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Of course, the free energy on the cylinder is not a measurable observable if you're given the theory on the infinite plane. But one can measure other observables which are proportional to the central charge, such as the two-point function of the stress-energy tensor. There are situations where that expression is an observable. If you have a one-dimensional ...

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A first-order time-translation-invariant Lagrangian $L(Q,\dot{Q},t)$ cannot have any explicit time-dependence. The claim in Ref. 1 that scale-invariance implies the Lagrangian (1.11) is strictly speaking wrong as written. Scale-invariance (that is assumed compatible with the free kinetic term) only implies that the Lagrangian is of the form $$L(Q,\dot{Q})~=~... 1 For the Ising model, we have the power of exact solutions. In particular, the one-dimensional transverse-field quantum Ising model, defined by the Hamiltonian$$ H = - J \sum_{i} \sigma^z_i \sigma^z_{i+1} - h \sum_i \sigma^x_i,  turns out to be exactly solvable. One can use the Jordan-Wigner transformation to map it to free fermions, after which you can ...

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The introduction to this answers a part of my question in a concise manner. The following points are necessary in order for the CFT to have a holographic dual. The first AdS/CFT papers (Maldacena, Witten, GKP) state that the three point functions like $\langle TTT \rangle$ must have specific structures for a holographic dual to exist. (This means that any ...

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Unitary scale invariant theories are either conformal or indistinguishable from conformal theories on $\mathbf{R}^4$. This means that, for all practical purposes, scale invariance and unitarity imply conformality. For review see slides "Scale and Conformal Invariance" by Zohar Komargodski.

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zzz's answer concerns QFTs as used in particle physics. However, OP also mentions phase transitions, thus statistical field theory, where the relationship between CFT and scale-invariant RG fixed points is more subtle. Indeed, for local QFTs where unitarity is assumed, there are theorems showing that scale invariance implies conformal invariance, see e.g. ...

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