Mott-insulator phase of the one-dimensional Bose-Hubbard model:A high-order perturbative study

Bogdan Damski1 and Jakub Zakrzewski21Theory Division, Los Alamos National Laboratory, MS-B213, Los Alamos, New Mexico 87545, USA2Instytut Fizyki imienia Mariana Smoluchowskiego and Mark Kac Complex Systems Research Center,

Uniwersytet Jagielloński, ulica Reymonta 4, PL-30-059 Kraków, Poland�Received 17 April 2006; revised manuscript received 21 August 2006; published 11 October 2006�

The one-dimensional Bose-Hubbard model at a unit filling factor is studied by means of a very high-ordersymbolic perturbative expansion. Analytical expressions are derived for the ground-state quantities such asenergy per site, variance of on-site occupation, and correlation functions: �aj

†aj+r� and �njnj+r�. These findingsare compared to numerics and good agreement is found in the Mott insulator phase. Our results provideanalytical approximations to important observables in the Mott phase, and are also of direct relevance to futureexperiments with ultracold atomic gases placed in optical lattices. We also discuss the symmetry of theBose-Hubbard model associated with the sign change of the tunneling coupling.

DOI: 10.1103/PhysRevA.74.043609 PACS number�s�: 03.75.Lm, 05.30.Jp, 64.60.�i, 73.43.Nq

I. INTRODUCTION

One of the most fascinating recent trends in physics ofcold gases concerns atomic gases in optical lattices �1–3�.These systems offer “atomic Hubbard toolbox” �4� that canbe used for studies of condensed-matter models in a uniquelycontrolled manner. Cold atoms in optical lattices can be usedfor investigation of high-Tc superconductivity �5�, disorderedsystems �6�, various spin models �7�, novel quantum magnets�8�, etc. Perhaps the most important version of the Hubbardmodel, which can be studied in optical lattices, is the Bose-Hubbard model �BHM� �9�. This model is a prototypical sys-tem on which understanding of quantum phase transitions�QPTs� in boson systems is based �10�, and it has importantapplications in construction of a quantum computer �11�.

Despite many theoretical studies on the BHM �12–17�,there is still a lack of analytical predictions about some basicexperimentally relevant quantities. This motivates us topresent here the results of a high-order perturbative expan-sion in the tunneling coupling. These results, providing ana-lytical approximations to different physical quantities in theMott phase, should be helpful in the interpretation of experi-mental data.

We focus on a one-dimensional �1D� homogeneous sys-tem at average density of one atom per site, i.e., on the sim-plest BHM undergoing a QPT. We calculate the followingground-state quantities: energy per site E, atom-atom corre-lations C�r�= �aj

†aj+r�, density-density correlations D�r�= �njnj+r�, and variance of on-site number operator var�n�= ��nj

2�− �nj�2�1/2.The Hamiltonian of interest, in terms of dimensionless

variables used through the paper, reads

H = − J�i=1

M

�ai+1† ai + H.c.� +

U

2 �i=1

M

ni�ni − 1� , �1�

where we denote the number of atoms, and the number oflattice sites as M. Since physics of the BHM depends on theJ /U ratio only, we set U�1 for convenience. With thischoice, the critical point between the Mott insulator �MI�

�0�J�Jc� and superfluid �SF� �J�Jc� is at Jc0.29 �16�.Though a great deal of attention was recently devoted to

cold atoms in inhomogeneous lattices �2,3,17�, the homoge-neous systems described by the Hamiltonian �1� can be real-ized in the near future in at least two setups. First, there wasa recent experiment done in the Raizen group �18�, where asingle, one-dimensional, homogeneous box was realized in aproper configuration of laser beams. After superposing astanding laser field on it, a 1D homogeneous Bose-Hubbardmodel �1� with open boundary conditions can be achieved.Since it was already demonstrated in �18� that one can loadthis 1D box with ultracold bosons and then count them veryefficiently, the studies of lattices with a desired number ofatoms per site should be available �as discussed below onspecific examples�. Another experimental opportunity showsup after realization of the ring-shaped optical lattice pro-posed recently in Ref. �19�. This time, a 1D homogeneouslattice with periodic boundary conditions should be availablefor experimental investigations.

