Orbital order in Mott insulators of spinless p-band fermions

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by  Erhai Zhao

Orbital order in Mott insulators of spinless p-band fermions
Erhai Zhao and W. Vincent Liu
Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA
A gas of strongly interacting spinless p-orbitalfermionic atoms in 2D optical lattices is proposed
and studied. Several interesting new features are found. In the Mott limit on a square lattice,
the gas is found to be described effectively by an orbital exchange Hamiltonian equivalent to a
pseudospin-1/2 XXZ model. For a triangular, honeycomb, or Kagome lattice, the orbital exchange
isgeometrically frustrated and described bya newquantum120◦ model. Wedeterminetheorbital
ordering on the Kagome lattice, and show how orbital wave fluctuations select ground states via
8 theorder by disorder mechanism for thehoneycomb lattice. We discuss experimental signatures of
0 various orbital ordering.
0
2
Theelectronorbitaldegreeoffreedomplaysanimpor- tential V(r r ). Atoms interact predominately in the
n 1 2
a tant role in correlated quantum materials such as tran- p-wave chann−el, i.e., in momentum space, V(k′ k)
− ≃
J sition metal oxides [1]. Many intriguing phases observed 3V (k)(kˆ kˆ). The actual form of V (k) is unimpor-
1 ′ · 1
1 in experiments are attributed to the coupling of elec- tant for our discussion [16], as long as it reproduces
1 tron d-orbitals, typically in the t2g or eg manifold, to the low energy p-wave scattering amplitude f1(k) =
] the electron charge, spin, and/or the lattice degree of k2/(−v1−1+c1k2 −ik3), where v1 is the scattering vol-
l freedom [2, 3]. Understanding the intricate interplay be- ume and c is the effective range parameter [17]. These
e 1
- tweenthemremainsatheoreticalchallenge. Itisthusde- parametersareknownfor40K[18]. Forexample,onecan
tr sirabletostudysimplersystemswheretheorbitaldegree useaseparablemodelpotential[16]orapseudopotential
s of freedom is disentangled from others, namely “plain [19] with coupling constant g = 12v π~2/m (m is the
. p 1
at vanilla” orbital ordering in a Mott insulator. fermion mass). The strength of V1 can be tuned using
m Rapidadvancesinloadingandcontrollingalkaliatoms a p-wave Feshbach resonance [20, 21, 22, 23]. We only
- on the excited bands of optical lattices [4, 5, 6, 7] have consider repulsive interaction here.
d
made itpossibleto investigateorbitalorderingofbosons We focus on a setup that captures the physics of the
n
o and fermions in new settings [8, 9, 10, 11, 12, 13, 14]. orbital symmetry and quantum degeneracy of fermionic
c Here we show that strongly interacting single species atoms. We first consider a strongly anisotropic 3D cubic
2 [ (izsepiMnloetsts)stpa-toersbcithaalrfaecrtmeriioznesdobnyooprbtiictaall-loantltyicmesocdaenlsr[1ea5l]-. wopitthicaVlxla=ttiVcyepaontdenVtzial,VVopx(,yr,)w=hPereµ=kxL,y,izsVtµhesinw2a(vkeLvreµc)-
v We consider the case where the degenerate p-orbitals of tor of the laser field a≫nd we set the lattice spacing to
9 theopticalpotentialwellarepartiallyoccupiedwhilethe be unit, a = π/k 1. In the deep lattice limit,
L
8 fully occupieds-orbitalactsas“closedshell”andremain V E ~2k2/2m≡(the recoil energy) for all three
05 inert at low energy scales. Thus, these atomic Mott in- diµre≫ctionRs, ≡the boLttom of the optical potential at each
. sulators differ significantly from their solid state coun- latticesitecanbeapproximatedbyaharmonicoscillator,
1
terparts of d electron systems in orbital symmetry. We and the lowest few energy levels are s,p ,p ,p orbital
0 x y z
8 derive the low energy effective orbital exchange Hamil- states. The px and py orbitals are degenerate, while
0 tonian of p-band fermions in the strong coupling regime s (p )isw|ellibelow|(abiove)them,separatedinenergy
z
: and commensurate filling for simple 2D lattices and de- b|yith|eiharmornic oscillator frequencies ~ω = 4V E .
