Mereological Quantum Phase Transitions

Abstract

We introduce the novel concept of mereological quantum phase transition (m-QPTs). Our framework is based on a variational family of operator algebras defining generalized tensor product structures (g-TPS), a parameter-dependent Hamiltonian, and a quantum scrambling functional. By minimizing the scrambling functional, one selects a g-TPS, enabling a pullback of the natural information-geometric metric on the g-TPS manifold to the parameter space. The singularities of this induced metric -- so-called algebra susceptibility -- in the thermodynamic limit characterize the m-QPTs. We illustrate this framework through analytical examples involving quantum coherence and operator entanglement. Moreover, spin-chains numerical simulations show susceptibility sharp responses at an integrability point and strong growth across disorder-induced localization, suggesting critical reorganizations of emergent subsystem structure aligned with those transitions.

Figures (2)

• Figure 2: The algebra susceptibility $g$ for the minimally scrambling symmetric bipartitions of the transverse field Ising model for various points in the Hamiltonian parameter space $h$. For $h=0$, the model can be mapped to free fermions and is nonintegrable otherwise. We observe that at the point of the integrability transition, the algebra susceptibility becomes sharply peaked, indicating the onset of an m-QPT.
• Figure 3: (a) The algebra susceptibility for the minimally scrambling symmetric bipartitions for the disordered transverse field Ising model using the method discussed in \ref{['app:loc']}. Zero disorder corresponds to the nonintegrable model, while otherwise the model is in a localized phase. We observe that the algebra susceptibility grows by an order of magnitude at the point of this localization transition, which can be seen as a response to the "emergent" integrability of the localized phase. (b) Visual representation of the parameter points sampling for the disordered model for $N=3$ and $N_\text{steps}=3$. The first point, $p_3$ is drawn randomly within the shaded cube. Then, the points $p_2$ and $p_1$ are chosen equidistantly on the line that connects $p_3$ and the center of the cube that represents the zero disorder point. The points $q_1$, $q_2$ and $q_3$ are then the symmetrically opposite points of increasing disorder strength.