II. THE METHOD

Our findings come from a high-order symbolic perturba-tive expansion in the tunneling coupling. This method wasfirst successfully applied to the calculation of a ground stateand excited states of the BHM in one- and two-dimensionalsystems by Elstner and Monien �15�. We compare perturba-tive expansions to numerical data obtained using the imagi-nary time evolution with the so-called Vidal algorithm �20�,which is equivalent to the density-matrix renormalization-group scheme �21�. This allows for verification of accuracyof our analytical predictions. Vidal’s algorithm calculationsassume open boundary conditions, which breaks the transla-tional invariance of the system. To minimize finite-size ef-fects during comparison between the numerics and perturba-tive expansions valid for infinite systems �where boundaryconditions are irrelevant�, we have calculated the correlationfunctions around the system center.

We aim at calculation of high-order perturbative correc-tions to different quantities of interest in the infinite Bose-

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Hubbard model. The perturbation theory is developed aroundthe Fock state 1,1 , . . . �, where the numbers are boson on-site occupations. The expansion is done in the J parameter�1�.

In principle, the calculations can be performed by hand byperturbative determination of the wave function up to a givenorder in the infinite system, and then subsequent calculationof expectation values in this wave function. The attainableorder of the expansion, however, is very limited �the wavefunction can be determined up to the J3–J4 terms� so thismethod is not an option here.

A better alternative is to perform a linked cluster expan-sion �LCE� �22� that has been used so far in spin systems�23� and the Bose-Hubbard model �15�. This method consistsof two steps. First, one has to generate all the clusters �sets oflattice sites in the BHM �15�� that contribute to a given orderof expansion. The size of the largest of these clusters is com-parable to the order of expansion. Then, one has to perform aperturbative expansion in the relevant clusters and sum upthe results properly. In the end, one gets perturbative expan-sion of different quantities valid for an infinite system fromanalysis of finite clusters. Since the whole procedure can beimplemented on a computer in a symbolic way, the high-order expansions become feasible.

The key for our calculation message from the LCE is thefollowing: all the information about the ith order perturbativeexpansions in the infinite system is encoded in the smallsubsystems of the size �i. In accordance with this statement,we observe for every nonextensive observable that we con-sider, say O, that when we do a perturbative expansionOM =�iOiMJi �M is the system size�, the following holds:

OiM � OiMc, M � Mc�i� , �2�

i.e., the perturbative corrections become size-independent forlarge enough systems. Naturally, Mc�i� grows with i, but thegrowth is “reasonably” slow: Mc�i�� i �see Table I�. We can-not, of course, check explicitly the relation �2� for arbitrarilylarge M, but it is clear from the LCE that the size-independent expansion terms �OiM�Mc

� correspond to infi-nite system predictions.

In our calculations, we make a direct use of the relation�2� avoiding, therefore, generation of the cluster states. Thissimplifies the computer implementation of the whole proce-dure, but probably leads to more stringent requirements onthe computer resources. To be more specific, we fix the sys-tem size M, assume periodic boundary conditions to keep thesystem translationally invariant, and do a standard perturba-tive expansion leading to determination of the ground-statewave function up to a given order �Appendix A�. Having the

ground-state wave function, we calculate perturbative correc-tions to the expectation values of different operators andstudy their dependence on the system size M. This way weeasily get Mc�i�, which guarantees Eq. �2�: Table I, as well asthe desired perturbative corrections valid for infinite systems.Notice that due to Eq. �2�, all finite-size contributions arefiltered out from the perturbative expansions for largeenough M. To illustrate these findings, we note that thesecond-order correction to the ground-state energy per site inthree sites and three-atom system is the same as in the infi-nite model at unit filling factor—a result that can be easilyverified analytically.

III. PERTURBATIVE EXPANSIONS

To start, the ground-state energy per site, E, satisfies

E4

= − J2 + J4 +68

9J6 −

1267

81J8 +

44171

1458J10 −

4902596

6561J12

−8020902135607

2645395200J14 + O�J16� . �3�

Before proceeding further, it should be stressed that the frac-tions come from a symbolic calculation, whose details arepresented in Appendix A.

