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University of Groningen Phonons, Charge and Spin in Correlated University of Groningen Phonons, charge and spin in correlated systems Macridin, Alexandru; Sawatzky, G.A IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below. Document Version Publisher's PDF, also known as Version of record Publication date: 2003 Link to publication in University of Groningen/UMCG research database Citation for published version (APA): Macridin, A., & Sawatzky, G. A. (2003). Phonons, charge and spin in correlated systems. s.n. Copyright Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons). The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license. More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne- amendment. Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons the number of authors shown on this cover page is limited to 10 maximum. Download date: 30-09-2021 Chapter 4 Hubbard-Holstein Bipolaron 4.1 Introduction Aside form dressing the charge carriers, the electron-phonon interaction also introduces an effective attraction between them. As the BCS theory shows [1], even a small attraction makes a Fermi liquid instable towards electron pairing, resulting in superconductivity. The pairs (Cooper pairs) are formed in momentum space, having a large spatial extent (of the order of several thousand lattice constants). On the other hand, when the electron- phonon interaction is large, two electrons (or holes) will form a small size pair in real space, called bipolaron. Bipolaronic effects (real space pairing) were found in many materials like transition metal-oxides [2–5], superconducting materials [6] and conjugated polymers [7]. The dis- covery of high Tc superconductors renewed the interest in the study of electron-phonon interaction at intermediate and large coupling and special theoretical attention was given to the scenarios where electrons form light bipolaron states which suffer Bose-Einstein condensation, leading to superconductivity [8]. In correlated systems both electron-electron and electron-phonon interactions are im- portant. The first is repulsive and instantaneous and the second induces a retarded at- traction between electrons. The retarded nature of the phonon mediated attraction is ex- tremely important, the time delay stabilizing the pair formation even when the Coulomb repulsion is large. The strength of the electron-phonon coupling in high Tc and other strongly correlated materials is situated in the intermediate region, where the applicabil- ity of weak or strong coupling perturbation theories is not valid. Therefore solutions for Hamiltonian models which consider both the electron-electron and the electron-phonon in- teraction in this difficult region of parameters space are desirable. The Hubbard-Holstein Hamiltonian, which we address in this chapter, is one of the most popular model studied in the literature. Even if it is not the best candidate to explain the physics of cuprates, producing bipolarons with extremely large effective mass, its solution is important for understanding the competition between the phonon induced electron-electron attraction and the electron-electron Coulomb repulsion. For the one dimensional case the problem was satisfactory solved by Bon˜ca et al. [9] with a method which uses the exact diagonal- ization technique on a variationally determined Hilbert subspace. Other one-dimensional calculations were based on variational methods [10] and density-matrix combined with 59 60 Chapter 4. Hubbard-Holstein Bipolaron Lancszos diagonalization technique [11, 12]. The two-dimensional case was investigated in the adiabatic approximation [13, 14] and with variational methods [15]. In this chapter we present the calculation of the two-dimensional Hubbard-Holstein bipolaron based on a Diagrammatic Quantum Monte Carlo algorithm. To our knowledge, this is the first two-dimensional calculation which considers dynamical phonons and does not imply any artificial truncation of the Hilbert space. The idea of the algorithm is to compute the imaginary time two-particle Green’s function and to extract information about the bipolaron state from the Green’s function behavior at long imaginary time, where the ground state of the system is projected out. An algorithm based on the same idea was used in [16] for exciton problem calculation, but in contrast to that situation where the calculation was done in momentum space, we work in direct space (site) rep- resentation, the basis consisting of Wannier orbitals and phonons at each site. In this way we manage to avoid the sign problem