lattice simulations for nuclei, ultracold atoms, and …...a tale of two interactions essential...
TRANSCRIPT
![Page 1: Lattice simulations for nuclei, ultracold atoms, and …...A tale of two interactions Essential elements for nuclear binding Quantum scale anomalies with trapped ions Discrete scale](https://reader036.vdocument.in/reader036/viewer/2022062402/5ecb1f57ae33b5606f63adc3/html5/thumbnails/1.jpg)
Dean Lee Facility for Rare Isotope Beams & Dept. of Physics and Astronomy Michigan State University
24th European Conference on Few-Body Problems in Physics University of Surrey September 3, 2019
Lattice simulations for nuclei, ultracold atoms, and ions
1
![Page 2: Lattice simulations for nuclei, ultracold atoms, and …...A tale of two interactions Essential elements for nuclear binding Quantum scale anomalies with trapped ions Discrete scale](https://reader036.vdocument.in/reader036/viewer/2022062402/5ecb1f57ae33b5606f63adc3/html5/thumbnails/2.jpg)
Outline
Lattice effective field theory
A tale of two interactions
Essential elements for nuclear binding
Quantum scale anomalies with trapped ions
Discrete scale invariance for two bosons
Time fractals
Summary and outlook
2
![Page 3: Lattice simulations for nuclei, ultracold atoms, and …...A tale of two interactions Essential elements for nuclear binding Quantum scale anomalies with trapped ions Discrete scale](https://reader036.vdocument.in/reader036/viewer/2022062402/5ecb1f57ae33b5606f63adc3/html5/thumbnails/3.jpg)
n
n
p
p
Lattice effective field theory
π
π
Review: D.L, Prog. Part. Nucl. Phys. 63 117-154 (2009) Springer Lecture Notes: Lähde, Meißner, “Nuclear Lattice Effective Field Theory” (2019)
3
![Page 4: Lattice simulations for nuclei, ultracold atoms, and …...A tale of two interactions Essential elements for nuclear binding Quantum scale anomalies with trapped ions Discrete scale](https://reader036.vdocument.in/reader036/viewer/2022062402/5ecb1f57ae33b5606f63adc3/html5/thumbnails/4.jpg)
Construct the effective potential order by order
4
N
N
N
N
π
N
N
N
N
N
N
N
N
ππ
N
N
N
N
Contact interactions
Leading order (LO) Next-to-leading order (NLO)
Chiral effective field theory
![Page 5: Lattice simulations for nuclei, ultracold atoms, and …...A tale of two interactions Essential elements for nuclear binding Quantum scale anomalies with trapped ions Discrete scale](https://reader036.vdocument.in/reader036/viewer/2022062402/5ecb1f57ae33b5606f63adc3/html5/thumbnails/5.jpg)
Related: See the talk by Lukas Bovermann on Friday afternoon at 14:55 in the Atoms and Molecules Session. He will discuss the spherical wall method and calculating phase shifts and mixing angles on the lattice for an arbitrary number of coupled channels.
5
![Page 6: Lattice simulations for nuclei, ultracold atoms, and …...A tale of two interactions Essential elements for nuclear binding Quantum scale anomalies with trapped ions Discrete scale](https://reader036.vdocument.in/reader036/viewer/2022062402/5ecb1f57ae33b5606f63adc3/html5/thumbnails/6.jpg)
6Li, Elhatisari, Epelbaum, D.L., Lu, Meißner, PRC 98, 044002 (2018)
Figures by Ning Li
![Page 7: Lattice simulations for nuclei, ultracold atoms, and …...A tale of two interactions Essential elements for nuclear binding Quantum scale anomalies with trapped ions Discrete scale](https://reader036.vdocument.in/reader036/viewer/2022062402/5ecb1f57ae33b5606f63adc3/html5/thumbnails/7.jpg)
7Li, Elhatisari, Epelbaum, D.L., Lu, Meißner, PRC 98, 044002 (2018)
Figures by Ning Li
![Page 8: Lattice simulations for nuclei, ultracold atoms, and …...A tale of two interactions Essential elements for nuclear binding Quantum scale anomalies with trapped ions Discrete scale](https://reader036.vdocument.in/reader036/viewer/2022062402/5ecb1f57ae33b5606f63adc3/html5/thumbnails/8.jpg)
π
π
8
Euclidean time projection
![Page 9: Lattice simulations for nuclei, ultracold atoms, and …...A tale of two interactions Essential elements for nuclear binding Quantum scale anomalies with trapped ions Discrete scale](https://reader036.vdocument.in/reader036/viewer/2022062402/5ecb1f57ae33b5606f63adc3/html5/thumbnails/9.jpg)
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.
