Table Of ContentIncommensurate quantum-size oscillations in acene-based molecular wires - effects of
quantum fluctuations
Peter Schmitteckert,1,∗ Ronny Thomale,1 Richard Koryt´ar,2,3 and Ferdinand Evers2
1Institute for Theoretical Physics and Astrophysics,
Julius-Maximilians University of Wu¨rzburg, Am Hubland 97074 Wu¨rzburg, Germany
2Institute of Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany
3Department of Condensed Matter Physics, Faculty of Mathematics and Physics,
Charles University in Prague, 121 16 Praha 2, Czech Republic
(Dated: January 18, 2017)
7 Molecular wires of the acene-family can be viewed as a physical realization of a two-rung lad-
1 der Hamiltonian. For acene-ladders, closed-shell ab-initio calculations and elementary zone-folding
0 arguments predict incommensurate gap oscillations as a function of the number of repetitive ring
2 units, N , exhibiting a period of about ten rings. Results employing open-shell calculations and a
R
n mean-field treatment of interactions suggest anti-ferromagnetic correlations that could potentially
a open a large gap and wash out the gap oscillations. Within the framework of a Hubbard model
J with repulsive on-site interaction, U, we employ a Hartree-Fock analysis and the density matrix
7 renormalization group to investigate the interplay of gap oscillations and interactions. We confirm
1 thepersistenceofincommensurateoscillationsinacene-typeladdersystemsforasignificantfraction
of parameter space spanned by U and NR.
]
l
l
a I. INTRODUCTION rowest graphene nano-ribbons with zig-zag terminated
h
edges. Due to recent progress in synthesis methodology,
-
s oligoacenesuptolengthNR =9(nonacene)havealready
e The question how properties of a macroscopic system been produced.10,11
m emerge when more and more atoms or molecules accu-
Previously,wehavefoundthatlinear(oligo)acenesex-
t. mulate exhibits many different facets and, for that rea- hibit oscillations in the NR (or length) dependence of
a son,reappearseveryonceinawhileindifferentcontexts. optical excitation gaps.12 Oscillations of electronic prop-
m Onthesimplestlevelof“emergence”, onecouldconsider
erties with the geometric size of a nano-system are per
- the appearance of band-structures in crystal growth. A se frequently encountered. An established example is
d first qualitative description of this phenomenon can al-
given by the threefold gap-oscillation of zigzag carbon
n
ready be given within a picture of non-interacting quasi-
o nanotubes (armchair-ribbons) as a function of tube di-
particles. A conceptually analogous situation arises in
c ameter (width).13–15 In this context, the three-fold peri-
[ mesoscopic physics: when a quantum dot couples to an odicity reflects the symmetry of the hexagonal Brillouin
electrode, the dot states hybridize with contact states, zone, with the Dirac points located at the corners. A re-
1
thereby acquiring the level broadenings, Γ, that sig-
v markable aspect of the acene oscillations is that they are
nal finite lifetime effects. Qualitatively new physics
1 incommensurate in the sense that the oscillation period
emerges upon accounting for interaction effects. For in-
6 is not dictated by lattice symmetry; it can reach periods
6 stance, inquantum-dots, theKondophenomenonresults of ten times the length of the unit cell. The oscillations
4 from the interplay of hybridization and the Coulomb-
come with an important consequence, namely, a close-to
0 energy. Another example displaying the appearance of
closing gap for molecules of certain (periodically repeat-
1. strong correlation effects with growing system size is ing) finite length.
0 superconductivity.1,2 There,oneisfacingacrossoversce-
We have demonstrated that similar to the case
7 nario, where the super-conducting gap, ∆ , competes
sc of the nanotubes, the acene-oscillations can likewise
1 with the single particle level spacing, ∆ (cid:29)Γ.
: 0 be described (nearly) quantitatively by a selection
v
In this paper, we investigate an incarnation of a sim- rule (zone-folding) for the wave-vectors applied to the
i
X ilar motif as it appears in molecular electronics: we ask band structure of the underlying extended tight-binding
howtheelectronicspectralpropertiesofamolecularwire model (Fig. 1). The procedure is analogous to the
r
a built out of N repetitive units (“rings”) evolve with in- threefold periodicity in band-gaps of nanotubes13 and
R
creasing N . Motivated by earlier experiments3–5, our nanoribbons14,15; in the acene case, however, incom-
R
specific example will be the acene-family such as ben- mensurate periodicity is reached because of the non-
zene, naphthalene, anthracene, and others. Due to their universal position of k labelling the longitudinal mo-
D
peculiar electronic structure, oligo-acenes have received mentumattheacene-analogueoftheDiracpoint. Based
a considerable amount of attention in organic chemistry on the computational analysis of the acene-like Hubbard
over the past 50 years.6–8 A recent technological interest model, we find that the zone-folding argument is not in-
isbasedonsuggestionstousethemaslightharvestersin validated by interactions beyond the independent quasi-
organicphoto-voltaicdevices.9 Besidesbeingasimplere- particlepicture;theoscillationspersistgiventheinterac-
alization of nano-wires, acenes can be viewed as the nar- tion strength U does not exceed a threshold of the order
2
3
ε(+,+)
2
ε(-,-)
t' t b) [t] 01
s,
t' (k -1
ε
t t" t" -2 ε(+,-)
⊥ ⨯ ε(-,+)
-3
-π -π/2 0 π/2 π
longitudinal momentum k
FIG.1: (Left)Partsoftheoligoacenebackboneinthetight-bindingrepresentationwithanearestneighborhopping(t,yellow),
hoppings to the second neighbor (t(cid:48), dashed blue) and third neighbor (t(cid:48)(cid:48), dotted black). On the left, we draw these hoppings
for the ”outer” carbon site, on the right for the ”inner” carbon. t(cid:48)(cid:48) denotes the second neighbor hooping of the outer sites, t(cid:48)(cid:48)
⊥ ×
of the inner sites. In the present study we adopt t(cid:48)(cid:48) =t(cid:48)(cid:48) =t(cid:48)(cid:48) and, in contrast to Ref.12, t(cid:48)=0
⊥ ×
(Right)Thetightbindingband-structurecorrespondingtothelong-wirelimitfort=1,t(cid:48) =0,andt(cid:48)(cid:48) =0.3. Thelinesarethe
analytical result according to (2) with t(cid:48)(cid:48) =t(cid:48)(cid:48). The crosses are the results from a tight binding system consisting of N =31
⊥ × R
rings and periodic boundary conditions. The valence- and conduction-band are crossing at the Γ−point k .