v µ p µ R
i termine their long range orbital ordering patterns. Due to the strong lattice potential in z-dimension, the
X
Mott states of p-band fermions, described by orbital- system is dynamically separated into a stack of approxi-
r onlymodels,representnewexamplesofcorrelatedstates mately independent layers with suppressed tunneling in
a
in condensed matter. Although we refer to the emergent between, each being two-dimensional (2D). Then, in or-
lowenergysymmetryofthedegeneratep-orbitalstatesas der to explore the orbital-related quantum dynamics in
pseudo-spinsymmetry,itisnotaninternalsymmetrylike such a lattice, one can fill fermions up to the p-band by
the true spin degree of freedom, but rather intrinsically havinganaveragenumberoffermionspersitebetween1
spatial. As we shall demonstrate, in general p orbital and2. Inthe atomiclimit, for twoparticles per site, one
x,y
states transform as the components of a spinor under ofthemoccupiesthes-orbitalandtheotheroneoccupies
space rotation, in a manner reminiscent of the Lorentz either px or py. We shall refer to this as half filling.
spinor. Thissetsthe orbitalexchangemodelsapartfrom
We expand the fermion field operator in the Wannier
the familiar spin exchange models.
basis, ψ(r)= w (r R )c , where R is the lattice
Pi,ℓ ℓ − i iℓ i
Spinless latticep-bandfermions. Weconsideragasof vector at site i in two dimensions and ℓ=s,p ,p ,p ,...
x y z
fermion atoms in a single hyperfine spin state loaded in is the band index. Then the interacting system is de-
an optical lattice interacting via a generic two-body po- scribed by a multi-band Hubbard model. Close to half
2
t
filling,withthefilledsbandandtheemptyp bandswell J
z z Neel
separated from the degenerate px and py bands, the low p
energy effective model for the fermions is the following y
2D p-band Hubbard Hamiltonian, J t
xy
Hp = X tµν[c†i,µci+ν,µ+h.c.]+UXnixniy. (1) t p
i;µ,ν=x,y i x
Here,t =t δ +t (1 δ )isthehoppingamplitude FIG.1: (Coloronline)Left: Virtualhoppingprocessesgiving
µν µν µ,ν
oforbitalµinktheν d⊥irec−tion[theorbitalswillbedenoted rise to the antiferro-orbital Ising exchange Jz and the ferro-
as(p ,p ) (x,y) ( , )interchangeablytolightenthe orbitalXYexchangeJxy. Thetransversehoppingt⊥ ismuch
x y
≡ ≡ ↑ ↓ smaller than the longitudinal hopping t and has opposite
notation],andU istheonsiterepulsionbetweenatomsin k
sign. ThereforeJxy ≪Jz. Dashlinesindicateemptyorbitals.
p and p orbitalstates. In the harmonicapproximation
x y Right: Neel orbital ordering on the squarelattice.
of the Wannier function, the transverse hopping t =
e (η/2)2V /2isnegativeandsmall,whilethelongit⊥udi-
− x
−
nalhoppingt =(1 η2/2)t ispositiveandmuchlarger model for pseudospin T = 1,
k − ⊥ 2
in magnitude. Here, α = (V /E )1/4k is the inverse
µ µ R L
oftheharmonicoscillatorlengthintheµ-dimension,and Horb ≃Jz X TizTjz. (3)
η αxaistypicallyalargenumber. Theonsiteenergyis <i,j>
≡
given by U =g α2α (22α2+α2)/32(2π)3/2 in the pseu-
p x z x z We neglect three body and ring exchange terms which
dopotentialapproach. Eq.(1)looksliketheordinaryone-
are of higher order in t /U. The Ising exchange favors
band Hubbard model for spinless fermions, albeit with a
antiparallelconfigurationk ofnearbypseudospins,i.e. per-
twist: thep (pseudo-spin )atomspreferhoppinginthe
x
↑ pendicular configuration of nearby p orbitals (one in p
x direction while the p ( ) atoms prefer hopping in the x
y
↓ and the other p ). Thus, the Mott state is antiferro-
y direction. Such anisotropy has dramatic consequences y
orbitally ordered on square lattice. The translational
in the strong coupling limit.
symmetry is broken with an alternative arrangement of
EffectiveorbitalexchangeHamiltonian. Athalffilling
p and p orbitals as shown in Fig. 1.
