Block Diagonalization using SRG Flow Equations

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by  E. Anderson

Block Diagonalization using SRG Flow Equations
E. Anderson,1 S.K. Bogner,2 R.J. Furnstahl,1 E.D. Jurgenson,1 R.J. Perry,1 and A. Schwenk3
1Department of Physics, The Ohio State University, Columbus, OH 43210
2National Superconducting Cyclotron Laboratory and Department of Physics and Astronomy,
Michigan State University, East Lansing, MI 48844
3TRIUMF, 4004 Wesbrook Mall, Vancouver, BC, Canada, V6T 2A3
(Dated: February 3, 2008)
By choosing appropriate generators for the Similarity Renormalization Group (SRG) flow equa-
tions, different patterns of decoupling in a Hamiltonian can be achieved. Sharp and smooth block-
diagonal forms of phase-shift equivalent nucleon-nucleon potentials in momentum space are gener-
ated as examples and compared to analogous low-momentum interactions (“V ”).
lowk
PACSnumbers: 21.30.-x,05.10.Cc,13.75.Cs
8
The Similarity Renormalization Group (SRG) [1, 2, be generated using SRG flow equations by choosing a
0
0 3] applied to inter-nucleon interactions is a continuous block-diagonal flow operator [9, 10],
2 series of unitary transformations implemented as a flow (cid:18) (cid:19)
PH P 0
n equation for the evolving Hamiltonian Hs, Gs = 0s QH Q ≡Hsbd , (2)
a s
dH
J dss =[ηs,Hs]=[[Gs,Hs],Hs]. (1) withprojectionoperatorsP andQ=1−P. Inapartial-
7 wave momentum representation, P and Q are step func-

] HeresisaflowparameterandtheflowoperatorGs spec- tions defined by a sharp cutoff Λ on relative momenta.
h ifiesthetypeofSRG[4]. Decouplingbetweenlow-energy This choice for Gs, which means that ηs is non-zero only
t andhigh-energymatrixelementsisnaturallyachievedin whereG iszero,suppressesoff-diagonalmatrixelements
- s
l a momentum basis by choosing a momentum-diagonal such that the Hamiltonian approaches a block-diagonal
c
flow operator such as the kinetic energy T or the diag- form as s increases. If one considers a measure of the
u rel
n onal of Hs; either drives the Hamiltonian toward band- off-diagonal coupling of the Hamiltonian,
[ diagonal form. Thisdecouplingleadstodramaticallyim-
Tr[(QH P)†(QH P)]=Tr[PH QH P](cid:62)0, (3)
proved variational convergence in few-body nuclear sys- s s s s
1
tems compared to unevolved phenomenological or chiral
v then its derivative is easily evaluated by applying the
EFT potentials [5, 6].
8 SRG equation, Eq. (1):
9 RenormalizationGroup(RG)methodsthatevolveNN
0 interactions with a sharp or smooth cutoff in relative d
Tr[PH QH P]
1 momentum, known generically as V , rely on the in- ds s s
lowk
1. variance of the two-nucleon T matrix [7, 8]. These ap- =Tr[PηsQ(QHsQHsP −QHsPHsP)]
0 proaches achieve a block-diagonal form characterized by +Tr[(PH PH Q−PH QH Q)Qη P]
s s s s s
8 acutoffΛ(seeleftplotsinFigs.1and2). Asusuallyim-
=−2Tr[(Qη P)†(Qη P)](cid:54)0. (4)
0 plementedtheysetthehigh-momentummatrixelements s s
:
v to zero but this is not required. Thus, the off-diagonal QH P block will decrease in gen-
s
Xi Block-diagonaldecouplingofthesharpVlowk formcan eral as s increases [9, 10].
r
a
FIG.1: (Coloronline)Comparisonofmomentum-spaceV FIG.2: (Coloronline)Comparisonofmomentum-spaceV
lowk lowk
(left) and SRG (right) block-diagonal potentials with Λ = (left) and SRG (right) block-diagonal potentials with Λ =
2fm−1 evolved from an N3LO 3S potential [11]. The color 2fm−1 evolved from an N3LO 3S potential [11]. The color
1 1
axis is in fm. and z axes are in fm.
2
FIG. 3: (Color online) Evolution of the 3S partial wave with a sharp block-diagonal flow equation with Λ=2fm−1 at λ=4,
1
3, 2, and 1fm−1. The initial N3LO potential is from Ref. [11]. The axes are in units of k2 from 0–11 fm−2. The color scale
ranges from −0.5 to +0.5fm as in Fig. 1.
