a r X i v :0803.3888v 2 [c o n d -m a t .s t a t -m e c h ] 1 A p r 2008
The e?ects of next to nearest neighbor hopping on Bose-Einstein condensation in
cubic lattices
G.K.Chaudhary and R.Ramakumar
Department of Physics and Astrophysics,University of Delhi,Delhi-110007,Delhi,India
(27March 2008)In this brief report,we present results of our calculations on the e?ects of next to nearest neighbor boson hopping (t ′)energy on Bose-Einstein condensation in cubic lattices.We consider both non-interacting and repulsively interacting bosons moving in the lowest Bloch band.The interacting bosons are studied making use of the Bogoliubov method.We ?nd that the bose condensation temperature is enhanced with increasing t ′for bosons in a simple cubic (sc)lattice and decreases for bosons in body-centered cubic (bcc)and face-centered cubic (fcc)lattices.We also ?nd that interaction induced depletion of the condensate is reduced for bosons in a sc lattice while is enhanced for bosons in bcc and fcc lattices.
PACS numbers:03.75.Lm,03.75.Nt,03.75.Hh,67.40.-w
I.INTRODUCTION
Studies of Bose-Einstein condensation in optical lat-tices and crystalline lattices is an active ?eld of research
both in atomic 1–3and condensed matter physics 4–10.In condensed matter physics,there have been extensive studies of bose condensation of bipolarons 4,exctions 5,exciton-polartions 6,7,and magnons 8–10.Studies of bosons in optical lattices may be said to have re-ceived a boost with the demonstration of bose condensed to Mott insulator transition 2predicted in theoretical studies 1,11–14of strongly interacting lattice bosons.In the presently available optical lattices,it has been shown 1that it is su?cient to include nearest-neighbor (NN)hop-ping of bosons in the kinetic energy part of the Hamil-tonian of the system.Nevertheless,considering the fast pace of developments in this ?eld,it may be useful to investigate the e?ects of the next to nearest-neighbor (NNN)hopping on bose condensation in optical and crys-talline lattices.Recently,we presented a study of the lattice symmetry e?ects on bose condensation in cubic lattices 15.In that work,we had con?ned to NN hop-ping of lattice bosons.In this note,we extend this work including NNN boson hopping.In the next section,we describe the models and methods used in our calculations along with a discussion of results.The conclusions are given Sec.III.
II.BOSE CONDENSATION IN CUBIC LATTICES
WITH NNN HOPPING
Non-interacting bosons :Consider bosons moving in cu-bic lattices.The energy eigen-functions of a single bo-son moving in a periodic optical or crystalline lattice po-tential are Bloch waves 16and energy eigen-values form bands.The Hamiltonian of non-interacting bosons in an energy band is:
H =
k
[?(k )?μ]c ?k c k ,
(1)
where ?(k )is the one-boson energy band structure,k is the boson quasi-momentum,μis the chemical potential,and c ?k is a boson creation operator.Within a tight-binding approximation scheme 17,including the NN and the NNN Wannier functions overlaps,the s -band struc-tures we consider for cubic lattices are:?sc (k x ,k y ,k z )=?2t
z
μ=x cos(k μ)
?2t
′
z μ=x z
μ=ν,ν=x
cos(k μ)cos(k ν),(2)
?bcc (k x ,k y ,k z )=?8t z
μ=x
cos
k μ
2
cos
k ν
N x N y N z
k x
k y
k z
1
k B T
?1
,(5)
where N s =N x N y N z is the total number of lattice sites,k B is the Boltzmann constant,T is the temperature,and
n is number of bosons per site.We have numerically solved the bosons number equation (Eq.5)to obtain the bose condensation temperature and ground state oc-cupancy.The results of these calculations for various lattices considered are shown in Fig.1.
0.3
0.4 0.5 0.6 0.7 0.8
0.9 1 0
0.02
0.04
0.06
0.08
0.1
K B T /W
t’
FIG.1.The bose condensation temperature vs NNN hop-ping t ′for noninteracting bosons in various cubic lattices.The curves are shown for:sc (red),bcc (green),and fcc (blue).These results are for n =0.4.In this and other ?gures W is the half-band-width.
We ?nd that t ′
increases the bose condensation temper-ature of bosons in a sc lattice.For bosons in bcc lattice the T B decreases with increasing t ′.For bosons in a fcc lattice also increasing t ′more or less leave T B unaltered.
0.2 0.4 0.6
0.8 1
0 0.2 0.4 0.6 0.8 1 1.2
C o n d e n s a t e F r a c t i o n
K B T/W
FIG.2.The variation of condensate fraction with temper-ature (T)for bosons in a sc lattice with NNN hoppings t ′:t ′=0(red),t ′=t/10(green).Here n =0.4.
0 0.2 0.4 0.6 0.8 1
1.2 1.4 0
0.2
0.4
0.6
0.8
1
K B T B /W
n
FIG.3.The bose condensation temperature vas n for bosons in a sc lattice with NNN hopping t ′:t ′=0(red),t ′=t/20(blue),t ′=t/10(green).