D
of the bandwidth - at least as long as interactions are with Coulomb interactions: acene-ladders with onsite
screened, i.e., short-ranged. Hubbardinteractionsexhibitanti-ferromagneticorderon
Building upon our previous work, we intend to give a themeanfieldlevelalongwithalargechargegap,thereby
comprehensivestudyofinteractionsandincommensurate diminishing IO. By microscopic DMRG analysis, how-
oscillations (IO) of acenes by contrasting Hartree-Fock ever, we show that also in short ladders quantum fluctu-
(HF) mean field treatments against microscopic numer- ations destroy the long-range order. In particular, they
ical studies. We model acene-type molecular wires by significantly reduce the HF excitation gap, restoring the
finite length Hubbard ladders. A principal physical in- IOsignatureanduptointermediateinteractionstrengths
centive is given by the potentially rich phase diagram of the order of the bandwidth.
of interacting acene wires16. In this context, one could
hope to manipulate acenes via chemical synthesis, or en-
gineering of the environment so as to explore some part II. MODEL AND METHODS
of this phase diagram. (Admittedly before this is exper-
imentally feasible, major technological difficulties have
A. Model definition
to be overcome that are realated to chemical stability.
Momentarily, long acene wires need to be kept within a
We study the acene Hubbard-ladder
noble-gas matrix to prevent further reactions11.)
Furthermethodologicalincentiveisprovidedbyrecent (cid:18) (cid:19)(cid:18) (cid:19)
open-shellscGW-calculationsfortheacene-series17;they Hˆ =(cid:88)t cˆ† cˆ +U(cid:88) nˆ − 1 nˆ − 1 (1)
ij iσ jσ i↑ 2 i↓ 2
indicated that in short wires the IO might disappear in
i(cid:54)=j i
the presence of anti-ferromagnetic order in the ground
state. At first sight, this result may look very plausi- at half filling, where the first term describes the tight-
ble. However, effects of quantum-fluctuations are ne- binding dynamics, and the second term in Eq. (1) the
glectedinitsderivationandinquasi-onedimensionalsys- onsite Hubbard repulsion with strength U. In the case
tems their effect typically is very strong. Specifically, in of periodic boundary conditions (PBC), we have a linear
the long-wire limit, quantum fluctuations tend to desta- length of L = 2N sites, where N gives the number of
R R
bilize mean-field ordered phases that break a continuous hexagon (benzene) rings. For hard wall boundary con-
symmetry18. In this (quantum-disordered) phase, the ditions (HWBC), we complete the last ring leading to
order-parameter exhibits significant short-range correla- L=N +1. Thetotalnumberofsitesis2L. Thenearest
R
tions that leave a characteristic signature in local probe neighborhoppingischaracterizedbyanamplitudet =
(cid:104)ij(cid:105)
measurements. −t, and the longer-range hoppings accordingly by t(cid:48), t(cid:48)(cid:48),
⊥
In this work, we clarify the fate of the anti- and t(cid:48)(cid:48) (Fig. 1). If not stated otherwise, t will be taken
×
ferromagnetic correlations in acenes-type ladders in the as the unit of energy and t(cid:48)(cid:48) = t(cid:48)(cid:48) = t(cid:48)(cid:48). Note that t(cid:48)(cid:48)
⊥ × ⊥
presence of quantum fluctuations, i.e., by providing a andt(cid:48)(cid:48) giverisetoanon-avoidedcrossingofconduction-
×
joint analysis from Hartree-Fock (HF) mean field and and valence band at non-commensurate wave-vectors k
D
density matrix renormalization group (DMRG). Within predicted for the case U=0.19 For t(cid:48)(cid:48),t(cid:48)(cid:48) =0, the system
⊥ ×
HF mean field analysis, we reproduce the observation features a quadratic band touching16. For U = 0, the
made in the ab-initio study Ref. 17 for acene-molecules energy spectrum of (1) reduces to
3
(cid:15) (k)=2t(cid:48)cos(k)+ 1s(cid:0)t+t(cid:48)(cid:48) +2t(cid:48)(cid:48) cos(k)(cid:1)+ sb(cid:113)(cid:0)t−t(cid:48)(cid:48) +2t(cid:48)(cid:48) cos(k)(cid:1)2+8(1+cos(k))(t+st(cid:48))2, (2)
sb 2 ⊥ × 2 ⊥ ×
where b,s=±1 denote a band and parity index, respec- Once the phase-diagram has been mapped out, it may
tively, yielding four bands resulting from the four-site bepossibletopredictthequalitativebehaviorforagiven
unit cell. experiment, even though the actual parameters repre-
sentative for this situation may not be known with high
Inthefollowing,weexcludenext-nearestneighborhop-
ping, t(cid:48) =0, so there is no hopping between sites sharing accuracy, at least in those situations where very stable
phases have been identified.
the same sublattice; the model becomes bipartite and a
chiral symmetry is imposed connected to the particle-
hole symmetry of the spectrum (Fig. 1). This choice can
be made without loss of generality regarding IO, as the B. Computational details
maineffectofbreakingthechiralsymmetryinmoregen-
eral models with non-vanishing t(cid:48) would be to shift the In this spirit, we continue our analysis and treat the
position of the Dirac point kD. The corresponding ana- model Hamiltonian (1) numerically within DMRG20,21.
lytical calculations are given in the Appendix A. Similar to previous authors22,23, we also observe that
General comments on Hubbard-models. We briefly the ground state is a spin singlet independent of the
comment about the applicability of Hubbard models to wire length. Thereby, we track the many-body Hamil-
real systems. Indeed, models along the lines of (1) have tonian at fixed particle number (equal to the number of
been investigated intensively over the last fifty years in lattice sites) and in the Sz =0 spin sector. By target-
tot
physics and in chemistry. They provide substantial sim- ing not only the ground state but also excited states,
plifications by (a) ignoring the coupling to molecular vi- we obtain the optical gap by taking the energy differ-
brations; (b) including only one orbital per atom and ence between the first excited and the ground state.