Coming back to Eq. �3�, we notice that so far the largestpublished order of E expansion was the sixth �15�. Naturally,the expansion of E from �15� matches the first three terms ofEq. �3�. Changing fractions into numbers, one gets approxi-mately the following sequence of nonzero coefficients�−1,1 ,8 ,−16,30,−747,−3032 , which shows that the serieshas a rather unpredictable form.

To estimate the accuracy of the expansion, we plot Eq. �3�versus numerical data for a fairly large system of 40 atoms in40 sites: Fig. 1. This plot shows that there is quite a goodagreement between Eq. �3� and numerics for J smaller thanabout 0.3, i.e., at least in a MI phase. The discrepanciespresent in Fig. 1 �and all other figures in this paper� maycome from the following sources. First, our numerics is donein a finite system with open boundary conditions, while per-turbative expansion yields results for infinite system. Second,we might need more expansion terms to get more accuratepredictions. Third, the full perturbative expansion may fail toconverge for large enough J’s, probably J�Jc. In the worst

TABLE I. Conditions on the system size M for getting size-independent perturbative predictions at ith order to E, C�r�, andD�r�—see Eq. �2�. These results are obtained for the system filledwith one atom per site.

E C�r� D�r�

M � i+1 M � i+1+r M � i+1 and 2r� i

0 0.10.1 0.20.2 0.30.3 0.40.4J

-0.5-0.5

-0.4-0.4

-0.3-0.3

-0.2-0.2

-0.1-0.1

0

ε

FIG. 1. The perturbative prediction for the ground-state energyper site from Eq. �3� �solid line� vs numerics for M =40 system�dots�.

BOGDAN DAMSKI AND JAKUB ZAKRZEWSKI PHYSICAL REVIEW A 74, 043609 �2006�

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case, the series might be of an asymptotic kind, as wasshown to be the case in some other systems �24�. Furtherdiscussion about convergence of expansions presented hereis beyond the scope of the present contribution.

The variance of the on-site number operator, var�n�, up toO�J15� terms satisfies

var�n��2

= 2J − 3J3 −1441

36J5 +

32045

648J7 −

3105413

5184J9

+6979423019

839808J11 +

207832615291307

5290790400J13, �4�

which is illustrated in Fig. 2. As for the E expansion, theagreement between perturbative prediction and numerics isgood for J smaller than �0.35. Thus, our result can be wellapplied to the system in the MI phase. Most interestingly, itshows that the variance of site occupation on the MI-SFboundary equals as much as 60% of its deep superfluid valuein the limit of J→� when the system is maximally delocal-ized �Eq. �B3��.

Expansion �4� is useful, e.g., because there is an ongoingexperiment that aims at var�n� determination in a 1Duntrapped setup �18�. The measurement of var�n� can bepossible due to the ability of a high efficiency single-atomdetection already shown in �18�. It can be performed onceextraction of atoms from a single lattice site is demonstrated.Extracted atoms can be counted, and then all the remainingatoms from a lattice can be released and counted. Averagingthe results of single site countings over the measurementswhere the total number of atoms is close to the number oflattice sites, one should get experimentally var�n�. Anotheraspect of this experiment is that the lattice is blocked at endswith laser beams, which corresponds to open boundary con-ditions used in all our numerical calculations.

Other quantities of interest are atom-atom correlationfunctions C�r� defined above. They were previously studiednumerically in the one-dimensional Bose-Hubbard model in�14� and perturbatively in two-dimensional Bose-Hubbardmodels in �15�. They are important because another measur-able quantity, the momentum distribution of atoms in a lat-tice, is expressed as ��rC�r�exp�ikr� �25�, with k beingatomic momentum. We list a few of the most important onesas follows:

C�1� = 4J − 8J3 −272

3J5 +

20272

81J7 −

441710

729J9

+39220768

2187J11 +

8020902135607

94478400J13 + O�J15� ,

C�2� = 18J2 −320

3J4 −

1826

9J6 +

234862

243J8 +

345809

2916J10

+4434868108963

220449600J12 + O�J14� ,

C�3� = 88J3 −8324

9J5 +

126040

81J7 +

7883333

486J9

−220980576341

1049760J11 + O�J13� . �5�

The comparison between expansions �5� and numerics �Fig.3� shows good agreement up to J equal to 0.25–0.3, i.e.,almost in an entire MI phase. Additionally, Eq. �5� and ourresults for C�4, . . . ,7� indicate that C�r�0�=O�Jr�.

