Afternoon Session: Tensor Network States

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Bela's Notes

Talks

  • Chair: Guifre Vidal
  • Speakers
    • Tomotoshi Nishino
      • How to optimize?
        • "Alternating least squares" or generalized eigenvalue problems
      • IRF vs vertex type - different ansatz?
        • Yes
      • Equilibration problems for strong first order transitions?
        • KT transitions can be hard
        • Finding ground state can be hard
    • Valentin Murg
      • Fidelity: how much can one trust it?
        • Gapless: good energy, bad overlap
        • Gapped (large): easier
    • Bela Bauer


Discussion

  • What causes translational symmetry breaking?
    • Finite bond dimension
    • Symmetry breaking states have lower entanglement
  • Why MERA-chi so much larger than PEPS-M?
    • Cut off height of MERA tree
    • Any spin is only correlated with a finite number of other spins
  • Question?
    • Convergence
    • Unbiased method
      • Example series expansion: one starts from a reference state
        • This might affect result of series expansion calculation
      • Example VMC: choice of basis biases answer
      • Kagome: multiplicity of unit cell
        • This seems the best ground state
        • Finite correlations biases towards crystal
        • Tradeoff: higher tree / smaller chi
      • Semantics: no a priori assumptions
      • Bias towards low-entanglement states
      • Kagome up and down some more
    • Classify systems b entanglement instead of "sign problem"
    • How to quantify entanglement?
      • von Neumann entropy
    • Area law is necessary to be able to address the thermodynamic limit
    • Finding Haldane gap for S=5?
    • Range of applicability <=> entanglement

Simon's Notes

Talks

  • Guifre Vidal:
    • introduces tensor network states
    • one perspective: generalization of mean-field approach to higher bond dimensions
  • Tomotoshi Nishino:
    • Probably the oldest reference to MPS: Kramers-Wannier approximation (1941)
    • Historical overview (classical models): Baxter, Nightingale & Bloete
    • Gives various examples for 2D stensor product state calculations for class. models:
      1. 3D Ising model, Tc (from MC) determined within 0.3% (0.6%) for bond dimension m=3 (m=2)
      2. 3D ANNNI model
    • Questions:
      • How do you optimize (variationally) the tensors?
      • Have you observed problems when trying to simulate "slowly equilibrating systems"?
        • One example: KT transitions
  • Valentin Murg:
    • PEPS (finite) for 2D quantum models
    • (scaling of) energy minimization vs. time evolution step
    • Applications: J1,J2,J3 model, ultracold fermions in optical lattices
    • Questions:
      • Quality of wavefunction overlap estimates
  • Bela Bauer:
    • Accuracy of PEPS on infinite lattices
    • Examples: Ising, 1st order transition, frustrated models
    • high-accuracy for 1st order transitions (because of low entanglement)
    • Questions:
    • Why does computation time depend on size of unit cell?

Discussion

  • Guifre shows results for kagome AFM and free fermions on square lattice
  • TNS are biased towards low-entanglement solutions