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arXiv:cond-mat/0104348 · Journal of the Physical Society of Japan

Magnetic and Metal–Insulator Transitions through Bandwidth Control in Two-Dimensional Hubbard Models with Nearest and Next-Nearest Neighbor Transfers

Tsuyoshi Kashima, Masatoshi Imada

Published Oct 1, 2001• 87 citations
arXivHTMLPDF

Abstract

Numerical studies on Mott transitions caused by the control of the ratio between bandwidth and electron-electron interaction ($U$) are reported. By using the recently proposed path-integral renormalization group(PIRG) algorithm, physical properties near the transitions in the ground state of two-dimensional half-filled models with the nearest and the next-nearest neighbor transfers ($-t$ and $t'$, respectively) are studied as a prototype of geometrically frustrated system. The nature of the bandwidth-control transitions shows sharp contrast with that of the filling-control transitions: First, the metal-insulator and magnetic transitions are separated each other and the metal-insulator (MI) transition occurs at smaller $U$, although the both transition interactions $U$ increase with increasing $t'$. Both transitions do not contradict the first-order transitions for smaller $t'/t$ while the MI transitions become continuous type accompanied by emergence of {\it unusual metallic phase} near the transition for large $t'/t$. A nonmagnetic insulator phase is stabilized between MI and AF transitions. The region of the nonmagnetic insulator becomes wider with increasing $t'/t$. The phase diagram naturally connects two qualitatively different limits, namely the Hartree-Fock results at small $t'/t$ and speculations in the strong coupling Heisenberg limit.

DOI
10.1143/jpsj.70.3052

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Numerical studies on Mott transitions caused by the control of the ratio between bandwidth and electron-electron interaction (UUU) are reported. By using the recently proposed path-integral renormalization group(PIRG) algorithm, physical properties near the transitions in the ground state of two-dimensional half-filled models with the nearest and the next-nearest neighbor transfers (−t-t−t and t′t't′, respectively) are studied as a prototype of geometrically frustrated system. The nature of the bandwidth-control transitions shows sharp contrast with that of the filling-control transitions: First, the metal-insulator and magnetic transitions are separated each other and the metal-insulator (MI) transition occurs at smaller UUU, although the both transition interactions UUU increase with increasing t′t't′. Both transitions do not contradict the first-order transitions for smaller t′/tt'/tt′/t while the MI transitions become continuous type accompanied by emergence of {\it unusual metallic phase} near the transition for large t′/tt'/tt′/t. A nonmagnetic insulator phase is stabilized between MI and AF transitions. The region of the nonmagnetic insulator becomes wider with increasing t′/tt'/tt′/t. The phase diagram naturally connects two qualitatively different limits, namely the Hartree-Fock results at small t′/tt'/tt′/t and speculations in the strong coupling Heisenberg limit.
A new numerical algorithm for interacting fermion systems to treat the grand-canonical ensemble is proposed and examined by extending the path-integral renormalization group method. To treat the grand-canonical ensemble, the particle-hole transformation is applied to the Hamiltonian and basis states. In the interaction-term projection, the Stratonovich-Hubbard transformation which hybridizes up and down spin electrons is introduced. By using this method, the phase diagram of the two-dimensional Hubbard model with next-nearest-neighbor transfer is accurately determined by treating the filling-control (FC) and bandwidth-control (BC) Mott transitions on the same ground. A V-shaped Mott insulating phase is obtained in the plane of the chemical potential and the Coulomb interaction, where the transitions at the corner (BC) and the edges (FC) show contrasted characters with large critical fluctuations near the edges coexisting with the first-order transition at the corner. This contrasted behavior is shown to be consistent with the V-shape structure of the phase boundary because of a general relation, in which the slope of the metal-insulator transition line in the phase diagram is expressed by thermodynamic quantities. The V-shaped opening of the Mott gap is favorably compared with the experimental results of the transition metal oxides.
Antiferromagnetic and metal to insulator transitions in the ground state of two-dimensional half-filled and nearly half-filled Hubbard models with the nearest neighbor and the second nearest neighbor transfers (- t and t ′ , respectively) are studied by using the Green's function approach with the self-energy up to the second order in the coupling constant U . For a choice of the parameter value t ′ / t =0.2 the half-filled model undergoes the first order phase transition between paramagnetic metallic (PM) phase and antiferromagnetic insulator (AFI) phase at U / t =4.0 while the first order transition between PM and antiferromagnetic metallic (AFM) phases takes place for nearly half-filled band. The one-electron spectral density is calculated for the PM, AFM, and AFI phases near the phase boundaries. In the PM phase the quasi-particle dispersion indicates the band narrowing from the Hartree-Fock results and the flat band behavior as was indicated by previous stadies and was observed in high- T c cuprates. The quasi-particle dispersion in the AFM phase is seen to be close to the Hartree-Fock results.
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{t}_{2}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.
The ground-state properties of the two-dimensional Hubbard model with nearest-neighbor and next-nearest-neighbor hoppings at half filling are studied by the path-integral-renormalization-group method. The nonmagnetic-insulator phase sandwiched by the the paramagnetic-metal phase and the antiferromagnetic-insulator phase shows evidence against translational symmetry breaking of the dimerized state, plaquette singlet state, staggered flux state, and charge ordered state. These results support that the genuine Mott insulator which cannot be adiabatically continued to the band insulator is realized generically by Umklapp scattering through the effects of geometrical frustration and quantum fluctuation in the two-dimensional system.
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Path‐integral renormalization group (PIRG) method has been developed for studying strongly correlated electron systems. In a wide range of systems including Hubbard‐type models, this method overcomes a number of difficulties known in the Monte‐Carlo‐type methods such as the negative sign problem. This method has been combined with a procedure of quantum number projection and grand canonical ensemble method as well, which contribute to a wider applicability. The quantum‐number projected PIRG enables calculations of excited spectra with a specified momentum or spin. We review recent numerical results on strongly correlated electron systems studied by this method. By using the methods, we determine the phase diagram of the two‐dimensional Hubbard model in the parameter space of the onsite interaction U, strength of the geometrical frustration effects defined by the ratio between next‐nearest to nearest neighbor transfer integrals, t′/t, and the chemical potential. It reveals severe competitions of various phases. The phase diagram contains a remarkable nonmagnetic Mott insulator phase with gapless and dispersionless spin excitations, sandwitched by the first‐order Mott transition and the antiferromagnetic transition. The first‐order character becomes more continuous one with increasing the frustration effect.