which would appear in the momentum repre- sentation when the Coulomb repulsion is introduced. The code can be easily adapted to include longer range electron-phonon or electron-electron interaction and to study models more suitable to the cuprates, as for example the extended Hubbard-Holstein model [17]. The disadvantage is that in this basis the momentum dependence of different quantities is difficult or sometimes impossible to compute. We work on a square lattice of 25 × 25 sites with periodic boundary conditions which is large enough for having negligible finite size errors. There are no other truncations of the Hilbert space. The Hubbard-Holstein Hamiltonian is X X † H = −t (ciσcjσ + H.c) + U ni↑ni↓ + hiji,σ i X X † † +ω0 bi bi + g niσ(bi + bi) (4.1) i i,σ † Here ciσ(ciσ) is the creation (annihilation) operator of an electron with spin σ at † site i, and analogues bi , bi are phonon creation and annihilation operators. The first term describes the nearest-neighbor hopping of the electrons, and the second the on-site Coulomb repulsion between two electrons. The lattice degrees of freedom are described by a set of independent oscillators at each site, with frequency ω0. The electrons couple † † through the density niσ = ciσciσ to the local lattice displacement xi ∝ (bi + bi) with the strength g. This Hamiltonian is a tight-binding model together with an on-site Coulomb repulsion term and an on-site electron-phonon interaction term. The Holstein and the Hubbard models are limiting cases for U = 0 and g = 0 respectively. We are going to address the pairing problem as a function of both Coulomb repulsion and electron-phonon interaction by studying two electrons on a two-dimensional lattice. 4.2 Analytical Approaches Similar to the Holstein polaron problem, perturbation theory can be used to calculate the weak and the strong electron-phonon interaction regimes. In this section we present the results for these two cases. 4.2. Analytical Approaches 61 k k−q q p p+q Figure 4.1: Effective phonon induced interaction in second order perturbation theory. 4.2.1 Weak-Coupling Perturbation Theory For g = 0, the ground state will be formed by two electrons with zero momentum moving freely through the lattice. When the electron-phonon interaction is switched on, two things happen. First, the electrons (at the bottom of the band) get lightly dressed which results in an increase of their effective mass, and second the electron-phonon interaction introduces an effective attraction between the electrons. Up to second order in g the effective interaction induced by the phonon is proportional to the phonon propagator 2 eff 2 2g ω0 Vph (ω) = g D(q, ω) = − 2 2 (4.2) ω0 − ω as can be seen from Fig. 4.1. This is a retarded interaction and attractive at small fre- quency (for ω < ω0). In a two-dimensional lattice an attractive interaction will cause the electrons to form a bound pair. A similar situation happens even in a three-dimensional case with a Fermi surface present, the instability towards electron pairing leading to the superconductivity [1]. In our model the attractive phonon induced interaction competes with the Coulomb repulsion, which results in a total effective interaction 2 eff 2g ω0 V (ω) = U − 2 2 (4.3) ω0 − ω When the electron-phonon coupling is increased the situation gets rapidly complicated, and any calculation should consider the renormalization of the electron-phonon interaction vertex. Migdal’s theorem [18] applied in classical superconductivity theory is not valid here because of the absence of the Fermi sea ∗. In the antiadiabatic limit (ω0 −→ ∞) where the ions are considered light and able to follow instantaneously the motion of the electrons, the effective interaction (Eq. 4.3) is instantaneous too 2g2 V eff = U − (4.4) ω0 and the situation can be described by a pure Hubbard model. ∗ Migdal’s theorem shows that the vertex corrections are of order ω0/εF due to Pauli exclusion principle which blocks the electron-phonon scattering inside the Fermi sea. 62 Chapter 4. Hubbard-Holstein Bipolaron 4.2.2 Strong-Coupling Perturbation Theory When g is large, one can apply perturbation theory with respect to the hopping part of the Hamiltonian. The last three terms in Eq. (4.1) are diagonal in the rotated basis obtained by applying the unitary operator eS [19] where g X S = − n (b† − b ) (4.5) ω iσ i i 0 i,σ The transformed phonon operators are X g b˜ = b + n (4.6) i i ω iσ σ This is the same transformation we applied for Holstein polaron model (Eq. 3.8), the only difference being the summation over the spin index σ.
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