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arXiv:1801.04208 · Physical review. D/Physical review. D.

Minkowski conformal blocks and the Regge limit for Sachdev-Ye-Kitaev-like models

Timothy G. Raben, Chung-I Tan

Published Oct 8, 2018• 11 citations
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Abstract

We discuss scattering in a conformal field theory (CFT) via the conformal partial-wave analysis and the Regge limit. The focus of this paper is on understanding an operator product expansion (OPE) with Minkowski conformal blocks. Starting with a t-channel OPE, it leads to an expansion for an s-channel scattering amplitude in terms of t-channel exchanges. By contrasting with Euclidean conformal blocks we see a precise relationship between conformal blocks in the two limits without preforming an explicit analytic continuation. We discuss a generic feature for a CFT correlation function having singular ${F}^{(M)}(u,v)\ensuremath{\sim}{u}^{\ensuremath{-}\ensuremath{\delta}}$, $\ensuremath{\delta}>0$, in the limit $u\ensuremath{\rightarrow}0$ and $v\ensuremath{\rightarrow}1$. Here, $\ensuremath{\delta}=({\ensuremath{\ell}}_{\mathrm{eff}}\ensuremath{-}1)/2$, with ${\ensuremath{\ell}}_{\mathrm{eff}}$ serving as an effective spin and it can be determined through an OPE. In particular, it is bounded from above, ${\ensuremath{\ell}}_{\mathrm{eff}}\ensuremath{\le}2$, for all CFTs with a gravity dual, and it can be associated with string modes interpolating the graviton in AdS. This singularity is historically referred to as the Pomeron. This bound is nearly saturated by SYK-like effective $d=1$ CFT, and its stringy and thermal corrections have piqued current interests. Our analysis has been facilitated by dealing with Wightman functions. We provide a direct treatment in diagonalizing dynamical equations via harmonic analysis over physical scattering regions. As an example these methods are applied to the SYK model.