(c) by including all other degrees of freedom in terms of Fornon-interactingsystems, theopticalgapcorresponds
effective parameters, only, such as a short range interac- to the HoMO/LuMO gap, as it is given by the lowest
tion that mimics screening. In fact, information about particle-hole excitation. Due to spin-rotational invari-
the molecular geometry (e.g. the relative postion of the ance, the excited states of the model in (1) come in
atoms) is absorbed in hopping parameters and interac- spin-multiplets according to irreducible representations
tion matrix elements. These parameters span a multi- of SU(2). By performing reference calculations in the
dimensional space which has a hyperplane that features spin-sectors Sz =1,2, we have ascertained that the low-
tot
those particular combination of parameters that can be estexcitationisatripletstate,andthatthegroundstate
reached in realistic situations. isindeedasingletstate. DetailsoftheDMRGimplemen-
tation are given in Appendix C. To investigate the effect
In return, because Hubbard models are in a certain
of quantum fluctuations, and to link with previous mean
sense minimal, they allow for a numerically exact treat-
field treatments, it is useful to compare the DMRG re-
ment of the many-body problem, with a size of the com-
sults to a HF description. We have also implemented a
putationalHilbertspacethatismuchlargerascompared
self-consistentHFcycletoobtainmeanfieldestimatesfor
to most other many-body treatments that attempt to be
the energy gaps, ground state, and magnetization (Ap-
more “first principles” on the Hamiltonian level.
pendix C). In HF theory, we look for the lowest lying
Keeping this in mind, Hubbard models like (1) are by (charge-neutral) excitation energy to determine the en-
no means as oversimplistic as they may appear at first ergy gap, for which we compute the difference in energy
sight. First, if properly parametrized such models have between the lowest unoccupied and the highest occupied
a good chance to give a realistic, sometimes even quan- eigenvalue of the Fock operator.
tative description of low-energy excitations also in those
situations where quantum-fluctuations are dominant, so
that most competing methods fail. Second, one often is
III. RESULTS AND DISCUSSION
interested in the system behavior under varying exter-
nal conditions, i.e. in the phase diagram of the model;
one would like to figure out what phases exist, what are A. Hartree-Fock analysis
their properties and, ideally, also where are the phase-
boundaries located at. Since the qualitative aspects of Before investigating finite size features of the spec-
phasediagramstendtobeveryrobustunderachangeof trum, we analyze the infinite wire limit. In Fig. 2 (top),
theHamiltonian, oneoftenadoptsconvenientchoicesfor wedisplaytheevolutionoftheHFgapforrepulsiveinter-
the model parametrization, even though they are some- action strength U. We extract the bulk gap from finite-
times far from realistic. size calculations by extrapolating the finite size gap vs.
4
the inverse linear length of the system (Appendix B).
While the non-interacting wire exhibits a band crossing 1
HF
at the Fermi-energy and therefore is metallic, the repul-
sive interaction U opens up a correlation gap. 10-1 DΔ1M,∞R G∝ U2
]
Within HF theory, the ground state is found to be un- U
stableagainstantiferromagneticorder;thecorresponding [∞10-2 Δ1,∞ = 0.5 t2/U
1,
staggeredmagnetizationisshowninFig. 2(bottom). As Δ 10-3
we will see in our DMRG-calculations below, this order
will be destroyed by quantum fluctuations. We here wit- 10-4
ness the established fact that mean-field theories overes-
timate the tendency for symmetry breaking, at least if 10-1 1 10 102
U [t]
the interactions are short ranged. In passing, we men-
tion that broken symmetries refer to the long-distance te 0.5
properties of a quantum state, which are not well de- r si ΔH1,F∞ / (2U)
e 0.4 MHF
scribed on the HF-level. In contrast, the total energy is p
F
predominantly dictated by the pair-correlation function HM 0.3
atshort-distances. Thelatterislargelyinsensitivetothe n
o 0.2
global symmetries. Therefore, with respect to total en- ti
a
ergy calculations, HF can often give useful answers even tiz 0.1
e
though the long-range physics is described qualitatively n
g 0
incorrect. (L¨owdin’s “symmetry breaking dilemma” of a
M 0 0.25 0.5 0.75 1
HF24)
Δ1,∞ [U]
In the perturbative limit of weak repulsion, U(cid:28)1, we
can restrict ourselves to states with AF ordering on top
of the non-interacting ground state. In this situation the FIG. 2: Top: Evolution of the excitation gaps ∆HF (U) (+)
1,∞
gap opens in long (but finite-length) wires as and ∆DMRG(U) (×) in units of the interaction U vs. the in-
1,∞
creasingstrengthoftherepulsiveHubbardinteractionU. For
(cid:113)
∆HF = ∆2+(MHFU)2, (3) large U we recover the expected asymptotic behavior. For
1,NR 0 U →0thenumericalextrapolationisimpairedbytheIO;the
asymptoticbehaviorisindicatedbythe(yellow)dashedline;
where∆ isthefinitesizegapofthenon-interactingwire,
0 see text and for comparison Fig. 9.
∆0 ≡ ∆U1,=N0R; MHF denotes the staggered magnetization Bottom: Corresponding staggered magnetization MHF (◦)
amplitude. We recall that at U(cid:28)1 the response of en- over the band-gap ∆HF (U) in units of U displaying a lin-
1,∞
ergies to the staggered field is linear in U, implying a ear dependence in the complete regime.
quadratic gap opening ∆HF (U) ∼ U2 in the thermody-
1,∞
namic limit (where ∆ →0).
