Hermitian Hamiltonian commutes with the group generators those correspond the group eigenvalues θ$_{i}$ where i runs over 1 , · · · , N . The invariance Haar measure is reduced to where − π ≤ θ$_{i}$ ≤ π for the circular ensemble. The variables ( θ$_{1}$, · · · , θ$_{N}$ ) are the diago- nalized group eigenvalues. The number of adjoint charges is reduced to N$_{adj}$ = N $^{2}$. The normalisation constant N$_{U}$ is determined by The saddle points of the group eigen-variables accumulate around the origin for the hot and dense nuclear matter under extreme conditions. Subsequently, the canonical ensemble can be written as the Gaussian-like ensemble in the following way, Therefore, the invariance Haar measure is approximated to The unimodular condition for the special unitary symmetry group SU ( N ) is imposed by Hence, the invariance Haar measure for SU ( N ) becomes 22 ∫ dµ$_{SU}$ ( θ$_{1}$ · · · θ$_{N}$ ) e − 1 $_{2}$a ∑ N i θ 2 i = N$_{U}$ 1 N ! ∫ ∞ −∞ N ∏ i =1 dθ$_{i}$ 2 π 2 πδ ( N ∑ i θ$_{i}$ ) × N ∏ i<j ( θ$_{i}$ − θ$_{j}$ ) 2 e − 1 $_{2}$a ∑ N i θ 2 i . (70) N ∑ i θ$_{i}$ = 0 . (69) ∫ dµ ( θ$_{1}$ · · · θ$_{N}$ ) ( · · · ) ≈ N$_{U}$ 1 N ! ∫ ∞ −∞ N ∏ i =1 dθ$_{i}$ 2 π N ∏ i<j 4 ( θ$_{i}$ − θ$_{j}$ 2 ) 2 ( · · · ) , ≈ N$_{U}$ 1 N ! ∫ ∞ −∞ N ∏ i =1 dθ$_{i}$ 2 π N ∏ i<j ( θ$_{i}$ − θ$_{j}$ ) 2 ( · · · ) . (68) ∫ dµ ( θ$_{1}$ · · · θ$_{N}$ ) e − a ∑ N c i θ 2 i . (67) 1 N$_{U}$ = 1 N ! ∫ π − π ( N ∏ i =1 dθ$_{i}$ 2 π ) N ∏ i<j 4 sin 2 ( θ$_{i}$ − θ$_{j}$ 2 ) . (66) ∫ dµ ( θ$_{1}$ · · · θ$_{N}$ ) = N$_{U}$ 1 N ! ∫ π − π ( N ∏ i =1 dθ$_{i}$ 2 π ) N ∏ i<j 4 ( e iθ$_{i}$ − e iθ$_{j}$ ) ( e iθ$_{i}$ − e iθ$_{j}$ ) ∗ = N$_{U}$ 1 N ! ∫ π − π ( N ∏ i =1 dθ$_{i}$ 2 π ) N ∏ i<j 4 sin 2 ( θ$_{i}$ − θ$_{j}$ 2 ) , (65)