The ferromagnetism of the checkerboard lattice Hubbard model at quarter filling is one of the few exact ferromagnetic ground states known in the family of Hubbard models. When the nearest neighbor hopping, t1, is negligible compared with the second neighbor one, t2, the system reduces to a collection of Hubbard chains. We find that the 1D character is surprisingly robust as long as t1 < t2. This phenomenon of dimensional reduction due to the geometrical frustration leads to peculiar magnetic orders with 1D character for intermediate U and is responsible for odd-frequency superconducting states close to the magnetic boundary.
The ferromagnetism of the checkerboard lattice Hubbard model at quarter filling is one of the few exact ferromagnetic ground states known in the family of Hubbard models. When the nearest neighbor hopping, t1, is negligible compared with the second neighbor one, t2, the system reduces to a collection of Hubbard chains. We find that the 1D character is surprisingly robust as long as t1 < t2. This phenomenon of dimensional reduction due to the geometrical frustration leads to peculiar magnetic orders with 1D character for intermediate U and is responsible for odd-frequency superconducting states close to the magnetic boundary.
Ferromagnetism in the one-dimensional Hubbard model with the next-nearest-neighbor hopping is explored by using the exact-diagonalization method in a small cluster and the equation-of-motion method in the thermodynamic limit with electron density n = 3/2. With these two complementary methods, it is found that an intermediate value of the next-nearest-neighbor hopping amplitude t1 tends to stabilize the fully polarized ferromagnetic state under the condition that the on-site coulomb interaction U is sufficiently large in our model. The ground-state phase diagram of the model is presented in the t1—U plane.
The one-dimensional Hubbard model with first- and second-neighbor hopping, t1 and t2, with t2≊t1/(2)1/2 and large repulsion U, is proposed as a testing ground of Anderson’s idea that the existence of an resonating valence bond ground state in the half-filled-band case will show superconductivity when a finite concentration of holes is introduced. Interestingly, our preliminary calculations in the U=∞ limit for small chains have revealed a new mechanism for enhancement of singlet pairing: We find that for t2<0 the ground state is a singlet, and, for t2<0 and ‖t2/t1‖≪1, the nearest-neighbor extended-singlet-pairing correlation function becomes enhanced appreciably as the temperature decreases below t2.
We have studied the two-dimensional Hubbard model on a square lattice with nearest-neighbor hopping. We first discuss the properties of the model within the mean-field approximation: Because of the form of the band structure, some peculiar features are found. We then discuss the simulation algorithm used and compare simulation results with exact results for 6-site chains to test the reliability of the approach. We present results for thermodynamic properties and correlation functions for lattices up to 8\ifmmode\times\else\texttimes\fi{}8 in spatial size. The system is found to be an antiferromagnetic insulator for all values of the coupling constant at zero temperature in the half-filled-band case, but the long-range order is much smaller than predicted by mean-field theory. We perform a finite-size-scaling analysis to determine the character of the transition at zero coupling. For non-half-filled-band cases, our results suggest that the system is always paramagnetic, in contradiction with Hartree-Fock predictions. The system does not show tendency to ferromagnetism nor triplet superconductivity in the parameter range studied. We also discuss some properties of the attractive Hubbard model in the half-filled-band case.
We study the magnetic properties of the two-dimensional Hubbard model with nearest-neighbor (t) and next-nearest-neighbor (t2) hopping and on-site repulsion U. We first calculate the mean-field phase diagram as a function of band filling and U. Because of the Van Hove singularity in the density of states, we find a ferromagnetic phase extending to zero U for certain band filling. For the half-filled band case, antiferromagnetism sets in at a finite value of U if t2\ensuremath{\ne}0. We study the behavior of spin-spin correlation functions for small lattices of up to N=64 atoms using Monte Carlo simulations, as well as exact diagonalization for N=4. Our results show enhanced ferromagnetic correlations in some regions, but apparently no ferromagnetic long-range order. In the half-filled case, our numerical results are consistent with a nonzero critical U. For a non-half-filled band our results suggest that there is no long-range order.