0
Atintermediateinteractions,theHFvaluesforthegap
opening allow for a phenomenological two-parameter fit,
For longer wires, however, the IO fade away, because
−D
∆H1,F∞(U)≈0.5UeU2+D1U, (4) the native band-gap is suppressed as ∼ 1/NR and even-
tually no longer overcomes ∆HF . As soon as ∆HF
whereDisoftheorderofthenon-interactingbandwidth; exceeds the maxima of the nati1v,NeRband gaps, the IO1d,NisR-
we obtain D ≈5.5±0.4 and D ≈1.0±0.1, provided U appear from the HF spectra. Indeed, the IO in ∆HF
is not too small. As expected,1in the large U limit, the are seen to become weaker with increasing N (and1,NUR)
R
HF particle-hole gap is set by U. Its evolution follows (Fig. 3). This explains why large systems sizes N (cid:29)1
R
closelytheoneoftheanti-ferromagneticorderparameter are needed in order to obtain the HF gap in the small U
MHF(U), Fig. 2 (bottom). regime: the finite size induced gap has to fall below the
We now turn to the HF analysis of acene wires with HF gap.
finite length. In this case, one expects that the open-
ing of the HF gap ∆HF decreases the ground-state en- Asthemagnetizationcloselyfollows∆HF ,itemerges
ergyonlyif∆HF exc1e,NeRdsthenon-interacting(“native”) preferably in acene wires that exhibit a1r,eNlaRtively small
1,NR
level-spacing∆ attheFermienergysignificantly. Hence, (native) band-gap (Fig. 3). Vice versa, the HF-ground
0
for shorter wires, the gap opening is most effective at state remains non-magnetized at wire-lengths, such as
particular values of N that have an anomalously small N =11,19,28,..., where the native gap is particularly
R R
single-particle gap – and ineffective at the others. In large; there mean-field magnetism only appears at larger
other words, only at N -values situated close to a mini- interaction values. In contrast, for large interaction val-
R
mum of the IO an AF-order develops and at the others uestheHFgapsurpassesthenon-interactinggapalready
the non-interacting ground-state prevails. In this situa- for small system size (Fig. 4), and the IO disappear in
tion IO remain visible even in ∆HF . that regime.
1,NR
5
12
HF [U]1,NR 00..23 UUU===000...808,,, MΔΔ11H,,NNFRR//U0.8 00..115 HFzation M [t]N1R108 UUU===011...205 HHHFFF,,, UUU===101...025
optical gap Δ 0.1 0.05 ered Magneti L Δ1, 246
gg 0
0 0 ta 5 10 15 20 25 30 35 40
s
20 40 60 80 100 120
Number of Rings N
R
Number of rings NR
FIG. 5: Evolution of the lowest lying excitation gap ∆
1,NR
for acene-ladders with increasing number of rings N , L =
FIG. 3: Evolution of the HF-magnetization MHF and HF- R
2N +1, for varying interaction strengths U (+ : 0.2,× :
gap∆HF withincreasingnumberofringsN atU=0.8and R
1,NR R 1.0;(cid:79):1.5). For comparison also the HF-result (◦) is shown.
HWBC.Asisseen,theHFmagnetizationMHF oftheacene-
IncontrasttothestronglyreducedIOonHF-level,theysur-
wire (×) is non-vanishing whenever the HF-gap (◦) exceeds
vive for not too large U in the DMRG calculations.
thenon-interactinggap(+). Forthisreason, eventhemean-
field magnetization exhibits the IO, in principle. They fade
away only for large system sizes, when the non-interacting
that have non-negligible weight in the reduced density
finite site gap falls below the interaction induced HF gap.
matrix.
By comparing to the HF result, one directly observes
the effect of strong quantum fluctuations in the DMRG-
B. DMRG calculations
trace shown in Fig. 2 (top). Fluctuation effects are vis-
ible by a reduction of the excitation gap, but also by a
We reignite the discussion from the viewpoint of modified gap evolution with U as compared to the HF
DMRG by the infinite wire limit. Within our DMRG- trajectory. The asymptotic power-law at large U seen in
implementation, this limit can only be achieved through Fig. 2 is readily understood in terms of the mapping to
finite size extrapolation: for every value of U, the ex- the Heisenberg spin-chain. It proceeds via the effective
trapolation is performed to yield one data point of the coupling constant J(U) = 4t2/U that sets the scale for
bluecurvefortheexcitationgapinFig. 2(top),whereit ∆DMRG in this limit.25–28 As is seen in Fig. 2 (top), the
1,∞
is compared against the HF result. We mention that at Heisenberg limit is reached at values of U exceeding the
small values of U numerical calculations are more chal- bandwidth, D, by about a factor of two.33
lenging to converge in DMRG, so that the low U asymp- Our final Fig. 5 displays the most important result of
toticsisnotfullyresolvedinFig. 2(top). Thedifficultyis our work. It confronts the mean-field data for the exci-
thatduetothemetalliccharacterofthewire, theentan- tation gaps with the results obtained from the DMRG-
glementcontentisspreadoveralargemanifoldofmodes calculations for finite-length wires. Specifically, the evo-
lutionoftheirDMRG-excitationgaps, ∆DMRG, isshown
1,NR
for increasing length of the wire N at different U. It is
R
0.25 F clearly seen there, that the DMRG-traces exhibit pro-
0.5 H
U=2.0 M nounced IO even for those values of U at which the
HFal gap Δ [U]1,NR 000...234 U=1.5 000...1125 Magnetization rawfldelueersnceiudtcaunyudadayote.lfiroosmVntseascernialyidlnna-mttfifiihoeunilnscidhtsreet-iihssnnuiezlot∆etrhiaH1ewe,sFsNisarRpteomisrh.ioatapnvToeiefnfheesatutyhlampetdoieleoosxsntnctirgofto-afawydtqiieuortdehannealgtiwmautepaminyst--,,,
optic 0.1 U=1.0 0.05 gered tuhnebrreobkyenrepsthoarsien.g the IO as a hallmark of the symmetry-
0 0 g
a
20 40 60 80 100 st
Number of rings NR IV. CONCLUSIONS
FIG.4: EvolutionoftheHF-magnetization(×)andHF-gap We have studied the acene Hubbard model for a large
∆HF (◦)withincreasingnumberofringsN ,L=2N +1, range of on-site repulsion U. The gap oscillations, that
1,NR R R
at U=1.0, 1.5, 2.0. As in Fig. 3 the IO vanish if the HF gap arecharacteristicoftheweakly-correlatedphase, survive
is larger than the non-interacting gap. Accordingly, at these as a periodic modulation also in the strongly-correlated
instances the staggered magnetization MHF stays finite. phase. The latter still persist for sizable interaction
6
strength, as quantum fluctuations significantly reduce symmetry in the ground state is enough to also re-
the formation of the correlation gap. storetheIO.Namely,evenassumingthatlong-rangeand
With an eye on acene-molecules, we expect that our short-range models of molecular wires share the same
results have the following implications. We have per- phases, it is still unclear whether the effective value of
formed model-calculations employing short-range inter- U in the gas-phase calculations is small or large as com-
actions. They may apply to acenes in the presence of pared to the (non-interacting) band-width.
