Optical and spectral properties of quantum domain-walls in the generalized Wigner lattice

0.21 MB·English

by  S. Fratini

Optical and spectral properties of quantum domain-walls in the
generalized Wigner lattice
S. Fratini1 and G. Rastelli1,2
1Institut N´eel, D´epartement Mati`ere Condens´ee et Basses Temp´eratures - CNRS
BP 166, F-38042 Grenoble Cedex 9, France and
2Istituto dei Sistemi Complessi, CNR-INFM
v. dei Taurini 19, 00185 Roma, Italy
7
0 (Dated: February 6, 2008)
0
Westudythespectralpropertiesofasystemofelectronsinteractingthroughlong-rangeCoulomb
2
potential on a one-dimensional chain. When the interactions dominate over the electronic band-

n width, the charges arrange in an ordered configuration that minimizes the electrostatic energy,
a forming Hubbard’s generalized Wigner lattice. In such strong coupling limit, the low energy ex-
J citations are quantum domain-walls that behave as fractionalized charges, and can be bound in

7 excitonic pairs. Neglecting higher order excitations, the system properties are well described byan
1 effective Hamiltonian in the subspace with one pair of domain-walls, which can be solved exactly.
The optical conducitivity σ(ω) and the spectral function A(k,ω) can be calculated analytically,

] andrevealuniquefeatures oftheunscreenedCoulomb interactions thatcanbedirectlyobserved in
l
e experiments.
-
r
t
s I. INTRODUCTION withpreviousvariationalestimates8 aswellaswithmore
.
t recent exact diagonalization data,10 indicating that the
a
m Quantum many-body systems often exhibit complex single pair approximation captures the essential physics
of the GWL phase. On the other hand, it clearly breaks
- behavior, arising from the strong interactions between
d the individualdegreesoffreedom. Insome cases,the ex- down close to the quantum melting transition, where a
n proliferation of domain-walls is expected.
oticphysicalpropertiescanbe simplyexplainedinterms
o of suitable renormalized collective excitations. Well- In this paper, we take advantage of the exact solu-
c
[ known examples are the charge-spin separation in the tion of the discrete Coulomb problem given in Refs.11,12
Luttinger liquid with gapless bosonic excitations,1 soli- to evaluate the optical conductivity of the generalized
1 tons on a CDW condensate,2 and fractional charges in Wigner lattice at simple commensurate fillings n = 1/s.
v
quantum Hall systems.3 The formation of excitons manifests through the emer-
3
1 Another system that exhibits emerging complex be- gence of a series of sharp peaks, followed by a strongly
4 havior is the generalized Wigner lattice (GWL), de- asymmetric absorption band due to continuum of scat-
1 fined as the classical charge pattern that minimizes the tering states. Remarkably, both the discrete peaks and
0 Coulomb repulsion on a discrete lattice. The GWL was the asymmetry of the absorption continuum are direct
7 introduced by Hubbard4 to explain the charge ordering consequences of the long-range interactions among the
0 observedin the TTF-TCNQ organic salts, and has since electrons in the original model, that disappear when the
/
t been invoked in several classes of narrow-band quasi- Coulombinteractionsarereplacedby short-rangepoten-
a
m one-dimensional compounds.5,6,7 Increasing the electron tials.
bandwidth smears the classical charge distribution, and An analogous formalism is used to calculate the spec-
-
d eventually drives the system towards a small amplitude tral function A(k,ω). However, since the addition or re-
n charge density wave.8 movalofanelectrontoaGWLatfillingn=1/sisequiv-
o However, in the strongly interacting limit, where the alent to the creation of s domain-walls,9 the single pair
c bandwidth is much smaller than the electrostatic repul- approximation only applies to the special case n = 1/2.
