Abstract In this work, a non-Gaussian cMERA tensor network for interacting quantum field theories (icMERA) is presented. This consists of a continuous tensor network circuit in which the generator of the entanglement renormalization of the wavefunction is nonperturbatively extended with nonquadratic variational terms. The icMERA circuit nonperturbatively implements a set of scale dependent nonlinear transformations on the fields of the theory, which suppose a generalization of the scale dependent linear transformations induced by the Gaussian cMERA circuit. Here we present these transformations for the case of self-interacting scalar and fermionic field theories. Finally, the icMERA tensor network is fully optimized for the λϕ4 theory in (1 + 1) dimensions. This allows us to evaluate, nonperturbatively, the connected parts of the two- and four-point correlation functions. Our results show that icMERA wavefunctionals encode proper non-Gaussian correlations of the theory, thus providing a new variational tool to study phenomena related with strongly interacting field theories.
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 wave functionals 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 $\ensuremath{\lambda}{\ensuremath{\phi}}^{4}$ 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.
The continuous multiscale entaglement renormalization ansatz (cMERA) [Haegeman et al., Phys. Rev. Lett. 110, 100402 (2013)] is a variational wave functional for ground states of quantum field theories. So far, only scalar bosons and fermions have been considered. In this paper we explain how to generalize the cMERA framework to gauge invariant quantum fields. The fundamental difficulty to be addressed is how to make the gauge constraints (local linear constraints in the Hilbert space) compatible with the UV structure of the cMERA wave functional (which is generated by a quasilocal entangler). For simplicity, we consider U(1) gauge theory in
d+1 spacetime dimensions, a noninteracting theory with massless Hamiltonian
HU(1) and Gaussian scale invariant ground state $|{\mathrm{\ensuremath{\Psi}}}_{\mathrm{U}(1)}⟩$. We propose a gauge invariant cMERA wave functional $|{\mathrm{\ensuremath{\Psi}}}_{\mathrm{U}(1)}^{\mathrm{\ensuremath{\Lambda}}}⟩$ that, by construction, accurately reproduces the long distance properties of $|{\mathrm{\ensuremath{\Psi}}}_{\mathrm{U}(1)}⟩$ while remaining somewhat unentangled at short distances. Moreover, $|{\mathrm{\ensuremath{\Psi}}}_{\mathrm{U}(1)}^{\mathrm{\ensuremath{\Lambda}}}⟩$ is the exact ground state of a gauge invariant, local Hamiltonian ${H}_{\mathrm{U}(1)}^{\mathrm{\ensuremath{\Lambda}}}$ whose low energy properties coincide with those of
HU(1). Our construction also extends the cMERA formalism to massive (nongauge invariant) vector boson quantum fields.
The generalization of the multiscale entanglement renormalization ansatz (MERA) to continuous systems, or cMERA [Haegeman et al., Phys. Rev. Lett. 110, 100402 (2013)], is expected to become a powerful variational ansatz for the ground state of strongly interacting quantum field theories. In this Letter, we investigate, in the simpler context of Gaussian cMERA for free theories, the extent to which the cMERA state $|{\mathrm{\ensuremath{\Psi}}}^{\mathrm{\ensuremath{\Lambda}}}⟩$ with finite UV cutoff $\mathrm{\ensuremath{\Lambda}}$ can capture the spacetime symmetries of the ground state $|\mathrm{\ensuremath{\Psi}}⟩$. For a free boson conformal field theory (CFT) in
1+1 dimensions, as a concrete example, we build a quasilocal unitary transformation
V that maps $|\mathrm{\ensuremath{\Psi}}⟩$ into $|{\mathrm{\ensuremath{\Psi}}}^{\mathrm{\ensuremath{\Lambda}}}⟩$ and show two main results. (i) Any spacetime symmetry of the ground state $|\mathrm{\ensuremath{\Psi}}⟩$ is also mapped by
V into a spacetime symmetry of the cMERA $|{\mathrm{\ensuremath{\Psi}}}^{\mathrm{\ensuremath{\Lambda}}}⟩$. However, while in the CFT, the stress-energy tensor ${T}_{\ensuremath{\mu}\ensuremath{\nu}}(x)$ (in terms of which all the spacetime symmetry generators are expressed) is local, and the corresponding cMERA stress-energy tensor ${T}_{\ensuremath{\mu}\ensuremath{\nu}}^{\mathrm{\ensuremath{\Lambda}}}(x)=V{T}_{\ensuremath{\mu}\ensuremath{\nu}}(x){V}^{\ifmmode\dagger\else\textdagger\fi{}}$ is quasilocal. (ii) From the cMERA, we can extract quasilocal scaling operators ${O}_{\ensuremath{\alpha}}^{\mathrm{\ensuremath{\Lambda}}}(x)$ characterized by the exact same scaling dimensions ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\alpha}}$, conformal spins ${s}_{\ensuremath{\alpha}}$, operator product expansion coefficients ${C}_{\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\gamma}}$, and central charge
c as the original CFT. Finally, we argue that these results should also apply to interacting theories.
We adapt the techniques of entanglement renormalization tensor networks to weakly interacting quantum field theories in the continuum. A key tool is ``quantum circuit perturbation theory,'' which enables us to systematically construct unitaries that map between wave functionals which are Gaussian with arbitrary perturbative corrections. As an application, we construct a local continuous multiscale entanglement renormalization ansatz (cMERA) circuit that maps an unentangled scale-invariant state to the ground state of ${\ensuremath{\varphi}}^{4}$ theory to one loop. Our local cMERA circuit corresponds exactly to one-loop Wilsonian renormalization group (RG) flow on the spatial momentum modes. In other words, we establish that perturbative Wilsonian RG on spatial momentum modes can be equivalently recast as a local cMERA circuit in ${\ensuremath{\varphi}}^{4}$ theory and argue that this correspondence holds more generally. Our analysis also suggests useful numerical ans\"atze for cMERA in the nonperturbative regime.
The continuous multiscale entanglement renormalization ansatz (cMERA) is a variational class of states for quantum fields. As originally formulated, the cMERA applies to infinite systems only. In this paper we generalize the cMERA formalism to a finite circle, which we achieve by wrapping the action of the so-called entangler around the circle. This allows us to transform a cMERA on the line into a cMERA on the circle. In addition, in the case of a Gaussian cMERA for noninteracting quantum fields, the method of images allows us to prove the following result: If on the line a cMERA state is a good approximation to a ground state of a local quantum field theory (QFT) Hamiltonian, then (under mild assumptions on their correlation functions) the resulting cMERA on a circle is also a good approximation to the ground state of the same local QFT Hamiltonian on the circle.