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arXiv:2111.04725 · Journal of High Energy Physics

Universal fine grained asymptotics of free and weakly coupled quantum field theory

Weiguang Cao, Tom Melia, Sridip Pal

Published May 3, 2024• 3 citations
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Abstract

A bstract We give a rigorous proof that in any free quantum field theory with a finite group global symmetry G, on a compact spatial manifold, at sufficiently high energy, the density of states ρ α ( E ) for each irreducible representation α of G obeys a universal formula as conjectured by Harlow and Ooguri. We further prove that this continues to hold in a weakly coupled quantum field theory, given an appropriate scaling of the coupling with temperature. This generalizes similar results that were previously obtained in (1 + 1)-D to higher spacetime dimension. We discuss the role of averaging in the density of states, and we compare and contrast with the case of continuous group G, where we prove a universal, albeit different, behavior.

DOI
10.1007/jhep05(2024)031

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A bstract We give a rigorous proof that in any free quantum field theory with a finite group global symmetry G, on a compact spatial manifold, at sufficiently high energy, the density of states ρ α ( E ) for each irreducible representation α of G obeys a universal formula as conjectured by Harlow and Ooguri. We further prove that this continues to hold in a weakly coupled quantum field theory, given an appropriate scaling of the coupling with temperature. This generalizes similar results that were previously obtained in (1 + 1)-D to higher spacetime dimension. We discuss the role of averaging in the density of states, and we compare and contrast with the case of continuous group G, where we prove a universal, albeit different, behavior.
We consider a ddd-dimensional unitary conformal field theory with a compact Lie group global symmetry GGG and show that, at high temperature TTT and on a compact Cauchy surface, the probability of a randomly chosen state being in an irreducible unitary representation RRR of GGG is proportional to $(\mathrm{dim}\text{ }R{)}^{2}\mathrm{exp}[\ensuremath{-}{c}_{2}(R)/(b{T}^{d\ensuremath{-}1})]$. We use the spurion analysis to derive this formula and relate the constant bbb to a domain wall tension. We also verify it for free field theories and holographic conformal field theories and compute bbb in these cases. This generalizes the result in 2109.03838 that the probability is proportional to (dimR)2(\mathrm{dim}R{)}^{2}(dimR)2 when GGG is a finite group. As a byproduct of this analysis, we clarify thermodynamical properties of black holes with non-Abelian hair in anti--de Sitter space.
Abstract We prove a recent conjecture by Harlow and Ooguri concerning a universal formula for the charged density of states in QFT at high energies for global symmetries associated with finite groups. An equivalent statement, based on the entropic order parameter associated with charged operators in the thermofield double state, was proven in a previous article by Casini, Huerta, Pontello, and the present author. Here we describe how the statement about the entropic order parameter arises, and how it gets transformed into the universal density of states. The use of the certainty principle, relating the entropic order and disorder parameters, is crucial for the proof. We remark that although the immediate application of this result concerns charged states, the origin and physics of such density can be understood by looking at the vacuum sector only. We also describe how these arguments lie at the origin of the so-called entropy equipartition in these type of systems, and how they generalize to QFT’s on non-compact manifolds.
It is a remarkable property of BCS theory that the ratio of the energy gap at zero temperature [Formula: see text] and the critical temperature [Formula: see text] is (approximately) given by a universal constant, independent of the microscopic details of the fermionic interaction. This universality has rigorously been proven quite recently in three spatial dimensions and three different limiting regimes: weak coupling, low density and high density. The goal of this short note is to extend the universal behavior to lower dimensions [Formula: see text] and give an exemplary proof in the weak coupling limit.
A bstract It is known that the asymptotic density of states of a 2d CFT in an irreducible representation ρ of a finite symmetry group G is proportional to (dim ρ ) 2 . We show how this statement can be generalized when the symmetry can be non-invertible and is described by a fusion category C \mathcal{C} C C . Along the way, we explain what plays the role of a representation of a group in the case of a fusion category symmetry; the answer to this question is already available in the broader mathematical physics literature but not yet widely known in hep-th. This understanding immediately implies a selection rule on the correlation functions, and also allows us to derive the asymptotic density.
A large class of conformal field theories can be studied by setting the mass to zero in a set of non-interacting fields that form a representation of the Lorentz group. We consider all such field theories defined on a general compact manifold. In the large volume limit, the density of states as a function of energy is shown to obey the same differential equation regardless of the manifold's geometry. From this equation, we compute the leading and next-to-leading behaviour for the density of states, and establish a method that can be iterated to obtain the asymptotic series to any desired precision.
A bstract We derive Cardy-like formulas for the growth of operators in different sectors of unitary 2 dimensional CFT in the presence of topological defect lines by putting an upper and lower bound on the number of states with scaling dimension in the interval [∆ − δ, ∆ + δ ] for large ∆ at fixed δ . Consequently we prove that given any unitary modular invariant 2D CFT symmetric under finite global symmetry G (acting faithfully), all the irreducible representations of G appear in the spectra of the untwisted sector; the growth of states is Cardy like and proportional to the “square” of the dimension of the irrep. In the Schwarzian limit, the result matches onto that of JT gravity with a bulk gauge theory. If the symmetry is non-anomalous, the result applies to any sector twisted by a group element. For c > 1, the statements are true for Virasoro primaries. Furthermore, the results are applicable to large c CFTs. We also extend our results for the continuous U(1) group.