A treatment of cooperative Jahn-Teller effect in interacting chains

1.6 MB·English

by  Ravindra Pankaj

A treatment of cooperative Jahn-Teller effect in interacting chains
Ravindra Pankaj1 and Sudhakar Yarlagadda1,2
1 CMP Division, Saha Institute of Nuclear Physics, Kolkata, India and
2Cavendish Lab, Univ. of Cambridge, Cambridge, UK
(Dated: January 27, 2015)
Studyingthenatureandconsequencesof electron-phononinteraction inmanganites isanarea of
intense ongoing research. Here, in an attempt to model charge and orbital ordering in manganites
displaying C-type antiferromaganetism, we study cooperative Jahn-Teller effect in two-band one-
dimensionalchainsintheregimesofbothstrongandweakelectron-phononcouplings. Thesechains
5 exhibit orbital ferromagnetism with only dz2 orbitals being occupied. At strong coupling and in
1 the antiadiabatic regime, using a controlled analytic nonperturbative treatment that accounts for
0
the quantum nature of the phonons, we derive the effective polaronic Hamiltonians for a single
2
chain as well as for interacting identically-long chains. Due to cooperative effects, these effective

n Hamiltonians manifest a dominant next-nearest-neighbor hopping compared to the usual nearest-
a neighbor hopping and a significantly enhanced nearest-neighbor repulsion. For densities up to half
J filling, upon tuning electron-phonon coupling, interacting-chain [single-chain] Jahn-Teller systems

6 undergo quantum phase transition from a charge disordered state to a conducting charge-density-
2 wave state characterized by a wavevector ~k = (π,π) [k = π]. On the other hand, up to half
filling,aweakcouplinganalysisrevealsatransitionfromadisorderedstatetoaninsulatingcharge-

] density-wavestatewith awavevectorthatdependslinearly onthedensity;theorderingisanalyzed
l
e withinaPeierlsinstabilityframeworkinvolvingthedynamicnoninteractingsusceptibilityatnesting
- wavevector and phonon frequency. Our analysis provides an opportunity to identify the regime of
r
t electron-phonon coupling in manganites through experimentally determining the charge-ordering
s wavevector.
.
t
a
PACSnumbers: 71.38.-k,71.45.Lr,75.47.Lx,71.38.Ht
m
-
d I. INTRODUCTION been developed only recently14. The situation is also
n
complicated because of lack of conclusive identification
o
of the strength of the electron-phonon interaction in ox-
c Understanding the wide variety of exotic long-range
[ orders in transition metal oxides (such as manganites) ides such as manganites. Evidence of strong local Jahn-
and designing artificial structures (such as heterostruc- Tellerdistortionspointingtostrongelectron-phononcou-
1
pling has been reported in manganites by direct tech-
v tures, quantum wires, and quantum dots) using these
niques such as EXAFS15, pulsed neutron diffraction16,
1 materials is of immense fundamental interest and also
or through direct evidence of orbital ordering using res-
2 of huge technological importance. Perovskite mangan-
4 ites R A MnO (R = La, Pr, Nd, etc., A = Sr, Ca) onant X-ray scattering17. On the other hand, weak cou-
1 x x 3
6 are sys−tems containing Mn3+ ions with one electron in pling wasinferredinthe overdopedregime(i.e., x>0.5)
0 using transmission electron microscopy18,19, orientation-
doubly degenerate e orbitals; this results in cooperative
1. Jahn-Teller (CJT) dgistortion (on adjacent sites) which dependent transport measurements20, terahertz spec-
0 troscopy(tomeasureconductivityandpermittivity)21,22,
lifts the orbital degeneracy and produces a cooperative
5 and coherent synchrotron source (to measure optical
occupation of orbitals. Consequently, a strong interplay
1 conductivity)23.
ensues between charge, spin, and orbital degrees of free-
:
v dom. This interplay leads to closely competing energy In the overdoped regime (x>0.5) of manganites such
i
X states with a variety of spin textures (such as metal- as La Ca MnO , at low temperatures, charge order-
1 x x 3
r lic ferromagnets and A-, C-, CE-, or G-type antiferro- ing res−ultsalongwith orbitalorderingandantiferromag-
a magnets), charge orders [such as density-dependent and netism. The nature of the charge order has been a sub-
density-independent chargedensity waves(CDWs)], and ject of much debate. Earlier on, charge order was de-
orbital orders (such as ferromagnetic and antiferromag- scribed in terms of a strong-coupling picture of stripes
netic orbital density waves)1–3. of localized charges where the mixed-valence manganese
Therehavebeennumerousstudiesofthemany-polaron ions split into Mn3+ ions and Mn4+ ions24,25. However,
effectsproducedbyquantumphononsinaone-bandone- based on the findings in Refs. 26–28, a phenomeno-
dimensional Holstein model4 which is a simple case of logical picture of an extended CDW (instead of local-
noncooperative electron-phonon interaction5–12. How- ized charges) was developed in Ref. 29. The experi-
ever,controlledmathematicalmodelingofCJTquantum ments indicating weak coupling18–20,23, for 0.5 x
systems that goes beyond modeling localized carriers13 0.75 in La Ca MnO , also seem to agree wi≤th th≤is
1 x x 3
hasremainedelusive(atleasttoourknowledge). Infact, picture of w−eakly-modulated CDW characterized by a
acontrolledanalytictreatmentofcooperative-breathing- concentration-dependent wavevector q = (1 x)a with
∗
mode effects in a single-band, many-polaron chain has a being the reciprocal lattice vector30. F−urthermore,
∗
2
in another manganite (i.e., La0.2Sr0.8MnO3), charge or- where d†x2;i,j,k,d†y2;i,j,k,d†z2;i,j,k are creation operators at
dering wavevector q = 0.2a∗ has been reported31. As the site (i,j,k) for dx2, dy2, and dz2 orbitals, respec-
regards magnetic ordering, in the overdoped regimes tively. The labelling indices i, j, and k run along the x-,
of La1 xSrxMnO3 (for 0.65 < x < 0.95)32–34 and y-, and z-axes, respectively. The electron-phonon inter-
La1 xC−axMnO3 (for 0.75 x 0.87),35–37 C-type anti- action term can be written as:
ferr−omaganetism has also b≤een≤reported.
