Table Of ContentTheoretical study of bosonic atoms with competing short- and global-range
interactions in an optical lattice
Renyuan Liao∗
Fujian Provincial Key Laboratory for Quantum Manipulation and New Energy Materials,
College of Physics and Energy, Fujian Normal University, Fuzhou 350117, China and
Fujian Provincial Collaborative Innovation Center for Optoelectronic
Semiconductors and Efficient Devices, Xiamen, 361005, China
7
(Dated: February 27, 2017)
1
0 We consider bosonic atoms loaded into optical lattices with cavity-mediated infinite-range inter-
2 actions. Competing short- and global-range interactions cultivates a rich phase diagram. With a
b systematicfield-theoreticalperspective,wepresentananalyticalconstructionofglobalground-state
e phase diagram. We derive an effective theory describing compressible superfluid and supersolid
F states. We construct a self-consistent mean-field theory and find numerical results consistent with
ourtheoretical analysis.
4
2
PACSnumbers: 67.85.Hj,05.30.JP,03.75.Hh,03.75.Lm
]
s
a Recent experimental realization of competing short- associated number operator, and M is the total number
g
and infinite-range interactions [1] for bosonic atoms in of lattice sites. The subscript e (o) refers to even (odd)
-
nt optical lattices has opened a new avenue for exploring latticesitesi=(ix,iy)ofsquarelattice potentialdefined
newphasesofmatter. Thisisachievedbytrappingquan- as i +i ∈ even (odd), and <ij> denotes pair of site
a x y
u tum gases in an optical lattice inside a high finesse opti- i and j. The first term represents the kinetic energy
q calcavity. Theinfinity-rangeinteractionsismediatedby resulting from the tunneling between sites with hopping
t. a vacuum mode of the cavity and can be independently amplitude tij and supports superfluidity. The second
a
controlledbytuningthecavityresonance[2–4]. Essential term describes the on-site repulsive interaction with
m
physics of this system falls into the category of the ex- strength U>0. It prefers balanced atom populations
-
tended Hubbard model[5], andhas been investigatedby on even and odd sites and vanishing spatial coherence.
d
n mean-fieldtheories and quantummonte carlosimulation The third term introduces the infinite-range interactions
o and various exotic phases have been predicted [6–13]. with strength K>0, which favors an overall even-odd
c Ultracold gases in optical lattices are one of the most imbalance. Interplay of three energy scales is expected
[
intriguingsystemsinwhichthepowerofatomicandlaser to give rise to a multitude of ground-state manifolds.
2 physics can be exploited to explore generic phenomena The chemical potential µ is introduce to fix the average
v of solid-sate physics [14]. They have been proven to be particle number N.
6
impressively successful in simulating strongly correlated
7
models like the Bose-Hubbard models, which features a
6
3 quantumphasetransitionfromasuperfluidtoaMottin- Within the framework of Euclidean functional inte-
0 sulating phase [15, 16]. Adding to the excitement is the gral, the partition function of the system may be cast
1. experimental realization of global-range interactions [1], as Z = D[b∗,b ]e−S with the action given by [24, 25]
i i
0 whichraisesthefascinatingpossibilityofexploringemer- S = βdτ [b∗∂ b +H(b∗,b )],hereβ =1/k T isthe
7 gent phenomena such as self-organizingstructures. The- 0 R i i τ i i i B
inversetemperature. Todecoupletheoff-sitetermsinthe
1
oretical understanding of this interesting system is still R P
: action,weintroducearealfieldθ(τ)andcomplexbosonic
v initsinfancy,andthiscallsforinsightsfromvariousper-
fields Ψ (τ) by performing Hubbard-Stratonovich trans-
i spectives. Inthis work,we shallinvestigatethis problem i
X formations, resulting in an equivalent representation of
fromafield-theoreticalapproach[17–23],whichallowsto
r the partition function
a treat both compressible and non-compressible phases.
