Table Of ContentQuantum phase transitions in the Fermi-Bose Hubbard model
L. D. Carr and M. J. Holland
JILA, National Institute of Standards and Technology and Department of Physics, University of Colorado, Boulder, CO 80309
(Dated: February 2, 2008)
Wepropose a multi-bandFermi-Bose Hubbardmodel with on-site fermion-boson conversion and
general fillingfactor in threedimensions. Sucha Hamiltonian models an atomic Fermigas trapped
5 in a lattice potential and subject to a Feshbach resonance. We solve this model in the two state
0 approximation for paired fermions at zero temperature. The problem then maps onto a coupled
0 Heisenberg spin model. In the limit of large positive and negative detuning, the quantum phase
2 transitionsintheBoseHubbardandPaired-FermiHubbardmodels arecorrectly reproduced. Near
resonance, the Mott states are given by a superposition of the paired-fermion and boson fields and
n
a theMott-superfluid borders go through an avoided crossing in the phasediagram.
J
PACSnumbers:
7
]
r The experimental investigation of cold atomic gases The effect of the conversion term is to lock the order
e
is proceeding rapidly. Both bosons and fermions have parameter of the fermions and bosons together. It thus
h
been brought to quantum degeneracy [1, 2]. They have leads to a reduction in the number of quantum phases
t
o been trapped in the sinusoidal lattice potential created fromfourtotwo. Themainreasonweintroduceabosonic
.
t by an optical standing wave of two counter-propagating field is to describe the the BCS-BEC crossover: the at-
a
m lasers [3, 4]. In the tightly bound regime of this po- tractive Fermi-Hubbard Hamiltonian, even in the paired
tential, the Bose-Hubbard Hamiltonian has proven to fermion limit, does not map simply onto the repulsive
-
d be a useful model to describe the transition from a su- Bose Hubbard Hamiltonian [15]. After proposing this
n perfluid, in which the atoms are delocalized, to a Mott new FBHH, we solve it in detail in the limit of on-site
o insulator, which has an integer number of atoms at paired fermions [15] in the two-state approximation at
c
each lattice site [5, 6, 7, 8]. Recently, the success- zerotemperatureforafillingoffromzerototwofermions
[
fulimplementationofFeshbachresonancesindegenerate per site. The on-site paired-fermionlimit correspondsto
1
Fermi gases has enabled the experimental study of the the experimentallyrealizablecaseofastronglyconfining
v
Bardeen-Cooper-Schrieffer (BCS) to Bose-Einstein con- potential and/or strong interactions. In this limit the
6
densate(BEC)crossoverinthe continuum,alongstand- FBHH maps isomorphically onto a coupled Heisenberg
5
1 ing theoretical problem [9, 10]. Initial evidence of a new spin model, or coupled magnets.
1 superfluid state has been found in the strongly interact- Consider the FBHH in the grand canonical ensemble,
0 ing regime [11]. It is a logical next step to study such
5 a crossover in an atomic Fermi gas trapped in a lattice H =Hf +Hb+Hfb, (1)
0
/ potential [4]. Hb ≡−JbX(b†ibj +bib†j)
t
a hi,ji
m In this Letter, we investigate the BCS-BEC crossover 1
- in the context of the Fermi-Bose Hubbard Hamiltonian + 2VbXnbi(nbi −1)−µbXnbi, (2)
d (FBHH), motivated by this vigorousexperimentalactiv- i i
on ity. Hubbard models have proven useful in experiments Hf ≡−Jf X (fi†smfjsm′ +fismfj†sm′)
on BEC’s [5, 8] and are expected to be equally relevant
c hi,ji,s,m,m′
for fermions [4, 12]. A phenomenological fermion-boson
: 1
iv conversionterminasimplifiedFBHHwasfirstsuggested − 2Vf X nfi↑mnfi↓m′ − X(µf −Em)nfism, (3)
X in the contextofhigh temperature superconductors[13], i,m,m′ i,s,m
ar wthheilceoFntBeHxtHo’sf wcoitldhoquutacnotnuvmergsiaosnesh[a1v4e].beUennlitkreeaitnedthiins Hfb ≡gX(b†ifi↑fi↓+bifi†↓fi†↑)+ V2fb X nbinfism. (4)
i i,s,m
earlierwork,themodelweshallstudyincludesthepossi-
bilityofbothclassicalandquantumphasetransitions[7] Equations (2)-(3) are the usual repulsive Bose-Hubbard
in the Fermi and Bose Hubbard limits, as well as a con- and multi-band attractive Fermi-Hubbard Hamiltonians
version term. In addition, we allow the fermions to oc- forauniformlatticeandEq.(4)isthefermion-bosoncou-
cupy multiple bands, so thatthe filling factoris notcon- pling. Thesymbolhi,jidenotesnearestneighbors,while
strained. In contrastto high-T [13], fermion-boson con- the indices s ∈ {↑,↓} and m denote the spin state and
c
version is a real physical process in cold quantum gases, band number. The hopping or tunnelling strengths J
f,b
whereaFeshbachresonanceisusedtocoherentlytransfer andtheon-siteinteractionstrengthsV aretakenasreal
f,b
fermionicatomsintoaboundtwo-atombosonicstate,as and positive definite. The band gap energy of the mth
illustrated in Fig. 1. band is E . The strength g of the interconversion term
m
2
assumption is that the pairing of fermions into bosons
closed channel occurs on-site. This is physically reasonable for present
experiments [18].
