# Afternoon Session: Quantum Monte Carlo

From ALPS

## Talks

- Topic, more specifically: Path Integral Quantum Monte Carlo

- Chair: Boris Svistunov and Synge Todo
- Speakers
- Synge Todo:
*Parallel Monte Carlo*- Why large load imbalance for parallel tempering?
- Dependence of CPU time on temperature requires fine-tuning

- Multi-canonical ensemble
- Weight depends on global properties of the system

- Why large load imbalance for parallel tempering?
- Simon Trebst:
*Extended statistical ensembles*- More than one variable? Possible, but no examples where it helps significantly
- Energy/temperature are good choices for thermal phase transitions
- Ising: energy allone suffices, cannot get better than N^2 behaviour

- More than one variable? Possible, but no examples where it helps significantly
- Anatoly Kuklov:
*Flowgram Method in QM*- Apply to known problems!

- Matthias Troyer:
*The Sign Problem*- Where does it appear?
- Bosonic systems: no
- Fermions: exchanging fermions leads to sign problems
- Bosons in a gauge field: phases
- Frustrated magnets: exchange leads to sign problem

- Can it be solved?
- Basis-dependent!
- In general: NP-hard & QMA-hard
- Scaling with system size: using an appropriate method away from criticality, the scaling should not be exponential
- In some cases, symmetries help to solve sign problem completely: Meron cluster algorithm
- In these cases, phases are close to classical phases

- Where does it come from?
- Sampling with absolute value of the weights is equivalent to sampling bosons to learn about fermions

- Where does it appear?
- Boris Svistunov

- Synge Todo:

## Discussion

- What models? What system sizes/temperatures?
- QMC: unfrustrated spin systems
- Why spin chains and Haldane conjecture?
- Because I can!
- Because Petaflop computers are there and this is an application: do what you can do
- First purpose of the petaflop machine is
**to be used**

- Create a list of open problems in physics?
- Quantum magnets: beta * Vol = 10^8 on standard clusters
- 10^7 spins or 10^5 lattice bosons or 10^4 continuous bosons on a single node
- 10^8 on a MPP: memory constraints
- Disorder etc: up to a few 100,000 CPUs

- Bosonic models without frustration (hopping matrix cannot trivially be mapped to positive definite matrix)
- No real time dynamics (only equilibrium statistics) (short time with diagrammatic MC)

- Quantum magnets: beta * Vol = 10^8 on standard clusters
- Challenges
- Better representations, better analytical/conceptual ideas are needed
- General challenges: solve the sign problem
- More specific ones: find a good representation for specific problems -> diagrammatic Monte Carlo
- Finite-size analysis
- First-order phase transitions
- Disrodered systems
- Correctly distinguishing second order from weakly first order transitions: flowgram technique

- Methods
- Loop algorithm
- Worm algorithm
- Directed loop
- Worm spin-offs
- Determinant Monte Carlo
- Under control?

- All of the above: path integral representation (and SSE)
- Discrete time (cont. space), continuous time (discr. space), SSE

- Treat
*representation*and*update strategy*separately - Are there more or less reliable methods?
- Some people call variational/fixed-node/... Monte Carlo QMC
- Reliable == controllable error (no systematic errors)
- Who's doing it?: Sufficient equilibration
- Diagrammatic MC: is it clear ow controlled it is?
- Path Integral MC is well-established
- Defining well-established
- Depend on mapping to classical system: d -> d+1
- Everything depends on having sufficiently good statistics
- Everything said about classical statistics maps into some quantum counterpart
- Many quantum problems map into well-behaved classical problems
- Slowly equilibrating problems are rarely considered in QMC calculations
- Counterexample: melting of solid Helium