We consider the system described by the following
canonical Hamiltonian realized very recently [1]
U
Hˆ =− t ˆb†ˆb +h.c. + nˆ (nˆ −1)
ij i j 2 i i
<Xij>(cid:16) (cid:17) Xi Z = D[Ψ∗i,Ψi] D[θ,b∗i,bi]e−SR, (2)
2 Z Z
K
− nˆ − nˆ − µnˆ . (1)
i i i
M
" #
i∈e i∈o i
X X X
Here ˆb†(ˆb ) are the bosonic creation (annhilation)
i i
operators of an atom at lattice site i, nˆ = ˆb†ˆb is the where the resultant action is given by S =
i i i R
2
βdτ Ψ∗t−1Ψ +S +S with if K/U ∈ (1/2,1), the system enters into a fully polar-
0 ij i ij j 0 I
ized CDW phase with n = int 1+2µ/U and n = 0.
R P e 2(1−K/U) o
S = βdτ Kθ2−2Kθ b∗b − b∗b Finally, if K/U > 1, the grounhd energy iis unbounded
0 Z0 "M M i∈e i i i∈o i i!# from below rendering that the system becomes unstable
X X toward collapse. The above discussions for the ground-
U
+ b∗(∂ −µ)b + b∗b∗b b , (3) state phase diagram are summarized in Fig. 1.
i τ i 2 i i i i
i (cid:20) (cid:21) The low temperature properties of the system may
X
β be captured by only considering the particle and hole
S =− dτ (Ψ∗b +b∗Ψ ). (4)
I i i i i excitations on one supercell [26], since tunneling be-
Z0 Xi tween sites are completely neglected and hence the sys-
tem consists of isolate pair sites. For brevity, let us
Before embarking on detailed analysis with field-
theoretical machinery, we make some comments. The define Cp = s=e,oe−βEsp and Ch = s=e,oe−βEsh,
where E and E is the particle and hole excitation
free part S is readily solvable since the corresponding sp sh
0 P P
Hamiltonian Hˆ (θ) = Hˆ (θ) can be diagonalized in on s = (e/o) site, respectively. To put it explicitly,
0 i 0i
E = E(n + 1,n ) − E(n ,n ) and E = E(n −
the occupation number representation. With the inter- ep e o e o eh e
P 1,n )−E(n ,n ), and similarly for E and E . The
actingpartS present,thephysicscouldnotbesolvedin o e o op oh
I
partition function on one supercell is therefore approx-
a close form, however, we can gain physical insights by
seeking perturbative expansion on top of S in terms of imated as z0 = e−βE(ne,no)(1 + Cp + Ch). The varia-
0
tion of density fluctuations on a supercell is given by
fieldsΨ ,whichservesasthesuperfluidorderparameter.
i
δn = (C −C )/(1+C +C ). Equivalently interest-
Now we subject the action S to a saddle point anal- p h p h
R ing is the square density fluctuations δn2 ≡ n2 −hni2,
ysis. Extremum of variation of the action with respect
to θ(τ) yields θ = b∗b − b∗b . The physi- which is related to the isothermal compressibility via
0 i∈e i i i∈o i i thermodynamic relation δn2 = hni2κ v /β(cid:10)wit(cid:11)h v be-
cal meaning is clear: θ counts the particle number dif- T 0 0
P P ing the volume of one supercell. We find that δn2 =
ference between even sites and odd sites, and θ could
0
(C +C +4C C )/(1+C +C )2. Thetemperaturede-
be regarded as an order parameter representing charge p h p h p h
pendenceofδnandκ isshowninFig.2. Atzerotemper-
degrees of freedom. It is easy to notice that quantum T
ature, the particle number fluctuation and thermal com-
fluctuation of θ field only gives a trivial factor to the
pressibility is frozen out, indicating its non-compressible
partition function. Therefore, we are satisfied to replace
nature. It clearly indicates that the larger K/U is, the
θ(τ) by θ .
0
larger thermal fluctuation it induces. We attribute this
Let’s considerthe atomiclimit wherethe hopping am-
fluctuation-enhancingbehaviortothedelocalizingeffects
plitude between sites is negligible (t /U = 0), and the
ij
brought about by the infinite-range interactions.
