In a recent article published in the journal Physical Review B, researchers highlighted the importance of understanding quantum phase transitions and gapless phases in fermionic condensed matter systems, key phenomena for advancing quantum materials and technologies. These include optical devices that depend on low-energy excitations and critical behavior. To tackle the complexity of these systems, the authors introduce topological holography, a conceptual framework that maps strongly correlated fermionic systems onto higher-dimensional topological field theories (TFTs).
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This approach enables a more structured way to classify and analyze different phases and phase transitions, particularly those protected by noninvertible symmetries. While the focus remains on fermionic quantum critical points and phases, the framework also offers potential insights into optical contexts. In particular, it sheds light on how gapless modes might shape light-matter interactions in these unconventional quantum states.
Background
The theoretical backbone of the study is rooted in symmetry-protected and symmetry-enriched topological phases, structures defined by unconventional, often noninvertible, symmetries that go beyond traditional group theory. Among these are fusion-category symmetries and their fermionic variants, which, while more complex to work with, are crucial for capturing the full landscape of topological phases. Central to this framework is the concept of a symmetry topological field theory (SymTFT), a higher-dimensional topological order that encodes the symmetry data of a lower-dimensional system. Within this setup, different phases and their transitions are represented through boundary conditions known as gapped boundaries. These boundaries serve as a key tool for identifying and analyzing distinct physical regimes. A notable aspect of the framework is its explicit inclusion of spin structures, which are essential when dealing with fermionic systems. This ensures the theory accurately reflects the nuanced relationship between topology, fermion parity, and phase dynamics.
The Current Study
This study uses the topological holography framework to investigate fermionic gapped phases and quantum critical points. At its core, the approach constructs symmetry topological field theories (SymTFTs) in higher dimensions, which capture the full symmetry and topological structure of lower-dimensional fermionic systems. A central element of the method is identifying gapped boundary conditions within the SymTFT. These are mathematically described by Lagrangian algebras in Modular Tensor Categories (MTCs). These algebras consist of direct sums of simple objects, each equipped with multiplication maps that satisfy consistency and commutativity conditions. Physically, they correspond to boundary phases, some preserving the underlying symmetries, others breaking them.
Fermionization is implemented via algebraic condensation procedures that convert bosonic topological models into fermionic ones, incorporating the fermion parity Z2F. This involves forming superfusion categories capturing fermionic signs and statistics. Once boundary conditions are established, the authors compute partition functions across various spin structures and symmetry backgrounds to differentiate trivial, symmetry-protected, and symmetry-breaking phases. These calculations utilize algebraic data (such as fusion rules and bilinear forms) to analyze phase transitions and emergent conformal field theories (CFTs).
The framework also extends to higher dimensions, allowing insights into gapless fermionic phases and anomalies. The method combines algebraic techniques with numerical evaluations to map phase diagrams and characterize critical points. Overall, the integration of fusion category theory, fermionic condensation, and partition function computations provides a robust, systematic approach to classifying and understanding fermionic topological phases and their transitions with potential experimental relevance, notably in optical systems probing topological phenomena.
Results and Discussion
The analysis reveals an extensive landscape of fermionic phases exhibiting exotic features such as noninvertible symmetries and emergent anomalies. Among the key findings is the classification of fermionic gapped and gapless phases, including symmetry-broken states, topological insulators, and fermionic SPT phases with higher symmetry groups like Z4F. These phases demonstrate unique response functions and excitation spectra, which could influence optical properties such as reflectivity, transmissivity, and absorption, especially near critical points where gapless modes dominate. The topological holographic approach exposes how certain gapless modes behave as emergent particles, akin to optical quasiparticles, whose properties are dictated by the underlying topological constraints and spin structures. These insights imply that phase transitions characterized within this framework could manifest in optical experiments as critical phenomena, anomalous light scattering, or enhanced nonlinear responses arising from the underlying fermionic critical excitations. The discussion emphasizes the importance of boundary conditions, fusion rules, and the role of spin structures in influencing observable optical signatures.
Conclusion
The study effectively showcases the potential of topological holography as a unifying framework for describing intricate fermionic quantum phases and their transitions. While the primary focus is on condensed matter systems, the broader implications for optics are particularly intriguing. The framework points to the possibility that emergent gapless modes and topologically protected excitations could be detected, or even utilized, through optical probes, especially near quantum critical points where system responses become highly sensitive and complex. As the theoretical tools continue to develop, they may offer guidance for designing optical materials and devices that leverage topological and fermionic critical behavior, paving the way for novel applications in quantum optics and photonics.
Looking ahead, the authors suggest further investigation into how topological boundary conditions, spin structures, and light-matter interactions interrelate. This could extend the holographic approach to optical systems that feature fermionic degrees of freedom, such as photonic platforms based on topological insulators.
Journal Reference
Huang S.-J. (2025). Fermionic Quantum Criticality and Topological Holography. Physical Review B, 111, 15, 155130. DOI: 10.1103/PhysRevB.111.155130, https://journals.aps.org/prb/abstract/10.1103/PhysRevB.111.155130