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Symmetries, signs, and complex actions in
lattice effective field theory
1
Dean Lee
North Carolina State University
ECT* Workshop on Signs and Complex Actions
March 3, 2009 – Trento, Italy
2
Outline
What is lattice effective field theory?
Computational strategies on the lattice
Dilute neutron matter at NLO
Studies of light nuclei at NNLO
Summary, future directions, and connections
Chiral effective field theory for nucleons
Auxiliary fields, signs, and complex actions
Phase shifts and unknown operator coefficients
How severe is the sign problem really?
p
n
p
n
p
n
3
Lattice EFT for nucleons
Accessible by
Lattice EFT
early
universe
gas of light
nuclei
heavy-ion
collisions
quark-gluon
plasma
excited
nuclei
neutron star core
nuclear
liquid
superfluid
10-110-3 10-2 1
10
1
100
rNr [fm-3]
T[M
eV
]
neutron star crust
Accessible by
Lattice QCD
4
Construct the effective potential order by order
…
Solve Lippmann-Schwinger equation non-perturbatively
Weinberg, PLB 251 (1990) 288; NPB 363 (1991) 3
5
Chiral EFT for low-energy nucleons
Ordonez et al. ’94; Friar & Coon ’94; Kaiser et al. ’97; Epelbaum et al. ’98,‘03;
Kaiser ’99-’01; Higa et al. ’03; …
6
Nuclear
Scattering Data
Effective
Field Theory
p
7
Leading order on lattice
p
p
p
p
. . .
p
p
8
Next-to-leading order on lattice
NNNLO
NNLO
NLO
LO
9
Computational strategy
NNNLO
NNLO
NLO
LO
NNNLO
NNLO
NLO
LONon-perturbative – Monte Carlo Perturbative corrections
“Improved LO”
10
11
Lattice formulations
Free nucleons:
Free pions:
Pion-nucleon coupling:
12
Euclidean-time transfer matrix
CI contact interaction:
C contact interaction:
13
… with auxiliary fields
p
p
14
15
Mij(s; sI; ¼I) = h~pijM(Lt¡1)(s; sI; ¼I) ¢ ¢ ¢M(0)(s; sI; ¼I) j~pji
hÃinitjM(Lt¡1)(s; sI; ¼I) ¢ ¢ ¢ ¢ ¢M(0)(s; sI; ¼I) jÃiniti = detM(s; sI; ¼I)
Euclidean-time projection Monte Carlo
¿2M¿2 =M¤
For A nucleons, the matrix is A by A. For the leading-order calculation,
if there is no pion coupling and the quantum state is an isospin singlet
then
This shows the determinant is real. Actually we can show that the
determinant is positive semi-definite
16
MÁ= ¸Á
Consider an eigenvector
~Á = ¿2Á¤
Let us define a new vector
M~Á=M¿2Á¤ = ¿2¿2M¿2Á
¤ = ¿2M¤Á¤
Note that
= ¿2 (MÁ)¤= ¿2 (¸Á)
¤= ¸¤¿2Á
¤ = ¸¤ ~Á
Note also that the two vectors are orthogonal
~ÁyÁ = (¿2Á¤)yÁ = ÁT¿
y2Á = Á
T¿2Á = 0
So complex eigenvalues come in conjugate pairs, and the real spectrum
is doubly-degenerate
17
With nonzero pion coupling the determinant is real for a spin-singlet
isospin-singlet quantum state
¾2¿2M¾2¿2 =M¤
but the determinant can be both positive and negative
Some comments about Wigner’s approximate SU(4) symmetry…
Theorem: Any fermionic theory with SU(2N) symmetry and two-body
potential with negative semi-definite Fourier transform
~V (~p) · 0
satisfies SU(2N) convexity bounds…
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2NK 2N(K+1)
E
A2N(K+2)
2NK 2N(K+1) 2N(K+2)
E
A
weak attractive potential
strong attractive potential
SU(2N) convexity bounds
19
RDÁ e¡S(Á) detM(Á)
S(Á) = ¡®t2
X
nt
X
~n;~n0
Á(~n; nt)V¡1(~n¡ ~n0)Á(~n0; nt)
Mk0;k(Á) = hfk0jM(Lt¡1)(Á)£ ¢ ¢ ¢ £M(0)(Á) jfkiRDÁ e¡S(Á) [detMn1£n1(Á)]
j[detMn2£n2(Á)]
2N¡j
Apply Hölder inequality
Chen, D.L. Schäfer, PRL 93 (2004) 242302;
D.L., PRL 98 (2007) 182501
