Nontrivial Local Conserved Quantities

Updated 29 July 2025

Nontrivial local conserved quantities are strictly local operators that remain constant under dynamics yet cannot be reduced to trivial forms.
Rigorous methods using operator expansions (e.g., Pauli strings) distinguish integrable models with a full tower of conserved charges from nonintegrable ones.
Advanced frameworks like fractal decompositions and MPO representations offer actionable insights into integrability, thermalization, and relaxation in quantum many-body dynamics.

Nontrivial local conserved quantities are operators or functionals that—while having strictly local or finite support—are constants of motion under a given unitary or nonunitary dynamics, yet cannot be reduced to trivial (e.g., identity or Hamiltonian) forms or simple sums thereof. Their existence, classification, and absence serve as sharp diagnostics for quantum integrability, ergodicity, and the nature of relaxation in both closed and open quantum systems. Considerable recent effort has yielded a comprehensive mathematical framework for rigorously analyzing the presence or absence of nontrivial local conserved quantities across lattice spin models, fermionic systems, field theories, and open-system dynamics.

1. Rigorous Absence and Classification in Lattice Models

For quantum lattice systems, especially spin-1/2 and higher, recent mathematically rigorous results have established a sharp dichotomy: in strictly local Hamiltonian models in one or higher dimensions, nontrivial $k$ -support conserved quantities (with $k \geq 3$ ) exist if and only if the model is integrable. Otherwise, all such quantities (except the Hamiltonian and sometimes a handful of two-site or onsite operators) vanish.

Consider a general translation- and parity-invariant nearest-neighbor Hamiltonian: $H = \sum_i \left( J_X X_i X_{i+1} + J_Y Y_i Y_{i+1} + J_Z Z_i Z_{i+1} + h_X X_i + h_Y Y_i + h_Z Z_i \right)$ Every $k$ -local operator $Q$ can be expanded in the “Pauli string” basis: $Q = \sum_{A^{(k)}} q_{A^{(k)}} A^{(k)}$ The requirement $[Q, H]=0$ yields a coupled set of linear equations for the expansion coefficients. When at least two $J_\alpha$ and at least one $h_\alpha$ are nonzero (i.e., the generic nonintegrable setting), this system is overdetermined, and all $q_{A^{(k)}}$ are forced to vanish for $k \geq 3$ , save for special fine-tuned points (XXX, XXZ, XYZ, or free-fermion chains). This establishes the generic absence of nontrivial local conserved quantities in nonintegrable spin-1/2 chains (Yamaguchi et al., 4 Nov 2024), with analogous results extended to the spin-1 bilinear-biquadratic model with uniaxial field (Hokkyo et al., 7 Nov 2024).

For higher-dimensional lattices (e.g., the $S=1/2$ XY, XYZ, or Ising models on a hypercubic lattice), the absence of nontrivial local conserved quantities is even more dramatic: regardless of a model’s solvability in $1d$, in $d\geq 2$ only the Hamiltonian itself (and, in rare finely tuned settings, one-site or two-site operators) commute with $H$ (Shiraishi et al., 24 Dec 2024, Chiba, 25 Dec 2024). Notably, this extends to the XX model, which is trivially solvable and integrable in 1d but not in higher dimensions.

The quantum compass model on the square lattice also exhibits complete absence of nontrivial local conserved quantities, save for $H$ , a fact established via the “Shiraishi shift” method which iteratively eliminates all non-standard local operator forms based on the recursive support truncation and extension via commutation with the Hamiltonian (Futami et al., 15 Feb 2025).

2. Integrable Models and Explicit Construction of Charges

Integrable models form a sharp exception, where a tower of nontrivial local conserved quantities with arbitrarily large but finite support exists. In such models:

Spin-1/2 XXZ chain: Local conserved charges $Q_k$ can be constructed explicitly as linear combinations of irreducible words in the Temperley–Lieb algebra, with coefficients determined by combinatorial and algebraic parameters (width, transpositions, vacancies, gaps). These $Q_k$ are proven to mutually commute and to commute with $H$ via the triangle equation for their generating combinatorics (Nienhuis et al., 2021).
Heisenberg chain (SU( $N$ ) cases): The MPO formalism provides a tractable, hierarchical representation; the bond dimension for the $k$ th charge grows linearly with $k$ , and the coefficients align with Catalan tree patterns, indicating an intrinsic combinatorial structure (Yamada et al., 2023).
Hubbard model in 1d: All local conserved quantities have been classified and shown to be exhausted by those generated from the logarithmic expansion of the transfer matrix. Diagrams for higher charges exhibit nontrivial coefficients determined recursively; completeness is rigorously established (Fukai, 2023, Fukai, 2023).

Table: Character of Local Conserved Quantities in Prototypical Lattice Models

Model	Nontrivial Local Conserved Quantities Exist?	Characterization/Reference
1d XXZ/XXX chain	Yes (tower/integrable)	(Nienhuis et al., 2021 Yamada et al., 2023)
1d XYZ, $h = 0$	Yes (Bethe ansatz)	(Shiraishi, 2018)
1d XYZ, $h \neq 0$	No (nonintegrable)	(Shiraishi, 2018)
2d/3d $S=1/2$ XY/XYZ/Ising	No (generic)	(Shiraishi et al., 24 Dec 2024 Chiba, 25 Dec 2024)
1d Hubbard	Yes (complete set)	(Fukai, 2023 Fukai, 2023)
2d/3d Hubbard	No (generic)	(Futami, 27 Jul 2025)
Compass model (2d)	No	(Futami et al., 15 Feb 2025)
Spin-1 BLBQ (AKLT etc)	No (nonintegrable)	(Hokkyo et al., 7 Nov 2024)

3. Open-System Dynamics and Lindbladian Conservation

The classification of local conserved quantities extends to open-system quantum dynamics governed by Lindblad master equations. A precise distinction emerges: for translationally invariant local Lindblad equations,

Any 1-local operator (e.g., density, magnetization) can be strictly conserved by local dissipators.
For 2-local operators (e.g., typical energy densities of interacting systems), only “Ising-like” operators (those reducible to sums of $1\otimes w$ , $u\otimes 1$ , $u\otimes w$ ) admit local conservation by Lindblad generators; all other nontrivial 2-local quantities (e.g., transverse Ising or Heisenberg energy densities) cannot be strictly locally conserved in this way (Znidaric et al., 2013).