resultant action reduces to a free one: S = S . The
R 0
eigenvaluecorrespondingto Hˆ for per “supercell”(with We proceed to take into account the effects of a finite
0
hopping amplitude. We can evaluate the partition func-
one oven and one odd sites) after taking account of the
tion by performing Taylor expansion in the exponent
self-consistency conditions for θ is given by
E(ne,no)= U4 (cid:20)(ne+no)−(1+ 2Uµ)(cid:21)2 ZZ0 =Z D[Ψ∗i,Ψi]e−RdτPijΨ∗it−ij1Ψj*Xl=0 Sl!Il+0,(6)
U 2K 2µ
+ (1− )(ne−no)2−(1+ )2 . (5) where hOi = ( e−S0O)/( e−S0). To the
4 U U 0
(cid:20) (cid:21) quadratic order in the fields Ψ , by transforming
i
R R
Here n (n ) represents the occupation number of one to momentum-frequency representation, we obtain
e o
even (odd) site. The ground state is achieved by mini- Z/Z0 = D[Ψ∗(k),Ψ(k)]e−Sg with the gaussian action
mizingtheeigenvalueE(n ,n )withrespecttoquantum given by
o e R
numbers n and n . Since the system enjoys an Ising-
e o M
typeZ symmetrycorrespondingtoexchangeofevenand S = Ψ†(k)G−1(k)Ψ(k), (7)
2 g 2
odd sites, we may choose ne ≥ no from now on. To fa- k=(Xk,iwn)
cilitate the analysis, we define 1+2µ/U = n+x, with
n=int[1+2µ/U]being the integerclosestto 1+2µ/U, where we have defined Ψ(k) = (Ψe(k),Ψo(k))T, and
and x ∈ (−1/2,1/2). Firstly, let’s consider the case used a shorthand notation k = (k,iwn), with wn being
K/U ∈(0,1/2), the system is in a Mott insulating (MI) the bosonic Matsubara frequencies. The inverse Green’s
phase with n =n =n/2 if n is even. If n is odd, then function assumes the form of 2×2 matrix
e o
the system is in the MI phase when K/U < |x|, and in
ne+1 + ne −t˜−1(q)
a partially polarized charge density wave (CDW) phase −G−1 = Eep−iwn Eeh+iwn (.8)
with ne = no + 1 = (n + 1)/2, vice versa. Secondly, −t˜−1(q) Eonpo−+i1wn + Eehn+oiwn!
3
FIG. 1. (color online). Global ground-state phase diagram spanned by µ/U and K/U at atomic limit (tij/U =0). The phase
diagram can be loosely divided into three regimes depending on the strength of infinite-range interaction: (1)K/U ∈(0,0.5),
thesystemiseitherinaMottinsulating(MI)phasewithne =no orinapartiallypolarizedchargedensitywave(CDW)phase
with ne−no = 1, where we always assume ne ≥ no as the system enjoys an Ising-type Z2 symmetry; (2)K/U ∈ (0.5,1), the
system is in a fully polarized CDW phase with no =0; (3)K/U >1, thesystem is unstable toward collpase.
In the above, t˜−1(q) is the Fourier transform of t−1. detG−1(0,0)=0, which yields
ij
For convenience we shall consider the nearest neigh-
bor hopping only with amplitude t, then t˜−1(q) = ne+1 ne no+1 no 1
+ + = . (9)
1/[2t dj=1cos(kjλ/2)], with d being the dimension of (cid:18) Eep Eeh(cid:19)(cid:18) Eop Eoh(cid:19) (zt)2
the system and λ being the wavelength of the laser field
P We show the phase boundary in Fig. 4. Evidently the
forming the optical lattices.
regime of MI phase diminishes as K/U increases. In
The excitation spectrum of the system corresponds
starkcontrast,theregimeofCDW phasegetsbroadened
to the poles of the Green’s function. It can be readily
as K/U increases. This is caused by the delocalizing ef-
found by seeking solutions ω for the secular equations
fect of infinite-range interaction, favoring the formation
detG−1(k,ω)=0. Itfeaturesfourbranchesofexcitation
ofdensity modulationinthe formofacheckerboardpat-
spectrum ω (i=1..4), as shown in Fig. 3 at k = (~0,0)
i tern with alternating site occupation.
