Effects of finite temperature on the robustness of the Mott insulator phase in pseudo one-dimensional Bose-Hubbard Model

2.9 MB·English

by  Motoyoshi Inoue

Effects of finite temperature on the robustness of the Mott insulator phase in pseudo
one-dimensional Bose-Hubbard Model
Motoyoshi Inoue,1,∗ Keita Kobayashi,2 Yusuke Nakamura,3 and Yoshiya Yamanaka3
1Department of Nano-Science and Nano-Engineering, Waseda University, Tokyo 169-8555, Japan
2Research Institute for Science and Engineering, Waseda University, Tokyo 169-8555, Japan
3Department of Electronic and Photonic Systems, Waseda University, Tokyo 169-8555, Japan
(Dated: January 25, 2011)
1
We study the superfluid-Mott insulator (SF-MI) transition in an one-dimensional optical lattice
1
system, and employ the Bose-Hubbard model in two dimension with a combined potential of an
0
optical lattice in one direction and a confining harmonic trap in the other direction, which we
2
referto asthepseudoone-dimension Bose-Hubbardmodel. There someexcited states with respect
n to the harmonic trap are considered. The Mott lobes shrink in the µ and J directions of the
a
µ-J phase diagram. The shrinkage occurs because the interactions involving the excited states
J
become weaker than that between particles in the ground state. The dispersion of the in-site

2 particle increases because theenergy spacing between theeigenstates of theHamiltonian decreases
2 atfinitetemperature. TheexcitedstatessignificantlyaffecttherobustnessoftheMIphaseatfinite

temperate.
]
s
a PACSnumbers: 67.85.Hj,67.85.Bc,03.75.Lm
g
-
t
n I. INTRODUCTION itself is deep, this system can be described in a good ap-
a proximationby the one-dimensional(1D) Bose-Hubbard
u Since the realization of Bose-Einstein condensates in model, restricted to the first band. While the 1D Bose-
q
thecoldatomicsystems[1],manystudiesbothfromthe- Hubbard model without higher bands has been investi-
.
t oretical and experimental sides have been done. Vari- gatedinsomedetailbothatzeroandfinitetemperatures
a
m ous parameters, say the strength of the inter-atomic in- [11], its validity is not clear at finite temperature. It is
teraction and the configuration of the confinement po- reported that the contributions of the higher excitations
-
d tential, are experimentally well-controllable in the cold of the harmonic trap cannot be neglected for the one-
n atomicsystems[2],whichoffermanyintriguingphenom- dimensional system with Bose-Einstein condensate at fi-
o enasuchasthesuperfluid-Mottinsulator(SF-MI)transi- nite temperature [18].
c
tion. Moreover,itispossibletorealizealowdimensional Ouraimofthispaperistoinvestigatethetemperature
[
systemsimplybytighteningtheconfinementpotentialin dependenceoftheSF-MItransitioninaone-dimensional

1 oneortwodirections,anditprovidesagoodtestingfield optical lattice system. To this end, we concentrate on a
v to study a low dimensional physics. coldatomicsystemintwodimensionwithacombinedpo-
2
Experimentally, the SF-MI transition in this system, tentialofanopticallatticeinx-directionandaconfining
8
whichisthequantumphasetransition,hasbeenobserved harmonic trap in y-direction, and suppose that the opti-
2
4 when the depth of the lattice potential is increased [3], callatticeissodeepthatthe1DBose-HubbardHamilto-
. anditwasfollowedbytheobservationsinseveralsystems nian without higher bands describes the dynamics in x-
1
suchas the multi-componentone [4], the multi-band one direction,butthatsomeexcitedstatesiny-directionhave
0
1 [5], and the low-dimensional ones [6, 7]. Theoretically, to be takenaccountofbecause the harmonicpotentialis
1 the strongly-correlated bosonic system in optical lattice not tight enough to forbid transitions into the excited
: iswelldescribedbythe Bose-Hubbardmodel[8],accord- states absolutely. We refer to this model as the pseudo
v
ing to which the possibility of the SF-MI transition was one-dimensional(P1D)Bose-Hubbardmodelthroughout
i
X expected before its experimental observation [9]. This thispaper. ThereasonforanalyzingtheP1Dmodelhere
r model has been analyzed in various situations, i.e., at is that while it is simple and easy to analyze, and it can
a finite temperature [10, 11], for the multi-component sys- berealizedexperimentallywithameritthatthestrength
tems [12–14], for the multi-band ones [15, 16], and for of the harmonic trap is variable. It is also pointed out
the low-dimensional one [17]. that the structure of the transition terms in the P1D
The one-dimensional optical lattice system is real- Bose-HubbardHamiltonianissimilartothoseinthespin
ized experimentally in a combined potential of a one- transition [13, 14] or the one between the Bloch-bands
dimensional optical lattice and a cigar-shaped harmonic [15, 16], implying that our model serves as a prototype
trap. When the harmonic trap is tight in the directions to understand the physics of the low dimensional SF-MI
perpendiculartotheopticallatticeandtheopticallattice transition in more general situations.