This result underscores the structural obstruction posed by nontrivial operator commutation relations, marking the boundary for which local conservation in dissipative quantum dynamics is feasible. It also constrains feasible engineering of open quantum systems for dissipative quantum state preparation, transport, or thermalization.

4. Fractal and Nonlocal Constructions: Partitioning Conserved Quantities

A fundamentally different approach to “nontrivial partitions” of conserved quantities arises in the mathematical framework of $p\lambda n$ fractal decomposition (Garcia-Morales, 2015). Any function $f(x)$ —in particular, a Hamiltonian or partition function—can be exactly and nontrivially decomposed as: $f(x) = \sum_{n=0}^{\lambda-1} F_{p\lambda n} f(x)$ Each $F_{p\lambda n} f(x)$ is a discontinuous, fractal function (constructed from the digit expansion and additional filter functions) whose sum precisely reconstructs $f(x)$ . Applied to Hamiltonians or partition functions, this construction demonstrates that global (even smooth) objects can be viewed as the sum of fractal, locally structured components, each with nontrivial support. The method links number theory, group-theoretic structure, and statistical ensembles, and reveals that “hidden” fine scale partitions are compatible with global symmetry and conservation, even when absent from operator algebra constraints. This decomposition is exact and analytic, distinct from approximate or coarse-grained notions of locality.

5. Nonlocal and Subregion-Quasilocal Conserved Quantities

Nonlocal conservation laws—such as those appearing in $\mathcal{PT}$ -symmetric open quantum systems—eschew locality entirely. In such systems, conserved observables are characterized by the intertwining property: $\hat{\eta} H_{PT} = H_{PT}^\dagger \hat{\eta}$ leading to time-invariant but generically nonlocal (e.g., antidiagonal pairing) correlators. The experimental and theoretical characterization of such observables demonstrates that conservation laws in non-Hermitian systems operate on a global scale and have no analog in closed, local-density-based continuity equations (Bian et al., 2019).

In quantum many-body systems, entanglement Hamiltonians of eigenstates can, for sufficiently large subregions, be approximated as linear combinations of subregionally (quasi)local operators that are approximately conserved up to boundary corrections proportional to the area-to-volume ratio. For free fermion and XYZ chain models, this leads to an extensive but subregion-limited set of approximate conservation laws with commutation errors scaling as the boundary area (Lian, 2021).

6. Field-Theoretic and Geometrical Generalizations

The connection between local conservation laws and field equations admits a broader geometric reformulation. On flat manifolds, the strong Poincaré lemma ensures that any locally conserved current (divergence-free multivector field) admits a local potential whose divergence recovers the conserved current: $D \cdot J = 0 \implies \exists\ F:\ DF = J$ This observation provides a geometric underpinning for the universality of Maxwell-type equations for any conserved quantity satisfying a continuity equation, not only for electromagnetism but also for mass, probability, or more general densities (Burns, 2019). Extensions to curved geometries are possible but require covariant generalizations of the integral formulas.

In Lagrangian field theory, “nonlocal constants”—quantities invariant along motion but dependent on spacetime-integrated history—can be incorporated via families of perturbed fields, leading to locally conserved currents through appropriate total divergence conditions. This approach generalizes Noether's theorem and extends the palette of conservation laws in classical and quantum field theories, including dissipative and nonlinearly interacting models (Scomparin, 2022).

7. Models with Exact Local Fragmentation

Certain exactly solvable models—such as the spin/fermion ladders with multipole phases—admit nontrivial local conserved quantities at the level of local dimer (bond) invariants. These project the total system onto sectors labeled by static $\mathbb{Z}_2$ -valued invariants, producing effective degrees of freedom (e.g., higher-order spins) within each sector. Such fragmentation underlies unconventional ordered phases (dipole, quadrupole, octopole, etc.) with zero net magnetization but robust dynamical features; the corresponding phases and transitions are classified by the values of these local conserved quantities (Fu, 7 Jul 2025). The exact mapping to transverse-field Ising models or their generalizations in each sector is a direct result of the extensive local conservation and its impact on the effective Hilbert space structure.

The rigorous nonexistence, precise algebraic construction, and classification of nontrivial local conserved quantities demarcate integrable and nonintegrable quantum many-body dynamics and shape fundamental properties such as thermalization, operator spreading, and hydrodynamical behavior. Recent work has not only established the scarcity of such quantities in higher-dimensional or generic spin/fermion systems but also developed a general operator-basis, commutator, and recursion-based toolkit for their systematic identification or elimination. In integrable cases, explicit combinatorial or MPO representations provide a powerful description of the conserved towers. Beyond these regimes, fractal decompositions and operator-based generalizations underscore hidden structural possibilities for partitions or approximate conservation, potentially relevant to future extensions in quantum algorithms, condensed matter, and gauge theories.