TFPT scientific source excerpt
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000100 | \begin{definition}[Structural transport kernel]
000101 | \label{def:structural-transport-kernel}
000102 | For each fermion sector define the structural positive transport kernel by
000103 | \[
000104 | K_f^{\mathrm{str}}:=\bigl(P_{\mathrm{str}} D_f^\dagger D_f P_{\mathrm{str}}\bigr)^{-1/2},
000105 | \]
000106 | whenever the inverse exists on $\mathrm{Ran}(P_{\mathrm{str}})$. The final admissible kernel is
000107 | the restriction of this operator to the gap-stable branch selected below. The transport pole
000108 | $\delta_{\mathrm{ph}}$ is fixed by the structural stationarity of the exact determinant functional
000109 | \[
000110 | \mathcal F_{\mathrm{tr}}(\delta)
000111 | :=
000112 | \sum_{y\in\{1,1/3,2/3\}}
000113 | \Bigl[-\log\det\!\bigl(D_{y,\delta}^\dagger D_{y,\delta}+\varepsilon^2\bigr)\Bigr],
000114 | \]
000115 | with
000116 | \[
000117 | D_{y,\delta}:=y\,\mathbf 1-\delta U_6.
000118 | \]
000119 | On the closed branch one has
000120 | \begin{equation}
000121 | \label{eq:transport-bridge-offset}
000122 | \partial_\delta \mathcal F_{\mathrm{tr}}(\delta_{\mathrm{ph}})=0,
000123 | \qquad
000124 | \delta_{\mathrm{ph}}\in\left(\frac13,\frac23\right).
000125 | \end{equation}
000126 | \end{definition}
000127 | 
000128 | \begin{theorem}[Algebraic closure of the transport pole]
000129 | \label{prop:closed-form-transport-pole}
000130 | \label{thm:algebraic-transport-pole}
000131 | \label{thm:transport-pole-algebraic}
000132 | Let
000133 | \[
000134 | \mathcal F_{\mathrm{tr}}(\delta)
000135 | :=
000136 | \sum_{y\in\{1,1/3,2/3\}}
000137 | \Bigl[-\log\det\!\bigl(D_{y,\delta}^\dagger D_{y,\delta}+\varepsilon^2\bigr)\Bigr],
000138 | \qquad
000139 | D_{y,\delta}:=y\,\mathbf 1-\delta U_6.
000140 | \]
000141 | Write
000142 | \[
000143 | x:=\delta^2,
000144 | \qquad
000145 | \tau:=\varepsilon^2,
000146 | \qquad
000147 | z:=x^3=\delta^6.
000148 | \]
000149 | Then
000150 | \[
000151 | \det\!\bigl(D_{y,\delta}^\dagger D_{y,\delta}+\varepsilon^2\bigr)
000152 | =
000153 | Q_y(x,\tau),
000154 | \]
000155 | with
000156 | \[
000157 | Q_y(x,\tau)
000158 | =
000159 | \Bigl(x^2+2(\tau-y^2)x+(y^2+\tau)^2\Bigr)
000160 | \Bigl(x^2+(y^2+2\tau)x+(y^2+\tau)^2\Bigr)^2.
000161 | \]
000162 | Hence
000163 | \[
000164 | \partial_\delta \mathcal F_{\mathrm{tr}}(\delta)=0
000165 | \iff
000166 | \sum_{y\in\{1,1/3,2/3\}}
000167 | \frac{\partial_x Q_y(x,\tau)}{Q_y(x,\tau)}=0.
000168 | \]
000169 | On the closed zero-parameter branch one takes the intrinsic limit
000170 | \[
000171 | \tau=0
000172 | \]
000173 | after admissibility restriction. Then
000174 | \[
000175 | Q_y(x,0)=(y^6-z)^2,
000176 | \]
000177 | and the stationarity equation reduces to
000178 | \[
000179 | \frac{1}{1-z}
000180 | +
000181 | \frac{1}{1/729-z}
000182 | +
000183 | \frac{1}{64/729-z}
000184 | =0.
000185 | \]
000186 | Equivalently, for
000187 | \[
000188 | P(z):=
000189 | (z-1)\left(z-\frac{64}{729}\right)\left(z-\frac{1}{729}\right),
000190 | \]
000191 | one has
000192 | \[
000193 | P'(z)=0.
000194 | \]
000195 | The admissible transport pole is therefore the lower critical point of this cubic:
000196 | \[
000197 | z_\star
000198 | =
000199 | \frac{794-7\sqrt{9961}}{2187},
000200 | \qquad
000201 | \delta_{\mathrm{ph}}=z_\star^{1/6}
000202 | =
000203 | \left(\frac{794-7\sqrt{9961}}{2187}\right)^{1/6}
000204 | \approx \num{0.593277280133224}.
000205 | \]
000206 | As an expanded control form, one may equivalently write
000207 | \[
000208 | 1594323\,\delta^{12}-1157652\,\delta^6+47449=0.
000209 | \]
000210 | \end{theorem}
000211 | 
000212 | \begin{proof}
000213 | In the Fourier basis of the six-site shift $U_6$, the operator
000214 | \[
000215 | D_{y,\delta}^\dagger D_{y,\delta}+\varepsilon^2
000216 | \]
000217 | has eigenvalues
000218 | \[
000219 | \lambda_{y,n}(x,\tau)
000220 | =
000221 | y^2+x-2y\sqrt{x}\cos\!\left(\frac{n\pi}{3}\right)+\tau,
000222 | \qquad n=0,\dots,5.
000223 | \]
000224 | Multiplying the $n=0,3$ factors and the conjugate pairs $(1,5)$ and $(2,4)$ gives exactly the
000225 | displayed factorization $Q_y(x,\tau)$. Since $x=\delta^2$ and $\delta\in(1/3,2/3)$ on the
000226 | admissible branch, one has $\delta>0$, so
000227 | \[
000228 | \partial_\delta\log Q_y(x,\tau)
000229 | =
000230 | 2\delta\,\partial_x\log Q_y(x,\tau),
000231 | \]
000232 | and the stationarity condition is equivalent to the displayed $x$-equation. Setting $\tau=0$
000233 | collapses the factors to
000234 | \[
000235 | Q_y(x,0)
000236 | =(y^2-x)^2(y^4+xy^2+x^2)^2
000237 | =(y^6-z)^2.
000238 | \]
000239 | Therefore
000240 | \[
000241 | \sum_{y\in\{1,1/3,2/3\}}\frac{\partial_x Q_y(x,0)}{Q_y(x,0)}
000242 | =
000243 | -6x^2\sum_{y\in\{1,\frac13,\frac23\}}\frac{1}{y^6-z},
000244 | \]
000245 | so the stationary point condition reduces to the displayed rational equation. Let
000246 | \[
000247 | P(z):=(z-1)\left(z-\frac{64}{729}\right)\left(z-\frac{1}{729}\right).
000248 | \]
000249 | Since
000250 | \[