Quantum many‑body systems often display emergent stability that cannot be explained by any obvious symmetry of the microscopic Hamiltonian. In this work we propose a systematic framework, the Hidden Conservation Law Search in Quantum Matter (HCLQM), for uncovering unknown conserved quantities $Q_h$ that may underlie such anomalous behavior. We develop an algebraic criterion that any operator $Q_h$ must satisfy in order to commute with the full Hamiltonian $H$ while not belonging to the algebra generated by known symmetries. Using this criterion we design a numerical search algorithm based on tensor‑network representations of the Hilbert space, enabling us to explore large Hilbert spaces efficiently. We apply HCLQM to three representative systems: a one‑dimensional spin‑1/2 XXZ chain, a two‑dimensional Chern insulator with disorder, and a Hubbard‑type strongly correlated lattice model. In each case we discover non‑trivial conserved operators that explain the presence of long‑lived excited states, protected edge modes, and anomalous transport properties. These findings demonstrate that hidden conserved quantities can be systematically identified and may provide a route to discovering new symmetries in quantum materials, potentially leading to Nobel‑level breakthroughs.Quantum many‑body systems often display emergent stability that cannot be explained by any obvious symmetry of the microscopic Hamiltonian. In this work we propose a systematic framework, the Hidden Conservation Law Search in Quantum Matter (HCLQM), for uncovering unknown conserved quantities $Q_h$ that may underlie such anomalous behavior. We develop an algebraic criterion that any operator $Q_h$ must satisfy in order to commute with the full Hamiltonian $H$ while not belonging to the algebra generated by known symmetries. Using this criterion we design a numerical search algorithm based on tensor‑network representations of the Hilbert space, enabling us to explore large Hilbert spaces efficiently. We apply HCLQM to three representative systems: a one‑dimensional spin‑1/2 XXZ chain, a two‑dimensional Chern insulator with disorder, and a Hubbard‑type strongly correlated lattice model. In each case we discover non‑trivial conserved operators that explain the presence of long‑lived excited states, protected edge modes, and anomalous transport properties. These findings demonstrate that hidden conserved quantities can be systematically identified and may provide a route to discovering new symmetries in quantum materials, potentially leading to Nobel‑level breakthroughs.

Aug 17, 20260 citations

Conservation Laws in Quantum Theory

Niels Henrik David Bohr

NatureJul 1, 193645 citations

Conserved Quantities in Linear and Nonlinear Quantum Search

DAVID A. MEYER, Thomas G. Wong

Abstract In this tutorial, which contains some original results, we bridge the fields of quantum computing algorithms, conservation laws, and many-body quantum systems by examining three algorithms for searching an unordered database of size N using a continuous-time quantum walk, which is the quantum analogue of a continuous-time random walk. The first algorithm uses a linear quantum walk, and we apply elementary calculus to show that the success probability of the algorithm reaches 1 when the jumping rate of the walk takes some critical value. We show that the expected value of its Hamiltonian H 0 is conserved. The second algorithm uses a nonlinear quantum walk with effective Hamiltonian H ( t ) = H 0 + λ | ψ | 2 , which arises in the Gross-Pitaevskii equation describing Bose-Einstein condensates. When the interactions between the bosons are repulsive, λ > 0, and there exists a range of fixed jumping rates such that the success probability reaches 1 with the same asymptotic runtime of the linear algorithm, but with a larger multiplicative constant. Rather than the effective Hamiltonian, we show that the expected value of H 0 + 1 2 λ | ψ | 2 {H_0} + {1 \over 2}\lambda |\psi {|^2} is conserved. The third algorithm utilizes attractive interactions, corresponding to λ < 0. In this case, there is a time-varying critical function for the jumping rate γ c ( t ) that causes the success probability to reach 1 more quickly than in the other two algorithms, and we show that the expected value of H ( t )/[ γ c ( t ) N ] is conserved.Abstract In this tutorial, which contains some original results, we bridge the fields of quantum computing algorithms, conservation laws, and many-body quantum systems by examining three algorithms for searching an unordered database of size N using a continuous-time quantum walk, which is the quantum analogue of a continuous-time random walk. The first algorithm uses a linear quantum walk, and we apply elementary calculus to show that the success probability of the algorithm reaches 1 when the jumping rate of the walk takes some critical value. We show that the expected value of its Hamiltonian H 0 is conserved. The second algorithm uses a nonlinear quantum walk with effective Hamiltonian H ( t ) = H 0 + λ | ψ | 2 , which arises in the Gross-Pitaevskii equation describing Bose-Einstein condensates. When the interactions between the bosons are repulsive, λ > 0, and there exists a range of fixed jumping rates such that the success probability reaches 1 with the same asymptotic runtime of the linear algorithm, but with a larger multiplicative constant. Rather than the effective Hamiltonian, we show that the expected value of H 0 + 1 2 λ | ψ | 2 {H_0} + {1 \over 2}\lambda |\psi {|^2} is conserved. The third algorithm utilizes attractive interactions, corresponding to λ < 0. In this case, there is a time-varying critical function for the jumping rate γ c ( t ) that causes the success probability to reach 1 more quickly than in the other two algorithms, and we show that the expected value of H ( t )/[ γ c ( t ) N ] is conserved.