DOI
10.1103/physrevd.98.086009

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We discuss scattering in a conformal field theory (CFT) via the conformal partial-wave analysis and the Regge limit. The focus of this paper is on understanding an operator product expansion (OPE) with Minkowski conformal blocks. Starting with a t-channel OPE, it leads to an expansion for an s-channel scattering amplitude in terms of t-channel exchanges. By contrasting with Euclidean conformal blocks we see a precise relationship between conformal blocks in the two limits without preforming an explicit analytic continuation. We discuss a generic feature for a CFT correlation function having singular F(M)(u,v)\ensuremath∼u\ensuremath−\ensuremathδ{F}^{(M)}(u,v)\ensuremath{\sim}{u}^{\ensuremath{-}\ensuremath{\delta}}F(M)(u,v)\ensuremath∼u\ensuremath−\ensuremathδ, \ensuremathδ>0\ensuremath{\delta}>0\ensuremathδ>0, in the limit u\ensuremath→0u\ensuremath{\rightarrow}0u\ensuremath→0 and v\ensuremath→1v\ensuremath{\rightarrow}1v\ensuremath→1. Here, \ensuremathδ=(\ensuremathℓeff\ensuremath−1)/2\ensuremath{\delta}=({\ensuremath{\ell}}_{\mathrm{eff}}\ensuremath{-}1)/2\ensuremathδ=(\ensuremathℓeff​\ensuremath−1)/2, with \ensuremathℓeff{\ensuremath{\ell}}_{\mathrm{eff}}\ensuremathℓeff​ serving as an effective spin and it can be determined through an OPE. In particular, it is bounded from above, \ensuremathℓeff\ensuremath≤2{\ensuremath{\ell}}_{\mathrm{eff}}\ensuremath{\le}2\ensuremathℓeff​\ensuremath≤2, for all CFTs with a gravity dual, and it can be associated with string modes interpolating the graviton in AdS. This singularity is historically referred to as the Pomeron. This bound is nearly saturated by SYK-like effective d=1d=1d=1 CFT, and its stringy and thermal corrections have piqued current interests. Our analysis has been facilitated by dealing with Wightman functions. We provide a direct treatment in diagonalizing dynamical equations via harmonic analysis over physical scattering regions. As an example these methods are applied to the SYK model.
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We analyze the operator spectrum of Conformal Field Theories at large conformal dimensions Delta\\DeltaDelta with and without global symmetries. General constraints of crossing symmetry and unitarity allow us to extract a number of universal properties of heavy operator spectrum perturbatively in 1overDelta1\\over \\Delta1overDelta.\nFirst, we consider four-point correlators and solve crossing equations in the deep Euclidean regime. Large scaling dimension Delta\\DeltaDelta tails of the weighted spectral density of primary operators of given spin in one channel are matched to the Euclidean OPE data in the other channel. Subleading 1overDelta1\\over \\Delta1overDelta tails are systematically captured by including more operators in the Euclidean OPE in the dual channel. We use dispersion relations for conformal partial waves in the complex Delta\\DeltaDelta plane, the Lorentzian inversion formula and complex tauberian theorems to derive this result. We make predictions for the 3d Ising model.\nSecond, we apply the methods of tauberian theory to modular invariance in 2d CFTs. We derive lower and upper bounds on the number of operators within a given energy interval. At high energies we rigorously derive the Cardy formula for the microcanonical entropy together with optimal error estimates for various widths of the averaging energy shell. We identify a new universal contribution to the microcanonical entropy controlled by the central charge and the width of the shell. We derive an upper bound on the spacings between Virasoro primaries. Analogous results are obtained in holographic 2d CFTs. \nThird, we consider unitary CFTs with continuous global symmetries in d>2d>2d>2. We consider a state created by the lightest operator of large charge Qgg1Q \\gg 1Qgg1 and analyze the correlator of two light charged operators in this state. Assuming the correlator admits a well-defined large QQQ expansion and, relatedly, that the macroscopic (thermodynamic) limit of the correlator exists, we find that the crossing equations admit a consistent truncation, where only a finite number NNN of Regge trajectories contribute at leading nontrivial order. We classify all such solutions to the crossing. For one Regge trajectory N=1N=1N=1, the solution is unique and given by the effective field theory of a Goldstone mode.
In this thesis we apply techniques from quantum information theory to study quantum gravity within the framework of the anti-de Sitter / conformal field theory correspondence (AdS/CFT). Through AdS/CFT, progress has been made in understanding the structure of entanglement in quantum field theories, and in how gravitational physics can emerge from these structures. However, this understanding is far from complete and will require the development of new tools to quantify correlations in CFT. This thesis presents refinements of a duality between operator product expansion (OPE) blocks in the CFT, giving the contribution of a conformal family to the OPE, and geodesic integrated fields in AdS which are diffeomorphism invariant quantities. This duality was originally discovered in the maximally symmetric setting of pure AdS dual to the CFT ground state. In less symmetric states the duality must be modified. Working with excited states within AdS3_33​/CFT2_22​, this thesis shows how the OPE block decomposes into more fine-grained CFT observables that are dual to AdS fields integrated over non-minimal geodesics. Additionally, this thesis contains results on the dynamics of entanglement measures for general quantum systems. Results are presented for the family of quantum Rényi entropies and entanglement negativity. Rényi entropies are studied for general dynamics by imposing special initial conditions. Around pure, separable initial states, all Rényi entropies grow with the same timescale at leading, and next-to-leading order. Mathematical tools are developed for the differentiation of non-analytic matrix functions with respect to constrained arguments and are used to construct analytic expressions for derivatives of negativity. We establish bounds on the rate of change of state distinguishability and the rate of entanglement growth for closed systems. Note: Abstract shortened.
A bstract In this work we address partial wave decompositions of thermal one-point functions in conformal field theories on S 1 × S d− 1 . With the help of Casimir differential equations we develop efficient algorithms to compute the relevant conformal blocks for an external field of arbitrary spin and with any spin exchange along the thermal circle, at least in three dimensions. This is achieved by identifying solutions to the Casimir equations with a special class of spherical functions in the harmonic analysis of the conformal group. The resulting blocks are then applied to study the decomposition of one-point functions of the scalar ϕ 2 and the stress tensor T for a three-dimensional free scalar field ϕ . We are able to read off averaged OPE coefficients into exchanged fields of high weight and spin for a complete set of tensor structures. We also extract an asymptotic behaviour of conformal blocks and use it to analyse the density of heavy-heavy-light OPE coefficients for spinning operators, comparing it with semi-classical predictions, such as the dimensions of operators at large charge.
Given a critical quantum spin chain described by a conformal field theory (CFT) at long distances, it is crucial to understand the universal conformal data. One most important ingredient is the operator product expansion (OPE) coefficients, which describe how operators fuse into each other. It has been proposed in [Zou and Vidal, Phys. Rev. B 105, 125125 (2022)] that the OPE coefficients can be computed from overlaps of low-energy wave functions of the spin chain. In this paper, we establish that all conformal data including central charge, conformal dimensions, and OPE coefficients are encoded in the wave-function overlaps, with universal finite-size corrections that depend on the operator content of the cyclic orbifold CFT. Thus this method allows us to numerically compute all the conformal data based solely on the low-energy eigenstates. The predictions are verified in the Ising and XXZ model. As an application, we study the recently proposed Haagerup model built from the Haagerup fusion category. We find that the CFT has central charge $c\ensuremath{\approx}2.1$ and the lowest spin-1 operator in the twisted sector has scaling dimension $1<{\mathrm{\ensuremath{\Delta}}}_{J}\ensuremath{\le}1.4$.