A new approach for studying large numbers of fermions in the unitary regime

0.14 MB·English

by  Michael G. Endres

A new approach for studying large numbers of
fermions in the unitary regime
1
1
0
2
n MichaelG. Endres∗
a PhysicsDepartment,ColumbiaUniversity,NewYork,NY10027,USAand
J
TheoreticalPhysicsLaboratory,RIKEN,Wako,Saitama351-0198,Japan

5
E-mail: [email protected]

]
t DavidB.Kaplan
a
l InstituteforNuclearTheory,UniversityofWashington,Seattle,WA98195-1550,USA
-
p E-mail: [email protected]
e
h Jong-WanLee
[
InstituteforNuclearTheory,UniversityofWashington,Seattle,WA98195-1550,USA

2 E-mail: [email protected]
v
9
Amy N.Nicholson
8
0 InstituteforNuclearTheory,UniversityofWashington,Seattle,WA98195-1550,USA
3 E-mail: [email protected]
.
1
1
Anovellatticeapproachispresentedforstudyingsystemscomprisingalargenumberofinteract-
0
1 ing nonrelativisticfermions. The constructionis ideally suited fornumericalstudy of fermions
:
v near unitarity–a strongly coupled regime corresponding to the two-particle s-wave scattering
Xi phase shift d 0 =p /2. Such systems may be achieved experimentally with trapped atoms, and
r providea starting point for an effective field theorydescription of nuclear physics. We discuss
a
theconstructionofourlatticetheory,whichallowsustostudysystemsofupto(butbynomeans
limited to) 38 fermions with high accuracy and modest computational resources, and offer an
overviewofseveralapplicationsofthetechnique.Amoredetaileddiscussionofapplicationsand
simulationresultswillbedescribedincompanionproceedingsbyA.N.N.andJ-W.L.
TheXXVIIIInternationalSymposiumonLatticeFieldTheory,Lattice2010
June14-19,2010
Villasimius,Italy
Speaker.
∗
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
Anewapproachforstudyinglargenumbersoffermionsintheunitaryregime MichaelG.Endres
1. Introduction
Simulatingtheoriesatfinitefermiondensityhasbeenalong-standing challengeinlatticefield
theory. One of the main issues faced in Monte Carlo studies of such theories is the fermion sign
problem: when a chemical potential is introduced, the action often becomes complex. A conse-
quenceofthisisthattheexponentialoftheactioncannotbeinterpretedasaprobabilitymeasureas
required by standard Monte Carlo algorithms. In the case of lattice QCD at finite baryon number
density, thesignproblemhasseverelylimitedtheexplorable regionsoftheQCDphasediagramin
thetemperature-chemical potential (T-m )planetotheregimewhere m /T .1.
B B
Alternatively, at zero temperature, finite densities can be achieved in a canonical ensemble
setting by considering multi-fermion correlation functions ina finite box. In this case, however, a
closelyrelatedproblememerges: correlatorsinvolvingfermionstypicallyhaveasignal/noisewhich
decays exponentially with the time separation of the source and sink. This makes identification
of effective mass plateaus difficult–if not impossible–for large numbers of fermions. In QCD,
standard Lepagearguments[1]suggestthatthesignal/noise formulti-baryon correlation functions
decay with a time constant tB−1 ∼B(mp−3/2mp ), where B is the baryon number, mp the proton
massandmp thepionmass.
In an effort to better understand the sign and signal/noise problems as well as their interrela-
tionship, we choose to study multi-fermion systems in a much simpler setting than QCD. One of
thesimplestnontrivialandinterestingtheoriesdescribesadilutetwo-componentsystemofnonrel-
ativisticfermionsintheunitaryregime. Thisregimecorrespondstoatwoparticles-wavescattering
phaseshiftneard =p /2,andwhenthislimitisachieved, thebounds 4p /p2 onthes-wave
