Afternoon Session: Tensor Network States

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