arXiv:1705.08474v2 [cond-mat.str-el] 29 Jun 2017 it ue-zln odslyasnlttilttransition singlet-triplet a display pre- to approaches numerical fused-azulene polariza- of magnetic kind dicts same display The not [12–15]. to tion reported theoreti- been and also been techniques has has many-body different length), by investigated bond cally [1–4]. symmetric rings with (benzene aromatic oligoacenes chain symmetric of monomers, number nine Beyond the (fi- oligoacene with increasing of promising conductivity a devices displays state chain) Solid fused-benzene reported nite is magnetic. and be 11] to [10, not monomers nine to up synthesized carbon of number odd as with molecules, rings atoms. azulene conjugated fused one- having from formed a so chain is second linear the a first edges is zigzag the with While stripe graphene dimensional fused. laterally hydrocarbons (C teto r us-ndmninlmlcls Exam- of deal (C fused-benzene great molecules. are attracted ples quasi-unidimensional Reli- has explored are that Ref.[9]). be attention systems see to be physical review to hard recent able atoms be a of (for can number computationally systems the these single on build- instead considered, Once depending molecules are then goal. materials atoms, such a of still blocks ma- is paramag- ing functional super ferromagnetism for as and such seeking netism be- properties the magnetic compounds [8], with magnetic reality terials organic [5– a purely application already of ing biomedical Despite on research 7]. to electronic of [1–4] engineering devices from spanning direction several ∗ [email protected] h ue-ezn eiso oeue a led been already has molecules of series fused-benzene The h td fcro ae aeil a rw to grown has materials based carbon of study The 8 n +2 H 4 n +4 epu odsrb h antcgon tt falre set larger a of state ground magnetic chain the these calculations describe of (DMRG) to helpful Group geometry Renormalization frustrated that Matrix the hypothesis Density from the comes supports oligomers results our fused-azulene For lo oacrtl ne bu h antcsaeo hs l these of state magnetic the about infer accurately perturbati to of correl allow means electronic by of treatment correlation perturbation electronic second-order low of range a in tnadmdlfrcnuae oeue,w conside we molecules, parametrize conjugated empiric for the model of standard symmet Instead of a magnetic. nian, composed no system always similar rep be A been to has size. that charac system molecule be with a to moment fused-azulene, hard are oligomers this long of makes ample that reasons the of one eododrprubto hoyt eemn h magneti the determine to theory perturbation Second-order lgmr ul rmplcci aromatic polycyclic from built oligomers ) ojgtdsse aecmlxbhvoswe nraigt increasing when behaviors complex have system Conjugated .INTRODUCTION I. 1 2 etoA´mc aioh n nttt asio NA 84 CNEA, Balseiro, Instituto and At´omico Bariloche Centro osj ainld netgcoe in´fia T´ecnic Cient´ıficas y Investigaciones de Nacional Consejo 4 n ffiiesz rmtchdoabn molecules hydrocarbons aromatic size finite of +2 H 2 n +4 n fuzed-azulene and ) .Valentim A. Dtd uut1,2021) 12, August (Dated: ,2, 1, ∗ n .J Garc´ıa J. D. and e sdsrbdb xeddmlclrobtlcom- orbital single molecular sys- a extended the by a of only azu- by physics posed the the described in and is hy- neglected link tem [17] are theory twisted atoms orbital a molecular drogen oligomers by Within these differ monomer. fig- of lene only at representation represented Fig.1(b-c)) like not ( ladder are The dou- (which bonds alternate ure). single by and linked compose ble hydrocarbons as unsaturated described by be can conventionally Fig.1(a-b)) fer- proprieties. displaying so ferromagnetic as and Fused- candidate [15], roelectricity a material 16]. as multiferroic presented organic few 15, been of a [9, also has only fused oligomers when linearly azulene oligomer are the molecules of azulene polarization spin with osa adrrpeettosdnt h oiin ftec the Th of positions structure. atoms. the polymer bon denote each representations ladder of at cell dots enclo unit brackets the square represents The ladd areas (d). respective and its (c) and representations (b) fused-azul like molecule a fused-benzene for a structure and Schematic (a) online) (color 1. FIG. oe nrytiltectto hnbneemolecules. a benzene have For than to excitation atoms. molecules triplet carbon azulene energy of expect lower number would we odd reason with this rings the to due on tion ( π obtl) thl le adteeetoi distribu- electronic the band filled half At -orbitals). hs lnrmlcls(ceaial hw at shown (schematically molecules planar These tosb en fRayleigh-Schr¨odinger of PT means by ations nter P) eso htasimple a that show We (PT). theory on re odsly nicesn magnetic increasing an displays to orted π h ihsi rudsaeo azulene on state ground high-spin the h ubr aitna oexplore to Hamiltonian Hubbard the r fcnuae molecules. conjugated of n complex ong obtl losmgei rsrto nazulene, on frustration magnetic allows -orbitals aie-arPpe(P)Hamilto- (PPP) Pariser-Parr-Pople d ex- An methods. numerical by terized u rcdr rcdr ol be could procedure procedure Our . .W aiaeti prahusing approach this validate We s. enme fmnmr,wihis which monomers, of number he i ue-ezn ig sreported is rings fused-benzene ric s(OIE) Argentina (CONICET), as ,2 1, 0Brlce Argentina Bariloche, 00 π cnuae molecules. -conjugated p z ria nec abnatom carbon each on orbital state c ene sed ar- er e 2
