The density of states approach at finite chemical potential: a numerical study of the Bose gas

0.35 MB·

by  L. Bongiovanni

The density of states approach at finite chemical
potential: a numerical study of the Bose gas.
Roberto Pellegrini∗
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HiggsCentreforTheoreticalPhysics,SchoolofPhysicsandAstronomy,Universityof
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0 Edinburgh,EdinburghEH93FD,UnitedKingdom
2 E-mail: [email protected]

n
a Lorenzo Bongiovanni
J
CollegeofScience,SwanseaUniversity,SingletonPark,SwanseaSA28PP,UnitedKingdom

2
E-mail: [email protected]
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Kurt Langfeld
]
at CentreforMathematicalSciences,PlymouthUniversity,Plymouth,PL48AA,UnitedKingdom
l E-mail: [email protected]
-
p
e Biagio Lucini
h
CollegeofScience,SwanseaUniversity,SingletonPark,SwanseaSA28PP,UnitedKingdom
[
E-mail: [email protected]
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v
Antonio Rago
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2 CentreforMathematicalSciences,PlymouthUniversity,Plymouth,PL48AA,UnitedKingdom
9 E-mail: [email protected]
2
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. Recently,anovelalgorithmforcomputingthedensityofstatesinstatisticalsystemsandquantum
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0 field theories has been proposed. The same method can be applied to theories at finite density
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affectedbythenotorioussignproblem,reducingahigh-dimensionaloscillatingintegraltoamore
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: tractableone-dimensionalone. AsanexampleweappliedthemethodtotherelativisticBosegas.
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The33rdInternationalSymposiumonLatticeFieldTheory
14-18July2015
KobeInternationalConferenceCenter,Kobe,Japan*
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommons
Attribution-NonCommercial-NoDerivatives4.0InternationalLicense(CCBY-NC-ND4.0). http://pos.sissa.it/
ShortTitleforheader RobertoPellegrini
1. Introduction
Non perturbative phenomena play a relevant role in many aspects of modern quantum field
theory. The most developed tool to probe quantitatively these phenomena is the Markov Chain
Monte-Carlo simulations of the theory discretised on a space-time lattice. However, this method
hasseverallimitations,oneofthemostseverebeingthesocalledsignproblem.
The method relies on the interpretation of the Euclidean path integral measure as a probability
measure; in which case it is possible to build a Markov Chain that will reproduce the Boltzmann
weightinthelongrun. However, inmanyinterestingtheoriestheEuclideanactionisnotrealand
themethodbecomesveryinefficient.
Recently,anewsimulationmethodbasedonthedensityofstateshasbeenproposed[1]andtested
successfullyinfewmodels[2,3,4,5].
We feel that the method deserves further testing in order to properly assess its performance. So
far, the method has been probed in discrete models and far from phase transitions. Our aim is to
study a model with continuous degrees of freedom and in the neighboring of a phase transition.
The relativistic Bose gas at finite density is very well suited for these kind of tests; it is known to
undergoasecondorderphasetransitionanditisrelativelyfasttosimulate.
2. Thedensityofstates
LetusconsideranEuclideanquantumfieldtheorydescribedbythevariablesφ,theproperties
ofthesystemcanbederivedfromthepartitionfunction
(cid:90)
Z(β)= [Dφ]eβS[φ]. (2.1)
Thedensityofstatesisdefinedbytheintegral
(cid:90)
ρ(s)= [Dφ]δ(s−S[φ]), (2.2)
and its geometrical interpretation is the volume of phase-space available to the system at fixed
action. Fromwhichitispossibletocomputepartitionfunctionsandobservables
(cid:90)
Z(β)= dsρ(s)eβs. (2.3)
An algorithm to compute the log-derivative of the density of states was proposed in [1], and the
convergenceofthealgorithmwasprovedin[2].
