In this paper, inspired by [1-3], we proposed a class of lattice
renormalization group (RG) operators, each operator determined by a topological
order $T$ in $D+1$ space-time dimensions. Taking the overlap between an
eigenstate $\langle\Omega|$ of the RG operator with the ground state
wave-function $|\Psi\rangle$ of $T$ (i.e. $\langle\Omega|\Psi\rangle$) gives
rise to partition functions of conformal (including topological) theories in
$D$ dimensions with categorical symmetry related to $T$, realizing a
holographic relation discussed in the literature explicitly. We illustrate this
in explicit examples at $D=1,2,3$. Exact eigenstates of the RG operator can be
solved explicitly from (higher) separable Frobenius algebra of the (higher)
input fusion category $\mathcal{C}$ defining the lattice model of $T$, and they
give the $D$ dimensional symmetric TQFTs. Eigenstates corresponding to actual
conformal theories describe phase transitions between these topological fixed
points. The critical points can be searched numerically and we demonstrate that
known critical couplings of $SU(2)_k$ integrable lattice models are numerically
recovered from our procedure, alongside a curious tricritical point that we
found at $k=4$. We demonstrate that the 2+1 D Ising model can also be obtained
as a strange correlator with the associated 4D topological order being the 4D
toric code. The numerical procedure that we devise to search for the 3D
critical temperature is a novel tensor renormalization group algorithm, that
fully harnesses the algebraic and geometric properties of the RG operator.
Finally since the RG operator is in fact an exact analytic holographic tensor
network, we compute “bulk-boundary” correlator and compare with AdS/CFT.
Promisingly, they are numerically compatible given our accuracy, although
further works will be needed to explore the precise connection to the AdS/CFT
correspondence.



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