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
  • 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

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