Morning Session: Exact Diagonalization and Series Expansion

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  • Chair: Simon Trebst
  • Speakers
    • Didier Poilblanc: Exact Diagonalization
      • Quantities that can be computed:
        • GS properties
        • Excitations
        • Dynamical correlations
        • Time evolution
      • What is it good for?
        • Benchmarks for DMRG, PEPS, QMC, ...
        • Hamiltonians with sign problems:
          • Fermions
          • Frustrated magnets
          • PBC
        • "Constrained models"
          • Hard core coverings of the lattice
          • Non-Abelian anyons
            • Finite-size scaling?
              • In principle applies to these models, even in presence of constraints
      • Efficiency of the method?
        • "tricks" depend on
          1. Hamiltonian
          2. Structure of Hilbert space
          3. Symmetries
        • Size of Hilbert Space is not the criterion (at least not the only one)
        • Fixed size of HS (this ranks coding difficulty):
          • (Fermi) Hubbard no symm. < (Fermi) Hubbard < Heisenberg < t-J < QDM < anyons
        • Symmetries can make coding much more complicated without computational advantage
      • What can be done currently? See table below
        • How long does 4x4 Hubbard take?
          • Can be done on a PC
        • Computation time depends strongly on how difficult it is to generate Hamiltonian/states
      • Use of symmetries
        • Reduce matrix to some block-diagonal form
        • Use only one representative from each irreducible representation
        • Identifying representatives can be expensive, but depends stronly on system
      • Limitations:
        • Memory problem for storing one wave-vector
        • CPU-time limitation with on-the-fly matrix generation
        • Disk size limitation of storing the full matrix
      • How to store on disk for ED/DMRG?
        • NAS RAID systems with parallel file systems
        • DMRG for FQH: 200 GB
        • ED: 100 GB
      • Approaching a quantum phase transition makes ED ~2 more difficult due to smaller separation of low-lying states
      • How to generate restricted Hilbert spaces?
        • Start from given state, apply Hamiltonian -> can be vectorized
        • Generates states within a given topological sector
      • Finite size scaling: what happens with those irregular lattices or very small sizes?
        • Classifying clusters due to neighbours: Betts (Canada)?
        • Small lattices has other symmetries than the infinite system: accidental degeneracies
      • Soft bosons?
        • Truncate occupation and check consistency of results
        • Adjust truncation
Models -> Hubbard no symm Hubbard Frustrated Heisenberg t-J model QDM Fibonacci anyon ladder Fibonacci anyon ladder
# states in HS (most symmetrical sector) 159 * 10^9 5.7 * 10^9 3.4 * 10^9 2.8 * 10^9 10^8 30*10^6 < 10^6
systems 16 fermions/24 sites 20 fermions, C_20 molecule 40 sites 4 holes, 32 sites 2 holes, 8x8 QDM 2x21 2x8
Computers? Earth Simulator (???) ... ... ... ... NEC SX8
  • Rajiv Singh
    • Milestone 1: superconductivity in HTE of t-J
    • high parallelizability
    • How does one find errors?
      • Subgraph subtraction
      • This does not help that one is correct at 0th order
    • How to quantify error in series order expansion?
      • Heuristic
      • Cross-check different resummation techniques, different Pade approximants
      • Even if convergence radius is 0, resummation can be well-behaved
    • Rigorous upper/lower bounds?
      • In general: No.
      • Ising model: coefficients are all of same sign, that makes bounds possible
      • Spin-S Heisenberg: convergent series
      • Bose-Hubbard: divergent series (convergence radius of bare series), but still good results
    • Milestone 2: HTE for Triangular Lattice Heisenberg Model
    • Milestone 3: Multiparticle Spectra in d=2 and higher
    • Other challenges: DCP, spin liquids
    • Kagome: exotic phase without order
      • Series expansion: good in ordered phases
      • Analyzing spectral weights with ED
      • Is there a good theory at this point?
        • VB phase is well-understood
        • Is that the right phase?
    • Conceptual challenge 1: Lifetime effects
    • Conceptual challenge 2: Stochastic Approaches
      • How does Monte Carlo error in diagram weights affect series extrapolation?
        • Badly in SSE
        • Diagrammatic MC: largest term comes with largest error, one can neglect errors on previous terms
    • Conceptual challenge 3: ...


  • Series expansion in higher-dimensional cases?
    • Leading-order HTE becomes exact at some point for classical models
    • 1/d expansion
  • Only good for lattice systems?
    • Classical systems: continuum was done
    • Quantum system: has not been done, like Virial expansion
    • Technically very difficult
  • Identify families of clusters of different order that can be related?
    • Cluster weight
    • Hardest cluster: largest number of embeddings
  • Largest cluster for Kagome?
    • Zero-temperature calculation
    • Very special problem due to lattice geometry (only few empty triangles)
  • HTE is unbiased
  • Zero-temperature calculations are specific to each problem -> no milestone challenges
  • Next milestone for ED?
    • Not only progress in technique, but progress in effective Hamiltonians with more and more terms
    • No sign problem?
  • Fundamental difference between ED and DMRG?
    • No bias
    • Not variational -> not possible to be stuck in local minima; could however look in wrong symmetry sector
    • Trapping can be just beyond ED; but where is this point?
    • Local updates restrict convergence of DMRG - non-local changes to MPS?