0 l=0
≤
scattering cross-section becomes saturated. Although these systems are dilute, they are strongly
interacting and therefore require a nonperturbative treatment for reliable study. Unitary fermions
areinterestingnotonlybecausetheycanbestudiedexperimentallyusingtrappedultra-coldatoms,
but also because they serve as a starting point for a lattice effective field theory (EFT) descrip-
tion of nuclear physics. The latter is due to the fact that the 1S and 3S scattering lengths for
0 1
nucleon-nucleon scattering areunnaturally largecomparedtotherangeofinteraction.
Inthiswork,wefocusonanewcanonicalensembleapproachforsimulatinglargenumbersof
nonrelativistic fermions in the unitary regime and at zero temperature. Using this new approach,
we consider two systems in particular: unitary fermions in a finite box and unitary fermions in
a harmonic trap. From numerical simulations of these systems, one may extract experimentally
measurable non-perturbative quantities such as the Bertsch parameter (i.e., the ratio of the energy
to that of the free gas energy in the thermodynamic limit) and pairing gap. Furthermore, moving
away from unitarity one may test a set of universal relations involving a quantity known as the
“integrated contactdensity”, firstdiscoveredbyTan[2,3,4].
These proceedings focus primarily on the details of our lattice construction, as well as the
simulation and parameter tuning methods used in our studies of unitary fermions. Many past
numerical simulations at zero temperature have been variational in nature, offering only an upper
bound on the Bertsch parameter and possessing unknown systematic errors on other quantities
[5, 6, 7]. Our approach is nonvariational and is therefore in principle free from such systematic
errors. However, as is common among all simulations of this type, obtaining reliable results for
thespectrum and matrixelements requires agoodchoice ofinterpolating operators whichpossess
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Anewapproachforstudyinglargenumbersoffermionsintheunitaryregime MichaelG.Endres
large overlap with the states of interest. Details ofhow weconstruct optimal sources and sinks as
well as consideration of systematic errors due to finite volume and lattice spacing effects are the
focus of our companion proceedings [8, 9]. In those proceedings we also present results of our
initial exploratory studies ofupto20trapped and 38untrapped fermions, andpresent preliminary
valuesfortheBertchparameterandpairinggap.
2. Lattice construction
Thestartingpointforourconstructionisahighlyimprovedvariantofthenonrelativisticlattice
actionfirstproposed in[10],givenby:
S=(cid:229) n (cid:20)y¯n(¶ t y )n−21My¯n((cid:209) 2y )n+f ny¯n(√Cy )n−e0(cid:21) . (2.1)
Thisaction describes twospecies ofinteracting fermions y =(y ,y )defined onaT L3 lattice
↑ ↓ ×
withopenboundary conditionsinthetimedirectionandperiodicboundaryconditions inthespace
directions. Thederivative¶ t representsaforwarddifference operatorintimeand(cid:209) 2 isanon-local
lattice gradient operator which we define so-as to give a “perfect” continuum-like single particle
dispersionrelationforfreefermions. Afour-fermioncontactinteractionisachievedviaaGaussian
orZ auxiliaryfieldf associated withthetime-likelinksofthelattice. TheoperatorC actsonlyin
2
space, andinprinciple mayinclude derivativeinteractions toanarbitrary evenorderinmomenta.
We may express Eq. 2.1 succinctly as S=y¯Ky , where the time components of the fermion
matrixK aregiveninblock-matrix formby:
D X(0) 0 0 ... 0
0 D X(1) 0 ... 0
 