a screening environment, such as substrates or electro- Interestingly, the authors of Ref. 8 propose an expo-
chemicalsolutions. WithscreenedCoulomb-interactions, nential decay of the lowest lying spin-triplet gap with
the effective value of U should be significantly smaller increasing N saturating near N =11 at roughly 0.13
R R
thanthemolecularbandwidth,implyingthatourweakly- eV without a trace of IO. If indeed correct, this finding
correlated scenario should apply. In contrast, for gas- could still be interpreted within the short-range model.
phase molecules screening is very weak and the applica- It could be taken as a hint that the effective interaction
bility of our model-calculation is not immediately guar- U as it appears in the study Ref. 8 should be consid-
anteed. ThisisthesituationinvestigatedinRefs. 7,8,17; ered large. However, the authors’ data underlying their
we conclude with a brief overview. exponentialfit(Table1inRef. 8)appearstobeinconsis-
Concerning Ref. 17, it remains to be seen if quantum- tent with an exponential asymptotics. In fact, the data
fluctuations can restore the spin-rotational symmetry forthespin-restrictedcasesexhibitsevennon-monotonic
also in the presence of a long-range Coulomb interac- behaviour (upturn at large wire length), which is quali-
tion. Very recent quantum-chemistry calculations on tatively consistent with the onset of IO as predicted by
the level of density-functional multi-reference theory us.
(DFT/MRCI)foroligoacenes(uptoN =9)andpp-RPA
R
(up to N =12) may give an indication that this indeed
R
could be the case.7,8
Acknowledgments
The authors of Ref. 7 confirm earlier findings29 that
best agreement between DFT/MRCI results and experi-
We express our gratitude to Holger Bettinger, Rein-
mental IR-excitation energies is achieved for closed-shell
hold Fink, Steve Kivelson and Jan Wilhelm for enspir-
referencestates(RB3LYP)eventhoughspin-unrestricted
ing discussions. Also, we thank Jan Wilhelm for shar-
calculations may produce a lower total energy. It is thus
ing a manuscript17 prior to publication. PS and RT
suggested that quantum-fluctuations are indeed strong
are supported by DFG-SFB 1170 and the ERC start-
in long acene-wires so that single-determinant approx-
ing grant TOPOLECTRICS (ERC-StG-336012). FE ac-
imations of the ground-state are not representative. In
knowledgessupportoftheDFGunderprojectsEV30/8-1
agreementwiththisinterpretation,theseauthorsobserve
andEV30/11-1. Mostofthecalculationswereperformed
that the weight of the reference-state in the DFT/MRCI
on the compute cluster of the YIG group of Peter Orth.
ground state is progressively decreasing with growing
wire length, so that more and more Slater determinants
have to be included to accurately represent this state.
The authors of Ref. 8 come to a qualitatively sim- Appendix A: Non-interacting band-structure and an
ilar conclusion. The pp-RPA method is an interest- effective model
ing variant of RPA-type approaches, since it incorpo-
rates vertex-corrections that are missed, e.g., in stan- We consider a non-interacting (U =0) polyacene wire
dardRPAbubble-summations. Thus,itallowstoinclude with the lattice and single particle Hamiltonian of (1)
a certain amount of quantum fluctuations when treat- explained in Fig. 1. For this model, we derive analytic
ing unscreened wires (i.e. with Coulomb-interactions) expressions for the wave number of the nodal point k ,
D
that would otherwise be too long for more accurate ap- Fermi velocity v , and the oscillation period of gaps of
F
proaches. A certain drawback of the method is is that it finite chains. The ladder-like lattice features a symme-
is uncontrolled: while it can singalize qualitatively that try with a line group D . An important member of
∞h
quantum fluctuations are likely to be important, there is the group is a mirror symmetry with respect to the in-
no way of telling with confidence what they actually do terchange of stringers which we call parity. The third-
in that case, quantitatively. neighbor hoppings can be divided in two groups: a hop-
On the qualitative level, the conclusions of Ref. 7 and ping that connects atoms related by the parity and the
Ref. 8 are consistent. In particular, also the authors of two hoppings that connect different rungs (Fig. 1). We
8 emphasize that the multi-determinant character of the denote the former by t(cid:48)(cid:48) and the latter by t(cid:48)(cid:48). We label
⊥ ×
groundstate increases rapidly with growing wire length. the 4 Bloch bands (cid:15) (k) by the wave-number in units
sb
Ithinderstheformationoflong-rangemagneticorderand of a−1 (the inverse lattice spacing), a parity s = ±1
implies that mean-field results (i. e. unrestricted open- and a remaining band index b = ±1. The explicit ex-
shell calculations) are to be interpreted with a grain of pression for the band dispersions is given in (2). The
salt. bands are particle-hole symmetric only if t(cid:48) =0, whence
It remains to be seen, however, if restoring the spin- (cid:15) (k)=−(cid:15) (k).
sb −sb
7
The two bands with b = −1 give rise to a linear dis-
0.2
persion bvF(k −kD) at the Fermi level if the condition U=2.5 U=10.0
tt(cid:48)(cid:48) > 0 is met, as we will show below. We derive ana- U=5.0 U=50.0
⊥ 0.15
lyticexressionsfork andv undertheconditiont(cid:48) =0.
D F
The nodal (“Dirac”) point k for t(cid:48) = 0 is the solution
D L
of the equation 1, 0.1
Δ
t2− 1tt(cid:48)(cid:48) 0.05
cos(π−k )= 2 ⊥. (A1)
D t2−t(cid:48)(cid:48)t(cid:48)(cid:48)
⊥ ×
0
Inthelimitof(π−k )smallthenodal(Dirac)pointand 0 0.025 0.05 0.075 0.1
D
the Fermi velocity are expressed by inverse length 1/L
π−kD =(cid:114)tt(cid:48)⊥(cid:48) (cid:18)1− tt(cid:48)×(cid:48) (cid:19)·(cid:2)1+O(t(cid:48)(cid:48)/t)2(cid:3) (A2) FvaIlGu.es6:U (E(cid:3)x:tra2c.5ti,o◦n:o5f.t0h,e(cid:79)D:M1R0.G0,g(cid:5)a:p5f0o.r0)lavrgiaeaintlienreaacrtiofint
of the finite site excitation gap ∆ vs. the inverse system
(cid:113) (cid:20) t(cid:48)(cid:48) 9t(cid:48)(cid:48) (cid:21) 1,L
v =2 tt(cid:48)(cid:48) · 1− × − ⊥ +O(t(cid:48)(cid:48)/t)2 . (A3) length 1/L, L=2NR+1.
