Fractionnally charged excitations in the charge density wave state of a quarter-filled t-J chain with quantum phonons

4 Pages·0.17 MB·English

by  Philippe Maurel

Fractionnally charged excitations in the charge density wave state of a quarter-filled
t-J chain with quantum phonons
Philippe Maurela, Marie-Bernadette Lepetita and Didier Poilblanca,b
aLaboratoire de Physique Quantique & UMR–CNRS 5626, Universit´e Paul Sabatier, F-31062 Toulouse, France
bTheoretische Physik, ETH-H¨onggerberg, CH-8093 Zu¨rich
(February 1, 2008)
1
0 Elementary excitations of the 4kF charge density wave state of a quarter-filled strongly correlated
electronic one-dimensional chain are investigated in the presence of dispersionless quantum optical
0
2 phonons using Density Matrix Renormalization Group techniques. Such excitations are shown to
betopologicalsolitonscarryingchargee/2andspinzero. Relevancetothe4kF chargedensitywave
n instabilityin(DI−DCNQI)2Agorrecentlydiscoveredin(TMTTF)2X(X=PF6,AsF6)isdiscussed.
a
J PACS: 75.10.-b, 75.50.Ee, 71.27.+a, 75.40.Mg

5
2
Itiswellknowthatone-dimensional(1D)Su-Shrieffer- kF =π/4,A4kF is the magnitude andφ the phase of the
] Heeger(SSH)1 orHubbard-SSHmodelsexhibitexoticel- chargeoscillation. Hence,duetotheequivalencebetween
l
e ementary excitations including neutral soliton with spin theevenandoddsites,theGSistwofolddegenerate(φ=
r- 21, chargedsoliton with spin zero (21-filled band)2 as well 0 and φ=π). A solitonic excitation can be described as
st as fractionally charged soliton (1- and 1-filled band)3,4. a state which interpolates between the two different GS
3 4
t. In these models, the phonons couple to the electrons via patternswithaslowlymonotonicallyvaryingphaseφ(ri)
a inter-site interactions which lead to an insulating Bond fromlet’ssay0atri →−∞toπwhenri →+∞. Simple
m
OrderWavegroundstate(GS).Infact,suchsolitonicex- countingargumentsshow,infact,thatsuchanexcitation
- citations are also generic in commensurate site-centered carriesachargeQ=±e and,therefore,canbegenerated
d 2
Charge Density Wave (CDW) states and, hence, should by doping the commensurate CDW GS.
n
o also exist (in the vicinity of commensurate fillings) in In this paper, we investigate the role of quantum lo-
c the case of strong short range electronic repulsion lead- cal phononic (optical) modes on the formation and on
[ ing to commensurate 4k charge instability. In addi- the stability of the solitonic excitations occuring in an
F
2 tion to strong electron-electron correlation, local on-site insulating 4kF CDW phase of a quarter-filled strongly
v electron-phononcoupling(tobecomparedwiththeinter- correlatedelectronic chain. This issue is ofparticularin-
8 sitevibrationintheSSHmodel)isofparticularrelevance terest since a coupling to local phonons might affect the
5 in systems where the “site” represents a complex struc- physics of the solitons (such as its width, its interaction,
3 ture with internal degrees of freedom. Molecular crys- etc...). NumericalresultswillbeobtainedbytheDensity
1 tals such as the quasi-1D charge transfer salts5 present MatrixRenormalizationGroup(DMRG)methodapplied
1
this type of characteristic. Interplay between electron- to open or cyclic chain segments carrying no, a single or
0
electronandelectron-phononinteractionsprovidesavery two solitonic excitations.