9
Auxiliary field method
![Page 10: Lattice simulations for nuclei, ultracold atoms, and …...A tale of two interactions Essential elements for nuclear binding Quantum scale anomalies with trapped ions Discrete scale](https://reader036.vdocument.in/reader036/viewer/2022062402/5ecb1f57ae33b5606f63adc3/html5/thumbnails/10.jpg)
10
![Page 11: Lattice simulations for nuclei, ultracold atoms, and …...A tale of two interactions Essential elements for nuclear binding Quantum scale anomalies with trapped ions Discrete scale](https://reader036.vdocument.in/reader036/viewer/2022062402/5ecb1f57ae33b5606f63adc3/html5/thumbnails/11.jpg)
A tale of two interactions
11
Two LO interactions, A and B, have nearly identical nucleon-nucleon phase shifts and well as three- and four-nucleon bound states
Elhatisari, Li, Rokash, Alarcon, Du, Klein, Lu, Meißner, Epelbaum, Krebs, Lähde, D.L., Rupak, PRL 117, 132501 (2016)
![Page 12: Lattice simulations for nuclei, ultracold atoms, and …...A tale of two interactions Essential elements for nuclear binding Quantum scale anomalies with trapped ions Discrete scale](https://reader036.vdocument.in/reader036/viewer/2022062402/5ecb1f57ae33b5606f63adc3/html5/thumbnails/12.jpg)
12
![Page 13: Lattice simulations for nuclei, ultracold atoms, and …...A tale of two interactions Essential elements for nuclear binding Quantum scale anomalies with trapped ions Discrete scale](https://reader036.vdocument.in/reader036/viewer/2022062402/5ecb1f57ae33b5606f63adc3/html5/thumbnails/13.jpg)
13
Bose condensate of alpha particles!
![Page 14: Lattice simulations for nuclei, ultracold atoms, and …...A tale of two interactions Essential elements for nuclear binding Quantum scale anomalies with trapped ions Discrete scale](https://reader036.vdocument.in/reader036/viewer/2022062402/5ecb1f57ae33b5606f63adc3/html5/thumbnails/14.jpg)
δ 0(d
egre
es)
ELab (MeV)
Afzal et al.
-60
-20
20
60
100
140
180
0 2 4 6 8 10
δ 0(d
egre
es)
ELab (MeV)
A (LO)A (LO + Coulomb)
B (LO)B (LO + Coulomb)
-60
-20
20
60
100
140
180
0 2 4 6 8 10
Alpha-alpha scattering
14
Calculated using the adiabatic projection method Elhatisari, D.L., Rupak, Epelbaum, Krebs, Lähde, Luu, Meißner, Nature 528, 111 (2015)
![Page 15: Lattice simulations for nuclei, ultracold atoms, and …...A tale of two interactions Essential elements for nuclear binding Quantum scale anomalies with trapped ions Discrete scale](https://reader036.vdocument.in/reader036/viewer/2022062402/5ecb1f57ae33b5606f63adc3/html5/thumbnails/15.jpg)
15
Control parameters: Sensitivity to interaction range and locality
Elhatisari, Li, Rokash, Alarcon, Du, Klein, Lu, Meißner, Epelbaum, Krebs, Lähde, D.L., Rupak, PRL 117, 132501 (2016)
![Page 16: Lattice simulations for nuclei, ultracold atoms, and …...A tale of two interactions Essential elements for nuclear binding Quantum scale anomalies with trapped ions Discrete scale](https://reader036.vdocument.in/reader036/viewer/2022062402/5ecb1f57ae33b5606f63adc3/html5/thumbnails/16.jpg)
Essential elements for nuclear binding
What is the minimal nuclear interaction that can reproduce the ground state properties of light nuclei, medium-mass nuclei, and neutron matter simultaneously with no more than a few percent error in the energies and charge radii? We construct an interaction with only four parameters.