andinthestrongcouplinglimit, U t ,orbitalfluctua- x y
tion is the only remaining low energ≫y dkegree of freedom. Frustrated 120◦ model on oblique lattices. We now
generalize the analysis to oblique 2D lattices. For any
It is well known that virtual hopping processes lead to
vector e directing at angle θ with the x axis, such as
direct and higher order orbitalexchange interactions [2]. θ
the bond direction indicated in Fig. 2, it is convenient
In our case, as shown in Fig. 1, the nearest neighbor
to introduce a local coordinate system with axis x˜ and
orbital exchange is strongly anisotropic as a direct con-
y˜ rotated from the global coordinate system by θ. The
sequence of the anisotropic shape of the p-orbitals. The
p-orbitalwavefunctions transformunder rotationas the
effectiveHamiltonianforH canbederivedfollowingthe
p components of a planar vector, i.e.,
standard canonical transformation method. Up to order
Rofetf2k.,2it4.isWgieveinntbroydHuc(e2)p=se−udUo−-s1pTin−(21)o,1peinratthoernTo+ta=tiocn†xcoyf ccxy →→cc˜˜yx == c−xccxossinθθ++cycsyincoθs,θ. (4)
andTz =(c c c c )/2,orequivalentlyT= 1c σ c
in the Carte†xsiaxn−ve†yctyor form with σ being the2st†µandµaνrdν Accordingly, the pseudospin operators transform as
Pauli matrices, and rewrite H(2) as, T T˜ =T sin2θ+T cos2θ,
z z x z
→
T T˜ =T cos2θ T sin2θ, (5)
Horb = X [Jxy(Ti+Tj−+h.c.)+JzTizTjz]. (2) Tyx → T˜yx =Tyx. − z
<i,j> →
Inother words,the pseudospinvectorT is rotatedby 2θ
ReplacingTwiththe usualspinSinamagneticsystem, aboutthey axisinthepseudospinspace. Theleadingor-
Eq.(2) corresponds to the familiar XXZ model. The der exchange interactionbetween site i and j, connected
antiferro-orbitalIsingexchangeJz =2(t2 +t2)/U results by bond eθ, is then given by JzT˜izT˜jz. Note the value of
⊥ k
from the longitudinal (and transverse)virtual exchange longitudinalhoppingalonge ,andconsequentlyJ ,fora
θ z
hopping. By contrast, the XY exchange J =2t t /U specificlatticedependsondetailsoftheopticalpotential.
xy
⊥ k
involves both longitudinal and transverse hopping, it is
ferro-orbital (J < 0) due to the opposite sign of t We are particularly interested in the triangular and
xy
k
and t . Because the transverse hopping amplitude of honeycomb lattice, both of which can be implemented
⊥
orbitals is small, we have J J , so the leading using interfering laser beams. For these lattices (lattice
xy z
| | ≪ | |
orderinteractiontakesthe formofantiferro-orbitalIsing spacing a=1), we introduce three basis vectors: eˆ xˆ,
1
≡
3
e
~2 T3 C C pseudospins at vertex A,B,C with angles φA,B,C with
x respect to the T axis (AB bond). The classical energy,
z
y~ θ T1 A B B E/T2 =f(0;A,B)+f(2π/3;A,C)+f(4π/3;B,C)where
e A f(γ;A,B) cos(γ φ )cos(γ φ ), is minimizedwhen
T y 1 ≡ − A − B
e3 T2 x x Tz C TφAhe−twφoB e=nerφgCy −miφnAima=m2aπn/i3fe,sφtAth=e C52π/s6ymomr e1t1rπy/o6f.
H , which is invariant under a π-rotation of T about
120
FIG. 2: (Color online) Left: The spatial coordinate system the y axis, i.e. T T with θ = π/2 in (5). The
x,z x,z
(inblue). Basisvectorsej areat120◦ fromeachother. Three corresponding orbita→l c−onfiguration is shown in Fig. 2,
pseudospins Tj (in black) defined on bond ej. The quantiza- where the orbital at A forms angle φ /2 = 75 with
A ◦
tionaxisTzisx(px). Right: Orbitalorderingonanindividual the AB bond. Permutation of (A,B,C) yields configu-
triangle ABC and long range order on the kagome lattice.
rationsof the same energy. This resultimmediately tells
us the classical orbital ordering pattern (see Fig. 2) on
eˆ , and eˆ , which are 120 from each other as shown in the Kagome lattice, which consists of corner sharing tri-
2 3 ◦
Fig. 2 and use T to denote the operator T˜ along bond angles.
