Nematic Ferromagnetism on the Lieb lattice

0.52 MB·English

by  Wei Zhang

Nematic Ferromagnetism on the Lieb lattice
Wei Zhang
Department of Physics, Renmin University of China, Beijing 100872, China
We discuss the properties of possible ferromagnetic orders on the Lieb lattice. We show that
thepresenceofaquadratic-flatbandcrossingpoint(QFBCP) athalffillingwill dramatically affect
the magnetic ordering. In the presence of a weak on-site repulsive interaction, we find the ground
state is a nematic ferromagnetic order with simultaneously broken of time-reversal and rotational
symmetries. Whentheinteraction strength increases, therotational symmetry will restore at some
2
criticalvalue,andthesystementersaconventionalferromagneticregime. Wealsopointoutthatthe
1
spin gap in both the nematic and conventional ferromagnetic phases is of the order of interaction.
0
2 This observation suggests that these magnetic orders can be realized and detected in cold atomic
systems with present technology.
n
a
PACSnumbers: 03.75.Ss,37.10.Jk,05.30.Fk,71.70.Di
J

3
The investigationonferromagnetismis oneofthe cen-

]
tral topics in condensed matter physics, and has at-
s
a tractedattentionfor nearlya century since the earlyage
g of quantum theory. Heisenberg showed that a system
-
t of localized spins would favor a fully polarized state by
n
gainingexchangeenergy.[1]However,thesameargument
a
cannotbesimplyappliedinitinerantfermions,wherethe
u
q kineticenergyoftheunderlyingsystemhastobeconsid-
. ered on an equal footing with the interaction effect.
t
a The discussion on the stability of itinerant ferromag-
m
netism in a lattice system dates back to 1960’s. Thou-
- less [2] and Nagaoka [3] pointed out that a ferromag-
d
netic ground state can be stabilized in any finite bi-
n
partite lattice with infinite on-site repulsive interaction.
o
Lieb showed that the stable region of ferromagnetism
c
[ can be extended to arbitrary repulsive interaction, pro-

vided that the number of sites are different for the com-
1
posite sublattices. [4] The key ingredient in Lieb’s argu-
v
2 ment is the existence of a non-dispersive, or briefly flat,
2 band. When the flat band is partially filled, fermions
7 tend to be spinpolarizedto minimize the interactionen-
0 ergy, without any cost in the kinetic energy. In other
1. words, since states in the flat band consists of localized
FIG. 1: (Color online) (a) 2D Lieb lattice consisting of two
0 Wannier functions, the ferromagnetic states can benefit
square sublattices A and B. There are three sites (1, 2, 3)
2 from the exchange interaction as pointed out by Heisen-
within a unit cell (dotted square). (b) The Lieb lattice can
1 berg. Subsequent studies confirm the stability of ferro-
be realized by arranging three square optical lattices [14]:
:
v magnetism in various models [5, 6], and generalize the V1 = 2V0[sin2(x/a)+sin2(y/a)], V2 = 2V0[sin2(x/2a+π)+
Xi idea to nearly-flat-band cases [7]. Experimental realiza- sin2(y/2a+π)],andV3=V0[cos2(x/2a+y/2a)+cos2(x/2a−
tion of the (nearly-) flat-band ferromagnetism has been y/2a)]. Here, a is the lattice spacing. (c-d) Band structure
ar proposed in a class of physical systems including atomic of the Lieb lattice with Vb/t = 1. Note that the flat band
quantum wires [8], quantum-dot super-lattices [9], and degenerates with the upper dispersive band at the M-point,
organic polymers [10]. leading to a quadratic-flat band crossing point.
