TFPT scientific source excerpt
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000032 | 
000033 | Let \(\Gamma=\Z^2\) with graph distance \(d\) and coordination \(z=4\).
000034 | For a finite \(\Lambda\Subset\Gamma\), put
000035 | \[
000036 |  \Hh_\Lambda=\bigotimes_{x\in\Lambda}\Hh_x,\qquad
000037 |  \A_\Lambda=\mathcal B(\Hh_\Lambda),
000038 |  \qquad
000039 |  \iota_{\Lambda,\Lambda'}(A)=A\otimes\id_{\Lambda'\setminus\Lambda}.
000040 | \tag{1.1}\label{eq:ambient}
000041 | \]
000042 | One cell contains the four representative physical edge modes, a mirror
000043 | Fock factor, the two positively oriented outgoing finite-group gauge links,
000044 | and a four-state seam clock.  Thus
000045 | \[
000046 |  \dim\Hh_x=16\cdot 2^{n_{\rm mir}}\cdot 2^2\cdot4,
000047 |  \qquad n_{\rm mir}\in\{4,16\}.
000048 | \tag{1.2}\label{eq:localdim}
000049 | \]
000050 | The DET4 numerical analogue has dimension \(4096\); the full DET16 cell has
000051 | dimension \(16\,777\,216\).  Only finiteness and a volume-independent upper
000052 | bound are used below.  The quasilocal algebra is
000053 | \[
000054 |  \A=\overline{\bigcup_{\Lambda\Subset\Gamma}\A_\Lambda}^{\norm{\cdot}}.
000055 | \tag{1.3}\label{eq:quasilocal}
000056 | \]
000057 | 
000058 | \begin{definition}[TFPT assembled interaction]
000059 | For finite \(X\Subset\Gamma\), let \(\Phi(X)=\Phi(X)^*\), with
000060 | \(\Phi(X)=0\) unless \(X\) is one of the supports in
000061 | Table~\ref{tab:terms}.  The free-boundary finite-volume Hamiltonian is
000062 | \[
000063 |  H_\Lambda=\sum_{X\subseteq\Lambda}\Phi(X).
000064 | \tag{1.4}\label{eq:H}
000065 | \]
000066 | Scalar multiples of the identity may be removed termwise, since they do
000067 | not change \(e^{itH_\Lambda}Ae^{-itH_\Lambda}\).
000068 | \end{definition}
000069 | 
000070 | \begin{table}[ht]
000071 | \centering
000072 | \small
000073 | \begin{tabularx}{\textwidth}{@{}lXccc@{}}
000074 | \toprule
000075 | term & frozen operator and coefficient & \(\norm{\Phi(X)}\) & \(|X|\) &
000076 | \(\diam X\)\\
000077 | \midrule
000078 | electric &
000079 | \(\frac{g^2}{2}E_\ell^2,\ g=1.2\), with each positive link assigned to
000080 | its tail cell & \(18/25=0.72\) & 1 & 0\\
000081 | plaquette &
000082 | \(\frac1{2g^2}(2-U_p-U_p^*)\); its dynamical representative is
000083 | \(-\frac1{2g^2}(U_p+U_p^*)\) & \(25/36\) & 4 & 2\\
000084 | hopping &
000085 | \(-t\sum_{a=1}^4(c_{y,a}^*U_{xy}c_{x,a}+\mathrm{h.c.}),\ t=0.6\)
000086 | & \(\leq4t=12/5\) & 2 & 1\\
000087 | DET16 &
000088 | \(2(\id-P_{\phi,x})\), where \(P_{\phi,x}\) is the rank-one
000089 | Spin(10)-invariant determinant projector & \(2\) & 1 & 0\\
000090 | seam clock &
000091 | \(-0.5(\Psi_x+\Psi_x^*)\), \(\Psi_x^4=\id\) & \(1\) & 1 & 0\\
000092 | seam charge &
000093 | \(0.2(\Psi_x+\Psi_x^*) (n_{Q,x}-1/2)/2\) & \(1/10\) & 1 & 0\\
000094 | staggered mass &
000095 | \(0.2\sum_{a=1}^4(-1)^x(n_{x,a}-n^{\rm bg}_{x,a})\) & \(\leq4/5\) & 1 & 0\\
000096 | \bottomrule
000097 | \end{tabularx}
000098 | \caption{Supports and norm bounds extracted from the frozen assembled
000099 | models.  The positive plaquette term itself has norm at most
000100 | \(2/g^2=25/18\); after deleting its scalar part, the norm relevant to
000101 | dynamics and commutators is \(1/g^2=25/36\).}
000102 | \label{tab:terms}
000103 | \end{table}
000104 | 
000105 | The determinant identity checked in
000106 | \texttt{verification/v1002\_spin10\_mirror\_projector.py} gives
000107 | \(P_{\phi,x}=P_{\phi,x}^*=P_{\phi,x}^2\), so its row has norm exactly
000108 | two.  The clock and charge bounds follow from
000109 | \(\norm{\Psi}=1\), \(\norm{(\Psi+\Psi^*)/2}=1\), and
000110 | \(\norm{n_Q-1/2}=1/2\).
000111 | 
000112 | \subsection{Uniform constants}
000113 | 
000114 | Define the finite-range interaction moment
000115 | \[
000116 |  \kappa:=\sup_{x\in\Gamma}
000117 |  \sum_{X\ni x}|X|\,\norm{\Phi(X)}.
000118 | \tag{1.5}\label{eq:kappa}