al ν We consider the limit in which Jf ≪ Vf, which cor-
nti responds to a strongly confining lattice [19]. Since the
e
ot open channel latticeheightisproportionaltotheintensityofthelasers
p
c creating the standing wave, this is straightforward to
mi
o obtain. Because the on-site interactions are attractive
rat (a) and s-wave, and the hopping is taken perturbatively,
e
nt the fermions form spin-up/spin-down pairs. We also re-
I
strict them to be in the lowest band. This is the typical
(b)
experiment case in three dimensions, where 105 to 106
fermions are distributed among 1003 sites. Thus m = 1
Internuclear distance andn∈[0,2],i.e.,therearefromzerototwofermions,or
zerotoonefermipair,persite. Thensecondorderdegen-
FIG. 1: Outer figure: pairs of fermionic atoms in the
erate perturbation theory maps H onto a new spin-1/2
f
open channel are coherently transferred into a closed chan-
system (a quantum XXZ model) [15]:
nel, bosonic state via a Feshbach resonance. Inset: second
order degenerate perturbation theory in the limit J ≪ V
leads to two hopping events on the lattice, (a) pair hfoppingf, Hf′ =−Jf′ X(τi+τj−−ninj)−µ′fXni, (7)
and(b)asinglefermionhoppingtoanadjacentsiteandback. hi,ji i
where J′ ≡8J2/V and µ′ ≡2µ +V /2. The operator
f f f f f f
τ+ ≡(τ−)† ≡f†f† isapaircreation/annihilationoper-
i i i↓ i↑
Tanhde Vcrfebatoifonthaenddenansintyihiclaotuipolninogpmeraaytorhsavfe†,efithaenrdsbig†,nb. amteonrtarnedsunltis≡in12(twnfio↑h+onppfi↓in−g-1t)y.pTeheevepnetrst,uarsbaistivskeettrcehaetd-
satisfy the usualcommutationrelationsfor fermions and in the inset of Fig. 1: the τ+τ− term corresponds to
i j
bosons, respectively. The number operators are defined pairhopping,whilethen n termcorrespondstoasingle
i j
as nbi = b†ibi, nfism = fi†smfism. In order to match the fermion hopping to an adjacent site and hopping back.
physical context of quantum degenerate gases in chemi- These operators obey the commutation relation
cal equilibrium, we require [τα,τβ] = 2iτγǫαβγδ , where α,β,γ ∈ {x,y,z}, τ± ≡
i j i ij i
µ =2µ +~ν, (5) τix±iτiy, and τiz ≡ni. Thus, despite the fact that τi± is
b f
a creation/annihilation operator for fermion pairs, the τ
where ν is the detuning associated with a Feshbach res- operatorsobeythePaulispincommutationrelations,not
onance and we set ~=1. The conserved quantity the bosonic commutation relations. This is one reason
why the attractive Fermi-Hubbard model does not map
n≡2Pinbi +Pi,s,mnfism (6) simply onto the repulsive Bose-Hubbard model, even in
thelimitofstronginteractions. Asecondreasonisthatin
is the total number of fermions. Eliminating µ by sub-
b ordertoachievesuchamapping,asumovermanybands
stitutingEq.(5)intoEqs.(1)-(4),onefindsthatµ mul-
f isrequired,sincetheinternalenergyofbosonscomposed
tiplies n. One canthustakeµ asthe chemicalpotential
f of two fermions is much greater than the band spacing.
