usepackage{amsbsy}
+ [mathcal{D}_2]_inftyleft(
begin{array}{c}
frac{1}{2}\
vdots\
0
end{array}
right)
+mathbb{E}left[mathrm{Tr}left(mathcal{tilde{D}}_2right)right],endaligned\$\$ where \$mathbb Eleft[Xright]\$ is the expectation of \$X\$ over the random variables \$X\$. The first term in the right hand side of Eq.Â ([eq:delta2]) is the leading term in Eq.([eq1]), and the second term is the next-to-leading term. The second term in this equation is the sum of the second and third terms. The third term is a constant that does not depend on \$X\$, and the fourth term is an additional constant that depends on \$N\$. This constant is the second-order correction to the leading order. The constant in the fourth line of Eqs.Â (1) and (2) is the same as the one in Eqs.(ref{eq:D2}), (ref {eq:tilde D2}). The constant is equal to \$1/2\$ if the number of sites in the system is even, and to \$0\$ if it is odd.
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The leading term of the density of states of the system can be calculated by using the fact that the density operator \$mathcal D_2\$ is diagonal in the basis of the eigenstates of the Hamiltonian. The density of the states of a system with \$N\$ sites is equal \$\$label{eq1}
n(omega)=sum_{i=1}^{N}delta(omeg_i-omega).\$\$ The density operator is diagonalized by the transformation \$\$mathbf{S}_i=left{
begin{aligned}
label {eq2}
\