regge trajectory The constraints can then be written as where fixed degree in are some polynomials of. The coeffs are variables. With only a single operator of each J, it follows that, thus. The J=1 operator is already present, it is a descendant of the lightest operator. This is the EFT result! 21
New solutions Analyticity in spin for spin at least 2 allows a systematic treatment with an arbitrary number, N, of Regge trajectories. The solution to the smoothness conditions are that the energies are roots of a polynomial of degree N. Some solutions look like the addition of new fields, other lack a quasi-particle interpretation. 22
Summary Assuming the smoothness of the spectrum as a function of the large charge, existence of the macroscopic limit with finite charge density, and a finite gap in the spectrum at large charge: We classified solutions to crossing at the first nontrivial order with finite numbers of Regge trajectories. The minimal solution is exactly the Goldstone EFT. 23
Future directions Develop the bootstrap analysis to higher orders which new solutions survive? Reproduce the universal EFT quantum corrections. Explore or rule out situations with vanishing gap at large Q. Explore cases with moduli spaces of vacua. What can one learn for generic heavy operators described by hydrodynamics? Are there any constraints on low dimension operators from consistency of high dimension ones? 24