of the coupled system, while ν determines the relative
Incontrast,theFBHHisasymptoticallyabletorepresent
number of bosons and fermions.
both the attractive Fermi-Hubbard and repulsive Bose-
The FBHH of Eqs. (1)-(4) models a pseudo-spin-1/2
Hubbard models in a simple way. It is therefore a good
systemoffermionswith s-waveinteractions,as inexper-
candidate for the study of the BCS-BEC crossover.
iments [2, 4, 11]. In practice, the index s ∈ {↑,↓} rep-
In general, a paired Fermi Hubbard Hamiltonian can
resents two hyperfine states in the level structure of an
act on all number states of the fermions. However, as
effectivelyfermionicalkaliatom,suchas40Kor6Li,scat-
in Eq. (7) we consider only n ∈ [0,2], the Hilbert space
tering near threshold in an open channel. The bosonic
on which it operates is restricted to two paired-number
field represents a bound closed-channel molecular state, states. Thus H′ is equivalent to the Heisenberg spin
6Li or 40K , which is coupled to the fermionic field via f
2 2 Hamiltonian, or a magnet,
aresonancewithanunboundopen-channelatomicstate,
called a Feshbach resonance. A schematic is shown in H =− J S~ ·S~ −~h· S~ , (8)
spin Pi,j ij i j Pi i
Fig. 1(a). Note that V and g are not functions of
f,b
ν. Methods for calculating the parameters V ,V , etc. where µ′ plays the role of the magnetic field h . One
f b f z
in Eqs. (2)-(4) from few-body atomic physics have been therefore expects paramagnetic and either ferro-or anti-
described in detail elsewhere [17]. Another important ferromagnetic phases. The former correspond to the su-
3
perfluid phase, while the latter are Mott and charge- (a) (b)
2
density wave (checkerboard) phases. Similarly, the re- 6 MI n=2 MI n=2
striction of the Hilbert space on which H operates to SF 1 SF
tHwaomnilutmonbiaerns(ttahteesqlueaandtsutmo aXnXisomtorodpeilc[7H]e)b.iseWnbeerfgorsmpuin- µ / Vff24 MI n=0 µ / Vff0 MI n=0
-1
late the two-state approximation for the coupled model 0
as superposition states of the form |ψi= |ψi , where 0 0.05 0.1 0.15 0.2 -20 0.05 0.1 0.15 0.2
Qj j J’ / V J’ / V
f f f f
(c) (d)
|ψij ≡|0ibj ⊗|0ifj cosθj +sin(θj)eiφj 2 2
MI n=2
×(|1ibj ⊗|0ifj cosχj +|0ibj ⊗|1ifj sinχjeiαj). (9) Vf1MI n=2 SF Vf-20 MI n=0 SF
The superscripts b and f refer to Fock states of bosons µ / f0MI n=0 µ / f-4
-1
and fermi pairs on the jth site. -6
The two state approximation is useful in determining -20 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2
J’ / V J’ / V
the Mott-superfluid borders in the phase diagram. The f f f f
Mottstateisasinglenumberstate,whilethelowestorder
FIG. 2: (color online) Shown is the phase diagram for de-
approximation of a superfluid is a superposition of two tunings (a) ν/V = −10, (b) ν/V = −1, (c) ν/V = 1/2,
f f f
number states. Therefore, Mott states occur in Eq. (9) (d) ν/V = 10. The blue solid curves show the Mott-
f
for θ ∈ {0,π/2,π}. The mixing angle χ is determined superfluid borders, while the red dashed curves show alter-
by the detuning ν in Eq. (5). To determine which phase nateextremawhicharemaxima(seeFig.3). SF≡superfluid,
MI≡Mott insulator, n≡ fermion filling factor.
is energetically favorable one evaluates E ≡ hψ|H|ψi.
