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

Abstract

The generalization of the multi-scale 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 paper we investigate, in the simpler context of Gaussian cMERA for free theories, the extent to which the cMERA state |Ψ Λ with finite UV cut-off Λ can capture the spacetime symmetries of the ground state |Ψ . For a free boson conformal field theory (CFT) in 1+1 dimensions as a concrete example, we build a quasi-local unitary transformation V that maps |Ψ into |Ψ Λ and show two main results. (i) Any spacetime symmetry of the ground state |Ψ is also mapped by V into a spacetime symmetry of the cMERA |Ψ Λ . However, while in the CFT the stress-energy tensor Tµν (x) (in terms of which all the spacetime symmetry generators are expressed) is local, the corresponding cMERA stress-energy tensor(ii) From the cMERA, we can extract quasi-local scaling operators O Λ α (x) characterized by the exact same scaling dimensions ∆α, conformal spins sα, operator product expansion coefficients C αβγ , and central charge c as the original CFT. Finally, we argue that these results should also apply to interacting theories. The study and numerical simulation of interacting quantum many-body systems is an extremely challenging task. Making progress often requires the use of a simplifying variational ansatz, such as the multi-scale entanglement renormalization ansatz (MERA) [1], which aims to describe the ground state of lattice Hamiltonians. The MERA can be visualized as the result of a unitary evolution, running from large distances to short distances, that maps an initial unentangled state into a complex many-body wavefunction by gradually introducing entanglement into the system, scale by scale. The success of the MERA in a large class of lattice systems, including systems with topological order [2] or at a quantum critical point [1,[3][4][5], teaches us that this entangling evolution in scale picture is a valid -and computationally powerful!-way of thinking about ground states and their intricate structure of correlations. With a built-in notion of the renormalization group [6], MERA is also actively investigated in several other contexts, from holography [7][8][9] (as a discrete realization of the AdS/CFT correspondence [10]) to statistical mechanics [11], error correction [12], and machine learning [13].The MERA formalism can also be applied to a quantum field theory (QFT), after introducing a lattice as a UV regulator. For instance, when applied to a conformal field theory (CFT) [14][15][16], corresponding to a critical QFT, lattice MERA has been seen to accurately reproduce the universal properties of the corresponding quantum phase transition (as given by the conformal data) [4, 5]. However, introducing a lattice has a devastating effect on the spacetime symmetries of the original QFT, * qhu@perimeterinstitute.ca with e.g. translatio...

Liouville action as path-integral complexity: from continuous tensor networks to AdS/CFT

“…Nevertheless, explicit formulations of cMERA are so far only available for free field theories [5] (see [7,8,20,21] for various studies) which is the opposite regime from the strongly interacting CFTs which possess gravity duals, the so-called holographic CFTs. A formal construction of cMERA for general CFTs can be found in [14,19].…”

“…We can find an explicit relation between ρ and r and the function φ as follows (up to the linear order of M ) 20) where the function f (r) depends on the dimension d…”

Discrete gravity on random tensor network and holographic Rényi entropy

“…G N is the Newton constant in d dimensions. S EE (A) satisfying RT formula is referred to as the holographic entanglement entropy (HEE) [16][17][18][19][20][21][22] The tensor network can be understood as a discrete version of the AdS/CFT correspondence [23][24][25][26][27][28][29][30]. The tensor network states approximates the CFT state at the boundary, while the structure of tensor network emerges an bulk dimension built by layers of tensors.…”

Complexity functionals and complexity growth limits in continuous MERA circuits

“…Finally, it would be worth to better understand what kind of universal data can be extracted from complexity [63]. One hint may come from the established connection between the complexity measure based on the cMERA circuit length and the amount of Left-Right Entanglement created along the RG-flow.…”