Exploring spin-orbital models with dipolar fermions in zig-zag optical lattices

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by  G. Sun

Exploring spin-orbital models with dipolar fermions in zig-zag optical lattices
G. Sun,1 G. Jackeli,2,∗ L. Santos,1 and T. Vekua1
1Institut fu¨r Theoretische Physik, Leibniz Universita¨t Hannover, 30167 Hannover, Germany
2Max-Planck-Institut fu¨r Festko¨rperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany
Ultra-cold dipolar spinor fermions in zig-zag type optical lattices can mimic spin-orbital models
relevant in solid-state systems, as transition-metal oxides with partially filled d-levels, with the
interestingadvantageofrevivingthequantumnatureoforbitalfluctuations. Wediscusstwodifferent
physicalsystemsinwhichthesemodelsmaybesimulated,showingthattheinterplaybetweenlattice
geometry and spin-orbital quantum dynamicsproduces a wealth of novelquantumphases.
2
1
t
0 Orbital degrees of freedom of electrons play an im- (a) (b)
n 2 pobosretarvnetdroinletrinanstihteionfo-mrmeatatiloonxiodfesvawriitohuspanrotviaelllyphfiallseeds y λ t px λ py t
a d-levels [1, 2]. In Mott insulators, they may enhance x
J thermal as well as quantum fluctuations [3] and lead t px y xpx py py
1
to spin/orbital liquid states [4–7], and to spontaneously
1
dimerized states without any long-range magnetic or- FIG.1: (Coloronline)Twodifferentsystemsdiscussedinthis

] der [8, 9]. In many solid-state systems, the orbital work. (a) Snake-like ladder with rungs (bonds) along y (x),
el dynamics is often quenched, due to the coupling of and one fermion per rung. The inter (intra)-rung hopping is
- the orbitals to Jahn-Teller phonons, and the study of indicated by t (λ); (b) Zig-zag lattice with one fermion per
tr the quantum nature of orbitals demands systems with site occupying either px or py orbital states. Along a bond
.s strong super-exchange coupling between spins and or- in x (y) direction only px (py) orbitals are connected by a
t finitehoppingamplitudet. Adeformation ofthelatticewells
a bitals. However,inrealmaterials,thecouplingstrengths
m are fixed by nature, being very difficult to modify, thus (wsihtahdaemdpalrietua)deleλa.ds to a mixing between px and py orbitals
- limiting the experimental access to a potentially vast
d
phase diagram.
n
o Ultra-coldspinorgasesinopticallatticesopennewfas- start by presenting two possible scenarios, A and B, in
c cinatingperspectivesfortheanalysisofthequantumna- which dipolar Fermi gases may allow for the simulation
[
ture of orbitals [10–12]. The coupling constants in these of the above-mentioned spin-orbital models.

2 systems canbe easily controlledby modifying the lattice A: First scenario is provided by a snake-like ladder
v parametersand/orbymeansofFeshbachresonances[13].
as that shown in Fig. 1(a), formed by bonds and rungs
2
Differentlatticegeometries,includingfrustratedlattices,
8 alongxandy directions,respectively. Thislatticegeom-
like triangular [14] and Kagom´e lattices [15], may be
0 etry may be achieved by using a combined blue-detuned
createdbycombiningdifferentcounter-propagatinglaser
5 ladder-like lattice and a properly aligned red-detuned
. beams, and superlattice techniques. In addition, not
2 zig-zag lattice [25]. The overimposed red-detuned lat-
only the physics in the lowest band but also that in
1 tice allows for enhancing the potential barrier at alter-
1 higher bands may be controllably studied [16]. More- nated bonds in the upper and lower legs. A sufficiently
1 over,recent experiments on Chromium [17] and Dyspro-
strong red-detuned lattice leads hence to broken bonds
: sium [18] atomic gases,and polar molecules [19], are un-
v as depicted in Fig. 1(a). A similar technique has been
i veilingtheexcitingphysicsofultra-colddipolargases,for
X recently employed for the realization of a Kagom´e lat-
which the dipole-dipole interactions may lead to exotic
tice [15]. The hopping within the same rung is denoted
r phases [20, 21].