in terms of the tuning parameter zt/U with z =4 being
To explore the physics of compressible superfluid
the coordination number of square lattices. In absence
and supersolid phases, we proceed even further by
of hopping (zt = 0), the incompressible MI phase
(1,1) evaluating the action to the quartic order in or-
possesses only one type of particle excitations and one
der parameter Ψ , S = S + S + S with S =
i 0 g 4 4
type of hole excitations, while incompressible CDW(2,1) ss′ k+l=m+nuss′Ψ∗s(k)Ψ∗s′(l)Ψs′(m)Ψs(n), where we
phase carrying charge order possesses two types of par-
may evaluate the coefficients uss′ on shell. By perform-
ticle excitations and two types of hole excitations. At a P P
ing derivative expansion,we keep only the most relevant
finite hopping, these two phases both accommodate two
termsinalong-wavelengthapproximation,andobtainan
branchesofparticle excitationsandtwo branchesofhole
effectiveactionofGinzburg-Landau-Wilsontype [28,29]
excitations. The minimal energy difference between one
particle excitation and one hole excitation corresponds S−S =
0
to the energy gap for density fluctuations. This excita- d2r
tiongapbecomessoftatthephaseboundarywherephase dτ 2 rs|Ψs|2+asΨ∗s∂τΨs+bs|∂τΨs|2+uss|Ψs|4
transition from a non-compressible phase to a compress- Z s
X(cid:0) (cid:1)
ible phase occurs. + dτd2r r Ψ∗(1− λ2∇2)Ψ +c.c+u |Ψ |2|Ψ |2 .
The phase boundary separating the superfluid phase 2 eo e 16 o eo e o
Z (cid:20) (cid:21)
and the non-compressible phase occurs [19, 20, 27] at (10)
4
0.8 0.5
0.2 (a) 0.4 (b) (a) (b)
K/U =0 ω
0.6 1
K/U =0.1 0.3 ω2
0.15 K/U =0.2 0.3 ωω12 ωω34
K/U =0.3 ω
T 3
n κ ω4 0.1
δ 0 0.2
v
ω
0.1 0.2 ω
MI(1,1) CDW(2,1)
−0.1
−0.2
0.05 0.1
−0.3
0 0 −0.6 −0.5
0 0.05 0.1 0.15 0 0.05 0.1 0.15 0 0.04 0.08 0.12 0 0.02 0.04 0.06
k T/U k T/U zt/U zt/U
B B
FIG. 2. (color online). Temperature dependence of (a) the FIG. 3. (color online). Four branches of excitation spectrum
variation of particle number δn on one supercell (with one ωi(i=1..4)atmomentumk=(~0,0)asafunctionoftunneling
even site and one odd site) and (b) isothermal compressibil- parameter zt/U for (a) the MI phase with µ/U = 0.50
(1,1)
ityκT forMI(1,1) phaseatdifferentinfinite-rangeinteraction andK/U =0.25;(b)theCDW(2,1)phasewithµ/U =1.0and
strengthK/U =0,0.1,0.2and0.3. Hereµ/U =0.60, andv0 K/U =0.40. Theuppertwobranchesareparticleexcitations
is the volumeof onesupercell. andthelowertwobranchesareholeexcitations. Forbothin-
sulatingphases,thereexistsanenergygapforaparticle-hole
excitation, manifesting their non-compressible nature. Here
To present the coefficients above in a succinct fash- z=4 is the coordination numberfor squarelattices.
ion, we define A = (n + 1)/E2 and B = n /E .
s s sp s s sh
Then the relevant coefficients are given as follows: r =
s
−(A E +B E ),a =A −B ,b =A /E +B /E , erally,therealizationofthephaseisdeterminedbyseek-
s ps s hs s s s s s sp s sh
r =1/(zt),u =(A +B )(A E +B E )−A (n + ing the global minimum of Ω. The saddle point condi-
eo ss s s s sp s sh s s
2)/E −B (n −1)/E , and tion ∂Ω/∂Ψ = 0 yields 2r Ψ + 2r Ψ + 4u Ψ3 +
s2p s s s2h s s s eo −s ss s
2u Ψ Ψ2 . Clearly, if Ψ = 0, then from the above
U E +E E +E eo s −s s
eo =A A ep op +B B eh oh equationweimmediatelyobtainΨ =0,namelyΨ and
e o e o −s e
K E +E +K E +E +K
ep op eh oh Ψ vanishesidenticallyatthetransitionpoint. Thephase
o
Esp+E−sh boundary is determined by r r =r2 , which reproduces
− AsB−s . (11) o e eo
Esp+E−sh−K Eq. (9). Typically close to the phase boundary, the or-
s=e,o
X der parameter field satisfies a simple scaling Ψ /Ψ =
s −s
Here Ee2p = E(ne + 2,no) − E(ne,no) is the “double −reo/rs = − r−s/rs. At this level the phase transi-
particles” excitation energy at even sites and E =
e2h tion is of a continuous one. However, when the system
E(ne −2,no)−E(ne,no) is the “double holes” excita- is deep into apsuperfluid phase with a crystalline order,
tionenergyatevensites,andsimilarexpressionsforEo2p theremayinduceastructuraltransition(wherequantum
andE . Theuniversalityclassandassociatedquantum
o2h numbersn andn change)fromasupersolid(SS)phase
e o
criticalityisintimatelyrelatedtotherelevantparameters to a superfluid (SF) phase, typically a first-order one,
given above.
