Cold atoms and nuclear physics:
Lattice calculations and the question of universality
1
Dean Lee (NC State U. / U. Bonn)
Evgeny Epelbaum, Hans-Werner Hammer, Hermann Krebs, Ulf-G.
Meißner (U. Bonn / FZ Jülich)
Recent Progress in Many-Body Theories
July 27 – 31, 2009
Ohio State University, Columbus, OH
2
Outline
Lattice calculations…
• Lattice effective field theory for nucleons
• Pion exchange and auxiliary fields
• Phase shifts on the lattice
• Results: Dilute neutron matter
… and the question of universality
• Low-energy universality
• Universality in higher partial waves
• Effective range for general d and L
• Causality and generalized Wigner bounds
Part I
Part II
p
n
p
n
p
n
a» 1¡5 fm
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
O(Q0)
LO
NLOO(Q2)
O(Q3)
N3LOO(Q4)
N2LO
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
1; ~¾1 ¢ ~¾2
VOPEP
7
Leading order on lattice
p
p
p
p
. . .
p
p
(~¾1 ¢ ~r1) (~¾2 ¢ ~r2)~r1 ¢ ~r2
V TPEP
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
exp
"¡12
³~r¼
´2¢t¡m
2¼
2¼2¢t
#
Free nucleons:
Free pions:
Pion-nucleon coupling:
exp
·1
2mNy~r2N¢t
¸
exp
"¡ gA2f¼
Ny¿~¾N ¢ ¢~r¼¢t#
11
Euclidean-time transfer matrix
(CI > 0)
CI contact interaction:
exp
·¡12CIN
y¿N ¢Ny¿N¢t¸
=1p2¼
ZdsI exp
·¡12sI ¢ sI + isI ¢Ny¿N
qCI¢t
¸
exp
·¡12CNyNNyN¢t
¸(C < 0)
C contact interaction:
=1p2¼
Zdsexp
·¡12s2+ sNyN
p¡C¢t
¸
12
… with auxiliary fields
t = 0
t = tf
p
pexp(¡Ht)
13
t = 0
t = tf
14
s(~r1)
¼I(~r3)
sI(~r2)
Mij(s; sI; ¼I)
jji
jii
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)
Auxiliary-field determinantal 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 can show the determinant
is positive semi-definite.
15
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
obeys SU(2N) convexity bounds (see next slide)
Corollary: It can be simulated without sign oscillations
16
Chen, D.L. Schäfer, PRL 93 (2004) 242302;
D.L., PRL 98 (2007) 182501
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
17
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
18
Schematic of projection calculations
jÃinitihÃinitj
jÃinitihÃinitj
Znt;NLO =
ZhOint;NLO
=
hOi0;NLO = limnt!1ZhOint;NLO
=Znt;NLO
19
A(VLO3) = C1S0f(~q 2)¡14¡ 1
4~¾1 ¢ ~¾2
¢¡34+ 1
4¿1 ¢ ¿2
¢LO3: Gaussian smearing only in even partial waves
+A(VOPEP)
LO1: Pure contact interactions
LO2: Gaussian smearing
+ C3S1f(~q2)¡34+ 1
4~¾1 ¢ ~¾2
¢¡14¡ 1
4¿1 ¢ ¿2
¢
A(VLO2) = Cf(~q 2) +CIf(~q 2)¿1 ¢ ¿2 +A(VOPEP)
A(VLO1) = C +CI¿1 ¢ ¿2 +A(VOPEP)
20
Unknown operator
coefficients
Physical
scattering data
Spherical wall imposed in the center of
mass frame
Spherical wall method
Borasoy, Epelbaum, Krebs, D.L., Meißner,
EPJA 34 (2007) 185
21
Energy levels with hard spherical wall
Rwall = 10a
a= 1:97 fm
Energy shift from free-particle values gives the phase shift
22
LO3: S waves a= 1:97 fm
23
LO3: P waves a= 1:97 fm
24
25
Results: Dilute neutron matter at NLO
N = 8, 12, 16 neutrons at L3 = 43, 53, 63, 73
Epelbaum, Krebs, D.L, Meißner, 0812.3653 [nucl-th], EPJA 40 (2009) 19926
a= 1:97 fm
27
Low-energy universality
Large disparity in length scales
d » k¡1F ÀR
d
R
range of interaction
average separation
¸T ÀR
¸Tthermal wavelength
f(~p 0; ~p) =
1X
L=0
fL(p)PL(cos µ)
Partial wave decomposition of the scattering amplitude
fL(p) =¡i2p
he2i±L(p) ¡ 1
i=
1
p [cot ±L(p)¡ i]
Ã(~r)! ei~p¢~r + f(~p 0; ~p)eipr
rr!1
S-wave effective range expansion
p cot ±0(p) = ¡a¡10 +1
2r0p
2 +
1X
n=0
(¡1)n+1P(n)0 p2n+4
f0(p) =1
p cot ±0(p)¡ ip
28
Dimensional analysis: Length dimensions of coefficients
a¡10 = [`]¡1; r0 = [`]; P(n)0 = [`]2n+3
For the S-wave case,
a¡10 »R¡1; r0 » R; P(n)0 » R2n+3
A parameter is of “natural size” if it is roughly the same order of
magnitude as the interaction range raised to the corresponding
length dimension.