We study the P1D Bose-Hubbard model both analyt-
ically and numerically in this paper, and show how the
transition terms affect the SF-MI phase diagram at zero
∗ [email protected] and finite temperatures. The main result at zero tem-
2
perature is the shrinkage of the Mott lobes, which origi- model throughoutthis paper for simplicity, an extension
nates from a decrease in the effective on-site interaction to three-dimensional systems, e.g. one with a combined
due to the transitions between the harmonic states in y- potentialofanopticallatticeinx-directionandaconfin-
direction. Onemayexpectthatsuchdeformationsofthe ing harmonic trap in yz-plane, is straightforward.
Mott lobes canbe fitted simply by adjusting aneffective The field operator ψˆ(x,y) is expanded as
on-site interaction in the 1D model, but it is not true
and they reflect the complex structure of interaction in ψˆ(x,y)= ψˆ (x)u (y), (2)
ℓ ℓ
theP1Dmodel. AlthoughthephasediagramfortheP1D Xℓ
modelreducestothatforthe1Doneinthetightconfine-
in the complete set of wave functions of harmonic oscil-
ment limit, we note that the shrinkage is non-negligible
lation,
even for comparativelytight confinement. The contribu-
tions of the transition in y-direction is more enhanced ~2 d2
at finite temperature. The Mott lobes disappear gradu- (cid:18)−2mdy2 +Vhar(y)(cid:19)uℓ(y)=Eℓuℓ(y), (3)
ally with an increase in temperature due to the thermal
fluctuation. While the disappearance in the 1D model where E = ~ω (ℓ+1/2) (ℓ = 0,1,2, ). Then we
ℓ y
occurs independently of the value of the chemicalpoten- expand the field operator ψˆ (x) in the c·o··mplete set of
ℓ
tial , it is not so in the P1D model. This is explained as
the Wannier functions w (x) with site indices i as
follows: the effective on-site interaction becomes weaker i
for particlesin the excitedstates, andthe fractionofthe ψˆ (x)= aˆ w (x). (4)
ℓ ℓ,i i
excited particles becomes larger as the chemical poten-
Xi
tial is increased, which implies that the hopping term
becomes relatively significant for the larger chemical po- We assume here and hereafter that the optical lattice
tential. Thiswaywehavethedependenceofthechemical potential is so deep that only the states in the lowest
potentialin the phase diagram,calculatedfromthe P1D band contribute, but take account of the excited states
model. We have confirmed through our analysis that it (ℓ=0) of harmonic oscillation in y-direction. Under the
6
is only at very low density and temperature that the 1D tight-binding approximationwe obtain the Hamiltonian,
Bose-Hubbardmodelwithouthigher bandsandthe P1D whichwerefertoastheP1DBose-HubbardHamiltonian,
one give almost the same results, and therefore that the
Hˆ =Hˆ +Hˆ , (5)
validity of the 1D model is restrictive. hop on-site