Expansions �5� reveal that atom-atom correlations takevery substantial values at the critical point, e.g., C�1� at Jc

equals about 0.8, i.e., 80% of its deep superfluid value �B3�,an interesting result showing that the system wave functiondeparts significantly from the 1,1 , . . . � state at the criticalpoint.

Finally, we discuss the density-density correlations. Theycan be determined in a counting experiment almost the sameas the one discussed for var�n� measurement, except for thefact that now atoms from two depleted sites have to becounted first. A perturbation theory predicts �6� with an ac-curacy of O�J16�. These data and our results for D�4, . . . ,7�reveal that D�r�0�=1−O�J2r�,

D�1� = 1 − 4J2 +136

3J4 −

2008

27J6 −

150638

81J8 +

4897282

729J10

−415922848153

14696640J12 +

1022120948444278027

7777461888000J14,

FIG. 2. The perturbative result for the variance of on-site num-ber operator from Eq. �4� �solid line� vs numerics for M =40 system�dots�.

0 0.10.1 0.20.2 0.30.3J

0

0.20.2

0.40.4

0.60.6

0.80.8

C(r)

C(r)

FIG. 3. Atom-atom correlations as a function of the tunneling J.Numerics for M =40 system: C�1� �dots�, C�2� �triangles�, C�3���’s�. Lines: solid, dashed, dotted represent perturbative expansions�5� of C�1�, C�2�, and C�3�, respectively.

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D�2� = 1 −100

3J4 +

2128

3J6 −

1156462

243J8 −

6848011

729J10

+10808763042127

44089920J12 −

5150051155340205251

3888730944000J14,

D�3� = 1 −13064

27J6 +

3727066

243J8 −

1588041877

8748J10

+1710030328933

2755620J12 +

2208787916976404357

370355328000J14.

�6�

A comparison between the expansion and the numerics ispresented in Fig. 4: there is a good agreement up to J equalto 0.2–0.225, which may be a little surprising concerning thehigh order of Eqs. �6�. One may attribute it to the presence ofopen boundary conditions introducing small inhomogeneitiesin atom density in our numerics. On the other hand, theeffects of open boundaries will always be present in an ex-perimental setup proposed in �18�. Thus, our numerical cal-culations might actually represent the experimental situationbetter than the expansion �6� derived for an idealized infinitelattice.

Our perturbative predictions can be tested for self-consistency, e.g., the following identity can be derived fromthe eigenequation for the ground-state energy:

E = − 2JC�1� +var�n�2

2.

Using Eqs. �3�–�5�, one easily shows that indeed it exactlyholds up to O�J16� terms, as expected from the orders ofexpansions presented in this paper.

Though it is known that mean-field theory works badly in1D, it is instructive at this point to compare our findings topredictions of the most popular mean field, i.e., a Gutzwillervariational wave function �1,13�. This approach predicts in1D, at unit filling factor, that the ground state is 1,1 , . . . � for0�J�1/11.6, which is interpreted as a MI phase. It impliesthat in this J range, E=0, var�n�=0, C�r�=0, and D�r�=1.Our results analytically quantify the amount of discrepancybetween these mean-field predictions and the exact ones. It

would also be instructive to compare our findings to predic-tions of the perturbatively improved Gutzwiller approach�26�.

IV. SYMMETRY OF PERTURBATIVE EXPANSIONS

Expansions of E, C�r�, and D�r� have well-defined paritywith respect to the J→−J transformation. That is explainedbelow, and provides an insight into the J→−J symmetry ofthe BHM. Additionally, it helps in determining the accuracyof our perturbative predictions. Indeed, the symmetry of per-turbative expansions proven below implies that if the lastcalculated nonvanishing term shows up in the rth order, theexpansion term in the r+1 order vanishes.