gs
We make the uniform approximation θ = θ,φ = φ,
j j
χ = χ,α = α. Then φ does not appear in the ground
j j
state energy, while α can only change the sign of g. Set- Then χ∈{0,π/2} and one obtains the Mott borders
ting g′ =min[gexp(iα)], neither φ norα need be consid-
µ /V = −1/4+(Z/2)(1−2σ )J′/V , (13)
ered to obtain the phase diagram. An important point f f f f f
is that the ground state is either paramagnetic (super- µb/Vb = −2σbZJb/Vb, (14)
fluid) or ferromagnetic (Mott). It can be proven that it
where σ ≡ ±1 gives the vacuum/one-fermi-pair and
is notantiferromagnetic(charge-densitywave),either by f
σ ≡ ±1 the vacuum/one-boson Mott states. Equa-
setting the angles to differ by π/2 on each site, or by b
tions (13)-(14) correspond to the solutions one finds for
making a spin rotation in the Hamiltonian [20].
g =0inthetwo-stateapproximation. Forν →+∞,con-
The Mott-superfluid borders are obtained as follows.
dition(c)showsthatEq.(13)isaminimumandEq.(14)
The ground state energy is expanded around the Mott
is a maximum. For ν → −∞, the inverse is the case.
angles θ ∈ {0,π/2,π}. The zeroth order term gives the
Thus the Bose Hubbard and paired-Fermi Hubbard lim-
energy. The first order term is zero, showing that the
its are obtained naturally from the ansatz of Eq. (9) in
Mott state is always an extremum. The sign of the sec-
the limits of large negative and positive detuning. The
ond order term determines whether the Mott state is a
FBHH we have proposed therefore correctly obtains the
maximum or a minimum. Setting this equalto zero, one
endpoints of the BCS-BEC crossover on a lattice.
obtains the Mott-superfluid borders. One must also ex-
Nextconsiderthecaseofthephysicallyreasonablepa-
tremize in the mixing angle χ and determine whether or
rameter set V = V , J = J′, with the scaling chosen
not it is a maximum. Thus there are three conditions: b f b f
such that V = 1. The quartic equation has four roots.
f
Two are complex and therefore physically extraneous.
∂2E /∂θ2 = 0, (10)
gs The other two represent an energy minimum and an en-
∂Egs/∂χ = 0, (11) ergy maximum. There is no saddle point. The phase
∂2E /∂χ2 > 0. (12) diagram is shown in Fig. 2 for ν = −10,−1,1/2,10 and
gs
g = 1. The results are qualitatively the same for all
Using conditions (10)-(11) to eliminate χ and Eq. (5) to g 6= 0. The point ν = 1/2 is the actual crossover in our
eliminate µb, one finds a quartic equation in µf. The model, i.e., the point at which the Mott borders become
coefficients are functions of Jf′, Jb, Vf, Vb, |g′| = |g|, degenerate. To illustrate this, in Fig. 3(a) is shown the
and ν. The solution to this quartic equation, though mixing angle χ as a function of ν. Note the appropriate
lengthy,canbewritteninclosedanalyticform. Itisbest ν →±∞ limits. In Fig. 3(b) are shown the y-intercepts
understood when evaluated in limits of the parameters of the Mott phases from the phase diagrams of Fig. 2 as
and for particular values of them. afunction ofν. These gothroughanavoidedcrossingat
First consider the case ν → ±∞. We assume a bi- ν = 1/2. Smaller values of |g| cause the avoided cross-
partide lattice with Z the number of nearest neighbors. ing to become narrower. Similarly, the width of χ(ν) in
4
0.5
(a)
0.4
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We thank Matthew Fisher, Markus Greiner, Walter
cond-mat/0407381 (2004).
Hofstetter, and especially Daniel Sheehy for useful dis-
[19] In the opposite limit, J ≫V , the fermionic field is al-
f f
cussions. WeacknowledgethesupportoftheDepartment
ways superfluid.The coupling introduces intersite corre-
ofEnergy,OfficeofBasicEnergySciencesviatheChemi- lationsintothebosonicfield,andthusdestroystheboson
calSciences,GeosciencesandBiosciencesDivision. LDC Mott state.
is grateful to the KITP for hosting him and thanks the [20] A. Auerbach, Interacting Electrons and Quantum Mag-
NSF for partial support under grants PHY99-0794 and netism (Springer,Berlin, 1994).
MPS-DRF 0104447.