:
v sive energy, the quantum charge distribution remains The interactions being repulsive, because the domain-
i veryclosetotheclassicalconfiguration. Inthislimit,the walls have equal charge, the low-lying excitations in the
X
low-lyingexcitationsarepairsofdomain-walls(kinksand spectral function are unbound scattering states, while
r
a anti-kinks), that carry fractional charge.4,9,10 It follows anti-bound states appear as a set of dispersive quasi-
thatthelowenergypropertiesofthesystemcanbedeter- particle peaks above the continuum.13
minedbysolvingtheproblemoftwointeractingdomain- The paper is organized as follows. In Sec. II we in-
walls, which is equivalent to the Coulomb problem on a troduce the one-dimensional model for spinless fermions
one-dimensional chain. Since kinks and anti-kinks have with long-range interactions, and solve it in the narrow-
oppositecharges,thelowestlyingexcitationsoftheGWL band regime, restricting to the subspace with one pair
willbeboundpairs,followedbyacontinuumofunbound of domain-walls. In Sections III and IV we use this ana-
domain-walls. Restricting to the subspace with only one lytical solution to calculate respectively the optical con-
kink/anti-kink pair, the quantum melting of the GWL ductivity and the single particle spectral function. The
was estimated in Ref.9 as the point where the gapin the results arebriefly discussedin Sec. V in connectionwith
excitationspectrumvanishes. The resultis inagreement existing experimental work.
2
II. MODEL AND SINGLE PAIR of a GWL becomes unfavorable at very low fillings. In
APPROXIMATION this caseanordinaryWigner crystalphase ismore likely
realized, where the electron wavefunctions are spreadon
WeconsiderthefollowingHamiltonianforfermionson lengthscalesmuchlargerthanthelatticespacing.8Asim-
a linear chain, interacting through long-range Coulomb ilar argument shows that the GWL phase is unfavored
forces: when moving away from the simple commensurate fill-
ingsn=1/s. Indeed,atgenericrationalfillingsn=r/s,
V (ni−n)(nj −n) it is easily shown that the gap still scales as ∼V/s3 (i.e.
H =−t c†ici+1+c†ici−1 + 2 |i−j| it is governed by the periodicity s of the classical pat-
Xi (cid:16) (cid:17) Xi6=j tern), which can be arbitrarily small regardless of the
(1)
value of n.14 In the following, we shall therefore restrict
Here t is the nearest-neighbor hopping amplitude and
to the special fillings n = 1/s, where the physics of the
V sets the energy scale of the long-range repulsion,
GWL is more relevant.
Vm =V/|m|. {c†i,ci} are, respectively,the fermionic cre-
ationandannihilationoperators,ni =c†ici istheoccupa-
tionnumberatsiteiandnistheaveragechargepersite. Let us denote |m,di the classical state corresponding
Thespindegreesoffreedomareexplicitlyneglected,cor- to a kink/antikink pair of length sd, with center of mass
responding to the limit of large on-site repulsionU. The at m. The total two-body wavefunction in the quantum
lattice parameter is set to unity. case can be written as
Since we are interested in the strong coupling regime,
L/s L/2s
it is useful to start by describing the classical solution, s
Ψ = eiKmϕ (d)|m,di (4)
obtained for t = 0. At simple commensurate fillings K,λ 2L K,λ
n = 1/s, the ground state takes one of the s equiv- r m=X−L/sd=X−L/2s
alent configurations that minimize the electrostatic en-
where the center of mass of the pair moves freely with
ergy, with one electron on every s-th site.23 The excited
momentum K, while the radial part obeys the following
configurations of lowest energy correspond to the inclu-
discrete Schr¨odinger equation:9,15
sion of a domain with a different (shifted) ground state,
delimited by a kink and an anti-kink.9 The energy cost
V/s3
to form a classical domain of finite lenght sd (d is the −2t(K)[ϕ(d+1)+ϕ(d−1)]+ Σ − −E ϕ(d)=0
s
d
number of perturbed unit cells) can be calculated as (cid:20) (cid:21)
(5)
d with the boundary condition ϕ(0) = 0. The factor
∞
∆ (d)= (V +V −2V ). (2) t(K) = tcos(Ks/2) reflects the fact that at each elec-
s sm+1 sm 1 sm
− tron hopping event, the center of mass of the defect pair
p=1m=p
X X
movesby±s/2latticesites. Thediscretehydrogenprob-
It defines an effective interaction between the domain- lemonthe semi-infinite chainadmits anexactanalytical
walls,andcanbeevaluatedexactlyintermsofDigamma solution11,12 in terms of Gauss-hypergeometricfunctions
functions. However, a very good approximation is ob- (or, equivalently, of incomplete beta functions). The ex-
tained by expanding in powers of 1/sd and keeping only citation spectrum consists, as in the ordinary hydrogen
the leading term atom,ofaninfiniteseriesofboundstates(kink/anti-kink
excitons), followed by a continuum of scattering states.