0 0 D X(2) ... 0
K =  , (2.2)
 0 0 0 D ... 0 
 ... ... ... ... ... X(T 1)
 − 
 0 0 0 0 ... D 
 
 
with
(cid:209) 2
D=1 , X(t )=1 f (t )√C . (2.3)
−2M −
NotethattheL3 L3matricesD,X andCactonlyinspaceandthatf (t )isadiagonalmatrixwith
×
independent random elements. In momentum space the specific expressions we use for D andC
are:
ep2/(2M)d p <L
hp|D|p′i= ¥ p,p′ |p| L , hp|C|p′i=C(p)d p,p′ (2.4)
(
| |≥
where L = p is a hard momentum cutoff imposed on the fermions and C(p) is some analytic
function of p2 which may be determined order by order in momenta from scattering data (see
Sec.5fordetails).
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Anewapproachforstudyinglargenumbersoffermionsintheunitaryregime MichaelG.Endres
Fermion propagators from timeslice zero to timeslices t maybe expressed asasequence of
applications ofDandX operators:
K 1(t ;0)=D 1X(t 1)D 1...D 1X(0)D 1 , (2.5)
− − − − −
−
whichprovidesasimplerecursiveapproachforcomputation. Inversionofthenon-localDoperator
andapplication ofX(t )maybeperformedefficientlywithfastFouriertransforms(FFTs);itisthis
feature that allows us to use a perfect dispersion relation and momentum dependent interactions.
Inthefree-fieldlimit,onemayexplicitly verifythatourdefinitionforDprovided inEq.2.4yields
aperfectdispersion relation forfermions. Inmomentumspacethefermionpropagator reducesto:
p2
p K 1(t ;0)p =e E(p)t d , E(p)= , (2.6)
− ′ − p,p
h | | i ′ 2M
formomentalessthanthecut-offL .
3. Simulationmethod andcost
SinceK isanupper tri-diagonal block matrix, thefermiondeterminant obtained by“integrat-
ingout”thefermionsisgivenbydetK=detDT. Weseethatthedeterminantisindependent ofthe
auxiliary field, and therefore the full numerical simulation of Eq. 2.1 is equivalent to a quenched
simulation. Byconstruction, our simulations are therefore free of the sign problem. Furthermore,
the decomposition offermion propagators as aproduct of random matrices in Eq.2.5 provides an
unusual interpretation for our approach: in essence we perform Euclidean time-evolution of sin-
gle particle wavefunctions over random background noise (either Z orGaussian). Multi-particle
2
sources and sinks may be constructed from a direct product of single particle wavefunctions, and
in the case of sinks, more elaborate constructions may be considered as well which incorporate
pairing correlations [8,9].
NumericalsimulationofEq.2.1consistsoffoursteps: 1)latticegeneration,2)propagatorgen-
eration, 3) projection of time-evolved states onto sinks and 4) computation of Slater determinants
(i.e., anti-symmetrization of initial and final states). The relative computational cost of each of
thesestepsonanL=32latticeisshowninFig.1asafunctionofthenumberofidenticalfermions.
To give meaning to the vertical axis, note that the O(N0) curve indicates the numerical cost of
generating O(L3) random numbers (i.e., lattice generation). The steps 1)-3) scale like T L3 or
×
T L3logL3, whereas step 4) is independent of the spatial volume. Although step 4) scales like
×
O(N4),1thecomputationalcostofthisstepisnegligibleevenforasmanyasN 50 100identical
∼ −
fermionsonthetypicalvolumesweconsider, whichrangefromL 12 64.
∼ −
4. Transfer matrices
Wemaytranslate our lattice action Eq.2.1into Hamiltonian language withasuitable reinter-
pretation ofthe expression formulti-fermion correlation functions. Suchcorrelation functions are
1ThecostofcomputingdeterminantsscaleslikeN3,however, wecomputedeterminantsofallN sub-matricesas
well,givingrisetoanadditionalpowerofN.
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Anewapproachforstudyinglargenumbersoffermionsintheunitaryregime MichaelG.Endres
150
1012
L=32 100
1010 N1
50
Cost110068 N2 Latticegen. N0 HLSΗ 0
-50
Propagatorgen.
104 N´N3 Sinkproj.
Slaterdet. -100
100
1 2 5 10 20 50 100 200 -150
-4 -2 0 2 4 6 8 10
Nidenticalfermions
Η
Figure 1: Cost ofnumericalsimulationsas a func- Figure2:Aplotofthethree-dimensionalz function
tionofthenumberoffermionsN. S(h )givenbyEq.5.2
obtainedfromanensembleaverageofdirectproductsofpropagators. Sincethepropagatordefined
inEq.2.5isitselfaproductofuncorrelated randommatrices(theauxiliaryfieldisaction-less), the
multi-fermioncorrelationfunctionwillfactorizeintoamatrixproductofensembleaverages. Ifwe
define:
T =D1/2(1 V)D1/2 , (4.1)
−
where
D =D 1/2 ... D 1/2 , (1 V )= X(t ) ... X(t ) (4.2)
− −
⊗ ⊗ − h ⊗ ⊗ i
N N
areVN dimensional m|atrices, th{ezntheN-}fermioncorrelator|maybew{zrittenas:}
K 1(t ;0) ... K 1(t ,0) =D 1/2T t D 1/2 , (4.3)
− − − −
h ⊗ ⊗ i
N
and we may identify T| as the trans{fzer matrix and}H = logT as the Hamiltonian of the N-
−
fermionsystem.
Inthecaseoftwofermions(oneupandonedown),thetransfermatrixevaluatedinmomentum
spaceisgivenby:2
pq T pq = d p,p′d q,q′+ C(p′) C(q′)d p+q,p′+q′ , C(p)= 4p (cid:229) C O (p), (4.4)
h ′ ′| | i e−(pp2+p′2+q2p+q′2)/(4M) M n 2n 2n
wherewehaveexpandedC(p)intheoperator basis:
n
O (p)=Mn 1 e p2/M p2n , for p <<1. (4.5)
2n −
− ≈ | |