In this paper, compared to the phenomenological the- H = gω 2Mω [n Q
ep 0 0 x2;i,j,k x;i,j,k
oryofRef. 29,weattemptatamicroscopictheoryinthe −
i,j,k
p X
overdoped regime. To reduce the daunting complexity,
+n Q +n Q ], (2)
y2;i,j,k y;i,j,k z2;i,j,k z;i,j,k
we assume C-type antiferromagnetism and invoke only
the charge and orbital degrees of freedom to model the
where g is the electron-phonon coupling, M is the
remaining possibilities of charge and orbital orderings.
mass of an oxygen ion, ω is the frequency of optical
0
We investigate the regimes of weak coupling and strong
couplingandshowhowthe regimeofcouplingcanbeas- phonons, and nx2(y2,z2);i,j,k =d†x2(y2,z2);i,j,kdx2(y2,z2);i,j,k
are the number operators. Furthermore, Q ,
certained experimentally. Since manganites with C-type x;i,j,k
Q and Q are defined in terms of the dis-
antiferromaganeticorderingcanbeeffectivelyconsidered y;i,j,k z;i,j,k
placements [u & u ; u & u ;
aschains,westudytheeffectofelectron-phononcoupling u & u x;i,j,k] of oxxy;gi−en1,ji,kons ayr;io,ju,knd (andy;ii,nj−1t,hke
in a single chain and in interacting chains. At strong dizr;ie,cj,tkion ofz);it,hj,ek−d1 , d , and d orbitals, respectively,
coupling, we find that the charge ordering wavevector is x2 y2 z2
as follows: Q = u u , Q =
always q = 0.5a∗ irrespective of the value of the doping u u x;i,,j,akndQ x;i,j,=k −u x;i−1u,j,k y;i.,jH,kere,
x;whereasatweakcoupling,theorderingwavevectorhas bye;sii,dj,eks−coyn;is,ijd−e1r,ikng the dzi;sip,jl,kacemezn;it,j,okf−thez;ii,oj,nks−,1we also
the density-dependent form q = (1 x)a . Thus, based
− ∗ consider their kinetic energy, thereby invoking quantum
on the experimental determination of the charge order-
nature of the phonons. Then, the lattice Hamiltonian is
ing wavevectorsatvarious dopingvalues x in C-type an-
given by
tiferromagnets, we believe that our theory offers an op-
portunity for identifying the regime of electron-phonon
M
coupling. Hl = 2 [u˙2x;i,j,k+u˙2y;i,j,k+u˙2z;i,j,k]
The paper is organized as follows. We derive a gen- i,j,k
X
eral Hamiltonian for CJT effect in three dimensions in K
+ [u2 +u2 +u2 ], (3)
Sec. II. In Sec. III, we derive the effective Hamiltonian 2 x;i,j,k y;i,j,k z;i,j,k
forCJTinteractioninonedimensionandstudyitsnature iX,j,k
at various electron-phonon couplings. In the subsequent
where u˙ , u˙ , and u˙ are the time deriva-
Secs. IVandV, weanalyzethe CJTeffect ininteracting x;i,j,k y;i,j,k z;i,j,k
tivesoftheoxygen-iondisplacementsu ,u ,and
chainsforthecaseofstrongcouplingintheantiadiabatic x;i,j,k y;i,j,k
u , respectively.
regimeandforthecaseofweakcouplingintheadiabatic z;i,j,k
The usual orthogonal basis states ψ and ψ are
regime, respectively. Before closing, we discuss the con- x2 y2 z2
related to the over-complete basis states−ψ , ψ , and ψ
nectionbetweenoursystemandothersystemsinSec.VI x y z
as follows:
and present our conclusions in Sec. VII.
1
ψ = (ψ ψ ),
II. GENERAL CJT HAMILTONIAN x2−y2 √3 x− y
ψ =ψ .
z2 z
The general Hamiltonian for the CJT system in man- (4)
ganitescanbewrittenasHG =H +H +H ,whereH
t ep l t
isthehoppingterm,H theelectron-phonon-interaction From Eq. (4) we get,
ep
term, and H the lattice term. We start with an over-
l
completebasisψx =3x2 r2,ψy =3y2 r2,ψz =3z2 r2 ψ = 1(√3ψ ψ ),
which satisfies the relati−on ψx+ψy +ψ−z =0. The b−asis x 2 x2−y2 − z2
1
sTthaetehψozppcoinrgrestperomndcsatno btheeedxzp2roesrsbeidtalindetphiectaebdoivneFbiags.1is. ψy =−2(√3ψx2−y2 +ψz2),
as: ψz =ψz2.
