Entanglement renormalization for interacting field theories

Abstract

A general method to build the entanglement renormalization (cMERA) for interacting quantum field theories is presented. We improve upon the well-known Gaussian formalism used in free theories through a class of variational non-Gaussian wavefunctionals for which expectation values of local operators can be efficiently calculated analytically and in a closed form. The method consists of a series of scale-dependent nonlinear canonical transformations on the fields of the theory under consideration. Here, the λ φ 4 and the sine-Gordon scalar theories are used to illustrate how nonperturbative effects far beyond the Gaussian approximation are obtained by considering the energy functional and the correlation functions of the theory.In recent years, tensor networks, a new and powerful class of variational states, have proved to be very useful in addressing both static and dynamical aspects of a wide number of interacting many-body systems. They represent a class of systematic variational ansätze which, through the Rayleigh-Ritz variational principle, provide an elegant approximation to the ground state of an interacting theory by systematically identifying those degrees of freedom that are actually relevant for observable physics. These variational ansätze are nonperturbative and can be applied both in the lattice and in the continuum. As an example, the Multiscale Entanglement Renormalization Ansatz (MERA), a variational real-space renormalization scheme on the quantum state, represents the wavefunction of the quantum system at different length scales [1].A continuous version of MERA, known as cMERA, was proposed in [2] for free field theories. It consists of building a scale-dependent representation of the ground state wavefunctional through a scale-dependent linear canonical transformation of the fields of the theory. Namely, the renormalization in scale is generated by a quadratic operator, and thus, the resulting state is given by a Gaussian wavefunctional. Despite this fact obviously limits the interest of this trial state for interacting quantum field theories (QFT), the Gaussian ansatz has been used in cMERA and correctly reproduces correlation functions and entanglement entropy in free field theories [3, 4]. Furthermore, as the Gaussian cMERA is currently studied as a possible realization of holography [5][6][7][8][9][10], it is timely to develop interacting versions of cMERA in order to advance in this program. In [11], the Gaussian cMERA was applied to interacting bosonic and fermionic field theories. In [12], authors developed some techniques to build systematic perturbative calculations of cMERA circuits but restricted to the weakly interacting regime.Our aim here is to provide a non-perturbative method to build truly non-Gaussian cMERA wavefunctionals for interacting QFTs. A justifiable way of doing so would be to formulate a perturbative expansion for which the Gaussian wavefunction appears in its first order [13][14][15][16].Unfortunately, with these methods, expectation values of operators cannot be cal...

Non-Gaussian entanglement renormalization for quantum fields

“…This is what could constitute a more satisfactory and natural continuum generalization of the lattice MERA algorithms. In the present work we refine our method and solve this issue using the firm ground provided by the conceptual framework presented in [21]. In addition, we carry out an exhaustive optimization process and show the efficiency and predictability of our method.…”

“…It is expected that this connection could yield cMERA wavefunctionals that are able to capture nonperturbative physics beyond the Gaussian approximation. In [21], authors presented a truly nonperturbative method to build non-Gaussian cMERA wavefunctionals for interacting…”

“…Let us remark the last proposal. The scale dependent NLCT in [21], which were constructed from the product of two unitary operators, obviously admit a realization through a unitary operator. This is what could constitute a more satisfactory and natural continuum generalization of the lattice MERA algorithms.…”

“…Being a nonperturbative operator, we have studied the non-Gaussian variational parameters and given the prescription for which the series of nested commutators truncates. Secondly, we have reviewed [21] as a primary method to introduce the scale dependence on the NLCT. This is a key construction for the final formulation of the circuit.…”

“…For example, entanglers containing only quadratic operators generate scale dependent Bogoliubov transformations on the field operators, and these transform Gaussian states into Gaussian states. Conversely, entanglers possessing higher order operators induce nonlinear transformations on the field operators which, when acting on Gaussian states, generate non-Gaussian states [18,21].…”