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arXiv:2510.18992 · Physical review. B./Physical review. B

Loop charges and fragmentation in pairwise difference conserving circuits

Anonymous, Cheryne Jonay, Tomaž Prosen

Published Aug 12, 2026• 1 citation
arXivHTMLPDF

Abstract

In this work, we introduce a broad class of circuits, or quantum cellular automata, which we call pairwise difference conserving (PDC) circuits. These models are characterized by local gates that preserve the pairwise difference of local operators (e.g., particle number). Such circuits can be defined on arbitrary graphs in arbitrary dimensions for both quantum and classical degrees of freedom. A key consequence of PDC construction is the emergence of an extensive set of loop charges associated with closed walks of even length on the graph. These charges exhibit a one-dimensional character reminiscent of 1-form symmetries and lead to strong Hilbert space fragmentation. As a case study, we analyze a quasi-one-dimensional ladder geometry, where we characterize all dynamically disconnected sectors by the loop-charge symmetries, providing a complete decomposition of the Hilbert space. For the ladder geometry, we observe clear signatures of nonergodic dynamics even within the largest symmetry sector.

DOI
10.1103/gwgn-vp7k

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•Sep 13, 2021•31 citations
In this work, we introduce a broad class of circuits, or quantum cellular automata, which we call pairwise difference conserving (PDC) circuits. These models are characterized by local gates that preserve the pairwise difference of local operators (e.g., particle number). Such circuits can be defined on arbitrary graphs in arbitrary dimensions for both quantum and classical degrees of freedom. A key consequence of PDC construction is the emergence of an extensive set of loop charges associated with closed walks of even length on the graph. These charges exhibit a one-dimensional character reminiscent of 1-form symmetries and lead to strong Hilbert space fragmentation. As a case study, we analyze a quasi-one-dimensional ladder geometry, where we characterize all dynamically disconnected sectors by the loop-charge symmetries, providing a complete decomposition of the Hilbert space. For the ladder geometry, we observe clear signatures of nonergodic dynamics even within the largest symmetry sector.
The Hamiltonian formulation of lattice gauge theories plays a central role in quantum simulations of gauge theories, and understanding their spectrum and other properties is expected to become crucial in the upcoming years. The relevant Hamiltonians in this framework possess local symmetry at each lattice site and may exhibit higher-form symmetries. There are then an exponentially large number of dynamically disconnected symmetry sectors, most of which are not translation-invariant. An exponential number of dynamically disconnected sectors, i.e., Hilbert space fragmentation, can also occur in systems in which no such symmetries have been identified. In this contribution, we describe an emergent gauge symmetry that is valid only in a subset of sectors of the fragmented S=1S=1S=1 dipole-conserving spin chain. These non-invertible symmetries can label exponentially many of the model's sectors. Simulating this Hamiltonian, which is not gauge-invariant, yields an exact quantum simulation of a gauge theory.
The Hamiltonian formulation of lattice gauge theories plays a central role in quantum simulations of gauge theories, and understanding their spectrum and other properties is expected to become crucial in the upcoming years. The relevant Hamiltonians in this framework possess local symmetry at each lattice site and may exhibit higher-form symmetries. There are then an exponentially large number of dynamically disconnected symmetry sectors, most of which are not translation-invariant. An exponential number of dynamically disconnected sectors, i.e., Hilbert space fragmentation, can also occur in systems in which no such symmetries have been identified. In this contribution, we describe an emergent gauge symmetry that is valid only in a subset of sectors of the fragmented S=1S=1S=1 dipole-conserving spin chain. These non-invertible symmetries can label exponentially many of the model's sectors. Simulating this Hamiltonian, which is not gauge-invariant, yields an exact quantum simulation of a gauge theory.
Quantum kinetically constrained models have recently attracted significant attention due to their anomalous dynamics and thermalization. In this work, we introduce a hitherto unexplored family of kinetically constrained models featuring conserved particle number and strong inversion-symmetry breaking due to facilitated hopping. We demonstrate that these models provide a generic example of so-called quantum Hilbert space fragmentation, that is manifested in disconnected sectors in the Hilbert space that are not apparent in the computational basis. Quantum Hilbert space fragmentation leads to an exponential in system size number of eigenstates with exactly zero entanglement entropy across several bipartite cuts. These eigenstates can be probed dynamically using quenches from simple initial product states. In addition, we study the particle spreading under unitary dynamics launched from the domain wall state, and find faster than diffusive dynamics at high particle densities, that crosses over into logarithmically slow relaxation at smaller densities. Using a classically simulable cellular automaton, we reproduce the logarithmic dynamics observed in the quantum case. Our work suggests that particle conserving constrained models with inversion symmetry breaking realize so far unexplored dynamical behavior and invite their further theoretical and experimental studies.
Interacting quantum systems may exhibit fragmentation of their Hilbert space into dynamically disconnected sectors leading to unusual properties like nonergodicity. Fragmentation usually stems from the simultaneous conservation of charge and dipole moment. The authors demonstrate that an alternative scenario occurs in the absence of dipole conservation via a minimal model of interacting spins where fragmentation, including a novel secondary fragmentation in an entangled basis, emerges. A Floquet version of the model yields detectable stroboscopic freezing of certain states at specific drive frequencies.
We introduce a large class of models exhibiting robust ergodicity breaking in quantum dynamics. Our work is inspired by recent discussions of ``topologically robust Hilbert space fragmentation'' but massively generalizes in two directions: First from states describable as ``loop-soups'' to a broader class of states reminiscent of string-nets and sponges and, second, from models restricted to square or cubic lattices to models defined on arbitrary lattices and graphs, including those without translation invariance. Our constructions leverage a recently proposed group-theory framework [Balasubramanian et al., Phys. Rev. X 14, 021034 (2024)] and identify a host of new phenomena arising from the interplay of ``group-model dynamics'' and lattice structure. We make crisp connections to gauge theories, and our construction generalizes Kitaev's quantum double to infinite groups.
We propose a new type of locally interacting quantum circuits—quantum cellular automata—that are generated by unitary interactions round-a-face (IRF). Specifically, we discuss a set (or manifold) of dual-unitary IRFs with local Hilbert space dimension d [DUIRF(d)], which generate unitary evolutions both in space and time directions of an extended 1+1 dimensional lattice. We show how arbitrary dynamical correlation functions of local observables can be evaluated in terms of finite-dimensional completely positive trace preserving unital maps in complete analogy to recently studied circuits made of dual-unitary brick gates (DUBGs). The simplest non-vanishing local correlation functions in dual-unitary IRF circuits are shown to involve observables non-trivially supported on two neighboring sites. We completely characterize the ten-dimensional manifold of DUIRF(2) for qubits (d=2) and provide, for d=3,4,…,7, empirical estimates of its dimensionality based on numerically determined dimensions of tangent spaces at an ensemble of random instances of dual-unitary IRF gates. In parallel, we apply the same algorithm to determine dimDUBG(d) and show that they are of similar order though systematically larger than dimDUIRF(d) for d=2,3,…,7. It is remarkable that both sets have a rather complex topology for d≥3 in the sense that the dimension of the tangent space varies among different randomly generated points of the set. Finally, we provide additional data on dimensionality of the chiral extension of DUBG circuits with distinct local Hilbert spaces of dimensions d≠d′ residing at even/odd lattice sites.