content_image/406813.jpg content_image/406814.jpg content_image/406815.jpg content_image/406816.jpg content_image/406820.jpg content_image/406821.jpg content_image/406811.jpg content_image/406812.jpg content_image/406810.jpg content_image/406807.jpg content_image/406808.jpg content_image/406809.jpg
tions of the t - J model (3) can be obtained as, 

<img src='content_image/406813.jpg'>

<img src='content_image/406814.jpg'>

where the charge carrier quasiparticle spectrum E$_{h}$$_{k}$ = √ ¯ ξ 2 $_{k}$+ | ¯ Δ$_{hZ}$( k ) | 2 with the renormalized d-wave charge carrier pair gap function ¯ Δ$_{hZ}$( k ) = ¯ Δ$_{hZ}$[cos k$_{x}$ − cos k$_{y}$ ] / 2, and the charge carrier quasiparticle coherence factors U 2 h k = (1 + ¯ ξ$_{k}$/E$_{h}$$_{k}$ ) / 2 and V 2 h k = (1 − ¯ ξ$_{k}$/E$_{h}$$_{k}$ ) / 2, while the charge carrier quasiparticle coherent weight Z$_{hF}$ and other notations are defined as same as in Ref. 25, and have been determined by the self-consistent calculation $^{17,25}$. For the convenience of the following dis- cussions, the full charge carrier Green functions (4) and (5) can also be expressed in the Nambu representation 

as, 

where τ$_{0}$ is the unit matrix, τ$_{1}$ and τ$_{3}$ are Pauli matrices. 

In the CSS fermion-spin representation $^{20,21}$, the elec- tron Green’s function in the Nambu representation, 

is a convolution of the spin Green’s function and charge carrier Green’s function (6), and its diagonal and off- diagonal components G$_{11}$ ( i − j, t − t $^{′}$) = 〈〈 C$_{iσ}$ ( t ); C † $_{jσ}$( t $^{′}$) 〉〉 and G$_{21}$ ( i − j, t − t $^{′}$) = 〈〈 C † i $_{↑}$( t ); C † j $_{↓}$( t $^{′}$) 〉〉 have been ob- tained as $^{25}$, 

<img src='content_image/406815.jpg'>

<img src='content_image/406816.jpg'>

<img src='content_image/406820.jpg'>

<img src='content_image/406821.jpg'>

respectively, where the electron quasiparticle coherent weight Z$_{F}$ = Z$_{hF}$ / 2, and the spin excitation spectrum ω$_{p}$ and B$_{p}$ have been given in Ref. 25. 

In the presence of a single point-like impurity scatter- ing potential, 

the unperturbed electron Green’s function in Eq. (7) is dressed by this impurity scattering, where the T ma- trix exactly accounts for multiple scattering off that impurity $^{32}$. Since translational invariance is broken by the impurity, the dressed electron Green’s function in the Nambu representation in real-space depends on two positions r and r $^{′}$, 

<img src='content_image/406811.jpg'>

with the impurity induced T matrix can be obtained as $^{32}$, 

<img src='content_image/406812.jpg'>

<img src='content_image/406810.jpg'>

where G ( ω ) = (1 /N ) ∑ $_{k}$G ( k , ω ). In this case, LDOS of the system can be obtained as, 

<img src='content_image/406807.jpg'>

where the homogeneous density of states ρ$_{0}$ ( ω ) = − Im G (0 , ω ) /π , and is uniform in real-space, therefore it reflects a homogenous background, while δρ ( r , ω ) is the modulation for the homogeneous density of states due to the presence of the impurity scattering potential (9), and can be obtained as, 

<img src='content_image/406808.jpg'>

and its Fourier transform is evaluated explicitly as, 

<img src='content_image/406809.jpg'>