Thispaperisorganizedasfollows. InSec.II,weintro-
with
duce the P1D Bose-Hubbardmodel. The phase diagram
in the zero hopping limit is obtained analytically. In Hˆ = J aˆ† aˆ +aˆ† aˆ , (6)
Sec. III, we show the phase diagram for non-zero hop- hop − Xℓ,i h ℓ,i ℓ,i+1 ℓ,i ℓ,i−1i
ping both at zero and finite temperatures by employing
the mean-field approximation. Section IV is devoted to Hˆon-site = hˆi, (7)
summary and discussion. Xi
hˆ = (E µ)aˆ† aˆ
i ℓ− ℓ,i ℓ,i
Xℓ
II. PSEUDO ONE-DIMENSIONAL
1
BOSE-HUBBARD MODEL + U aˆ† aˆ† aˆ aˆ , (8)
2 ℓ1ℓ2ℓ3ℓ4 ℓ1,i ℓ2,i ℓ3,i ℓ4,i
ℓ1ℓX2ℓ3ℓ4
We start with the following Hamiltonian to describe
where
the cold Bose atomic gas system in two dimension with
a combined potential of an optical lattice in x-direction ~2 d2
J = dxw∗(x) +V (x) w (x),
and a confining harmonic trap in y-direction as follows: −Z i (cid:18)−2mdx2 opt (cid:19) i+1
(9)
~2 ∂2
Hˆ = dxdy ψˆ†(x,y) +V (x) ψˆ(x,y)
Z (cid:20) (cid:16)− 2m∂x2 opt (cid:19) gℓ1ℓ2ℓ3ℓ4 =gZdyu∗ℓ1(y)u∗ℓ2(y)uℓ3(y)uℓ4(y), (10)
~2 ∂2
+ψˆ†(x,y) +V (y) µ ψˆ(x,y)
(cid:18)− 2m∂y2 har − (cid:17) Uℓ1ℓ2ℓ3ℓ4 =gℓ1ℓ2ℓ3ℓ4Zdxwi∗(x)wi∗(x)wi(x)wi(x). (11)
g
+ ψˆ†(x,y)ψˆ†(x,y)ψˆ(x,y)ψˆ(x,y) , (1)
Note that the on-site interaction coefficient U de-
2 (cid:21) ℓ1ℓ2ℓ3ℓ4
pends on ℓ (the quantum number of the harmonic oscil-
where V (x), V (y) = mω2y2/2, µ, and g represent lation) while the hopping coefficient J is independent of
opt har y
the optical lattice potential, the harmonic potential, the ℓ. In Table I, the ratios of the on-site interaction coef-
chemical potential, and the coupling constant, respec- ficients to the ground state interaction are shown. All
tively. Note that no harmonic potential in x-direction is the on-site interaction coefficients involving the excited
considered. Althoughwestudyonlythistwo-dimensional states are smallerthan the interactioncoefficient U .
0000
3
at least, and will take this minimal sum throughout this
TABLE I. The ratios of the on-site interaction coefficients
paper.
U tothegroundstateinteractionU .Thecoefficients
ℓ1ℓ2ℓ3ℓ4 0000 Before full numerical calculations, let us turn to the
U have the symmetric property, for example, U =
Uℓ1ℓ2ℓ3=ℓ4U , etc. 0011 zero hopping limit, J = 0. Then the hopping Hamilto-
0101 1100 nian Hˆ in Eq. (5) is absent, and the total Hamilto-
hop
Uℓ1ℓ2ℓ3ℓ4/U0000 Uℓ1ℓ2ℓ3ℓ4/U0000 nian Hˆ = Hˆ is entirely separable with respect to
U 1/2 U √2/4 on-site
U0011 3/4 U0002 −3/8 the site index i. Note that hˆi in Hˆon-site commutates
1111 0022
with the in-site particle number operator at the i-site,
U √2/8 U √2/32
U10112122 7/16 U20222222 41/64 nˆi = ℓaˆ†ℓ,iaˆℓ,i, and hence the groundstate of the total
systemPis obtained simply by the diagonalization with a
fixed in-site particle number. Hereafter, the trivial sub-
script i is omitted.
