Quantum Fields Adrin Franco-Rubio Joint work with Qi Hu and Guifr - - PowerPoint PPT Presentation

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Quantum Fields Adrin Franco-Rubio Joint work with Qi Hu and Guifr - - PowerPoint PPT Presentation

May 25, 2023 642 likes •919 views

Continuous Tensor Network Renormalization for Quantum Fields Adrin Franco-Rubio Joint work with Qi Hu and Guifr Vidal arXiv:1809.05176 Quantum Information and String Theory 2019 YITP, Kyoto The renormalization group [Kadanoff 66,

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Continuous Tensor Network Renormalization for Quantum Fields

arXiv:1809.05176

Joint work with Qi Hu and Guifré Vidal Adrián Franco-Rubio

Quantum Information and String Theory 2019 YITP, Kyoto

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The renormalization group

[Kadanoff ‘66, Wilson ‘71,…]

Study and comparison of the behaviour of a physical system at different scales Formalized as RG flow: RG step = Coarse-graining + Rescaling Fixed points of RG flow constitute conformal field theories (CFTs):

  • Describe universality classes (second order phase transitions)
  • Characterized by conformal data
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The renormalization group

Interesting problem: computational implementation of RG flow

Why?

Phase classification problems Computational efficiency Holography 29 May 2019 Adrián Franco Rubio - Perimeter Institute 3

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Tensor networks

Computational tools (+ physical insight!)

Including RG flow! quantum states statistical partition functions Euclidean path integrals

…on the lattice.

[MERA: Vidal ‘06] [TRG: Levin, Nave ‘06] [TNR: Evenbly, Vidal ‘14]

Allow for an efficient representation and manipulation of (e.g.)

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Continuous tensor networks: A research program

Aim: import ideas and techniques from (lattice) tensor networks to be applied in the realm of quantum field theory.

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Remark: Two main approaches to continuum limit

  • “N → ∞” limit: tensors become closer to a trivial tensor as their number

diverges

  • “Conceptual” limit: imitate tensor network constructions directly in a

continuum setting

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Continuous tensor networks: A research program

quantum states statistical partition functions Euclidean path integrals

In particular, RG flow:

29 May 2019 Adrián Franco Rubio - Perimeter Institute 6 [cMERA: Haegeman et al. ‘11] This talk

(See related work: Caputa et al., 2017, Bhattacharyya et al., 2018)

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This talk

✓ Introduction and motivation

  • Review of (lattice) Tensor Network Renormalization
  • Our proposal: continuous Tensor Network Renormalization
  • Proof-of-principle example: free boson
  • Conclusion

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Lattice TNR - Setting

[Evenbly, Vidal ‘14]

  • Bonds represent degrees of freedom we sum over
  • Tensors contain information of local Boltzmann

weights

  • Lattice spacing provides UV cutoff

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Lattice TNR - Algorithm

[Evenbly, Vidal ‘14] 29 May 2019 Adrián Franco Rubio - Perimeter Institute 9 RG flow in the space of tensors!

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Lattice TNR - Algorithm

[Evenbly, Vidal ‘14] RG flow in the space of tensors! Disentanglers and isometries (chosen variationally) provide a local rearrangement of DOF At each step, the lattice needs rescaling 29 May 2019 Adrián Franco Rubio - Perimeter Institute 10

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Lattice TNR - Flow

Each phase flows to a fixed, scale-invariant tensor. At criticality, conformal data are retrievable from the fixed point tensor!

[Evenbly, Vidal ‘14]

  • rdered phase

disordered phase 29 May 2019 Adrián Franco Rubio - Perimeter Institute 11

Example: phase diagram of the 2D lattice Ising model

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This talk

✓ Introduction and motivation ✓ Review of (lattice) Tensor Network Renormalization

  • Our proposal: continuous Tensor Network Renormalization
  • Proof-of-principle example: free boson
  • Conclusion

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cTNR - Setting

Euclidean Path Integral… [Hu, A.F.-R., Vidal ‘18] 29 May 2019 Adrián Franco Rubio - Perimeter Institute 13

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cTNR - Setting

Euclidean Path Integral… …with smeared fields [Hu, A.F.-R., Vidal ‘18] 29 May 2019 Adrián Franco Rubio - Perimeter Institute 14

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cTNR - Setting (Example)

We work with the free Klein-Gordon field: [Hu, A.F.-R., Vidal ‘18] 29 May 2019 Adrián Franco Rubio - Perimeter Institute 15

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cTNR - Setting (Example)

We work with the free Klein-Gordon field: The two-point function goes to a constant at scales smaller than the cutoff [Hu, A.F.-R., Vidal ‘18] 29 May 2019 Adrián Franco Rubio - Perimeter Institute 16

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cTNR - Algorithm

Don't forget goal: implement RG flow at the level of Euclidean path integrals

Scaling

Don't forget the lattice: local rearrangement of DOF + rescaling

Disentangling [Hu, A.F.-R., Vidal ‘18] 29 May 2019 Adrián Franco Rubio - Perimeter Institute 17

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cTNR - Algorithm

Lattice – continuum analogy RG flow of the Euclidean path integral 29 May 2019 Adrián Franco Rubio - Perimeter Institute 18

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Linear Ansatz for the disentangler: Massless free boson is a CFT, so we impose an RG fixed point condition:

cTNR – Algorithm (Example)

(+ the disentangler does not depend on scale) 29 May 2019 Adrián Franco Rubio - Perimeter Institute 19

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Success! The fixed point condition translates into

cTNR – Algorithm (Example)

Choosing appropriately quasilocal functions, we find an explicit realization for which the regularized free boson CFT is a fixed point of cTNR:

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cTNR – Algorithm (Example)

Understanding the transformations: The net effect of both generators on the cutoff is to leave it invariant

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cTNR – Algorithm (Example)

If we add a mass, we obtained the expected flow to a massive fixed point: We can also recover correct conformal data from a fixed point!

Free boson case → Smeared version of the original CFT scaling operators (rotation generator) 29 May 2019 Adrián Franco Rubio - Perimeter Institute 22

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This talk

✓ Introduction and motivation ✓ Review of (lattice) Tensor Network Renormalization ✓ Our proposal: continuous Tensor Network Renormalization ✓ Proof-of-principle example: free boson

  • Conclusion

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Take-home messages

  • Tensor networks are able to realize the RG flow on the lattice.
  • We propose a general scheme for a similar implementation of the

RG flow for continuous partition functions / Euclidean path integrals.

  • This scheme can be applied to the free boson theory, yielding the

correct fixed point behavior, and the correct conformal data.

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Outlook

  • Extension to fermions and gauge theories (19XX.XXXX)
  • Move past analytic solutions: truly variational algorithm (probably in

parallel to cMERA)

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Thanks!

To know more: arXiv 1809.05176 (or let’s talk!)

Acknowledgements