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arXiv:cond-mat/9811280 · Physical review. B, Condensed matter

Phase diagram of the two-chain Hubbard model

Young‐Ho Park, Shoudan Liang, T. K. Lee

Published Jan 15, 1999• 11 citations
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Abstract

We have calculated the charge gap and spin gap for the two-chain Hubbard model as a function of the on-site Coulomb interaction and the interchain hopping amplitude. We used the density matrix renormalization group method and developed a method to calculate separately the gaps numerically for the symmetric and antisymmetric modes with respect to the exchange of the chain indices. We have found very different behaviors for the weak and strong interaction cases. Our calculated phase diagram is compared to the one obtained by Balents and Fisher using the weak coupling renormalization group technique.

DOI
10.1103/physrevb.59.2587

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We have calculated the charge gap and spin gap for the two-chain Hubbard model as a function of the on-site Coulomb interaction and the interchain hopping amplitude. We used the density matrix renormalization group method and developed a method to calculate separately the gaps numerically for the symmetric and antisymmetric modes with respect to the exchange of the chain indices. We have found very different behaviors for the weak and strong interaction cases. Our calculated phase diagram is compared to the one obtained by Balents and Fisher using the weak coupling renormalization group technique.
The T=0 phase diagram for two Hubbard chains with interchain hopping ${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}$ and on-site repulsion U is derived within a renormalization-group (RG) approach. We explicitly allow for renormalization of the Fermi momenta. A nonzero ${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}$ drives the chains towards strong coupling, and only for large ${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}$ is a Luttinger liquid recovered. By use of weak-coupling RG, a qualitative picture of the phase diagram can be obtained. Our calculations show two different strong-coupling regimes: one with nonvanishing effective interchain hopping (〈${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}$〉\ensuremath{\ne}0), not confining, for small U, and another with 〈${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}$〉=0, confining, for large U. Numerical diagonalizations for a small cluster support the RG results.
We study the phase diagram of the one-dimensional Hubbard model with next-nearest-neighbor hopping using exact diagonalization, the density-matrix renormalization group, the Edwards variational ansatz, and an adaptation of weak-coupling calculations on the two-chain Hubbard model. We find that a substantial region of the strong-coupling phase diagram is ferromagnetic, and that three physically different limiting cases are connected in one ferromagnetic phase. At a point in the phase diagram at which there are two Fermi points at weak coupling, we study carefully the phase transition from the paramagnetic state to the fully polarized one as a function of the on-site Coulomb repulsion. We present evidence that the transition is second order and determine the critical exponents numerically. In this parameter regime, the system can be described as a Luttinger liquid at weak coupling. We extract the Luttinger-liquid parameters and show how their behavior differs from that of the nearest-neighbor Hubbard model. The general weak-coupling phase diagram can be mapped onto that of the two-chain Hubbard model. We exhibit explicitly the adapted phase diagram and determine its validity by numerically calculating spin and charge gaps using the density-matrix renormalization group.
We study the phase diagram of a one-dimensional extended Hubbard model with antiferromagnetic exchange interaction analytically and numerically. The bosonization and transfer-matrix renormalization group methods are used in the corresponding coupling regimes. At half-filling, the system is a Mott insulator with a finite spin excitation gap if the on-site Coulomb repulsion is fairly smaller than the antiferromagnetic exchange JJJ. This Mott insulator is characterized by the bond-charge-density-wave order or spontaneously dimerization. In the weak-coupling regime where the spin-charge separation holds approximately, the critical point separating the gapless and gapped spin liquid phases is ${U}_{c}\ensuremath{\sim}J∕2$. However, as JJJ increases, the spin-charge couplings become important and the critical point Uc{U}_{c}Uc​ is significantly suppressed and eventually tends to zero as $J\ensuremath{\rightarrow}\ensuremath{\infty}$. Away from half-filling, the charge gap completely collapses but the spin gap persists.
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 present a general method for determining the phase diagram of systems of a finite number of one-dimensional Hubbard-like systems coupled by single-particle hopping with weak interactions. The technique is illustrated by detailed calculations for the two-chain Hubbard model, providing controlled results for arbitrary doping and interchain hopping. Of nine possible states which could occur in such a spin-1/2 ladder, we find seven at weak coupling. We discuss the conditions under which the model can be regarded as a one-dimensional analog of a superconductor. \textcopyright{} 1996 The American Physical Society.
The spin gap of a one-dimensional repulsive Hubbard model is numerically calculated with the density-matrix renormalization group, with a special emphasis on the effect of a next-nearest-neighbor hopping ${(t}^{\ensuremath{'}})$ and the nearest-neighbor ferromagnetic exchange (J)(J)(J) interaction. At half filling, a significant spin gap opens if $|{t}^{\ensuremath{'}}|\ensuremath{\simeq}|t|$ and J=0J=0J=0, in agreement with the weak-coupling theory, while the gap is strongly suppressed by the introduction of JJJ. On the other hand, the quarter-filled system has very small spin gaps regardless of the values of ${t}^{\ensuremath{'}}$ and JJJ. Implications for the CuO2{\mathrm{CuO}}_{2}CuO2​ chain in Sr14Cu24O41{\mathrm{Sr}}_{14}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}Sr14​Cu24​O41​ and related materials are discussed.