The harmonic frequency ω representing the strength
y Explicitly we set up the following eigenequation for
of the harmonic potential comes into the dynamics of
each fixed in-site particle number n and seek the lowest
our model in two ways, namely through the energy-level
spacing ∆E = E E = ~ω and the on-site inter- energy eigenstate,
y ℓ+1 ℓ y
−
action U . The latter dependence is expressed as
ℓ1ℓ2ℓ3ℓ4 hˆ Ψ =E Ψ , (12)
Uℓ1ℓ2ℓ3ℓ4 ∼ √ωy universally for any ℓ, and is absorbed where the state Ψ |isienxpandne|dians
intorescalingthephasediagrambyusingtheparameters n
| i
µ/U and J/U . Note that it does not mean that
0000 0000
the P1D model reduces to the 1D model by such rescal- |Ψin = δn,n0+n1+n2 g(n0,n1,n2)|n0,n1,n2i,
ing. Wepaylittleattentiontothistrivialωy dependence, n0Xn1n2
but focus on the nontrivial effect of the transitions be- (13)
tween the harmonic states, which is absent in the 1D withthedirectproductoftheparticlenumberstates nℓ
| i
model. withℓ=0,1,2. Forillustrationwewritedownthematrix
We assume that the confinement harmonic potential eigenequations for n=1 and n=2:
is relatively tight but is not so tight as all the excited
states can be neglected. Then, the infinite sum over ℓ 0 0 0
canbeapproximatedbyafinitesumofsomelowerenergy  µ+0 ∆Ey 0 Φ1 =E1Φ1, (14)
−
states. Becauseofthe even-oddselectionrulefor the on- 0 0 2∆Ey
  
site interaction coefficient U , the transition of one
ℓ1ℓ2ℓ3ℓ4
particle from the ground state to the first excited one is with
forbidden. The transition with the lowest energy from
the ground state is either the simultaneous one of two g(1,0,0)
particlestothefirstexitedstateortheoneofoneparticle Φ1 =g(0,1,0) , (15)
to the second excited state. Hence, in order to estimate g(0,0,1)
 
effects of the excited states at a minimum, we have to
sum up the states not with ℓ = 0,1 but with ℓ = 0,1,2 and
U √2U U U 0 0
0000 0002 0022 0011
 √2U0002 2∆Ey+2U0022 √2U1122 √2U0112 0 0 
U √2U 4∆E +U U 0 0
 2µ+ 0002 1122 y 2222 1122 Φ =E Φ ,
−  U √2U U 2∆E +U 0 0  2 2 2
  0011 0112 1122 y 1111 
  0 0 0 0 ∆E +U 2U 
  y 0011 0112 
  0 0 0 0 2U 3∆E +2U 
  0112 y 1122
(16)
with Equation (14) implies that the lowest eigenvalue of E
1
is equal to E = µ. One can calculate the the low-
1,min
−
est eigenvalue of E , denoted by E , from Eq. (16).
2 2,min
g(2,0,0) Similarly, the lowest energy and its eigenstate for each
g(1,0,1) n are obtained. The results allow us to draw the phase
g(0,0,2) diagram as indicated in Fig. 1. There the boundary be-
Φ = . (17)
2 g(0,2,0) tween n=1 and n=2 regions, for example, is a line on
g(1,1,0) which E1,min =E2,min.
g(0,1,1)
 
4
TABLE II.The squared coefficient g(n0,n1,n2)2 for ∆Ey/U0000 =3.0.
| |
in-siteparticle number g(n,0,0)2 g(n 1,0,1)2 g(n 2,2,0)2 others
n=2 9|.84 10−|1 | 7.88− 10−3| | 7.47− 10−3| 8.37 10−4
n=3 9.35×10−1 4.23×10−2 2.11×10−2 1.49×10−3
n=4 8.57×10−1 1.07×10−1 3.48×10−2 1.15×10−3
× × × ×
Obviously,thedispersionofthein-siteparticlenumber
is equal to zero as a result of its conservation, and the
state of the whole system is in a Mott phase. But the
particle number in each ℓ-state can fluctuate. The dis-
persionsofthe particlenumberineachℓ-stateareshown
in Fig. 2, where they varies stepwise as functions of µ.
For comparison, we recall the corresponding 1D Bose-
Hubbard model,
Hˆ = J aˆ†aˆ +aˆ†aˆ + hˆ , (18)
1D − Xi h i i+1 i i−1i Xi i,1D FIG. 2. (Color online) Thedispersion of theparticle number
1 in the ground, first excited, and second excited states ℓ =
hˆi,1D =−µaˆ†iaˆi+ 2U0000aˆ†iaˆ†iaˆiaˆi. (19) 0,1,2 for the P1D Bose-Hubbard model with ∆Ey/U0000 =
3.0.