1. Strength of the two-nucleon S-wave interaction 2. Range of the two-nucleon S-wave interaction 3. Strength of three-nucleon contact interaction 4. Range of the local part of the two-nucleon interaction
Lu, Li, Elhatisari, D.L., Epelbaum, Meißner, arXiv:1812.10928, PLB in press 16
![Page 17: Lattice simulations for nuclei, ultracold atoms, and …...A tale of two interactions Essential elements for nuclear binding Quantum scale anomalies with trapped ions Discrete scale](https://reader036.vdocument.in/reader036/viewer/2022062402/5ecb1f57ae33b5606f63adc3/html5/thumbnails/17.jpg)
Related: See the talk by Alejandro Kievsky that took place on Monday afternoon at 14:30 in the Few-Nucleons Session, and also the talk by Sebastian König on Thursday afternoon at 12:05 in the Young Researcher Award Ceremony.
17
![Page 18: Lattice simulations for nuclei, ultracold atoms, and …...A tale of two interactions Essential elements for nuclear binding Quantum scale anomalies with trapped ions Discrete scale](https://reader036.vdocument.in/reader036/viewer/2022062402/5ecb1f57ae33b5606f63adc3/html5/thumbnails/18.jpg)
SU(4)-invariant pionless interaction
lattice spacing = 1.32 fm
18
S-wave parameters:
Triton binding energy:
![Page 19: Lattice simulations for nuclei, ultracold atoms, and …...A tale of two interactions Essential elements for nuclear binding Quantum scale anomalies with trapped ions Discrete scale](https://reader036.vdocument.in/reader036/viewer/2022062402/5ecb1f57ae33b5606f63adc3/html5/thumbnails/19.jpg)
14 16 18 20
15
20
25
30
35NL40
NL45
NL50
NL55
Lattice Mac-Mic FRLDM
mean field
surfa
ce-e
nerg
y co
nsta
nt aS
(MeV
)
volume-energy constant aV (MeV)
NL60
19
![Page 20: Lattice simulations for nuclei, ultracold atoms, and …...A tale of two interactions Essential elements for nuclear binding Quantum scale anomalies with trapped ions Discrete scale](https://reader036.vdocument.in/reader036/viewer/2022062402/5ecb1f57ae33b5606f63adc3/html5/thumbnails/20.jpg)
0 10 20 30 40 50
-400
-300
-200
-100
0
CaArS
Si
Mg
Ne
O
C
BeHe
Lattice Exp.
Bin
ding
ene
rgy
(MeV
)
Mass number A
H Pion-less EFT Leading Order
20Lu, Li, Elhatisari, D.L., Epelbaum, Meißner, arXiv:1812.10928, PLB in press
![Page 21: Lattice simulations for nuclei, ultracold atoms, and …...A tale of two interactions Essential elements for nuclear binding Quantum scale anomalies with trapped ions Discrete scale](https://reader036.vdocument.in/reader036/viewer/2022062402/5ecb1f57ae33b5606f63adc3/html5/thumbnails/21.jpg)
21Lu, Li, Elhatisari, D.L., Epelbaum, Meißner, arXiv:1812.10928, PLB in press
![Page 22: Lattice simulations for nuclei, ultracold atoms, and …...A tale of two interactions Essential elements for nuclear binding Quantum scale anomalies with trapped ions Discrete scale](https://reader036.vdocument.in/reader036/viewer/2022062402/5ecb1f57ae33b5606f63adc3/html5/thumbnails/22.jpg)
0 2 4 6
0.00
0.05
0.10
0 2 4 6
0.00
0.05
0.10
Empirical Leading Order Leading Order + Coulomb (pert.)
