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Franco-Rubio, Bochniak and Cirac Map Defects in Matrix-Product States
Adrián Franco-Rubio, Arkadiusz Bochniak and J. Ignacio Cirac develop a framework for symmetry defects in matrix-product states with matrix-product unitary symmetries. Their gauging method preserves bond dimension when block independence hol
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Gauging a symmetry usually means imposing local constraints that commute with one another. But for some symmetries in matrix-product states, the standard construction cannot meet that condition without changing the state’s structure. Adrián Franco-Rubio, Arkadiusz Bochniak and J. Ignacio Cirac propose a way to track the defects behind that mismatch—and a more limited alternative when conventional gauging fails.
Defects move from bonds to lattice sites
The authors study one-dimensional matrix-product states (MPS), a tensor-network representation of quantum many-body states, with symmetries described by matrix-product unitaries (MPUs). For familiar onsite symmetries, a defect can be represented by inserting an operator on an internal bond of the MPS. The paper extends that picture to MPU symmetries, defining defect tensors on lattice sites instead.
Those site-based defects can be created, moved along the chain and fused using local unitary operators. “In the MPU case, we can define analogous defect tensors, this time sitting on lattice sites, that can be created, moved, and fused by local unitary operators,” the authors write in the paper. The framework gives researchers a tensor-network description of symmetry twists even when the symmetry acts across neighboring sites rather than independently at each site.
Block independence sets the terms
The defect construction feeds into a gauging procedure: the global symmetry is promoted to local gauge degrees of freedom. Under a condition the authors call block independence, the resulting MPS keeps the original bond dimension and has local symmetry constraints that commute. That combination connects the tensor-network construction to the usual picture of gauging, where compatible local constraints define a gauge-invariant space of states.
Block independence is not guaranteed for every symmetric MPS. The paper says it always holds for injective MPS, but can fail for noninjective states; anomalous MPU symmetry representations also fall outside the condition. In nonanomalous cases where block independence fails, the authors note that a different, projection-based gauging remains possible, though it comes at the cost of increased bond dimension.
When constraints stop commuting
For cases beyond block independence, the authors introduce “state-level gauging.” It promotes defects to gauge degrees of freedom and produces a locally symmetric MPS with the same bond dimension, but the resulting gauge constraints need not commute across the full Hilbert space. They do share a common +1 eigenspace containing the constructed state; the paper cautions that the size and physical significance of that space need further study.
The examples include an ordinary non-onsite Z2 symmetry built from a product of neighboring controlled-Z gates, as well as the anomalous Z2 symmetry in the CZX model. Those cases illustrate the distinction between keeping the full commuting-constraint structure and constructing a locally symmetric state when that stronger requirement is unavailable.
A method, with a defined boundary
The paper’s result is a formalism for handling symmetry defects and constructing gauged tensor-network states, not a report of a new device or experimental measurement. Its central trade-off is explicit: block-independent systems permit defect promotion with commuting constraints and unchanged bond dimension; beyond that boundary, state-level gauging retains the state-level symmetry but relaxes commutation across the full space.
That boundary matters for using tensor networks to study phases protected by non-onsite symmetries. The authors also identify an open question: whether the gauge-invariant subspace in the state-level construction supports nontrivial physics or grows with system size. For now, the paper establishes how to build the locally symmetric MPS and specifies what its constraints do—and do not—guarantee.