A vacuum in conformal theory can be defined as a state where correlation functions of primary operators take a particularly simple form. It is also known that entanglement entropy of an interval in conformal vacuum has a universal logarithmic form. Can a mixed state with similar properties exist? In particular, can one find an example of a density matrix $\rho$ for which some basis of local fields $V_\Delta$ has the usual form of correlators? For example for two-point functions this should be something like $$Tr\left(\rho V_{\Delta}(z_1)V_{\Delta}(z_2)\right)\propto \frac1{(z_1-z_2)^{2\Delta}}$$ and similarly for 3-point functions etc. Potentially the definitions of the primary operators and/or their dimensions might be different from the original theory. If such a state exists, can it be diagnosed by entanglement entropy of an interval?

Any hints and pointers are welcome!

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    $\begingroup$ 1. An obvious candidate would be to take a (uniform) mixture of two or more pure vacuum states. Have you tried looking at that? 2. What does the phrase "entanglement entropy of an interval" mean? An interval of what? $\endgroup$
    – ACuriousMind
    Commented Jul 28, 2021 at 22:55
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    $\begingroup$ @ACuriousMind 1. I do not know an example where the conformal vacuum is not unique. I would be interested to know if one exists, but I had a more generic situation (non-degenerate vacuum) in mind, 2. Entanglement entropy of an interval is a standard setting in 1+1 QFT. One chooses an interval on a spatial axis and traces out degrees of freedom outside. The resulting density matrix has von Neumnamm entropy proportional to the logarithm of the interval length. $\endgroup$ Commented Jul 28, 2021 at 23:15
  • $\begingroup$ @WeatherReport I haven't thought about this carefully, but it seems that if you want your density matrix to be trace-class (so that it makes sense to say that it is properly normalized), then this might be tricky. This is because trace-class operators are in particular Hilbert-Schmidt, and so they live in the tensor product $H^*\otimes H$, where $H$ is the Hilbert space. Now, you know that $H=\oplus_i D_i$ where $D_i$ are the parabolic Verma modules of local operators. If I am not mistaken, there are no singlets in $D_i^*\otimes D_j$, even for $i=j$, although I can't give you a reference. $\endgroup$ Commented Sep 13, 2021 at 14:36
  • $\begingroup$ (This is because $D_i$ are infinite-dimensional; for finite-dimensional irreps there's always a singlet in $D_i^*\otimes D_i$. The logic is that the only singlet in $D_i^*\otimes D_i$ would be the inner product matrix, but it has infinite trace, because in an orthonormal basis it is just the identity matrix.) $\endgroup$ Commented Sep 13, 2021 at 14:37
  • $\begingroup$ @PeterKravchuk Wouldn't you say that thermal state is actually an example, since it is conformally equivalent to a vacuum on a cylinder physics.stackexchange.com/a/234083/8910 ? $\endgroup$ Commented Sep 21, 2021 at 6:17


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