Itwastestedin4dcompactU(1)latticegaugetheory,whichisknowntoundergoafirstorderphase
transition with remarkable results. In fig.(1) a plot of the log-derivative of the density of states is
shown,andintab.(1)wereportacomparisonbetweenthecriticalvalueofthecouplingcomputed
usingthedensityofstatesmethodandatraditionalMonte-Carlo.
3. Generalizeddensityofstates
Thecaseofcomplexactioncanbetreatedinthesamefashion;eq.(2.1)becomes
(cid:90)
Z(β,µ)= [Dφ]eβSR[φ]+iµSI[φ] (3.1)
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ShortTitleforheader RobertoPellegrini
-0.99
-1
-1.01
ai
-1.02 L=8
L=10
L=12
-1.03 L=14
L=16
L=18
-1.04 L=20
3.5 3.6 3.7 3.8 3.9 4 4.1
E/V
Figure1: Thelog-derivativeofthedensityofstatesfordifferentvolumesin4dU(1)LGT
Lmin kmax βCv(∞) χr2ed
14 1 1.011125(3) 0.91
12 1 1.011121(3) 2.42
12 2 1.011129(4) 0.67
10 1 1.011116(5) 7.44
10 2 1.011127(3) 0.60
8 1 1.011093(5) 90.26
8 2 1.011126(2) 0.62
Table 1: Comparison between the critical value of β computed using the density of states and using a
traditionalMonte-CarloCITAZ
where S ,S are the real and imaginary parts of the action and µ is the Lagrange multiplier cor-
R I
respondent to the complex part of the action. The generalized density of states [3] is introduced
by
(cid:90)
P (s)= [Dφ]δ(s−S [φ])eβSRe[φ]. (3.2)
β Im
Ifβ =0,P (s)isthevolumeofphase-spaceavailabletothesystematgivenimaginarypartofthe
0
action. Atfiniteβ thereisanadditionalweightproportionaltoeβSR[φ]. Thepartitionfunctionisthe
Fouriertransformofthegeneralizeddensityofstates
(cid:90)
Z(β,µ)= dsP (s)eiµs. (3.3)
β
Itisworthtonoticethattheintegrandineq.(3.2)isrealandthatthedifficultiesduetotheoscillating
phase are present only in the 1-dimensional integral in eq.(5.4). Using the LLR algorithm it is
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ShortTitleforheader RobertoPellegrini
possible to compute P (s) with very high precision over many order of magnitude, thus the sign
β
problemisreducedtothecomputationofa1-dimensionalfastlyoscillatingintegral.
4. RelativisticBosegasatfinitechemicalpotential
The Bose gas at finite chemical potential is a perfect toy-model to test new algorithms for
simulating systems affected by a sign problem. It has been studied extensively with different ap-
proaches i.e. complex Langevin dynamics, Lefschetz thimble, dual formulation and mean field
theory[6,7,8,9].
Inthecontinuumformulationtheactionhastheform
(cid:90)
S[φ]= d4x∂ φ∂ φ+(m2−µ2)|φ|2+µ(φ∗∂ φ−φ∂ φ∗)+λ|φ|4. (4.1)
µ µ 4 4
Thediscretizationofthisactionposesnodifficultiesaslongasthechemicalpotentialistreatedas
avectorpotential. Thisleadsto
4 (cid:16) (cid:17)
S[φ]=∑(2d+m2)φ∗φ +λ(φ∗φ )2− ∑ φ∗e−µδν,4φ +φ∗ eµδν,4φ . (4.2)
x x x x x x+ν x+ν x
x ν=1
Thelattercanbewrittenintermoftherealandimaginarypartofthefieldsφ = √1 (φ +iφ )
a 1 2
2
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S[φ]=∑(2d+m2)φ2 φ +λφ4 −∑ φa,xφ −coshµφ φ +isinhµε φ φ . (4.3)
a,x x a,x a,x+ν a,x a,x+4ˆ ab a,x b,x+4ˆ
x ν=1
InthediscretecasetheLagrangemultiplierµ iscoupledonboththerealandimaginarypartofthe
action,andasaconsequencethegeneralizeddensityofstateswilldependalsoonµ,
(cid:90)
P (s)= [Dφ]δ(s−S [φ])eSRe(m,λ,µ)[φ] (4.4)
m,λ,µ Im
Weareparticularlyinterestedintheexpectationvalueoftheoscillatingphase,whichquantifiesthe
severityofthesignproblem
(cid:104)e−SIm(cid:105)=e−VF, (4.5)
whereV isthevolumeandF isthevariationofthefreeenergyduetotheoscillatingphase.
Anotherinterestingobservableisthedensityofparticles(cid:104)n(cid:105),givenby
dlogZ
(cid:104)n(cid:105)= . (4.6)
dµ
5. Theoscillatingintegral
It is fairly easy to compute the generalized density of states using the LLR algorithm; an
example is shown in fig.(2). The LLR algorithm delivers a very high precision over different
ordersofmagnitude. Infig.(2)itisclearthattheDOSdeviatesfromapureGaussiandistribution.
DeviationsfromaGaussianfunctionareclearlyvisibledespitethesearesuppressedbyafactorof
ordere−100;whichiswellbeyondtheprecisionofastandardMonte-Carloprocedure.
