Continuous Tensor Network States for Quantum Fields

@article{Tilloy2019ContinuousTN,
  title={Continuous Tensor Network States for Quantum Fields},
  author={Antoine Tilloy and Juan Ignacio Cirac},
  journal={Physical Review X},
  year={2019}
}
We introduce a new class of states for bosonic quantum fields which extend tensor network states to the continuum and generalize continuous matrix product states (cMPS) to spatial dimensions $d\geq 2$. By construction, they are Euclidean invariant, and are genuine continuum limits of discrete tensor network states. Admitting both a functional integral and an operator representation, they share the important properties of their discrete counterparts: expressiveness, invariance under gauge… 

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References

SHOWING 1-10 OF 77 REFERENCES
Continuum tensor network field states, path integral representations and spatial symmetries
A natural way to generalize tensor network variational classes to quantum field systems is via a continuous tensor contraction. This approach is first illustrated for the class of quantum field
Renormalization and tensor product states in spin chains and lattices
We review different descriptions of many-body quantum systems in terms of tensor product states. We introduce several families of such states in terms of the known renormalization procedures, and
Tensor Networks for Lattice Gauge Theories and Atomic Quantum Simulation
We show that gauge invariant quantum link models, Abelian and non-Abelian, can be exactly described in terms of tensor networks states. Quantum link models represent an ideal bridge between
Holographic quantum states.
We show how continuous matrix product states of quantum fields can be described in terms of the dissipative nonequilibrium dynamics of a lower-dimensional auxiliary boundary field by demonstrating
Fermionic Projected Entangled Pair States and Local U(1) Gauge Theories
Tensor networks, and in particular Projected Entangled Pair States (PEPS), are a powerful tool for the study of quantum many body physics, thanks to both their built-in ability of classifying and
Calculus of continuous matrix product states
We discuss various properties of the variational class of continuous matrix product states, a class of Ansatz states for one-dimensional quantum fields that was recently introduced as the direct
Entanglement renormalization for weakly interacting fields
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
Anyons and matrix product operator algebras
Quantum tensor network states and more particularly projected entangled-pair states provide a natural framework for representing ground states of gapped, topologically ordered systems. The defining
Class of quantum many-body states that can be efficiently simulated.
  • G. Vidal
  • Physics, Medicine
    Physical review letters
  • 2008
We introduce the multiscale entanglement renormalization ansatz, a class of quantum many-body states on a D-dimensional lattice that can be efficiently simulated with a classical computer, in that
Holographic geometry of entanglement renormalization in quantum field theories
A bstractWe study a conjectured connection between AdS/CFT and a real-space quantum renormalization group scheme, the multi-scale entanglement renormalization ansatz (MERA). By making a close contact
...
1
2
3
4
5
...