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hÃinitjM(Lt¡1)(Á) ¢ ¢ ¢ ¢ ¢M(0)(Á) jÃiniti = detM(Á)
jÃiniti ! fj °avors occupation n1;2N ¡ j °avors occupation n2g
0
-10
-20
-30
-40
-50
-60
-70
-80
-90
-100
-110
-120
-130
1H 2H 3He4He 5Li 6Li 7Be 8Be 9B 10B 11C 12C 13N 14N 15O 16O
(MeV)
E
nucleus
n nn 3H 5He 6He7Li 9Be10Be11B 13C 14C 15N
SU(4) convexity bounds
21
jÃinitihÃinitj
=MLO =MSU(4) =Oobservable
jÃinitihÃinitj
Znt;LO =
ZhOint;LO
=
e¡E0;LOat = limnt!1Znt+1;LO=Znt;LO
hOi0;LO = limnt!1ZhOint;LO
=Znt;LO
=MNLO =MNNLO
Hybrid Monte Carlo sampling
22
jÃinitihÃinitj
(1¡¢E0;NLOat)e¡E0;LOat = hMNLOi0;LO
jÃinitihÃinitj
Znt;NLO =
ZhOint;NLO
=
hOi0;NLO = limnt!1ZhOint;NLO
=Znt;NLO
23
LO3: Gaussian smearing only in even partial waves
LO1: Pure contact interactions
LO2: Gaussian smearing
24
Unknown operator
coefficients
Physical
scattering data
Spherical wall imposed in the center of
mass frame
Spherical wall methodBorasoy, Epelbaum, Krebs, D.L., Meißner,
EPJA 34 (2007) 185
25
LO3: S waves
26
LO3: P waves
27
N = 8, 12, 16 neutrons at L3 = 43, 53, 63, 73
Close to unitarity limit result with
scattering length, effective range
corrections, etc.
Epelbaum, Krebs, D.L, Meißner, 0812.3653 [nucl-th]
28
Dilute neutron matter at NLO
jÃinitihÃinitjnt » E¡1F
Sign oscillation scaling by far most significant
hsigni » 2¡(kFNneutrons)=(400MeV)
Cost of generating new configurations
» c1NneutronsL3nt + c2N3neutrons + ¢ ¢ ¢
29
How bad is the sign problem really?
Current (0.05 Teraflop years)
6% normal nuclear matter density for 16 neutrons
Near future (1 Teraflop years)
15% normal nuclear matter density for 16 neutrons
kFNneutrons = 2000MeV
kFNneutrons = 2800MeV
30
a= 1:97fm
a= 1:6 fm
Petascale (100 Teraflop years)
15% normal nuclear matter density for 24 neutrons
Exascale (100 Petaflop years)
50% normal nuclear matter density for 24 neutrons
kFNneutrons = 4200MeV
kFNneutrons = 6100MeV
31
a= 1:6 fm
a= 1:2 fm
32
?
P-wave pairing?
Fit cD and cE to spin-1/2 nucleon-deuteron
scattering and 3H binding energyE D
3H binding energy
Fitting point
Spin-1/2 nucleon-deuteron
33
Three-body forces at NNLO
Spin-3/2 nucleon-deuteron scattering
34
Alpha-particle energy
35
Promising but relatively new tool that combines the framework of
effective field theory and computational lattice methods
Potentially wide applications to zero and nonzero temperature
simulations of cold atoms, light nuclei, neutron matter
Improve accuracy – higher order, smaller lattice spacing, larger
volume, more nucleons
Include Coulomb effects and isospin breaking
Compute nucleon-nucleus scattering, nucleus-nucleus scattering
Summary
Future directions
36
Finite volume matching for two-nucleon states
For the same periodic volume, compute two-nucleon energies in
Lattice QCD and match to two-nucleon energies Lattice EFT
Pion mass dependence?
p
n n
p
E(L) E(L)
37
Connecting lattice QCD and lattice EFT
Calculate gA for the two-neutron state at finite volume
For given lattice spacing in lattice EFT, use the value of gA
obtained via Lattice QCD at the same volume to fix cD
n
n
cD
gA(L) gA(L)
38
Connecting lattice QCD and lattice EFT
early
universe
gas of light
nuclei
heavy-ion
collisions
quark-gluon
plasma
excited
nuclei
neutron star core
nuclear
liquid
superfluid
10-110-3 10-2 1
10
1
100
rNr [fm-3]
T[M
eV
]
neutron star crust
39
Connecting lattice QCD and lattice EFT