(5)
Ht =−t [{d†x2;i+1,j,kdx2;i,j,k+d†y2;i,j+1,kdy2;i,j,k
iX,j,k Next, using Eq. (5), we express the general Hamilto-
+d†z2;i,j,k+1dz2;i,j,k}+H.c.], (1) nian in the orthogonalbasis ψx2−y2 and ψz2 as follows:
3
t 1 √3 d t
Ht =− 4i,j,k{(d†z2;i+1,j,k,d†x2−y2;i+1,j,k)(cid:18)−√3 −3 (cid:19)(cid:18)dx2z−2y;i2,j;i,,kj,k(cid:19)+H.c.}− 4i,j,k{(d†z2;i,j+1,k,d†x2−y2;i,j+1,k)
X X
1 √3 d 1 0 d
×(cid:18)√3 3 (cid:19)(cid:18)dx2z−2y;i2,;ji,,kj,k(cid:19)+H.c.}−ti,j,k{(d†z2;i,j,k+1,d†x2−y2;i,j,k+1)(cid:18)0 0(cid:19)(cid:18)dx2z−2y;i2,j;i,,kj,k(cid:19)+H.c.}, (6)
X
1
H = gω 2Mω
ep 0 0
− 4
p Q +Q +4Q √3Q +√3Q d
×i,j,k(d†z2;i,j,k,d†x2−y2;i,j,k)(cid:18) −x;√i,j3,kQx;i,jy,k;i,+j,k√3Qy;iz,;ji,,kj,k − 3Qxx;i;,ij,,jk,k+3Qy;i,jy,;ki,j,k(cid:19)(cid:18)dx2z−2y;i2,;ji,,kj,k(cid:19), (7)
X
and H is again given by Eq. (3). Here, it should be mentioned that an expression for HG in an alternate basis has
l
been derived in Ref. 38; however,these authors consider classical phonons.
In the subsequent sections, we shall study special cases of our above general CJT Hamiltonian, namely, the single
chain and the interacting chains.
III. SINGLE CHAIN CJT MODEL AT STRONG and
COUPLING
HCJT
ep
gω √2Mω
We consider a one-dimensional Jahn-Teller chain with 0 0
1
cdoiroepcetiroantivaendelecntorno-nc-opohpoenroantivienteelreaccttrioonn-pahloonnognthinetezr-- =− (uz;k−uz;k−1)+ 4Q+xy;k d†z2;kdz2;k
action (of the Holstein-type4,5) along the x- and y- Xk (cid:20)n3 o
directions as shown in Fig. 1(b). The lattice term given +4Q+xy;kd†x2−y2;kdx2−y2;k
by Eq. (3) can be written for this case as follows: √3
− 4 Q−xy;k d†z2;kdx2−y2;k+H.c. . (11)
M (cid:16) (cid:17)(cid:21)
HCJT = [u˙2 +u˙2 +u˙2 +u˙2
l 2 x;0,k x;1,k y;0,k y;1,k Next,wenotethatthecenter-of-massdisplacementterms
+u˙2zX;kk]+ K2 [u2x;0,k+u2x;1,k+u2y;0,k tQe′xrm;ksanQ˙d′x;Qk ′ya;kndasQ˙w′ye;klloafsEthqe. c(e8n)tedr-oofn-motascsoumpolemetontuthme
electrons[ascanbeseenfromEqs. (10)and(11)]. Hence,
k
X
+u2 +u2 ]. (8) for our single chain case, Eq. (9) simplifies to be
y;1,k z;k
1 1
HCJT = Mu˙2 + Ku2
WQxe;kdefinuexQ;1,′xk;k ≡uxu;x0,;k1,,ka+nduxQ;0,yk;k, Q′y;kuy≡;1,kuy;1,uky+;0,uky;a0n,kd, l Xk (cid:20)2 z;k 2 z;k(cid:21)
≡ − ≡ − 1M 1K
incorporate these definitions in Eq. (8) to obtain + Q˙+2 + Q+2
2 4 xy;k 2 4 xy;k
k (cid:20) (cid:21)
X
HlCJT + 21M4 Q˙−xy2;k+ 21K4 Q−xy2;k . (12)
= M2 12{Q˙′x2;k+Q˙′y2;k}+ 14{Q˙+xy2;k+Q˙−xy2;k}+u˙2z;k Xk (cid:20) (cid:21)
k (cid:20) (cid:21) ThegeneralHamiltonianforthepresentsinglechainCJT
X
K 1 1 case can be expressed as follows by adding Eqs. (10),
+ 2 2{Q′x2;k+Q′y2;k}+ 4{Q+xy2;k+Q−xy2;k}+u2z;k , (11), and (12):
k (cid:20) (cid:21)
X
(9)
HCJT =HCJT +HCJT +HCJT. (13)
t ep l
where Q±xy;k ≡Qx;k±Qy;k. For the present single chain Ntaetxiotn, boyf tuhseinvgartihoeusfodllioswpliancgemseecnotndop-qeuraatnotrizse:d represen-
case, Eqs. (6) and (7) reduce to the following equations:
HtCJT =−t (d†z2;k+1dz2;k+H.c.), (10) uz;k = a√†z;k2M+aωz0;k,Q+xy;k = b†k2+Mbωk ,Q−xy;k = c†k2+Mcωk ,
k 4 0 4 0
X
q q
4
FIG. 1. (Color online) Depiction of (a) one-dimensional single-band chain with cooperative breathing mode; (b) cooperative
Jahn-Teller chain involving dz2 and dx2−y2 orbitals; and (c) interacting cooperative Jahn-Teller chains with dz2 and dx2−y2
orbitals. For simplicity, only dz2 orbitals are displayed in (b) and (c).
