Magic entanglement renormalization for quantum fields

Abstract

Continuous tensor networks are variational wavefunctions proposed in recent years to efficiently simulate quantum field theories (QFTs). Prominent examples include the continuous matrix product state (cMPS) and the continuous multi-scale entanglement renormalization ansatz (cMERA). While the cMPS can approximate ground states of a class of QFT Hamiltonians that are both local and interacting, cMERA is only well-understood for QFTs that are quasi-local and non-interacting. In this paper we propose the magic cMERA, a concrete realization of cMERA for a free boson QFT that simultaneously satisfies four remarkable properties: (i) it is the exact ground state of a strictly local Hamiltonian; (ii) in the massless case, its spectrum of scaling operators is exactly soluble in real space; (iii) it has the short-distance structure of a cMPS; (iv) it is generated by a quasi-local entangler that can be written as a continuous matrix product operator. None of these properties is fulfilled by previous cMERA proposals. Properties (iii)-(iv) establish a firm connection between cMERA and cMPS wave-functionals, opening the path to applying powerful cMPS numerical techniques, valid for interacting QFTs, also to cMERA calculations.

On the Entanglement Entropy in Gaussian cMERA

“…The cut off functions, which act as alternative UV regularization schemes of the cMERA formalism, will determine Σ(u). In Table 1, it is shown the resulting Σ(u) for some cutoff functions that have been proposed in the literature [9,18]. We observe that the result is similar to the one obtained through sharp cut off function up to some numerical factors.…”

Renormalization group and approximate error correction

“…[25] For concreteness, consider free massive boson field φ(x) and its momentum conjugate π(x) in one spatial dimension. Following [26] we choose an entangler independent of scale…”

“…This state is the ground state of a massive free boson deformed by a non-relativistic irrelevant term [26][27]:…”

Continuous entanglement renormalization on the circle

“…≤ (s). (30) This result (see [A2] for a proof) follows quite simply from the fact that the QFT correlators on the line and on the circle obey relations analogous to Eqs. ( 27)- (28) .…”

“…The ultimate goal of the cMERA program is to become a nonperturbative approach to strongly interacting QFTs (see [25][26][27] for steps in this direction), and in this way mirror the success of MERA on the lattice. In the meantime, however, the Gaussian cMERA for non-interacting QFTs [24,[28][29][30][31] has already proven its worth. Besides providing a valuable proof of concept that the lattice MERA formalism can be carried over to the continuum, Gaussian cMERA has been used as a toy model also in the context of the AdS/CFT correspondence and quantum gravity [33][34][35][36][37][38][39][40].…”