First, the ground-state energy per site, Eq. �3�, satisfiesE�J�=E�−J�. To see it, one performs the canonical transfor-mation

ai → �− 1�iai, �7�

which is a bosonic equivalent of the Shiba transformationused to prove a similar symmetry property of the Fermi-Hubbard model: Sec. 2.2.4 of �27�. After the transformation,Hamiltonian �1� reads

H = J�i=1

M

�ai+1† ai + H.c.� +

U

2 �i=1

M

ni�ni − 1�

iff �i� the system is infinite, �ii� the system consists of aneven number of sites and periodic boundary conditions areapplied, and �iii� open boundary conditions are chosen. In therest of this paper, we assume that one of these conditionsholds, and that there exists a unique ground state. The trans-formed Hamiltonian is a J→−J version of Eq. �1�, so we getthe desired prediction about the symmetry of E�J�.

Second, we focus on C�r�J�0= �+aj†aj+r+ �, where

+ � = ��ni

C�ni �a1

†�n1¯ �aM

† �nMvac� �8�

is a ground state of Hamiltonian �1� for J�0 �C�ni are some

coefficients and aivac�=0�. Using Eq. �7�, one gets that theground state of the Hamiltonian �1� with J�0 is �up to aphase factor�

− � = ��ni

�− 1��jnjC�ni �a1

†�n1¯ �aM

† �nMvac� ,

where C�ni are the same as in Eq. �8�. An explicit calculation

shows that C�r�J�0= �−aj†aj+r−�= �−1�rC�r�J�0, therefore

C�r�→ �−1�rC�r� as J→−J: see Eq. �5� for an example when�ni=M→�.

Third, following the above calculation, one shows thatD�r� and var�n� does not change under J sign inversion. Thelatter result is shadowed by explicit assumption of J�0 dur-ing Taylor expansion leading to Eq. �4�.

Finally, we note that the ground state of the two-component Bose-Einstein condensate in an optical latticewas found to change after inversion of J sign �28�. Our paperprovides tools for a closer inspection of this interesting find-ing.

0 0.050.05 0.10.1 0.150.15 0.20.2 0.250.25J

0.90.9

0.920.92

0.940.94

0.960.96

0.980.98

1

D(r)

D(r)

FIG. 4. Density-density correlations as a function of the tunnel-ing J. Numerics for the M =40 system: D�1� �dots�, D�2� �tri-angles�, D�3� ��’s�. Perturbative expansions �6�: solid, dashed, anddotted lines represent D�1�, D�2�, and D�3�, respectively.

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V. SUMMARY

In this paper, we have derived and discussed analyticalpredictions on important quantities describing the physics ofthe Mott phase: variance of on-site number operator, atom-atom, and density-density correlation functions. We havealso explored, for the first time to our knowledge, the funda-mental J→−J symmetry of the BHM. We expect that ourfindings can be useful in interpretation of ongoing experi-ments performed in homogeneous lattices filled with ultra-cold atoms.

ACKNOWLEDGMENTS

We would like to acknowledge insightful discussions withDominique Delande, Mark Raizen, and Eddy Timmermans,as well as the financial support from the U.S. Department ofEnergy. Calculations based on Vidal’s algorithm were donein ICM UW under Grant No. G29-10. J.Z. was supported byPolish Government scientific funds.

APPENDIX A: PERTURBATIVE EXPANSION

Here we sketch some of technical details of the nondegen-erate perturbative expansions that we perform.

To introduce the notation: the Hamiltonian �1� is rewritten

as H0+�V, where �=−J, H0= 12�i=1

M ni�ni−1�, V=�i=1M �ai+1

† ai

+H.c.�.The ground-state wave function and eigenenergy up to a

given order are

� = �r=0

S

�r�r��, E = �r=0

S+1

�rEr, �A1�

�r�� = �i

ai�r�i�, H0i� = eii�, �ij� = �ij . �A2�

To keep the wave function �A1� normalized to unity, wemust satisfy � �=1+O��S+1�. This implies that��0� �0��=1 and �29�

��0��r�� = −1

2�i=1

r−1

��i��r − i��, r = 1, . . . ,S .

�A3�

The wave function �A1� follows the eigenequation

�H0 + �V�� = E� , �A4�

which in the �0 order implies that �0��=a0�0�0�= 0� �after

proper choice of basis�, and E0=e0. In higher orders a0�r�

= �0 �r�� is found from Eq. �A3�.The coefficients ai�0

�r�1� �A2� are obtained from projectionof Eq. �A4� onto basis states �i�0� �A2�,

ai�r��ei − e0� + �

j

�iVj�aj�r−1� = �

k=1

r−1

Ekai�r−k�. �A5�

The eigenenergies are found from a projection of Eq. �A4�onto 0� �29�

Er = �j

�0Vj�aj�r−1� − �

k=1

r−1

Eka0�r−k�, �A6�

where r=1, . . . ,S+1. Equations �A3�, �A5�, and �A6� have tobe solved order by order starting from r=1.

