Surface Topological Quantum Criticality Demonstrates Conformal Manifolds with Infinite Fermion Colors

The behaviour of matter at the edge of stability, where new phases of matter emerge, continues to fascinate physicists, and recent work explores the intricate relationship between entanglement and critical points on surfaces. Saran Vijayan and Fei Zhou, both from the University of British Columbia, alongside their colleagues, investigate the emergence of conformal manifolds, smooth landscapes of interacting systems, and their connection to topological phases. This research demonstrates that these manifolds arise naturally when considering systems with a large number of fundamental particles, revealing a surprising link between the geometry of these manifolds and the behaviour of entanglement. The team identifies specific pathways along these manifolds that lead to stable critical points, and importantly, finds that increased entanglement accompanies these flows, suggesting a fundamental role for entanglement in defining the properties of surface phase transitions and potentially unlocking new universality classes in condensed matter physics.

Surface Quantum Criticality and Conformal Manifolds

This research proposes the existence of conformal manifolds, smooth mathematical spaces describing interacting quantum systems in two dimensions. These manifolds represent a fundamentally new type of quantum criticality, extending beyond the conventional understanding of isolated fixed points. The team demonstrates that these manifolds naturally emerge at the surface of certain topological quantum materials, specifically those exhibiting symmetry-protected topological order. A detailed analysis of the surface Hamiltonian reveals a one-parameter family of scale-conformal field theories, smoothly connecting different symmetry-breaking patterns and realising a continuous manifold of fixed points.

The investigation of these surface states reveals a logarithmic violation of the area law for entanglement, a characteristic feature of critical systems. Calculations of entanglement entropy reveal universal scaling functions dependent on the central charge of the underlying conformal field theory. Importantly, the research shows that the central charge varies continuously along the conformal manifold, providing a direct measure of the system’s effective dimensionality. This continuous variation distinguishes these findings from conventional quantum phase transitions, where the central charge remains fixed.

This work establishes a novel connection between topological order, conformal symmetry, and quantum criticality. The surface of topological quantum matter provides a natural setting for realising conformal manifolds, offering a new avenue for exploring exotic quantum phenomena. The ability to continuously tune the effective dimensionality of the system via external parameters opens up possibilities for designing novel quantum devices with tailored properties. This research significantly advances our understanding of quantum criticality and provides a framework for exploring new states of matter beyond the conventional paradigm.

Fermion Colors Drive Conformal Manifold Structure

This research demonstrates the existence of conformal manifolds, smooth mathematical spaces describing interacting quantum systems in two dimensions. The team established that these manifolds emerge as exact solutions when considering a large number of fermion colours, revealing distinct operators that uniquely define the manifold’s structure. By examining the system with a finite, yet substantial, number of colours, the researchers found that quantum fluctuations induce a predictable flow within the manifold, ultimately leading to stable fixed points. These fixed points arise from a spontaneous symmetry breaking within the flow as the number of colours increases.

Notably, the direction of this flow correlates with an increase in EPR-like entanglement entropy, suggesting a fundamental link between entanglement and the system’s behaviour. The findings indicate that these manifolds provide a framework for understanding complex topological critical points with numerous interacting parameters, potentially housing fixed points with varying degrees of stability, and highlight the crucial role of entangled operators in defining universality classes in surface topological phase transitions. The authors acknowledge that the conformal manifold breaks down into isolated fixed points when the number of fermion colours is finite, due to quantum fluctuations that disrupt the symmetry present in the infinite-colour limit. Future research directions include further exploration of the relationship between entanglement entropy and the direction of flow within the manifold, as well as investigations into the potential emergence of supersymmetric fixed points, which were found to be unstable under certain conditions.

Topological Matter, Entanglement and Quantum Field Theory

This research explores a broad range of topics in physics and mathematical physics, including topological phases of matter, conformal field theory, quantum field theory, and holographic entanglement entropy. The research focuses on classifying topological phases, such as topological insulators and superconductors, and investigating their protected surface states and associated quantum phenomena. A central theme is the F-theorem, which relates the central charge of a conformal field theory to its underlying ultraviolet completion. Entanglement entropy plays a crucial role in understanding quantum correlations and probing the structure of quantum systems.

The research explores the connection between entanglement entropy and holographic duality, a correspondence between gravity and conformal field theory. The investigation of renormalization group flow, beta functions, and fixed points is central to understanding the stability of quantum systems. The team also explores emergent phenomena, such as emergent symmetries and supersymmetry, which can arise in strongly interacting systems. The research aims to identify new phenomena that emerge in complex systems and to uncover underlying principles that govern their behaviour. The application of holographic duality provides a powerful tool for studying systems that are difficult to analyse using traditional methods.

Key concepts explored include topological invariants, renormalization group flow, fixed points, beta functions, and central charge. Potential research directions include investigating non-equilibrium dynamics, the effects of disorder, applications in quantum information, and the use of machine learning techniques. The team also suggests exploring experimental tests of these theoretical predictions. This is a rich and diverse collection of papers that represents a cutting-edge research program in theoretical physics, highlighting the importance of topology, entanglement, and emergent phenomena in understanding the behaviour of complex quantum systems and providing new insights into the fundamental laws of nature.