in the above Hamiltonian of Eq. (13), we obtain where nz2;k ≡ d†z2;kdz2;k and nx2−y2;k ≡
HCJT =−t (d†z2;k+1dz2;k+H.c.) d†x2−y2;kdx2−y2;k.
k We will now adapt the well-known Lang-Firsov
X
transformation39 for the above Hamiltonian so that we
−gω0 (a†z;k+az;k)(nz2;k−nz2;k+1) canperformperturbationin the polaronic(Lang-Firsov)
k (cid:20)
X frameofreference. ThetransformedHamiltonianisgiven
1
+ 2(b†k+bk)(nz2;k+3nx2−y2;k)
√3
− 2 (c†k+ck)(d†z2;kdx2−y2;k+H.c.)
(cid:21)
+ω0 (a†z;kaz;k+b†kbk+c†kck), (14)
k
X
5
by H˜CJT =exp(S)HCJT exp( S) where In Eq. (19), the contribution of cross terms involving
− HI and HII is zero because the phonons do not match;
1 1
S =−g [(a†z;k−az;k)(nz2;k−nz2;k+1) hence, we get
k
X
1
+ 2(b†k−bk)(nz2;k+3nx2−y2;k)]. (15) H(2) = h0|phH1I|miphhm|phH1I|0iph
Here, in our modified Lang-Firsov transformation, it Xm E0ph−Emph
should be noted that we have included only the density 0 HII m m HII 0
terms and ignored the orbital-flip terms (d†z2;kdx2 y2;k + h |ph 1 |EpihphhE|pphh 1 | iph. (20)
anditsHermitianconjugate)appearingintheinterac−tion Xm 0 − m
part of the above equation (14). This choice is dictated
by mathematical expediency to arrive at an analytic ex-
pression. Then, the Lang-Firsov transformed Hamilto- WewillfirstevaluatetheterminvolvingH1II intheabove
nian is given by H˜CJT =H +H where equation. After some algebra, we get the following ex-
0 1
pression:
H0 =ω0 (a†z;kaz;k+b†kbk+c†kck)
k
X
9 0 HII m m HII 0
−4g2ω0 (nz2;k+nx2−y2;k) h |ph 1 |EpihphhE|pphh 1 | iph
−32g2ω0Xk nz2;knx2−y2;k+2g2ω0 nz2;knz2;k+1 Xm≈−ω40 n0z2;k−+nmx2−y2;k−2nz2;knx2−y2;k .(21)
Xk Xk Xk h i
−te−143g2 (d†z2;k+1dz2;k+H.c.), (16)
k
X Wenotethatthecoefficientsofthetermsn ,n ,
z2;k x2 y2;k
where the term 2g2ω0 knz2;knz2;k+1 arises because of andnz2;knx2 y2;k intheaboveequationaremuchsm−aller
the cooperative nature of the interaction; furthermore, than the coe−fficients of the same terms in Eq. (16); con-
the attractiveinteractioPnterm 3g2ω n n sequently, we ignore the contribution from Eq. (21) in
will be negated by a much lar−ge2r rep0ulskivez2C;kouxlo2−myb2i;kc the expression for the effective Hamiltonian of the CJT
P
term U n n because of which no site can chain.
k z2;k x2 y2;k
have both the orbita−ls occupied simultaneously. The re-
mainingPterm of H˜CJT is given by H HI +HII with Next, after performing some tedious algebra (using
1 ≡ 1 1 considerations similar to those in Ref. 14), the effective
H1I =−te−143g2 [d†z2;k+1dz2;k{T+k†T−k−1}+H.c.],(17) Hamiltonian can be obtained as:
k
X
awnhdereT±k ≡exp[±g(2az;k−az;k−1−az;k+1)±g2(bk−bk+1)] HeCfJfT =−te−143g2 (d†z2;k+1dz2;k+H.c.)
k
X
H1II = √23gω0e−32g2 (c†k+ck) d†z2;kdx2−y2;k − ωt20e−123g2G3(cid:18)2,2,14(cid:19)
k (cid:20)
×eg(a†z;k−1−a†zX;k+b†k)e−g(az;k−1−az;k+bk)+H.c. .(18) ×Xk hd†z2;k−1(1−2nz2;k)dz2;k+1+H.c.i
(cid:21) t2 1 1
Now, to perform perturbation theory, we note that the +2 g2ω0+ ω e−123g2G5 4,1,1,4,4
" 0 (cid:18) (cid:19)#
eigenstatesofH aregivenby n,m = n m with
0 el ph
0,0 being the ground state.| Weicon|sidier⊗t|heistrong- nz2;knz2;k+1, (22)
| i ×
coupling case g2 >> 1 and the antiadiabatic regime k
X
t/ω < 1; consequently, the coefficients of the pertur-
0
bation terms HI and HII in Eqs. (17) and (18), re-
spectively, satisfy the conditions te−143g2 << ω0 and where
√23gω0e−32g2 << ω0. Now, the second-order perturba-
tionterm[obtainedusingSchrieffer-Wolfftransformation
1 1 1
as mentioned in Eq. (6) of Ref. 14] is expressed as
G 2,2, F 2,2, +F (2,2)+2F 2,
3 3 2 2
4 ≡ 4 4
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
H(2) = h0|phH1|miphhm|phH1|0iph. (19) +2F (2)+F 1 , (23)
Xm E0ph−Emph 1 1(cid:18)4(cid:19)
6
and effective Hamiltonian14:
1 1 HCBM
G 4,1,1, , eff
5(cid:18) 4 14(cid:19)1 1 1 1 =−te−3g2 (d†z2;k+1dz2;k+H.c.)
