Ground State Phases of the Half-Filled One-Dimensional Extended Hubbard Model

Abstract

Using quantum Monte Carlo simulations, results of a strong-coupling expansion, and Luttinger liquid theory, we determine quantitatively the ground state phase diagram of the one-dimensional extended Hubbard model with on-site and nearest-neighbor repulsions U and V . We show that spin frustration stabilizes a bond-ordered (dimerized) state for U ≈ V /2 up to U/t ≈ 9, where t is the nearest-neighbor hopping. The transition from the dimerized state to the staggered charge-densitywave state for large V /U is continuous for U < ∼ 5.5 and first-order for higher U . The one-dimensional Hubbard model, which describes electrons on a tight-binding chain with single-particle hopping matrix element t and on-site repulsion U , has a charge-excitation gap for any U > 0 at half-filling [1]. In the spin sector, the low-energy spectrum maps onto that of the S = 1/2 Heisenberg chain; the spin coupling J = 4t 2 /U for U → ∞. The spin spectrum is therefore gapless and the spin-spin correlations decay with distance r as (−1) r /r [2]. Hence, the ground state is a quantum critical staggered spin-density-wave (SDW). In the simplest extended Hubbard model, a nearest-neighbor repulsion V is also included. The Hamiltonian is, in standard notation and with t = 1 hereafter,The low-energy properties for V < ∼ U/2 are similar to those at V = 0. For higher V the ground state is a staggered charge-density-wave (CDW), where both the charge and spin excitations are gapped. The transition between the critical SDW and the long-rangeordered CDW has been the subject of numerous studies [3,4,5,6,7,8,9,10,11,12,13,14]. Until recently, it was believed that the SDW-CDW transition occurs for all U > 0 at V > ∼ U/2 and that it is continuous for small U ( < ∼ 5) and first-order for larger U . However, based on a study of excitation spectra of small chains, Nakamura argued that there is also a bond-order-wave (BOW) phase [10], where the ground state has a staggered modulation of the kinetic energy density (dimerization), in a narrow region between the SDW and CDW phases for U smaller than the value at which the transition changes to first order. Previous studies [6,7,8,9] had indicated an SDW state in this region. Nakamura's BOW-CDW boundary coincides with the previously determined SDW-CDW boundary. The presence of dimerization and the accompanying spin gap were subsequently confirmed using quantum Monte Carlo (QMC) simulations [11,12]. The BOW phase now also has a weak-coupling theory [13].The existence of an extended BOW phase has recently been disputed. Jeckelmann argued, on the basis of density-matrix-renormalization-group (DMRG) calculations, that the BOW exists only on a short segment of the first-order part of the SDW-CDW boundary [14], i.e., that the transition always is SDW-CDW and that BOW order is only induced on part of the coexistence curve. However, QMC calculations demonstrate the existence of BOW order well away from the phase boundary [12].Although several studies agree on the existence of an extended BOW phase [10,11,12,13], the shape of ...

Ground-state properties of the one-dimensional extended Hubbard model at half filling

“…1). As for the phase boundaries, our results agree quantitatively with the renormalization-group results [9] in the weakcoupling regime (U 2t), with the perturbation results [7] in the strongcoupling regime (U 6t), and with the quantum Monte Carlo results [13] in the intermediate-coupling regime. In addition, we obtained the tricritical point (U t , V t ) = (5.89t, 3.10t) where the BOW-CDW transition changed from continuous to first order, and the critical end point (U c , V c ) = (9.25t, 4.76t) where the BOW phase disappears.…”

“…The ground-state phase diagram of the 1D half-filled EHM is still controversial, though there are a number of analytical [5][6][7][8][9][10] and numerical [11][12][13][14][15] studies. Quite recently, in order to put an end to the controversy, we have reexamined the phase diagram using the density-matrix renormalization group (DMRG) method with considerable accuracy [16]; we determined the SDW-BOW and BOW-CDW phase boundaries based on the results of various physical quantities such as the charge gap, spin gap, Luttinger exponents, and BOW order parameter (see Fig.…”

Entanglement of indistinguishable particles as a probe for quantum phase transitions in the extended Hubbard model

“…The precise location of the BOW phase is then still a subject of debate. To the best of our knowledge, the best estimates for the transitions, taking U/t = 4, correspond to a CDW-BOW transition at V /t ≈ 2.16 [11,12,15,16], and to a BOW-SDW transition in the range V /t ≈ 1.88 − 2.00 [11,12,15,16], or V /t = 2.08 ± 0.02 [13].…”

“…Many efforts have been devoted to the investigation of the EHM's phase diagram at half-filling, using both analytical and numerical methods [9][10][11][12][13][14][15][16]. Despite the apparent simplicity of the model, it exhibits a very rich phase diagram which includes several distinct phases: chargedensity wave (CDW), spin-density wave (SDW), phase separation (PS), singlet (SS) and triplet (TS) superconductors, and a controversial bond-order wave (BOW).…”

Bond-order wave phase of the extended Hubbard model: Electronic solitons, paramagnetism, and coupling to Peierls and Holstein phonons

“…The key features are broken inversion symmetry, doubly degenerate ground state (gs) and finite magnetic gap E m in a regular (equally spaced) array. Competition among electron delocalization t, on-site repulsion U > 0 and nearest-neighbor repulsion V > 0 stabilizes the BOW phase over a narrow range whose boundaries motivated subsequent studies [3][4][5][6][7][8][9]. The BOW phase has V ≈ U/2 and substantial t. The quantum transition to the charge density wave (CDW) phase at large V is first order [4,5] for U > U * ≈ 7t, continuous for U < U * .…”