Lattice Effective Field Theory and
Impurity Lattice Monte Carlo
Dean Lee (NC State)
Advances in Diagrammatic Monte Carlo
for Quantum Field Theory Calculations in
Nuclear, Particle, and Condensed Matter Physics
ECT*, Trento, Italy
October 5 – 9, 2015
1
Outline
Lattice effective field theory
Next-generation nuclear lattice simulations
New leading-order chiral EFT lattice action
Attractive Fermi polarons
Impurity lattice Monte Carlo method
Polarons in 2D zero-range theory
2
Lattice effective field theory
n
n
p
n
p
3 Low energy nucleons: Chiral effective field theory
Construct the effective potential order by order
N
N
N
N
N
N
N
π
π
N
N
N
N
N
N
N
N
π
N
Contact interactions
Leading order (LO)
Next-to-leading order (NLO)
4 Spherical wall method
Spherical wall imposed in the center of mass frame
Carlson, Pandharipande, Wiringa, NPA 424 (1984) 47
Borasoy, Epelbaum, Krebs, D.L., Meißner, EPJA 34 (2007) 185
Rwall
5 Energy levels with spherical wall
Energy shift from free-particle values gives the phase shift
6 Nucleon-nucleon phase shifts
S waves
7 P waves
8 Euclidean time projection
π
π
9 Auxiliary field method
We can write exponentials of the interaction using a Gaussian integral
identity
We remove the interaction between nucleons and replace it with the
interactions of each nucleon with a background field.
10 11 Theorem: Any fermionic theory with SU(2N) symmetry and two-body
potential with negative semi-definite Fourier transform obeys SU(2N)
convexity bounds.
Corollary: System can be simulated without sign oscillations
E
weak attractive potential
Chen, D.L. Schäfer, PRL 93 (2004) 242302;
D.L., PRL 98 (2007) 182501
2NK
2N(K+1)
2N(K+2)
A
A
2NK
2N(K+1)
2N(K+2)
E
strong attractive potential
12 Number of nucleons
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
0
-10
-20
-30
Ground State Energy (MeV)
-40
-50
-60
-70
-80
-90
-100
-110
-120
-130
-140
-150
-160
-170
13 Schematic of lattice Monte Carlo calculation
Hybrid Monte Carlo sampling
14 15 See Serdar Elhatisari’s talk at this workshop for recent results obtained
using nuclear lattice effective field theory and new work on ab initio
scattering of clusters using the adiabatic projection method.
16 Next-generation nuclear lattice simulations
Any successful product in the consumer market must balance higher
performance with lower price.
higher
performance
lower
price
Making real progress on both higher performance and lower price
requires some new disruptive technology that changes the equilibrium
point.
17
In nuclear lattice simulations, we must balance accurate nuclear forces
with reducing the sign problem in the many-body Monte Carlo
simulations.
accurate
nuclear forces
reduced
sign problem
It has long been said that non-local forces cannot be realized with
Monte Carlo simulations. However this is not true.
18
New leading-order chiral EFT lattice action
We use non-local contact interactions which are produced by smearing
the creation and annihilation operators.
These interactions can be reproduced with auxiliary fields without sign
oscillations by coupling the smeared density to an auxiliary field.
19
We also define the one-pion exchange potential in momentum space
using fast Fourier transforms on the lattice. We include a Gaussian
regulator in momentum space to reduce rotational symmetry breaking
due lattice artifacts.
See also Klein, D.L., Liu, Meißner, Phys. Lett. B747 511 (2015)
We compute phase shifts and mixing angles using the new method that
using spherical harmonic projection and auxiliary potentials in addition
to the spherical wall.
