Variance minimization variational Monte Carlo method

0.2 MB·English

by  Imran Khan

Variance minimization variational Monte Carlo method
∗
Imran Khan and Bo Gao
Department of Physics and Astronomy, University of Toledo, MS 111, Toledo, Ohio 43606
(Dated: February 2, 2008)
We present a variational Monte Carlo (VMC) method that works equally well for the ground
and the excited states of a quantum system. The method is based on the minimization of the
7 variance of energy, as opposed to the energy itself in standard methods. As a test, it is applied to
0
theinvestigation of theuniversalspectrum at thevanderWaalslength scale fortwoidenticalBose
0
atoms in a symmetricharmonic trap, with results compared to thebasically exact results obtained
2
from a multiscale quantum-defecttheory.

n
a PACSnumbers: 02.70.Ss,03.75.Nt,21.45.+v,31.15.-p
J

9
1 I. INTRODUCTION for multiple energy levels canbe obtainedindependently
usingothermethods [11,12,13,14],including,inpartic-

] ular,amultiscalequantum-defecttheory(QDT)[15,16].
h MonteCarlomethodshaveplayedanimportantrolein
Conclusions are given in Sec. IV. We point out that in
p ourunderstandingofavarietyofquantumsystems,espe-
the process of writing this article, we have discovered
- ciallyfew-andmany-bodyquantumsystemswithstrong
p that an equivalent approach has been developed earlier
interactionsthataredifficulttotreatotherwise(see,e.g.,
m by Umrigar et al. [17]. The derivation of our method,
Refs. [1, 2, 3, 4, 5, 6]). It is also well-known, however,
o that most quantum Monte Carlo methods [2, 3] are for- and the applications presented here and earlier [10], are
c however different.
mulated in such a way that they are strictly applicable
.
s only to the ground state of a quantum system, a restric-
c
tion that has severely limited their applicability. Con-
si sider,forexample,thegaseousBose-Einsteincondensates II. VARIANCE MINIMIZATION VARIATIONAL
y MONTE CARLO METHOD
(BEC) of alkali-metal atoms (see, e.g., [7]). Any theory
h
p that intends to treat the real atomic interaction has to
Consider the time-independent Schr¨odinger equation
[ dealwiththefactthatthe gaseousBECbranchofstates

are in fact highly excited states of a many-atom system.
1 There are many branches of states of lower energies, in- H|Ψni=En|Ψni , (1)
v
3 cluding the first branch of liquid states as suggested and where the energy ebigenstates |Ψni form a complete, or-
2 studied recently by one of us [8]. thonormal basis.
2 In this paper we present a variational Monte Carlo Existing quantum Monte Carlo methods are mostly
1 (VMC) method that works the same way for either the basedonthefactthatforanarbitrarytrialwavefunction
0 ground or the excited states of a quantum system. It is satisfying proper boundary conditions, we have
7
basedonthe minimizationofthe varianceofenergy,and
0
/ is the method underlying a recent investigation of the ΨT H ΨT
s universal equation of state at the van der Waals length E [Ψ ]≡ D (cid:12) (cid:12) E ≥E , (2)
c scale [8, 9] for few atoms in a trap [10]. The details of T T hΨ(cid:12)(cid:12)Tb|Ψ(cid:12)(cid:12)Ti 0
i
s the method were skipped in the earlier article [10], both which means that the ground state wave function is the
y
h because the focus there was on a single gaseous BEC one that minimizes the energy functional ET[ΨT]. The
p state, which was not the best example illustrating the proofcanbefoundinstandardquantummechanicstext-
: method, and because there were no other independent books (see, e.g., [18]).
