Morning Session: DMRG

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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