FIG. 4: (Color online) Same as Fig. 3 but for the 1P partial wave.
1
TherightplotsinFigs.1and2resultfromevolvingthe for the relative kinetic energy, giving an explicit solution
N3LO potential from Ref. [11] using the block-diagonal
Gs of Eq. (2) with Λ = 2fm−1 until λ ≡ 1/s1/4 = hpq(s)≈hpq(0)e−s((cid:15)p−(cid:15)q)2 (8)
0.5fm−1. TheagreementbetweenV andSRGpoten-
lowk
tialsformomentabelowΛisstriking. Asimilardegreeof with (cid:15)p ≡ p2/M. Thus the off-diagonal elements go to
universalityisfoundintheotherpartialwaves. Deriving zero with the energy differences just like with the SRG
√
an explicit connection between these approaches is the with Trel; one can see the width of order 1/ s = λ2 in
topic of an ongoing investigation. the k2 plots of the evolving potential in Figs. 3 and 4.
The evolution with λ of two representative partial While in principle the evolution to a sharp block-
waves (3S and 1P ) are shown in Figs. 3 and 4. The diagonalformmeansgoingtos=∞(λ=0), inpractice
1 1
evolution of the “off-diagonal” matrix elements (mean- we need only take s as large as needed to quantitatively
ing those outside the PH P and QH Q blocks) can be achieve the decoupling implied by Eq. (8). Furthermore,
s s
roughly understood from the dominance of the kinetic it should hold for more general definitions of P and Q.
energy on the diagonal. Let the indices p and q run To smooth out the cutoff, we can introduce a smooth
over indices of the momentum states in the P and Q regulator f , which we take here to be an exponential
Λ
spaces, respectively. To good approximation we can re- form:
place PH P and QH Q by their eigenvalues E and E
in the SRGs equationss, yielding [9, 10] p q fΛ(k)=e−(k2/Λ2)n , (9)
d with n an integer. For V potentials, typical values
lowk
h ≈η E −E η =−(E −E )η (5)
ds pq pq q p pq p q pq used are n = 4 and n = 8 (the latter is considerably
sharper but still numerically robust). By replacing Hbd
s
and with
η ≈E h −h E =(E −E )h . (6)
pq p pq pq q p q pq G =f H f +(1−f )H (1−f ), (10)
s Λ s Λ Λ s Λ
Combining these two results, we have the evolution of
we get a smooth block-diagonal potential.
any off-diagonal matrix element: ArepresentativeexamplewithΛ=2fm−1andn=4is
showninFig.5. Wecanevolvetoλ=1.5fm−1withouta
d
dshpq ≈−(Ep−Eq)2hpq . (7) problem. ForsmallerλtheoverlapoftheP andQspaces
becomes significant and the potential becomes distorted.
In the NN case we can replace the eigenvalues by those This distortion indicates that there is no further benefit
3
FIG.5: (Coloronline)Evolutionofthe3S partialwavewithasmooth(n=4)block-diagonalflowequationwithΛ=2.0fm−1,
1
starting with the N3LO potential from Ref. [11]. The flow parameter λ is 3, 2, 1.5, and 1fm−1. The axes are in units of k2
from 0–11 fm−2. The color scale ranges from −0.5 to +0.5fm as in Fig. 1.
to evolving in λ very far below Λ; in fact the decoupling andhigh-momentumblocksandtheregionthatisdriven
worsens for λ<Λ with a smooth regulator. tozeroconsistsofseveralrectangles. Resultsfortwopar-
Another type of SRG that is second-order exact and tial waves starting from the Argonne v potential [12]
18
yields similar block diagonalization is defined by are shown in Fig. 6. Despite the strange appearence,
these remain unitary transformations of the original po-
η =[T,PV Q+QV P], (11)
s s s tential, with phase shifts and other NN observables the
same as with the original potential. These choices pro-
which can be implemented with P → f and Q →
Λ
vide a proof-of-principle that the decoupled regions can
(1−f ),withf eithersharporsmooth. Wecanalsocon-
Λ Λ
be tailored to the physics problem at hand.