v
i results to directly compared with, except in the shape- The variance minimization variational Monte Carlo
X independent limit [10]. method (VMVMC), as proposed here, is based on the
r We present here, in Sec. II, the details of the vari- functional
a
ational Monte Carlo method based on the minimization
2
ofthevarianceofenergyandshowsthatitappliesequally Ψ H2 Ψ Ψ H Ψ
T T T T
wsyesltletmo.thIengSreocu.ndIIIa,nwdethpereesxecnitteadbsetattteers iollfuastqrautainotnumof η[ΨT]≡ D hΨ(cid:12)(cid:12)(cid:12)Tb|Ψ(cid:12)(cid:12)(cid:12)Ti E −D hΨ(cid:12)(cid:12)(cid:12)Tb|Ψ(cid:12)(cid:12)(cid:12)Ti E ≥0. (3)
 
the method through the universal spectrum at the van
der Waals length scale for two identical Bose atoms in a TheproofofEq.(3)anditsphysicalmeaningcanbebest
symmetricharmonictrap. Itisanexamplewhereresults understood by expanding the trial wave function using
the complete basis defined by Eq. (1) to write η[Ψ ] as
T
|hΨ |Ψ i|2(E −E )2
η[Ψ ]= m m T m T . (4)
∗Email:[email protected];Homepage:http://bgaowww.physics.utoledo.edu T P |hΨ |Ψ i|2
m m T
P
2
∗
From Eq. (4), it is clear that zero is the minimum of = Ψ |H|Ψ Ψ |H|Ψ
m T m T
the functionalη[ΨT], andthis minimum is reachedwhen Xm D E D E
and only when E = E and hΨ |Ψ i = 0 for m 6= n, b b
T n m T ′ ′ ∗ ′ ∗ ′
namely, only when |ΨTiis aneigenstate ofenergy as de- = Z dτdτ [Ψm(τ )ΨT(τ )ELoc(τ )
fined by Eq. (1). This statement is equally applicable to Xm
∗
the ground and the excited states of a quantum system. ×Ψm(τ)ΨT(τ)ELoc(τ)] . (11)
The implementation of VMVMC, based on the mini-
Using the completeness relation
mization of the variance of energy η[Ψ ], is straightfor-
T
ward. It does not require much more than the standard
′ ∗ ′
Ψ (τ )Ψ (τ)=δ(τ −τ), (12)
VMC,asweillustratehereusingtheexampleofidentical m m
particles. Xm
Consider N identical particlesin an externalpotential we obtain
and interacting via pairwise interactions. It is described
by a Hamiltonian: Ψ |H2|Ψ = dτΨ∗(τ)Ψ (τ)|E (τ)|2 , (13)
D T TE Z T T Loc
N N b
H = hˆ + v(r ), (5) and therefore
i ij
Xi=1 i<Xj=1
b ΨT|H2|ΨT dτΨ∗(τ)Ψ (τ)|E (τ)|2
with D E = T T Loc . (14)
hΨTb|ΨTi R dτΨ∗T(τ)ΨT(τ)
hˆi = −2¯hm2 ∇2i +Vext(ri). (6) The computation of the vRariance of energy, Eq. (3), has
thus been reduced to two integrals, Eqs. (7) and (14),
Here Vext(r) is the external “trapping” potential, and bothofwhichinvolvingthesamelocalenergy,ELoc,that
v(r) is the interaction between particles. oneencountersinstandardVMC.Itisclearthatthefor-
For the evaluation of the energy functional, we have mulationandthe equationsinthis sectionareapplicable
to both bosons and fermions.
1
hΨT|H|ΨTi=hΨT|Nhˆ1+ N(N −1)v12|ΨTi One can easily show that our method is equivalent to
2
that of Umrigar et al. [17]. However, we believe that
= bdτΨ∗Ψ 1 {Nhˆ + 1N(N −1)v(r )}Ψ our derivation provides a more rigorous foundation and
Z T TΨT 1 2 12 T shows more explicitly why it works for both the ground
∗ and the excited states.
= dτΨ Ψ E (τ), (7)
Z T T Loc
where τ represents an N particle configuration specified
III. SAMPLE RESULTS FOR IDENTICAL BOSE
by their 3N coordinates. E is the so-called local en-
Loc ATOMS IN A SYMMETRIC HARMONIC TRAP
ergy, and is given by
¯h2 1 1 The VMVMC, as outlined in Sec. II, was first applied
E =N − ∇2Ψ +NV (r )+ N(N−1)v(r ).