Rapid progress in cold atom experiments has paved
a new route toward the exploration of ferromagnetism
in itinerant fermions. Thank for the extraordinary con- consists of two square lattices (sublattice A and B), and
trollability of lattice potentials and interaction, several has three sites per unit cell, as shown in Fig. 1. With
proposalshasbeenmadetorealize(nearly)flat-bandfer- only nearest-neighbor hopping, the lattice topology sup-
romagnetism [11–13]. In particular, Noda et al. inves- ports a flat band in the middle of two dispersive bands,
tigated two-component cold fermions loaded into a two- hence can stabilize ferromagnetism when it is partially
dimensional(2D) generalizedLieb lattice, andsuggested filled.
that a ferromagnetic order can be stabilized in a wide Apparently,asole flatbanddoesnotexistinanyreal-
parameter region. [12] The 2D generalized Lieb lattice istic physical system. In all proposed model lattices for
2
flat-band ferromagnetism, the flat band is always asso- cases of V 6= 0, depending on the sign of V , one of the
b b
ciated with dispersive bands, and the effect of their ac- two massive bands breaks the M-point degeneracy, and
company is not fully understood. As a typical example, the flat band only touches the other dispersive band as
if the two composite sublattices of the Lieb lattice have illustrated in Fig. 1(d). As a consequence, the M-point
the same depth, the flat band degenerates with the two becomes a quadratic-flat band crossing point (QFBCP).
linearlydispersivebandsattheM=(π,π)point. Onthe Around this point, the effective two-band Hamiltonian
otherhand,ingeneralcaseswherethedepthofsublattice reads
B is shifted from that of sublattice A by an amount of
2t2 |δk|2 δk2
Vbab,ndthseatflathtebMan-dpotoinutc,haessiollnulsytroanteedoifnthFeigt.w1o(dd)i,slpeeardsiinvge H0eff = Vb (cid:18) δk2− |δk+|2 (cid:19)+O(δk4), (2)
toIanqthuiasdmraatnicu-sflcartipbta,nwdecsrhooswsinthgaptotihnetf(eQrrFoBmCaPgn).eticor- where δk = k−M and δk± = δkx ±iδky. From this
effectiveHamiltonian,itisclearthatthecasesofpositive
der is dramatically affected by the existence of the QF-
and negative chemical potential offset V are equivalent
BCP. In the non-interacting level, the QFBCP is pro- b
via a particle-hole transformation. Thus, we focus on
tected by the time reversal(TR) and C4 rotationalsym- systems with V > 0 without loss of generality in the
metries. When a repulsive on-site interaction is present, b
following discussion.
since the 2D density of state (DOS) is singular, the QF- The presence of a QFBCP at the M-point is the
BCP becomes marginally unstable, leading to a sponta-
central feature of the Lieb lattice. This band cross-
neousbrokenoftheTRand/orC4rotationalsymmetries. ing point (BCP) is protected by the time reversal (TR)
We findthat inthe weakcoupling limit the groundstate
andC4rotationalsymmetriesinthenon-interactingcase,
is in a nematic ferromagnetic (NFM) order, character-
and is characterized with a nontrivial topological index
ized by a broken of C4 point group down to C2. The 2π. Such a putative topologically stable BCP becomes
C4 rotational symmetry will restore with increasing in- marginally unstable against infinitesimal repulsive inter-
teraction, via a second or first order phase transition,
action U [16], leading to a spontaneous broken of TR
dependingonthevalueofV . Withinamean-fieldcalcu-
b and/or C4 rotational symmetries, which drives the sys-
lation,wefurthermapoutthe phasediagramathalffill-
tem toward a magnetic and/or nematic phase. Besides,
ing, and identify three phases including: (i) a semimetal
sincetheBCPconsistsofanon-dispersiveband,theinfi-
with NFM order; (ii) a band insulator with NFM order,
nitedensityofstate(DOS)allowsthepossibilityoffilling
and (iii) a band insulator with conventional FM order.
the high momentum states with one single spin species
These magnetic orders have the potential to be realized
without gaining any kinetic energy. As a consequence, if
and detected in cold fermions loaded in optical lattices
the on-site interaction U is repulsive, the system could
with present technology.
easily favor a ferromagnetic phase for filling factors be-
We consider spin 1/2 fermions loaded in the 2D Lieb
tween 1/3 and 2/3.
lattice
Next,wefocusonthecaseofrepulsiveHubbardU >0
H =−t c† c +V n +U n n , (1) with half filling, and investigate the magnetic and ne-
iα jα b i i↑ i↓
matic magnetic order within a mean-field (MF) level.
<iX,j>,α iX∈B Xi
The magnetic order is characterized by on-site magne-
where t is the hopping matrix, c†iα(ciα) is the creation tizationB1 andB2 forsublatticesAandB,respectively.