arXiv:2503.06423Quantum Information and ComputationAug 1, 20251 citation

CONSERVATION LAWS IN QUANTUM MECHANICS

L.D. LANDAU, Efrat Lifshitz

Elsevier eBooksJan 1, 19748 citations

A Simple Necessary and Sufficient Condition for Yang--Baxter Integrability

Mizuki Sanatani, Naoto Shiraishi, Fuga Ishii

Quantum integrability is a cornerstone of the exact theory of interacting quantum spin chains. In its standard formulation, however, one starts from R-matrices satisfying the Yang--Baxter equation, rather than from the Hamiltonian itself. It has therefore remained unclear how Yang--Baxter solvability can be characterized directly at the Hamiltonian level, and how it is related to the existence of local conservation laws. Here we prove that, in a broad standard setting, the Reshetikhin condition is not only necessary but also sufficient for Yang--Baxter integrability, thereby reducing the hidden algebraic structure of integrability to a Hamiltonian-level conservation law. Since the Reshetikhin condition is equivalent to conservation of the total energy current, this Hamiltonian-level criterion is also experimentally accessible. This result establishes a quantum counterpart of the Liouville--Arnold theorem for isotropic spin chains, stating that Yang--Baxter solvability is equivalent to an infinite hierarchy of local conserved quantities. Our result also simplifies substantially the search for integrable spin chains by replacing the search for R-matrices with a direct criterion on local Hamiltonians.Quantum integrability is a cornerstone of the exact theory of interacting quantum spin chains. In its standard formulation, however, one starts from R-matrices satisfying the Yang--Baxter equation, rather than from the Hamiltonian itself. It has therefore remained unclear how Yang--Baxter solvability can be characterized directly at the Hamiltonian level, and how it is related to the existence of local conservation laws. Here we prove that, in a broad standard setting, the Reshetikhin condition is not only necessary but also sufficient for Yang--Baxter integrability, thereby reducing the hidden algebraic structure of integrability to a Hamiltonian-level conservation law. Since the Reshetikhin condition is equivalent to conservation of the total energy current, this Hamiltonian-level criterion is also experimentally accessible. This result establishes a quantum counterpart of the Liouville--Arnold theorem for isotropic spin chains, stating that Yang--Baxter solvability is equivalent to an infinite hierarchy of local conserved quantities. Our result also simplifies substantially the search for integrable spin chains by replacing the search for R-matrices with a direct criterion on local Hamiltonians.

cond-mat.stat-mechCross-listedarXiv:2607.29660Aug 3, 20260 citations

Symmetry and Conservation Law

Tao Xiang

Corresponding to each symmetry, there is a conservation law with a characteristic conserving quantity which is invariant under that transformation. One of the aims of physical research is to search for symmetries and the corresponding observables that are invariant under certain classes of transformation. The Stone theorem ensures the existence of a Hermitian infinitesimal generator for an Abelian group of unitary transformation. The spatial translation operator is a specific example of this theorem and the momentum operator is the generator of spatial translation. This Galilean principle of relativity must be true in quantum mechanics as well. Time-reversal symmetry is a symmetry of the system under the transformation of time reversal.Corresponding to each symmetry, there is a conservation law with a characteristic conserving quantity which is invariant under that transformation. One of the aims of physical research is to search for symmetries and the corresponding observables that are invariant under certain classes of transformation. The Stone theorem ensures the existence of a Hermitian infinitesimal generator for an Abelian group of unitary transformation. The spatial translation operator is a specific example of this theorem and the momentum operator is the generator of spatial translation. This Galilean principle of relativity must be true in quantum mechanics as well. Time-reversal symmetry is a symmetry of the system under the transformation of time reversal.

CRC Press eBooksMar 31, 20220 citations

Q_h

that may underlie such anomalous behavior. We develop an algebraic criterion that any operator

Q_h

must satisfy in order to commute with the full Hamiltonian

H

while not belonging to the algebra generated by known symmetries. Using this criterion we design a numerical search algorithm based on tensor‑network representations of the Hilbert space, enabling us to explore large Hilbert spaces efficiently. We apply HCLQM to three representative systems: a one‑dimensional spin‑1/2 XXZ chain, a two‑dimensional Chern insulator with disorder, and a Hubbard‑type strongly correlated lattice model. In each case we discover non‑trivial conserved operators that explain the presence of long‑lived excited states, protected edge modes, and anomalous transport properties. These findings demonstrate that hidden conserved quantities can be systematically identified and may provide a route to discovering new symmetries in quantum materials, potentially leading to Nobel‑level breakthroughs.