In the limit of J = 0, we consider the eigenequation
hˆ Ψ =E Ψ for eachfixedn, andits lowesteigen-
1D| in n| in contributions of the states with higher energies, is negli-
value is found to have a very simple form of
giblysmallwithinourparameterregion. Thisguarantees
the validity of our restriction to the energy levels up to
n(n 1)
En =−nµ+ 2− U0000, (20) ℓ = 2 when the energy-spacing ∆Ey/U0000 is relatively
large.
whichimpliesthatthephaseboundariesareatµ/U =
0000
1, 2, [19]. We can see from Fig. 1 that the existence
of th·e··excited states ℓ = 1,2 in the P1D model pushes III. MEAN-FIELD APPROXIMATION AND
NUMERICAL RESULTS
down the phase boundaries towards lower µ. It comes
from the fact that any on-site interaction involving the
excited states U is smaller than the interaction Inthissection,weshownumericalresultsoftheSF-MI
ℓ1ℓ2ℓ3ℓ4
between ground state particles U . transitionand the µ-J phase diagramfor the P1DBose-
0000
In Table II, we show the squared coefficients Hubbardmodelwith the hopping Hamiltonian(the non-
g(n ,n ,n )2forthein-citeparticlenumbersn=2,3,4. vanishing J) at zero and finite temperatures. To deal
0 1 2
| |
Theprobabilityofthegroundstate n,0,0 isthegreatest with the hopping Hamiltonian, we resort to the mean-
| i
foreachn,andthenextleadingprobabilities, n 1,0,1 field approximation[11].
| − i
and n 2,2,0 ,arenonnegligible. Ontheotherhand,the Let us introduce the order parameter
| − i
value in the column “others”, corresponding to a sum of
Tr[aˆ e−βHˆ]
ℓ
Φ aˆ = , (21)
ℓ ≡h ℓi Tr[e−βHˆ]
where β is the inverse temperature β = 1/k T with
B
the Boltzmann constant k . Neglecting the fluctuation
B
terms of the second order, we have an approximate ex-
pression of
¹=U 0000 aˆ† aˆ ⋍Φ∗ aˆ +Φ aˆ† Φ∗ Φ . (22)
ℓ,i ℓ,j ℓ,i ℓ,j ℓ,j ℓ,i− ℓ,i ℓ,j
Then, the approximate hopping Hamiltonian becomes
Hˆ(M) = J (Φ∗ +Φ∗ )aˆ
FIG. 1. Thephase diagram in thezero hoppinglimit, J =0, hop − Xℓ,i h ℓ,i+1 ℓ,i−1 ℓ,i
for the P1D Bose-Hubbard model. The values of µ/U on
the phase boundary line approach in the tight confin00e0m0ent +(Φℓ,i+1+Φℓ,i−1)aˆ†ℓ,i
limit,∆Ey/U0000 →∞,totheonesforthe1DBose-Hubbard Φ∗ Φ Φ∗ Φ , (23)
model, which are µ/U0000 =1, 2, ···. − ℓ,i ℓ,i+1− ℓ,i ℓ,i−1i
5
FIG.4. (Coloronline)(a)Thedispersionofthein-siteparticle
FIG. 3. (Color online) (a) The dispersion of the in-site par-
number and (b) the absolute square of the order parameter
ticle number for the 1D model. (b) The dispersion of the
with βU = 20.0 for the 1D model. Figures (c) and (d)
in-siteparticle number. (c)That ofthegroundstateparticle 0000
represent those with βU =8.0, respectively.
number. (d) The absolute square of the ground state order 0000
parameter Φ0,i 2 for the P1D model with ∆Ey/U0000 = 3.0
| |
atzerotemperature. Theblackcoloredregionswherethedis-
numberinthe Mott lobes,asinFigs. 3-(c)and(d). The
persions are zero in (a) and (b) denote the Mott lobes, and
the n-th lobe from the bottom represents the MI-phase with nonzero dispersion of the ground state particle number
n in-site particles. The regions where the order parameter shows that the particle number of each state is not fixed
vanishes, as depicted in (d), correspond to the Mott lobes in and can fluctuate, but only the in-site particle number
(b). is fixed. This is attributed to the decrease of the ef-
fective on-site interactionbecause the on-site interaction
coefficients aresmaller than the groundstate interaction
andthe totalHamiltonianunder the mean-fieldapproxi- coefficient U . The numerical results show that the
0000
mationisgivenbyH(M) =Hˆ(M)+Hˆ . Wecalculate analysis in Sec. II can be extended even to the case of
hop on-site
the lowestenergystatebydiagonalizingthetotalHamil- J =0.
