# Afternoon Session: Tensor Network States

From ALPS

# 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

- How to optimize?
- Valentin Murg
- Fidelity: how much can one trust it?
- Gapless: good energy, bad overlap
- Gapped (large): easier

- Fidelity: how much can one trust it?
- Bela Bauer

- Tomotoshi Nishino

## 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

- Example series expansion: one starts from a reference state
- 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:
- 3D Ising model, Tc (from MC) determined within 0.3% (0.6%) for bond dimension m=3 (m=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