Lu, Lähde, D.L., Meißner, arXiv:1506.05652
20
Benchmark calculations using Gaussian interaction with strong tensor-forces
phase shifts δ / mixing angles ε (degrees)
30
3
δ( P0)
0
-30
-60
-90
0
30
30
60
90
3
δ( P1)
60
160
80
0
40
0
30
60
90 120
δ( S1)
120
30
90 120
3
0
0
20
30
60
90 120
3
0
20
0
30
60
90 120
-30
δ( F2)
0
30
ε3(D-G)
0
0
30
60
90 120
-10
60
90 120
0
6
3
δ( G4)
20
10
30
ε2(P-F)
30
60
90 120
0
30
60
90 120
90 120
3
δ( F3)
0
30
60
90 120
-30
12
4
0
0
0
30
60
90 120
-20
15
0
60
90 120
3
60
-10
0
30
60
90 120
ε4(F-H)
30
60
90 120
-20
3
δ( G3)
0
30
60
90 120
lattice, V0 = 0
lattice, V0 = -25 MeV
lattice, V0 = -20 MeV
continuum
0
90 120
0
0
3
5
δ( H4)
30
30
δ( D2)
0
δ( D3)
10
0
3
ε1(S-D)
8
10
-10
0
-2
60
0
2
0
30
3
δ( F4)
4
10
0
10
20
-20
20
0
3
-20
40
0
-10
30
-10
δ( P2)
-10
-30
10
10
3
δ( D1)
-20
0
20
10
0
0
30
60
90 120
pCM (MeV)
Lu, Lähde, D.L., Meißner, arXiv:1506.05652
21
3
160
20
20
40
100
[deg]
60
120
80
0
60
15
40
5
0
-5
0
10
15
Pcm (MeV)
3S
1
1
PWA93 (np)
LO
3S
1
4
0
2
3S
1
50
100
PWA93 (np)
Pcm (MeV)
LO
00
180
6
3P
2
50
33D PWA93 (np)
LO
24
150
100
Pcm (MeV)
200
150
150
50Pcm (MeV)
100
0
Pcm (MeV)
1
150
2.5
0
200
0
5 0
0
0
12
0
-10
-10
-5
-10
8
-10
1.5
2
-4
8
200
3P 0.5
0
20
2
00
50
-1
4
10
6
-0.5
0
3P
3S 8
a =1 41.97
2fm
2
0
2.5
-2
120
-2
100
150
50
1
100
Pcm (MeV)
200
150
14
4
0
-6
(3D2)(degrees)
0
(3D1)(degrees)
(1D2)(degrees)
[deg]
1
0
-10
-8
2-12
212
6
2
Work2in2progress, Li, Alarcon, Du, Elhatisari,
Klein,
Lu, Lähde, Epelbaum, D.L., Meißner
-4
1.5
-2
-4
-4
0
0
200
-2
-1
1P
1
-20
-10
-1
0
Pcm (MeV)
Pcm
-6
-5 -6
1 -8
6
41
0
50
1D
2
2
PWA93 (np)
LO
10 2
10
8
-10 2
0
-5
P0
10
4
3D-10
1
PWA93 (np)
-5
LO
3 3
14
-2
20
10
100
20
3.5
1S
0 2
-2
10
0
PWA93 (np)
LO
50
1
4
0
-4
3D 8
2
0
-10
-5
200
0
14 PWA93 (np)
LO
12
1P-5
1 6
-5
0
-5
-5
-10
-10
40
0
-4
3.5
5
10
200
-2
40
0 -10
)
[deg]
150
140
60
[deg]
100
0
60
160
(1P1)(degrees)
(degrees)
[deg]
[deg]
2
0
120
180
180
50
60
3
4
3S
1
200
PWA93 (np)
LO
0
180
10
0
1P
1
0
5
0
3P
100 0
( P20)(degrees)
(1D
)(degrees)
10
120
140
120
0
5
80
20 0
150
1S
0
30
0
0
3 (np)
LO
(degrees)
[deg]
(degrees)
[deg]
10
20
10
PWA93 (np)
LO
1P
1
P0
50
40
40
60
5
5
(3P0)(degrees)
1S
0
3P
0
3S
1
180
(1P1)(degrees)
60
1S 60
0
20
(3D
)(degrees)
(3P11)(degrees)
1S
0
20
70
-5
-1
(np)
3PWA93
LO
D3
0PWA93
50
(np)
LO
Pcm
0 PWA93 (np)
50
LO
Pcm
10
8
6
22
0
-5
-5
-10
-5
-2
2
PWA93 (np)
LO
0
-4
-10
-10
10
10
0
0
0
50
-10
100
150
0 200
0
Pcm
3.5
200 3
S
0
3.5
0
1 -2
(3P1)(degrees)
MeV)
-4
-51
P1
200
0
-12
0.40
0.2
2
14
12
10
0
-0.2
-0.5
0
10
1D 8
2
50
PWA93 (np)
LO
8
2
-5
-1
-4
150 200
Pcm (MeV)
3
100
