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arXiv:1907.09494 · Physical review. D/Physical review. D.

Classification of nonlocal actions: Area versus volume entanglement entropy

Bora Basa, Gabriele La Nave, Philip Phillips

Published May 6, 2020• 14 citations
arXivHTMLPDF

Abstract

From the partition function for two classes of classically nonlocal actions containing the fractional Laplacian, we show that as long as there exists a suitable (nonlocal) Hilbert-space transform the underlying action can be mapped onto a purely local theory. In all such cases the partition function is equivalent to that of a local theory and an area law for the entanglement entropy obtains. When such a reduction fails, the entanglement entropy deviates strongly from an area law and can in some cases scale as the volume. As these two criteria are coincident, we conjecture that they are equivalent and provide the ultimate test for locality of Gaussian theories rather than a simple inspection of the explicit operator content.

DOI
10.1103/physrevd.101.106006

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From the partition function for two classes of classically nonlocal actions containing the fractional Laplacian, we show that as long as there exists a suitable (nonlocal) Hilbert-space transform the underlying action can be mapped onto a purely local theory. In all such cases the partition function is equivalent to that of a local theory and an area law for the entanglement entropy obtains. When such a reduction fails, the entanglement entropy deviates strongly from an area law and can in some cases scale as the volume. As these two criteria are coincident, we conjecture that they are equivalent and provide the ultimate test for locality of Gaussian theories rather than a simple inspection of the explicit operator content.
Entanglement entropy in the vacuum state of local field theories exhibits an area law. However, nonlocal theories at large N and strong coupling violate this area law. In these theories, the leading divergence in the entanglement entropy is extensive for regions smaller than the effective nonlocality scale and proportional to this effective nonlocality scale for regions larger than it. This raises the question: is a volume law a generic feature of nonlocal theories, or is it only present at strong coupling and large N ? This paper investigates entanglement entropy of large regions in weakly coupled non-local theories, to leading order in the coupling. The two theories studied are ϕ 4 theory on the noncommutative plane and ϕ 4 theory with a dipole type nonlocal modification using a fixed nonlocality scale. Both theories are found to follow an area law to first order in the coupling, hence no evidence is found for a volume law. This indicates that, perturbatively the nonlocal interactions considered are not generating sufficient entanglement at distances of the nonlocality scale to change the leading divergence, at least to first order in the coupling. An argument against volume laws at higher orders is also presented.
We compute holographic entanglement entropy in two strongly coupled nonlocal field theories: the dipole and the noncommutative deformations of SYM theory. We find that entanglement entropy in the dipole theory follows a volume law for regions smaller than the length scale of nonlocality and has a smooth cross-over to an area law for larger regions. In contrast, in the noncommutative theory the entanglement entropy follows a volume law for up to a critical length scale at which a phase transition to an area law occurs. The critical length scale increases as the UV cutoff is raised, which is indicative of UV/IR mixing and implies that entanglement entropy in the noncommutative theory follows a volume law for arbitrary large regions when the size of the region is fixed as the UV cutoff is removed to infinity. Comparison of behaviour between these two theories allows us to explain the origin of the volume law. Since our holographic duals are not asymptotically AdS, minimal area surfaces used to compute holographic entanglement entropy have novel behaviours near the boundary of the dual spacetime. We discuss implications of our results on the scrambling (thermalization) behaviour of these nonlocal field theories. ar X iv
To produce a fermionic model exhibiting an entanglement entropy volume law, we propose a particular version of nonlocality in which the energy-momentum dispersion relation is effectively randomized at the shortest length scales while preserving translation invariance. In contrast to the ground state of local fermions, exhibiting an entanglement entropy area law with logarithmic corrections, the entropy of nonlocal fermions is extensive, scaling as the volume of the subregion and crossing over to the anomalous fermion area law at scales larger than the locality scale, $\ensuremath{\alpha}$. In the 1D case, we are able to show that the central charge appearing in the universal entropy expressions for large subregions is simply related to the locality scale. These results are demonstrated by exact diagonalizations of the corresponding discrete lattice fermion models. Within the Ryu-Takayanagi holographic picture, the relation between the central charge and the locality scale suggest a dual spacetime in which the size of the flat UV portion and the radius of anti--de Sitter in the IR are both proportional to the locality scale, $\ensuremath{\alpha}$.
Entanglement entropy obeys area law scaling for typical physical quantum systems. This may naively be argued to follow from the locality of interactions. We show that this is not the case by constructing an explicit simple spin chain Hamiltonian with nearest-neighbor interactions that presents an entanglement volume scaling law. This non-translational model is contrived to have couplings that force the accumulation of singlet bonds across the half-chain. This configuration of the couplings is suggested by real-space renormalization group arguments. Computation of the entanglement entropy is performed by mapping the system to free fermions and diagonalizing numerically its correlation matrix. An analytical relationship between the entanglement entropy and the Frobenius norm of the correlation matrix is also established. Our result is complementary to the known relationship between non-translational invariant, nearest-neighbor interacting Hamiltonians and quantum Merlin–Arthur (QMA)-complete problems.