≡F5 4,1,1,4,4 +2F4 4,1,1,4 +2F4 4,1,4,4 t2 Xk
(cid:18) 1 1 (cid:19) (cid:18) (cid:19) 1(cid:18) (cid:19) −ω e−6g2G2(2,2) d†z2;k 1(1−2nz2;k)dz2;k+1+H.c.
+F4 1,1,4,4 +F3(4,1,1)+4F3 4,1,4 0 Xk h − i
+F3(cid:18)4,41,14 (cid:19)+2F3 1,1,41 +2F3(cid:18) 1,14,(cid:19)14 +2(cid:20)g2ω0+ ωt20e−6g2G3(4,1,1)(cid:21)Xk nz2;knz2;k+1. (27)
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
1 1 OnusingtherelationshipofEq. (25),theeffectiveHamil-
+2F (4,1)+2F 4, +F (1,1)+4F 1,
2 2 4 2 2 4 tonian for cooperative-breathing-mode chain [given by
(cid:18) (cid:19) (cid:18) (cid:19)
Eq. (27)] simplifies to be
1 1 1
+F , +F (4)+2F (1)+2F , (24)
2 1 1 1
4 4 4 HCBM
(cid:18) (cid:19) (cid:18) (cid:19) eff
te 3g2
with −
≈− (d†z2;k+1dz2;k+H.c.)
∞ ∞ (α1g2)m1...(αng2)mn Xk
Fn(α1,...,αn)≡ ... m !...m !(m +...+m ). teg2
mX1=1 mXn=1 1 n 1 n −4g2ω d†z2;k 1(1−2nz2;k)dz2;k+1+H.c.
0 −
Xk h i
AexpgreenseseradlatserGmn(oαf1t,hαe2,f.o.r.m,αGn)n(=α1F,nα(2α,1..,.α,2α,n..).,cαann)b+e + 2g2tω0 + 3g2tω e3g2 nz2;knz2;k+1. (28)
kn=−11 cFk(αc1,αc2,...,αck) where the summation (cid:20) 0(cid:21) Xk
over c represents summing over all possible nC com-
m ComparingEq.(26)withEq.(28),wefindthattheco-
P P
binations of m arguments chosen from the total set of n
efficients of next-nearest-neighbor(NNN) hopping terms
arguments α ,α ,...,α . We then obtain the follow-
{ 1 2 n} areapproximatelyequalandsoarethecoefficientsofthe
ing useful relationship (derived in Appendix A):
nearest-neighbor(NN) repulsion terms. Hence, both the
CBM chain and the CJT chain should exhibit similar
Gn(α1,α2,...,αn) behavior up to half-filling. We diagonalize these effec-
n 1 tive Hamiltonians using a modified Lanczos algorithm40
−
=Fn(α1,α2,...,αn)+ Fk(αc1,αc2,...,αck) [with antiperiodic (periodic) boundary conditions for
k=1 c even (odd) number of fermions] and calculate the struc-
XX
= ePni=1αig2 −1dg2 = ∞ ( ni=1αig2)m turefactorS(k)attheorderingwavevectorπ. Upontun-
g2 mm! ing the electron-phonon coupling g in the antiadiabatic
Z mX=1 P regime,earlier we found that the CBM model undergoes
forg2>>1 ePni=1αig2 . (25) a second-order quantum phase transition from a Lut-
≈ n α g2 tingerliquid to a CDW state atstrongcoupling14. From
i=1 i
Fig. 2, for both the CBM model and the CJT model, we
Then, on Pusing the above approximation for seethattheS(π)curvescoincidemoreorless. Hence,up
G (α ,α ,...,α ) at large g2, Eq. (22) simplifies tohalffilling,boththemodelsexhibitsimilarCDWtran-
n 1 2 n
as follows: sitionupontuningelectron-phononcouplingg. However,
above half filling in the CJT chain, particles will occupy
HCJT the d orbitals of the remaining sub-lattice because
eff x2 y2
te−143g2 there is−no repulsion between electrons in the dz2 and
d orbitals on NN sites [see Eq. (16)]. Thus, in con-
≈− (d†z2;k+1dz2;k+H.c.) trxa2s−ty2to the CBM model, the particle-hole symmetry is
k
X broken for the CJT model!
4 teg2
In the subsequent sections, we shall study interacting
−17g2ω0 d†z2;k−1(1−2nz2;k)dz2;k+1+H.c. CJT chains at couplings that are strong and weak.