0
/ rich physics. For example, severalsystems have been re- The following analysis is based on the 1D t−J −V-
at cently observedto presenttransitionstowardschargeor- Holstein model at quarter filling. This model describ-
m deredphaseswherethemoleculesoftheconductingstack ingstronglycorrelatedelectronscoupledtodispersionless
- supportunequalelectrondensities,andinsomecasesas- phonons can be written as H =He+He−ph with
d sociated relaxation of their interal geometry. This is for
on instance the case for the most strongly 1D system of the He =tXc†i+1,σci,σ+h.c.+JXSi·Si+1+V Xnini+1
c Bechgaard-Fabre salts familly, namely (TMTTF)2PF6 i,σ i i
: and (TMTTF)2AsF6; below the Mott localisation tem- H =g n b†+b +ω b†b +1/2
v perature T evidences for an additional transition to- e−ph X i(cid:16) i i(cid:17) X(cid:16) i i (cid:17)
ρ
i i i
X wards a 4kF (site-centered) CDW state have been re-
ar NceMntRly7 parnodvidaendombayloduiselteecmtrpicerraetsuproensdeepmenedaseunrceemoefnttsh6e, where c†i,σ, ci,σ are projected creation and annihilation
operators of electrons of spin σ at sites i (doubly occu-
X-ray Bragg peaks8. Similar transitions have been seen
pied sites are projected out, the strong correlation limit
in (BEDT−TTF)2X9 as well as in (DI−DCNQI)2Ag10 is therefore assumed), n is the electron number and S
i i
which exhibits below 220K a 4kF CDW associated with is the spin operator at site i. b† and b are the local
geometry modulations of the DI−DCNQI molecules. i i
phononscreationandannihilationoperators. Theenergy
Apictorialdescriptionofasolitonicstatecanbesimply
scale is fixed by t = 1. Note that the phononic part can
given assuming e.g. a quarter-filled strongly correlated
chain, in a 4kF CDW state, provided a doubling of the be re-written as ωh(cid:16)b†i +niωg(cid:17)(cid:0)bi+niωg(cid:1)i (apart from
unit cell. In that case,the GS chargemodulation canbe constant terms) showing that the coupling of the on-site
parametrized as hnii = 12 +A4kFcos(4kFri +φ) where vibrations to the electrons induces displacements of the
1
oscillator proportional to the site charge. In fact, this 44
ω=0.1
term mimics the relaxationof the internalgeometry of a saturated
site as a function of its ionicity. 33..55 4k−CDW <1
adiabatic F
Before proceeding further, it is interesting to examine limit <1
the adiabaticlimitω →0ofourmodel. Inthatcase,ab- 33 <1
sorbing the electron-phonon coupling g in the definition 0.7 <1 ω=0.2
tohfethpeh(ocnloanssiiccapla)rotnt-askiteesdthisepflaocrmemoefnatic.lea.ssgi(cba†il+elbais)ti→ceδni-, 22..55 00..52 1.6 <11.4 <1 <14k<−1CDW <1 ω=0.3
ergy 1K δ2. The magnitude of the electron-phonon g/ω 22 F
2 Pi i 0.01 1.4
coupling is then given by a single parameter namely the
inverse lattice stiffness K−1 = 2g2/ω. Hence, the adi- 11..55 ω=0.5
abatic limit is reached assuming the following limits; ∆
ω →0,g →0andK−1 →cst. Thephasediagramofthis 11 ρ
model has been investigated recently by Riera and Poil-
U
blanc12. It is well know that a quarter-filled infinite-U 00..55
(i.e. J =0) model exhibits a 4k CDW (Mott-Hubbard
F
like)instability only when the nearestneighbor(NN) re-
00
pulsionV exceeds211. Thisinstabilityisinfactenhanced 00 0.5 11 1.5 22 2.5 33 3.5
by the lattice (ω =0) coupling and the 4kF CDW phase K−1=2g2/ω
becomes stable even when V <2 (and J finite) for K−1
exceeding a V-dependant critical value12.
The numerical study of the model with quantum FIG. 1. Schematic phase diagram of a 14-filled t-J chain
phonons using the infinite system DMRG method13 re- for J = 0.3 and V = 1 as a function of K−1 = 2g2 and
t t ω
quires an approximate (but reliable) treatment of the g. Shaded regions correspond to the insulating CDW phase
phonondegreesoffreedom14–16. Indeed,aninfinitenum- wωhile the uniform phase is labelled by U. The darker region
ber ofphononic quantumstates lives oneachsite. In or- correspondstoanearlyfullysaturatedphasei.e. A4kF ≃1/2.
der to render the calculations feasible, the basis set has Thesolid long dashed line correspond toiso-gaps curves,the
been truncated on each site to the two lowest vibronic dot-dashed line correspond to iso-frequencies curves. The
states. This choice is physically reasonable as long as numberscorrespond to thewidths of the solitons ξ.
