Additive spectral Hamiltonian and normalized ensembles

Fix $q\ge 1$ and real level parameters $e_1,\dots ,e_q$. Use the real exponential of The real exponential function and the number $e$ by a power series and natural logarithm of The natural logarithm as the inverse of the exponential function. For $N\ge 1$, define the one-site diagonal matrix $h\mathrm{\mid }j⟩=e_j\mathrm{\mid }j⟩$ and, for $a=1,\dots ,N$, $h^{(a)}=I^{\otimes (a-1)}\otimes h\otimes I^{\otimes (N-a)},H_N={\sum }_{a=1}^N{h}^{(a)}\mathrm{.}$ Thus the product basis vector $\mathrm{\mid }{j}_1,\dots ,j_N⟩$ has eigenvalue $E(j_1,\dots ,j_N)={\sum }_{a=1}^N{e}_{j_a}$; repeated level values remain distinct basis labels and therefore count as degeneracy.

For finite $\beta \in \mathbb{R}$, put $Z_1={\sum }_{j=1}^q{e}^{-\beta e_j},p_j=\frac{e^{-\beta e_j}}{Z_1},E={\sum }_{j=1}^q{p}_j{e}_j\mathrm{.}$ All terms in $Z_1$ are positive, so $Z_1>0$. Define the one-site canonical density $\gamma =\mathrm{diag}(p_1,\dots ,p_q)$. For $N$ sites put $Z_N={\sum }_{(j_1,\dots ,j_N)\in \{1,\dots ,q{\}}^N}{e}^{-\beta {\sum }_{a=1}^N{e}_{j_a}},{\gamma }_N=\mathrm{diag}{\left(\frac{e^{-\beta {\sum }_{a=1}^N{e}_{j_a}}}{Z_N}\right)}_{(j_1,\dots ,j_N)\in \{1,\dots ,q{\}}^N}\mathrm{.}$ The mathematical definition permits $\beta =0$ and negative $\beta$ as well as positive $\beta$; a physical positive-temperature interpretation is a separate assumption.

For a width ${\delta }_N\ge 0$, define the spectral band label set and its projection by ${\mathrm{\Gamma }}_N=\{(j_1,\dots ,j_N):\mid {\sum }_{a=1}^N{e}_{j_a}-NE\mid \le {\delta }_N\},{\mathrm{\Pi }}_N={\sum }_{\mathbf{j}\in {\mathrm{\Gamma }}_N}\mathrm{\mid }\mathbf{j}⟩⟨\mathbf{j}\mathrm{\mid }\mathrm{.}$ Its rank is ${\mathrm{\Omega }}_N=\mathrm{Tr}{\mathrm{\Pi }}_N=\mathrm{\mid }{\mathrm{\Gamma }}_N\mathrm{\mid }$, which counts all basis vectors in the band, including degeneracy. When ${\mathrm{\Omega }}_N>0$, define the normalized band density ${\sigma }_N={\mathrm{\Pi }}_N\mathrm{/}{\mathrm{\Omega }}_N$. If ${\mathrm{\Omega }}_N=0$, this normalized density is undefined.