The thermal partition function, Z Z , of a CFT_d C F T d on S^{d-1} S d − 1 is parameterized by the inverse temperature \beta β along with \lfloor d/2\rfloor ⌊ d / 2 ⌋ angular velocities \omega_i ω i . In this paper, we investigate the behaviour of this partition function when n n of the \omega_i ω i are scaled to unity (the largest allowed value) at fixed values of the other (\lfloor d/2\rfloor-n) ( ⌊ d / 2 ⌋ − n ) angular velocities. We argue that \ln Z ln Z develops a simple pole in (1-\omega_i) ( 1 − ω i ) for each \omega_i ω i that is scaled to unity. The residue of this product of poles is a theory-dependent (so non-universal) function of \beta β and the fixed angular velocities. The inverse Laplace transformation of this partition function constrains the functional form of the field theory entropy as a function of charges in a limit in which angular momenta and the twist are scaled as follows. While n n special angular momenta J_1\ldots J_n J 1 …
The thermal partition function, Z Z , of a CFT_d C F T d on S^{d-1} S d − 1 is parameterized by the inverse temperature \beta β along with \lfloor d/2\rfloor ⌊ d / 2 ⌋ angular velocities \omega_i ω i . In this paper, we investigate the behaviour of this partition function when n n of the \omega_i ω i are scaled to unity (the largest allowed value) at fixed values of the other (\lfloor d/2\rfloor-n) ( ⌊ d / 2 ⌋ − n ) angular velocities. We argue that \ln Z ln Z develops a simple pole in (1-\omega_i) ( 1 − ω i ) for each \omega_i ω i that is scaled to unity. The residue of this product of poles is a theory-dependent (so non-universal) function of \beta β and the fixed angular velocities. The inverse Laplace transformation of this partition function constrains the functional form of the field theory entropy as a function of charges in a limit in which angular momenta and the twist are scaled as follows. While n n special angular momenta J_1\ldots J_n J 1 …
We investigate the equilibrium partition function of non-relativistic conformal field theories in harmonic quantization. We first analyze the hydrodynamic regime and show that, at leading order, the partition function exhibits a universal structure determined by the equation of state: the logarithm of the partition function develops simple poles in ω2−Ωa2, where ω is the harmonic trapping frequency and Ωa are angular velocities acting as chemical potentials for angular momentum. The corresponding residue is determined by a single-variable function of μ/T, with μ the particle-number chemical potential and T the temperature. We then study the large-angular-momentum limit Ωa→ω. In this regime centrifugal effects nearly cancel the trapping potential, and the logarithm of the partition function again exhibits simple poles in ω2−Ωa2, but with a less universal residue depending separately on μ/T and ω/T. As explicit examples we analyze superfluid systems realizable in cold-atom experiments, in particular fermions at unitarity confined in a harmonic trap.
A bstract We prove the conjecture proposed by Hartman, Keller and Stoica (HKS) [1]: the grand-canonical free energy of a unitary 2D CFT with a sparse spectrum below the scaling dimension 12c c 12 + ϵ and below the twist 12c c 12 is universal in the large c limit for all β L β R ≠ 4 π 2 . The technique of the proof allows us to derive a one-parameter (with parameter α ∈ (0, 1]) family of universal inequalities on the unitary 2D CFT partition function with general central charge c ⩾ 0, using analytical modular bootstrap. We derive an iterative equation for the domain of validity of the inequality on the ( β L , β R ) plane. The infinite iteration of this equation gives the boundary of maximal-validity domain, which depends on the parameter α in the inequality. In the c → ∞ limit, with the additional assumption of a sparse spectrum below the scaling dimension 12c c 12 + ϵ and the twist 12αc αc 12 (with α ∈ (0, 1] fixed), our inequality shows that the grand-canonical free energy exhibits a universal large c behavior in the maximal-validity domain. This domain, however, does not cover the entire ( β L , β R ) plane, except in the case of α = 1. For α = 1, this proves the conjecture proposed by HKS [1], and for α < 1, it quantifies how sparseness in twist affects the regime of universality. Furthermore, this implies a precise lower bound on the temperature of near-extremal BTZ black holes, above which we can trust the black hole thermodynamics.
We show that for a d-dimensional CFT in flat space, the Rényi entropy S q across a spherical entangling surface has the following property: in an expansion around q = 1, the first correction to the entanglement entropy is proportional to C T , the coefficient of the stress tensor vacuum two-point function, with a fixed d-dependent coefficient. This is equivalent to a similar statement about the free energy of CFTs living on S 1 × $ \mathbb{H} $ d−1 with inverse temperature β = 2πq. In addition to furnishing a direct argument applicable to all CFTs, we exhibit this result using a handful of gravity and field theory computations. Knowledge of C T thus doubles as knowledge of Rényi entropies in the neighborhood of q = 1, which we use to establish new results in 3d vector models at large N.
In this paper, we propose a novel expansion to compute the large interval limit of the R\'enyi entropy of 2D conformal field theory (CFT) at high temperature. Via the replica trick, the single interval R\'enyi entropy of 2D CFT at finite temperature could be read from the partition function on n-sheeted torus connected with each other along a branch cut. We calculate the partition function by inserting a complete basis across the branch cut. Because of the monodromy condition across the branch cut in the large interval limit, the basis of the states should be the ones in the twist sector. We study the twist sector of a general module of CFT and find that there is a one-to-one correspondence between the twist sector states and the normal sector states. As an application, we revisit the noncompact free scalar theory and discuss the large interval limit of the R\'enyi entropy of this theory by using our proposal. We find complete agreement in the leading and next-leading orders with direct expansion of the exact partition function. Moreover, we prove the relation (10) between thermal entropy and the entanglement entropy for a generic CFT with discrete spectrum.
A bstract We show that thermal effective field theory controls the long-distance expansion of the partition function of a d -dimensional QFT, with an insertion of any finite-order spatial isometry. Consequently, the thermal partition function on a sphere displays a fractal-like structure as a function of angular twist, reminiscent of the behavior of a modular form near the real line. As an example application, we find that for CFTs, the effective free energy of even-spin minus odd-spin operators at high temperature is smaller than the usual free energy by a factor of 1/2 d . Near certain rational angles, the partition function receives subleading contributions from “Kaluza-Klein vortex defects” in the thermal EFT, which we classify. We illustrate our results with examples in free and holographic theories, and also discuss nonperturbative corrections from worldline instantons.
Abstract We consider rigorous consequences of modular invariance for two-dimensional unitary non-rational CFTs with c>1 c > 1 . Simple estimates for the torus partition function can lead to remarkably strong results. We show in particular that the spectral density of spin- J operators must grow like exp(π32(c−1)J)/2J exp π 2 3 ( c - 1 ) J / 2 J in any twist interval at or above (c−1)/12 ( c - 1 ) / 12 , with a known twist-dependent prefactor. This proves that the large J spectrum becomes dense even without averaging over spins. For twists below (c−1)/12 ( c - 1 ) / 12 we establish that the growth must be strictly slower. Finally, we estimate how fast the maximal gap between two spin- J operators must go to zero as J becomes large.