The growth of the condensate fraction and the number dependence of T B for bosons in a sc lattice shown in Fig.2and Fig.3are similar to that found for the case of t ′=015.Similar results are obtained (not shown)for bosons in bcc and fcc lattices.
Interacting bosons :The Hamiltonian of interacting bosons is H =
k
[?(k )?μ]c ?k c k +
U
2N s
′ k
1+
ξ(k )+Un 0
e E (k )
2N s
′ k
1?
ξ(k )+Un 0
e
?E (k )
ξ2(k )+2Un 0ξ(k ).The primes on
the summation signs indicates that the sums excludes the k =0state in to which the bosons condense.The con-densate fraction for bosons in various cubic lattices are shown in Fig.4-6.
0.8
0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1 0
0.5
1
1.5
2
C o n d e n s a t e F r a c t i o n
U/W FIG.4.The variation of Condensate fraction (for T=0and n=0.4)with U/W of weakly interacting bosons in a sc lattice with NNN hoppings t ′:t ′=0(red),t ′=t/20(blue),t ′=t/10(green).For bosons in a sc lattice,we ?nd that interaction in-duced depletion of the condensate is reduced with in-creasing t ′as shown in Fig.4.For bosons in a bcc lattice (Fig.5),increasing t ′is found increase the interaction in-duced depletion.In the case of bosons in an fcc lattice,the e?ects of increasing t ′does not have much e?ect on condensate fraction as shown in Fig. 6.The bose con-densation temperature is una?ected by the interaction in the Bogoliubov method.
0.8
0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1
0.5
1
1.5
2
C o n d e n s a t e F r a c t i o n
U/W
FIG.5.The variation of Condensate fraction (for T=0and n=0.4)with U/W of weakly interacting bosons in a bcc lat-tice with NNN hoppings t ′:t ′=0(green),t ′=t/20(blue),t ′=t/10(red).
0.8
0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1
0.5
1
1.5
2
C o n d e n s a t e F r a c t i o n
U/W
FIG.6.The variation of condensate fraction (for T=0and n=0.4)with U/W of weakly interacting bosons in a fcc lattice with NNN hoppings t ′:t ′=0(green),t ′=t/20(blue),t ′=t/10(red).
III.CONCLUSIONS
In this brief report,we investigated the e?ects of NNN hopping of non-interacting and interacting bosons in cubic lattices on bose condensation temperature and ground state occupancy.We ?nd that the bose con-densation temperature is enhanced with increasing t ′for bosons in a simple cubic (sc)lattice and decreases for bosons in body-centered cubic (bcc)and face-centered cubic (fcc)lattices.We also ?nd that interaction induced depletion of the condensate is reduced for bosons in a sc lattice while it is enhanced for bosons in bcc and fcc lat-tices.
IV.ACKNOWLEDGEMENT
Gopesh Kumar Chaudhary thanks the University Grants Commission (UGC),Government of India for pro-viding ?nancial support through a Junior Research Fel-lowship.
3I.Bloch,J.Dalibard,W.Zwerger,arXiv:0704.3011v1 [cond-mat.other].
4A.S.Alexandrov and N.F.Mott,Rep.Prog.Phys.57, 1197(1994).
5For a review see:D.Snoke,Science298,1368(2002).
6J.Kasprzak,M.Richard,S.Kundermann,A.Baas,P. Jeambrun,J.M.J.Keeling,F.M.Marchetti,M.H.Szy-manska,R.Andr′e,J.L.Staehli,V.Savona,P.B.Lit-tlewood,B.Deveaud,and Le Si Dang,Nature443,409 (2006).
7R.Balili,V.Hartwell,D.Snoke,L.Prei?er,and K.West, Science316,1007(2007).
8T.Nikuni,M.Oshikawa,A.Oosawa,and H.Tanaka,Phys. Rev.Lett84,5868(2000).
9Ch.R¨u egg,N.Cavadini,A.Furrer,H.-U.Gdel,K.Krmer, H.Mutka,A.Wildes,K.Habicht,and P.Vorderwisch,Na-ture423,62(2003).10For a review see:T.Giamarchi,Ch.R¨u egg,and O.Tch-ernyshyov,Nature Phys.4,198(2008).
11M.P.A.Fisher,P.B.Weichman,G.Grinstein,and D.S. Fisher,Phys.Rev.B40,546(1989).
12W.Krauth and N.Trivedi,Europhys.Lett.14,627(1991). 13K.Sheshadri,H.R.Krishnamurthy,R.Pandit,and T.V. Ramakrishnan,Europhys.Lett.22,257(1993).
14J.K.Freericks and H.Monien,Phys.Rev.B53,2691 (1996).
15R.Ramakumar and A.N.Das,Phys.Rev.B72,094301 (2005).
16F.Bloch,Z.Phys.52,555(1928).
17N.W.Ashcroft and N.D.Mermin,Solid State Physics (Harcourt Asia,Singapore,2001).
18N.N.Bogoliubov,J.Phys.(Moscow)11,23(1947).