V 1
∆ (d)≃Σ − . (3) Observing that the two subspaces of different polarity
s s s2sd d > 0 and d < 0 are disconnected when only nearest-
neighbor electron hopping is allowed,10 the solution of
The above expression can be directly interpreted as the
the Coulomb problem Eq.(5) on the infinite chain is ob-
Coulomb attraction between two defects of charge ±1/s
at distance sd, with Σ = V 1− π cot π the energy tainedstraightforwardlybyconstructingcombinationsof
s s s s evenoroddparityofthewavefunctionsonthehalf-chain.
cost for creating two isolated defects at infinite distance.
(cid:2) (cid:0) (cid:1)(cid:3)
The approximate result Eq.(3) is quite accurate even at
a. Domain-wall excitons. The bound states have
shortdistances.24 Unlessotherwisespecified,weshallne-
energies:
glectthe higherorderterms(dipolar,multipolarinterac-
tions) in the following.
2
According to the above discussion, the gap in the ex- V
E =Σ − 16t(K)2+ (6)
citationspectrumisgivenclassicallyby∆s(1). Actually, K,n s s (cid:18)s3(n+1)(cid:19)
this can be viewed as the effective coupling parameter
of the model, that determines the stability of the classi- with n = 0,1,2,...,∞. Introducing the parameter
cal chargepattern againstquantum fluctuations, so that sinh(k ) = V/[4s3t(K)(n+1)], the normalized radial
n
the melting of the GWL is expected when t >∼ ∆s(1).8 wavefunctions read:
Since the gap scales as ∼ V/s3 ∼ Vn3, we see that for
a given value of the hopping integral t, the formation ϕ (d)=A de knd F (1−d,−n;2;1−e2kn) (7)
K,n n − 2 1
3
where F is one of the Gauss hypergeometric functions the excitation spectrum is obtained from Eq. (6) setting
2 1
and A is a normalization constant: K =0 and n=0:
n
An =2sinh(kn) tanh(kn)e−nkn (8) ∆opt =Σ −4t 1+ V 2. (13)
p s s s 4ts3
Theexcitonradius,definedasrn = ∞d=1d|ϕK,n(d)|2, (cid:18) (cid:19)
can also be evaluated:12 Thisvalueislowerthantheclassicalvalue∆ (1)because
s
P
oftheenergygainduetochargedelocalization. Thecon-
2+cosh(2k )
rn =(n+1) n (9) dition ∆ospt →0 can be used to locate the melting of the
sinh(2kn) GWL as the region where charge defects are expected
to proliferate.9 Forthe quarter-filledcases=2,the esti-
In the strong coupling regime, r ≈ (n + 1) and the
n matet/V ≃0.12isingoodagreementwiththenumerical
first bound state (n = 0) is strongly localized on one results of Ref.10.
site. At large quantum numbers the radius increases as
r ≃ (6s3t/V)(n +1)2 approaching the free scattering
n
states.