In our calculations,

�0�� ª 0� = 1,1, . . . � �A7�

and

i� = n1i,n2i, . . . ,nMi�, ei =1

2�k

�nki − 1�nki. �A8�

Efficient manipulations on Fock states �A8� are crucial forgetting high-order expansions on a computer. We haveachieved it by using a fast mapping �n1i ,n2i , . . . ,nMi → i,which was provided by the hash functions—see �30� for theirgeneral description.

Since ei �A8� is an integer number and the action of Vonto a Fock state produces factors of the form �nk±1+1�nk�nk is the atom occupation of the kth lattice site�, we see fromEqs. �A5� and �A6� that every ai

�r� coefficient will be in gen-eral equal to

ai�r� = �

s=1

smax b��r,i ,s�c��r,i ,s�

��k=1

kmax

p��r,i ,s,k� , �A9�

where b��r , i ,s� and c��r , i ,s� are integer numbers andp��r , i ,s ,k� are prime numbers satisfying p��r , i ,s ,k��r+1—notice that when we develop the expansion around theFock state �A7�, the highest on-site occupation in the rthorder is at most r+1 because it originates from r consecutive

actions of V operator onto Eq. �A7�. Similarly, one finds thatkmax��r , i ,s��r.

We have done the symbolic calculations by writing aproper C code. Each set of �p��r , i ,s ,k� :k=1, . . . ,kmax hasbeen encoded in bits of a single integer number, while everyb��r , i ,s� /c��r , i ,s� ratio has been allocated as a Gnu MultiPrecision rational number �31�. The first choice minimizesthe memory requirements on storage of the prime numberpart, while the latter one allows for avoiding overflows oc-curring at high-order expansions when b��r , i ,s� andc��r , i ,s� are allocated as integer numbers. To reduce usageof memory, the smax parameter has to be set as small aspossible during allocation process. By running the program,we found that smax=2 is sufficient for all the calculationsperformed in this paper.

The results of the symbolic calculations were directlyverified by calculation of the perturbative expansion in anumeric �as opposed to a symbolic� way. There we treat ai

�r�

as double-precision numbers avoiding complications of sym-bolic manipulations. Results obtained numerically are inagreement with the symbolic ones within accuracy providedby the double-precision format.

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APPENDIX B: NONINTERACTING SYSTEM

It is instructive to briefly summarize here the predictionsfor various observables in the limit of negligible atom inter-actions, say the U�0 case.

The N-particle ground state of the noninteracting M-siteHubbard model �1� with periodic boundary conditions isgiven by

SF� =1

�N!MN��i=1

M

ai†�N

vac� , �B1�

where aivac�=0 and �SF SF�=1—notice that Eq. �B1� ismerely the N-particle zero-momentum ground state �32�. Theground-state eigenenergy equals −2JN.

After some algebra, Eq. �B1� can be rewritten in the form

SF� =� N!

MN��ni

n1, . . . ,nM��n1! ¯ nM!

, �B2�

where the sum goes over all sets of boson on-site occupation�integer� numbers �ni such that �ini=N and 0�ni�N. The

wave function �B2� allows for an easy calculation of differ-ent expectation values. Indeed, the summation can be ex-pressed as ��ni

=�n1=0N

¯�nM=0N �n1+¯+nM,N, the Kronecker

delta can be represented as �l,m= 12��0

2�d�exp�i��l−m, andfinally the integration over can be replaced by the integra-tion over z=exp�i� on the unit circle surrounding the originof the complex z plane. Usage of the residue theorem allowsfor getting that

var�n� =�N

M−

N

M2 , C�r� =N

M, D�r� =

N�N − 1�M2 ,

for any integer N�0 and M �r.In this paper, we are interested in the case of N /M =1 and

M 1, i.e.,

var�n� = 1, C�r� = 1, D�r� = 1. �B3�

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