since it involves a discontinuous change of the free en-
At low energy, we seek field configurations that are
ergy. In broken-symmetry phases, the order parameters
spatiallyandtemporallyhomogenous. The grandpoten- aredeterminedbythecoefficients. Thegrandpotentialis
tialΩ=−lnZ/β ofthe systemreducestoasimple form
fully determined as Ω=Ω (n ,n ,µ)+δΩ(n ,n ,µ,zt).
0 e o e o
as follows Givenµ andzt,the globalminimum ofthe grandpoten-
Ω=Ω + r |Ψ |2+r (Ψ∗Ψ +c.c) tial is achieved by minimizing over non-negative integer
0 s s eo e o
of n and n . However, it should be noted that such
s=e,o e o
X perturbative treatment only give qualitatively sensible
+ u |Ψ |4+u |Ψ |2|Ψ |2. (12)
ss s eo o e physics for the regime deep into the compressible phase.
s=e,o
X
We notice that r >0, so Ψ and Ψ is out of phase, Self-consistent mean-field theory. The mean-field
eo e o
and we may choose both of them to be real. Quite gen- Hamiltonian for a supercell can be constructed as fol-
5
conditions Ψ =< ˆb > and Ψ =< ˆb >. The numeri-
e e o o
cal results from this self-consistent theory are shown in
0.15 (a) 0.15 (b)
Fig. 5, which confirms our field-theoretical analysis car-
Compressible Phase Compressible Phase ried out previously.
(SF or SS) (SF or SS)
RLacknowledgesfundingfromtheNSFCunderGrants
No. 11674058,11274064and NCET-13-0734.
K/U =0.10
0.1 0.1 K/U =0.25
U U
/ /
t t
z z
∗
[email protected]
0.05 0.05 [1] R. Landig, L. Hruby, N. Dogra, M. Landini, R. Mottl,
T. Donner,and T. Esslinger, Nature532, 476 (2016)
[2] K.Baumann,C.Guerlin,F.Brennecke,andT.Esslinger,
MI(1,1) Nature 464, 1301 (2010)
MI
(2,2) MI(1,1) MI(2,2) [3] R.Mottl,F.Brennecke,K.Baumann,R.Landig,T.Don-
ner, and T. Esslinger, Science336, 1570 (2012)
0 0
0 1 2 0 1 2 [4] H. Ritsch, P. Domokos, F. Brennecke, and T. Esslinger,
µ/U µ/U Rev. Mod. Phys. 85, 553 (2013)
[5] O. Dutta, M. Gajda, P. Hauke, M. Lewenstein,
D. Lu¨hmann, B. A. Malomed, T. Sowin´ski, and J. Za-
FIG. 4. (color online). The phase boundary separating com-
krzewski, Rep.Prog. Phys. 78, 066001 (2015)
pressible (SF or SS) and non-compressible phases (MI or
[6] Y.Li,L.He,andW.Hofstetter,Phys.Rev.A87,051604
CDW)fortwotypicalinfinite-rangeinteractionstrengths: (a)
(2013)
K/U = 0.10 and (b) K/U = 0.25. Increasing K/U leads to
[7] H. Habibian, A. Winter, S. Paganelli, H. Rieger, and
thebroadeningoftheregion ofCDW phasesandtheshrink-
G. Morigi, Phys. Rev.Lett. 110, 075304 (2013)
ing of the region of MI phases. Here a SF phase stands for
[8] Y. Chen, Z. Yu, and H. Zhai, Phys. Rev. A 93, 041601
a superfluid phase which has off-diagonal long-range order,
(2016)
and aSS phasestands for asupersolid phase which has both
[9] N. Dogra, F. Brennecke, S. D. Huber, and T. Donner,
diagonal and off-diagonal long-range orders.
