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arXiv:cond-mat/9709170 · Physical review. B, Condensed matter

Aspects of the ground state of the U=∞ Hubbard ladder

Masanori Kohno

Published Dec 15, 1997• 26 citations
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Abstract

We consider two aspects of the ground state of the $U=\ensuremath{\infty}$ Hubbard ladder: ferromagnetism and the metal-insulator transition at quarter-filling. First, we present rigorous results for the $U=\ensuremath{\infty}$ Hubbard ladder in the limit of the large interchain hopping ${(t}_{\ensuremath{\perp}}/t\ensuremath{\rightarrow}\ensuremath{\infty})$. In this limit, the total spin $S$ of the ground state is shown to be zero for the electron density $n<~0.5$ and its maximum ${(S=S}_{\mathrm{max}})$ for $n>0.5$. The charge gap at quarter-filling is ${2t}_{\ensuremath{\perp}}.$ We extend these results to finite ${t}_{\ensuremath{\perp}}/t$ by means of the density-matrix renormalization group method. We estimate the phase boundaries with respect to spontaneous magnetization and the charge gap at quarter-filling for finite ${t}_{\ensuremath{\perp}}/t$. Applying the extended Aharonov-Bohm method, we give numerical evidence that the critical ratio ${t}_{\ensuremath{\perp}}/t$, above which the charge gap opens, is less than 0.001. Ferromagnetism in the $t$-$J$ ladder is briefly discussed.