For the remainder of this talk we look for low-energy universality
and consider momenta small enough such that
pR¿ 1
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p cot ±0(p) = ¡a¡10 +1
2r0p
2 +
1X
n=0
(¡1)n+1P(n)0 p2n+4
f0(p) =1
p cot ±0(p)¡ ip
Assuming parameters of natural size, the terms contributing to the
amplitude have a simple hierarchy
¡a¡10¡ip 1
2r0p
2
¡P(0)0 p4
pR
more powers of
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At low momenta the unitarity limit is reached when the real part of the
denominator in the scattering amplitude can be neglected
Unitarity limit
p cot ±0(p) = ¡a¡10 +1
2r0p
2 +
1X
n=0
(¡1)n+1P(n)0 p2n+4
f0(p) =1
p cot ±0(p)¡ ip
¡a¡10¡ip 1
2r0p
2
¡P(0)0 p4
a¡10 ! 0; a0!§1must be fine-tuned
31
d
d0E0
E00
E0d2 = E00d
02
Scale-invariant physics
f0(p) ¼i
p
32
E0A= » ¢ E
free0A
= » ¢ 35EF
ξ is a dimensionless number
Neutron matter close to unitarity limit for kF » 80 MeV
Dilute neutrons and the unitarity limit
» = 0:31(1)
»1 ¼ 0:8E0Efree0
= » ¡ »1kFa0
+ ckF r0 + :::
f(kFa0)D.L., EPJA 35 (2009) 171;
PRC 78 (2008) 024001;
PRB 75 (2007) 134502
33
34
fL(p) =¡i2p
he2i±L(p) ¡ 1
i=
1
p [cot ±L(p)¡ i]
=p2L
p2L+1 cot ±L(p)¡ ip2L+1
p2L+1 cot ±L(p) = ¡a¡1L +1
2rLp
2 +
1X
n=0
(¡1)n+1P(n)L p2n+4
Universality in higher partial waves
For higher partial waves the effective range expansion is
35
Cold atoms: P-wave Feshbach experiments
Nuclear physics: P-wave alpha-neutron interactions in halo nuclei
Regal, PRL 90 (2003) 053201, …
Dimensional analysis: Length dimensions of coefficients
a¡11 = [`]¡3; r1 = [`]¡1; P(n)1 = [`]2n+1
p3 cot ±1(p) = ¡a¡11 +1
2r1p
2 +
1X
n=0
(¡1)n+1P(n)1 p2n+4
f1(p) =p2
p3 cot ±1(p)¡ ip3
For P-waves we have*
We again start with the case where the coefficients are of natural size
36
*caveat for van der Waals interactions in alkali atomsGao, PRA (1998) 1728
¡a¡11
¡ip3
1
2r1p
2
¡P(0)1 p4
For natural size parameter, the P-wave low-momentum hierarchy is
a¡11 »R¡3; r1 »R¡1; P(n)1 » R2n+1
37
p3 cot ±1(p) = ¡a¡11 +1
2r1p
2 +
1X
n=0
(¡1)n+1P(n)1 p2n+4
f1(p) =p2
p3 cot ±1(p)¡ ip3
pRmore powers of
p3 cot ±1(p) = ¡a¡11 +1
2r1p
2 +
1X
n=0
(¡1)n+1P(n)1 p2n+4
f1(p) =p2
p3 cot ±1(p)¡ ip3
As in the S-wave case we look for an analogy to the unitarity limit
where the real part of the denominator in the scattering amplitude can
be neglected
Unitarity limit for P-waves?