III. OPTICAL CONDUCTIVITY
b. Scattering states. The energy of the scattering
states is given by the sum of the energies of the indi- The finite frequency absorption at T = 0 can be ex-
vidual domain-walls, with k the relative momentum of pressed in terms of the current-current correlation func-
the two-body system: tion through the Kubo formula:
EK,k =Σs−2t[cos(Ks/2+k)+cos(Ks/2−k)]. (10) σ(ω)= 1 Im |hΨK,λ|ˆj|ψGSi|2 (14)
Lω ω−E +i0+
K,λ
The corresponding wavefunction is:12 XK,λ
where the sum runs over all eigenstates, and the ground
ϕ (d)=B 1(k)deikd F (1−d,1−iη(k);2;1−e 2ik)
K,k − 2 1 − state energy is set to 0. For nearest-neighbor hopping,
(11) the current operator on the lattice is defined as ˆj =
where η(k) = V/[2s3t(K)sink]. The normalization fac- it (c+c −c+ c ). The above dimensionless expres-
tor is determined by the asymptotic behavior at large d, i i i+1 i+1 i
sionshouldbe multiplied byaprefactorσ =e2a2/¯hv to
and is given by P 0
restorethe properunits, whereais the intersitedistance
on the chain and v is the volume of the unit cell.
4s πη(k)
B−2(k)= L sinh[πη(k)]sin2(k) e−(2k−π)η(k) (12) Approximating|ψGSiintheGWLphaseatfillingsn=
1/swiththe classicalstate|ψ i= ic†si|0i,we seethat
in the case of an open chain of length L/2s. theeffectoftheoperatorˆjisto∞createkink-antikinkpairs
Q
of length d = ±1 at any of the m possible sites. The
The excitation spectrum is illustrated in Fig. 1 for
matrix elements with the single pair states of Eq.(4) are
t/V = 0.05 at fillings n = 1/2 and n = 1/3. The gap in
thus given, in terms of the eigenfunctions of the semi-
infinite chain, as
0.8
n=1/2 n=1/3 |hψ |ˆj|Ψ i|2 =t2Lδ |ϕ (1)|2. (15)
K,λ K,0 K,λ
∞ s
0.2
0.6
A. Coulomb potential
V
y/
erg 0.4
En 0.1 Theopticalabsorptioncanbedividedintwopartsσ =
σ +σ , whichrepresentrespectivelythe sharptransi-
exc sc
0.2 tionsfromthegroundstatetotheexcitonicstatesandan
absorptionbanddue to transitionsto the kink/anti-kink
continuum. The corresponding energies are obtained
0 0
-π 0 π -π 0 π from Eq.(6) and Eq.(10) by setting K = 0. The exci-
K K tonic part is an infinite series of delta function peaks
πt2
FIG.1: Excitationspectrumvs. centerofmassmomentumK σ = |ϕ (1)|2δ(ω−E ) (16)
exc 0,n 0,n
in the single pair subspace for t/V =0.02 at fillings n=1/2 sE
0,n
n
(left) and n = 1/3 (right). The full lines correspond to the X
3 lowest bound states, while the dashed lines indicate the where the spectral weights are givenby Eqs. (7) and (8)
boundaries of the kink/anti-kinkcontinuum.
above, making use of the property F = 1 at d = 1.
2 1
4
0.01
analytical
Setting k(ω)=arccos[−(ω−Σ )/4t] and converting the ED
s
sum over states in Eq. (14) into an integral, we obtain
0.01
thefollowinganalyticalexpressionforthecontinuousab-
sorption band:
V k(ω) σω () 0.005
−2s3tsink(ω) 0
πV e 0.4 0.5
σ (ω)= (17)
sc 4s4ω V π
(cid:18) (cid:19) −
2s3tsink(ω)
1−e
0
Note that since we are assuming t <∼ V/s3, the denom- 0.3 0.35 0.4 0.45 0.5 0.55 0.6
inator can be set equal to 1 up to exponentially small ω/V
corrections(thesebecomeimportantclosetothemelting 1
e
oftheGWL,whereanywaythesinglepairapproximation m rul b)
breTahkesodpotwicna)l.absorptiondeterminedaboveis illustrated al su 0.5
in Fig. 2.a and agrees remarkably well with the exact parti 0
diagonalizationresultsofRef.10 The maindiscrepancyis 0 0.02 0.04 0.06 0.08 0.1
ashiftinthepositionofthefirstexcitonicpeak,thatcan t/V
be entirely ascribed to the the use of the pure Coulomb
potential in Eq. (3). A transfer of spectral weight FIG. 2: a) Optical conductivity of the generalized Wigner
latticeinthesinglepairapproximation asgivenbyEqs. (16)
from the excitons to the kink/anti-kink continuum takes
and (17) at n = 1/2 and t/V = 0.02. A finite broadening
place as t/V increases, as shown in Fig. 2.b, where we
hasbeenintroducedforgraphical purposes. Theopen circles
have plotted the quantities dωσexc(ω)/ dωσ(ω) and aretheexact diagonalization dataofRef.10. Intheinset,the
dωσ (ω)/ dωσ(ω). Note that formulas(16) and (17)
sc samecurveiscomparedwiththeresultobtainedwiththefull
R R
obey the following property, that can be checked by di- potential of Eq.(2) (dashed lines).