a byλ,whereastheinter-runghoppingforunblockedbonds
InthisLetter,weshowthatdipolarspinorFermigases isdenotedast[Fig.1(a)]. Theon-siterepulsionU results
in appropriate zig-zag lattice geometries allow for the fromthecombinationofdipolarandcontactinteractions.
quantum simulation of spin-orbital models for a family The dipoles are oriented in the xz-plane in such a way
of Mott insulating compounds, including systems with thatfermionsatthesamerungexperiencemaximalnon-
weak [8, 22, 23] as well as pronounced [24] relativis- localrepulsionV anddipolesonneighboringrungsinter-
tic spin-orbit couplings. Moreover, these models, which act identically, irrespective whether they occupy upper
are relevant for realmaterials,as pyroxene titanium and orlowersites,withtheinteractionstrengthmuchweaker
layered vanadium oxides, may be explored with dipolar thanintra-rungrepulsion. Asaresult,inter-rungdipolar
Fermi gases in parameter regimes which are hardly ac- interaction plays no role in the discussion below (inter-
cessible for solid-state compounds, allowing for the ob- action between next-nearest rungs is considered negligi-
servation of novel quantum phases. bly small). We assume U,V ≫ t,λ, and consider the
Physical realizations and effective Hamiltonian.– We case of one fermion per rung. The system is then in the
2
Mott-insulatorregimewithonefermionlocalizedoneach repulsion[27]. Thus,withoutdipolarcouplings,theMott
rung. The sites belonging to the i-th rung of the ladder phaseofonefermionperwellwouldnotbestable. Inthe
are distinguished by the pseudo-orbital quantum num- Mott-insulator regime, one can still vary ∆ in a wide
ber σz, with the convention that σz = +(−)1 when the range by changing the relative ratio of the strengths of
i i
upper (lower) site on a given rung is occupied. Defin- the contact and dipolar interactions.
ing α ≡ U/V, and setting t2/2U as the energy unit we The model (1), at α≃1, describes the spin-orbitalin-
arriveto the effective spin-orbitalHamiltonian ofKugel- terplay in Mott insulating transition-metal compounds.
Khomskii type [26, 27], At λ = 0, it describes a zig-zag chain of spin one-half
Ti3+ ions, with active d and d orbitals, in pyroxene
xy yz
N
1 titanium oxides ATiSi O (A=Na,Li) [22, 23]. On the
H = [2S S +α− ] 1+(−1)iσz][1+(−1)iσz 2 6
i i+1 2 i i+1 other hand, for ∆ = 0 the Hamiltonian (1) represents
Xi (cid:2) (cid:3)
the 1D counterpart of the 2D model for Sr VO [24].
2 4
N N
In the latter, the role of spins in Eq. (1) is played
− ∆ 2S S 1−σzσz −λ σx, (1)
i i+1 i i+1 i by an isospin variable discerning the Kramers partners,
Xi (cid:2) (cid:3) Xi
while the pseudo-orbital variables distinguishes two low-
where S is the spin-1 operator (stemming from the est Kramers doublets of V4+ ion, and λ term in Eq. (1)
i 2
spinor nature of the Fermi gas), and σz,x are Pauli ma- represents relativistic spin-orbit coupling.
i
trices describing the pseudo-orbital variables. We have In the following we study the ground-state properties
added an additional term proportional to the coupling of the model (1) in the wide parameter range by analyt-
constant ∆. Although ∆ = 0 for realization A, it plays ical and complementary numerical approaches.
