[Physics] Modular Hamiltonians and modular invariance


In the literature you will often see the use of modular Hamiltonians in e.g. entanglement entropy calculations in CFT's. The modular Hamiltonian $H$ is given in terms of the density matrix
$$\rho = e^{-H}$$
It is my understanding that for a CFT to be well defined on a general Riemann surface, modular invariance of the partition function (on the torus, say) is a requirement.

In this case then, my overall question is: how, if at all, is the modular Hamiltonian related to modular invariance?

More specifically, I understand that the modular Hamiltonians generate 'modular flow', and although I don't have a clean definition of what this means, it seems to just imply that acting on some operator generates a unitary transformation in some 'modular time', and the term modular just seems redundant. What does 'modular' mean in this context?

Best Answer

In QFT, the reduced density matrix can be written as,

\begin{equation} \rho = \frac{e^{-\beta H}}{\mathrm{Tr} (e^{-\beta H})} \end{equation}

where $H$ is the modular Hamiltonian often used in QFT literature while it is called entanglement Hamiltonian in the condensed matter theory literature. See the first few minutes of this talk (https://www.perimeterinstitute.ca/videos/modular-hamiltonians-2d-cft) by John Cardy.

The denominator is included just to ensure that $\mathrm{Tr} \rho = 1$.

The origin of the name "modular Hamiltonian" goes back to the Tomita-Takesaki modular theory where an operator of the form $ \Delta = e^{-K}$ is called the modular operator and $K$ is called the modular Hamiltonian. In general, $K$ is a complicated nonlocal operator.

See Definition 1.21 of https://arxiv.org/abs/1301.1836 or the book "Local Quantum Physics: Fields, Particles, Algebras" by R. Haag for details (Chapter 5).

In https://arxiv.org/abs/1102.0440, it was shown that in some cases one can write this modular Hamiltonian in terms of stress-energy tensor of a CFT.

(Extra note: The author is also part of the famous Haag–Lopuszański–Sohnius theorem which was a foundation work in supersymmetry introducing the non-trivial extension of the Poincare algebra, namely the supersymmetric algebra)

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