WORM ALGORITHM APPLICATIONS

1
WORM ALGORITHM APPLICATIONS
Nikolay Prokofiev, Umass, Amherst
Boris Svistunov, Umass, Amherst
Many thanks to collaborators
Igor Tupitsyn, PITP
Vladimir Kashurnikov, MEPI, Moscow
Massimo Boninsegni, UAlberta, Edmonton
Matthias Troyer, ETH
Lode Pollet, ETH
Anatoly Kuklov, CSI, CUNY
NASA
Les Houches, June 2006
2
Worm Algorithm
No critical slowing down (efficiency)
New quantities, more theoretical tools to
address physics
Grand canonical ensemble Off-diagonal
correlations Single-particle and/or condensate
wave functions Winding numbers and
Better accuracy Large system size Finite-size
scaling Critical phenomena Phase diagrams
Reliably!
Examples from superfluid-insulator transition,
spin chains, helium
solid glass, deconfined criticality,
holes in the t-J model, resonant
fermions,
3
Superfluid-insulator transition in disordered
bosonic system
For any finite the sequence is always SF
- Bose glass - Mott insulator
Fisher, Fisher, Weichman,Grinstein 89
Not found in helium films Disproved in
numerical simulations (many, 1D and 2D) New
theories to support direct SF-MI transition have
emerged
?
4
- The data look as a perfect direct SF-MI
transition ( )
- Up to the data look as a
direct SF-MI transition, but
40
160
160
10
10
For small the Bose glass state is
dominated by rare (exponentially)
statistical fluctuations resulting in hole-rich
and particle-rich regions
5
Wave function of the added particle
Complete phase diagram
Gap in the Ideal system
It is a theorem that for the
compressibility is finite
6
Quantum spin chains magnetization curves, gaps,
spin wave spectra
S1/2 Heisenberg chain
MC data
Bethe ansatz
7
Line is for the effective fermion theory with
spectrum
deviations are due to magnon-magnon interactions
Lou, Qin, Ng, Su, Affleck 99
8
Energy gaps
One dimensional S1 chain with
Spin gap
Z -factor
9
Spin waves spectrum
One dimensional S1 Heisenberg chain
10
Superfluid (XY) insulator transition in the one
dimensional S1 Heisenberg chain
Kosterlitz-Thouless scaling
Is (red curve) an exact answer
?
11
First principles simulations of helium
Density matrix
close to
Superfluid hydrodynamics (Bogoliubov)
64
2048
Finte-size scaling
12
64
2048
calculated
experiment
Better then 1 agreement at all T after
finite-size scaling
13
Insulating hcp crystals of He-4
Exponential decay of the single-particle density
matrix
near melting
14
Activation energies for vacancies and
interstitials
, of course
Melting density, N800, T0.2 K E(N1)-E(N) can
not be done with this accuracy
Large activation energies at all Pressures
(thermodynamic limit)
In fact, the vacancy gas, even if introduced by
hand, is absolutely unstable and phase
Separates (grand canonical simulations with
)
15
Superglass state of He-4
Monte Carlo temperature quench from normal liquid
Single-particle density matrix
density-density correlator
ODLRO,
16
Superglass state of He-4
Condensate wave function maps reveal broken
translation symmetry
density of points
10 slices across the z-axis
A rough estimate of metastability
17
Superfluid ridges and interfaces in He-4
Each of the 8 cubes is a randomly oriented
crystallite (24
interfaces)
the 4 slices
Condensate maps
across x-axis
simulation box
across y-axis
across z-axis
18
Worm Algorithm
- Extended configuration space for
- All updates exclusively through
local moves of source/drain or
etc. operators
New quantities, more theoretical tools to
address physics
No critical slowing down (efficiency)
Better accuracy Large system size Finite-size
scaling Critical phenomena Phase diagrams
Reliably!
Grand canonical ensemble Off-diagonal
correlations Single-particle and/or condensate
wave functions Winding numbers and
classical stat. mech. models Ising, lattice
field theories, polymers, quantum lattice spin
and particle systems, continuous space quantum
particle systems (high-T series, Feynman diagrams
in either momentum or real space, path-integrals,
whatever loop-like )