an importantrole in the alternativecase B discussedbe- Ground-state phase diagram for ∆ = 0.– We first dis-
low. Note that the ratio α may be modified basically cuss the case ∆=0 relevant for realization A. At λ=0
at will, since U and V may be independently controlled the orbitals become classicaland the ground-state phase
using Feshbachresonances[13], altering the lattice spac- diagram is easily mapped. For α > 2, the ground-
ings, or modifying the transversalconfinement [20]. state is 2×2N degenerate: there is anti-ferro (AF) or-
B: Second possible scenario is provided by a zig- der in orbitals hσzi = ±(−1)i (± refers to two degen-
i
zag lattice in the xy-plane, loaded with spinor dipolar erate AF states), whereas the spin part is completely
fermions in p-bands [16] [Fig. 1(b)], where the dipoles degenerate (bold line on Fig. 2(a)). For α > 2 and
are oriented along z-axis. Assuming a strong confine- λ→0,thesystemisdescribedbyaneffectivespinmodel,
ment along z, we retain two degenerate orthogonal px HS ∼ λ2 iSiSi+1, an isotropic Heisenberg antiferro-
and p orbitals per lattice site. In this realization, t de- magnet (iHP). Thus, quantum fluctuations of orbitals, in-
y
notesthehoppingbetweensimilarorbitalsatneighboring duced by λ-term, lifts immediately the infinite degen-
wells [Fig. 1(b)]. An in-plane deformation of the lattice eracy of the ground state, resembling order from dis-
wells (e.g. by an additional weak tilted lattice) leads to order. We denote this phase (iH,AF), where the first
a mixing of the p and p orbitals within the same well term denotes iH spin phase, and the second term AF
x y
with an amplitude λ. orbital phase. We employ a similar notation from now
Theinteractionparametersfortwofermionswithinthe on. Higher order terms in λ/α cannot break the SU(2)
same well are on-site repulsions U (within the same or- spin and translational symmetries, and thus for λ → 0
bitals) and V (among different orbitals), and Hund’s ex- the iH phase in spin degrees is stable. As shown be-
change J [27]. Two fermions occupying the same or- low,theiHisrecoveredforstrongλindependently ofthe
H
bital may form a symmetric or an antisymmetric state value of α. Thus, we can expect that for α > 2 there
with respect to the orbital index with corresponding en- is an unique iH phase in spin degrees of freedom for any
ergiesU+J andU−J ,whicharesplitbytheso-called λ6=0. Onthe contrary,withincreasingλthe orbitalde-
H H
pair-hoppingtermwith anamplitude J [27]. The over- grees of freedom experience an Ising transition from AF
H
all energy scale will now be modified to t2/2U˜, where to paramagnetic (P) phase with hσzi=0.
i
U˜ = (U2 − J2)/U. When two fermions occupy dif- For λ = 0 and 0 < α < 2, the exact ground state
H
ferent orbitals, they may form a spin-singlet or a spin- is two-fold degenerate and represents a direct product
triplet state with corresponding energies V + J and of ferro (F) orbital order, hσzi = +1 (−1), and spon-
H i
V − J , which are split by Hund’s exchange. In the taneously dimerized Majumdar-Ghosh (MG) state [28]
H
strong coupling limit U ±J ,V ±J ≫ t,λ and with in spins, with spin-singlets located on odd (even) bonds.
H H
one particle per lattice well the system is in the Mott- Wecallthisphaseadimer-ferro(D,F).Aninfinitesimalλ
insulator regime, and we arrive at Hamiltonian (1) with generatesanexchangebetweenthedisconnectednearest-
∆ = J U˜/(V2 −J2) [27]. Now, α also gets modified neighbour dimers, ∼ λ4 S S . With increasing
H H 2i+1 2i+2
accordingly, α=U˜(V +J /2)/(V2−J2). λ, the dimerizationorderPin spins, D = 1 |hS S −
H H N i i i+1
NotethatforapurelycontactinteractionV =J and S S i|, disappears together with the orPbital ferro or-
H i i−1
inthetripletchanneltwofermionsdonotexperienceany der at a Kosterlitz-Thouless (KT) phase transition [29].