the frequency ω is nottoo smallsince only the lowestvi-
bronic states are expected to be involved16. In all cases,
in this case, the truncation of the phonon basis is no
we kept m = 216 states per renormalized block. We
more adequate and more phonon states are expected to
have chosen parameters like V = 1 and J = 0.3 which beexcited. Indeed,K−1 ≃1.1obtainedintheadiabatic
aregenericforstronglycorrelated1Dmaterials. Forsuch crit
approximation12 does not seem to appear as an asymp-
parameters,theadiabaticgroundstateisa4k CDWfor
K−1 > K−1 ∼ 1.1. Ground states and solitFonic states totic limit for the metal-insulatorboundary when ω →0
crit and g/ω → +∞. Within our treatment of the phonons,
of the system have been investigated as a function of ω K−1 tends towards zero, which seems inconsistent with
and K−1. crit
the finite valueK−1 ≃1.1 obtainedin the adiabatic ap-
In order to determine the phase diagram at quarter- crit
proximation12 for the same J and V values. Therefore,
filling, we have computed the charge gap ∆ =
ρ
weshallrestrictinthefollowinganalysistoω >0.1where
E0(2N,N + 1) + E0(2N,N − 1) − 2E0(2N,N) (where
we expect our results to be fully reliable.
E0(Ns,Ne) is the ground state energy of Ne electrons
Letusnowdiscusstheeffectofthefrequencyωatfixed
on N sites). For that purpose, we have used open
s K−1: for values of K−1 such as the adiabatic ground
boundary conditions (OBC), systems size up to 50 sites
state is in the insulating phase, we found that, by in-
andextrapolatedtheresultstothethermodynamiclimit.
creasingω, the system becomes metallic. As one can ex-
Note that OBC are used in gap calculations because
pect, increasing phonon quantum fluctuations suppress
the DMRG method performs better in this case than
long-range CDW order. The opening of the charge gap
with periodic boundary conditions (PBC). In addition,
characteristic of the metal-insulator transition is fully
we have also calculated the charge correlation function
consistent with the formation of the CDW as shown in
cn(j)=h(ni−hni)(ni+j +hni)i where hni=Ne/Ns.
figure 2 and in figure 1 where the iso-gap curves are re-
Figure1showstheschematicphasediagramasafunc-
tion of g/ω and K−1 exhibiting a 4k CDW insulating ported. At intermediate frequencies (let’s say ω ≥ 0.3),
F
weobserveasmoothopeningofthegap(exponentiallike)
phase and a uniform metallic phase (U). The insulat-
with a saturationfor largevalue of K−1, whereasfor de-
ing 4k CDW phase was characterized both by an ex-
F
creasingfrequenciesthetransitionseemstobecomemore
trapolated finite charge gap and long range staggered
abrupt. Thisbehaviorisinfactcompatiblewiththefirst
charge correlations associated to the finite order param-
eter (−1)jc (j). Special care is needed for ω → 0 since, ordercharacterofthe metal-insulator transitions seenin
n the adiabatic limit12. In contrast, our calculation at fi-
2
0.4
nite phononfrequency wouldrather be consistentwith a (a) 2g2/ω=1.5 g/ω=1.58 ω=0.3
0.2
Kosterlitz-Thouless type of gap opening.
0.8 C(j)ne 0
ω=0.1 −0.2
ω=0.2 −0.4 0.4
ω=0.3 (b) 2g2/ω=2 g/ω=1.82
0.6 ω=0.5 0.2
0 C(j)ne
−0.2
∆
ρ 0.4
0.4 (c) 2g2/ω=2.5 g/ω=2.04 −0.4
0.2
0.2 C(j)ne 0
−0.2
−0.4
0 0 5 10 15 20 25 30 35 40 45
0 0.5 1 1.5 2 2.5 3 3.5 4 j
K−1=2g2/ω
FIG. 3. Charge correlation function of an odd chain
FIG. 2. Charge gap versus K−1 = 2ωg2 for different fre- Ns = 43, Ne = 22 for several couplings g. For convenience
quencies (as indicated on theplot). data of the same (opposite) sign as (−1)j are shown as full
(open) symbols. For clarity, the soliton has been shifted to
By doping (e.g. in electrons) the chain away from the thecenter of thechain.
