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Lattice gauge theories with Tensor Networks Luca Tagliacozzo Based - - PowerPoint PPT Presentation

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Lattice gauge theories with Tensor Networks Luca Tagliacozzo Based on: L. Tagliacozzo G. Vidal Entanglement renormalization and gauge symmetry Phys. Rev. B 83, 115127 (2011) L. Tagliacozzo, A. Celi, M. Lewenstein Tensor Networks for

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Lattice gauge theories with Tensor Networks

Luca Tagliacozzo Based on:

  • L. Tagliacozzo G. Vidal

“Entanglement renormalization and gauge symmetry”

  • Phys. Rev. B 83, 115127 (2011)
  • L. Tagliacozzo, A. Celi, M. Lewenstein “Tensor

Networks for Lattice Gauge Theories with continuous groups”, arXiv:1405.4811

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07/13/16 Luca Tagliacozzo, LGT and TN 2

Outline

Gauge theories in HEP 5 min Lattice gauge theory 5 min Motivation for TN and LGT 1 min Symmetries and superposition 15 min BB Exotic phases of matter 5 min Intro to Tensor Networks 5 min Intro to LGT (Z2) 20 min BB TN for Gauge theories (Z2) 20 min BB Generalization 10 min Example of results (2D MERA + PEPS) 5 min

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Outline

Gauge theories in HEP Lattice gauge theory Motivation for TN and LGT Symmetries and superposition Exotic phases of matter Intro to Tensor Networks Intro to LGT (Z2) Tensor Networks for Gauge theories (Z2) Generalization Example of results (2D MERA + PEPS)

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Gauge Theories

→ HEP, form QED, QCD, Standard Model, elementary gauge bosons → COND-MAT spin liquids, dimers (electrons in a material), emerging gauge bosons → Lattice allows for non-perturbative formulation of QCD

Wilson, K. G. Confinement of quarks.

  • Phys. Rev. D 10, 2445–2459 (1974).
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Outline

Gauge theories in HEP Lattice gauge theory Motivation for TN and LGT Symmetries and superposition Exotic phases of matter Intro to Tensor Networks Intro to LGT (Z2) Tensor Networks for Gauge theories (Z2) Generalization Example of results (2D MERA + PEPS)

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Lattice gauge theories

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Achievements LGT

  • Evidences of mass-gap in Yang Mills from

first principles.

  • Precise determination
  • f the lowest excitations

(agreement with experiments)

  • Matrix elements input for

phenomenology of Standard model

Fodor, Z. & Hoelbling, C. Light Hadron Masses from Lattice QCD.

  • Rev. Mod. Phys. 84, 449–495 (2012).

Aoki, S. et al. Review of lattice results concerning low energy particle physics. ArXiv:1310.8555

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Limitations LGT

  • Classification of phases
  • QCD at non-zero temperature and density

(nuclear matter)?

  • Real time dynamics (experiments at RICH and

CERN)

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Outline

Gauge theories in HEP Lattice gauge theory Motivation for TN and LGT Symmetries and superposition Exotic phases of matter Intro to Tensor Networks Intro to LGT (Z2) Tensor Networks for Gauge theories (Z2) Generalization Example of results (2D MERA + PEPS)

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Achievements in TN/Quantum Many Body

  • Study of frustrated and fermionic systems
  • Out of equilibrium dynamics
  • Characterization of topological phases

Corboz, P., Evenbly, G., Verstraete, F. & Vidal, G. Simulation of interacting fermions with entanglement renormalization.

  • Phys. Rev. A 81, 010303 (2010).

SEE PHILIPPE/TAO

  • Vidal, G. Efficient Classical Simulation of Slightly

Entangled Quantum Computations.

  • Phys. Rev. Lett. 91, 147902 (2003).
  • White, S. R. & Feiguin, A. E.

Real time evolution using the density matrix renormalization group.

  • Phys. Rev. Lett. 93, (2004).
  • Kitaev, A. & Preskill, J.

Topological Entanglement Entropy. Phys. Rev. Lett. 96, 110404 (2006).

  • Levin, M. & Wen, X.-G.

Detecting Topological Order in a Ground State Wave Function.

  • Phys. Rev. Lett. 96, 110405 (2006).