(annihilation) fermionic operator, V is the chemical po- By minimizing the MF energy functional, we find that
b
tentialoffsetforthesublatticeB(i.e.,therelativeshiftof the system is stabilized by a staggeredferromagneticor-
the two sublattices), and U is the on-site interaction. In derwithB1 andB2 bothorientedalongtheout-of-plane
the context of cold atoms, V can be easily controlled by z-axis. IntheweaklyinteractinglimitU ≪t,themagne-
b
varyingtherelativeintensityofopticallattices,andare- tizationon sublattice A is quadraticallydependent onU
pulsive interaction U >0 can be achieved and tuned via withB1z ≈α2U2/9t,whereα2 =(2/π2) d2k[cos2(kx)+
an adiabatic ramping to the upper branch on the BEC cos2(k )]/[V2/t2 + 4cos2(k ) + 4cos2(kR)]3/2, and the
y b x y
side of a Feshbach resonance [15]. magnetization on sublattice B is linearly dependent on
Inthenon-interactingcase,theHamiltoniancanbedi- the interaction with B2z ≈ −U/6. In the strongly cou-
agonalizedin momentum space, leading to a band struc- pling limit U ≫ t, the interaction turns to be the only
tureconsistingofthreebandsasshowninFig.1(d). One relevantenergy scale, and both B1z and B2z are linearly
of the three bands is completely flat, as required by the dependentonU withB1z,2z =±U/3,respectively. Here,
bipartitenessoftheLieblattice. TheflatbandhasBloch we set B1 to be along the positive z-axis without loss of
wavefunction ∝ [0,−cos(k ),cos(k )] on the three sites generality.
x y
within a unit cell, indicating the presence of local Wan- Tostudythepossibilityofnematicmagneticorderwith
nier functions residing on the sublattice B and having both TR and C4 rotationalsymmetries broken,we allow
opposite amplitudes between sites 2 and 3. When the the magnetization on sublattice B can be different for
chemical potential offset V =0, the three bands are de- sites 2 and 3, and map out the zero-temperature phase
b
generateat the M=(π,π) point, where the two linearly diagram as shown in Fig. 2(a). We find that the op-
dispersive bands intersect with the flat band. In general timized magnetization Bi=1,2,3 are always along the z-
3
FIG. 3: Band structure for (a) semimetal with NFM order,
(b) band insulator with NFM order, and (c) band insulator
with FM order. The solid (dashed) curves represent major-
ity (minority) spin. These three cases are represented in the
phase diagram Fig. 2(a) with red dots, and correspond to
Vb/t = 2.3, U/t = 1.5, 5.0 and 8.0, respectively. The varia-
tion ofspin gap ∆s andbandgap ∆b for Vb/t=2.3 isshown
in (d). Note that the spin gap is always in the order of the
interaction U.
FIG. 2: (a) Zero temperature phase diagram for spin 1/2
Fermi system on the Lieb lattice at half filling. The nematic
ferromagnetic (NFM) order is favored for weak interaction,
the corresponding metastable states.
andisseparatedfromtheferromagnetic(FM)orderbyasec-
ondorder(thinsolid)orfirstorder(thicksolid)transitionline. WithintheNFMphase,theC4 rotationalsymmetryis
WithintheNFMregime,thesystemisasemi-metal(SM)for spontaneousbrokendowntoC2 bysplittingthe QFBCP
small U and Vb, and undergoes a second order (thin dashed) intotwoDiracpointslocatedalongthedirectionofoneof
or first order (thick solid) transition to become a band insu-
the principal axes for weak interactions. The two Dirac
lator(BI)withincreasingU andVb. Anexampleoftheband pointshavethesameBerryfluxπ,inclearcontrasttothe
structure for different phases (red dots) are shown in Fig. 3
case of graphene where the two Dirac points have Berry
(a-c). For two typical values of Vb/t = 1.5 and 2.3 (dotted
fluxesπand−π. Inthiscase,thesystemisananisotropic
lines), the magnetization on the three sites are shown in (b)
and (c), respectively. semimetal (SM) at half filling, with the Fermi surface
shrinks to the two Dirac points as shown in Fig. 3(a).