Cha
rge
dens
ity ρch
(e fm
-3)
16O
Radius (fm)
40Ca
22Lu, Li, Elhatisari, D.L., Epelbaum, Meißner, arXiv:1812.10928, PLB in press
![Page 23: Lattice simulations for nuclei, ultracold atoms, and …...A tale of two interactions Essential elements for nuclear binding Quantum scale anomalies with trapped ions Discrete scale](https://reader036.vdocument.in/reader036/viewer/2022062402/5ecb1f57ae33b5606f63adc3/html5/thumbnails/23.jpg)
Quantum scale anomalies with trapped ions
In quantum mechanics and quantum field theory, scale invariance can be spoiled by quantum scale anomalies. This happens when there are bound states, which necessarily correspond to discrete energy levels.
Nevertheless it may happen that a discrete subgroup of the scale symmetry is preserved for the dynamics of certain sectors of the Hilbert space.
This phenomenon was first noted by Efimov for bound states of three bosons when the two-body interactions are pointlike and the interaction strength is tuned to produce a zero-energy two-body resonance.
Efimov, Sov. J. Nucl. Phys. 12, 589 (1971); Efimov, Phys. Rev. C47 1876 (1993)
23
![Page 24: Lattice simulations for nuclei, ultracold atoms, and …...A tale of two interactions Essential elements for nuclear binding Quantum scale anomalies with trapped ions Discrete scale](https://reader036.vdocument.in/reader036/viewer/2022062402/5ecb1f57ae33b5606f63adc3/html5/thumbnails/24.jpg)
Wikipedia
Realization with trapped ions
24
![Page 25: Lattice simulations for nuclei, ultracold atoms, and …...A tale of two interactions Essential elements for nuclear binding Quantum scale anomalies with trapped ions Discrete scale](https://reader036.vdocument.in/reader036/viewer/2022062402/5ecb1f57ae33b5606f63adc3/html5/thumbnails/25.jpg)
Zhang et al., Nature 543, 217 (2017), Zhang et al., Nature 551, 601 (2017) 25
![Page 26: Lattice simulations for nuclei, ultracold atoms, and …...A tale of two interactions Essential elements for nuclear binding Quantum scale anomalies with trapped ions Discrete scale](https://reader036.vdocument.in/reader036/viewer/2022062402/5ecb1f57ae33b5606f63adc3/html5/thumbnails/26.jpg)
26
We can write our Hamiltonian in terms of hardcore boson annihilation and creation operators.
Let us now take
This choice ensures that a zero-momentum boson has zero energy. We now consider the dispersion relation for one boson.
We define the state with all spins pointing up as the vacuum state. Then down-spin sites can be viewed as identical hardcore boson excitations.
![Page 27: Lattice simulations for nuclei, ultracold atoms, and …...A tale of two interactions Essential elements for nuclear binding Quantum scale anomalies with trapped ions Discrete scale](https://reader036.vdocument.in/reader036/viewer/2022062402/5ecb1f57ae33b5606f63adc3/html5/thumbnails/27.jpg)
27
For a boson with momentum p, the energy is
At low momenta, this can be simplified as
α = 2.5
α = 2.0
α = 1.5
0.2 0.4 0.6 0.8 1.0p
1
2
3
4
5E(p)
![Page 28: Lattice simulations for nuclei, ultracold atoms, and …...A tale of two interactions Essential elements for nuclear binding Quantum scale anomalies with trapped ions Discrete scale](https://reader036.vdocument.in/reader036/viewer/2022062402/5ecb1f57ae33b5606f63adc3/html5/thumbnails/28.jpg)
We now introduce a single-site deep trapping potential that traps one boson at some site i0
We choose the position of site i0 to be r = 0. We subtract a constant from the Hamiltonian so that the energy of this state is exactly zero.
