PHYSICAL REVIEW B 83, 033104 (2011) Variational cluster approximation study of the one-dimensional Holstein-Hubbard model at half filling Alexandre Payeur and David S´ en´ echal D´ epartement de physique and Regroupement qu´ eb´ ecois sur les mat´ eriaux de pointe, Universit´ e de Sherbrooke, Sherbrooke, Qu´ ebec J1K 2R1, Canada (Received 24 September 2010; revised manuscript received 23 November 2010; published 31 January 2011) The half-filled one-dimensional Holstein-Hubbard model presents, at zero temperature, a charge-density-wave (CDW) phase and a Mott insulator phase. Recent results have shown that the transition from one phase to the other might proceed through an intermediate metallic phase. In this work, we determine the CDW phase boundary using the variational cluster approximation. Using exact diagonalization and cluster perturbation theory, we study both the pair susceptibility and the spectral gaps in the non-CDW part of the phase diagram. We cannot rule out the existence of an intermediate metallic phase. DOI: 10.1103/PhysRevB.83.033104 PACS number(s): 71.27.+a, 71.10.Fd, 63.22.−m Questions such as the impact of phonons on the super- conductivity of high-T c cuprates motivate studies of models containing both electron and phonon degrees of freedom. Specifically for these materials, Angle-resolved photoemission spectroscopy (ARPES) experiments have shown that phonons might be involved in the electron dynamics. 1 More litigious is the existence of an isotope effect (see Refs. 2 and 3). Of course, the example of high-T c cuprates is far from being unique: A complete understanding of a material’s properties often requires careful studies of lattice degrees of freedom. In this work, we study the half-filled one-dimensional (1D) Holstein-Hubbard model (HHM) at zero temperature. This model includes two local interactions: one purely electronic (the Hubbard interaction 4 ) and the other coupling electrons to underlying optical phonons (the Holstein interaction). This type of electron-phonon interaction was first put forth when studying the polaron problem in molecular crystals. 5 As such, the 1D HHM can depict quasi-1D systems in which the energy scale of intramolecular vibrations is larger than intermolecular ones, such as organic superconductors. The Hamiltonian of the 1D HHM reads H =−t  〈i,i ′ 〉,σ c † iσ c i ′ σ + U  i n i ↑ n i ↓ − μ  i,σ n iσ − g  i,σ (a i + a † i )n σ + ω D  i a † i a i , (1) where c iσ annihilates an electron of spin σ at site i of the lattice, a i destroys a phonon at the same site, and n iσ = c † iσ c iσ . It contains five parameters, namely, the electron hopping integral t , the repulsive electron-electron (e-e) interaction U , the electron-phonon (e-p) coupling constant g, the frequency of the optical phonons ω D , and the chemical potential μ. Below, we will set t = 1, thus fixing the units of energy. The half-filling constraint will further reduce the number of parameters to three. In the large ω D limit, it can be shown that the HHM maps to an effective all-electron model, the mapping being valid for weak, 6 intermediate, or large 7 e-p coupling. The effective local e-e interaction is given by U eff = U − 2g 2 /ω D . U eff can be used to tentatively identify the phases of the model. For U eff > 0, it is likely a Mott insulator; 8 for U eff < 0 it is likely a charge-density wave (CDW) with wave vector 2k F = π , because the system is half-filled. In higher dimensions, U eff < 0 could also give rise to superconductivity. 9–12 The case U eff = 0 could correspond to a metallic phase. A larger ω D would favor a metallic state. Using spectral functions calculated from cluster perturbation theory (CPT), 13 Ning et al. 14 have shown that a Hubbard model (g = ω D = 0) with U =−2 is very nearly equivalent to a Holstein model (U = 0) with g = √ 8 and ω D = 8 at half-filling. Despite the large number of studies of the 1D HHM, there is still no unanimity on whether there is actually an intermediate metallic phase at half-filling. On the one hand, studies using a density matrix renormalizaton group (DMRG), 15,16 stochastic series expansion quantum Monte Carlo (QMC) method, 17 or a modified Lang-Firsov transformation 7 do find this phase; on the other hand, it has not been obtained using multiscale functional RG. 18 In this work, we use the variational cluster approximation (VCA) 19 to study the CDW phase in the 1D HHM. Doing so, we circumscribe the region of parameter space, presenting this phase at half-filling. In addition, a study of spectral gaps and of the positions of the lowest pole in the pair susceptibility will provide hints about the nature of the system in the remaining part of the phase diagram. To circumvent the burden of dealing with an infinite phonon state space, we select a single mode, q = π , out of the N s possible modes (N s is the number of lattice sites) and we set an upper limit on the phonon number in this mode. The first point is justified by the instability of the 1D system at 2k F = π ; the second one is sound as long as the average phonon number is small compared to this upper limit. The VCA is rooted in the so-called self-energy approach, 20 a variational method based on the existence of a functional [] of the self-energy that is stationary at the physical self-energy, and whose value at that point is the grand potential . That value of that functional can be calculated exactly at the physical self-energies of a family of “reference” Hamiltonians, H ′ , sharing with H the same two-body interactions, but dif- ferent one-body terms, provided these reference Hamiltonians can be exactly solved. In VCA, H ′ is obtained from H by tiling the original lattice into identical clusters of L sites, by severing the inter-cluster hoppings and by adding Weiss terms intended to probe possible broken symmetries. These Weiss 033104-1 1098-0121/2011/83(3)/033104(4) © 2011 American Physical Society