F ⊥ t 8 t
For oligoacenes within the tight-binding approximation havebeenabletoinvestigate. AtvaluesU ∼1theband-
parameterized as above, the band crossing implies gap structure deviates significantly from the massive Dirac-
oscillations with the unit cell period of shape and Eq. (B1) no longer applies. (For instance we
find a stronger damping of the oscillations as expected
(cid:113) (cid:18) t(cid:48)(cid:48) (cid:19) from Eq. (B1) indicating an interaction induced flatten-
(1−k /π)−1 =π tt(cid:48)(cid:48) 1+ × +O(t(cid:48)(cid:48)/t)2 . (A4)
D ⊥ t ing of the band.)
InthelimitoflargeinteractionU thefinitesizeoscilla-
Intheequations(A2-A4)thesymbolO(t(cid:48)(cid:48)/t)2 denotes tionsarecompletelywashedoutandthethermodynamic
all terms that are of second order in t(cid:48)(cid:48) and t(cid:48)(cid:48). We see limitoftheinteractioninducedgapcanbeextractedfrom
⊥ ×
that the band crossing sets in with (cid:112)t(cid:48)(cid:48), provided that a linear fit in the inverse system size 1/L, see Fig. 6.
⊥
the signs of t and t(cid:48)(cid:48) are equal. However note, that for U =2.5 an oscillatory part is al-
⊥
Thehoppingt(cid:48) preservesthebandcrossingandweakly ready visible for not too large system sizes. Therefore
we take an heuristic approach modeling the decay of the
renormalizes k and v . We remark that the low-energy
D F
oscillations with a power law decay
behavior of the non-interacting polyacene model used in
thepresentpapercanbemappedontoasimplereffective ∆ (L)=∆ +a/L+bcos(k L+η)/L1+α.. (B3)
model with t(cid:48) = t(cid:48)(cid:48) = 0 and t and t(cid:48)(cid:48) non-zero, i.e. a 1 ∞ D
× ⊥
ladder model with alternating rungs. This model has as shown in Fig. 7. Here, a/L describes the decay of the
been introduced originally by Kivelson and Chapman19. non-oscillatorypart,whilethecosinetermdescribesade-
cay of the oscillation faster than 1/L, i.e. α>0. b gives
theamplitudeoftheoscillations,k givesperiodicityand
D
Appendix B: Extracting the bulk gap from η allows for the shifting the phase of the oscillation. α
finite-size wires is an heuristic exponent modelling the damping of the
oscillations. In our fits we always find α > 0 and α ap-
Assuming that the gap opening only affects the band proaches zero for U →0.
structureclosetotheDiracpoint,onewouldexpectthat The gaps shown in Fig. 2 are obtained from the linear
the gaps are given by a minimal model of a type fits for U >2.0, see Fig. 6, and from fits of Eq. (B3) for
U ≤2.0. We mention that the fit for U =1.0 is still sen-
(cid:113) sitivetothefitrange,andtheresultshouldbetakenwith
∆DMRG(N ) = ∆(N )2+v2∆k2(N ) (B1)
R R F R care. For this reason we refrained from including results
π for U < 1.0. In addition, in that regime it is not suffi-
∆k(N ) = cos(k N +φ), (B2)
R 2N +2 D R cienttofitasinglecosine, astheresultsaregettingclose
R
to the saw tooth like behavior as in the non-interacting
that follows the zone-folding argument of Ref. 12. The
case. In order to fit the results for small U we introduce
sqrtterminEq.(B1)accountsforthelevelcrossingwith
a (truncated) saw tooth function
parameters ∆(N ), v , and φ; ∆(N ) and φ are fit-
R F R
parameters that accommodate additional finite-size ef- (cid:80)n cos((cid:96)x)/(cid:96)2
g(x)= (cid:96)=1,3,5,···, (B4)
fects. Somewhat unexpectedly, our calculations indicate (cid:80)n (cid:96)−2
(cid:96)=1,3,5,···
that the band-structure underlying Eq. (B1) does not
provide a faithful description of the true bandstructure and use it instead of the cosine in Eq. (B1). Here we
of the infinite wire with the range of U-values that we imposed a limit of n=17. The results for U = 0.2, 0.5
8
where HˆU=0 denotes the non-interacting system accord-
0.2
U=1.0 U=1.5 U=2.0 ing to Eq. (1), σ denotes up and down spins, −σ the
opposite spin to σ, and n¯ the expectation value of the
σ,x
0.15 local density operator nˆσ,x at site x with spin σ. We
have checked that by breaking the remaining Sz con-
tot
servation, therefore allowing for Fock terms of the form
L
1, 0.1 cˆ† cˆ ,weonlyfindsolutionswherethespinquantiza-
Δ σ,x −σ,x
tion axis is rotated with respect to the Hartree solution.
Note that in this case the solution depends on the ini-
0.05
tialization of the self consistency loop.
Since up and down spins do not mix in (C1), we can
0 represent the up and down spin sectors by independent
0 0.025 0.05 0.075 0.1 matrices. WestarttheHFcalculationsbyfirstdiagonal-
inverse length 1/L izing the non-interacting system in order to obtain the
ground state energy E as a reference. Note that in this
0
FIG. 7: Extraction of the DMRG gap ∆1,∞ for not too case we obtain a homogeneous solution n¯σ,x = 0.5. We
large interaction values U (×: 1.0, +: 1.5, (cid:79): 2.0). The then explicitly introduce a staggered magnetic order by
extrapolation is performed using Eq. (B3) for ∆ vs. the setting n¯ = 0.5 ± 0.01σ, where we take ’+’ on one
1,L σ,x
inverse length 1/L, L = 2NR +1, and displayed by lines, sublattice of the bipartite system and ’−’ for the other
while the symbols correspond to the DMRG results. sublattice (Fig. 1). With these initial values we diago-
nalize HHF and calculate the locate densities n¯ which
σ,x
we then iteratively insert into (C1). We perform at least
0.25
U=0.2 U=0.5 fiveiterationsandcontinueuntilthegroundstateenergy
EHF of Hamiltonian HHF changes by less then 10−10t.