6
tonian,whentheorderparameterΦ isdeterminedself- We see from Figs. 3-(a) and (b) that the Mott lobes
ℓ,i
consistently by minimizing the energy at zero tempera- for the P1D model also shrink in the J direction. The
ture or the free energy reason for the shrinkage in the J direction is as follows:
thesystemisfavorabletotheSF-phasewhentheratioof
F = β−1lnZ(β), (24) the hoppingtothe on-siteinteractionislarge. Theeffec-
−
Z(β)=Tr[e−βHˆ(M)], (25) tive interaction is smaller than that of the ground state
particles, which means that the particles in the excited
atfinitetemperature. Inthiscalculation,weconsiderthe statescaneasilybe inthe SF-phasethanthe particlesin
excited states up to the second excited state as stated in the ground state because then the effect of the hopping
Sec. II. is relatively large compared to the on-site interaction.
As a result, in comparison with the system for the 1D
model,thesystemsfortheP1Dmodelisfavorabletothe
A. Results at Zero Temperature SF-phase.
We emphasize that the P1D model is essential in the
above considerations since the particles in the excited
InFig.3,weshowthephasediagramsforthe1Dmodel
states play a crucial role.
and for the P1D model with the energy-level spacing
∆E /U = 3.0 at the zero temperature. Comparing
y 0000
the resultfor the P1Dmodel withthatfor the 1Dmodel
B. Results at Finite Temperature
in Fig. 3-(a), one can find the following two features.
First, the Mott lobes for the P1D model shrink in the µ
direction in Fig. 3-(b). The rate of the shrinkage of the In the previous subsection, we found that the effects
MottlobesfortheP1Dmodelincreaseasthein-sitepar- of the excited states become more conspicuous for larger
ticle number increases. Secondly, the dispersion of the in-site particle numbers. It is therefore anticipated that
ground state particle number in the Mott lobes for the the excited state is more important at finite tempera-
P1D model is nonzero and also increases as the in-site ture. In this subsection, we analyze the effect of finite
particle number increases, as in Fig. 3-(c). The order temperature for the P1D model.
parameter does not necessarily become nonzero even in InFig.4,weshowtheparticlenumberdispersionsand
thenonzerodispersionregionofthegroundstateparticle the order parameters for the 1D model at temperatures
6
FIG. 6. (Color online) The dispersions of the in-site particle
numbersin the zero hopping limits J =0 for (a) the1D and
(b) theP1D models with ∆Ey/U0000 =3.0.
FIG.5. (Coloronline)(a)Thedispersionofthein-siteparticle
number and (b) the absolute square of the order parameter
with βU = 20.0 for the P1D model. Figures (c) and (d)
0000
represent thosewith βU =8.0, respectively.
0000
βU = 20.0 and βU = 8.0. It is seen from Fig. 4
0000 0000
thatthe Mottlobesdonotfitthe regionofthe vanishing
orderparameter,the latteris widerthanthe former. We
have some regions where the dispersion is non-vanishing
but the order parameter vanishes, Φ =0, and interpret
0
that they are in the normal-liquid phase[11].
The phase diagram for the P1D model is shown in
Fig. 5. There the order parameter is that for the ground
state. Asinthe1Dmodel,thevanishingorderparameter
region covers the vanishing dispersion one. The normal-
liquid region expands as temperature goes up. In a sim-
FIG. 7. (Color online) The dispersions of the in-site particle
ilar fashion to the zero temperature case, the shrinkage
numbers (a) and the absolute square of the order parameter
of the Mott lobes for the P1D model also is found at fi-
(b) with J/U = 0.02 for the 1D model. Figures (c) and
0000
nite temperature. The shape of the lobes for the order (d) represent those for theP1D model respectively.
parameter in Fig. 5-(b) and (d) is lost at finite temper-
ature more clearly than for the 1D model in Fig. 4-(b)
and (d).
particle number nˆ . We also illustrate the temperature
To investigate the temperature dependence of the dis- dependence ofthhe iincreasingratesoftheexcitedparticle
persionin more detail, we show the dispersions for small numbersforℓ=1, 2inFig.8-(b). OnecanfindinFig.8
J, namely for J = 0 in Fig. 6 and for J/U0000 = 0.02 thattheincreaseintheexcitedparticlenumberssynchro-
in Fig. 7. Comparing the results for the P1D model nizeswiththatinthedispersions. Thissuggeststhatthe
in Figs. 6-(b) and 7-(c) with those for the 1D model in excitedparticlescontributetothelargedispersionsinthe
Figs. 6-(a) and 7-(a), we see that the dispersion for the P1D model.