Low electronic excitations, understood as arising II. MODEL AND METHOD from delocalized electrons in the molecular orbital (π- electrons), have been appropriately described by the We used the Hubbard model [29] to describe the elec- empiric Pariser-Parr-Pople Hamiltonian [18, 19] for un- tronic interactions in conjugated systems shown at Fig.1. saturated π-conjugated organic molecules. For sym- The model Hamiltonian contains a non-interacting part metric fused-azulene molecules the ground state of Hˆ0 and a term that incorporates the on-site electron- the PPP model − with inter-site Coulomb interaction electron interaction Hˆ1: parametrized by the Ohno [20] formula and fixed on-site ˆ ˆ ˆ Hubbard interaction 4.69 (in units of transfer integral H = H0 + H1. (1) between bonded sites)− is singlet up to 5-azulene. Be- The non-interacting part is a tight-binding Hamilto- yond this number of monomers and up to 11-azulene the nian, oligomers have a triplet state [15]. The fuzed-azulene N geometry has been also explored, in the strong corre- ˆ † † lated regime, by using the spin-1/2 antiferromagnetic H0 = −t cˆiσcˆjσ +ˆcjσ cˆiσ , (2) model Heisenberg model [21]. The magnetic ground state
At next section we briefly describe the model and nu- (0) merical approaches used, in section III the results are being |k i a state of the unperturbed system. Notice ˆ (0) presented and our final remarks are given in section IV. that H1 is not diagonal at the base of states of |k i. 3
Details on perturbation theory for the Hubbard model state while fused-azulene presents a increasing ground at low correlated regime are presented in [31]. Values of state with the chain length. DMRG calculations vali- E(0), α and β to different oligoacene and fused-azulene dates this analysis, with the agreement between different oligomers can be find at Table I. approaches increasing with the size of the system. We < take U/t ∼ 2 as the upper limit of validity for the results obtain from perturbation theory, as for larger U/t values B. Density Matrix Renormalization Group H1 becomes the most relevant term in the Hamiltonian [37, 38]. Although the DMRG algorithm was first presented To estimate Uc we calculate the singlet-triplet spin gap to treat an unidimensional problem [32], different low- defined as ∆E(MS )= E(MS = 1) − E(MS = 0) and de- dimensional systems can also be accurately treated with fine Uc as the value where this gap closes ∆E(MS) = 0. this approach [33, 34]. For the ladder like representa- This analysis is illustrated in Fig.2(a-b) for two linearly tions of oligoacene and fused-azulene in Fig.1 (c-d), we fused 2-azulene molecule ( 18 atoms). Panel (a) shows can map the Hubbard Hamiltonian with first-neighbors interactions, into a unidimensional Hamiltonian with up to second-neighbors interactions, easily implemented on a DMRG algorithm. FOPT MS=0 Oligomers described by Hamiltonian Eq.1 have SU(2) -25 FOPT MS=1 symmetry so we can take advantage of the degeneracy SOPT MS=0 of spin projections MS, for a total spin S, and compute SOPT MS=1 the low-lying energy of the system in a chosen subspace. DMRG MS=0 -30 DMRG M =1 The ground state energy in MS subspace is find when S
|MS| ≤ S and E(S) ≤ E(S + 1), where E(S) is the E/t lowest energy belonging to a given S value [35, 36]. To establish the accuracy of our DMRG simulations we -35 have compared DMRG results for the energy at U = 0 with the exact solution of the tight-binding Hamiltonian. (a) We compute the ground state-energy with different sizes -40 of the Hilbert space, and we kept a typical cutoff of m = 0 1 2 3 800 to a maximum of m = 1200 states per block at final U/t iteration. In these conditions the energy precision is in the fourth decimal digit. The relative energy error is 0.6 comparable to the DMRG weight lost kept lower than (b) 10−5 in the worst cases. 0.5
III. RESULTS AND DISCUSSIONS )/t S 0.4 E(M