Xk h i
2g2ω 4 t
+ t 0 + 13g2ω e143g2 nz2;knz2;k+1. (26)
(cid:20) 0(cid:21) k IV. INTERACTING IDENTICALLY-LONG CJT
X
CHAINS AT STRONG COUPLING
In contrast to the effective Hamiltonian of Eq. (22),
the relatively simpler case of single-band cooperative In the interacting CJT chains depicted in Fig. 1(c),
breathing mode [shown in Fig. 1(a)] yields the following we have C-type antiferromagnetism (i.e., each ferromag-
7
4 8 and
CBM chain CBM chain
CJT chain CJT chain
3.5 7 HICJT = gω 2Mω (u u )
ep − 0 0 j,k "(cid:26) z;j,k− z;j,k−1
6 p X
3
1
πS() 2.5 5 + 4(Qx;j,k+uy;j,k−uy;j−1,k)(cid:27)d†z2;j,kdz2;j,k#.(32)
4 Next, we observe that the displacement operator Q
2 ′x;j,k
does not couple to the electrons in the above Eqs. (31)
3 and (32); therefore, we drop terms involving Q from
′x;j,k
1.5 Eq. (30). Then,thetotalHamiltonianfortheinteraction
2 CJT chains is given by
(a) (b)
1
1.5 2 g 2.5 3 1.5 2 g 2.5 3 HICJT =HICJT +HICJT +HICJT. (33)
t ep l
FIG. 2. (Color online) Structure factor S(π) showing CDW
transition in CBM chain and CJT chain when t =0.1 and Now,representingtherelevantdisplacementoperatorsin
ω0 second quantized form as
for (a)N =16, Np =4; and (b)N =18, Np =6.
a†z;j,k+az;j,k b†y;j,k+by;j,k
u = , u = ,
netic chain is antiferromagnetically coupled to the adja- z;j,k √2Mω y;j,k √2Mω
0 0
centones);consequently,thereisnointer-chainhopping.
Inthesechains,hoppingofelectronsoccursonlybetween Q = c†x;j,k+cx;j,k, (34)
x;j,k
thedz2 orbitalsalongthez-axis. Furthermore,thechains 2M2 ω0
are assumed to have cooperative electron-phonon cou-
q
pling along the y- and the z-axes, whereas, along the
and substituting in the total Hamiltonian of Eq. (33),
x-axis the electron-phonon coupling is non-cooperative
we obtain
and is of the Holstein-type4,5. Up to half filling, only
the d orbitals would be occupied so that the system
lowersz2its energy through the mobile electrons. Here, we HICJT =−t (d†z2;j,k+1dz2;j,k+H.c.)
consider the interacting chains for only up to half filling; Xj,k
therefore,weneglecttheoccupationofd orbitalsin
our analysis. Hence, the lattice term ofx2E−qy.2(3) for the −gω0 (a†z;j,k+az;j,k)(nz2;j,k−nz2;j,k+1)
j,k (cid:20)
present case reduces to X
1
M + 4(b†y;j,k+by;j,k)(nz2;j,k−nz2;j+1,k)
HICJT = [u˙2 +u˙2 +u˙2 +u˙2 ]
l 2 x;0,j,k x;1,j,k y;j,k z;j,k 1
Xj,k + 2√2(c†x;j,k+cx;j,k)nz2;j,k
K (cid:21)
+ 2 [u2x;0,j,k+u2x;1,j,k+u2y;j,k+u2z;j,k].(29) +ω0 (a†z;j,kaz;j,k+b†y;j,kby;j,k+c†x;j,kcx;j,k),
Xj,k Xj,k
(35)
WedefineQ u +u andQ u
′x;j,k ≡ x;1,j,k x;0,j,k x;j,k ≡ x;1,j,k−
u and recast Eq. (29) as follows:
x;0,j,k
where nz2;j,k ≡ d†z2;j,kdz2;j,k. To obtain an effec-
HICJT tive Hamiltonian by performing perturbation theory, we
l device a relevant Lang-Firsov transformation for the
M 1
= 2 Xj,k (cid:20)2nQ˙′x2;j,k+Q˙2x;j,ko+u˙2y;j,k+u˙2z;j,k(cid:21) aenbcoev,eH˜HIaCmJTilto=nieaxnp.(SI)nHItChJeTpeoxlpa(r−onSic) wfrhaemree tohferaenfetri--
Hermitian operator S is identified as
K 1
+ Q2 +Q2 +u2 +u2 .(30)
2 2 ′x;j,k x;j,k y;j,k z;j,k
j,k (cid:20) (cid:21)
X (cid:8) (cid:9) S =−g (a†z;j,k−az;j,k)(nz2;j,k−nz2;j,k+1)
For the present system of interacting identically-long j,k (cid:20)
X
CJT chains, Eqs. (6) and (7) simplify to be 1
+ 4(b†y;j,k−by;j,k)(nz2;j,k−nz2;j+1,k)
HtICJT =−tXj,k (cid:16)d†z2;j,k+1dz2;j,k+H.c.(cid:17), (31) + 2√12(c†x;j,k−cx;j,k)nz2;j,k(cid:21). (36)
8
Subsequently, the Lang-Firsov transformed Hamiltonian perturbation HICJT is given by
1
is given by H˜ICJT =HICJT +HICJT with
0 1
H1ICJT =−te−143g2 d†z2;j,k+1dz2;j,k{T+j,k†T−j,k−1}
j,k (cid:20)
X
+H.c. , (38)
(cid:21)
H0ICJT = ω0 (a†z;j,kaz;j,k+b†y;j,kby;j,k+c†x;j,kcx;j,k) where
Xj,k j,k exp g(2a a a )
− 94g2ω0 nz2;j,k T± ≡ g(hb± z;j,+k−b z;j,k−b1− z;j,k+b1 )
Xj,k ±4 y;j−1,k+1 y;j,k− y;j−1,k− y;j,k+1
1 g
+g2ω0 2nz2;j,knz2;j,k+1+ 8nz2;j,knz2;j+1,k ±2√2(cx;j,k−cx;j,k+1) . (39)
Xj,k (cid:16) (cid:17) i
−te−143g2 j,k(d†z2;j,k+1dz2;j,k+H.c.), (37) Swiencgeetwtehceocnosniddeirtiosntrotne−g14c3ogu2p<lin<gωa0n.dTanhteina,disaimbailtaicrittyo,
X
the single chain CJT case [on identifying the eigenstates
of HICJT as n,m = n m with 0,0 being
0 | i | iel ⊗ | iph | i
the ground state], the second-order perturbation term
is given as:
0 HICJT m m HICJT 0
wg2hωe0re ntermsn 2g2ω0arisje,knbze2c;ja,kunsze2;j,okf+1coopearnad- H(2) =Xm h |ph 1 |E0pihph−hE|mpphh 1 | iph.(40)
8 j,k z2;j,k z2;j+1,k P
tive nature of the Jahn-Teller interaction along the z- Aftertediousalgebra,wegetthethefollowingexpression
P
and the y-directions, respectively. Furthermore, the for the effective Hamiltonian for interacting CJT chains:
HeIfCfJT =−te−143g2 (d†z2;j,k+1dz2;j,k+H.c.)
