Z. Phys. B 104, 27–32 (1997) ZEITSCHRIFT FUR¨ PHYSIK B c Springer-Verlag 1997 Spin density wave and ferromagnetism in a quasi-one-dimensional organic polymer Shi-Dong Liang1, Qianghua Wang1;2, and Z. D. Wang1 1 Department of Physics, University of Hong Kong, Pokfulam Road, Hong Kong 2 Department of Physics, Nanjing University, Nanjing 210093, People’s Republic of China Received: 5 February 1997 Abstract. The spin density wave(SDW) – charge density impurities, in which unpaired localized d or f electrons inter- wave(CDW) phase transition and the magnetic properties in act with the itinerant π electrons. Fang et al. [3] investigated a half-filled quasi-one-dimensional organic polymer are in- the stability of the ferromagnetic state and the spin configu- vestigated by the world line Monte Carlo simulations. The ration of the π- electron in a quasi-one-dimensional organic itinerant π electrons moving along the polymer chain are polymer by a mean field approach(the Hatree-Fock approx- coupled radically to localized unpaired d electrons, which imation). They found that not only the interaction between are situated at every other site of the polymer chain. The the π electrons but also the interaction between the π elec- results show that both ferromagnetic and anti-ferromagnetic trons and unpaired d electrons play an important role for radical couplings enhance the SDW phase and the ferromag- the ferromagnetic order, and the spin density wave(SDW) net order of the radical spins, but suppress the CDW phase. states is closely related to the ferromagnetism of this sys- By finite size scaling, we are able to obtain the phase tran- tem. However, the nearest neighbor interaction between the sition line in the parameter space. The ferromagnetic order π electrons has not been taken into account so that the inter- of the radical spins are observed to coexist with the SDW esting CDW-SDW transition as well as the effect of the π π phase. As compared to the system being free of the radical and π d electron interactions on the ferromagnetism cannot− coupling, the phase transition line is shifted upward in the be examined− in their work. On the other hand, the mean field U-V parameter space in favor of larger V , where U is the treatment on the total Hamiltonian introduces some uncertain on-site repulsion and V is the nearest-neighbor interaction approximation due to ignorance of the correlation effect at between the π electrons. All of these findings can be un- the outset. The world-line quantum Monte Carlo simulation derstood qualitatively by a second-order perturbation theory method [4] are, however, almost approximation-free aside starting from the classical state at zero temperature in the from the statistical error. In retrospection, Hirsh [4] used strong coupling limit. We also address the consequences of this technique to study the CDW-SDW transition in the half- the radical coupling for the persistent current if the polymer filled extended Hubbard model and the Su-Schrieffer-Heeger chain is fabricated as a mesoscopic ring. model. When the radical coupling is introduced, the inter- play between the π-π electron interaction and π-d electron PACS: 71.20.Rv; 75.30.Fv; 75.30.Hx; 75.40.Mg interaction is by no means trivial, and is in fact important to the ferromagnetism in the organic chain. [3] In this paper, we investigate the CDW-SDW transition and the magnetism of quasi-one-dimensional organic polymer by the world line Monte Carlo technique [6]. We focus on the dependence of I. Introduction the the CDW-SDW phase transition, magnetism and other relevant properties on the interaction between the localized With the rapid development of experimental technology, d electrons and the itinerant π electrons. In Sect. II, we de- low-dimensional systems, such as organic polymers, show scribe the model and Monte Carlo method. The results are attractive potential applications. In particular, the discovery discussed in Sect. III. A summary and some discussion can of organic polymer superconductivity and ferromagnetism be found in the Sect. IV. makes it possible to search superconductors and ferromag- nets based on organic molecules, and brings a challenge to theorists. A quasi-one-dimensional π-conjugated struc- II. Model and Monte Carlo method ture of the organic polymer shows interesting electric and ferromagnetic properties. Ovchinnikov et al. [1] and Cao A typical structure of organic polymer, such as poly-BIPO, et al. [2] discovered the ferromagnetism of poly-BIPO[1,4- consists of a main zigzag chain containing π-conjugated