P1D model is larger than that for the 1D model. In ad-
dition, the dispersion for the 1D model is independent
of the chemical potential µ, but that for the P1D model
depends on µ in such a manner that it is largerat larger TABLEIII.Inthistable,Em(1D) andEm(P1D) representthem-
µ. The order parameter for the P1D model begins to be theigenvaluesoftheHamiltonianforthe1DandP1Dsystems
with nˆ =3 and βU =10. These values are respectively
non-vanishinginsmallerµregionthanforthe 1Dmodel, h i 0000
theeigenvaluesofthegrandcanonicalHamiltonianHˆ nµ,so
asinFig.6-(b)and(d). Athighertemperature,theorder −
parameterbecomessmallerandtheregioninthenormal- thevaluescanbenegative. Thesymbol∆Em−1,m represents
liquidphaseexpandsequallyforthe 1DandP1Dmodels thespacing between energy levels, ∆Em−1,m =Em−Em−1.
in Fig. 7. m 0 1 2 3
Figure 8-(a) shows the dispersions of the average in- Em(1D)/U0000 4.50 4.00 4.00 2.50
− − − −
site particle number with nˆ = 2,3,4 and J = 0. In ∆E(1D) /U 0.500 0.00 1.50
h i m−1,m 0000
bothofthe1DandP1Dmodels,thedispersionsincrease Em(P1D)/U0000 3.54 3.19 3.19 2.15
as the temperature rises, but the increasing rate of the ∆E(P1D) /U − −0.359 −0.00 −1.04
m−1,m 0000
dispersion for the P1D model depends on the average
7
the 1D model is valid for very low density temperature,
the effects of the excited states can not be neglected for
describing more general situations.
IV. SUMMARY AND DISCUSSION
In this paper, we analyzed SF-MI transition at zero
andfinitetemperaturefortheP1DBose-Hubbardmodel.
The P1D model, which differs from the 1D model, con-
tains the transition terms, which are similar to the
FIG. 8. (Color online) (a) The dispersions of the average
spin transition and the transition between the Bloch
in-site particle number with hnˆi = 2,3,4 and ∆Ey/U0000 = bands. Specifically, the form of the multiband Bose-
3.0 for the P1D and 1D models. The dispersion for the 1D
Hubbard Hamiltonian corresponds to that of the P1D
model is independent of the average in-site particle number
Bose-HubbardHamiltonian, aside from the values of the
nˆ . (b) The ratios of the excited particle numbers at finite
themiperature ni to those at zero temperature ni,T=0 in the on-site interaction coefficients Uℓ1ℓ2ℓ3ℓ4 and the hopping
average in-site particle number nˆ = 3 for the P1D model coefficients. We expect that our analysis is also useful in
h i
with ∆Ey/U0000 =3.0. analyzing the multiband model.
Inzerotemperatureanalysis,theSF-MIphasediagram
fortheP1Dmodelisdifferentfromthatforthe1Dmodel
To clarify the rapid increase of the dispersions for the evenfor the relativelytight confinement. The major dif-
P1D model, we investigate the eigenvalues of the Hamil- ference is the shrinkage of the Mott lobes in the µ direc-
tonian, tion. The effect is attributed to decreasing the effective
on-site interaction by the transition between the ground
Hˆ(M) Ψm =Em Ψm , (26) and excited states. The transition produces the fluctu-
| i | i
ation of the particle number in each ℓ-state in the Mott
for the 1D and P1D models. In Table.III, the eigenval-
lobes. The decrease of the on-site interaction enhances
ues ofthe Hamiltonianand the energy-levelspacingsare
thehoppingrelatively,whichleadstotheshrinkageofthe
shown. One finds that the energy-level spacing for the
Mott lobes in the J direction as that in the µ direction.