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ShortTitleforheader RobertoPellegrini
Figure2: Thelog-derivativeofthegeneralizedDOSwithm=λ =1andV =84. Astraightlinewouldbe
apurelyGaussianDOS,itisveryremarkablethatusingtheLLRispossibletoclearlyseedeviationsfrom
gaussianity.
In order to extract physical quantities from the theory we need to carry out the Fourier trans-
formofthegeneralizedDOS,eq.(5.4). ThenaturalapproachwouldbetouseafastFouriertrans-
form algorithm, however this method is too naive and would break down at very low chemical
potential. Inordertoillustratewhythisapproachisnotadequateletusassumethat,infirstapprox-
imation,theDOSisaGaussianplussomewhitenoiseη(s),
P (s)∼e−s2σ(m,λ)+η(s). (5.1)
m,λ,µ
ThepartitionfunctionistheFouriertransformofthelatter
− µ2π2
Z(m,λ,µ)∼e σ(m,λ,µ) +c, (5.2)
wherecistheFouriertransformofthenoise. Therefore,thesignalisexponentiallysuppressedwith
µ2 whilethenoisegiveaconstantcontributiontothepartitionfunction. Amuchbetteralternative
istofilterthenoisebyinterpolatingthelog-derivativeoftheDOSwithapolynomialofdegree2n,
P (s)∼e∑icis2i. (5.3)
m,λ,µ
In this case the statistical errors affect the coefficients and the Fourier transform of P is a fast
decaying function. We found our results to be very stable whenever we avoid over-fitting; in
practice we use a Bayesian evidence criterion [11] to decide the degree of the polynomia in order
tolimitover-fitting.
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ShortTitleforheader RobertoPellegrini
Once a polynomial interpolant of the DOS is obtained, we are left with the computation of the
followingfastlyoscillatingintegral
(cid:90)
Z(m,λ,µ)= dse∑icis2i+iµs. (5.4)
The most straightforward approach is to use a simple numerical integration routine with multiple
precision. The number of digits needed for the calculation increases with volume and chemical
potentialasthecancellationsbecomemoreviolent. Itwilleventuallygrowtoaverylargenumber.
Togiveanidea,weneeded50digitstocomputethepartitionfunctionwithvolume84 and µ =1.
Another approach is the numerical steepest descent or Lefschetz Thimble [10] method. Let us
consideranintegralofthetype
(cid:90) +∞
Z= e−S(x)dx, (5.5)
−∞
whereS(x)isholomorphyic. Itispossibletoshowthat
(cid:90)
Z=∑m e−iSIm(zk) dze−SRe(z), (5.6)
k
k Jk
(cid:82)
herez arecriticalpoints∂ S=0. And areintegralsoverthecurvesofsteepestdescent. These
k z Jk
areparametriccurvesgivenbytheordinarydifferentialequation
x˙=−Re{∂ S(z)}, y˙=+Im{∂ S(z)}. (5.7)
z z
m isanintegernumberthatcountsthenumberofintersectionbetweenthecurvesofsteepestascent
k
andtheoriginaldomainofintegration. Noticethatthesignproblemdidnotdisappearcompletely,
there is a residual oscillating phase coming from the Jacobian of the transformation dz and each
dt
curve of steepest descent contributes with a global phase e−iSIm(zk). This residual sign problem is
verymildandcomputationindoubleprecisionaresufficient.
InFig(5)weshowtheaveragefreeenergydifferencebetweenthefullandthequenchedtheory
eq.(4.5). Our results are compared with a mean field calculation in the framework of complex
Langevindynamics[8],courtesyoftheauthor.
Theresultsareinverygoodagreementwitheachother.
6. Conclusions
WepresentedanapplicationoftheLLRalgorithmtoasystemaffectedbyaseveresignprob-
lem. Wefoundthatthemethodisabletoextractmeaningfulresultsatfinitedensityeveninregions
of the parameter space where the sign problem is severe. The main drawback is that it relies on a
polynomial fit of the log-derivative of the DOS, this might be difficult if the shape of the density
of states is not very regular. On the other hand, in all the system studied with this approach the
density of states turned out to be a remarkably smooth function, which gives hope that this is the
casealsoforphysicallymorerelevantsystems.
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ShortTitleforheader RobertoPellegrini
Figure3: ComparisonbetweenLLRmethodandmeanfieldtheoryforµ =λ =1
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