# Morning Session: DMRG

From ALPS

## Talks

- Chairs: Adrien Feiguin & Ian McCulloch
- Speakers
- Adrien Feiguin
- Is there a way to argue for the optimality of the basis truncation?
- Maximal amount of information
- Reduced DM eigenvalues are probability distribution governing the mixed state
- ...

- Errors under control?
- GS DMRG: scaling of energy error is related to energy error (empirical, typical problems)
- In 2d class/1d quant gapped, Baxter proved exact expression of spectrum: exponential first, algebraic later
- In 2d quant, decay is slow (as in: algebraic)

- Local minima?
- Algorithm can be trapped, tricks exist

- Time evolution size:
- Up to 100 sites, or infinite

- PBC?
- Take OBC as limit of PBC where boundary states are uncorrelated
- Introducing correlations between boundary states approximates PBC

- When single-site (with perturbation to DM) or two-site DMRG?
- Single site: less states, one diagonalizes a smaller system
- Homogeneous translational invariant: two-site works well
- Inhomogeneous (local basis different on the sites) or ladder: single site algorithm works better
- Differences are minor

- Is there a way to argue for the optimality of the basis truncation?
- Ian McCulloch
- Energy variance plot can be regarded as very accurate tool to estimate ground state convergence, but it cannot be proven
- Convergence problems are related to non-local excitations which cannot be relaxed due to constraints
- Close to level crossings, convergence depends on initial states
- Level crossing can occur as basis size is grown

- Tomotoshi Nishino
- Long wavelength pattern in DMRG - related to convergence issues?
- Cross-check at different system sizes and temperatures

- Maximum system size?
- 2d: 32..118
- 3d: thousands of sites

- Long wavelength pattern in DMRG - related to convergence issues?

- Adrien Feiguin

## Discussion

- Questions
- Single-site vs double-site
- Excited states
- OBC/PBC
- Start with large number of states early?
- How to estimate error
- SUV spectrum (exact results)
- Finite matrix size imposes finite-size cutoff on correlations
- Haldane conjecture
- What systems work?
- Long-range interactions are difficult
- How many states in density matrix? -> Entanglement
- Weakly coupled ladders: hard
- More coupling between ladders: easier

- Next inspiration?
- "Mostly discovered by now" (Ian)

- Longest time scale for time evolution?
- Entropy scales (more or less)
- Not very long
- Hubbard chain, local quench: 50-60

- Large entanglement states go to thermal state at infinite temperature: Interesting for simulations?
- How do we approach random noise regime?

- Calculating gap in TEBD?
- Can use quantum numbers: this is a property of the ansatz, not the algorithm

- Not using symmetry may be better for infinite systems