This is equivalent to fine-tuning all coefficients in the effective range
expansion with negative length dimensions
38
a¡11 ! 0; a1!§1¡a¡11
¡ip3
1
2r1p
2
¡P(0)1 p4
r1! 0
A first guess might be…
39
pR
more powers of
Effective range integral formula
u(p)
L (r) =pr cos ±L(p) ¢ jL(pr)¡ pr sin ±L(p) ¢ yL(pr)
sin ±L(p)
Generalization of result of Bethe (1949)
+¼
¡(L+ 32)¡(L+ 5
2)22L+2
R2L+3
a2L¡2 lim
p!0p2L
Z R
0
hu(p)
L (r)i2dr
The wavefunction normalization is chosen so that for r ¸ R
40
Hammer, D.L., 0907.1763 [nucl-th]
Since the integral term is less or equal to zero,
If we fine-tune to be unnaturally small
¯a¡1L
¯¿R¡2L¡1
then
+¼
¡(L+ 32)¡(L+ 5
2)22L+2
R2L+3
a2L
41
a¡1L
S-wave case was derived by Phillips and Cohen
Cohen, Phillips, PLB390 (1997) 7
The upper bound for is much more severe. Cannot fine-
tune to zero.
negative power of R for L ≥ 1
negative coefficient for L ≥ 1
Hammer, D.L., 0907.1763 [nucl-th]
L ¸ 1
42
For , we conclude that must be negative and natural size rLL ¸ 1
This upper bound is related to the causality bound derived by Wigner
(1954). Here is the heuristic argument…
Consider a wavepacket that is sharply peaked in momentum space
f(r) =1p2¼
Zdpeipr ~f(p)
~f(p) ¼ c±(p¡ ¹p)
Consider what happens when multiplying by a momentum-dependent
phase shift
43
g(r) =1p2¼
Zdpeipr~g(p)
We perform the inverse Fourier transform
~g(p) = e2i±(p) ~f(p)
¼ e2i±(¹p)e¡2i±0(¹p)¹pf[r+2±0(¹p)]
There is an overall phase multiplication and the packet is translated
backward in space an amount proportional to the derivative of the
phase shift with respect to momentum.
Similar relation for time delay and the derivative of the phase shift
with respect to energy.
44
time
r = 0
45
time
r = 0
¡2±0(¹p)
46
time
r = 0 r =R
Causality bound
¡±0(¹p) <R¡±0(¹p)
¡2±0(¹p)
47
a¡11 ! 0; a1!§1¡a¡11
¡ip3
1
2r1p
2
¡P(0)1 p4
Universality for P-waves
p3 cot ±1(p) = ¡a¡11 +1
2r1p
2 +
1X
n=0
(¡1)n+1P(n)1 p2n+4
f1(p) =p2
p3 cot ±1(p)¡ ip3
48
cannot be fine-tuned
simultaneously
Shallow P-wave bound state
E ¼ 1
a1r1¹
Normalization of tail controlled by r1
49
P>(r)!2
¡r1r
In the limit of zero binding energy
r
u2(r)
r ¸ R
Generalization to arbitrary dimension
fL;d(p) /p2L
p2L+d¡2 cot ±L;d(p)¡ ip2L+d¡2
uL;d(r) = r(d¡1)=2RL;d(r)
¡u00L;d(r) +"¡L+ d¡1
2
¢ ¡L+ d¡3
2
¢
r2+ 2¹V (r)
#uL;d(r) = 2¹E ¢ uL;d(r)
Radial Schrödinger equation
Partial wave scattering amplitude
50
p2L+d¡2·cot ±L;d(p)¡ ±dmod2;0
2
¼log(p½L;d)
¸
= ¡a¡1L;d +1
2rL;dp
2 +
1X
n=0
(¡1)n+1P(n)L;dp2n+4
Effective range expansion
a¡1L = [`]2¡2L¡d; rL = [`]4¡2L¡d; P(n)L = [`]6+2n¡2L¡d
51
Dimensional analysis:
even d only
Depends on the combination 2L+ d
2L+ d= 3 The unitarity limit reachable – non-trivial fixed point.
2L+ d ¸ 4 Causality prevents reaching the non-trivial fixed point
with finite-range interactions. Effective range emerges
as a second relevant parameter that cannot be fine-
tuned.
Although the interactions can be non-perturbative (i.e.,
perturbative expansion in scattering parameter breaks
down), the interactions remain weak at low energies.
Only trivial fixed point.
52
Summary
53
Causal wave propagation has significant consequences for low-energy
universality
For higher partial waves in the zero-energy resonance limit, a second
dimensionful parameter appears that cannot be fine-tuned to zero. This
second parameter is the effective range.
Lattice EFT – relatively new and promising tool that combines framework
of effective field theory and computational lattice methods
Applications to zero and nonzero temperature simulations of light nuclei,
neutron matter, cold atoms, etc.
Currently working on light nuclei at NNLO, including Coulomb
corrections, isospin-symmetry breaking, storing lattice configurations for
correlation functions, scattering, transitions, etc.
Part I
Part II
Effective range expansion: van der Waals
p3 cot ±1(p) = ¡a¡11 + c1;1p+1
2r1p
2 + : : :
a¡11 ! 0 =) c1;1! 0
For P-waves cannot in general define an effective range parameter
However in the zero-energy resonance limit where the scattering
parameter is infinite
54