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, 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 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}$, 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, where G ( ω ) = (1 /N ) ∑ $_{k}$G ( k , ω ). In this case, LDOS of the system can be obtained as, 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, 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 $^{′}$, with the impurity induced T matrix can be obtained as $^{32}$, and its Fourier transform is evaluated explicitly as, ρ ( r , ω ) = − 1 π Im ˜ G ( r , ω ) = ρ$_{0}$ ( ω ) + δρ ( r , ω ) , (12) δρ ( r , ω ) = ρ ( r , ω ) − ρ$_{0}$ ( ω ) = − 1 π Im[ G ( r , ω ) ˜ T ( ω ) G ( − r , ω )]$_{11}$ , (13) δρ ( q , ω ) = − 1 π Im ( 1 N ∑ k [ G ( k + q , ω ) ˜ T ( ω ) G ( k , ω )]$_{11}$ ) . (14) ˜ T ( ω ) = 1 1 − G ( ω ) V$_{0}$τ$_{3}$ V$_{0}$τ$_{3}$, (11) ˜ G ( r , r $^{′}$, ω ) = G ( r − r $^{′}$, ω ) + G ( r , ω ) ˜ T ( ω ) G ( − r $^{′}$, ω ) , (10) ˜ V = V$_{0}$δ ( r ) τ$_{3}$, (9) g$_{11}$ ( k , ω ) = Z$_{hF}$ ( U 2 h k ω − E$_{h}$$_{k}$ + V 2 h k ω + E$_{h}$$_{k}$ ) , (4) g$_{21}$ ( k , ω ) = − Z$_{hF}$ ¯ Δ$_{hZ}$( k ) 2 E$_{h}$$_{k}$ ( 1 ω − E$_{h}$$_{k}$ − 1 ω + E$_{h}$$_{k}$ ) , (5) g ( k , ω ) = Z$_{hF}$ ωτ$_{0}$ + ¯ ξ$_{k}$τ$_{3}$ − ¯ Δ$_{hZ}$( k ) τ$_{1}$ ω 2 − E 2 h k , (6) G ( k , ω ) = ( G$_{11}$ ( k , ω ) , G$_{12}$ ( k , ω ) G$_{21}$ ( k , ω ) , G$_{22}$ ( k , ω ) ) , (7) 3 as, tions of the t - J model (3) can be obtained as, G$_{11}$ ( k , ω ) = 1 N ∑ p Z$_{F}$ B$_{p}$ 2 ω$_{p}$ { coth[ 1 2 βω$_{p}$ ] ( U 2 h p + k ω + E$_{h}$$_{p}$$_{+}$$_{k}$ − ω$_{p}$ + U 2 h p + k ω + E$_{h}$$_{p}$$_{+}$$_{k}$ + ω$_{p}$ + V 2 h p + k ω − E$_{h}$$_{p}$$_{+}$$_{k}$ + ω$_{p}$ + V 2 h p + k ω − E$_{h}$$_{p}$$_{+}$$_{k}$ − ω$_{p}$ ) + tanh[ 1 2 βE$_{h}$$_{p}$$_{+}$$_{k}$ ] ( U 2 h p + k ω + E$_{h}$$_{p}$$_{+}$$_{k}$ + ω$_{p}$ − U 2 h p + k ω + E$_{h}$$_{p}$$_{+}$$_{k}$ − ω$_{p}$ + V 2 h p + k ω − E$_{h}$$_{p}$$_{+}$$_{k}$ − ω$_{p}$ − V 2 h p + k ω − E$_{h}$$_{p}$$_{+}$$_{k}$ + ω$_{p}$ )} , (8a) G$_{21}$ ( k , ω ) = 1 N ∑ p Z$_{F}$ ¯ Δ$_{hZ}$( p + k ) 2 E$_{h}$$_{p}$$_{+}$$_{k}$ B$_{p}$ 2 ω$_{p}$ { coth[ 1 2 βω$_{p}$ ] ( 1 ω − E$_{h}$$_{p}$$_{+}$$_{k}$ − ω$_{p}$ + 1 ω − E$_{h}$$_{p}$$_{+}$$_{k}$ + ω$_{p}$ − 1 ω + E$_{h}$$_{p}$$_{+}$$_{k}$ + ω$_{p}$ − 1 ω + E$_{h}$$_{p}$$_{+}$$_{k}$ − ω$_{p}$ ) + tanh[ 1 2 βE$_{h}$$_{p}$$_{+}$$_{k}$ ] ( 1 ω − E$_{h}$$_{p}$$_{+}$$_{k}$ − ω$_{p}$ − 1 ω − E$_{h}$$_{p}$$_{+}$$_{k}$ + ω$_{p}$ − 1 ω + E$_{h}$$_{p}$$_{+}$$_{k}$ + ω$_{p}$ + 1 ω + E$_{h}$$_{p}$$_{+}$$_{k}$ − ω$_{p}$ )} , (8b)