We consider DMRG simulations and a conventional ∆ PT up to second-order to compute the energy and 0.3 spin gap excitations of fused-azulene and fused-benzene- FOPT molecules with up to 74 π-orbital (9 azulene and 18 ben- SOPT DMRG zene monomers). When referring to the number of π- 0.2 orbital or number of atoms we are counting only carbon atoms. We explore the Hubbard’s model phase space 0 1 2 3 U/t with correlations ranging from the noninteracting limit, U = 0, to U = 3t and different systems size. We calcu- (0) late the unperturbed lowest energy E (MS), the first- FIG. 2. (color online) Panel (a) shows low-lying energies order α(MS) and second-order β(MS ) coefficients of the E(MS = 0) and E(MS = 1), and panel (b) shows the singlet- Rayleigh-Schr¨odinger PT for the system with spin pro- triplet spin gap ∆E(MS), both as a function of U for a fused- jection MS = 0 and MS = 1. These results, for different azulene molecule with 2 monomers (18 atoms). The DMRG oligomer sizes, and the value for Uc obtain by first- and data has a error bar which is smaller than the symbols size. second-order PT and by DMRG are shown at Table I. We find that by using fist-order PT both systems un- the energy computed on the MS = 0 and MS = 1 dergoes a singlet-triplet state transition in a Uc which spaces. We observe that the curve for E(MS = 0) lies depend of the oligomer size. However by means of second- slightly below E(MS = 1) curve, what indicates a sin- order PT we qualitatively reproduce the numerical pre- glet state with a finite gap between the ground state en- dictions, that oligoacene always has a singlet ground ergy, E(MS = 0), and the energy of the first magnetic 4
(0) (0) (1) (2) NE (0) α(0) β(0) E (1) α(1) β(1) Uc Uc Uc 2-azulene 18 -24.521 -4.449 -0.284 -24.016 -4.557 -0.271 4.709 - - 4-azulene 34 -46.807 -8.414 -0.532 -46.586 -8.507 -0.517 2.362 - - 5-azulene 42 -57.947 -10.396 -0.656 -57.787 -10.485 -0.641 1.796 - 2.6 6-azulene 50 -69.088 -12.378 -0.780 -68.967 -12.464 -0. 765 1.409 2.376 2.0 7-azulene 58 -80.228 -14.360 -0.904 -80.133 -14.443 -0. 889 1.134 1.593 1.8 9-azulene 74 -102.509 -18.324 -1.153 -102.447 -18.404 -1.137 0.779 0.965 1.0 4-acene 18 -24.930 -4.500 -0.294 -24.340 -4.604 -0.275 5.636 - - 10-acene 42 -58.601 -10.500 -0.696 -58.451 -10.559 -0.658 2.518 - - 18-acene 74 -103.492 -18.500 -1.242 -103.439 -18.537 -1.183 1.415 - -
(0) TABLE I. Non-interacting lowest energy E (MS), first- α(MS) and second-order β(MS ) coefficients of perturbation theory and critical value of electronic correlation Uc obtain by FOPT, SOPT, and DMRG calculations respectively (all in units of t) for different number of monomers (N atoms) in oligoacene and fused-azulene molecules.
excited state, E(MS = 1). As expected, DMRG is the lowest energy and second-order PT (SOPT) data is closer to DMRG results than first-order PT (FOPT). At panel (a) (b) we plot the singlet-triplet spin gap as a function of electronic correlations. A finite gap indicating a singlet state is found in agreement with Refs.[9, 15, 16]. FOPT results points out to a transition from singlet to triplet 0.1 ground state for values of Uc above the validity of the approximation (not displayed at the plot). In Fig.3, we plot the singlet-triplet spin gap as a func- tion of electronic correlations for two different sizes of )/t fused-azulene molecules. FOPT and DMRG calculations S FOPT displays a finite gap in both cases, for small values of 0 SOPT
E(M DMRG
U/t, indicating a singlet state. SOPT estimate Uc be- ∆ low 2t when the chain is larger than 6-azulene. In Ta- ble I we show that the agreement of the value of Uc be- tween the different approaches increases with the num- 0.1 ber of monomers. These results on fused-azulenes could lead to the incorrect assumption that first-order treat- ment of electronic correlations is enough to describe these molecules. We show that for fused-benzene FOPT in- correctly predicts a singlet-triplet transition while SOPT correctly predicts that the ground state for fused-benzene (b) is always a singlet. 0 0 1 2 3 At Fig.4 we repeat the analysis of the singlet-triplet spin gap for ten fused monomers of benzene (10-acene U/t with 42 atoms). DMRG simulations and SOPT cal- culus exhibit a significant spin gap while using FOPT FIG. 3. (color online) Singlet-triplet spin gap ∆E(MS) as a the spin gap goes to zero. Second-order PT predicts a function of U/t for (a) 5-azulene and (b) 6-azulene oligomers singlet ground state for all values of U/t in agreement with 42 and 50 atoms respectively. Arrows points the values with DMRG. We checked these results up to 18 fused of Uc estimated with each approach. The DMRG data has a monomers (see Table I) and we can reproduce the ex- error bar which is smaller than the symbols size. pected behavior of spin excitations of oligocene chains, i.e. a finite spin gap which indicates that the ground state of the system is a