Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2103.07463 · Physical review. B./Physical review. B

Entanglement renormalization for quantum fields with boundaries and defects

Adrián Franco-Rubio

Published Sep 20, 2021• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.104.125131

Related papers

Entanglement renormalization for gauge invariant quantum fields

Adrián Franco-Rubio, Guifré Vidal

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 ${H}_{\mathrm{U}(1)}$ 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 ${H}_{\mathrm{U}(1)}$. Our construction also extends the cMERA formalism to massive (nongauge invariant) vector boson quantum fields.

arXiv:1910.11815•Physical review. D/Physical review. D.•Jan 13, 2021•5 citations

Continuous entanglement renormalization on the circle

Ling-Yan Hung, Guifré Vidal

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.

arXiv:2101.03674•Physical Review Research•Oct 15, 2021•3 citations

Spacetime Symmetries and Conformal Data in the Continuous Multiscale Entanglement Renormalization Ansatz

Qicheng Hu, Guifré Vidal

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.

arXiv:1703.04798•Physical Review Letters•Jul 7, 2017•38 citations

Entanglement and correlations in the continuous multi-scale renormalization ansatz

Adrián Franco-Rubio, Guifré Vidal

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 $1/\Lambda$ 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.

•Jun 9, 2017•2 citations

Entanglement and correlations in the continuous multi-scale entanglement renormalization ansatz

Adrián Franco-Rubio, Guifré Vidal

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.

arXiv:1706.02841•Journal of High Energy Physics•Dec 1, 2017•28 citations

Renormalization, Entanglement and Continuous Tensor Networks

Adrián Franco-Rubio

The study of the ground states of local Hamiltonians in quantum many-body and quantum field theoretic systems is a source of many research problems of great complexity. Almost in its entirety, this thesis deals with constructions related to the field of tensor networks, which are efficient parametrizations of mathematical objects from quantum many-body theory (such as quantum states and operators) that exploit our knowledge of their entanglement structures in order to obtain improvements in our computational abilities. In the past decade, the new research program of continuous tensor networks has arisen with the goal of reproducing, in the setting of quantum field theory, the success that tensor network techniques have enjoyed on the lattice. First, we present a number of results from research performed on non-interacting examples of the continuous Multiscale Entanglement Renormalization Ansatz, or cMERA, a continuous tensor network that provides a variational wavefunctional for a quantum field theoretic ground state. Initially, we study the correlations and entanglement structure of cMERA states, providing evidence that such states can be interpreted as UV regularized versions of the physical states they approximate. After that, we study what modifications of the formalism are necessary in order to consider systems with gauge symmetry, or in the presence of boundaries and defects. We obtain prescriptions for the treatment of these cases which, we expect, may also hold in the interacting setting. We then abandon briefly the area of continuous tensor networks to present a piece of research within a different though related research program: that of the study of manifestations of universal emergent behaviour in critical lattice systems. In particular we bring attention to the existence of an approximate representation of the Virasoro algebra supported on the eigenvectors of the reduced density matrices of critical lattice systems. We call this the entanglement algebra, and study its accuracy in the particular case of the Ising model. Lastly, we present a proposal for a continuous version of the Tensor Network Renormalization (TNR) algorithm, which we dub continuous TNR, or cTNR. cTNR operates on UV regularized classical partition functions or quantum Euclidean path integrals, and generates a real-space renormalization group flow via continuous coarse-graining and rescaling operations. We show that a UV regularized version of the free boson path integral can be made a fixed point of such a renormalization flow, in a way that allows for recovery of the conformal data associated to the free boson conformal field theory.

•Jul 7, 2020•0 citations
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+11+11+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.
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+1d+1d+1 spacetime dimensions, a noninteracting theory with massless Hamiltonian HU(1){H}_{\mathrm{U}(1)}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){H}_{\mathrm{U}(1)}HU(1)​. Our construction also extends the cMERA formalism to massive (nongauge invariant) vector boson quantum fields.
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 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+11+11+1 dimensions, as a concrete example, we build a quasilocal unitary transformation VVV 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 VVV 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 ccc 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, 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/Λ1/\Lambda1/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 (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.
The study of the ground states of local Hamiltonians in quantum many-body and quantum field theoretic systems is a source of many research problems of great complexity. Almost in its entirety, this thesis deals with constructions related to the field of tensor networks, which are efficient parametrizations of mathematical objects from quantum many-body theory (such as quantum states and operators) that exploit our knowledge of their entanglement structures in order to obtain improvements in our computational abilities. In the past decade, the new research program of continuous tensor networks has arisen with the goal of reproducing, in the setting of quantum field theory, the success that tensor network techniques have enjoyed on the lattice. First, we present a number of results from research performed on non-interacting examples of the continuous Multiscale Entanglement Renormalization Ansatz, or cMERA, a continuous tensor network that provides a variational wavefunctional for a quantum field theoretic ground state. Initially, we study the correlations and entanglement structure of cMERA states, providing evidence that such states can be interpreted as UV regularized versions of the physical states they approximate. After that, we study what modifications of the formalism are necessary in order to consider systems with gauge symmetry, or in the presence of boundaries and defects. We obtain prescriptions for the treatment of these cases which, we expect, may also hold in the interacting setting. We then abandon briefly the area of continuous tensor networks to present a piece of research within a different though related research program: that of the study of manifestations of universal emergent behaviour in critical lattice systems. In particular we bring attention to the existence of an approximate representation of the Virasoro algebra supported on the eigenvectors of the reduced density matrices of critical lattice systems. We call this the entanglement algebra, and study its accuracy in the particular case of the Ising model. Lastly, we present a proposal for a continuous version of the Tensor Network Renormalization (TNR) algorithm, which we dub continuous TNR, or cTNR. cTNR operates on UV regularized classical partition functions or quantum Euclidean path integrals, and generates a real-space renormalization group flow via continuous coarse-graining and rescaling operations. We show that a UV regularized version of the free boson path integral can be made a fixed point of such a renormalization flow, in a way that allows for recovery of the conformal data associated to the free boson conformal field theory.