Topology and Geometric Structure of Branching MERA Network

TL;DR

This paper explores the topological and geometric properties of branching MERA networks at finite temperatures, revealing their nonorientable manifold structures and their implications for quantum critical systems and spin-charge separation.

Contribution

It demonstrates that branching MERA networks form nonorientable manifolds like Möbius strips and Klein bottles, and shows their dimensionality-dependent branching behavior in quantum critical systems.

Findings

Networks are nonorientable manifolds such as Möbius strips and Klein bottles.

Branching occurs in 1D quantum critical systems but not in higher dimensions.

Surface twists influence phase strings between spinon and holon excitations.

Abstract

We examine a bulk-edge correspondence of branching MERA networks at finite temperatures in terms of algebraic and differential topology. By using homeomorphic mapping, we derive that the networks are nonorientable manifolds such as a M$\ddot{\rm o}$bius strip and a Klein bottle. We also examine the stability of the branch in connection with the second law of black hole thermodynamics. Then, we prove that the MERA network for one-dimensional quantum critical systems spontaneously separates into multiple branches in the IR region of the network. On the other hand, the branch does not occur in more than two dimensions. The result illustrates dimensionality dependence of spin-charge separation / coupling. We point out a role of twist of the surfaces on the phase string between spinon and holon excitations.