The Hubbard model has attracted considerable interest due to its prototypical role in describing strongly interacting electronic systems, such as high-critical-temperature superconductors as well as many novel quantum materials. By introducing next-nearest-neighbor (NNN) hoppings to the Hubbard model, the phase diagram becomes richer, and fascinating phenomena arise in both, one-dimensional chains and square lattices, such as: antiferromagnetism, ferromagnetism, superconductivity (SC), as well as charge orders, among others. Moreover, NNN hoppings play a fundamental role in understanding effects of doping on magnetism and pairing orders in strongly interacting regimes. In this article, we review the recent progress in understanding the different competing phases of this model in one and two dimensions from a computational perspective. We comment on the pressing technical challenges, illustrate the controversial results concerning the emergence of the SC phase, and conclude with our perspectives on future explorations.
The origin of metallic ferromagnetism is a long-standing problem in condensed matter physics, because it is a strong coupling phenomenon essentially. It was also known that one dimensional systems have a lesser chance of showing ferromagnetism in the ground states. In fact, for a class of one-dimensional correlated electron systems including the simple Hubbard chain, the ground state was proved to be nonmagnetic. Several authors have been studied whether ferromagnetism is stabilized or not by taking neglected terms in the simple Hubbard chain into account. It was shown that a positive next-nearest neighbor hopping leads to ferromagnetism for less-than-half filled systems in strong correlation regime on the basis of DMRG calculations. We call this model the railroad–trestle-lattice Hubbard model below. The railroad–trestle lattice is graphically shown in Fig. 1. In this article, we treat the problem of ferromagnetism of the nearly half-filled extended Hubbard ladder with an additional intersite repulsion V . The Hamiltonian is H 1⁄4 t P hi; ji; ðc y i cj þ c y j ci Þ þU P i nini# þV P hi; ji ninj, where ci represent electron creation operators, ni 1⁄4 P c y i ci , and hi; ji denotes nearest-neighbor pairs of the sites on the two-leg ladder. Vojta et al. made a DMRG study of this model in strong coupling region. Tsuchiizu and Suzumura recently studied this model by using the weakcoupling bosonization method, motivated by the experimental observation of a charge-density-wave in the selfdoped ladder compound Sr14 xCaxCu24O41. 7,8) We first mention the half-filled case, i.e., electron density n 1⁄4 1. When U < 3V , the ground state has the checkerboard-type charge-ordering shown in Fig. 2 due to the intersite repulsion. (The sublattice consisting of doubly occupied sites is defined as A-sublattice.) Our main concern is what happens when holes are doped into this chargeordered insulating state. By using the DMRG method, Vojta et al. calculated the checkerboard-type charge-ordering correlation function from quarter-filling to half-filling, 1=2 n 1, where U 1⁄4 8t was chosen. They confirmed that the long-range charge-order correlation is obtained for V VcðnÞ ’ U=3. They also found that the total spin of the doped charge-ordering state is not zero, i.e., the charge ordering coexists with ferromagnetism. However, the origin of this ferromagnetism was not clarified. The main purpose of this paper is to get intuitive understanding of the ferromagnetism in the doped charge-ordered state. In order to attack this issue, we assume t U, V and use strong coupling expansion. Then, we calculate the effective Hamiltonian for holes doped into the charge-ordered state at half-filling. When holes are introduced, they enter the A-sublattice sites [see Fig. 3(a)]. If we totally neglect the bare hopping term in H, then we have the ground state degeneracy concerned with the way of arrangement of doped holes. The introduction of the bare hopping term resolves this degeneracy and is expected to induce an ordering of the doped holes. In order to write down the second-order effective Hamiltonian with respect to t, it is convenient to carry out the particle–hole transformation for the A-sublattice according to c‘ 1⁄4 a‘ , where a y ‘ are the creation operators of the A-sublattice holes. Within the two-hole approximation, we obtain the following effective Hamiltonian: