Charge-density-wave and superconducting states in the Holstein model on a square lattice
H. Zheng, S. Y. Zhu
DOI 10.1103/PhysRevB.55.3803 · Physical Review B
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Abstract
The physics of the charge-density-wave (CDW) and superconducting states (and the competition between them) in the two-dimensional Holstein model is studied by means of a unitary transformation method. The nonadiabatic effect due to a finite phonon frequency ω0>0 is treated through two energy-dependent electron-phonon scattering functions δ1(k|IH,k) and δ2(k|IH,k) introduced in the transformation. This leads to a weakening of the effective potential stabilizing the CDW state and results in the four-fermion terms, which lead to umklapp scattering terms in the CDW state and to Cooper pairing terms in the superconducting state. For the CDW state our calculated transition temperature Tp is much lower than the adiabatic one. In particular, when ω0/t≪1 the ratio 2λmp/Tp is much larger than the BCS value 3.53, because Tp is suppressed by the thermal lattice fluctuations in the finite temperature case. For larger ω0, however, the calculated Tp is higher than that of Monte Carlo simulations. The calculated density of states (DOS) of electrons has a gap. When ω0 is small the DOS is peaked above the true gap edge. For the superconducting state the gap equation is solved by a numerical method. The critical temperature Tc and the DOS of electrons are calculated from small to large phonon frequencies. Finally, we discuss the competition between the CDW and superconducting correlations by comparing Tp and Tc (as functions of ω0) for different band fillings n.
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