j,k
X
t2 1 1 1
−ω0e−123g2G5(cid:18)2,2,8,16,16(cid:19)Xj,k hd†z2;j,k−1(1−2nz2;j,k)(1−nz2;j−1,k)(1−nz2;j+1,k)dz2;j,k+1+H.c.i
t2 1 1 1 1 1 1 1
+2 g2ω0+ ω e−123g2G9 4,1,1,8,8,16,16,16,16 nz2;j,knz2;j,k+1+ 8g2ω0 nz2;j,knz2;j+1,k,(41)
" 0 (cid:18) (cid:19)# j,k j,k
X X
where G (α ,α ,...,α ) are the same as those defined in the previous section. For g2 >>1 (basedon derivations in
n 1 2 n
the Appendix A), the above equation reduces to
HICJT
te−ef14f3g2 =− j,k(d†z2;j,k+1dz2;j,k+H.c.)
X
4 teg2
− 17g2ω0 d†z2;j,k−1(1−2nz2;j,k)(1−nz2;j−1,k)(1−nz2;j+1,k)dz2;j,k+1+H.c.
Xj,k h i
2g2ω 4 t 1g2ω
+ t 0 + 13g2ω e143g2 nz2;j,knz2;j,k+1+ 8 t 0e143g2 nz2;j,knz2;j+1,k, (42)
(cid:20) 0(cid:21) j,k j,k
X X
which is exactly the same as the single chain CJT re- factors (1 n ) [appearing in the NNN hopping
z2;j 1,k
sult [see Eq. (26)] except for the inter-chain repulsion term] arise−due to±the fact that the reduced hopping in-
due to cooperative effects along the y-axis. Further- tegralte−143g2 issignificantlysmallerthantheinter-chain
more, in the above equation, the occupancy-projection repulsion strength g2ω0.
8
9
4
0.5 5
0.2
0.25 4 3
W(1,m) 0 π)S(,kz 23 W(1,m) 0 π)S(,kz 2
-0.25 1
1 -0.2
(a) (c) (a) (c)
-0.5
0 0
1
0.75 g = 1.30 (b) (d) g = 1.30 (b) (d)
0.6 1
g = 2.47 0.8 g = 2.61
0.5 g = 3.40 g = 3.40 0.8
W(0,m) 0.2 05 S(0,k)z 00..46 W(0,m) 0.3 S(0,k)z 0.6
0.4
0
-0.25 0.2 0.2
-0.5 0 -0.3 0
0 1 2 3 0 1 2 3 0 1 2 3 4 0 1 2 3
m k m k
z z
FIG.3. (Coloronline)Plotsof(a)density-densityinter-chain FIG. 4. (Color online) Plots of (a) inter-chain correla-
correlation function W(1,m); (b) density-density intra-chain tion function W(1,m); (b) intra-chain correlation function
correlation function W(0,m); (c) structure factor S(π,kz); W(0,m); (c) structurefactor S(π,kz); and (d) structure fac-
and (d) structure factor S(0,kz) evaluated at adiabaticity tor S(0,kz) at t/ω0 =0.1, N =16 and Np =4.
t/ω0 =0.1, system size N =12 and particle numberNp =4.