car- bis-(2,2,6,6- tetramethyl-4-oxyl)-4-piperidyl-butadiin]. They bon atoms and a kind of side radical containing an unpaired attributed this kind of ferromagnetism to the transition-metal electron, which may be regarded as a quasi-one-dimensional 28 system. The interaction between electrons dominates the be described in terms of the CDW or SDW phases, which are magnetic properties of the system. Although the electron- related to the ferromagnetism of the system, we will study a phonon interaction can suppress the SDW state of the sys- half-filled chain and assume that the spin-up and spin-down tem, [3] our attention will be focused on the effect of corre- π electron number is equal. The other filling fraction may be lation between the itinerant π electron and the local unpaired addressed elsewhere. In order to investigate the CDW-SDW electron on the CDW-SDW phase transition and the mag- transition, we shall need the CDW order parameter netic properties. Thus, we shall omit the electron-phonon 1 interaction, the effect of which will be argued in the last m = ( 1)i n ; (6) c N − h ii section. The simplified system can be described by a Hamil- i tonian (cf. [3, 4]) X and the CDW structure factor H = t (Ci,σ† Ci+1,σ +H:c:)+U ni; ni; 1 ↑ ↓ iq(Ri Rj ) − S(q)= e − n n ; (7) i,σ i N h i ji X X i;j +V n n +1 J δ S S ; (1) X i i − i i;R · i i i as well as the SDW order parameter X X where the first-three terms on the right side belong to the 1 i ms = ( 1) ni; ni; ; (8) usual extensive Hubbard Hamiltonian, i labels the carbon N − h ↑ − ↓i atoms along the chain, σ = 1 labels spin states (up and i ± X down), t>0 is the hopping matrix element, Ci† and Ci are and the SDW zero-frequency susceptibility the fermion creation and annihilation operators, respectively, β for π electrons with spin σ at the i’th site, U is the coupling 1 χ(q)= dτ (ni; (τ) ni; (τ))(nj; (0) strength between two π electrons at the same carbon atom, N 0 h ↑ − ↓ ↑ i;j Z † X and ni = Ci Ci;, V is a measure of the nearest neighbor iq(Ri Rj ) nj; (0)) e − ; (9) Coulomb interaction between the π electrons, and ni = ni; + − ↓ i ↑ ni; . The last term in (1) denotes the ferromagnetic(J>0) where R ( ) is the space coordinate of the i(j)’th site, τ is the ↓ i j or antiferromagnetic(J<0) coupling between the spin Si imaginary time along the Trotter direction of the checker- of π electrons in the chain and the residual spin Si;R.Asin board [4]. The symbol in (3)–(8) represents the statistical [3], we assume that every side radical has a noncompensated average in the canonicalh·i ensemble. spin Si;R and the radical connects with every other carbon If the organic chain is fabricated to form a mesoscopic atom, ring, under an applied magnetic flux, there should be an equilibrium persistent current in the ring [9]. This has been 1; for odd i δ = an interesting topic in recent years [10]. The maximum am- i 0; for even i: plitude of the persistent current is proportional to the charge The radical coupling term can be written as stiffness of the system [7, 8]. It is also interesting in our case to study the influence of the radical coupling on the charge 1 z z + + shiftiness of the system. Physically, the charge stiffness is Si;R Si = Si;RSi + (Si;RSi + Si;R− Si−): (2) · 2 a measure of the modulus of the free energy of the system As usual, by neglecting the spin-flip interaction, [3, 5] we against the change in the twist boundary condition, or in may write other words, the applied magnetic flux. According to [7, 8] the charge stiffness can be simply obtained in the world-line z ni; ni; Si;R Si S [ ↑ − ↓ ]: (3) Monte Carlo simulations by the zero-frequency limit of the · ' i;R 2 Fourier transform of the ‘current-current’ correlation func- 1 Here Sz = becomes an Ising-like spin state on the side tion ( )= (0) ( ) , where the pseudocurrent ( )is i;R ± 2 C τ J J τ J τ radical. defined by [7,h 8] i Interestingly, substituting (3) into (1), we find that the resultant Hamiltonian is invariant under the transformation J(τ)= R [n (τ + ∆τ) n (τ)]: (10) i i,σ − i,σ σ σtogether with J J. Due to this symmetry, i,σ the→− magnetic properties of the→− system is independent of the X sign of the coupling constant J. Therefore in the following it suffices to consider only the ferromagnetic coupling (i.e., III. Results and discussion J 0). ≥ The magnetic properties are reflected by the magnetization In our Monte Carlo simulations, we measure all energies 1 z 1 in units of t and set the imaginary time step ∆τ =0:25 in Mz = Si;Rδi + (ni; ni; ) ; (4) N h 2 ↑ − ↓ i the Trotter decomposition [4].