sider bizarre choices for f in Eq. (10), such as defining
Λ
it to be zero out to Λ , then unity out to Λ, and then
lower
zeroabovethat. Thismeansthat1−f definesbothlow 125
Λ
unevolved, uncut
-1
100 l = 4.0 fm cut
-1
l = 3.0 fm cut
-1
] l = 2.0 fm cut
g
e -1
d 75 l = 1.0 fm cut
[
ft
hi
s
e 50
s
a
h
p
3
25 S
1
3
N LO (500 MeV)
0
0 100 200 300 400
E [MeV]
lab
FIG. 7: (Color online) Phase shifts for the 3S partial wave
1
from an initial N3LO potential and the evolved sharp SRG
block-diagonal potential with Λ = 2fm−1 at various λ, in
each case with the potential set identically to zero above Λ.
Definitive tests of decoupling for NN observables are
nowpossibleforV potentialssincetheunitarytrans-
lowk
formation of the SRG guarantees that no physics is
lost. For example, in Figs. 7 and 8 we show 3S phase
1
shifts from an SRG sharp block diagonalization with
Λ=2fm−1 for two different potentials. The phase shifts
arecalculatedwiththepotentialscutsharplyatΛ. That
is, the matrix elements of the potential are set to zero
FIG. 6: (Color online) Evolved SRG potentials starting from above that point. The improved decoupling as λ de-
Argonne v in the 1S and 1P partial waves to λ=1fm−1 creases is evident in each case. By λ=1fm−1 in Fig. 7,
18 0 1
usingabizarrechoiceforG (seetext). Thecolorandz axes the unevolved and evolved curves are indistinguishable
s
are in fm. to the width of the line up to about 300MeV.
4
125 InFig.9weshowaquantitativeanalysisofthedecou-
unevolved, uncut pling as in Ref. [13]. The figure shows the relative error
100 l = 12.0 fm-1 cut of the phase shift at 100MeV calculated with a poten-
-1
l = 4.0 fm cut tial that is cut off by a smooth regulator as in Eq. (9)
75 l = 3.0 fm-1 cut at a series of values Λ . We observe the same universal
g] -1 cut
e l = 1.5 fm cut decouplingbehaviorseeninRef.[13]: ashoulderindicat-
d
ft [ 50 ing the perturbative decoupling region, where the slope
hi matchesthepower2nfixedbythesmoothregulator. The
s
e 25 onsetoftheshoulderinΛ decreaseswithλuntilitsat-
s cut
a
h urates for λ somewhat below Λ, leaving the shoulder at
p
0 Λ ≈Λ. Thus, as λ→0 the decoupling scale is set by
3 cut
S the cutoff Λ.
1
-25
AV18 In the more conventional SRG, where we use ηs =
[T,H ] = [T,V ], it is easy to see that the evolution of
-50 s s
0 100 200 300 400 the two-body potential in the two-particle system can
E [MeV]
lab be carried over directly to the three-particle system. In
FIG. 8: (Color online) Same as Fig. 7 but with Argonne v particular, it follows that the three-body potential does
18
as the initial potential [12]. not depend on disconnected two-body parts [4, 14]. If
we could implement η as proposed here with analogous
s
properties, we would have a tractable method for gener-
0
10
3 atingV three-bodyforces. Whileitseemspossibleto
N LO (500 MeV) lowk
defineFock-spaceoperatorswithprojectorsP andQthat
10-1 3S will not have problems with disconnected parts, it is not
1
-2 yet clear whether full decoupling in the few-body space
10
can be realized. Work on this problem is in progress.
)
E
(d10-3
)/
E E = 100 MeV
(Dd10-4 lab
-1
l=4.0 fm
-5
10 -1 Acknowledgments
l=3.0 fm
-1
l=2.0 fm
-6
10 -1
l=1.5 fm This work was supported in part by the National Sci-
-7 ence Foundation under Grant Nos. PHY–0354916 and
10
1 2 3 4 5 6 7
PHY–0653312, the UNEDF SciDAC Collaboration un-
-1
L [fm ]
cut der DOE Grant DE-FC02-07ER41457, and the Natural
FIG. 9: (Color online) Errors in the phase shift at E = Sciences and Engineering Research Council of Canada
lab
100MeV for the evolved sharp SRG block-diagonal potential (NSERC).TRIUMFreceivesfederalfundingviaacontri-
withΛ=2fm−1forarangeofλ’sandaregulatorwithn=8. butionagreementthroughtheNationalResearchCouncil
of Canada.
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