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Towards an exact density of states in a free conformal field theory

Connor Behan

Published Oct 20, 2012• 0 citations
OpenAlex

Abstract

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.

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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.
Results from spectral geometry such as Weyl's formula can be used to relate the thermodynamic properties of a free massless field to the spatial manifold on which it is defined. We begin by calculating the free energy in two cases: manifolds posessing a boundary and spheres. The subextensive contributions allow us to test the Cardy-Verlinde formula and offer a new perspective on why it only holds in a free theory if one allows for a change in the overall coefficient. After this we derive an expression for the density of states that takes the form of a Taylor series. This series leads to an improvement over known results when the area of the manifold's boundary is nonzero but much less than the appropriate power of its volume.
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 present a new framework for studying conformal field theories deformed by one or more relevant operators. The original CFT is described in infinite volume using a basis of states with definite momentum, P , and conformal Casimir, C. The relevant deformation is then considered using lightcone quantization, with the resulting Hamiltonian expressed in terms of this CFT basis. Truncating to states with C≤Cmax⁡ \mathcal{C}\le {\mathcal{C}}_{\max } C≤Cmax​ , one can numerically find the resulting spectrum, as well as other dynamical quantities, such as spectral densities of operators. This method requires the introduction of an appropriate regulator, which can be chosen to preserve the conformal structure of the basis. We check this framework in three dimensions for various perturbative deformations of a free scalar CFT, and for the case of a free O(N ) CFT deformed by a mass term and a non-perturbative quartic interaction at large-N . In all cases, the truncation scheme correctly reproduces known analytic results. We also discuss a general procedure for generating a basis of Casimir eigenstates for a free CFT in any number of dimensions.
Defining complexity in quantum field theory is a difficult task, and the main challenge concerns going beyond free models and associated Gaussian states and operations. One take on this issue is to consider conformal field theories in 1+1 dimensions and our work is a comprehensive study of state and operator complexity in the universal sector of their energy-momentum tensor. The unifying conceptual ideas are Euler-Arnold equations and their integro-differential generalization, which guarantee well-posedness of the optimization problem between two generic states or transformations of interest. The present work provides an in-depth discussion of the results reported in arXiv:2005.02415 and techniques used in their derivation. Among the most important topics we cover are usage of differential regularization, solution of the integro-differential equation describing Fubini-Study state complexity and probing the underlying geometry.
Defining complexity in quantum field theory is a difficult task, and the main challenge concerns going beyond free models and associated Gaussian states and operations. One take on this issue is to consider conformal field theories in 1+1 dimensions and our work is a comprehensive study of state and operator complexity in the universal sector of their energy-momentum tensor. The unifying conceptual ideas are Euler-Arnold equations and their integro-differential generalization, which guarantee well-posedness of the optimization problem between two generic states or transformations of interest. The present work provides an in-depth discussion of the results reported in arXiv:2005.02415 and techniques used in their derivation. Among the most important topics we cover are usage of differential regularization, solution of the integro-differential equation describing Fubini-Study state complexity and probing the underlying geometry.
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.