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arXiv:cond-mat/0602327 · Physical Review B

Electron-phonon coupling in the self-consistent Born approximation of the t−J model

O. Gunnarsson, O. Rösch

Published May 23, 2006• 24 citations
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Abstract

We study an undoped $t\text{\ensuremath{-}}J$ model with electron-phonon interaction using the self-consistent Born approximation (SCBA). By neglecting vertex corrections, the SCBA solves a boson-holon model, where a holon couples to phonons and magnons. Comparison with exact diagonalization results for the $t\text{\ensuremath{-}}J$ model suggests that the SCBA describes the electron-phonon interaction fairly accurately over a substantial range of $J∕t$ values. Exact diagonalization of the boson-holon model shows that the deviations are mainly due to the neglect of vertex corrections for small $J∕t$ and due to the replacement of the $t\text{\ensuremath{-}}J$ model by the boson-holon model for large $J∕t$. For typical values of $J∕t$, the electron-phonon part ${\ensuremath{\Sigma}}_{\mathrm{ep}}$ of the electron self-energy has comparable contributions from the second-order diagram in the electron-phonon interaction and a phonon-induced change of magnon diagrams. A very simple approximation to ${\ensuremath{\Sigma}}_{\mathrm{ep}}$ gives a rather accurate effective mass. Using this approximation, we study the factors influencing the electron-phonon interaction. Typically, we find that the magnons nominally have a stronger coupling to the holon than the phonons. The phonons, nevertheless, drive the formation of small polarons (self-localization) due to important differences between the character of the phonon and magnon couplings.

DOI
10.1103/physrevb.73.174521

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We study an undoped t\ensuremath-Jt\text{\ensuremath{-}}Jt\ensuremath-J model with electron-phonon interaction using the self-consistent Born approximation (SCBA). By neglecting vertex corrections, the SCBA solves a boson-holon model, where a holon couples to phonons and magnons. Comparison with exact diagonalization results for the t\ensuremath-Jt\text{\ensuremath{-}}Jt\ensuremath-J model suggests that the SCBA describes the electron-phonon interaction fairly accurately over a substantial range of J∕tJ∕tJ∕t values. Exact diagonalization of the boson-holon model shows that the deviations are mainly due to the neglect of vertex corrections for small J∕tJ∕tJ∕t and due to the replacement of the t\ensuremath-Jt\text{\ensuremath{-}}Jt\ensuremath-J model by the boson-holon model for large J∕tJ∕tJ∕t. For typical values of J∕tJ∕tJ∕t, the electron-phonon part \ensuremathΣep{\ensuremath{\Sigma}}_{\mathrm{ep}}\ensuremathΣep​ of the electron self-energy has comparable contributions from the second-order diagram in the electron-phonon interaction and a phonon-induced change of magnon diagrams. A very simple approximation to \ensuremathΣep{\ensuremath{\Sigma}}_{\mathrm{ep}}\ensuremathΣep​ gives a rather accurate effective mass. Using this approximation, we study the factors influencing the electron-phonon interaction. Typically, we find that the magnons nominally have a stronger coupling to the holon than the phonons. The phonons, nevertheless, drive the formation of small polarons (self-localization) due to important differences between the character of the phonon and magnon couplings.
The spin bipolaron in the t–J model, i.e., two holes interacting with an antiferromagnetic spin background, is treated by an extension of the self-consistent Born approximation (SCBA), which has proved to be very accurate in the single-hole (spin polaron) problem. One of the main ingredients of our approach is the exact form of the bipolaron eigenstates in terms of a complete set of two-hole basis vectors. This enables us to eliminate the hole operators and to obtain the eigenvalue problem solely in terms of the boson (magnon) operators. The eigenvalue equation is then solved by a procedure similar to Reiter's construction of the single-polaron wave function in the SCBA. As in the latter case, the eigenvalue problem comprises a hierarchy of infinitely many coupled equations. These are brought into a soluble form by means of the SCBA and an additional decoupling approximation, whereupon the eigenvalue problem reduces to a linear integral equation involving the bipolaron self-energy. The numerical solutions of the integral equation are in quantitative agreement with the results of previous numerical studies of the problem. The d-wave bound state is found to have the lowest energy with a critical value J/t| c ≈ 0.4. In contrast to recent claims, we find no indication for a crossover between the d-wave and p-wave bound states.
We derive sum rules for the phonon self-energy and the electron-phonon contribution to the electron self-energy of the Holstein-Hubbard model in the limit of large Coulomb interaction UUU. Their relevance for finite UUU is investigated using exact diagonalization and dynamical mean-field theory. Based on these sum rules, we study the importance of vertex corrections to the electron-phonon interaction in a diagrammatic approach. We show that they are crucial for a sum rule for the electron self-energy in the undoped system while a sum rule related to the phonon self-energy of doped systems is satisfied even if vertex corrections are neglected. We provide explicit results for the vertex function of a two-site model.
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