commensurate density of hni = 1/2 one can introduce
chargedsoliton-antisolitonpairs. Note that solitons nat-
solitonistotallyconfinedto,letsay,asinglesite. Onthe
urally appear in pairssince they are intrinsic topological
contrary, in the uniform phase, strictly speaking A ≃ 0
excitations. However, in a finite chain, it is possible to
and the charge correlation function decays as a power
enforce the existence of a single soliton in the GS by as-
law. It has been argued that, although phononic quan-
suminganoddnumberofsites. Forthispurpose,weshall
tum fluctuations are present, such state still belongs to
deal with odd-length chain with N = 2N +1 sites and
s the LuttingerLiquiduniversalityclass14,16. Inthatcase,
N = N +1 electrons (typically we choose N = 2p+1)
e the extra Q = +e charge is totally spread over the full
andPBC.Chains withsize up to 43sites havebeen con- 2
chain. Around the (infinite size) phase transition line
sidered. On the other hand, even-length periodic chains
between the uniform and CDW phases, the soliton will
with N = 2N sites (typically choosing N = 2p + 1)
s
appearspreadoutoverthe entirefinite systemwhenever
doped with 1 extra electron (N = N +1) of size up to
e
its size (in the infinite systemlimit) becomes largerthan
42 sites have been considered to study the behavior of a
the actual system size.
soliton-antisoliton pair.
In the CDW phase, the DMRG procedure introduces
The charge-charge correlation in an odd-length chain
(despiteofthePBC)asmalltranslationsymmetrybreak-
carryingasinglesolitonisshowninFigures3(a-c)fordif-
ing(containedintheinitialstate)whichleadstoalocal-
ferent values of the parameter g/ω and fixed frequency
ω = 0.3. For increasing electron-phonon coupling (or ization of the soliton around some arbitrary location x0
equivalently in this case for increasing K−1), the sys- along the chain. This indicates that, in real materials,
this type of excitation could very easily get pinned by
tem evolves from a delocalized state with no soliton (cf.
impuritiesordefects. Howeverwestillexpectthe soliton
fig.3a)toastatewithasolitonconfinedonasmallnum-
to be mobile in a perfectly pure system.
ber of sites. (cf. fig. 3c). Figure 3b shows the interme-
So far, we have imposed the presence of a single soli-
diate regime where a soliton exists and spreads over a
ton in the GS by a geometricalmean. However,in order
large number of sites (in fact over all the 43 sites of the
tofully provethe stabilityofsuchexcitationsoneshould
largestchainconsideredinthiswork). Infact,asolitonic
also consider a situation where at least two of them can
excitation becomes stable and acquires a width and an
scatter with each other. For this purpuse, a full extra
amplitude at the phase transition point where the CDW
charge Q = +e has been added to a cyclic ring with an
order parameter starts to grow. This width decreases
and the amplitude increases as K−1 increases up to sat- even number of site on top of the CDW GS. The charge
correlation function shown in figure 4a indicates that a
uration(one inter-sitedistance for the width and1/2for
solitonandanantisolitonwellseparatedfromeachother
the amplitude). In order to estimate the width of the
appear. For a fixed system size and different runs, we
soliton,onecanfitthestaggeredchargecorrelationfunc-
tion with a usual solitonic function Atanh(x−x0) where have found that the location of the soliton-antisoliton
ξ “center of mass” is arbitrary, but the calculated mean
A is the long-range CDW amplitude, ξ is the width of
soliton (reported on figure 1) and x0 is the location of distance ds−s¯ between the two solitons stays the same.
the center of the soliton. When the gap is saturated the ForincreasingsystemsizeNs,thedistanceds−s¯increases
3
0.4 a site-centered 2k CDW state19. Recent calculations20
(a) F
suggestthat,inthe caseofacoexisting2k CDWorder,
0.2 F
C(j)ne 0 ds−s two charge 2e solitons would bind.
−0.2
−0.4 0.4
ω=0.2 (b)
2g2/ω=2.25 0.2
0 ZS>j 1W.P.Su,J.R.SchriefferandA.J.Heeger,Phys.Rev.Lett.
< 42, 1698 (1979); Phys.Rev.B 22, 2099 (1980).
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j
(1982).
