Citation: Vidal, Guifre, et al. "Entanglement in quantum critical phenomena." Physical review letters 90.22 (2003): 227902.
Web: https://arxiv.org/abs/quant-ph/0211074
Tags: Physical, Conformal-field-theory, Spin-chains
In this paper, the authors present some numerics on entanglement in critical quantum spin chains and some interpretation of their results. In particular, they compute the von Neumann entropy of the reduced density matrices of the ground states on intervals of length L. They do their numerics for a family of quantum spin systems which includes the Ising chain and the isotropic Heisenberg chain. They can do these numerics effectively be exactly solving the models using Majorana operators (in the case of the Ising chains) and by using a more numerically-heavy procedure for the Heisenberg chain.
In all of their examples, they find that the entropy S_L of the system on L sites is constant when the system is not-critical, and grows like log(L) when the system is critical. In fact, it grows like S_L = c/6 log_2(L) + k where c is the chiral central charge and the appropriate critical CFT and k is some universal constant correction. This formula recovers a result of Holzhey-Larsen-Wilczek which was derived in the context of field theory, which is now being seen in a more grounded way through these spin chains:
> Holzhey, Christoph, Finn Larsen, and Frank Wilczek. "Geometric and renormalized entropy in conformal field theory." Nuclear physics b 424.3 (1994): 443-467.
The authors also talk about an application of their philosophy to DMRG. A key requirement in the DMRG algorithm is tensor network ansatz, which requires an area law. CFTs, as seen here, do not satisfy area laws and hence DMRG does not apply.
Another application applies to the philosophy of the renormalization group more generally. There is a theorem, known as Zamolodchikov's c-theory, which says that the chiral central charge of a QFT is non-increasing under the action of the renormalization group:
> Zamolodchikov, Alexander B. "Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory." JETP lett 43.12 (1986): 730-732.
Now, in this paper they've related the chiral central charge to entanglement. So, composing these results, what you get is that for 1D CFTs, renormalization group action will decrease entanglement entropy as you flow to the IR. The authors posit this as a (reasonable but non-obvious) general feature of the renormalization group action: it decreases entanglement entropy.