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arXiv:2410.06760 · Physical review. B./Physical review. B

Integrability is generic in homogeneous U(1)-invariant nearest-neighbor qubit circuits

Marko Žnidarič, Urban Duh, Lenart Zadnik

Published Jun 23, 2025• 10 citations
arXivHTMLPDF

Abstract

Integrability is an exceptional property believed to hold only for systems with fine-tuned parameters. Contrary, we explicitly show that in homogeneous nearest-neighbor qubit circuits with a U(1) symmetry, i.e., circuits that repeatedly apply the same magnetization-conserving two-qubit gate, this is not the case. There, integrability is generic: All such brickwall qubit circuits are integrable, even with a randomly selected gate. We identify two phases with different conservation laws, transport properties, and strong zero edge modes. Experimentally important is the fact that varying any one of the parameters in the generic U(1) gate, one will typically cross the critical manifold that separates the two phases. Finally, we report on an unconventional time-reversal symmetry causing the system with open boundary conditions to be in the orthogonal class, while the one with periodic boundary conditions is in the unitary class.

DOI
10.1103/tqy8-ynpd

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Integrability is an exceptional property believed to hold only for systems with fine-tuned parameters. Contrary, we explicitly show that in homogeneous nearest-neighbor qubit circuits with a U(1) symmetry, i.e., circuits that repeatedly apply the same magnetization-conserving two-qubit gate, this is not the case. There, integrability is generic: All such brickwall qubit circuits are integrable, even with a randomly selected gate. We identify two phases with different conservation laws, transport properties, and strong zero edge modes. Experimentally important is the fact that varying any one of the parameters in the generic U(1) gate, one will typically cross the critical manifold that separates the two phases. Finally, we report on an unconventional time-reversal symmetry causing the system with open boundary conditions to be in the orthogonal class, while the one with periodic boundary conditions is in the unitary class.
Symmetries are important for understanding equilibrium as well as nonequilibrium properties like transport. In translationally invariant extended systems one might expect symmetry generators to also be homogeneous. Studying qubit circuits with nearest-neighbor U(1) gates we show that this needs not be the case. We find new inhomogeneous screw SU(2) and Uq(sl2){{{{\rm{U}}}}}_{q}({{\mathfrak{sl}}}_{2})Uq​(sl2​) symmetries whose generators exhibit a spatial quasi-momentum modulation. They can be viewed as a parameter-dependent generalization of the standard rotational symmetry of the Heisenberg model and can be identified by the Ruelle-Pollicott spectrum of a momentum-resolved propagator. Rich integrability structure is reflected also in transport: picking an arbitrary U(1) gate and varying the gate duration one will transition through different phases, including fractal ballistic transport, Kardar-Parisi-Zhang superdiffusion at the critical manifold that also contains helix states, diffusion, and localization. To correctly explain transport the non-local SU(2) symmetries do not matter, while the inhomogeneous local ones that almost commute with the propagator do. Symmetries play a vital role in many-body physics, and in homogeneous systems one expects symmetries that are translationally invariant. Here, by studying magnetization conserving integrable quantum circuits, researchers find new spatially modulated symmetries that have consequences for quantum transport.
We present a complete classification of integrable Yang-Baxter quantum circuits with open boundary conditions and arbitrary circuit geometries. Starting from the standard transfer-matrix construction with two types of staggered inhomogeneities, we derive a general mapping that determines the arrangement of circuit gates in terms of the inhomogeneities and the system size. We conjecture that time-periodic quantum circuits are integrable whenever the local bulk and boundary gates satisfy the Yang-Baxter equation and the same bulk gate is applied exactly once per period to every nearest-neighbor pair of spins. Our construction also provides an algorithm to detect Yang-Baxter integrability for circuits with arbitrary geometries. Furthermore, we introduce a third type of inhomogeneity, denoted by ρρρ, and demonstrate that the minimum possible circuit depth is four. We show that when these ρρρ-inhomogeneities are placed at the endpoints and in their immediate neighborhood, the resulting boundary gates can be interpreted as single gates acting on multiple sites. Our construction is fully general and applies to regular RRR-matrices, both of difference and non-difference type, together with their associated boundary matrices. As an application, we consider two-qubit gates corresponding to 6- and 8-vertex RRR-matrices of non-difference form satisfying the Yang-Baxter equation, and we construct the associated reflection matrices that generate integrable quantum circuits.