R R
rect evaluation: b) Fraction of thespectral weight carried respectively bythe
excitons(fulllines)andthekink/anti-kinkcontinuum(dashed
hωi= dωσ(ω)ω =πt2/s. (18) line).
Z
This relation shows that the “average” absorption fre-
quency is independent on the coupling constant and can B. Screened potentials
be used to measure directly the value of the hopping in-
tegral in an optical absorption experiment.
The strong asymmetry of the optical lineshape Eq.
Intheabovederivationitwasassumedthatthepoten-
(17)ultimately followsfromthe long-rangenatureofthe
tial follows a pure Coulomb law, Eq.(3), which allowed
Coulomb potential. Indeed, although the energies of the
us to obtain explicit analytical formulas for the optical
scatteringstatesaregivenbythenon-interactingexpres-
absorption. However, the discrete Schr¨odinger equation
sion(10),theirwavefunctionsstronglydifferfromthefree
(5)withtheexactpotential∆ (d)ofEq.(2)canbesolved
s plane waves especially at short distances (the scattering
to arbitrary accuracy with small numerical effort. The
statesarerequiredtobeorthogonaltotheboundstates).
result is presented in the inset of Fig. 2.a. We see that
Thisisreflectedintheopticalabsorption,whichinvolves
the analytical expression (17) agrees quite well with the
the wavefunctionprecisely at d=1, where the condition
numericalresult. Discrepanciesariseatverysmallvalues
of orthogonality with the excitonic states is most strin-
oft/V,wheretheexcitonsbecomelocalizedonveryshort
gent. Toillustratethisissue,wehaveconsideredthecase
lengthscales,whichis where the potential∆ (d) sensibly
s ofa screenedpotentialV/d→V exp(−d/ℓ)/d. This po-
deviates from the Coulomb law.
tentialis convexandensures the stability ofthe classical
Note that an expression similar to Eq. (17) was de- configurationconsideredhere,4 whichispreferabletothe
rived by Gallinar16 in the half-filled case n = 1 for elec- case of arbitrarily truncated Coulomb interactions, for
trons with spin, in the limit of a large on-site repulsion which the ground state is not always univocally deter-
U. In that case the lowest-lying excitations correspond mined.
to pairs of doubly occupied/empty sites.17 The energy The results obtained for generic values of the screen-
cost for creating such pairs is set by U, while the long- ing length are reported in Fig. 3 at filling n = 1/2, for
range tail of the Coulomb potential acts to bind the op- t/V = 0.02. Upon reducing the screening length, the
posite charges together.25 Similar effects have also been optical absorption shifts to lower frequencies, the asym-
discussedinthecontextofconjugatedpolymers,inwhich metry of the lineshape becomes less marked and the ex-
casethe energyscaleofanelectron-holeexcitonis setby citonic peaks progressively disappear (below a critical
the Peierls dimerization gap18 (see also Ref.17). value, no bound states are possible and σ = 0). In
exc
5
two kinks (antikinks) are far apart. The creation energy
of each kink is Σ /2=V/4 (cf. Eq. (3)).
2
0.06 0.5 Inthe quantumcase,wecanwritethe1D-Schr¨odinger
∞
equation for the two-body problem, which is formally
5
identical to the one presented in Sec. II, except for a
2
0.04 change of sign in the interaction potential, which results
ω)
σ( 1 inasymmetricallyreflectedexcitationspectrum: thelow
energy part now corresponds to the domain-wall contin-
0.02 uum, while the discrete levels are moved to higher ener-
gies. Such anti-bound states form, as for the excitonic
states encountered previously, due to the long-range na-
ture of the repulsive potential.