Phys. Rev.A 94, 023632 (2016)
[10] A. E. Niederle, G. Morigi, and H. Rieger, Phys. Rev. A
94 (033607)
2 (a) [11] B. Sundar and E. J. Mueller, Phys. Rev. A 94, 033631
Ψ|
| (2016)
1 Ψ
Ψe [12] J. Panas, A. Kauch, and K. Byczuk, arXiv:
o 1607.00671(2016)
00 0.2 0.4 0.6 0.8 1 [13] T. Flottat, L. F. Parny,F. Hebert,V. G. Rousseau, and
zt/U G. G. Batrouni, arxiv: 1701.01624(2017)
3 [14] I. Bloch, J. Dalibard, and W.Zwerger, Rev.Mod. Phys.
(b) 80, 885 (2008)
2
n1 nneo [15] PD..ZJoalklesrc,hP, hCy.sB. Rruedve.rL,eJt.t.I.8C1,ir3a1c0,8C(.1W99.8)Gardiner, and
[16] M.Greiner,O.Mandel,T.Esslinger, T.W.H¨ansch,and
00 0.2 0.4 0.6 0.8 1 I. Bloch, Phys.Rev. Lett.95, 260401 (2005)
zt/U
[17] M.P.A.Fisher,P.B.Weichman,G.Grinstein,andD.S.
Fisher, Phys. Rev.B 40, 546 (1989)
FIG. 5. (color online). (a) The magnitude of the order pa- [18] J. K. Freericks and H. Monien, Phys. Rev. B 53, 2691
rameters |Ψe| and |Ψo| in the supersolid phase as a function (1996)
of zt/U; (b) The density at even site ne and odd site no [19] R. A. Lehrer and D. R. Nelson, Phys. Rev. B 58, 12385
as a function of zt/U. Here the parameters used here are (1998)
µ/U =0.98 and K/U =0.45. [20] D. vanOosten, P. vanderStraten, and H. T. C. Stoof,
Phys. Rev.A 63, 053601 (2001)
[21] K. Sengupta and N. Dupuis, Phys. Rev. A 71, 033629
lows: (2005)
[22] F. E. A. dos Santos and A. Pelster, Phys. Rev. A 79,
U K 013614
HˆMF = nˆ (nˆ −1)−µn − (n −n )2
2 s s s 2 e o [23] B. Bradlyn, F. E. A. d. S. dosSantos, and A. Pelster,
sX=e,o(cid:20) (cid:21) Phys. Rev.A 79, 013615 (2009)
−zt Ψ ˆb†+Ψ∗ˆb −Ψ Ψ∗ +h.c. . (13) [24] H.T.C.Stoof,K.B.Gubbels,andD.B.M.Dickercheid,
o e e o o e Ultracold Quantum Fields (Springer, Bristol, UK,2009)
h(cid:16) (cid:17) i [25] A.AltlandandB.Simons,Condensed Matter FieldThe-
We may diagonalize HMF in the basis spanned by ory, 2nd ed. (CUP, Cambridge, UK,2010)
|ne> |no>by simultaneouslyimposingself-consistency [26] F. GERBIER,Phys. Rev.Lett. 99, 120405 (2007)
N
6
[27] D.B.M.Dickerscheid,D.vanOosten,P.J.H.Denteneer, [30] S. Gopalakrishnan, B. L. Lev, and P. M. Goldbart, Nat.
and H. T. C. Stoof, Phys. Rev.A 68, 043623 (2003) Phys. 5, 845 (2009)
[28] I. Herbut, A Modern Approach to Critical Phenomena [31] S.Ostermann,F.Piazza,andH.Ritsch,Phys.Rev.X6,
(Cambridge University Press, New York,2007) 021026
[29] S. Sachdev, Quantum Phase Transitions, 2nd ed. (Cam- [32] J. Leonard, A. Morales, P. Zupancic, T. Esslinger, and
bridge UniversityPress, New York,2011) T. Donner,arxiv: 1609.09053(2016)
