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arXiv:1803.02637 · Europhysics Letters (EPL)

Proof of the absence of local conserved quantities in the XYZ chain with a magnetic field

Naoto Shiraishi

Published Nov 25, 2019• 49 citations
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Abstract

We rigorously prove that the spin-(1/2) XYZ chain with a magnetic field has no local conserved quantity. Any nontrivial conserved quantity of this model is shown to be a sum of operators supported by contiguous sites with at least half of the entire system. We establish that the absence of local conserved quantity in concrete models is provable in a rigorous form.

DOI
10.1209/0295-5075/128/17002

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We rigorously prove that the spin-(1/2) XYZ chain with a magnetic field has no local conserved quantity. Any nontrivial conserved quantity of this model is shown to be a sum of operators supported by contiguous sites with at least half of the entire system. We establish that the absence of local conserved quantity in concrete models is provable in a rigorous form.
Abstract We rigorously prove that the S=1/2S=1/2S=1/2 S = 1 / 2 anisotropic Heisenberg chain (XYZ chain) with next-nearest-neighbor interaction, which is anticipated to be non-integrable, is indeed non-integrable in the sense that this system has no nontrivial local conserved quantity. Our result covers some important models including the Majumdar–Ghosh model, the Shastry–Sutherland model, and many other zigzag spin chains as special cases. These models are shown to be non-integrable while they have some solvable energy eigenstates. In addition to this result, we provide a pedagogical review of the proof of non-integrability of the S=1/2S=1/2S=1/2 S = 1 / 2 XYZ chain with Z magnetic field, whose proof technique is employed in our result.
Absence of local conserved quantities is often required, such as for thermalization or for the validity of response theory. Although many studies have discussed whether thermalization occurs in the Ising chain with longitudinal and transverse fields, rigorous results on local conserved quantities of this model have still been lacking. Here, we rigorously prove that, if all coupling constants are nonzero, this model has no conserved quantity spanned by local operators with support size up to half of the system size other than a trivial one, i.e., a linear combination of the Hamiltonian and the identity. The proof is given not only for the periodic boundary condition but also for the open boundary condition. We also discuss relation to the integrability of the model where the longitudinal field is set to zero. Our results provide the second example of spin models whose nonintegrability is rigorously proved.
We provide a rigorous proof of the absence of nontrivial local conserved quantities in all spin-1/2 chains with symmetric nearest-neighbor interaction, except for known integrable systems. This result shows that there are no further integrable system that awaits to be discovered. Our finding also implies that there is no intermediate systems with a finite number of nontrivial local conserved quantities. In addition, we clarify all short-support conserved quantities in non-integrable systems, which we need to take into account in analyses of thermalization and level statistics.
Absence of local conserved quantities, or nonintegrability, is often assumed when discussing various phenomena in quantum many-body systems, such as thermalization and transport. However, no concrete proof of this property is known in electron--phonon coupled systems, a typical setting for condensed matter physics. In this paper, we show that the one-dimensional Holstein model has no nontrivial local conserved quantities other than the Hamiltonian itself and the total fermion number operator. We further show that the absence of nontrivial local conserved quantities also holds for the more general Holstein--Hubbard model. We also establish the absence of nontrivial local conserved quantities in these models on higher-dimensional hypercubic lattices. Our result has accomplished an advance in nonintegrability proofs by expanding their scope to systems in which particles with different statistical properties are mixed.
Absence of nontrivial local conserved quantities in a class of U(1)U(1)U(1)-symmetric spin-1 chains
We prove the absence of nontrivial local conserved quantities in a class of U(1)U(1)U(1)-symmetric spin-111 chains with nearest-neighbor interactions in which some of the quadrupolar couplings vanish, a class that is not covered by previous studies. Applying the technique of Shiraishi to these systems, we show that, for every model in this class on a periodic chain of NNN sites, there is no kkk-local conserved quantity for any 3≤k≤N/23\le k\le N/23≤k≤N/2. In particular, for a frustration-free spin-111 chain that exhibits spontaneous U(1)U(1)U(1) symmetry breaking at zero temperature in one spatial dimension, we prove that every local conserved quantity with support up to half of the system size is a linear combination of the identity, the total magnetization SzS^zSz, and the Hamiltonian itself. This rigorously establishes that, unlike the Heisenberg ferromagnet, the model admits no local order parameter commuting with the Hamiltonian, so that its continuous symmetry breaking is enabled by the frustration-free structure rather than by a conserved order parameter. We also prove the absence of kkk-local conserved quantities for 3≤k≤N/23\le k\le N/23≤k≤N/2 in the periodic Motzkin chain, a frustration-free spin-111 chain closely related to the original Motzkin chain, for which spontaneous U(1)U(1)U(1) symmetry breaking at zero temperature has also been reported.