Subdimensional Entropy Distinguishes Geometric and Topological Responses in Many-Body Systems

Entanglement entropy, a crucial measure for understanding phases of matter and their transitions, often struggles to fully distinguish between geometric and topological properties, limiting its diagnostic power. Meng-Yuan Li and Peng Ye introduce a new approach, termed subdimensional entropy, to overcome this challenge, offering a more nuanced characterisation of many-body systems. This innovative method examines lower-dimensional subsystems and their response to virtual deformations, revealing distinct signatures that clearly differentiate geometric and topological contributions, as demonstrated through calculations on various complex states of matter. By connecting the behaviour of these subsystems to mixed-state symmetries and establishing a correspondence between different symmetry classes, the researchers uncover robust composite symmetries and a novel form of spontaneous symmetry breaking, ultimately providing a unified framework for exploring the fundamental principles of topological order and correlated materials.

Pauli Stabilizers And Strong System Symmetries

This research investigates the symmetries within quantum states, particularly those found in topological phases of matter, and how these symmetries can change through spontaneous symmetry breaking. Scientists define Pauli stabilizer states as fundamental building blocks for understanding these phases and quantum error correction, distinguishing between strong symmetries, which act on the entire system, and weak symmetries, which appear locally. A key phenomenon studied is spontaneous strong-to-weak symmetry breaking, where a strong symmetry is broken in the ground state, revealing a residual weak symmetry. Researchers examine subsystem encoding, a method for encoding quantum information within a limited part of the quantum system, and its relation to these symmetry changes.

They introduce ‘t-patch operators’ as local operators that generate the weak symmetry after the strong symmetry breaks, and propose a conjecture linking the structure of the stabilizers to the predictable behavior of these symmetries. Demonstrations show that a one-dimensional subsystem exhibits spontaneous symmetry breaking, confirmed by measurements indicating a broken strong symmetry with a remaining weak symmetry, and similar results extend to two-dimensional systems. The team further demonstrates this breaking occurs for a specific type of symmetry known as a 1-form symmetry, using a unique diagnostic based on the Choi state and an order parameter. This work demonstrates the universality of spontaneous symmetry breaking in subsystem encoded states and provides a framework for understanding the connection between strong and weak symmetries.

Subdimensional Entropy Distinguishes Geometry and Topology

Scientists have developed a new method, termed subdimensional entropy (SEE), to characterize many-body quantum systems and differentiate between geometric and topological properties. This approach analyzes systems through lower-dimensional subsystems, revealing responses to virtual deformations of their geometry and topology. Analytical calculations performed on cluster states, discrete Abelian gauge theories, and fracton orders demonstrate that SEE reveals distinct subleading terms, effectively differentiating geometric and topological responses, and establishes a correspondence between stabilizers and mixed-state symmetries. Researchers demonstrate that weak symmetries act as transparent patch operators for strong ones, forming robust ‘transparent composite symmetries’ (TCSs) that remain invariant under finite-depth quantum circuits.

These TCSs lead to strong-to-weak spontaneous symmetry breaking (SW-SSB), a novel phenomenon observed within these subsystems. Measurements confirm that SEE provides a framework connecting entanglement, mixed-state symmetries, and holographic principles of topological order, and experiments with two-dimensional models, including the plaquette Ising model and the cluster-state model, reveal that SEE’s subleading terms depend on the orientation of the subsystem. For example, the 2D cluster state exhibits a specific subleading term dependent on diagonal orientations, while the plaquette Ising model shows a different angular dependence. In contrast, the 2D toric code, a topologically ordered state, shows that SEE depends solely on topological properties. Further investigations into three-dimensional systems, including the 3D toric code and the X-cube model, demonstrate the power of SEE to distinguish between topological and geometric contributions, establishing SEE as a unifying framework for understanding correlated quantum matter.