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000120 | Sixteen Majorana copies produce the carrier central charge
000121 | $16\cdot\frac12=8$.  We write $\mathfrak h_N$ for the one-particle space,
000122 | $H_N^{(r)}$ for the corresponding one-particle matrix, and
000123 | \[
000124 |  C_N^{(r)}
000125 |  =\chi_{(-\infty,0)}(H_N^{(r)})
000126 |   \frac12\chi_{\{0\}}(H_N^{(r)})
000127 | \tag{2.3}\label{eq:covariance}
000128 | \]
000129 | for the particle--hole symmetric covariance.  The finite observable
000130 | algebra is a matrix CAR algebra.  Its low-energy GNS space is denoted
000131 | $\Hh_N$, and $L_{0,N}$ denotes the shifted, nonnegative edge-energy
000132 | operator in the scaling normalization.
000133 | 
000134 | \begin{remark}[Necessary correction: there is no uniform full gap]
000135 | \label{rem:gapcorrection}
000136 | The topological collar has chiral edge levels with spacing $O(N^{-1})$.
000137 | Thus a claim of a spectral gap
000138 | $\inf_N\operatorname{gap}(H_N)>0$ is false.  The usable hypothesis is a
000139 | \emph{uniform bulk gap after removing the edge band}.  If $Q_N$ is the
000140 | edge-band projection, the correct statement is
000141 | \[
000142 |  \operatorname{spec}\bigl((1-Q_N)H_N(1-Q_N)\bigr)
000143 |  \cap(-\delta_{\rm bulk},\delta_{\rm bulk})=\varnothing
000144 | \tag{2.4}\label{eq:bulkgap}
000145 | \]
000146 | for one $\delta_{\rm bulk}>0$ independent of $N$.  This controls bulk
000147 | resolvents but does not control the massless longitudinal Green function
000148 | by itself.
000149 | \end{remark}
000150 | 
000151 | \subsection{The exact order-four vector}
000152 | 
000153 | In rational coordinates,
000154 | \[
000155 |  \omega_s=\left(\frac12,\frac12,\frac12,\frac12,\frac12\right),
000156 |  \qquad
000157 |  \omega_f=\left(\frac34,-\frac14,-\frac14,-\frac14\right),
000158 |  \qquad
000159 |  \lambda=(\omega_s,\omega_f).
000160 | \tag{2.5}\label{eq:lambda}
000161 | \]
000162 | The exact finite calculation gives
000163 | \[
000164 |  \norm{\omega_s}^2=\frac54,\quad
000165 |  \norm{\omega_f}^2=\frac34,\quad
000166 |  \norm{\lambda}^2=2,\quad h_\lambda=1,
000167 | \tag{2.6}\label{eq:lambda-norm}
000168 | \]
000169 | and
000170 | \[
000171 |  [\lambda]^4=[0],\qquad [\lambda]^2=[v].
000172 | \tag{2.7}\label{eq:lambda-order}
000173 | \]
000174 | The norm-two vectors in
000175 | $D_5\oplus A_3+k\lambda$, $k=0,1,2,3$, have census
000176 | \[
000177 |  [52,64,60,64],
000178 | \tag{2.8}\label{eq:census}
000179 | \]
000180 | so the four cosets contain $240$ roots and the two odd grades contain the
000181 | full $128$-dimensional spinor contribution.
000182 | 
000183 | \subsection{The finite charged field}
000184 | 
000185 | Let $\phi_N=(\phi_N^1,\ldots,\phi_N^8)$ be the bosonic current obtained
000186 | from the sixteen Majorana copies by edge-projected collar bosonization,
000187 | with zero mode fixed by the $\Z_4$ charge sector.  Each complex-fermion
000188 | pair gives one boson.  The longitudinal derivative uses the frozen
000189 | lattice symbol
000190 | \[
000191 |  \omega_N(m)=\frac{N}{\pi}\left|\sin\frac{\pi m}{N}\right|,
000192 |  \qquad 0<|m|\le\frac N2.
000193 | \tag{2.9}\label{eq:latticesymbol}
000194 | \]
000195 | The edge projection is the analytic top (or bottom) eigenbranch through
000196 | the zero-energy crossing; massive bulk-complement modes are not included
000197 | in $\phi_N$.  For a fixed normalized
000198 | two-cocycle $\varepsilon_\lambda$, define on the finite-energy vectors
000199 | \[
000200 |  \Psi_{\lambda,N}(f)
000201 |  =\sum_{m}\widehat f_m\,
000202 |    \varepsilon_\lambda
000203 |    :\!\exp(i\lambda\cdot\phi_N)\!:_m .
000204 |  \tag{2.10}\label{eq:chargedfield}
000205 | \]
000206 | Normal ordering is with respect to $C_N^{(0)}$.  Formula
000207 | \eqref{eq:chargedfield} is meaningful at every finite $N$.  The theorem
000208 | does not follow merely from that fact: the normal-ordering constant and
000209 | the graph norm of the exponential can grow with $N$.
000210 | 
000211 | \begin{definition}[Scaling convergence used below]
000212 | Let $V_N:\Hh_N\to\Hh_\infty$ be the canonical isometries of the chosen
000213 | operator-algebraic renormalization system.  A sequence of closable
000214 | operators $X_N$ converges on the finite-energy core
000215 | \[
000216 |  \Dfin=\bigcup_{E<\infty}\id_{[0,E]}(L_0)\Hh_\infty
000217 |  \tag{2.11}\label{eq:core}
000218 | \]
000219 | if for every $\xi\in\Dfin$, represented at sufficiently large scale,
000220 | \[
000221 |  V_NX_NV_N^*\xi\longrightarrow X\xi,\qquad
000222 |  V_NX_N^*V_N^*\xi\longrightarrow X^*\xi.
000223 |  \tag{2.12}\label{eq:graphconv}
000224 | \]