singlet independently of the elec- accumulates states in the Fermi level, but only fused- tronic correlation magnitude and system size [15]. azulene shows a transition to a high spin state, ruling The possibility for ferromagnetism in odd-membered out flat-band as the source for the magnetism on azulene rings oligomers (fused-azulene geometry) could be related oligomers. Geometric frustration on n-azulene increases to flat-band magnetism [39, 40], a ferromagnetic behavior as the chain length increases (more twisted links as in due to the accumulation of degenerated states at Femi Fig.1c) and the Coulomb repulsion needed for a S = 0 level. As shown at Fig.5 both n-acene and n-azulene to S = 1 transition diminish. These results points to 5
state for 2-azulene and 3-azulene with a decreasing of the singlet-triplet spin gap as a function of the chain length. They argue this would indicate a tendency to FOPT a magnetic ground state, however simulations on bigger 0.3 SOPT oligomer sizes are need in order to confirm the predic- DMRG tion by DMRG [15]. The authors also conjecture the increasing frustration percentage (number of frustrated )/t S 0.2 links, n, divided by the number of non-frustrated links, 9n + 1) as the origin of a possible higher ground state for E(M
∆ long azulene oligomers [9]. In agreement with Ref.[15] and our results, density functional theory methods also 0.1 reports a triplet state for 6-azulene [16]. In summary, second-order PT allowed us to explore the low correla- < tion regime (U/t ∼ 2) and find the same magnetic transi- 0 0 1 2 3 tions shown by more sophisticated and computationally U/t demanding numerical methods. In the specific case of fused-azulene our second-order PT results show that a magnetic transition in this weak interaction regime hap- FIG. 4. (color online) Singlet-triplet spin gap ∆E(MS) as a pens for oligomers with more than four monomers. function of U/t for 10-benzene oligomer with 42 atoms. The DMRG data has a error bar which is smaller than the symbols size. IV. CONCLUSION frustration (absent on oligoacene) as the source of the magnetic transition. We employ Rayleigh-Schr¨odinger perturbation theory and DMRG technique to study low-lying spin excitations for two geometrically related conjugated systems. We in- vestigate the ground state of these conjugated molecules by using the Hubbard model with weak on-site coulomb 6 interaction, ranging from the non-interacting limit to 4 3 U/t = 3, and different monomer length. We show that second-order perturbation theory is a reliable tool to 6 characterize magnetic transitions on complex molecules, 3 3 using as a non trivial example fused-benzene and fused- azulene oligomers. Our results support that for n-azulene -2 -1 0 1 2 there is a magnetic transition from singlet to triplet as 2 E/t the size the system increases. This transition is not re- lated to flat-band magnetism, instead, the comparison with oligoacene suggest that the origin of magnetism in 1 fused-azulene oligoacene fused-azulene oligomers comes from the frustrated geom- etry of these chains. For fused-azulene the value of Uc Number of states at Fermi level in which a transition from a singlet to a triplet state 0 10 20 30 40 50 60 70 80 90 take place decrease with the size of the system, being lower than 2t for oligomers bigger than 4-azulene. The Number of carbon atoms increasing agreement between SOPT and DMRG simula- tions, for longer oligomers, makes this approach a useful FIG. 5. (color online) Accumulation of the number of states tool to the investigation of conjugated organic molecules, near the Fermi level for oligoacene and azulene molecules as a at weak correlated regime, to system-sizes that would be function of oligomer size. The number of states is computed impossible by means of numeric techniques. within a range of 0.1t around the Fermi level for the tight- binding model. The inset displays the number of states over the whole energy spectrum for fused-azulene and oligoacene chains with 42 atoms. V. SUPPLEMENTARY MATERIAL
DMRG results on antiferromagnetic spin-1/2 Heisen- See supplementary material (1) for a detailed deriva- berg Hamiltonian for both azulene and benzene oligomers tion of perturbation theory for the Hubbard model at low shows that the magnetic state does not require the ex- correlated regime and supplementary material (2) for a istence of flat energy bands [15]. Configuration inter- Maxima [41] script to compute the fist- and second-order action on PPP model also reports a S = 0 ground quantities presented in Table I. 6
VI. ACKNOWLEDGMENTS
We acknowledge a fruitful conversation with Suchi Guha. This work has been supported by CONICET and PICT 2012/1069.
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