r = 0
28
![Page 29: Lattice simulations for nuclei, ultracold atoms, and …...A tale of two interactions Essential elements for nuclear binding Quantum scale anomalies with trapped ions Discrete scale](https://reader036.vdocument.in/reader036/viewer/2022062402/5ecb1f57ae33b5606f63adc3/html5/thumbnails/29.jpg)
29
We now add one more boson to the system. We regard the immobile boson at r = 0 as a static source.
where we have dropped terms of O(p2). We will consider the case where both J0 and V0 are negative. In order that the Hamiltonian have classical scale invariance, we take
The low-energy effective Hamiltonian for the mobile boson is
Discrete scale invariance for two bosons
Therefore
D.L., Watkins, Frame, Given, He, Li, Lu, Sarkar, PRA 100, 011403(R) (2019)
![Page 30: Lattice simulations for nuclei, ultracold atoms, and …...A tale of two interactions Essential elements for nuclear binding Quantum scale anomalies with trapped ions Discrete scale](https://reader036.vdocument.in/reader036/viewer/2022062402/5ecb1f57ae33b5606f63adc3/html5/thumbnails/30.jpg)
30
In the limit of zero energy, the bound-state wave functions have the following forms for even and odd parity
where
The case corresponds to a Hamiltonian of the form
![Page 31: Lattice simulations for nuclei, ultracold atoms, and …...A tale of two interactions Essential elements for nuclear binding Quantum scale anomalies with trapped ions Discrete scale](https://reader036.vdocument.in/reader036/viewer/2022062402/5ecb1f57ae33b5606f63adc3/html5/thumbnails/31.jpg)
31
We can rewrite the zero-energy bound-state solutions as
Under the scale transformations
For the case
![Page 32: Lattice simulations for nuclei, ultracold atoms, and …...A tale of two interactions Essential elements for nuclear binding Quantum scale anomalies with trapped ions Discrete scale](https://reader036.vdocument.in/reader036/viewer/2022062402/5ecb1f57ae33b5606f63adc3/html5/thumbnails/32.jpg)
32
The bound state energies form a geometric progression
The wave functions exhibit discrete scale invariance when the scale factors are
The general formulae are
![Page 33: Lattice simulations for nuclei, ultracold atoms, and …...A tale of two interactions Essential elements for nuclear binding Quantum scale anomalies with trapped ions Discrete scale](https://reader036.vdocument.in/reader036/viewer/2022062402/5ecb1f57ae33b5606f63adc3/html5/thumbnails/33.jpg)
33
0 5 10 15 20 25 30 35 40 45 50r
-0.4
-0.2
0
0.2
0.4
0.6
0.8
(r)
The first twelve even-parity bound-state wave functions:
D.L., Watkins, Frame, Given, He, Li, Lu, Sarkar, PRA 100, 011403(R) (2019)
![Page 34: Lattice simulations for nuclei, ultracold atoms, and …...A tale of two interactions Essential elements for nuclear binding Quantum scale anomalies with trapped ions Discrete scale](https://reader036.vdocument.in/reader036/viewer/2022062402/5ecb1f57ae33b5606f63adc3/html5/thumbnails/34.jpg)
34
![Page 35: Lattice simulations for nuclei, ultracold atoms, and …...A tale of two interactions Essential elements for nuclear binding Quantum scale anomalies with trapped ions Discrete scale](https://reader036.vdocument.in/reader036/viewer/2022062402/5ecb1f57ae33b5606f63adc3/html5/thumbnails/35.jpg)
35
We use a phase convention where all of the bound-state wave functions are real valued. Let us construct a coherent superposition of the first N even-parity bound states, where N is large.
We could have just as easily chosen odd-parity bound states. We now consider the amplitude
Aside from corrections of relative size 1/N from endpoint terms at n = 0 and n = N – 1, the amplitude is invariant under the discrete rescaling of time.