0
0 0.1 0.2 0.3 0.4 0.5
ω/V
FIG.3: Opticalabsorptionforascreenedpotentialatn=1/2
and t/V =0.02. The labels indicate the value of the screen-
ing length ℓ. The dashed line corresponds to the unscreened
Coulomb potential.
π
ω)
k,
the limit ℓ → 0, a truncated nearest-neighbor poten- A(
tial is recovered. In that case the effective interaction
0
∆(d) = Ve 1/ℓ ≡ V is constant for all d and the scat-
− 1
tering states reduce to free-particle states. The corre-
spondingcurrent-currentcorrelationfunctionhasasemi-
circular shape and the resulting absorption band is π
-
0 0.5 1
2 ω/V
t ω−V
1
σ (ω)= 1− . (19)
nn
2ωs (cid:18) 4t (cid:19) FIG. 4: Spectral function A(k,ω) for s = 2 and t/V =0.05.
The different curves correspond to different values of k, as
To conclude this section, let us stress that the exper- indicated on the left axis.
imental observation of an asymmetric lineshape of the
form shown in Fig.2.a can be taken as a signature of the The spectral function for ω > 0, which describes the
importanceoflong-rangeelectron-electroninteractionsin inverse photoemission processes, can be calculated as
a givenmaterial,andcalls fortheoreticalmodels thatgo
beyondtheintersiterepulsiontermusuallyconsidered.19 A(q,ω)= |hΨK,λ|c+q|ψGSi|2δ(ω−EK,λ) (20)
K,λ
X
where c+ = 1 eiqmc+ and|Ψ i are the solutions
IV. SPECTRAL FUNCTION AT HALF FILLING q √L m m K,λ
of the repulsive Schr¨odinger equation (5). An analogous
P
formula holds for the spectral function at ω < 0, with
Inaphotoemissionexperiment,anelectronisaddedor
A(q,−ω)=A(q,ω). Replacing asusualthe groundstate
removedfrom the system in a sudden process that is de-
withtheclassicalconfigurationatt=0enforcesthecon-
scribedby the singleparticlespectralfunctionA(k,ω)=
straint that electrons can only be created on the unoc-
−1ImG(k,ω). In the spirit of the strong coupling ap-
π cupied sites. The resulting expression for the spectral
proachpresentedintheprecedingsection,addinganelec-
function is
tron to the classical configuration at n = 1/s creates s
1
kinks of charge 1/s. Analogously, removing an electron A(q,ω)= |ϕ |2δ(ω−E ) (21)
q,λ q,λ
creates s anti-kinks of negative charge−1/s. In the spe- 2
λ
cial case s=2, two defects are created and the problem X
becomes equivalent to the one treated previously, albeit where the sum runs over both the continuous spectrum
withrepulsiveinteractions(the analogyisno longertrue (λ ≡ k) and the discrete anti-bound states (λ ≡ n),
for larger values of s, where the creation or removal of whose energies are given by Eqs. (10) and (6), that we
an electron take the system away from the single-pair reproducehere(inthelatter,thequantityE−Σschanges
subspace). sign due to the repulsive nature of the interactions):
Classically,since the interactionpotential is repulsive,
2
thestateoflowestenergyinthesystemwithoneparticle V V
E = + 16t2cos2(q)+ (22)
added(removed)correspondstoconfigurationswherethe q,n 2 s 8(n+1)
(cid:18) (cid:19)
6
The results are reported in Fig. 4. The width of the well as the sharp quasi-particle peaks in the single par-
continuous spectrum is modulated by q and vanishes at ticle spectralfunction, areall robustfeatures that follow
q =±π. The anti-bound states are clearly observable as fromthelong-rangenatureoftheCoulombpotential,and
well defined quasi-particle peaks in A(k,ω), that follow that are lost in models with strongly screened or trun-
the roughly sinusoidal dispersion relations Eq.(22). cated interactions. These features should be clearly ob-