We study Floquet systems with translationally invariant nearest-neighbor two-site gates. Depending on the order in which the gates are applied on an N -site system with periodic boundary conditions, there are factorially many different circuit configurations. We prove that there are only N−1 different spectrally equivalent classes, which can be viewed either as a generalization of the brick wall or of the staircase configuration. Every class, characterized by two integers, has a nontrivial space-time symmetry with important implications for the level-spacing distribution—a standard indicator of quantum chaos. Namely, in order to study chaoticity one should not look at eigenphases of the Floquet propagator itself, but rather at the spectrum of an appropriate root of the propagator. Published by the American Physical Society 2024
We study critical properties of the entanglement and charge-sharpening measurement-induced phase transitions in a nonunitary quantum circuit evolving with a U(1) conserved charge. Our numerical estimation of the critical properties of the entanglement transition at finite system sizes appears distinct from the generic nonconserving case and percolation. We provide two possible interpretations of this observation: (a) these two transitions occur at different measurement rates in the thermodynamic limit, but at finite system sizes their critical fans overlap and the critical exponents we probed here show a combination of both criticalities. Nonetheless, the multifractal properties of the entanglement transition remain distinct from the generic case without any symmetry, indicating a unique universality class due to the U(1) symmetry. (b) These two transitions occur at the same measurement rate at any lengthscale. Within this interpretation, our estimation of all the critical exponents is sharply different from the nonconserving case, again confirming the presence of a new universality class due to U(1) symmetry. We compute entanglement critical exponents and correlation functions via various ancilla measures, we use a transfer matrix for multifractality, and we compute correlators associated with charge sharpening to explain these findings. Through these correlators, we also find evidence consistent with the charge-sharpening transition being of the Berezinskii-Kosterlitz-Thouless type (including the predicted ``jump'' in stiffness), which simultaneously argues for a broad critical fan for this transition. As a result, attempts to measure critical properties in this finite-size system will see anomalously large exponents predicted by our numerical analysis.
Abstract We revisit the integrability of quantum circuits constructed from two-qubit unitary gates U that satisfy the Yang–Baxter equation. A brickwork arrangement of U typically corresponds to an integrable Trotterization of some Hamiltonian dynamics. Here, we consider more general circuit geometries which include circuits without any nontrivial space periodicity. We show that any time-periodic quantum circuit in which U is applied to each pair of neighbouring qubits exactly once per period remains integrable. We further generalize this framework to circuits with time-varying two-qubit gates. The spatial arrangement of gates in the integrable circuits considered herein can break the space-reflection symmetry even when U itself is symmetric. By analysing the dynamical spin susceptibility on ballistic hydrodynamic scale, we investigate how an asymmetric arrangement of gates affects the spin transport. While it induces nonzero higher odd moments in the dynamical spin susceptibility, the first moment, which corresponds to a drift in the spreading of correlations, remains zero. We explain this within a quasiparticle picture which suggests that a nonzero drift necessitates gates acting on distinct degrees of freedom. To the memory of Marko Medenjak, a dear friend and a brilliant colleague.
Random quantum circuits are paradigmatic models of minimally structured and analytically tractable chaotic dynamics. We study a family of Floquet unitary circuits with Haar random U(1)U(1)U(1) charge conserving dynamics; the minimal such model has nearest-neighbor gates acting on spin-1/2 qubits, and a single layer of even/odd gates repeated periodically in time. We find that this minimal model is not robustly thermalizing at numerically accessible system sizes, and displays slow subdiffusive dynamics for long times. We map out the thermalization dynamics in a broader parameter space of charge conserving circuits, and understand the origin of the slow dynamics in terms of proximate localized and integrable regimes in parameter space. In contrast, we find that small extensions to the minimal model are sufficient to achieve robust thermalization; these include (i) increasing the interaction range to three-site gates, (ii) increasing the local Hilbert space dimension by appending an additional unconstrained qubit to the conserved charge on each site, or (iii) using a larger Floquet period comprised of two independent layers of gates. Our results should inform future studies of charge conserving circuits, which are relevant for a wide range of topical theoretical questions.