# Morning Session: Exact Diagonalization and Series Expansion

From ALPS

## Talks

- 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

- Finite-size scaling?

- Efficiency of the method?
- "tricks" depend on
- Hamiltonian
- Structure of Hilbert space
- 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

- "tricks" depend on
- 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

- How long does 4x4 Hubbard take?
- 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

- Quantities that can be computed:

- Didier Poilblanc: Exact Diagonalization

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

- How does Monte Carlo error in diagram weights affect series extrapolation?
- Conceptual challenge 3: ...

## Discussion

- 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?