200
150
200
0200
Pcm (MeV)
p3cms [MeV/c]
D3
-6
200
-12
-10
100
0
00
a = 1.97
fm
1
0.8
0.6
50
PWA93 (np)
-1
LO
200
50 100
100 150
150 200
PWA93 (np)
50
LO
0
Pcm
100 50
1D
2
8
6
4
0
[MeV/c]
Pcm (MeV)
0
pcms [MeV/c]Pcm
2 10
-6
-8 0
PWA93 (np)
LO
14 PWA93 (np)
LO
12
3D
1
PWA93 (np)pcms
-10
-10
LO
Pcm (MeV)
3P
2
Pcm (MeV)
150
0
pcms [MeV/c]
6
4
-4
200
-2
100 150
50
100
100
-2
4
-6
-6
-5-10
-10
0
0
100
50 100
50
PWA93
0 (np)
LO
6
1.5 0
PWA93
(np)
1
-3
LO
-2
-8
0.8
(3D3)(degrees)
3D
1
1503
Pcm (MeV)
-10
-10
0 -4(np)
0.6 PWA93
LO
s)
200
MeV)
-10
-4
-4
-8
-8
-1
-5
10
-3
2000
10
100
1
2
2 -2
2000.5
pcms [MeV/c]
150 200
0
1P
1
0
0
2
50
5
1
6
0.5
4
pcms [MeV/c]
-1
4PWA93 (np)
LO
2.5
0
0 0
-6
100
150
3
2
8
1.5
-0.5
50 100 100
Pcm (MeV)
0
-2
3D -2
3
1D
2
2
(3D2)(degrees)
00
-4
150
0
0
10
2
0
2
-10 10
PWA93 (np)
-10
LO
(1D2)(degrees)
00
2
-6
0
-5
-2
-2 0
[MeV/c]
3P
0
(3D1)(degrees)
200
-5
(3P2)(degrees)
100
-1
20
s)
(3P2)(degrees)
1S
0 4
0
2.5
1
(1P1)(degrees)
6
3D
1
00
1
D1)(degrees)
(3P(1)(degrees)
PWA93 (np)
3P LO
2
8
14 PWA93
PWA93 (np)
(np)
LO
LO
3
12
D2)(degrees)
(3P(0)(degrees)
5
50
Pcm (MeV)
2
0
200
1
Pcm (MeV)
0
50
Pcm
3D
3
2
0
23
-12
200
-5 0
MeV)
100
150
Pcm (MeV)
10
8
8
(3P2)(degrees)
6
0
MeV)
150
150
200
4
2
V)
0.4
cm
0
0
3 8
-2 D3
0
V)
-1
200
1
0
-10
3P
2
4
100
0 50
PWA93
20 (np)Pcm (MeV)
-8
LO
0
100
200
[MeV/c]
pcms [MeV/c]
0
200
2
0
50
0
-5
3D
1
2 -0.5
-2
1D
2
-1
-10
10
-6
50
100
100
2
p3cms [MeV/c]
P2 Pcm (MeV)
PWA93 (np)
0
0
6
LO
-10
-0.5 0
4
250
150
200
-3
200
0
-1
1
100
150
100
pcms [MeV/c]
-2
-2.5
200
Pcm (MeV)
-1
-1
2
-1.5
0
0
2
PWA93 (np)
LO
3
0 D3
50
Pcm
-2
0
0
150
200
100
200
pcms
[MeV/c]
100
150
100
0
-1.5
200
Pcm (MeV)
100
-10 0.5
-1
0.5
8
-10
-8
(MeV)
0
-5
-4
-4
0
200
5
6
-4
-6
-0.4
200
200
150
10
10
0.2
-0.2
150
100
0
1
-2
-2
-0.2
PWA93 (np)
LO
0
1(degrees)
(3D3)(degrees)
0.6
[deg]
1
100
50
-10
-5
-10
1
-0.4
P2
2
200
0
0
0
0
2
p Pcm100
[MeV/c]
-450 3cms
D (MeV) 150
0
150
Pcm (MeV)
-10
P
22
2
0.4
0
0
-2
15
0
0
-2
0
0
-4
0
0
-5
4
100
4 0.2
1
0
2
1
6
60
4
0.8
0.6
PWA93 (np)
LO
0
-6
200
5
3S 8
1
120
4
1D
22
50
PWA93 (np)
LO
0.83
2(degrees)
3S
1
10
6
2
0
1
(3D3)(degrees)
3D
1
10
10
[deg]
(3D2)(degrees)
[deg]
14 PWA93 (np)
-10
3D 0 LO
12
2
-5
0
-4
0
180
0
15
-6
200
0
-5
1P
1
-4
-2
20
10 50
10
5
1 (degrees)
150
0 -2
3P
1
3P
0
2 (degrees)
PWA93 (np)
LO
60
0
40
[de
[de
-8
0
1S
0-10
100
200
pcms [MeV/c]
pcms [MeV/c]
200
-2
-2
-2.5
-3
0 PWA93 (np)
LO
0
100
pcms [MeV/c]
100
150
50
200
-3
0
-1
100
pcms [M
200
Pcm (MeV)
0
a = 1.97 fm
100
pcms [MeV/c]
200
0
100
pcms [M
24
Sign problem for 6He simulations
1
dh = 1.00
dh = 0.85
