The existing studies of the temperature effect of the spin-boson model show that the tunneling splitting will increase with temperature, no matter how weak the couplings are between the bath and the spin. However, the small polaron theory said that in the weak coupling and low temperature regime, the tunneling is, in fact, dominated by the diagonal transitions whereas this diagonal contribution to the tunneling should be faded away with increasing temperature. Taking advantage of the analysis originated from the Feynman-Vernon's influence functional theory, the influence on the tunneling by the phonon bath can be expressed as a product of the effects of every single phonon mode, which can be studied by numerical exact diagonalization. We find that, in the weak coupling and low temperature regime, all the spin-single-mode systems exhibit the same decreasing dependence of tunneling on increasing temperature. In turn, with the conclusion of influence functional theory, the tunneling of the spin-boson model should decrease and it is independent of the bath structure. In the strong coupling regime, however, the temperature effect reverses from suppressing to enhancing the tunneling with the increase of temperature. Discrepancies between the old theories and the small polaron theory are also explained.
Effects of $τ_1$ Scattering on Fourier-Transformed Inelastic Tunneling Spectra in High-$T_c$ Cuprates with Bosonic Modes
研究了d波超导体中$τ_1$杂质诱导的$\mathbf{q}$空间能量导数局域态密度（LDOS）谱.我们的动机部分是最近的扫描隧道显微镜（STM）观察到的强间隙不均匀性与弱的电荷密度变化在Bi $_{2}$ Sr $_2$ CaCu $_2$ O $_{8+δ}$（BSCCO）。我们的假设是，间隙的不均匀性可能是由对电位的无序触发的。我们着重讨论了d波超导能隙移动时，电子耦合对各种玻色子模的影响。由于电子与$B_{1g}$声子模式的高度各向异性耦合的图案与BSCCO中的傅里叶变换非弹性电子隧道光谱（FT-IETS）STM实验的初步结果相似。我们讨论了我们的研究结果的背景下，在ARPES实验中看到的带重整化效应的影响，并提出了进一步探讨高$T_c$铜酸盐中的电子玻色子耦合的方法。