P1D model ∆Em(P−1D1,)m is narrower than that for the 1D Wenotethatthe1Dmodelcannottakeaccountofthese
model ∆E(1D) . The narrow energy-level spacing en- effects by adjusting the parameters.
m−1,m
hancesthecontributionsofthe higherenergyeigenstates At finite temperature, the dispersion of the in-site
m = 1,2, at finite temperature, so that the disper- particle number increases as the temperature becomes
···
sions become large then. In addition, it turns out for higher. Inaddition,theregionofthenormal-liquidphase
nˆ =3 thatthe eigenstatesbelongingto the eigenvalues surroundingtheMottlobesexpands. Theseeffectsoccur
h i
E and E are those with the in-site particle numbers both for the 1D and P1D models due to the thermal
1 2
n = 4 and n = 2, respectively, while the eigenstate to fluctuation. The result for the 1D model corresponds to
belonging to E is that with n = 3. The number of Ref. [11]. We note that the increase of the dispersion
0
the excited particles grows as the in-site particle num- for the P1D model is enhanced more than that for the
ber does according to Table II. Thus the increase of the 1D model. The tendency is more noticeable at higher
excited particles in Fig. 8-(b) reflects the contributionof temperature due to the increase of thermal fluctuations.
the eigenstate with E , equivalently of the state of the These results are attributed to the weaker on-site inter-
1
in-site particle number n=4. action coefficients involving the excited states.
The narrow energy-level spacing arises from the fact It have been seen that the robustness of the MI phase
that the on-site interaction coefficients involving the ex- is affected at finite temperature by the presence of the
cited particles are small. This is explained in the follow- excited particles. From the view of the local density
ing way. As the fraction of the excited particle number approximation in Fig. 5, the P1D Bose-Hubbard model
becomes large at the large average in-site particle num- predicts that the Mott shell collapses more quickly than
ber nˆ , the effective repulsive interactionbecomes small the 1D model does. The validity of the 1D model is re-
h i
and so are the eigenvalues E (m=1,2, ). strictedtoverylowdensity andtemperature forrealistic
m
···
Theabovediscussionsindicatethatthepresenceofthe cold atomic gases. At high density and/or temperature,
excited states affects the robustness of the Mott phase onehastotoconsidertheexcitedstatescomingfromthe
definitelyandessentially. Althoughtheapproximationof confinement potential.
[1] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. [3] M.Greiner,O.Mandel,T.Esslinger,T.W.Hansch,and
Wieman, and E. A.Cornell, Science 269, 198 (1995). I. Bloch, Nature(London) 415, 39 (2002).
[2] I.Bloch, NaturePhys. 1, 23 (2005). [4] J.Catani,L.DeSarlo,G.Barontini,F.Minardi,andM.
8
Inguscio, Phys. Rev.A 77, 011603 (2008). [13] S. Tsuchiya, S. Kurihara, and Takashi Kimura, Phys.
[5] T. Mu¨ller, S. F¨olling, A. Widera, and I. Bloch, Phys. Rev. A 70, 043628 (2004).
Rev.Lett. 99, 200405 (2007). [14] M.YamashitaandM.W.Jack,Phys.Rev.A76,023606
[6] I. B. Spielman, W. D. Phillips, and J. V. Porto, Phys. (2007).
Rev.Lett. 98, 080404 (2007). [15] J. Larson, A. Collin, and J.-P. Martikainen, Phys. Rev.
[7] T. St¨oferle, H. Moritz, C. Schori, M. K¨ohl, and T. A 79 033603 (2009).
Esslinger, Phys. Rev.Lett. 92, 130403 (2004). [16] A. Collin, J. Larson, and J.-P. Martikainen, Phys. Rev.
[8] M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. A 81, 023605 (2010).
S.Fisher, Phys.Rev.B 40, 546 (1989). [17] N. Elstner and H. Monien, Phys. Rev. B 59, 12184
[9] D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and (1999).
P.Zoller, Phys.Rev.Lett. 81, 3108 (1998). [18] E. Arahata and T. Nikuni, Phys. Rev. A 77, 033610
[10] K. Sheshadri, H. R. Krishnamurthy, R. Pandit, and T. (2008).
V.Ramakrishnan, Europhys.Lett. 22, 257 (1993). [19] C.J.PethickandH.Smith,Bose-EinsteinCondensation
[11] X.Lu and Y.Yu,Phys. Rev.A 74, 063615 (2006). in Dilute Gases Second Edition, (Cambridge University
[12] P.Buonsante,S.M.Giampaolo,F.Illuminati,V.Penna, Press, 2008).
and A. Vezzani,Phys. Rev.Lett. 100, 240402 (2008).