Loc (cid:18) 2m(cid:19)Ψ 1 T ext 1 2 12 in Ref. [10] to study the universal equation of state at
T
the van der Waals length scale [8, 9] for few identical
(8)
Bose atoms (N = 3-5) in a trap. To better illustrate
The averageenergy is therefore
and to further test the method, we investigate here the
∗
E = dτΨTΨTELoc(τ) . (9) universalspectrum at the van der Waals length scale for
T R dτΨ∗Ψ two identical Bose atoms in a symmetric harmonic trap.
T T
Itisaproblemforwhichaccurateresultscanbeobtained
R
This is the standard integral in VMC, and can be eval- independentlyusingavarietyofmethods[11,12,13,14],
uated using standard Monte Carlo methods such the including a multiscale QDT [15, 16].
Metropolis method (see, e.g., [19]). Two identical Bose atoms in a symmetric harmonic
In order to calculate the variance of energy, one must traparedescribedbytheHamiltonian,Eqs.(5)-(6),with
also determine the average of H2. This can be done by N =2, and
first noting that, similar to Eq. (7), we have
b 1
V (r )= mω2r2 , (15)
∗ ext i 2 i
Ψ |H|Ψ = dτΨ Ψ E (τ), (10)
D m TE Z m T Loc
where m is the mass of an atom, and ω is the trap fre-
b
where |Ψ i is an eigenstate of energy as defined by quency.
m
Eq. (1). We have therefore For the trap states of interest here, we take the trial
wave function to be of the form of
Ψ |H2|Ψ = Ψ |H|Ψ Ψ |H|Ψ
D T TE Xm D T mED m TE ΨT =[φ1(r1)φ2(r2)+φ1(r2)φ2(r1)]F(r12), (16)
b b b
3
where φ and φ are independent-particle orbitals, and
1 2
F istheatom-atomcorrelationfunctionthatisdiscussed
in more detail in Ref. [10]. Specifically, we use
4
Au (r)/r , r <d
F(r)= λ , (17)
(cid:26)(r/d)γ , r ≥d
where u(r) satisfies the Schr¨odinger equation:
ω)
¯h2 d2 h 3
(cid:20)−m dr2 +v(r)−λ(cid:21)uλ(r)=0, (18) cle (
arti
P
for r < d. γ is the parameter characterizing the long- er
range correlation between atoms in a trap, with γ = 0 P
y
(meaning F = 1 for r > d) corresponding to no long- erg 2
range correlation. Both d and γ are taken to be varia- En
tional parameters, in addition to the variational param-
eters associated with the descriptions of φ and φ . The
1 2
parametersAandλarenotindependent. Theyaredeter-
minedbymatchingF anditsderivativeatd. Ourchoice
β /a =0.001
ofF differsfromtraditionalchoices(see,e.g. Ref.[6])not 1 6 ho
only in its treatment of the short-range correlation, but
especially in its allowance for the long-range correlation -10 -5 0 5 10
characterized by parameter γ. This was first suggested a / a
0 ho
by a multiscale QDT treatment of two atoms in a sym-
metric harmonic trap [15, 16], and was later found to be
the key for treating N trapped atoms in cases of strong FIG. 1: The universal spectrum at length scale β6 for two
coupling, namely when the s wave scattering length a Bose atoms in a symmetric harmonic trap as a function of
0
becomes comparable to or greater than the trap length a0/aho for β6/aho = 0.001. Solid line: results from a multi-
scale a =(h¯/mω)1/2 [10]. scale QDT[16]. Symbols: results of VMVMC.
