Transcript Document

Quantum Monte Carlo methods for charged
systems
N
H  
i 1
2
2mi
 
2
i
i j
ei e j
ri  rj
•Charged systems are the basic model of condensed matter physics
How can this be done with cold atoms?
ÞHow can they be simulated with Quantum Monte Carlo?
Phase diagram of the “one component plasma” in 2D
(as important as hubbard model?)
Many different quantum Monte Carlo methods
Which Hamiltonian?
•Continuum
•Lattice (Hubbard, LGT)
Which ensemble?
T=0
• VMC (variational)
• DMC/GFMC (projector)
or T>0?
• PIMC (path integrals)
Which basis?
Particle
• Coordinate Space
• Sz representation for
spin
Occupation Number
• Lattice models
Wave functions
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•
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Hartree-Fock
Slater-Jastrow
Backflow, 3 body
Localized orbitals in
crystal
Imaginary-time path integrals
The thermal density matrix is:
ˆ  e   (T V )
• Trotter’s theorem (1959):
ˆ  lim M  e

 T
e
 V
   /M
Z   dR 1...dR M e

M
 S(R i ,R i1 ; )
i 1
(R 0  R1 ) 2 τ
S ( R0 , R1 ; ) 
  V(R 0 )  V(R 1 ) 
4λτ
2


M
“Distinguishable” particles within PIMC
•
•
•
•
•
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Each particle is a ring
polymer; an exact
representation of a
quantum wavepacket in
imaginary time.
Integrate over all paths
The dots represent the
“start” of the path. (but all
points are equivalent)
The lower the real
temperature, the longer the
“string” and the more
spread out the wavepacket.
Path Integral methods can
calculate all equilibrium
properties without
uncontrolled
approximations
We can do ~2000 charges
with ~1000 time slices.
Bose/Fermi Statistics in PIMC
• Average by sampling
over all paths and over
connections.
• At the superfluid
transition a “macroscopic”
permutation appears.
• This is reflection of bose
condensation within
PIMC.
• Fermion sign problem: -1
for odd permutations.
Projector Monte Carlo (T=0)
aka Green’s function MC, Diffusion MC
• Automatic way to get better wavefunctions.
• Project single state using the Hamiltonian
 (t )  e (HE)t (0)
• This is a diffusion + branching operator.
• Very scalable: each walker gets a processor.
• This a probability for bosons/boltzmanons since ground state can
be made real and non-negative.
• Use a trial wavefunction to control fluctuations and guide the random
walk; “importance sampling”
• For a liquid we use a Jastrow wavefunction, for a solid we also use
Wannier functions (Gaussians) to tie particles to lattice sites.
• More accurate than PIMC but potentially more biased by the trial
wavefunction.
How can we handle charged systems?
• If we cutoff potential :
– Effect of discontinuity never disappears: (1/r) (r2) gets bigger.
– Will not give proper plasmons because Poisson equation is not
satisfied
• Image potential solves this: VI =  v(ri-rj+nL)
– But summation diverges. We need to resum. This gives the
Ewald image potential.
– For one component system we have to add a background to
make it neutral (background comes from other physics)
– Even the trial function is long ranged and needs to be resummed.
Computational effort
•
•
r-space part same as short-ranged potential
O(N3/2)
k-space part:
1. Compute exp(ik0xi) =(cos (ik0xi), sin (ik0xi)), k0=2 /L.
2. Compute powers exp(i2k0xi) = exp(ik0xi )*exp(ik0xi)
O(N)
etc. Get all values of exp(ik . ri) with just multiplications.
O(N3/2)
3. Sum over particles to get k for all k.
4. Sum over k to get the potentials.
O(N3/2)
O(N1/2)
Constant terms to be added.
O(1)
Table driven method used on lattices is O(N2).
For N>1000 faster methods are known.
9
Fixed-node method
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•
Impose the condition:
This is the fixed-node BC
•
Will give an upper bound to the exact
energy, the best upper bound consistent
with the FNBC.
 ( R)  0 when  T ( R)  0.
EFN  E0
EFN  E0 if 0 ( R ) ( R )  0 all R
•f(R,t) has a discontinuous gradient at the nodal location.
•Accurate method because Bose correlations are done exactly.
•Scales well, like the VMC method, as N3.
•Can be generalized from the continuum to lattice finite temperature,
magnetic fields, …
Dependence of energy on wavefunction
3d Electron fluid at a density rs=10
Kwon, Ceperley, Martin, Phys. Rev. B58,6800, 1998
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Wavefunctions
– Slater-Jastrow (SJ)
– three-body (3)
– backflow (BF)
– fixed-node (FN)
Energy < |H| > converges to ground
state
Variance < [H-E]2 > to zero.
Using 3B-BF gains a factor of 4.
Using DMC gains a factor of 4.
-0.107
-0.1075
Energy
•
-0.108
FN -SJ
-0.1085
FN-BF
-0.109
0
0.05
Variance
0.1
The 2D one component plasma
PRL 103, 055701 (2009); arXiv:0905.4515 (2009)
•Electrons or ions on liquid helium
•Semiconductor MOSFET
•charged colloids on surfaces
We need cleaner experimental systems!
Bryan Clark, UIUC & Princeton
Michele Casula, UIUC & Saclay, France
DMC UIUC
Support from: NSF-DMR 0404853
Phase Diagram for 2d boson OCP
(up to now)
rs 
a
aB
Hexatic phase
G ~124
e2
G
kTa
Classical
plasma
T (mR)
Quantumclassical
crossover
Electrons on helium
Wigner
crystal
1 / rs =(density)1/2
Quantum
fluid
electrons
rs ~ 60 ~ 3x1012 cm2
De Palo, Conti, Moroni, PRB 2004.
Inhomogenous phases
Maxwell construction
• Cannot have 2-phase coexistence at first
order transition! the background forbids it
•
Jamei, Kivelson and Spivak [Phys. Rev.
Lett. 94, 056805 (2005)] “proved” (with
mean field techniques) that a 2d charged
system cannot make a direct transition
from crystal to liquid
• a stripe phase between liquid and crystal
has lower energy
• Does not prove that stripes are the lowest
energy state, only that the pure liquid or
crystal is unstable at the transition
assume Boltzmann statistics – no
fermion sign problem.
F
area
snapshots
quantum
122<G<124
classical
Triangular lattice forms
spontaneously in PIMC
Structure Factors
PIMC
Exper.
Keim, Maret, von
Grunberg.
Classical
MC
Hexatic order
r
g 6 (r )   *  r    0 
  r      ri  r    rij  rc e
i 6ij
j ,i
He,Cui,Ma,Liu,Zou PRB 68,195104 (2003).
Muto , Aoki PRB 59, 14911(1999)
r
2d OCP Phase Diagram
PRL 102,055701 (2009)
Clausius-Clapeyron relation :
First order transition is on “nose”
dT
K

