Although Hamiltonians govern wave-matter interactions, conventional characterizations of topological phases of matter require more than the Hamiltonian alone, relying on global invariants defined on closed reciprocal-space manifolds or local commutation relations with position operators. This spatial dependence precludes a unique characterization of topological phases on an abstract platform, whose Hamiltonian can be defined independently of its spatial embedding. Here we establish a framework for characterizing universal topological phases, grounded in the invariance of topological features under arbitrary Hamiltonian-preserving deformations. We construct a Hamiltonian-based spatial embedding that groups a system's constituents according to their connectivity. This embedding, which reproduces the results of conventional topological diagnostics in established systems, enables a coherent extension of topological phases to complex networks using the Hamiltonian alone, further highlighting exceptional robustness in small-world networks. Our results reshape the conceptual foundations of topological physics by decoupling topological characterization from prescribed physical platforms.
Although Hamiltonians govern wave-matter interactions, conventional characterizations of topological phases of matter require more than the Hamiltonian alone, relying on global invariants defined on closed reciprocal-space manifolds or local commutation relations with position operators. This spatial dependence precludes a unique characterization of topological phases on an abstract platform, whose Hamiltonian can be defined independently of its spatial embedding. Here we establish a framework for characterizing universal topological phases, grounded in the invariance of topological features under arbitrary Hamiltonian-preserving deformations. We construct a Hamiltonian-based spatial embedding that groups a system's constituents according to their connectivity. This embedding, which reproduces the results of conventional topological diagnostics in established systems, enables a coherent extension of topological phases to complex networks using the Hamiltonian alone, further highlighting exceptional robustness in small-world networks. Our results reshape the conceptual foundations of topological physics by decoupling topological characterization from prescribed physical platforms.
This paper explores the rigorous formulation of entanglement spectra within the framework of C*-algebras, proposing their utility as robust topological invariants for classifying quantum phases of matter. Entanglement spectra, derived from the eigenvalues of reduced density matrices, have emerged as powerful diagnostic tools for characterizing topological order, often revealing hidden patterns beyond conventional symmetry-breaking paradigms. By embedding quantum systems and their subsystem decompositions into the abstract mathematical structure of C*-algebras, we provide a generalized and mathematically rigorous approach to define and analyze these spectra. This C*-algebraic perspective allows for a unified treatment of finite and infinite quantum systems, including those arising in quantum field theory, and naturally incorporates the algebraic notion of locality and quantum states. We demonstrate how specific features of the entanglement spectrum, such as degeneracy patterns or gaps, can be viewed as invariants under C*-algebraic deformations that preserve the underlying topological order. This framework suggests a deep connection between the algebraic properties of observables, the structure of entanglement, and the classification of topological phases, potentially linking to established mathematical tools like K-theory. The work establishes a theoretical foundation for leveraging entanglement spectra as fundamental topological indicators within a highly abstract yet physically relevant algebraic setting.
The standard approach to characterizing topological matter, computing topological invariants, fails when the symmetry protecting the topological phase is preserved only on average in a disordered system. Because topological invariants rely on enforcing the symmetry exactly, they can overcount phases by incorrectly identifying certain non-robust features as robust. Moreover, in intrinsic statistical topological insulators, enforcing the symmetry exactly is guaranteed to destroy the topological phase. We define a mapping that addresses both issues and provides a unified framework for describing disordered topological matter.
Topological phases are generally characterized by topological invariants denoted by integer numbers. However, different topological systems often require different topological invariants to measure, such as geometric phases, topological orders, winding numbers, etc. Moreover, geometric phases and its associated definitions usually fail at critical points. Therefore, it's challenging to predict what would occur during the transformation between two different topological phases. To address these issues, in this work, we propose a general definition based on fidelity and trace distance from quantum information theory: manifold distance. This definition does not rely on the berry connection of the manifolds but rather on the information of the two manifolds - their ground state wave functions. Thus, it can measure different topological systems (including traditional band topology models, non-Hermitian systems, and topological order models, etc.) and exhibit some universal laws during the transformation between two topological phases. Our research demonstrates that when the properties of two manifolds are identical, the distance and associated higher-order derivatives between them can smoothly transition to each other. However, for two different topological manifolds, the higher-order derivatives exhibit various divergent behaviors near the critical points. For subsequent studies, we expect the method to be generalized to real-space or non-lattice models, in order to facilitate the study of a wider range of physical platforms such as open systems and many-body localization.
Topological phases of matter exhibit robustness, quantized response, and protected boundary modes that cannot be explained by local symmetry breaking or microscopic Hamiltonian details alone. We present a structural account of topological phases based on coarse-grained projection and global admissibility of local effective descriptions. Phases correspond to admissible basins of projected dynamics, and topological phases arise precisely when locally well-defined effective descriptions fail to admit a globally consistent completion. Topological invariants appear as obstruction classes of overlap structures, explaining quantization and robustness. Bulk–boundary correspondence follows inevitably: protected boundary modes arise as compensating degrees of freedom required to restore local consistency at interfaces. The framework applies to interacting and disordered systems with a spectral gap or mobility gap and does not rely on band topology or free-fermion assumptions. All results are formulated at the level of effective descriptions.
Matter waves, as described by the complex wave function ψ(r, t), have traditionally been viewed as smooth, continuous objects whose physical properties are determined solely by their amplitude and phase. In this work we introduce the concept of an intrinsic topological structure Tψ associated with each matter wave. The internal topology is defined through a mapping from the configuration space of the particle to a manifold of internal degrees of freedom, which can carry nontrivial topological invariants such as winding numbers, Chern indices, or homotopy classes. We derive a set of general equations that relate Tψ to observable quantities, including interference patterns and transport coefficients. The theory predicts that matter waves with nontrivial internal topology will exhibit robust features that are insensitive to local perturbations, akin to topological insulators but in the context of single-particle quantum mechanics. We discuss the implications for large-mass quantum interference experiments and suggest that the internal topology could provide a new route to enhance coherence times and to detect subtle quantum gravitational effects. The proposed framework opens up a novel direction in quantum foundations and could ultimately lead to the discovery of new quantum states of matter with engineered topological properties.
This paper develops a rigorous C*-algebraic framework for understanding and classifying topological quantum phases through the lens of spectral entanglement. Topological quantum phases represent a frontier in condensed matter physics, characterized by robust properties that are insensitive to local perturbations and often manifest exotic excitations. Concurrently, entanglement, a quintessential quantum mechanical phenomenon, serves as a crucial resource for quantum information processing and a powerful diagnostic tool for quantum phases of matter. By integrating the formalisms of C*-algebras and spectral theory, we propose a robust mathematical language to describe the algebraic structure of observables in systems exhibiting topological order. Our approach leverages the spectral properties of density matrices and entanglement Hamiltonians within this algebraic setting to define and quantify spectral entanglement, revealing its intrinsic connection to topological invariants. We demonstrate how this framework provides a unified perspective on the interplay between local algebras of observables, their spectral characteristics, and the non-local correlations that define topological order. The paper outlines theoretical constructs for computing entanglement measures and topological indices directly from the C*-algebraic properties, thereby offering new avenues for classifying these phases and exploring their underlying quantum information content with potential implications for fault-tolerant quantum computing and novel electronic devices.