servableinexperiments,allowingtoaddresstherelevance
of long-range interactions in the charge-ordered insulat-
V. DISCUSSION AND CONCLUSIONS ing phases of narrow-bandone-dimensional systems.
As an example, the mid-infrared optical absorption
Inthiswork,wehaveexaminedthespectralproperties band recently measured20 in the organic quarter-filled
of spinless electrons interacting through the long-range (n=1/2)salt(DI-DCNQI)2Ag—aprototypicalsystem
Coulomb potential on a one-dimensional lattice, which exhibiting Wigner crystal ordering5 — resembles very
constitutesaminimalmodeltoaddresstheeffectsofelec- closely the theoretical spectra obtained from Eq. (16)
tronic correlations in narrow-bandsolids, away from the and(17),andasatisfactoryfitcanbeobtainedwithphys-
most studied half-filled case. In the strongly interacting ically sound parameters (V = 1.1eV and t = 80meV).
regime, the charges form a generalized Wigner lattice, Absorption shapes similar to the ones presented in this
whose elementary excitations are fractionally charged work have also been reported in other classes of organic
domain-walls,themselves interacting throughlong-range compounds, such as DBTSF-TCNQF4, TTF-TCNQ21
forces. Taking advantage of the exact solution of the andCs2(TCNQ)3,22 correspondingrespectivelyton=1,
Coulomb problem on a one-dimensional chain, we have n ≃ 1/2 and n = 1/3, whose detailed analysis is post-
derived an analytical expression for the optical conduc- poned to future work.
tivityatsimplecommensuratefillingsn=1/s,aswellas
forthesingleparticlespectralfunctioninthespecialcase
n=1/2. Both are in good agreement with the available Acknowledgments
exact diagonalization data at n=1/2.10,13
The sharp peaks emerging in the optical conductiv- Foroneofus,G.R.,thisworkwasfundedbytheSwiss-
ity, signaling the formation of domain-wall excitons, the Italianfoundation“AngeloDellaRiccia”. S.F.gratefully
asymmetric lineshape of the absorption continuum, as acknowledgesenlighteningdiscussionswithD.Baeriswyl.
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m=1
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193 (2004). ∆s(d)=Σs+ s2 [Ψ(d−1/s)−Ψ(d+1/s)]
10 M. Mayr, P.Horsch, Phys. Rev.B 73 195103 (2006).
11 J.-P. Gallinar, Phys. Lett. 103A, 72 (1984). d
12 A. A.Kvitsinski, J. Phys. A: Math. Gen. 25, 65 (1992). +V s[2Ψ(d)−Ψ(d+1/s)−Ψ(d−1/s)]
13 M. Daghofer, P.Horsch, cond-mat/0608062 (2006). Thelargest discrepancyoccursfor s=2andd=1, where
14 V. Slavin,Phys. Stat. Sol. 242 2033 (2005) theaboveexpressiongives∆2(1)≃0.386V whilethevalue
15 W. Barford, Phys. Rev.B 65, 205118 (2002). for thepureCoulomb law is 0.375V.
16 J.-P. Gallinar, Phys. Rev.B 48, 5013 (1993). 25 In the antiferromagnetic phase the problem is formally
17 F. B. Gallagher, S. Mazumdar, Phys. Rev. B 56, 15025 equivalent to the one treated here [set x = 1 in Eq.(3.1)
(1997). of Ref.16]. For more general magnetic arrangements, the
18 S. Abe,J. Phys.Soc. Japan 58, 62 (1989). excitation spectrum must be modified to account for the
19 H. Seo, J. Merino, H. Yoshioka, M. Ogata, J. Phys. Soc. probabilityoffindingneighboringsiteswithoppositespins,
Jpn. 75, 051009 (2006) leadingtoamorecomplexscenariofortheopticalabsorp-
20 K. Yamamoto, T. Yamamoto, K. Yakushi, C. Pecile, M. tion.
Meneghetti, Phys.Rev. B 71 045118 (2005)