Researchers at the Max-Planck-Institut für Quantenoptik are examining a new framework for understanding symmetry in one-dimensional quantum systems. Adrián Franco-Rubio, Arkadiusz Bochniak and J. Ignacio Cirac detail how defect tensors, residing on lattice sites, can be manipulated locally. The team explains this process allows them to gauge symmetries, potentially constructing new quantum systems and better understanding phases of quantum matter.
Matrix Product Unitary Symmetries in Quantum Spin Chains
Unlike previously understood symmetry defects localized to the bonds connecting lattice sites within a quantum system, this work details how defects now reside directly on those lattice sites. This distinction is critical because it allows for local manipulation of these defects, offering a new degree of control over the quantum states themselves. This localized nature contrasts with earlier approaches and opens possibilities for constructing novel quantum systems with tailored properties.
A key condition governing the successful “gauging” of these symmetries, essentially the ability to consistently define a quantum system with the symmetry, is block independence. Should block independence fail, the resulting system, while locally symmetric as described by a matrix product state, exhibits limitations in how those symmetries interact across the entire system. The researchers found that the existence of these MPU symmetries has implications for classifying quantum phases.
Specifically, they connect their work to established theoretical frameworks, citing references such as the work of Chen, Gu, and Wen on classifying gapped symmetric phases in one-dimensional spin systems, published in Rev. B 83, 035107 (2011). The implications extend beyond theoretical classification, touching upon the broader field of gauge theories, as explored in D 11, 395-408 (1975), and the historical development of gauge invariance as explored by Yang and Mills in Rev. 96, 191-195 (1954).
Defect Creation and Manipulation via Local Unitary Operators
This localized control stems from the framework of matrix product unitary (MPU) symmetries, which generalize the well-understood picture of onsite unitary symmetries. The ability to manipulate these defects locally supports the process of “gauging,” where symmetry defects are transformed into gauge degrees of freedom. J. Ignacio Cirac detailed this interplay in a recent publication, noting the implications for maintaining symmetry during the gauging process. This work builds upon earlier investigations into the relationship between symmetry and quantum states, including work by J.I.
Block Independence and the Gauging of MPU Symmetries
The ability to manipulate symmetry defects residing on lattice sites, rather than within the bonds of a quantum system, represents a shift in how researchers approach controlling quantum states. Unlike previous methods, this framework defines these defects as localized tensors that can be created, moved, and fused using only local unitary operators, offering a new level of precision. This localized manipulation is central to the team’s investigation into “gauging” symmetries, essentially promoting these defects into independent degrees of freedom within the quantum system.
When block independence holds, the symmetry defects can be transformed into gauge degrees of freedom without creating non-commuting gauge constraints, resulting in a system that maintains local symmetry. However, the absence of block independence, particularly when symmetry representations are anomalous, still allows for a locally symmetric system, but at the cost of gauge constraints that do not commute across different sites.
This means that while symmetry is preserved locally, its global consistency is compromised. Williamson, and Laurens Lootens detailed related concepts in “From gauging to duality in one-dimensional quantum lattice models,” published in SciPost Physics 20 4, 113 (2026), as well as the development of quantum circuits specifically designed for matrix-product unitaries, as described by Georgios Styliaris, Rahul Trivedi, and J.
Transforming Global Symmetries into Local Gauge Constraints
Researchers are using this formalism to explore the process of “gauging” symmetries, effectively transforming global symmetries into local gauge constraints within the quantum system. Earlier work laid the foundation for this research, notably the exploration of connections between symmetry and quantum states as detailed in publications from 2015 and 2017. The current approach builds upon these findings, extending the framework to encompass matrix product unitary symmetries. These symmetries, a generalization of established onsite symmetries, offer a more versatile tool for analyzing and manipulating quantum systems. B 107, 075137 (2023).
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