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arXiv:cond-mat/0509663 · Physical Review B

Quantum phase transition in the one-dimensional extended Peierls-Hubbard model

H. Benthien, Fabian H. L. Eßler, Anja Grage

Published Feb 10, 2006• 13 citations
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Abstract

We consider the one-dimensional extended Hubbard model in the presence of an explicit dimerization $\ensuremath{\delta}$. For a sufficiently strong nearest neighbor repulsion we establish the existence of a quantum phase transition between a mixed bond-order wave and charge-density wave phase from a pure bond-order wave phase. This phase transition is in the universality class of the two-dimensional Ising model.

DOI
10.1103/physrevb.73.085105

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We consider the one-dimensional extended Hubbard model in the presence of an explicit dimerization \ensuremathδ\ensuremath{\delta}\ensuremathδ. For a sufficiently strong nearest neighbor repulsion we establish the existence of a quantum phase transition between a mixed bond-order wave and charge-density wave phase from a pure bond-order wave phase. This phase transition is in the universality class of the two-dimensional Ising model.
We revisit the ground-state phase diagram of the one-dimensional half-filled extended Hubbard model with on-site (U) and nearest-neighbor (V) repulsive interactions. In the first half of the paper, using the weak-coupling renormalization-group approach (g(g(g-ology) including second-order corrections to the coupling constants, we show that bond-charge-density-wave (BCDW) phase exists for $U\ensuremath{\approx}2V$ in between charge-density-wave (CDW) and spin-density-wave (SDW) phases. We find that the umklapp scattering of parallel-spin electrons disfavors the BCDW state and leads to a bicritical point where the CDW-BCDW and SDW-BCDW continuous-transition lines merge into the CDW-SDW first-order transition line. In the second half of the paper, we investigate the phase diagram of the extended Hubbard model with either additional staggered site potential $\ensuremath{\Delta}$ or bond alternation $\ensuremath{\delta}.$ Although the alternating site potential $\ensuremath{\Delta}$ strongly favors the CDW state (that is, a band insulator), the BCDW state is not destroyed completely and occupies a finite region in the phase diagram. Our result is a natural generalization of the work by Fabrizio, Gogolin, and Nersesyan [Phys. Rev. Lett. 83,83,83, 2014 (1999)], who predicted the existence of a spontaneously dimerized insulating state between a band insulator and a Mott insulator in the phase diagram of the ionic Hubbard model. The bond alternation $\ensuremath{\delta}$ destroys the SDW state and changes it into the BCDW state (or Peierls insulating state). As a result the phase diagram of the model with $\ensuremath{\delta}$ contains only a single critical line separating the Peierls insulator phase and the CDW phase. The addition of $\ensuremath{\Delta}$ or $\ensuremath{\delta}$ changes the universality class of the CDW-BCDW transition from the Gaussian transition into the Ising transition.
We present an unbiased numerical density-matrix renormalization group study of the one-dimensional Bose-Hubbard model supplemented by nearest-neighbor Coulomb interaction and bond dimerization. It places the emphasis on the determination of the ground-state phase diagram and shows that, besides dimerized Mott and density-wave insulating phases, an intermediate symmetry-protected topological Haldane insulator emerges at weak Coulomb interactions for filling factor one, which disappears, however, when the dimerization becomes too large. Analyzing the critical behavior of the model, we prove that the phase boundaries of the Haldane phase to Mott insulator and density-wave states belong to the Gaussian and Ising universality classes with central charges c=1c=1c=1 and c=1/2c=1/2c=1/2, respectively, and merge in a tricritical point. Interestingly we can demonstrate a direct Ising quantum phase transition between the dimerized Mott and density-wave phases above the tricritical point. The corresponding transition line terminates at a critical end point that belongs to the universality class of the dilute Ising model with c=7/10c=7/10c=7/10. At even stronger Coulomb interactions the transition becomes first order.
We study extended Hubbard and Peierls-Hubbard models with up to N=10 sites and half-filled bands with modified boundary conditions. For the extended Hubbard model, the finite-size dependence of the transition from the charge-density-wave regime (CDW) to the spin-density-wave (SDW) regime and of the condensation transition is critically examined, clarifying some previous results. For the Peierls-Hubbard model and N\ensuremath{\le}10 with modified periodic boundary conditions, we always see a decrease of dimerization with U. Extrapolating these results to infinite N, we find that dimerization is enhanced by U for small electron-phonon coupling \ensuremath{\alpha}\ifmmode \tilde{}\else \~{}\fi{}0.75, whereas it decreases with U for 0.75\ensuremath{\le}1. By extending this model by a nearest-neighbor repulsion V, the amplitude of dimerization reaches a maximum at the same value U\ensuremath{\le}2V, where the CDW-to-SDW transition occurs in the extended Hubbard model.
We examine the phase diagram of the half-filled one-dimensional extended Hubbard model using quantum information entropies within the density-matrix renormalization group. It is well known that there is a charge-density-wave phase at large nearest-neighbor and a small on-site Coulomb repulsion and a spin-density wave at small nearest-neighbor and large on-site Coulomb repulsion. At intermediate Coulomb interaction strength, we find an additional narrow region of a bond-order phase between these two phases. The phase transition line for the transition out of the charge-density-wave phase changes from first order at strong coupling to second order in a parameter regime where all three phases are present. We present evidence that the additional phase-transition line between the spin-density-wave and bond-order phases is infinite order. While these results are in agreement with recent numerical work, our study provides an independent unbiased means of determining the phase boundaries by using quantum information analysis, yields values for the location of some of the phase boundaries that differ from those previously found, and provides insight into the limitations of numerical methods in determining phase boundaries, especially those of infinite-order transitions.
We use a stochastic series-expansion quantum Monte Carlo method to study the phase diagram of the one-dimensional extended Hubbard model at half-filling for small to intermediate values of the on-site U and nearest-neighbor V repulsions. We confirm the existence of a novel, long-range-ordered bond-order-wave (BOW) phase recently predicted by Nakamura [J. Phys. Soc. Jpn. 68, 3123 (1999)] in a small region of the parameter space between the familiar charge-density-wave (CDW) state for $V\ensuremath{\gtrsim}U/2$ and the state with dominant spin-density-wave (SDW) fluctuations for $V\ensuremath{\lesssim}U/2.$ We discuss the nature of the transitions among these states and evaluate some of the critical exponents. Further, we determine accurately the position of the multicritical point, ${(U}_{m}{,V}_{m})=(4.7\ifmmode\pm\else\textpm\fi{}0.1,2.51\ifmmode\pm\else\textpm\fi{}0.04)$ (in energy units where the hopping integral is normalized to unity), above which the two continuous SDW-BOW-CDW transitions are replaced by one discontinuous (first-order) direct SDW-CDW transition. We also discuss the evolution of the CDW and BOW states upon hole doping. We find that in both cases the ground state is a Luther-Emery liquid, i.e., the spin gap remains but the charge gap existing at half-filling is immediately closed upon doping. The charge and bond-order correlations decay with distance r as ${r}^{\ensuremath{-}{K}_{\ensuremath{\rho}}},$ where ${K}_{\ensuremath{\rho}}$ is approximately 0.50.50.5 for the parameters we have considered. We also discuss advantages of using parallel tempering (or exchange Monte Carlo)---an extended ensemble method that we here combine with quantum Monte Carlo---in studies of quantum phase transitions.
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.