0.2
In order to avoid getting stuck in an oscillation between
two solutions, we damp the HF self consistency loop by
0.15
taking an average of 0.7 times the new density and 0.3
L
Δ1, times the densities of the preceding iteration.
0.1 Finally,wecheckwhetherthegroundstateenergyEHF
including the double counting corrections
0.05
UM (cid:88)
EHF =ESC − + U n¯ n¯ (C2)
0 4 σ,x −σ,x
0 0.025 0.05 0.075 0.1 j
inverse length 1/L is smaller than the ground state energy E of the non-
0
interacting system. Indeed we always find EHF ≤ E
0
FIG.8: ExtractionoftheDMRGgap∆1,∞forsmallinterac- The spin-averaged local density n¯↑,j + n¯↓,j of our HF
tionvaluesfrom∆1,Lvs. theinverselength1/L,L=2NR+1. solution is homogenous and the total staggered magneti-
For not too large U (×: 0.2, ◦: 0.5). The fits correspond to sation MHF is given by the difference between the mag-
Eq.(B1)wherethecosineisreplacedbyasawtoothEq.(B4)
netization of the two sub lattices
1
m = (n¯ −n¯ ), (C3)
are displayed in Fig. 8. While the fits do capture the j 2 ↑,j ↓,j
oscillationquietwell,thefinitesizeoscillationsareorders (cid:88) (cid:88)
MHF = m − m . (C4)
of magnitudes larger than the extracted gaps. In return j j
we do not trust the extracted values and did not include j∈A j∈B
them in Fig. 2.
2. Hartree-Fock: commensurate case
Appendix C: Methodological details
The main problem of obtaining the U (cid:28) 1 HF solu-
1. Numerical Hartree-Fock tions in Fig. 2 roots in the finite size oscillations being
incommensurable. Therefore we need very large system
sizes in order to start with an U = 0 finite size gap ∆
Due to SU(2) symmetry combined with a local on- 0
beingsmallerthantheinteractioninducedgapintheHF
site interaction, we can restrict our HF equations to the
calculations. This extraction of the HF gap gets simpli-
Hartree terms
fiedifweswitchtothecaseofcommensurateoscillations
HHF =HˆU=0 + U(cid:88)(n¯ −0.5)nˆ , (C1) by choosing a fine-tuned t(cid:48)(cid:48) from our solution Eq. (A1).
σ,x −σ,x
InFig.9wedisplaytheresultsinthecaseofaperiodicity
x,σ
9
in Fig. 2.
1
COHF
0.00323 U
10-1
exp( -D0/( U + D1U2))
10-2
3. DMRG
]
U
[ 10-3
∞
HFΔ1,10-4 Within our DMRG calculations we keep up to 8000
D0=3.01 statesperblock; thedimensionofthetargetspacegrows
10-5 D1=0.397 up to 4.6·107, the discarded entropies are considerably
10-6 below 10−4 for U >1.0, while for U ≤1.0 the discarded
entropies reach 4.2·10−2 for systems with N > 20 in
R
10-4 10-3 10-2 10-1 1 10 eachDMRGstep,typicallysignificantlysmaller. Weem-
ploy at least 7 finite lattice sweeps with an optimized in-
U
finite lattice warm up for the smallest system sizes. In
order to reach large system sizes we iteratively restart a
FIG.9: TheHFgap∆HF asshowninFig.2,left,inthecase
of commensurate oscilla1t,∞ions t(cid:48)(cid:48) =0.13 where the periodicity NR simulationinordertoinvestigateaNR+1ring. Note
is given by 20 rings. The gap is obtained by HF calculations that for this we only have to deactivate the wave func-
for N =20,40,60,··· ,200. tionpredictionintheinitialstepofarestart. Inorderto
R
initialize the simulations we apply a sliding environment
block approach, that we already employed successfully
oftwentyrings. Weobtainedthecorrespondingt(cid:48)(cid:48) =0.13 tofractionalquantumhallsystems.30–32 Insteadofusing
by solving Eq. (A1) numerically.34 Since we now always a reflection or a growing right block during the DMRG
hit a minimum of the gap oscillations precisely, i.e. a infinite lattice warm up sweeps, we employ a right block
zero in the non-interacting case, we can extract the HF of up to 5 sites, which we can always build exactly in
gapwithoutdifficulties. Asexpected,theHFgapindeed such a way that we always have 4N sites for PBCs and
R
opens ∆HF ∼ U2. In addition, we see the same expo- 4N +2sitesforHWBCs. Withthisinitialwarmup,we
1,NR R
nential behavior for intermediate to large U as displayed avoid running into excited states in the initial sweep.
∗ Electronic address: [email protected] 12:305, 2016.
wuerzburg.de 8 YangYang,ErnestR.Davidson,andWeitaoYang. Nature
1 Jan Von Delft and Daniel C Ralph. Spectroscopy of dis- of ground and electronic excited states of higher acenes.
crete energy levels in ultrasmall metallic grains. Physics Proc. Nat. Acad. Sci., 113:E5098, 2016.
Reports, 345(2):61–173, 2001. 9 I.Dabo,A.Ferretti,Cheol-HwanPark,N.Poilvert,Y.Li,
2 I.S.Beloborodov,A.V.Lopatin,V.M.Vinokur,andK.B. M. Cococcioni, and N. Marzari. Donor and acceptor lev-
Efetov. Granular electronic systems. Rev. Mod. Phys., elsoforganicphotovoltaiccompoundsfromfirstprinciple.
79:469–518, Apr 2007. Phys. Chem. Chem. Phys., 15:685, 2013.