By increasing U, the two Dirac points move toward the
boundaryoftheBrillouinzone,andeventuallydisappear
axis, hence we consider only axial magnetic order in the when the magnetization difference δ = |B2z −B3z| be-
following discussion. tween sites 2 and 3 exceeds the bandwidth of the first
Inthe weaklyinteractinglimit, the nematic FM phase excited dispersive band. As a result, a full gap is open
is alwaysfavorablewith anexponentially smallmagneti- and the system becomes an anisotropic band insulator
zation difference |δ|≡|B2z−B3z|∼exp(−γt/U), where (BI) as depicted in Fig. 3(b). When the system enters
γ isapositiveparameterdependingonV . Thisobserva- the FM regime, the magnetization is large enough such
b
tionisalsoconfirmedbyaperturbativeanalysisbytreat- that a finite gap is always present, and the system is a
ingthe magnetizationdifferenceasaperturbationtothe band insulator as shown in Fig. 3(c).
existingFMorder. WhenU isincreasingfromtheweakly Note that in both the NFM and FM phases, the spin
interacting limit, the NFM phase remains stable up to a gap remains in the order of the interaction strength U
critical value Uc, above which the C4 rotational symme- [See Fig. 3(d)]. This observation suggests that the mag-
try restores and the system enters the conventional FM netic order on the Lieb lattice is robust against ther-
regime. For small values of V < V ∼ 1.7, the phase mal fluctuations. Although the Mermin-Wigner theo-
b c
transition between NFM and FM is of the second order, rem excludes the possibility of any 2D ferromagnetic or-
asidentifiedbytheconditionVb =B1z−B2z. Byincreas- der at finite temperature in the thermodynamic limit,
ing V > V , the NFM-FM phase boundary becomes of the existence of such an order in a finite size system
b c
the first order, resulting from the competition between is perfectly allowed, provided that the coherence length
4
is comparable or exceeding the system size. Specif- against infinitesimal repulsive interaction, given an infi-
ically, the coherence length of the magnetic order is nite density of state of the flat band. In the weak inter-
ξ ∼ ~cexp(ρ /T)/(k T), where c is the spin-wave ve- actinglimit, the groundstate is a nematic ferromagnetic
s B
locity at zero temperature, and the phase stiffness ρ (NFM) order with time-reversal and rotational symme-
s
is in the same order of the MF transition temperature tries broken. Within the NFM regime, the spontaneous
TMF. Since the spin gap ∆s increases linearly with U, generated magnetizations are different on sublattice B,
and the interaction can be tuned to be fairly large via and the QFBCP is broken into two Dirac points along
ans-waveFeshbachresonance,the temperaturerequired one of the principal axes. In the strong coupling limit,
to observe the FM and NFM phases could be reachable theinteractionU becomestheonlyrelevantenergyscale,
within present technique. [17] and a conventionalferromagnetic(FM) phase with rota-
The detection of the ferromagnetic orders can be im- tional symmetry restored is favored. We then map out
plemented via an in-situ measurement [18, 19], which is the zero-temperature phase diagram within a mean-field
able to extract single site density distribution for differ- analysis,andidentifythreeregionsincludingasemimetal
entspinspecies,andhence the localmagnetizationB ∝ with NFM order,a band insulator with NFM order,and
i
n −n . If the system is prepared with equally pop- a band insulator with FM order. We point out that the
i↑ i↓
ulated two-component Fermi gas, we expect to see FM spin gap for all three phases is in the same order of in-
or NFM domains with opposite magnetizations, which teraction strength, which can be tuned via a Feshbach
are both resolvable within present experimental technol- resonance. Thus, we expect these magnetic phases can
ogy. Anotherpossibledetectionschemeistomeasurethe be realized in two-component Fermi gases loaded in op-
single-particledispersionwithBraggspectroscopy[20]or tical lattices at experimentally reachable temperatures,
angle-resolvedphotoemissionspectroscopy(ARPES)[21] and can be distinguished via a species selective in-situ
to extract the spin gap. measurement.
In conclusion, we discuss the effect of a quadratic- We would like to thank NSFC (10904172), the Fun-
flat band crossing point (QFBCP) on the ferromagnetic damental Research Funds for the Central Universities,
(FM) order. Taking the 2D Lieb lattice as an exam- the Research Funds of Renmin University of China
ple, we show that the QFBCP is marginally unstable (10XNL016), and the NCET programfor support.
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