The Lieb theorem is a cornerstone of quantum magnetism theory in condensed matter. In this work, we investigate the instability of the Lieb insulating ferrimagnetic phase in the extended Hubbard model on a trimer chain at half-filling, with one electron per site, under increasing the nearest-neighbor Coulomb coupling $V$. Our results show that despite a noticeable increase in doublon density with $V$, the ferrimagnetic insulating phase remains robust up to the phase separation (PS) line, which is observed at $V\ensuremath{\gtrsim}U/4$, where $U$ is the local Coulomb repulsion. Above the PS line, one of the coexisting phases is primarily populated by doublons on one of the two sublattices of the chain. This phase coexists with a metallic, unsaturated ferromagnetic phase for $U\ensuremath{\gtrsim}t$, and with a singlet phase for $U\ensuremath{\lesssim}t$, where $t$ is the intratrimer hopping amplitude. We estimate the PS and the crossover lines with the help of density matrix renormalization group calculations.
The Lieb theorem is a cornerstone of quantum magnetism theory in condensed matter. In this work, we investigate the instability of the Lieb insulating ferrimagnetic phase in the extended Hubbard model on a trimer chain at half-filling, with one electron per site, under increasing the nearest-neighbor Coulomb coupling V. Our results show that despite a noticeable increase in doublon density with V, the ferrimagnetic insulating phase remains robust up to the phase separation (PS) line, which is observed at V\ensuremath≳U/4, where U is the local Coulomb repulsion. Above the PS line, one of the coexisting phases is primarily populated by doublons on one of the two sublattices of the chain. This phase coexists with a metallic, unsaturated ferromagnetic phase for U\ensuremath≳t, and with a singlet phase for U\ensuremath≲t, where t is the intratrimer hopping amplitude. We estimate the PS and the crossover lines with the help of density matrix renormalization group calculations.
Flat band systems have recently attracted significant attention due to their instability under small perturbations, which can lead to the stabilization of many exotic quantum phases. We study a trimer ladder which shows a middle flat band in the absence of onsite Coulomb interaction. We investigate the quantum phases of the Hubbard model on this geometry using exact diagonalization, density matrix renormalization group, and perturbation theory. We construct a quantum phase diagram in the plane of the next-nearest-neighbor hopping parameter t2 and onsite Coulomb interaction U, revealing five distinct quantum phases. At low U and moderate to high magnitude of t2, the system exhibits metallic behavior, while at large U and small magnitude of t2, it transitions to a ferrimagnetic insulator phase, similar to those observed in certain trimer materials. In the small t2 limit, the Fermi energy is in the flat band, leading to localization of the electrons within the trimer. At low U and small magnitude of t2, the flat band mechanism favors insulating ferrimagnetism, whereas at large U, ferrimagnetic states emerge from singlet dimer formation between neighboring sites of a trimer and an isolated corner spin, which connect ferromagnetically. The insulating cell spin density wave phase displays an up-up--down-down spin configuration due to competing nearest-neighbor hopping t1. Interestingly, in moderate U and ∣t2∣>0.3, the ground state behaves like metallic Tomonaga-Luttinger liquid.
We investigate the ground-state phase diagram of the half-filled one-dimensional Hubbard model with next-nearest-neighbor hopping using the density-matrix renormalization-group technique as well as an unrestricted Hartree-Fock approximation. We find commensurate and incommensurate disordered magnetic insulating phases and a spin-gapped metallic phase in addition to the one-dimensional Heisenberg phase. At large on-site Coulomb repulsion U, we make contact with the phase diagram of the frustrated Heisenberg chain, which has spin-gapped phases for sufficiently large frustration. For weak U, sufficiently large next-nearest-neighbor hopping t2 leads to a band structure with four Fermi points rather than two, producing a spin-gapped metallic phase. As U is increased in this regime, the system undergoes a Mott-Hubbard transition to a frustrated antiferromagnetic insulator.
Motivated by recent experiments on ultracold fermionic spin-1/2 6Li atoms in a Lieb lattice at various Hubbard repulsion U and filling fractions n (Lebrat et al., arXiv:2404.17555), we conduct a density matrix renormalization group (DMRG) analysis of the Hubbard model on a two-leg Lieb ladder. At half-filling, we find a ferrimagnetic Mott insulating ground state, consistent with Lieb's theorem. Away from half-filling, a state with finite total spin S2=0 and vanishing charge gap persists down to filling nc≈2/3. For lower, incommensurate fillings, the system is described by a Luttinger liquid with one charge and one spin mode. Intriguingly, in a narrow window near nc=2/3, close to the onset of ferromagnetic order, we identify a superconducting Luther-Emery phase with dominant sxy-wave pairing.
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:
The density-matrix renormalization group is used to study the phase diagram of the one-dimensional half-filled Hubbard model with on-site (U) and nearest-neighbor (V) repulsion and hopping t. A critical line V(c)(U) approximately U/2 separates a Mott insulating phase from a charge-density-wave phase. The formation of bound charge excitations for V>2t changes the phase transition from continuous to first-order at a tricritical point U(t) approximately 3.7t, V(t)=2t. A frustrating effective antiferromagnetic spin coupling induces a bond-order-wave phase on the critical line V(c)(U) for U(t)<U less, similar 7t.
The noninteracting electronic structures of tight-binding models on bipartite lattices with unequal numbers of sites in the two sublattices have a number of unique features, including the presence of spatially localized eigenstates and flat bands. When a uniform on-site Hubbard interaction U is turned on, Lieb proved rigorously that at half-filling ($\ensuremath{\rho}=1$) the ground state has a nonzero spin. In this paper we consider a ``CuO2 lattice'' (also known as ``Lieb lattice,'' or as a decorated square lattice), in which ``d orbitals'' occupy the vertices of the squares, while ``p orbitals'' lie halfway between two d orbitals; both d and p orbitals can accommodate only up to two electrons. We use exact determinant quantum Monte Carlo (DQMC) simulations to quantify the nature of magnetic order through the behavior of correlation functions and sublattice magnetizations in the different orbitals as a function of U and temperature; we have also calculated the projected density of states, and the compressibility. We study both the homogeneous (H) case, Ud=Up, originally considered by Lieb, and the inhomogeneous (IH) case, ${U}_{d}\ensuremath{\ne}{U}_{p}$. For the H case at half-filling, we found that the global magnetization rises sharply at weak coupling, and then stabilizes towards the strong-coupling (Heisenberg) value, as a result of the interplay between the ferromagnetism of like sites and the antiferromagnetism between unlike sites; we verified that the system is an insulator for all U. For the IH system at half-filling, we argue that the case ${U}_{p}\ensuremath{\ne}{U}_{d}$ falls under Lieb's theorem, provided they are positive definite, so we used DQMC to probe the cases Up=0,Ud=U and Up=U,Ud=0. We found that the different environments of d and p sites lead to a ferromagnetic insulator when Ud=0; by contrast, Up=0 leads to to a metal without any magnetic ordering. In addition, we have also established that at density $\ensuremath{\rho}=1/3$, strong antiferromagnetic correlations set in, caused by the presence of one fermion on each d site.