3
7 7 increasing ∆ the system necessarily enters first into a
(a) (iH,P) (b)
6 6 dimerized state via a KT phase transition at ∆KT ≃
(F,P) 1−1/2λ. At larger∆, a MG state will be approximated
5 (iH,P) 5
(D,F) at ∆MG ≃ 1+1/2λ, where dimerization will reach the
4 4
λ λ value D ≃ 3/4. Increasing further ∆, at ∆′ ≃ 1+3/2λ,
3 3 nearest-neighborcouplinginEq.(2)vanishes,j (∆′)=0,
1
2 2 and the 1/λ2 terms, j and Ω, are the leading ones that
(D,F) 3
1 (D,F) (iH,AF) 1 (F,AF) coupletwosub-chains[27]. Bosonizationshows[27]that,
inspiteoftheseterms,inthelargeλlimit,thesystembe-
0 1 α2 3 4 0 1 ∆2 3 4 havesatlow energies as two decoupledspin-1 chains,al-
2
beitat∆∗ =∆′+O(λ−2). Hencearoundthe∆=∆∗line
FIG.2: Ground statephasediagram of themodel(1)for (a) the effective spin model in strong coupling is described
∆=0and(b)α=1. Firstandsecondphasesinparentheses by a j −j model, where j changes sign from antiferro
1 2 1
refer to spin and orbital sectors, respectively. See text, for a (for ∆ < ∆∗) to ferro (for ∆ > ∆∗), whereas j stays
2
description of phases and phase transitions. positive. Bosonization, supported by recent numerical
studies,predictsthatthegroundstateoftwoweaklycou-
pledspin-1 chainsisdimerizedirrespectiveofthe signof
Anumericalground-statephasediagramofthemodel(1) 2
j coupling [31–33]. Hence, in our case there is a special
for ∆=0 is presentedin Fig. 2(a) (details ofthe numer- 1
fine-tuning line bisecting the dimerized phase, ∆ = ∆∗
ical simulations are discussed later).
line, describedbydouble KTphase transition[34] where
Ground-state phase diagram for ∆6=0.– We now turn
spindimerizationandferroorbitalorderbothvanish. Fi-
tothecaseoffinite∆. Wefocusontheregimeα=1rele-
nally,thereisafirstorderphasetransitionlineseparating
vanttorealisticcondensed-mattersystems[8,22–24]. For
(D,F) and (F,P) states at ∆ ≃1+7/2λ.
λ = 0, a simple calculation shows that the ground state F
This sequence of phases and phase transition curves
is (F,AF) for ∆>2(2−α) and (D,F) for ∆<2(2−α).
(∆ ,∆∗,∆ ), which has been established analytically
Note, that once the spin dimerization pattern is sponta- KT F
for large λ with the help of effective spin model (2), has
neously chosen, the direction of orbitals becomes unam-
been confirmed for λ& 4 by numerical simulations (dis-
biguously selected, thus Ising Z orbital order is ’slaved’
2 cussed below) of the original model (1).
bytranslationsymmetrybreaking. Whereasfor∆=0an
Numerical procedures.– The phase diagrams depicted
infinitesimal λ induces AF spin-exchange ∝ λ4 on inter-
in Figs. 2(a,b) were obtained by means of a combination
dimer (weaker) bonds in the (D,F) phase, for finite ∆
of Lanczos exact diagonalization, density matrix renor-
the leading spin-exchange along the weaker bonds is in-
steadferromagnetic∼−λ2∆ S S . Hence there malization group simulations based on matrix product
2i+1 2i+2
states (MPS) [35] and the infinite time-evolving block
is a competition between thPe ∆ and λ terms promot-
decimation (iTEBD) algorithm [36], confirming the ana-
ing, respectively, a F and AF character of the weaker
lytical predictions for λ→0 and λ→∞.
bonds. The characterofspincorrelationsonweakbonds
hS S i changes from AF (small ∆ region) to F The KT transition between (D,F) and (iH,P) was ex-
2i+1 2i+2
(larger ∆ region) across ∆ = λ2/4+O(λ4) line, re- tractedbyLanczosmethodfromtheextrapolationofthe
MG
semblingthebehaviouracrosstheMGpointinthespin-1 level crossing[37] between the first excited singlet of the
2 (D,F) phase and the first excited triplet of the (iH,P)
j −j model [30].