j z
e (j =1,2,3). Explicitly, T =T , T = 1T √3T , Orbital order by disorder. For the bipartite hon-
j 1 z 2 −2 z − 2 x eycomb lattice, we introduce transformation T
and T3 = −21Tz + √23Tx, all lying within the xz plane. T¯x,z = Tx,z, with + ( ) for sites withixn,z t→he
Then the orbital exchange Hamiltonian can be put into ± −
A (B) sublattice. Then the orbital exchange be-
a simple form, come ferro-orbital in terms of the new T¯ operators,
H = J T¯ (R)T¯ (R+eˆ ). Itcanberewritten
H120 =JzXTj(R)Tj(R+eˆj), (6) (a1p2a0rt f−romzPaRco∈nAs,tjanjt factojr) as j
R,j
J
where the sum over R includes all lattice sites (or the H¯120 = z X [T¯j(R) T¯j(R+eˆj)]2,j =1,2,3. (7)
2 −
sites within the A sublattice) for the triangular (or hon-
R A,j
eycomb) lattice. We call Eq. (6) the quantum 120 ∈
◦
model, as a formally similar Hamiltonian was proposed Therefore, the classical energy is minimized for any ho-
to describe the two-fold degenerate e electrons in cu- mogeneous configuration (T¯z,T¯x)(R)=(T cosφ,T sinφ),
g
bic perovskites [15, 25, 26, 27]. The underlying physics independent of the polar angleφ. This implies Neel (an-
is however very different. There, e label the cartesian tiferro)orbitalordering. NotethatthiscontinuousSO(2)
j
basis vectors of the cubic lattice, while the 120 configu- degenerate manifold of classical ground states is not an
◦
rationofthree Tj’s arisesfroma fundamentally different inherent property of H120 which is only invariant under
transformationpropertyoftwo d-orbitals, 3z2 r2 and finitepointgrouprotations. Orbitalwaveexcitationsare
| − i
x2 y2 , under the permutation of x,y,z. important quantum fluctuations beyond mean field the-
| − i
The antiferro-orbital exchange favors perpendicular ory. To the leading order in 1/T, their correction to the
alignmentsofnearestneighbororbitalsalongbonds. Ap- groundstateenergycanbecomputedfollowingthespirit
parently, it is impossible to achieve this for all three ofHolstein-Primakovspinwavetheory[30]. The algebra
bonds on an elementary triangle. Nor is it possible for iscumbersomebutstraightforward. The centralresultis
three bonds joining at a site on the honeycomb lattice. the quantum correction to the ground state energy per
The orbital exchange is thus geometrically frustrated on unit cell (containing one A site and one B site),
thetriangularandhoneycomblattice. Byfrustration,we
E (φ) 1 3
mean the energy on each bond cannot be minimized si- c = Xωλ(k) . (8)
multaneously [28]. As we shall show below, the classical TJz N − 2
k,λ
ground state possesses a continuous degeneracy for the
honeycomb lattice. Solving the quantum 120 model is Here, index λ = 1 labels the two branches of
◦
±
a nontrivial task, e.g., it remains an open problem even the orbital wave excitations with dispersion ω (k) =
λ
for the cubic lattice [15, 25, 26, 27, 29]. Here, in search- (√3/4)p3+2λβk(φ). The form factor βk(φ) =
ing for the groundstate orderingpattern, weemploy the sin2(φ)eikeˆ1 +s|in2(φ | π/3)eikeˆ2 +sin2(φ+π/3)eikeˆ3.
· · ·
−
semiclassical analysis which proves fruitful in the study N isthe numberofsiteswithintheAsublattice,andthe
of frustrated magnets [30]: first find the classicalground k sum is within the first Brillouin zone of the triangular
state, then consider the effect of leading order orbital sublattice. E (φ) is plotted in the right panel of Fig. 3
c
wave fluctuations. Formally, this is done by generalize for φ = 0 to π, and E (φ+π) = E φ). We see that or-
c (
H to arbitrary psuedospin T 1 and consider the bital fluctuation lifts the SO(2) degeneracy and chooses
120 ≥ 2
limit of large T. ground state φ = nπ/3 (n is any integer), where func-
n
First we determine the classical ground state of H tion E (φ) reaches minimum. This mechanism is well
120 c
onatriangleclustershowninFig.2. Weparametrizethe known in frustrated spin systems as “order by disorder”
4
Here q = mr/~t+G is the quasimomentum folded into
the first Brillouin zone of the underlying lattice, G is
B -0.13
the reciprocal wave vector, and µ,ν are the orbital in-
A
dices. The closed-shell s-fermions contribute the usual
Jz
T anti-bunching dips at G, reflecting the Fermi statistics
B A E/c [32, 34]. The p-fermions on the other hand contribute
B additional terms such as structure factor TzTz and
A -0.15 φ/π hTq+Tq−′i, which lead to new dips at Gr, thheqrecqi′pirocal
latticewavevectoroftheenlargedunitcellintheordered
0 0.2 0.4 0.6 0.8 1
state. For example, G = (π,π), (0,π), (2π/√3,0), for
r
thesquare,kagome,andhoneycomblattice,respectively.
FIG.3: (Coloronline)Left: Neelorbitalorderingonthehon-
eycomb lattice. The orbital at site A forms an angle of 30◦ Note Added. After the submissionofourmanuscript,
with the horizontal AB bond. Global rotation of all orbitals there appeared Ref. [35] which independently proposed
by nπ/6 yields degenerate configurations. Right: The quan- and studied a similar quantum 120 model.
◦
tumfluctuationcorrectiontothegroundstateenergyperunit
We thank C. Ho and V. Stojanovic for helpful discus-
cell, Ec/TJz, which reaches minima at φn =nπ/3. This lifts
thecontinuous degeneracy of theclassical ground states. sions. This workis supportedin partby ARO W911NF-
07-1-0293.
[28, 30]. The Left panel of Fig. 3 shows a representative
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