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Progress of Theoretical Physics Supplement

Quantum Phase Transitions in One-Dimensional Peierls-Hubbard Model with Next-Nearest-Neighbor Hopping Integrals

Hiromi Otsuka

Published Jan 1, 2000• 0 citations
Publisher

Abstract

Two kinds of phase transitions observed in the half-filled Peierls-Hubbard chain with the next-nearest-neighbor hopping integral are investigated. In the uniform case, according to the conformal field theory prediction, we numerically determine a phase boundary between the spin-fluid and dimer states. Then, the criticality of the spin-Peierls transition (i.e., power-law dependences of the energy gain and spin gap on the lattice dimerization parameter) is argued on the phase boundary.

DOI
10.1143/ptps.138.139

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•Nov 25, 2008•0 citations
Two kinds of phase transitions observed in the half-filled Peierls-Hubbard chain with the next-nearest-neighbor hopping integral are investigated. In the uniform case, according to the conformal field theory prediction, we numerically determine a phase boundary between the spin-fluid and dimer states. Then, the criticality of the spin-Peierls transition (i.e., power-law dependences of the energy gain and spin gap on the lattice dimerization parameter) is argued on the phase boundary.
We investigate two kinds of quantum phase transitions observed in the one-dimensional half-filled Peierls-Hubbard model with the next-nearest-neighbor hopping integral in the strong-coupling region $U\ensuremath{\gg}t$, ${t}^{\ensuremath{'}}$ [ttt ${(t}^{\ensuremath{'}})$, nearest- (next-nearest-) neighbor hopping; UUU, on-site Coulomb repulsion]. In the uniform case, with the help of the conformal field theory prediction, we numerically determine a phase boundary ${t}_{\mathrm{c}}^{\ensuremath{'}}(U/t)$ between the spin-fluid and the dimer states, where a bare coupling of the marginal operator vanishes and the low-energy and long-distance behaviors of the spin part are described by a free-boson model. To exhibit the conformal invariance of the systems on the phase boundary, a multiplet structure of the excitation spectrum of finite-size systems and a value of the central charge are also examined. The critical phenomenological aspect of the spin-Peierls transitions accompanied by the lattice dimerization is then argued for the systems on the phase boundary; the existence of logarithmic corrections to the power-law behaviors of the energy gain and the spin gap (i.e., the Cross-Fisher scaling law) are discussed.
We investigate critical aspects of the spin-Peierls transition in the one-dimensional half-filled Peierls-Hubbard model. The bosonized Hamiltonian shows that low-energy excitations of the spin and the charge degrees of freedom are coupled through a cross term arising from the transfer-integral alternation. According to the renormalization-group theory, the cross term becomes relevant, which brings about the energy gain and the singlet-triplet gap described by the so-called Cross-Fisher scaling law. To confirm the prediction numerically, the density-matrix renormalization group method is employed. We find that the transition obeys the scaling law; the critical exponents satisfy the hyperscaling relation, whereas a logarithmic correction to the power-law behaviors is suggested. Based on these results, we give a universality argument among the dimerization transition and the criticalities of the two-dimensional qqq-state Potts models (i.e., q=2q=2q=2 and 4) in connection with a role of the on-site Coulomb interaction. Some related experiments on CuGeO3{\mathrm{CuGeO}}_{3}CuGeO3​ are also discussed.
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.
We study the one-dimensional Hubbard model with nearest-neighbor and next-nearest-neighbor hopping integrals by using the density-matrix renormalization group method and Hartree-Fock approximation. Based on the calculated results for the spin gap, total-spin quantum number, and Tomonaga-Luttinger-liquid parameter, we determine the ground-state phase diagram of the model in the entire filling and wide parameter region. We show that, in contrast to the weak-coupling regime where a spin-gapped liquid phase is predicted in the region with four Fermi points, the spin gap vanishes in a substantial region in the strong-coupling regime. It is remarkable that a large variety of phases, such as the paramagnetic metallic phase, spin-gapped liquid phase, singlet and triplet superconducting phases, and fully polarized ferromagnetic phase, appear in such a simple model in the strong-coupling regime.
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 investigate three different types of quantum phase transition occurring in quasi one-dimensional systems theoretically and numerically. First, we study the band-insulator to Mott-insulator transition occurring in charge-transfer complexes, for which the half-filled one-dimensional ionic Hubbard model is considered to be the prototype model. The study is carried out by first deriving an effective spin-one model, and then studying the model numerically using the density matrix renormalization group. We perform a careful finite-size scaling analysis of the mass gaps, order-parameters, and relative susceptibility. We confirm the existence of two quantum critical points. Analysis of the critical exponents confirms that the band-insulator-to-spontaneously-dimerized phase transition belong to the 2D Ising class. The spontaneously dimerized phase undergoes a phase transition to the Mott-insulator which is an infinite-order. Second, we investigate the Mott metal-insulator transition for the half-filled Hubbard model with both nearest-neighbor t and next-nearest-neighbor t′ hopping terms. We study the model using the bosonization approach and density matrix renormalization group simulations. An effective low-energy Hamiltonian that describes the insulator-metal transition is derived. We present results of density matrix renormalization group calculations of spin and charge distribution in various sectors of the phase diagram. The numerical results support the picture derived from the effective theory and give evidence for the complete separation of the transitions involving the spin and the charge degrees of freedom. Finally, we investigate quantum phase transitions phases in low-dimensional fermionic and spin models that go from uniform to spatially inhomogeneous, i.e., dimerized, trimerized, or incommensurate, phases. We propose a new approach based on studying the length dependence of the von Neumann entropy and its corresponding Fourier spectrum for finite segments in the ground state of finite chains. Peaks at a nonzero wave vector are indicators of oscillatory behavior in decaying correlation functions and also provide significant information about certain relevant features of the excitation spectrum; in particular, they can identify the wave vector of soft modes in critical models.