1 2
phase for systems of up to N = 12 rungs/wells. Ising
For λ → ∞ the orbital degrees of freedom are
transitionlineswereobtained,byMPSsimulations,from
quenched. The system becomes equivalent to a SU(2)-
symmetricspin-1 chain,whichuptoO(λ−2)isdescribed the peak in the fidelity susceptibility [38], and that on
2 Fig. 2(b) between (F,AF) and (F,P) accurately follows
by the effective Hamiltonian,
the analytical line λ = ∆/2+α. The first order phase
H = [j S S +Ω(S S )(S S )] (2) transition into the (F,AF) or (F,P) states [bold line in
S n i i+n i−1 i i+1 i+2
X Fig. 2(b)] was obtained by MPS method, from the jump
i,n
of the ground-state total spin from 0 (singlet state) to a
where j =2−2∆+O(λ−1), j =λ−1, j ,Ω are both of fullypolarizedN/2(ferrostate)[39]. InMPSsimulations
1 2 3
orderλ−2,andthelongerrangeexchangesaresuppressed we have used periodic boundary conditions and system
as j ∼O(λ1−n) [27]. There are two clear phases in this sizes of up to N = 32 rungs/wells were considered. For
n
regime, (iH,P) for 1−∆ ≫ λ−1 and (F,P) for 1−∆ ≪ λ&4, the double KT phase transition line bisecting the
−λ−1. (D,F) phase [see Fig. 2(b)] was determined from vanish-
One may suspect for large λ a direct (iH,P) to (F,P) ingdimerorderusingiTEBDmethod. Forsmallervalues
transition with growing ∆. This, however, is not the of λ (1 < λ . 4), around the dashed line of Fig. 2(b),
case, as can be shown by a bosonization analysis of the our simulations indicate an intermediate quadrumerized
effectivespinmodel(2). Startingfrom(iH,P)state,with phase [40].
4
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5
SUPPLEMENTARY MATERIAL TO B: p-band zig-zag lattice
”EXPLORING SPIN-ORBITAL MODELS WITH
DIPOLAR FERMIONS IN ZIG-ZAG OPTICAL In the system B, the pseudo orbital index a = 1 (2)
LATTICES”
refers to p (p ) orbitalin a given well. The kinetic part
x y
ofthe Hamiltonianremainsthe sameasinEq.(3), while
In this supplementary material, we provide addi-
the interaction part, in addition to the terms shown in
tionaldetailsconcerningthederivationofthespin-orbital
Eq. (4), acquires the following supplementary terms:
Hamiltonian and the bosonization procedure.
n n
H′ = −2J S S + 1i 2i (5)
int HXh 1i 2i 4 i
THE EFFECTIVE SPIN-ORBITAL i
HAMILTONIANS + J [c† c† c c +c† c† c c ],
H 1i↑ 1i↓ 2i↓ 2i↑ 2i↑ 2i↓ 1i↓ 1i↑
X
i
In this section we outline the essential steps of the
where S and n are, respectively, spin and den-
derivation of the effective spin-orbital Hamiltonian for 1(2)i 1(2)i
the two systems, A and B, discussed in the Letter. sity operators for a fermion in px(py) orbital state. The
first line stands for Hund’s exchange and the second one
describes the so-called pair-hopping [1].
A: Snake-like lattice The coupling constants U, V, and J , entering in
H
Eqs. (4,5), can be expressed in terms of a two-body po-
Weintroducethefermionicannihilationoperatorc , tential and single-particle wave functions [1]. One finds:
ai,s
where a = 1 (2) indicates the up (down) sites on i-th
roufntgh,eaLnedttser=,↑syosrte↓mreAferistdoetshceribspeidn.byUtshinegHthuebbnaortdatliiokne U=Z dr1dr2Px2(y)(r1)V(r1−r2)Px2(y)(r2), (6)
Hamiltonian H =Hkin+Hint, where V =Z dr1dr2Px2(r1)V(r1−r2)Py2(r2), (7)
Hkin = −λ (c†1i,sc2i,s+c†2i,sc1i,s) (3) and
X
i,s
− 2t 1+(−1)i+a c†ai,scai+1,s+c†ai+1,scai,s , JH=Zdr1dr2Px(r1)Py(r1)V(r1−r2)Px(r2)Py(r2). (8)
iX,a,s(cid:2) (cid:3)(cid:2) (cid:3)
Above P and P are orbital wavefunctions of the same
accounts for single-particle processes, and x y
well, and V(r1 −r2) is the total interparticle potential
H = U c† c c† c (4) (including both contact as well as dipolar interactions).