dh = 0.75
dh = 0.65
dh = 0.55
dh = 0.35
0.9
0.8
0.7
0.6
ei
new lattice action
0.5
0.4
0.3
0.2
0.1
0
previous lattice action
0
2
4
6
8
10
12
14
16
18
20
22
Nt
Original plot from Lähde, Luu, D.L., Meißner, Epelbaum, Krebs, Rupak, Eur. Phys. A51 92 (2015)
25
Opens the possibility of simulations of neutron-rich nuclei with up to eight
neutron excess by direct simulation and up to twenty neutron excess by sign
extrapolation.
Lähde, Luu, D.L., Meißner, Epelbaum, Krebs, Rupak, Eur. Phys. A51 92 (2015)
For N = Z = even nuclei, may be possible to going up to A = 80.
26
Test of Galilean invariance
The non-local interaction produces some breaking of Galilean invariance. It
tends to reduce the effective mass of bound states.
6.5
2
k /2mα
E(latt)
n
E (MeV)
5.5
4.5
3.5
2.5
1.5
0.5
−0.5
−30
−20
−10
0
10
20
30
n
At lattice spacing a = 1.97 fm, the alpha-particle effective mass is about
10% larger than physical. The non-local interaction is actually improving
the effective mass of the alpha particle.
27
Current work and outlook
We are now looking at the performance of the new lattice action and Monte
Carlo codes for structure of medium mass nuclei, ground state energies of
neutron-rich nuclei, and scattering using the adiabatic projection method.
We are also working on making similar improvements to the higher order
corrections and doing many-body simulations with smaller lattice spacings.
28
Attractive Fermi polarons
Fermi gas
29
fermionic polaron
30
bosonic molecule
31
Impurity lattice Monte Carlo
Impurity lattice Monte Carlo is a hybrid method which treats the impurity
using diffusion Monte Carlo and all other particles using auxiliary field
Monte Carlo. We demonstrate using two-component fermions in d spatial
dimensions with zero-range interactions. We use lattice units where the
lattice spacing a = 1.
32
We use the transfer matrix formalism with temporal lattice spacing
at The :: symbols indicate normal ordering. Using our lattice Hamiltonian,
we get We now consider a system where we have N up-spin particles and just
one down-spin particle. We consider any worldline for the down-spin
particle. For simplicity we show the case for one spatial dimension. 33
at
a
34
at
a
Elhatisari, D.L., PRC 90 064001 (2014)
Bour, D.L, Hammer, Meißner, arXiv:1412.8175, PRL in press
35
at
a
36
at
a
37
Polarons in 2D zero-range theory
We consider one down-spin particle and N up-spin particles with
attractive zero-range interactions in 2D tuned to according the twobody binding energy EB or binding momentum κB.
We consider two possible sets of initial states.