ho
For atoms in their ground state, the atom-atom inter-
actionisofthevanderWaalstypeof−C /rn withn=6
n As in Ref. [10], the universal spectrum at length scale
at large interatomic separations, i.e.,
β ,namelytheΩ ’sinEq.(20),canbecomputedbyusing
6 i
v(r)r−→→∞−C /r6 . (19) a correlation function, Eq. (17), with uλ(r) as given by
6 the angular-momentum-insensitive quantum-defect the-
ory (AQDT) [21],
This interaction has an associated length scale of β =
6
(mC /¯h2)1/4, and a corresponding energy scale of s =
(h¯2/m6 )(1/β6)2 [20]. OverawiderangeofenergiesthaEtis uλs(rs)=B[fλc(s6l=)0(rs)−Kcgλc(s6l)=0(rs)]. (21)
hundreds of s aroundthe threshold [21, 22], the details
E c(6) c(6)
of atomic interactions of shorter range than β6 are not Here B is a normalization constant. fλsl and gλsl are
important, andcanbe characterizedby a single parame- universal AQDT reference functions for −C6/r6 type of
terthatcanbetheswavescatteringlengtha ,theshort potentials [9, 20]. They depend on r only through a
0
range K matrix Kc, or some other related parameters scaled radius rs = r/β6, and on energy only through a
[21, 23, 24]. In this range of energies, the spectrum of scaledenergyλs =λ/sE. Kc istheshort-rangeKmatrix
twoatomsinatrapfollowsauniversalpropertythatcan [21] that is related to the s wave scattering length a0 by
be characterized by [10, 15, 16] [24, 25]
E¯hi/ωN =Ωi(a0/aho,β6/aho), (20) a0/βn =(cid:20)b2bΓΓ((11−+bb))(cid:21)KKcc−+ttaann((ππbb//22)) , (22)
and is called the universal spectrum at length scale β . where b=1/(n−2), with n=6.
6
Here Ω are universal functions that are uniquely deter- Figure 1 shows a portion of the universal spectrum at
i
mined by the number of particles, the exponent of the length scale β for two Bose atoms in a symmetric har-
6
van der Waals interaction (n = 6), and the exponent monic trap. Specifically, it gives the energies of the first
of the trapping potential (2 for the harmonic trap). The threeswavetrapstatesasafunctionofa /a . Thecor-
0 ho
strengthsofinteractions,characterizedbyC andω,play responding φ s usedin Eq.(16) are independent-particle
6 i
a role only through scaling parameters such as β and orbitals based on standard solutions for a single particle
6
a . in a symmetric harmonic potential (see, e.g., [26]). For
ho
4
4 4
ω) ω)
h 3 h 3
e ( e (
cl cl
arti arti
P P
er er
P P
y y
g g
er 2 er 2
n n
E E
1 β6/aho=0.1 1 ββ66 // aahhoo == 00..0101
-10 -5 0 5 10 -10 -5 0 5 10
a0 / aho a0 / aho
FIG. 2: The same as Fig. 1 except for β6/aho =0.1. FIG. 3: A comparison of the spectra for two different values
of β6/aho, illustrating the shape-dependent correction that
becomes moreimportant for greater valuesof β6/aho andfor
the lowest s wave trap state, they are taken to be more highly excited states.
φ (r)=exp(−α x2), i=1,2. (23)
i i
They are taken to be
φ (r) = exp(−α x2),
1 1
3 butareinanycaselessthan1.8×10−3forallparameters
φ2(r) = −x2 exp(−α2x2), (24) considered. The results shownin Figure 1, which are for
(cid:18)2 (cid:19)
a smallβ /a =0.001,illustrate the shape-independent
6 ho
for the first excited s wave trap state, and limit of β6/aho → 0 for states with Ei/2 ∼ ¯hω ≪ sE
[10, 16]. They agree, in this limit, with the results ob-
3 tained using a delta-function pseudopotential [11]. For
φ (r)= −x2 exp(−α x2), i=1,2, (25)
i (cid:18)2 (cid:19) i greater β6/aho, the effects of the van der Waals inter-
action become gradually more important, especially for
for the second excited s wave trap state. Here x is a strong coupling (a /a ∼ 1 or greater) and for more
0 ho
scaled radius defined by x = r/a . The variational pa- highly excited states [12, 16]. This is illustrated in Fig-
ho
rameters are d, γ, α , and α in all three cases. The ure 3, which compares the results for β /a = 0.1 with
1 2 6 ho
variance of energy is calculated accordingto Sec. II, and those for β /a = 0.001. We note that even the low-
6 ho
the minimization is carried out using a type of genetic est trap state is itself a highly excited diatomic state.
