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Can we use the density matrix renormalization group (DMRG) method to understand problems in conformal field theory? I have been trying to find some connections, but nothing is coming up when I search. Is it particularly hard/impossible to represent a CFT in matrix product states, or is it not a useful question? This link might be helpful:

What is the connection between Conformal Field Theory and Renormalization group in QFT?

However, it just talks about the connections between CFT and the renormalization group--I don't see how it could be connected to DMRG.

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    $\begingroup$ Well they are both useful techniques in 1+1 dimensions. You might be interested in continuous matrix product states, which can represent quantum field states in 1D, presumably including those corresponding to a CFT. $\endgroup$
    – Mark Mitchison
    Apr 19, 2015 at 16:29

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The name of DMRG is a bit misleading: the modern view is that it does not really have much to do with the idea of real-space renormalization (although that might be a motivation for Steve White to develop it, following the success of Wilson's numerical RG). DMRG is essentially a variational method based on matrix product state(MPS) representation of the ground state of 1D Hamiltonians, and any MPS with a finite bond dimension represents a state with finite correlation length. But CFT is gapless and has long-range correlations. In terms of entanglement, CFT has diverging entanglement entropy which MPS can not represent unless you take the bond dimension to infinity. Of course, in practice one can numerically approximate CFT ground state using MPS, and it works extremely well if the central charge is not too large. So practically DMRG is still the method of choice to study CFT in 1D systems in numerical simulations. However, conceptually these two are not naturally related. You may want to look at another technique, so-called MERA(multi-scale entanglement renormalization ansatz), which was designed to handle ground states like CFT which contain a lot of entanglement (more than area law). The connection to CFT there is quite explicit.

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  • $\begingroup$ Can MERA be used in 2D, or is it usually only used in 1D, like DMRG? $\endgroup$
    – Joshuah Heath
    Apr 19, 2015 at 16:42
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    $\begingroup$ In principle it can be used in 2D. However computationally it becomes intractable, and I don't think anyone including G. Vidal whose proposed MERA and has been working on developing the technique, has seriously attempted the 2D case numerically ever. But conceptually there is no problem. $\endgroup$
    – Meng Cheng
    Apr 19, 2015 at 16:51

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