The first is an initial state which is the ground state of the noninteracting (N, 1) system,
where
38
We also consider the molecular initial state
where the pair length is adjustable. It is convenient to define the
quantity
39
0.14
0.075
E(t) [latt. unit]
_
_
= 0.43h /a, L=30, N=21
B
= 0.43h /a, L=40, N=21
B
0.07
0.135
Initial state
Initial state
0.13
0.065
0.125
0.06
0
200
400
600
800
-0.002
0
200
400
800
-0.016
_
E(t) [latt. unit]
600
B
_
= 0.53h /a, L=70, N=20
Initial state
-0.003
B
= 0.62h /a, L=70, N=20
Initial state
-0.017
-0.004
-0.018
0
100
200
300
Lt
400
500
0
100
200
300
400
500
Lt
40
-0.2
-0.6
Solid: [5] Polaron variational
calculation including two
particle-hole pairs
( p+| B|)/
F
-0.4
Dashed: [4] Polaron variational
calculation including one
particle-hole pair
[1] [1] [2][3]
-0.8
molecule, diag MC
polaron, diag MC
experimental data
_
=0.22h /a, N=21
B
_
=0.31h /a, N=21
B
_
=0.43h /a, N=21
B
_
=0.53h /a, N=20
B
_
=0.62h /a, N=20
B
-1
-1
1.
2.
3.
4.
5.
0
1
2
Dot-dashed: [5] Molecule
variational calculation
including one particle-hole
pairs
Calculations done on Jülich
supercomputer JUQUEEN
3
Vlietinck, Ryckebusch, Van Houcke, PRB 89 (2014) 085119
Koschorreck, et al., Nature 485, 619 (2012); M. Köhl talk at APS 2012
Levinsen, Baur, PRA 86 (2012) 041602 – discusses correction from quasi-2D to pure 2D
Parish, PRA 83 (2011) 051603; see also Zhang et al., PRL 108 (2012) 235302
Parish, Levinsen, PRA 87 (2013) 033616
41
Fermionic polaron versus bosonic molecule
-0.2
( p+| B|)/
F
-0.4
-0.6
molecule, diag MC
polaron, diag MC
experimental data
_
=0.22h /a, N=21
B
_
=0.31h /a, N=21
B
_
=0.43h /a, N=21
B
_
=0.53h /a, N=20
B
_
=0.62h /a, N=20
B
-0.8
-1
-1
0
1
2
3
Smooth crossover transition at
Bour, Lee, Hammer, Meißner, arXiv:1412.8175, PRL in press
42
Diagrammatic Monte Carlo calculations find a transition at
Vlietinck, Ryckebusch, Van Houcke,
PRB 89 (2014) 085119
Kroiss, Pollet,
PRB 90 (2014) 104510
Analysis of experimental data with conversion from quasi 2D to
pure 2D gives a transition at
Koschorreck, et al., Nature 485, 619 (2012);
M. Köhl talk at APS 2012;
Levinsen, Baur, PRA 86 (2012) 041602
For discussion of polarons and density profiles in 2D spin-imbalanced
systems see
Ong, Cheng, Arakelyan, Thomas, PRL 114 (2015) 110403
43
Density-density correlations
η=+1.5
L=40,N=8
L=40,N=14
L=40,N=20
η=+0.5
L=40,N=8
L=40,N=14
L=40,N=20
η=-0.8
L=40,N=8
L=40,N=14
L=40,N=20
η=-1.0
L=40,N=8
L=40,N=14
L=40,N=20
_2
2
log [ρ↑↓h /kF ]
2
1
0
-1
_2
2
log [ρ↑↓h /kF ]
2
1
0
-1
-5
0 _
xkF / h
5
-5
0 _
xkF / h
5
Smooth crossover transition
Bour, Lee, Hammer, Meißner, arXiv:1412.8175, PRL in press
Current work and outlook
We are studying 3D polaron system using impurity lattice Monte Carlo
calculations to measure the ground state energy and density correlations.
We also starting calculations which apply impurity lattice Monte Carlo
to impurities in a paired superfluid.
These simulations should have numerous applications to cold atomic
systems as well as alpha particles in a superfluid neutron gas.
45
Paired superfluid
46
auxiliary
field s at
every
point in
space and
time
at
a
47
End. 48