Web of Non-invertible Dualities for (2+1) Dimensional Models Enables Mapping Between Symmetry-Broken and Topological Phases

Recent research explores a surprising connection between different states of matter in two-dimensional systems, revealing a network of relationships previously understood only in simpler, one-dimensional models. Avijit Maity and Vikram Tripathi, both from the Tata Institute of Fundamental Research, alongside Andriy H. Nevidomskyy from Brookhaven National Laboratory, demonstrate how seemingly distinct arrangements of matter, spontaneously broken symmetry and topologically protected states, can be linked through a process called non-invertible duality. This work extends the concept of duality to systems with ‘subsystem symmetries’, where symmetry is only present in parts of the material, and crucially shows that these dualities are sensitive to the boundaries of the system. By establishing these connections, the team provides new tools for understanding and classifying complex quantum phases of matter, potentially paving the way for the design of novel quantum materials with tailored properties.

Topological Order and Symmetry Protected Phases

This collection of research papers details a comprehensive exploration of theoretical condensed matter physics, quantum computation, and related fields. The work focuses on understanding topological order, symmetry-protected topological phases, and their potential applications. Early research laid the groundwork for understanding topological order beyond the fractional quantum Hall effect, exploring different phases and the role of symmetry in stabilizing them. Subsequent studies introduced the concept of fractons, quasiparticles with restricted mobility, and explored subsystem symmetry-protected topological order, where symmetry acts on a subspace of the quantum system.

More recent work continues to investigate these phases, their connections to fractons, and their potential for realizing exotic quantum states. Alongside these developments, researchers have investigated the use of topological phases and symmetries for robust quantum computation. Foundational papers detail measurement-based quantum computation using entangled states, and explore how symmetry can protect quantum computations from errors. Further studies explore the use of fractal symmetry-protected cluster phases and topological quantum error correction codes. A rapidly developing area focuses on non-invertible symmetries and defects, exploring their potential to create novel quantum phases and for quantum computation.

Recent research establishes the algebraic properties of these symmetries and explores how defects can construct gapless symmetry-protected states. Researchers are also investigating symmetry topological field theories to describe the behavior of these symmetries and defects. The research also encompasses studies of gauge theory, string and particle braiding in topological order, and the anisotropic quantum orbital compass model. A key trend is the increasing complexity of the systems under investigation, moving from simple models to those with subsystem symmetries, non-invertible symmetries, and defects. The work requires a high level of mathematical sophistication, drawing on concepts from topology, category theory, and quantum field theory, and is increasingly interdisciplinary, drawing on ideas from condensed matter physics, quantum information theory, and mathematics. This body of work represents a cutting-edge exploration of topological phases of matter, symmetry-protected topological order, and their applications to quantum technologies.

Subsystem Duality Connects Quantum Phases of Matter

Scientists have significantly advanced our understanding of quantum phases of matter by extending the concept of duality from one-dimensional systems to two dimensions. This work constructs a network of dualities for lattice models possessing subsystem symmetries, revealing deep connections between different quantum states. The team developed two complementary transformations: one mirroring the well-known Kramers-Wannier duality, and a generalized Kennedy-Tasaki transformation that maps systems exhibiting spontaneous subsystem symmetry breaking to those with subsystem symmetry-protected topological order. Crucially, the Kennedy-Tasaki transformation successfully transfers the spontaneous ground-state degeneracy of the symmetry-broken state onto the protected boundary degeneracy characteristic of the symmetry-protected topological phase.

Researchers confirmed this non-invertibility through matching ground-state degeneracies, analyzing symmetry-twist sectors, and examining the fusion algebra of the duality operator. By expanding the Hilbert space to include twisted sectors, the Kramers-Wannier map could be expressed as a projective unitary, preserving probabilities. These results provide a concrete lattice realization of non-invertible subsystem dualities, highlighting the central role of symmetry-twist sectors in characterizing generalized symmetries and exotic phases of quantum matter.

Subsystem Duality and Topological Phase Transitions

Researchers have extended the concept of duality, traditionally applied to one-dimensional systems, to two dimensions by establishing a network of dualities for lattice models possessing subsystem symmetries. They constructed two complementary transformations: a subsystem Kramers-Wannier duality, which maps a spontaneously broken symmetry state to a trivial state, and a generalized subsystem Kennedy-Tasaki transformation, which maps spontaneous symmetry breaking to a symmetry-protected topological phase. These dualities exhibit a sensitivity to boundary conditions, acting as standard transformations on open lattices but becoming non-invertible when applied to closed systems. Notably, the Kennedy-Tasaki transformation successfully transfers the spontaneous ground-state degeneracy of the symmetry-broken state onto the protected boundary degeneracy characteristic of the symmetry-protected topological phase.

Researchers confirmed this non-invertibility through matching ground-state degeneracies, analyzing symmetry-twist sectors, and examining the fusion algebra of the duality operator. While local repair operators transform into nonlocal objects under the duality, the essential algebraic relationships and the connection between bulk and edge properties remain consistent, demonstrating a faithful transmission of key invariants. These findings contribute to a deeper understanding of quantum phases of matter and provide new tools for classifying and relating different phases, particularly those exhibiting symmetry-protected topological order.