DOI
10.1103/physrevb.56.15015

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We consider two aspects of the ground state of the U=\ensuremath∞U=\ensuremath{\infty}U=\ensuremath∞ Hubbard ladder: ferromagnetism and the metal-insulator transition at quarter-filling. First, we present rigorous results for the U=\ensuremath∞U=\ensuremath{\infty}U=\ensuremath∞ Hubbard ladder in the limit of the large interchain hopping (t\ensuremath⊥/t\ensuremath→\ensuremath∞){(t}_{\ensuremath{\perp}}/t\ensuremath{\rightarrow}\ensuremath{\infty})(t\ensuremath⊥​/t\ensuremath→\ensuremath∞). In this limit, the total spin SSS of the ground state is shown to be zero for the electron density n< 0.5n<~0.5n< 0.5 and its maximum (S=Smax){(S=S}_{\mathrm{max}})(S=Smax​) for n>0.5n>0.5n>0.5. The charge gap at quarter-filling is 2t\ensuremath⊥.{2t}_{\ensuremath{\perp}}.2t\ensuremath⊥​. We extend these results to finite t\ensuremath⊥/t{t}_{\ensuremath{\perp}}/tt\ensuremath⊥​/t by means of the density-matrix renormalization group method. We estimate the phase boundaries with respect to spontaneous magnetization and the charge gap at quarter-filling for finite t\ensuremath⊥/t{t}_{\ensuremath{\perp}}/tt\ensuremath⊥​/t. Applying the extended Aharonov-Bohm method, we give numerical evidence that the critical ratio t\ensuremath⊥/t{t}_{\ensuremath{\perp}}/tt\ensuremath⊥​/t, above which the charge gap opens, is less than 0.001. Ferromagnetism in the ttt-JJJ ladder is briefly discussed.
We investigate the ground-state phase diagram of the quarter-filled Hubbard ladder with nearest-neighbor Coulomb repulsion VVV using the density matrix renormalization-group technique. The ground state is homogeneous at small V,V,V, a ``checkerboard'' charge-ordered insulator at large VVV and not too small on site Coulomb repulsion U, and is phase separated for moderate or large VVV and small U. The zero-temperature transition between the homogeneous and the charge-ordered phase is found to be second order. In both the homogeneous and the charge-ordered phases the existence of a spin gap mainly depends on the ratio of interchain to intrachain hopping. In the second part of the paper we construct an effective Hamiltonian for the spin degrees of freedom in the strong-coupling, charge-ordered regime that maps the system onto a frustrated spin chain. The opening of a spin gap is thus connected with spontaneous dimerization.
We study two (very) weakly coupled Hubbard chains in the half-filled case, and especially the situation where the intrachain Mott scale m is much larger than the (bare) single-electron interchain hopping ${t}_{\ensuremath{\perp}}.$ First, we find that the divergence of the intrachain umklapp channel at the Mott transition results in the complete vanishing of the single-electron interchain hopping: this is significant of a strong confinement of coherence along the chains. Excitations are usual charge fermionic solitons and spinon-(anti)spinon pairs of the Heisenberg chain. Then, we show rigorously how the tunneling of spinon-(anti)spinon pairs produces an antiferromagnetic interchain exchange of the order of ${J}_{\ensuremath{\perp}}{=t}_{\ensuremath{\perp}}^{2}/m.$ In the ``confined'' phase and in the far infrared, the system behaves as a pure spin ladder. The final result is an insulating ground state with spin-gapped excitations exactly as in the opposite ``delocalized'' limit (i.e., for rather large interchain hoppings) where the two-leg ladder is in the well-known insulating $D\ensuremath{-}\mathrm{Mott}$ phase. Unlike materials with an infinite number of coupled chains (Bechgaard salts), the confinement/deconfinement transition at absolute zero is here a simple crossover: no metallic phase is found in undoped two-leg ladders. This statement might be generalized for N-leg ladders with N=3,4,…N=3,4,\dots{}N=3,4,… (but not too large).
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:
We investigate the charge-order transition at zero temperature in a two-leg Hubbard ladder with additional nearest-neighbor Coulomb repulsion VVV using the density matrix renormalization group technique. We consider electron densities between quarter and half filling. For quarter filling and U=8t,U=8t,U=8t, we find evidence for a continuous phase transition between a homogeneous state at small VVV and a broken-symmetry state with ``checkerboard'' [wave vector $\mathbf{Q}=(\ensuremath{\pi},\ensuremath{\pi})$] charge order at large V.V.V. This transition to a checkerboard charge-ordered state remains present at all larger fillings, but becomes discontinuous at sufficiently large filling. We discuss the influence of U/tU/tU/t on the transition and estimate the position of the tricritical points.
We examine the critical behavior near the metal-insulator transition (MIT) in the one-dimensional extended Hubbard model with the on-site and the nearest-neighbor interactions UUU and VVV at quarter filling using a combined method of the numerical diagonalization and the renormalization group (RG). The Luttinger-liquid parameter ${K}_{\ensuremath{\rho}}$ is calculated with the exact diagonalization for finite size systems and is substituted into the RG equation as an initial condition to obtain ${K}_{\ensuremath{\rho}}$ in the infinite size system. This approach also yields the charge gap $\ensuremath{\Delta}$ in the insulating state near the MIT. The results agree very well with the available exact results for $U=\ensuremath{\infty}$ even in the critical regime of the MIT where the characteristic energy becomes exponentially small and the usual finite size scaling is not applicable. When the system approaches the MIT critical point $V\ensuremath{\rightarrow}{V}_{c}$ for a fixed UUU, ${K}_{\ensuremath{\rho}}$ and $\ensuremath{\Delta}$ behave as ${\ensuremath{\mid}\mathrm{l}\mathrm{n}\ensuremath{\Delta}\ensuremath{\mid}}^{\ensuremath{-}2}={c}_{\ensuremath{\Delta}}(V∕{V}_{c}\ensuremath{-}1)$ and ${({K}_{\ensuremath{\rho}}\ensuremath{-}1∕4)}^{2}={c}_{K}(1\ensuremath{-}V∕{V}_{c})$, where the critical value Vc{V}_{c}Vc​ and the coefficients ${c}_{\ensuremath{\Delta}}$ and cK{c}_{K}cK​ are functions of UUU. These critical properties, which are known to be exact for $U=\ensuremath{\infty}$, are observed also for finite UUU case. We also observe the same critical behavior in the limit of the MIT critical point $U\ensuremath{\rightarrow}{U}_{c}$ when UUU is varied for a fixed VVV.
We investigate a ladder system with two inequivalent legs, namely, a Hubbard chain and a one-dimensional electron gas. Analytical approximations, the density-matrix renormalization group method, and continuous-time quantum Monte Carlo simulations are used to determine ground-state properties, gaps, and spectral functions of this system at half-filling. Evidence for the existence of four different phases as a function of the Hubbard interaction and the rung hopping is presented. First, a Luttinger liquid exists at very weak interchain hopping. Second, a Kondo-Mott insulator with spin and charge gaps induced by an effective rung exchange coupling is found at moderate interchain hopping or strong Hubbard interaction. Third, a spin-gapped paramagnetic Mott insulator with incommensurate excitations and pairing of doped charges is observed at intermediate values of the rung hopping and the interaction. Fourth, the usual correlated band insulator is recovered for large rung hopping. We show that the wave numbers of the lowest single-particle excitations are different in each insulating phase. In particular, the three gapped phases exhibit markedly different spectral functions. We discuss the relevance of asymmetric two-leg ladder systems as models for atomic wires deposited on a substrate.