0.2 3
Although in the present work we focus on interacting
0.1
cionhbtateairnianscetidninsgtiwmCoiJ-ladTrimlyc.ehnasiniosnsinonthlyr,eee-ffdeicmtievnesiHonasmcilatnonbiaenaflosor W(1,m) 0 πS(,k)z 2
1
Using the modified Lanczos algorithm of Ref. 40, in
-0.1
order to study the nature of the system as the electron- (a) (c)
phonon coupling g is tuned, we diagonalizethe Hamilto- 0
nianinEq.(41)andcalculatequantitiessuchasdensity- 0.6 g = 1.30 (b) 1 (d)
density correlation function, structure factor, fidelity,
g = 2.71
and fidelity susceptibility in the following subsections. 0.4 g = 3.40 0.8
FCuJrTthcehraminosrei,satatkweon-dtoimbeensmioimnaiclkseydstbemy awtiwtho-icnhtaeirnacCtiJnTg W(0,m) 0.2 S(0,k)z 00..46
system with periodic boundary conditions. Although
0
we perform our calculations for a conservative value of 0.2
t/ω =0.1,ourresultsarevalidforfairlylargervaluesof
0 -0.2 0
t/ω0 <1 as demonstrated in the Appendix B. 0 1 2 3 4 5 0 1 2 3
m k
z
FIG. 5. (Color online) Plots of (a) correlation function
A. Density-density correlation function, structure
W(1,m); (b)correlation function W(0,m);(c) structure fac-
factor, and order parameter
tor S(π,kz); and (d) structure factor S(0,kz) at t/ω0 = 0.1,
N =20 and Np =4.
The two-point correlation function, for density fluctu-
ations of electrons that are apart by a vector (l,m), is
of W(l,m), i.e., the structure factor S(k ,k ), is given
defined as y z
by
4
W(l,m)= N [hnz2;j,knz2;j+l,k+mi S(ky,kz)= ei(kyl+kzm)W(l,m), (44)
j,k
X l,m
n n ], (43) X
z2;j,k z2;j+l,k+m
−h ih i where for our two-chain CJT system, on assuming pe-
ftoortaalnfiullminbgerfroafctsiiotensνin=thhenszy2s;jt,ekmi =andNNpN,pwthheerteoNtalinsutmhe- nriyod=ic1,b2o;uannddarkyzc=on2dnNiztπion=s,4nkNzyπ =with2n2nyπz ==1,n2y,π...w,iN2th.
2
beroftheparticlesinthesystem. TheFouriertransform We consideronlyevenvaluesofN/2soastoobtainsub-
10
0.2 1
N=12, N = 4
p
0.9 N=16, N = 4
p
2 N=20, N = 4
0.1 0.8 p
N=24, N = 4
W(1,m) 0 π)S(,kz 1 π) 00..67 p
π, 0.5
*S( 0.4
-0.1 (a) (c)
0.3
0
1 0.2
0.5 g = 1.30 (b) (d)
0.1
0.4 g = 2.80 0.8
g = 3.40 0
W(0,m) 000...123 S(0,k)z 00..46 1.5 2 g 2.5 3
∗
0 0.2 FIG.8. (Coloronline)OrderparameterS (π,π)asafunction
-0.1 of the coupling g when t/ω0 =0.1.
0
0 1 2 3 4 5 6 0 1 2 3
m k
z
To begin with, we observe that
FIG. 6. (Color online) Display of (a) two-point correla-
tion function W(1,m); (b) two-point correlation function
S(π,π)= [ W(1,m)+W(0,m)].
W(0,m);(c) structurefactor S(π,kz); and (d) structure fac- − ! −
tor S(0,kz) at t/ω0=0.1, N =24 and Np =4. mXeven mXodd
Basedonthe derivationofEq.(C11)inthe Appendix C,
we can write
6
4
0.2 5 S(π,π)= [(Nˆ Nˆ )+(Nˆ Nˆ )]2 , (46)
1,e 1,o 2,o 2,e
Nh − − i
4
W(1,m) 0 π)S(,kz 23 wnuhmerbeeNrˆjo,pee=ratorkefvoenrnthz2e;jt,kot(cid:16)alNˆnju,om=ber okofdpdanrzt2i;cjl,eks(cid:17)iinstthhee
P P
even (odd) sites of the j-th chain. Next, again from Eq.
-0.2 1
(a) (c) (45), we find
0
g = 1.30 (b) 1.2 (d)
0.6 S(0,π)= [W(1,m)+W(2,m)].
g = 2.60 1 − !
W(0,m) 00..24 g = 3.40 S(0,k)z 00..68 AC,sSsh(o0w,πn)bcyanthbeemdXeeevxreinpvraetsmiXsoeondddoafsEq.(C12)intheAppendix
0 0.4
0.2 4
-0.2 S(0,π)= [(Nˆ Nˆ )+(Nˆ Nˆ )]2 . (47)
1,e 1,o 2,e 2,o
0 Nh − − i
0 2 4 6 0 1 2 3
m kz Furthermore, Eq. (45) also yields
FIG. 7. (Color online) Depiction of (a) correlation function
W(1,m);(b) correlation function W(0,m);(c) structurefac- S(π,0)= + [ W(1,m)+W(2,m)].
tNor=S2(π4,aknzd);Nanpd=(6d.) structure factor S(0,kz) at t/ω0 = 0.1, mXeven mXodd! −
Then, from Eq. (C13) in the Appendix C, we observe
that
lattice structure. We will now derive exact relations for S(π,0)= 4 [(Nˆ +Nˆ ) (Nˆ +Nˆ )]2 . (48)
the structure factor in a few special cases. To this end, Nh 1,e 1,o − 2,e 2,o i
the above equation is simplified to be
Since hopping is permitted only along the chain, the
particles will distribute themselves equally between the
4πn N/2 4n πm chains in order to minimize the energy. Hence, for the
S nyπ, z = cos z [( 1)nyW(1,m) full span of g
N N −
(cid:18) (cid:19) m=1 (cid:18) (cid:19)
X
+W(0,m)].(45) S(π,0)=0, (49)