algorithm. There are other “molecular” states that are lower in en-
Both Figs. 1 and 2 show that the results of VMVMC ergy[15,16]. Thisfactdoesnot,however,leadtoanydif-
are in excellent agreements with those of a multiscale ficultiesbecauseVMVMCworksthesamefortheground
QDT [15, 16], which gives basically exact results for two and the excited states. It is for the same reason that we
atomsinasymmetricharmonictrap. (Thescaledenergy were able to investigate the gaseous BEC state for few
per particle, E /(2h¯ω), used here is related to the scaled atomsinatrap[10],whichisagainahighlyexcitedstate.
i
center-of-mass energy, e = ǫ/¯hω, used in Ref. [16], by More detailed discussions of the universal spectrum at
E /(2h¯ω)=(e+3/2)/2.) The agreements are all within length scale β for two atoms in a symmetric harmonic
i 6
thevariancesofenergy,whicharesmallerforweakercou- trap, including the molecular states and the spectra for
pling (smaller a /a ) and greaterfor stronger coupling, nonzero partial waves, can be found elsewhere [16].
0 ho
5
IV. CONCLUSIONS lent agreements with the basically exact results derived
independently from a multiscale QDT [15, 16].
We have presented a variationalMonte Carlo method,
VMVMC, that works the same for the excited states as
it does for the ground state. The method is tested here Acknowledgments
throughtheuniversalspectrumatlengthscaleβ fortwo
6
identical Bose atoms in a symmetry harmonic trap, for This work was supported by the National Science
whichtheresultsfromVMVMCarefoundtobeinexcel- Foundation under Grant No. PHY-0457060.
[1] M. H. Kalos, D. Levesque, and L. Verlet, Phys. Rev. A [14] E. L. Bolda, E. Tiesinga, and P. S. Julienne, Phys. Rev.
9, 2178 (1974). A 66, 013403 (2002).
[2] D.M. Ceperley, Rev.Mod. Phys. 67, 279 (1995). [15] Y.ChenandB.Gao, Bull.Am.Phys.Soc.50,No.3,80
[3] W. M. C. Foulkes, L. Mitas, R. J. Needs, and G. Ra- (2005).
jagopal, Rev.Mod. Phys. 73, 33 (2001). [16] Y. Chen and B. Gao, cond-mat/0701384.
[4] S. Giorgini, J. Boronat, and J. Casulleras, Phys.Rev. A [17] C. J. Umrigar, K. G. Wilson, and J. W. Wilkins, Phys.
60, 5129 (1999). Rev. Lett.60, 1719 (1988).
[5] D. Blume and C. H. Greene, Phys. Rev. A 63, 063601 [18] J. J. Sakurai, Modern Quantum Mechanics (Addison-
(2001). Wesley, New York,1994).
[6] J. L.DuBois and H.R.Glyde, Phys.Rev.A 63, 023602 [19] J.M.Thijssen,Computational Physics (CambridgeUni-
(2001). versity Press, Cambridge, 1999).
[7] F.Dalfovo,S.Giorgini,L.P.Pitaevskii,andS.Stringari, [20] B. Gao, Phys.Rev.A 58, 1728 (1998).
Rev.Mod. Phys. 71, 463 (1999). [21] B. Gao, Phys.Rev.A 64, 010701(R) (2001).
[8] B. Gao, Phys. Rev.Lett. 95, 240403 (2005). [22] B. Gao, E. Tiesinga, C. J. Williams, and P. S. Julienne,
[9] B. Gao, J. Phys. B 37, L227 (2004). Phys. Rev.A 72, 042719 (2005).
[10] I.Khan and B. Gao, Phys. Rev.A 73, 063619 (2006). [23] B. Gao, Phys.Rev.A 58, 4222 (1998).
[11] T.Busch,B.-G.Englert,K.Rzazewski,andM.Wilkens, [24] B. Gao, Euro. Phys. J. D 31, 283 (2004).
Found.Phys.28, 549 (1998). [25] B. Gao, J. Phys. B 36, 2111 (2003).
[12] E. Tiesinga, C. J. Williams, F. H. Mies, and P. S. Juli- [26] N. Zettili, Quantum Mechanics (John Wiley and Sons,
enne,Phys. Rev.A 61, 063416 (2000). New York,2001).
[13] D. Blume and C. H. Greene, Phys. Rev. A 65, 043613
(2002).