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ponent (cid:10)SiZ(cid:11) (b) for an even chain with Ns = 38, Ne = 20, Lett. 58, 2134 (1987).
frequency ω = 0.2 and 2g2/ω = 2.25. For convenience, the 5T. Ishiguro, K. Yamaji, and G. Saito, in Organic Super-
twosetsofdatacorrespondingtothetwosublattices(“even” conductors, 2nd ed., Springer Series in Solid-State Sci-
and “odd” sites) are shown separately as full and open sym- ences, Vol. 88 (Springer-Verlag, Berlin, 1998); C. Bour-
bols. bonnais and D. Jerome, in Advances in Synthetic Metals,
TwentyYearsofProgressinScienceandTechnology,ed.by
P. Bernier, S. Lefrant, and G. Bidan (Elvesier, New York,
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pendentsolitonsandanti-solitonsarestableintheCDW 6F. Nad et al., Phys. Rev. B 62, 1753 (2000); ibid
phaseanddo notbind intochargee quasiparticles. Note J. Phys. Condens. Matter 12, L435-L440 (2000).
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sured” by integrating the excess charge over the soliton 8S.Ravy,privatecommunication.
width. It is alsointeresting to notice that previousstud- 9R. Chiba, H. Yamamoto, K. Hiraki, T. Takahashi and T.
iesonspin 1 solitonsinspin-Peierlschainswithdynami- Nakamura, J. Phys. Chem. Solids, in press ; T. Takanao,
2
calphonons17 havedemonstratedthatsoliton-antisoliton K.Hiraki, H.Yamamoto, T. NakamuraandT. Takahashi,
bound states cannot exist unless a two-dimensional cou- J. Phys. Chem. Solids, in press.
pling is considered. Quite generally Ref. 4 predicts soli- 10K. Hiraki and K. Kanoda, Phys. Rev. Lett. 80, 4737
tons to have either spin zero or spin-1/2. In fact, the (1998) ; K. Kanoda, K. Miyagawa, A. Kawamoto and K.
on-site average spin plotted in figure 4b shows that, in Hiraki,J.Phys.IVFrance9,pr10-353(1999);Y.Nogami,
K. Oshima, K. Hiraki and K. Kanoda, J. Phys. IV France
the present case, each soliton carries no spin.
9, pr10-357 (1999) ; Y. Nogami et al., Synth.Metals 102,
We finish this paper by a brief discussion on the
1778 (1999).
possible relevance to experimental systems. It is clear
11F.D.M. Haldane, J. Phys. C 14, 2585 (1981).
that with on-site CDW associated with differential ge-
12J. Riera and D. Poilblanc, Phys. Rev.B 59, 2667 (1999).
ometry relaxation of the DI − DCNQI molecules, the 13S.R.White,Phys.Rev.lett.69,2863(1992);S.R.White,
(DI−DCNQI)2Ag compoundistheperfectcandidatefor Phys.Rev.B 48, 10345 (1993).
the present study. Note that under 5.5K18 this com- 14R.Fehrenbacher,Phys.Rev.B29,12230(1994);ibidPhys.
pound undergoes a second phase transition towards an Rev.Lett. 77, 2288 (1996)
4kF CDW, 2kF SDW mixed state. As already men- 15A similar truncation procedure has also been used in the
tioned by different authors16,19, on-site 4k CDW have
F case of a magneto-elastic coupling; for details see eg.
thepropertytoallowsimultaneaous2kF SDW.Although D.Augier et al., Physica B 259, 1015 (1999).
the metal-insulator instability in (TMTTF)2X (X=PF6, 16P. Maurel and M. B. Lepetit, Phys. Rev. B 62, 10744
AsF6, etc...) is believed to be of the Mott-Hubbard (2000).
type, the recent experiments6–8 revealing, at lower tem- 17D. Augier, D. Poilblanc, E. Sørensen and I. Affeck, Phys.
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gether with the coupling to the anions) might play a 19J.RieraandD.Poilblanc,Phys.Rev.B62,R16243(2000);
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the molecularstacks)andmightbe revealedine.g. opti- 20R.T. Clay, S. Mazumdar and D.K. Campbell, cond-
cal experiments. Note that it has also been theoretically mat/0010272.
suggested that the low temperature spin-Peierls phase
would exhibit, in addition to the lattice tetramerization,
4