On the lowest energy excitations of one-dimensional strongly correlated electrons

It is proven that the lowest excitations Elow(k) of one-dimensional halfinteger spin generalized Heisenberg models and half-filled extended Hubbard models are π-periodic functions. For Hubbard models at fractional fillings Elow(k + 2kf ) = Elow(k), where 2kf = πn, and n is the number of electrons per unit cell. Moreover, if one of the ground states of the system is magnetic in the thermodynamic limit, then Elow(k) = 0 for any k, so the spectrum is gapless at any wave vector. The last statement is true for any integer or half-integer value of the spin. Typeset using REVTEX 1 Low-energy excitation spectra of correlated systems have long been in the center of the theoretical condensed matter research (see [1] and references therein). In 1932 Hans Bethe proposed his Ansatz method [2] for correlated wave functions, that was then used to solve many one-dimensional quantum and also two-dimensional classical models (see [3], [4], [5]). Using the Bethe Ansatz (BA) method des Cloizeaux and Pearson [6] derived the energy of low lying S = 1 (triplet) states of the spin-1/2 antiferromagnetic Heisenberg model (AFHM). The spectrum of the excitations was found to be Et (k) = 1/2Jπ | sin k |. Et (k) is a πperiodic function of k. des Cloizeaux and Pearson assumed the excitations Et (k) to be the elementary excitations of the system, therefore assigning to an elementary excitation spin 1. Later investigations [7] have shown that the spectrum of [6] was incomplete. The full spectrum of the S = 1 magnon excitations is characterized by two continuous quantum numbers. Fixing the wave vector k one still has a continuous one-parametric family of excitations. The des Cloizeaux and Pearson spectrum is only a lower bound of these excitations. As a result S = 1 states can not be considered as elementary indivisible particles having only one wave number k. They are rather two-particle compositions. The problem was finally clarified by Faddeev and Takhtajan in Ref. [8]. It was shown that all excitations of the spin-1/2 AFHM are superpositions of spin-1/2 elementary excitations (called kinks or spinons). For periodic rings having odd/even number of sites only odd/even number of kinks are possible in the system. Kinks do not form bound states. Their interaction manifests itself only through a scattering amplitude. Energies and momenta of many-kink states add, as for independent particles. The dispersion relation of a kink is 1/2Jπ sin k, with 0 ≤ k ≤ π. The Brillouin zone is therefore only half of the original one. The S = 1 states of the model are pairs of kinks. The lowest excitations are found by minimizing E (k1) +E (k2) for k1 + k2 = k. The result is k1 = k and k2 = 0 (or vice versa), so the lowest excitations probe the one-spinon excitations spectrum. For the ferromagnetic case the one magnon spectrum of the onedimensional spin-1/2 ferromagnetic Heisenberg model (FHM) has the form 2J(1 − cos k), possessing a gap at nonzero k and not being π-periodic. Careful analysis shows, that onemagnon states are not the lowest energy states of the system. Magnons can form bound 2 states and the interaction is so strong,, that the two-magnon states have energy less than the energy of one magnon. It was shown, using Bethe Ansatz [9], that the elementary excitations of the system (excitations having one quantum number) are magnons and strings. Strings are complexes of 2M + 1 spins, having a dispersion relation J 2 2M+1 (1− cos k). The lowest bound for the spectrum of excitations is formed, when M goes to infinity. Therefore the spectrum is gapless for all wavevectors k. As will be shown below, this property characterizes all Heisenberg-like one-dimensional models (integer-spin or half-integer-spin), which are magnetic in the thermodynamic limit. The excitations spectrum of the half-filled one-dimensional Hubbard model was derived in [10]. The lowest-lying triplet states of the model are given by equation (16) of [10]. The function Et (k) is π-periodic and reaches its maximum at q = π/2. For a quarter filling the spectrum of the lowest energy excitations is periodic with a period of π/2 = 2kf , where kf is the Fermi wave vector of the noninteracting system. For the attractive Hubbard model the excitations spectrum is calculated in [11]. The elementary excitations of the model at half-filling are bound pairs and ”free” electrons (instead of spinons and holons in the repulsive case). The energy-momentum dispersion relation of excitations is the same, as in the repulsive case, the lowest excitations being again π-periodic. It is proven below that the π-periodicity of the lowest-lying excitations is a model independent feature and holds for all one-dimensional isotropic Heisenberg models having a half-integer value of the spin per unit cell, and for all one-dimensional isotropic Hubbard models with an odd number of electrons per unit cell. For Hubbard models at fractional fillings the spectrum of lowest excitations is periodic with a period πn where n is the number of electrons per unit cell. For systems with a nonzero groundstate magnetization we are able to prove a stronger statement. We call a 1-D system magnetic in the thermodynamic limit, if there exist such N0, that, for all N > N0, one has SN/N ≥ ǫ > 0, where SN is the