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Operator-Algebraic Renormalization and Wavelets

Alexander Stottmeister, Vincenzo Morinelli, Gerardo Morsella, and Yoh Tanimoto
Phys. Rev. Lett. 127, 230601 – Published 1 December 2021
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  • Introduction.—
  • Operator-algebraic renormalization.—
  • Wavelets and the scalar field.—
  • Scaling limits of harmonic lattice…
  • Conclusions and outlook.—
  • ACKNOWLEDGMENTS
    • References

    Abstract

    We report on a rigorous operator-algebraic renormalization group scheme and construct the free field with a continuous action of translations as the scaling limit of Hamiltonian lattice systems using wavelet theory. A renormalization group step is determined by the scaling equation identifying lattice observables with the continuum field smeared by compactly supported wavelets. Causality follows from Lieb-Robinson bounds for harmonic lattice systems. The scheme is related with the multiscale entanglement renormalization ansatz and augments the semicontinuum limit of quantum systems.

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    • Received 13 February 2020
    • Revised 26 May 2021
    • Accepted 15 October 2021

    DOI:https://doi.org/10.1103/PhysRevLett.127.230601

    © 2021 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Lattice field theoryMathematical physicsRenormalization group
    1. Techniques
    Functional analytical methodsSecond quantization
    Statistical Physics & ThermodynamicsParticles & FieldsGeneral PhysicsQuantum Information, Science & Technology

    Authors & Affiliations

    Alexander Stottmeister1, Vincenzo Morinelli2, Gerardo Morsella3, and Yoh Tanimoto3

    • 1Institute of Theoretical Physics, University of Hannover, Appelstraße 2, 30167 Hannover, Germany
    • 2Dipartimento di Matematica, Sapienza Università di Roma, Piazzale Aldo Moro 5, 00185 Roma, Italy
    • 3Department of Mathematics, University of Rome “Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Roma, Italy

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    Issue

    Vol. 127, Iss. 23 — 3 December 2021

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    Images

    • Figure 1
      Figure 1

      Illustration of the decomposition of lattice sites for d=1 by an RG step determined by the scaling equation (5): On the left: The block-spin RG and its weights. On the right: The wavelet-based RG with weights determined by the low-pass filter of Daubechies’s D4 scaling function.

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    • Figure 2
      Figure 2

      Wilson’s triangle of renormalization: Vertical lines represent renormalization steps, either by coarse graining states (E’s) or by refining fields (α’s). Horizontal lines represent sequences of renormalized states considered on the algebra generated by fields and momenta at a fixed scale (right column).

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    • Figure 3
      Figure 3

      Illustration of the analytic MERA in d=1 induced by the wavelet scaling maps. From bottom to top: The first layer represents the isometric embedding IN+1N, and the second layer represents the action of the (dis)entangler UN+1 at scale N+1.

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