int ai,↑ ai,↑ ai,↓ ai,↓
Xi,a The energy of the state with two fermions occupy-
+ V c†1i,sc1i,sc†2i,s′c2i,s′. tinegrmthmeosvaemsetwoorbpitaarlticislesUf.roHmowtheevedro,utbhleypoacicru-hpoiepdpionrg-
iX,s,s′
bital to an empty orbital with the amplitude J , and
H
corresponds to two-particle interactions. In the limit true eigenstates become orbital-symmetric and orbital-
U,V ≫ t, and retaining one particle per ladder rung antisymmetricstateswithcorrespondingenergiesU+J
H
(quarter filling), the effective Hamiltonian up to the sec- and U −J .
H
ond order in t acquires the form of Eq. (1) of the Let- Two fermions occupying different orbitals may form
ter (at ∆=0), where a spin-singlet or a spin-triplet state with corresponding
Sz = Sz = 1 (c† c −c† c ) energies V +JH and V −JH [2].
i ai 2 ai,↑ ai,↑ ai↓ ai,↓ In the limit U ±JH,V ±JH ≫ t, and up to second
X X
a a order in t, the effective Hamiltonian acquires the form
S+ = S+ = c† c , H = H ,whereH isthetwo-siteHamiltonian:
i ai ai,↑ ai,↓ i i,i+1 i,i+1
Xa Xa P
t2
are the spin-1 operators on the i-th rung, and H =− P (ST=0)(1+(−1)iσz)(1+(−1)iσz )
2 i,i+1 U˜ i,i+1 i i+1
σix = Xs (c†1i,sc2i,s+c†2i,sc1i,s) − 2(V t+2J )Pi,i+1(ST=0)(1−σizσiz+1)
H
σiz = Xs (c†1i,sc1i,s−c†2i,sc2i,s) − 2(V t−2J )Pi,i+1(ST=1)(1−σizσiz+1)
H
are the Pauli matrices corresponding to the pseudo- λ
orbital degrees of freedom. − 2(σix+σix+1), (9)
6
j S1 S2 = S . Using bosonization identification, the spins
2 i i 2i+1
along each chain are represented by smooth and stag-
j
1 2i S2 gered parts Sai → JaL +JaR +(−1)ina [4]. The coupling
i
of the two chains induces marginal operators. Apart
2i−1 2i+1 from the scalar (dimerization) operators, there are spin-
full marginal terms, so-called twist operators [5]. Up to
FIG. 3: Two sub-chains with intrachain coupling j2 (bold irrelevant and marginal couplings that induce only ve-
lines)coupledbyzig-zaginterchaincouplingj1 (dashedline). locity renormalization, the effective bosonized Hamilto-
niandensitydescribingthespindegreesoffreedominthe
wherewe haveintroducedthe operatorsP (ST=0)= large-λ limit is of the form
i,i+1
−SiSi+1+1/4,andPi,i+1(ST=1)=SiSi+1+3/4,which H= πvJaJa+D (J1J2 +J1J2)+D JaJa (11)
projectonto two-fermionstates onsites i and i+1 with, 2 ξ ξ 1 L R R L 2 L R
respectively, total spin ST = 0 and ST = 1, and U˜ = +T ε (ǫa∂ ǫb+na∂ nb)+T ε (3ǫa∂ ǫb−na∂ nb),
1 ab x x 2 ab x x
(U2−J2)/U.
H
The first line in Eq. (9) accounts for the situation where v ∼ j , a summation convention is assumed for
2
in which both fermions occupy same orbitals on sites repeated indices a,b = (1,2), and ξ = (L,R), ε is the
ab
i and i+1. In that case, when a fermion hops to the antisymmetricsymbol,andǫa standforthe dimerization
neighboring site [3], the orbital-symmetric or orbital- operators, such that SaSa ∼ (−1)iǫa(x)+ a less rele-
i i+1
antisymmetric states of the spin-singlet pair is reached vant smooth part.
