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.
The continuous multiscale entanglement renormalization ansatz (cMERA) [J. Haegeman et al., Phys. Rev. Lett. 110, 100402 (2013)] gives a variational wave functional for ground states of quantum field-theoretic Hamiltonians. A cMERA is defined as the result of applying to a reference unentangled state a unitary evolution generated by a quasilocal operator, the entangler. This makes the extension of the formalism to the case where boundaries and defects are present nontrivial. Here we show how this generalization works, using the (
1+1)-dimensional free boson cMERA as a proof-of-principle example, and restricting ourselves to conformal boundaries and defects. In our prescription, the presence of a boundary or defect induces a modification of the entangler localized only to its vicinity, in analogy with the so-called principle of minimal updates for the lattice tensor network MERA.
Continuous tensor networks give a variational ansatz for the ground state of the quantum field theories (QFTs). Notable examples are the continuous matrix product state (cMPS) and the continuous multiscale entanglement renormalization ansatz (cMERA). While cMPS is just adapted to the nonrelativistic QFTs, only the Gaussian cMERA is well understood, which we cannot use to approximate the ground state of the interacting relativistic QFTs. But, instead, cMERA also corresponds to a real-space renormalization group flow in the context of the wave functions. In this paper, we investigate the backward Gaussian cMERA renormalization group flow of the class of cMPS by putting the standard cMPS at the IR scale. At the UV scale, for the bosonic systems in the thermodynamic limit, we achieve the variational class of states that has been proposed recently, as the relativistic cMPS (RCMPS) is adapted to the relativistic QFTs without requiring one to introduce of any additional IR or UV cutoff. We also extend the RCMPS to fermionic systems and theories on a finite circle.
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 investigate the entanglement structure of the continuous multi-scale entanglement renormalization ansatz (cMERA) [Haegeman et al., Phys. Rev. Lett. 110 (2013) 100402] for ground states of quantum field theories (QFTs). The cMERA, proposed as an extension to QFTs of the lattice MERA, is defined directly in the continuum but is nevertheless naturally equipped with a short-distance scale 1/Λ that acts as a UV regulator for quantum fluctuations. We consider the simplified setting of Gaussian cMERA for free QFTs, where explicit calculations can be performed. For relativistic free massless bosonic and fermionic QFTs in both 1+1 and 2+1 spacetime dimensions, we show that the cMERA state indeed displays no UV divergences in two-point correlation functions or entanglement entropy, in sharp contrast with the exact ground states.
We investigate the entanglement structure of the continuous multi-scale entanglement renormalization ansatz (cMERA) [Haegeman et al., Phys. Rev. Lett., 110, 100402 (2013)] for ground states of quantum field theories (QFTs). The cMERA, proposed as an extension to QFTs of the lattice MERA, is defined directly in the continuum but is nevertheless naturally equipped with a short-distance scale
1/Λ that acts as a UV regulator for quantum fluctuations. We consider the simplified setting of Gaussian cMERA for free QFTs, where explicit calculations can be performed. For relativistic free massless bosonic and fermionic QFTs in both 1+1 and 2+1 spacetime dimensions, we show that the cMERA state indeed displays no UV divergences in two-point correlation functions or entanglement entropy, in sharp contrast with the exact ground states.