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

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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

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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

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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

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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

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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

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jÃinitihÃinitj

(1¡¢E0;NLOat)e¡E0;LOat = hMNLOi0;LO

jÃinitihÃinitj

Znt;NLO =

ZhOint;NLO

=

hOi0;NLO = limnt!1ZhOint;NLO

=Znt;NLO

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LO3: Gaussian smearing only in even partial waves

LO1: Pure contact interactions

LO2: Gaussian smearing

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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

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LO3: P waves

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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 + ¢ ¢ ¢

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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

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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

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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)

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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