(note that the formation of the spin-triplet pair is for- Bare values of the twist couplings T , and the inter-
1,2
bidden by Pauli exclusion). chain dimerization D depend on j ,j and Ω, in linear
1 1 3
The second and third lines of Eq. (9) account for the order (with non-universal proportionality coefficients),
configuration in which two fermions occupy orthogonal whereas the intra-chain dimerization amplitude D de-
2
orbitals on neighboring sites. When a fermion hops to pends on j and Ω.
2
a (singly) occupied neighboring site (remembering that One-loop RG equations corresponding to the effective
inter-site hopping t does not change orbital quantum model (11) are identical to those obtained in [5] for the
number),dependingonthetotalspinofthetwofermions SU(2) symmetric case. RG flow is dominated by inter-
the interaction may take place either in the spin-singlet chaindimerizationD ,whichintheinfraredlimitalways
1
channel, with energy cost V +JH, or in the spin-triplet scales to strong coupling, except for the initial condition
channel, with energy cost V −JH. A proper re-ordering T1 =T2,correspondingtothefinetuning∆=∆∗,which
ofthetermsinEq.(9)leadstotheHamiltonianofEq.(1) we interpret as an infrared decoupling point of the two
of the Letter. chains. In contrast D scales to zero at low energies for
2
λ → ∞. Thus for λ → ∞ there is a special line in
(D,F) phase where spin dimerization (and consequently
BOSONIZATION ANALYSIS
the ferro orbital order) vanishes.
In fact, bosonization is only needed to capture the in-
In the following we provide some details on our fluence of j and Ω terms close to j = 0. Away of
3 1
bosonizationanalysisofthelargeλregime(λ→∞),and that region, for λ → ∞, one just needs to retain j and
1
in particular on how this approach provides an under- j terms, and borrow known results from the frustrated
2
standing of the special fine-tuning line ∆ = ∆∗ cutting j −j spin-1 chain. This gives us an estimate of other
1 2 2
the (D,F) phase at which spin dimerization and orbital phase transition lines (we use the notations of the Let-
ferro orders vanish. ter)∆ and∆ whichfollowrespectivelyfromj ≃4j
KT F 1 2
Integrating out the orbitals in large λ limit, to the and j = −4j conditions. Whereas ∆ (which does
1 2 MG
secondorderin 1/λwe obtainthe effective spin-1 model not represent a phase transition line) follows from the
2
given in Eq. (2) of the Letter with, condition j =2j .
1 2
4+∆(1+∆)
j = 2(1−∆)+ +O(λ−2),
1
2λ
1 1+∆
j = +O(λ−2), j = +O(λ−3),
2 λ 3 2λ2 ∗ Also at AndronikashviliInstituteof Physics, 0177 Tbilisi,
Ω = 2 +O(λ−3) , (10) Georgia.
λ2 [1] See, e.g., C. Castellani, C. R. Natoli, and J. Ranninger,
Phys. Rev.B 18, 4945 (1978).
whereweusethesamenotationsasintheLetter. λ→∞
[2] For the spin-triplet state, the coordinate wave-function
is a convenient limit for bosonization, in particular for
must have a node when the positions of the two fermions
j1 ≪j2. In that case, one can consider two weakly cou- coincide. Hence, for the spin-triplet state the contact s-
pledchains(seeFig.3),whichwedenoteasS1i =S2i and wave interaction does not play any role. This is reflected
7
in Eqs. (7) and (8) where for V(r1 −r2) ∼ δ(r1 −r2), [4] A.O. Gogolin, A.A. Nersesyan, and A.M. Tsvelik,
V = JH and hence scattering in the spin-triplet channel BosonizationandStronglyCorrelatedSystems,Cambridge
vanishes. University Press (1998); T. Giamarchi, Quantum Physics
[3] Note that hopping from a site i to a site i+1 is only in One Dimension, Oxford University Press (2003).
possible if the fermion occupies on site i the orbital that [5] A.A. Nersesyan, A.O. Gogolin, and F.H.L. Eßler, Phys.
isdirected towardsthesite i+1, asdepictedon Fig. 1(b) Rev.Lett. 81, 910 (1998).
of theLetter.