Time fractals
![Page 36: Lattice simulations for nuclei, ultracold atoms, and …...A tale of two interactions Essential elements for nuclear binding Quantum scale anomalies with trapped ions Discrete scale](https://reader036.vdocument.in/reader036/viewer/2022062402/5ecb1f57ae33b5606f63adc3/html5/thumbnails/36.jpg)
36
Given the self-replicating behavior of the amplitude under time rescaling, we call it a time fractal.
We choose an integer time scaling factor
by taking the parameters
![Page 37: Lattice simulations for nuclei, ultracold atoms, and …...A tale of two interactions Essential elements for nuclear binding Quantum scale anomalies with trapped ions Discrete scale](https://reader036.vdocument.in/reader036/viewer/2022062402/5ecb1f57ae33b5606f63adc3/html5/thumbnails/37.jpg)
37
0 10 20 30 40 50 60 70 80t
-8
-6
-4
-2
0
2
4
6
8
10
A(t)
0 20 40 60 80 100 120 140 160t
-8
-6
-4
-2
0
2
4
6
8
10
A(t)
0 50 100 150 200 250 300t
-8
-6
-4
-2
0
2
4
6
8
10
A(t)
0 100 200 300 400 500 600t
-8
-6
-4
-2
0
2
4
6
8
10
A(t)
D.L., Watkins, Frame, Given, He, Li, Lu, Sarkar, PRA 100, 011403(R) (2019)
![Page 38: Lattice simulations for nuclei, ultracold atoms, and …...A tale of two interactions Essential elements for nuclear binding Quantum scale anomalies with trapped ions Discrete scale](https://reader036.vdocument.in/reader036/viewer/2022062402/5ecb1f57ae33b5606f63adc3/html5/thumbnails/38.jpg)
38
This is a particular case of the Weierstrass function,
In our case we take a and truncate after a finite number of terms. The next slide shows a picture of the Weierstrass function for a = 0.5, b = 3.
![Page 39: Lattice simulations for nuclei, ultracold atoms, and …...A tale of two interactions Essential elements for nuclear binding Quantum scale anomalies with trapped ions Discrete scale](https://reader036.vdocument.in/reader036/viewer/2022062402/5ecb1f57ae33b5606f63adc3/html5/thumbnails/39.jpg)
39
The Weierstrass function has fractal dimension
Hardy, Trans. Amer. Math. Soc. 17, 301 (1916) Hunt, Proc. Amer. Math. Soc. 126, 791 (1998)
![Page 40: Lattice simulations for nuclei, ultracold atoms, and …...A tale of two interactions Essential elements for nuclear binding Quantum scale anomalies with trapped ions Discrete scale](https://reader036.vdocument.in/reader036/viewer/2022062402/5ecb1f57ae33b5606f63adc3/html5/thumbnails/40.jpg)
Summary and Outlook
40
We have constructed a minimal nuclear interaction that can reproduce the ground state properties of light nuclei, medium-mass nuc le i , and neutron matter simultaneously with no more than a few percent error in the energies and charge radii. This SU(4)-invariant interaction has consequences for the computational reach of future nuclear lattice simulations. Opens the possibility of simulations with as many as one or two hundred nucleons.
![Page 41: Lattice simulations for nuclei, ultracold atoms, and …...A tale of two interactions Essential elements for nuclear binding Quantum scale anomalies with trapped ions Discrete scale](https://reader036.vdocument.in/reader036/viewer/2022062402/5ecb1f57ae33b5606f63adc3/html5/thumbnails/41.jpg)
41
Trapped ion quantum simulators offer a new way to produce quantum scale anomalies and do so quite naturally. Using several adjustable parameters, one can study a broad class of systems with discrete scale invariance and fractal-like time dependence. There are many interesting questions that remain to be explored. For example, the N-boson systems exhibit multi-halo structures with heterogeneous discrete scale invariance. We are using the quantum scale anomaly system as a testing ground for new quantum computing algorithms.