3 ManabuKiguchi,OrenTal,So¨renWohlthat,FabianPauly, 10 Chr. To¨nshoff and H. F. Bettinger. Photogeneration of
Michael Krieger, Darko Djukic, Juan Carlos Cuevas, and octacene and nonacene. Angew. Chem. Int. Ed., 49:4125,
JanMvanRuitenbeek. Highlyconductivemolecularjunc- 2010.
tionsbasedondirectbindingofbenzenetoplatinumelec- 11 H.BettingerandChr.T¨onshoff. Thelongestacene. Chem.
trodes. Physical review letters, 101(4):046801, 2008. Rec., 15:2015, 2015.
4 TamarYelin,RanVardimon,NataliaKuritz,RichardKo- 12 Richard Koryta´r, Dimitra Xenioti, Peter Schmitteckert,
rytar,AlexeiBagrets,FerdinandEvers,LeeorKronik,and M´ebarek Alouani, and Ferdinand Evers. Signature of the
OrenTal. Atomicallywiredmolecularjunctions: Connect- dirac cone in the properties of linear oligoacenes. Nature
ing a single organic molecule by chains of metal atoms. communications, 5, 2014.
Nano letters, 13(5):1956–1961, 2013. 13 S. Reich, Chr. Thomsen, and J. Maultzsch. Carbon nan-
5 Tamar Yelin, Richard Koryta´r, N. Sukenik, Ran Vardi- otubes: Basic concepts and physical properties. 2004.
mon, B. Kumar, C. Nuckolls, Ferdinand Evers, and Oren 14 L. Brey and H. A. Fertig. Electronic states of graphene
Tal. Conductancesaturationinhighlyconductivemolecu- nanoribbons studied with the dirac equation. Phys. Rev.
lar junctions. submitted, 2015. B, 73:235411, Jun 2006.
6 H. Bettinger. Electronic structure of higher acenes and 15 A. Dasgupta, S. Bera, F. Evers, and M. J. van Setten.
polyacene: Theperspectivedevelopedbytheoreticalanal- Quantumsizeeffectsintheatomisticstructureofarmchair
yses. Pure Appl. Chem., 82:905, 2010. nanoribbons. Phys. Rev. B, 85:125433, Mar 2012.
7 H. Bettinger, Chr. To¨nshoff, M. Doerr, and E. Sanchez- 16 G.Karakonstantakis,L.Liu,R.Thomale,andS.A.Kivel-
Garcia. Electronically excited states of higher acenes up son. Correlations and renormalization of the electron-
to nonacene: A density functional theory/multireference phonon coupling in the honeycomb hubbard ladder and
configurationinteractionstudy.J.Chem.TheoryComput., superconductivity in polyacene. Phys. Rev. B, 88:224512,
10
Dec 2013. change interactions. Phys. Rev., 115:2–13, Jul 1959.
17 Jan Wilhelm, Mauro Ben Del, and Ju¨rg Hutter. Gw in 27 L. N. Bulaevskii. Zh. Eksp. Teor. Fiz., 51:230, 1966.
the gaussian and plane waves scheme with application to 28 M Takahashi. Half-filled Hubbard model at low tempera-
linear acenes. J. Chem. Theo. Comp., 12:3623, 2016. ture.JournalofPhysicsC:SolidStatePhysics,10(8):1289,
18 N.D.MerminandH.Wagner. Absenceofferromagnetism 1977.
orantiferromagnetisminone-ortwo-dimensionalisotropic 29 R. Mondal, Chr. To¨nshoff, D. Khon, D. C. Neckers, and
heisenberg models. Phys. Rev. Lett., 17:1133–1136, Nov H. F. Bettinger. Synthesis, stability, and photochemistry
1966. ofpentacene,hexacene,andheptacene: Amatrixisolation
19 S. Kivelson and O. L. Chapman. Polyacene and a new study. J. Am. Chem. Soc., 131:14281, 2009.
class of quasi-one-dimensional conductors. Phys. Rev. B, 30 Zi-Xiang Hu, Z. Papic, S. Johri, R. N. Bhatt, and Peter
28:7236–7243, Dec 1983. Schmitteckert. Phys. Lett. A, 376(30-31):2157–2161, JUN
20 StevenR.White. Densitymatrixformulationforquantum 18 2012.
renormalization groups. Phys. Rev. Lett., 69(19):2863– 31 SonikaJohri,Z.Papic,R.N.Bhatt,andP.Schmitteckert.
2866, November 1992. Phys. Rev. B, 89(11), MAR 24 2014.
21 Steven R. White. Density-matrix algorithms for quan- 32 SonikaJohri,Z.Papic,P.Schmitteckert,R.N.Bhatt,and
tum renormalization groups. Phys. Rev. B, 48(14):10345– F. D. M. Haldane. NJP, 18, FEB 5 2016.
10356, October 1993. 33 We adopt the ratio ∆HF/∆DMRG taken at a cross-over
∞ ∞
22 Zexing Qu, Dawei Zhang, Chungen Liu, and Yuansheng scale,D,toquantifythestrengthofquantum-fluctuations
Jiang. Open-shell ground state of polyacenes: a va- inaparametricfashion.AnaturalchoiceforDisthe(non-
lence bond study. The Journal of Physical Chemistry A, interacting) bandwidth that according to Fig. 1 (right) is
113(27):7909–7914, 2009. given by D ≈ 6t. Furthermore, ∆DMRG(U≈D) ∼ t2/D
∞
23 JohannesHachmann,JonathanJDorando,MichaelAvil´es, while ∆HF(U≈D)∼D so that we expect a large ratio
∞
and Garnet Kin-Lic Chan. The radical character of the
acenes: Adensitymatrixrenormalizationgroupstudy.The ∆H∞F (cid:12)(cid:12)(cid:12) ∼ D2;
Journal of chemical physics, 127(13):134309, 2007. ∆DMRG(cid:12) t2
24 P. Lykos and G. W. Pratt. Discussion on the hartree-fock ∞ U≈D
approximation. Rev. Mod. Phys., 35:496–501, Jul 1963. ittakesvaluesabout36broadlyconsistentwiththefactor
25 P. W. Anderson. Antiferromagnetism. theory of superex- 35±10 seen in Fig. 2(top).
change interaction. Phys. Rev., 79:350–356, Jul 1950. 34 For reference, the precise value used in the numerics is
26 P. W. Anderson. New approach to the theory of superex- t(cid:48)(cid:48) =0.130064666287555.