(cid:16) (cid:17)
Thistransfer matrixcanbediagonalized exactly, withthezerocenterofmomentum energy eigen-
values(e E)givenbysolutions totheintegralequation:
−
(cid:229) 1 (cid:229) O2n(q)
C I (p)=1, I (p)= , (4.6)
2n 2n 2n V e( p2+q2)/M 1
n q BZ −
∈ −
2WehaverecentlyimplementedanewHermitian,Galileaninvariantandanalyticversionofthisinteractionwhich
corresponds to replacing: C(p) C(q) C(p p); results using this improved interaction will be presented in
′ ′ → − ′
detailinafuturepublication.
p p
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Anewapproachforstudyinglargenumbersoffermionsintheunitaryregime MichaelG.Endres
where p=√ME andqaremomentawithinthefirstBrillouinzone(BZ).
5. Lattice parameter tuning andthe continuum limit
Inthecontinuum,Luscher’sformulaallowsustorelatethetwoparticlescatteringphaseshifts
at infinite volume to the discrete energies of the same system at finite volume [11, 12]. Having
solved the twobody problem exactly on the lattice atfinite volume, wemay now relate the lattice
couplings ofourtheorytocontinuum scattering data. Startingfromtheeffectiverangeexpansion
1 1
pcotd = + r p2+... (5.1)
0 0
−a 2
one can relate the scattering length (a), effective range (r ) and higher order shape parameters to
0
the s-wave scattering phase shift. Given an expression for pcotd one may then determine the
0
continuum energy eigenvalues inafiniteboxfromthesolutions to[13]:
pcotd = 1 S(h ), S(h )= lim (cid:229) 1 4p L , (5.2)
0 p L L ¥ n2 h −
→ "n<L − #
| |
with h = (pL/2p )2. Finally one may tune the lattice couplings C (for n = 1,...,k) defined
2n
in Eq. 4.5 by matching the lattice eigenvalues predicted by Eq. 4.6 to the lowest k continuum
energies predicted by Luscher’s formula. In this way, we may absorb all temporal and spatial
latticediscretization errorsintoourdefinitionofthecouplings.
Thecontinuum limitforourlatticetheorycorresponds tothelimitofinfinitescattering length
as measured inunits ofthe lattice spacing. In order tomaintain afinitephysical scattering length,
onemusttakethislimitwhilekeeping otherphysical quantities measuredinunitsofthescattering
length held fixed. In the case of unitary fermions, however, we need not concern ourselves with
such complications and simply take pcotd =0. We therefore tune the couplings C so that the
0 2n
latticeeigenvalues matchthelowestkrootsofS(h )showninFig.2.
In Fig. 3 we plot the percent deviation between the roots of S(h ) and the lattice eigenvalues
predicted byEq.4.6foruptoseventunedC-valuesforunitaryfermions. Foreigenvalues lessthan
the k-th, the deviation is exactly zero by construction. For eigenvalues beyond the k-th, we find
thatthepercentdeviation remainsquitesmallevenforeigenvalues ashighas27,corresponding to
three filled shells. Given a tuned set ofC-values, we may also take the eigenvalues predicted by
Eq.4.6and insert them back into Luscher’s formula, giving alattice prediction for pcotd . Fig. 4
0
shows a plot of the predicted pcotd for up to seven tuned C-values. We find pcotd 1 for a
0 0
≪
widerangeofmomenta,extending wellbeyondthatofthek-theigenvalue wetunedto.
6. Conclusion
Wehave developed anew approach for simulating alarge number ofnonrelativistic fermions
in the unitary regime. In these proceedings we have described some of the details of our lattice
construction, an efficient numerical implementation of the theory and a method for tuning the
lattice couplings to scattering data. Application of these ideas to unitary fermions in a finite box
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Anewapproachforstudyinglargenumbersoffermionsintheunitaryregime MichaelG.Endres
5
0.00006
LΗ
HS 0.1 4 0.00004
of 0.00002
Out[76]= omroots0.001 No321.Cs pcot∆23 0 5 10 15 20 25-00.00002 No321.Cs
v.fr10-5 54 1 54
de 6 6
% 7 7
10-7 0
0 5 10 15 20 25 0 50 100 150 200 250
Eigenvalueno. Η
Figure3: Percentdeviationinh betweenexactlat- Figure 4: Implied pcotd obtainedfrom exactlat-
0
ticeeigenvalues(usingM=5andL=32)andcon- tice eigenvalues (using M = 5 and L = 32) and
tinuumLuschereigenvalues. Luscher’sformula.
[8]andinaharmonictrap3 [9]arediscussedingreaterdetailinourcompanionproceedings. There
the issue of finding optimal sinks/sources as well as finite volume and cutoff effects are explored,
andpreliminary resultsfortheBertschparameter andpairinggapreported.
7. Acknowledgments
Thisworkwassupported byU.S.DepartmentofEnergygrantsDE-FG02-92ER40699(toM.
G.E.)andDE-FG02-00ER41132(toD.B.K.,J-W.L.andA.N.N.). M.G.Eissupported bythe
ForeignPostdoctoral Researcherprogram atRIKEN.
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[9] A.N.Nicholson,M.G.Endres,D.B.Kaplan,andJ.-W.Lee, (2010),arXiv:1011.2804.
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[12] M.Luscher, Nucl.Phys.B354,531(1991).
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3Detailsregardingtheintroductionofexternalpotentialswillbepresentedin[9].
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