Renormalization flows for 1D mixed states and a quantum Goursat lemma

• URL: http://arxiv.org/abs/2607.08568v1
• Date: Thu, 09 Jul 2026 14:56:47 GMT
• Title: Renormalization flows for 1D mixed states and a quantum Goursat lemma
• Abstract summary: We study the boundary states of non-chiral two-dimensional topologically ordered models.<n>Data consist of renormalization fixed points built from representations of finite-dimensional $C*$-Hopf algebras.<n>We formulate and prove a quantum generalization of Goursat's lemma for finite-dimensional $C*$-Hopf algebras.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Renormalization provides a framework for relating microscopic models of physical systems to effective descriptions at larger length scales. This procedure is studied for the boundary states of non-chiral two-dimensional topologically ordered models. The initial data consist of renormalization fixed points built from representations of finite-dimensional $C^*$-Hopf algebras, which are then perturbed by uniform on-site noise quantum channels and repeatedly coarse-grained. The resulting flows admit an intrinsic algebraic description in terms of completely positive maps on the $C^*$-Hopf algebra or, equivalently, positive linear functionals on its enveloping $C^*$-Hopf algebra. Their iteration is governed by convolution powers, and convergent trajectories yield new matrix product density operator fixed points, described by finite $*$-quantum hypergroups. This provides a concrete physical interpretation of such structures. For finite group algebras and their duals, we provide explicit classifications via Goursat's lemma for groups. Finally, we formulate and prove a quantum generalization of Goursat's lemma for finite-dimensional $C^*$-Hopf algebras, a result of independent interest, which gives an explicit structural description of all convergent renormalization trajectories.

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