See also FRANK/NORBERT/FRANK

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Outline

Gauge theories in HEP Lattice gauge theory Motivation for TN and LGT Symmetries and superposition Exotic phases of matter Intro to Tensor Networks Intro to LGT (Z2) Tensor Networks for Gauge theories (Z2) Generalization Example of results (2D MERA + PEPS)

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Symmetry and superpositon

  • We can try to construct local H whose ground

state has large superpositions

  • One possibility is Hamiltonian with a

symmetry

PRODUCT GROUND STATE ENTANGLED GROUND STATE

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Fate of large superpositions

  • If there is a global discrete symmetry, it is

spontaneously broken in the ground state (Absence of macroscopic cat states)

  • If there is a local discrete symmetry the

symmetry is not broken in the ground state (Presence of long range entanglement and short correlations)

  • Phase transition without symmetry breaking....
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Outline

Gauge theories in HEP Lattice gauge theory Motivation for TN and LGT Symmetries and superposition Exotic phases of matter Intro to Tensor Networks Intro to LGT (Z2) Tensor Networks for Gauge theories (Z2) Generalization Example of results (2D MERA + PEPS)

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Outline

Gauge theories in HEP Lattice gauge theory Motivation for TN and LGT Symmetries and superposition Exotic phases of matter Intro to Tensor Networks Intro to LGT (Z2) Tensor Networks for Gauge theories (Z2) Generalization Example of results (2D MERA + PEPS)

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Notation

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Quantum Many Body

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Tensor Networks for LGT

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What do TN describe

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Outline

Gauge theories in HEP Lattice gauge theory Motivation for TN and LGT Symmetries and superposition Exotic phases of matter Intro to Tensor Networks Intro to LGT (Z2) (Blackboard) Tensor Networks for Gauge theories (Z2) Generalization Example of results (2D MERA + PEPS)

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Constructing Z2 LGT

Discussed by Kogut & Susskind, M. Creutz 70s

  • Definition of a group
  • Group algebra
  • Building regular representation

matrices

  • Irreducible representations
  • The local symmetry
  • Interactions
  • Hamiltonian
  • Phases
  • TN ansatz
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Outline

Gauge theories in HEP Lattice gauge theory Motivation for TN and LGT Symmetries and superposition Exotic phases of matter Intro to Tensor Networks Intro to LGT (Z2) (Blackboard) Tensor Networks for Gauge theories (Z2) Generalization Example of results (2D MERA + PEPS)

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Hamiltonian LGT

Discussed by Kogut & Susskind, M. Creutz 70s

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Constructing a LGT

  • Constituents on links
  • Local symmetry operators
  • Left right rotations of the state

Tagliacozzo, L., Celi, A. & Lewenstein, M. TN for LGT with continuous groups. ArXiv:1405.4811

Notion of symmetry

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Tensors

a) b)

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Orthogonality theorem

Matrix representation of g in irrep r:

Serre, J.-P. Linear representations of finite groups. (Springer-Verlag, 1977).

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LR multiplication

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Generalized cross

  • perators
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Generalized disentanglers

  • U operators
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Gauge invariant Hilbert space

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Dynamic on Hp

Kogut, J. & Susskind, L. Phys. Rev. D 11, 395 408 (1975). – Creutz, M. Phys. Rev. D 15, 1128 (1977).

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Outline

Gauge theories in HEP Lattice gauge theory Motivation for TN and LGT Symmetries and superposition Exotic phases of matter Intro to Tensor Networks Intro to LGT (Z2) (Blackboard) Tensor Networks for Gauge theories (Z2) Generalization Example of results (2D MERA + PEPS)

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The two ways

MERA,Hierarchical TN

Tagliacozzo, L. & Vidal, G.

  • Phys. Rev. B 83, 115127 (2011)

Tagliacozzo, L., Celi, A. & Lewenstein, M. ArXiv:1405.4811

TPS/PEPS

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Variational Ansatz for gauge invariant states

  • Phys. Rev. B 83, 115127 (2011)
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Low energy spectrum MERA

Z2 LGT 8x8 torus

  • Phys. Rev. B 83, 115127 (2011)
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Disorder parameter MERA

Z2 LGT 8x8 torus

  • Phys. Rev. B 83, 115127 (2011)
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Topological fidelities MERA

  • Phys. Rev. B 83, 115127 (2011)
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Topological QPT with TPS

From the ground state of to the ground state of

Through a wave function modification

ArXiv:1405.4811

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Topological Entropy

ArXiv:1405.4811

Stéphan et. al. Phys. Rev. B 80, 184421 (2009). Stéphan et. al. J. Stat 2012, P02003 (2012).

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Schmidt-gap

Does not detect the topological phase transition

ArXiv:1405.4811

Li, H. & Haldane, F. D. M. Phys. Rev. Lett. 101, 010504 (2008). De Chiara et. al Phys. Rev. Lett. 109, (2012).

  • A. Läuchli, arXiv:1303.0741

Luitz, D. et al. J. Stat. 2014, P08007 (2014).

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Conclusions

  • I have justified the need of TN framework to

analyze LGT

  • It is suited both for theoretical analysis

and to design numerical ansatz

  • Discrete, continuous Abelian and Non-Abelian

model can be considered

  • Both hierarchical TN and TPS/PEPS
  • Already have benchmark numerical results in

2D

  • Easily extended to include matter
  • Interesting time to come...

THANKS FOR THE ATTENTION !!!