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Mott Transition Mott Physics Frustrated Magnet Spin Chain Metallic Ferromagnetism Magnetic Transition Theoretical Approaches

 Mott gap and high-energy states

The band gap in a conventional band insulator opens due to multiple sites (or multiple orbitals) in a unit cell (bonding and antibonding bands). On the other hand, because the Mott gap is due to Coulomb interactions between electrons, it opens even if there is only one site (or one orbital) in a unit cell. Namely, a magnetic long-range order (sublattice structures accampanied by it) is not required for the Mott gap. This can be understood from the facts that the Mott gap opens in the 1D Hubbard model without antiferromagnetic long-range order and that the upper Hubbard band (UHB) persists even in the large-doping regime in any dimensions. Thus, the origin of the Mott gap essentially differs with that of the conventional band gap.

 Quasiparticle characterizing the upper Hubbard band (doublon) [1]
The charge excitation gap in a Mott insulator can be intuitively understood as the energy const due to Coulomb repulsion when two electrons simultatneously sit on a site. But, because doubly occupancy also exists in the lower Hubbard band (LHB) as long as the Coulomb repulsion is finite, we cannot distinguish the UHB from the LHB simply by the presence of double occupancy. Also, although double occupancy is frequently refered to as a doublon, the double occupancy does not behave as a quasiparticle that characterizes the UHB. This can be unserstood by noting that even the ground state of a Mott insulator has double occupancy and that the number of double occupancy is not an integer in each eigen state. In addition, we can easily understand it by considering the non-interacting limit where double occupancy occurs just when an electron passes another. Then, the questions arising are whether there is a quasiparticle that characterizes the UHB and how the quasiparticle can be defined if it exists. The answers to the questions were derived using exact solutions [1]. By comparing the intensity distribution in the single-particle spectral function calculated using the DDMRG method and the dispersion relations calculated using exact solutions, it turned out that the LHB can be explained in terms of spinons, holons, and antiholons only, whereas the UHB can be identified as states with one k-Λ string in addition to the spinons, holons, and antiholons [1]. Namely, we can classify the UHB and LHB as the states with one k-Λ string and no k-Λ string, respectively [1]. Furthermore, by noting that the k-Λ string can be regarded as a pair of electrons, we can call the quasiparticle characterizing the UHB defined by the k-Λ string the doublon. Note that although there are solutions with multiple k-Λ strings, they have much higher energies (E ≈ 2U, 3U, …) in the large-U regime. Also, their spectral weights are much smaller than those of solutions with up to one k-Λ string (the sum rule is almost satisfied by taking into account the solutions with up to one k-Λ string) [1,2].

 Quasiparticle characterizing high-energy states in quantum spin systems [3]
Similar quasiparticles appear in Heisenberg antiferromagnets in a magnetic field [3]. In zero magnetic field, it is known that 2-spinon gapless excitations well explain the excitation spectra of the spin-1/2 Heisenberg chain. In a magnetic field, the 2-spinon continuum splits, and a high-energy continuum with an energy gap emerges [3]. This continuum is due to 2-string solutions where the 2-string can be regarded as a pair of anti-psinons [3]. At first glance, there is nothing to do with the spectral features of the Hubbard model, the high-energy states of a Heisenberg chain in a magnetic field are the same kind of states as the UHB in a Hubbard chain from the veiwpoint of the structure of solutions [1,2,3]. We can understand the relationship more intuitively, by mapping the spin-1/2 Heisenberg chain to the hard-core boson model with nearest neighbor repulsions, where the nearest neighbor repulsions play a role similar to the on-site repulsion of the Hubbard model, which causes the high-energy states corresponding to the UHB [2]. The relationship of the high-energy states of a Heisenberg chain in a magnetic field with the Mott physics is consistent with the fact that they appear without sublattice structures in the ground state. In fact, the high-energy states appear by applying a magnetic field, althoguh the excitation is gapless in zero field where the ground state has almost antiferromagnetically ordered spin correlations.

 Extention to higher dimensions [4,5]
We can generalize the above features of the doublon and Mott insulators to higher dimensions. By using a weak-interchain-coupling approach combined with exact solutions of a 1D Heisenberg chain in a magnetic field, an anisotropic-2D frustrated Heisenberg antiferromagnet has been shown to exhibit a high-energy mode in a mgnetic field. The mode can be regarded as bound states of the quasiparticle for a 2-string and the psinon, which behaves as a quasiparticle in the anisotropic-2D frustrated Heisenberg antiferromagnet[4]. This means that, even in 2D, the descendant of the quasiparicle defined by a 2-string in 1D behaves as the quasiparticle characterizing the high-energy states. Hence, we can expect that, in general, the descendant of the quasiparicle defined by a string with length of two in 1D (a pair of single-particles defined by the k-Λ string or 2-string) behaves as the quasiparticle characterizing the UHB and high-energy states in higher dimensions. The quasiparticle does not exist in low-energy states (LHB), in contrast with the conventional doublon defined by double occupancy.


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© 2011 Masanori Kohno