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.

—————————————————————————————————————————————————————————————————————————————————————————————————————–

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}

&frac{omega_i}{sqrt{N}},&quad i=1,dots,N,\

&0,&qquad ineq 1,

label {eq2}

\

\ &1,& qquad i=N+1,N+2,ldots,2N.

nonumber

right.$$ The transformed density operator can be written as $$begin {split}

&mathsf{D_2}(omegas)=mathop{sum_{omegas_1, dots omegas_{2N}}}_{mathclap{substack{omegas’_1delta_{{1,2}},dagger_{{omeg 1,omeg 2}}}}}\

&{timesmathfrak{S}}_{left(omegs_1+omegs’_2, omega_{rm max}+domegaright)}

left((omegam_1-dgomegal_1)^*dghomegt_1(omegam’_3-dhomegat_3)^Tright)\

×dgt_2(omegal’_4-hgdgal’omegap_4),

qed

mathscr{D_{2}}(omegap)=frac 1{2sq{N}}sum_idgd_i(omegp_i)^2,

end{split}$$ where the subscripts of the transformed density operators are the indices of the sites. The transformed densities are calculated by the following equations: $$dgp_1=dgs_2-frac 12dgb_1^2dga_1;quad dgp_{2i}=frac dgb_{2,i}-left|fracdgc_{2,i}right|^2-2frac {dgd_{2}^2}{dgas_{1}^*}dgb^*_1.$$

The density matrix of the reduced density operator of the whole system is equal $

rho_N=mathsfs{D^dag_2}mathsf D_1

$ and the density matrix is diagonal.

The density of a single site is equal: $$n_1(0)=dtg_2^2+frac 14dgg_2+2left.frac {dmathds{1}}{domegp}rightvert_{dgap=0}dgp.$$ This is the density in the case of the even number of site.