1

1
E
Trs drs
Hexatic phase
Classical
plasma
T (mR)
Wigner
crystal
Quantum
fluid
1 / rs
Transition order differs from KT?
Internal energy vs T
?
1st order
T (mR)
1 / rs
T(mR)
2nd order
Unusual structure in peak of S(k)
Could be caused by many small
crystals
rs=65, N=2248
Structures exist in transition region
•Are structures real? Or an
ergodic problem
•Are they different from a liquid?
•Can we make a ground state
model? Not one that is
energetically favorable.
2D Bose OCP
Normal fluid
hexatic
T (mR)
Wigner crystal
superfluid
2dOCP fermion Phase diagram
UNKNOWN
Quantum
Fluid
Superconductor?
• 2d Wigner crystal is a spin
liquid.
• Magnetic properties are
nearly divergent at melting
(2d) and (nearly) 2nd order
melting.
• But sign problem?
Phase Diagram of 3DEG
• second order partially polarized
transition at rs=52 like the Stoner
model (replace interaction with a
contact potential)
• Antiferromagnetic Wigner Crystal
at rs>105
Polarization
transition
Conclusions
• Long-ranged interactions are not an intractable problem for
simulation.
• We have established the outlines of the OCP phase diagram for
boltzmannons and bosons.
• Evidence for intervening inhomogeneous phases is weak
• Future work: fermi statistics – but the “sign problem” makes the fluid
phases challenging (not hopeless).
• The OCP is a good target for a “quantum emulator.”
• With optical lattice+disorder one can reach some of the most
important problems in CMP.