Finite-size generators for weak integrability breaking in the Heisenberg chain: asymptotic scaling, quasi-locality, and exact recovery of the boost-generated adiabatic gauge potential

The spin-1/2 Heisenberg chain perturbed by special 'weak integrability-breaking' perturbations thermalizes far more slowly than Fermi's golden rule predicts. In infinite systems such perturbations are generated by extensive-local, boost, and bilocal operators arising from long-range deformations of integrable chains, but only the extensive-local generators are well defined on finite periodic chains, where the boost operator's position weighting is incompatible with periodicity. Vanovac, Surace, and Motrunich [Phys. Rev. B 110, 144309 (2024)] showed numerically that the regularized adiabatic gauge potential (AGP) serves as the finite-size proxy for these generators, finding near-linear growth of the squared AGP norm for boost-generated perturbations and exponentially suppressed weights on long Pauli strings. The problem asks for the missing mathematics: the exact large-L asymptotic scaling of the boost-AGP norm (present data at $L\leq 18$ cannot distinguish $L\log L$ from a weak power law $L^{1+p}$ with $p\approx 0.03$), a proof or counterexample for the emergent quasi-locality of finite-size boost AGPs, and an algorithmic characterization of how many quasi-conserved-charge corrections the generalized AGP must include to recover the exact finite-size generators.

Background

An integrable quantum spin chain, such as the spin-1/2 isotropic Heisenberg (XXX) chain $H_0=\sum_j \vec\sigma_j\cdot\vec\sigma_{j+1}$ with periodic boundary conditions (PBC), is special among many-body systems: it carries an extensive family of mutually commuting conserved quantities, the integrals of motion (IoMs) $\{Q_\alpha\}$ with $[H_0,Q_\alpha]=[Q_\alpha,Q_\beta]=0$ (the convention is $H_0=2Q_2$, with $Q_{\alpha\geq 3}$ the nontrivial charges). Adding a generic perturbation $\lambda V$ breaks integrability strongly, and the system obeys the eigenstate thermalization hypothesis (ETH) — the statement that individual highly excited eigenstates appear thermal — with relaxation rates of order $\lambda^2$ as predicted by Fermi's golden rule. A special class of 'weak integrability-breaking' perturbations thermalizes parametrically more slowly: there exists an operator $X$ with $i[X,H_0]=V-V_{\rm diag}$, where $V_{\rm diag}$ is the part of $V$ block-diagonal in the degenerate eigenspaces of $H_0$, and corrected charges $Q_\alpha+\lambda Q_\alpha^{(1)}$ with $Q_\alpha^{(1)}=i[X,Q_\alpha]$ remain conserved up to $O(\lambda^2)$ — quasi-IoMs. A prominent example is the next-neighbor SU(2)-symmetric interaction $V=\sum_j \vec\sigma_j\cdot\vec\sigma_{j+2}$, whose anomalously slow transport has been known since Jung, Helmes, and Rosch (2006).

A systematic construction of such generators comes from long-range deformations of integrable spin chains (Bargheer, Beisert, and Loebbert 2008; Pozsgay 2020; Surace and Motrunich 2023): unitary rotations that turn short-range chains into integrable models with increasingly long-range interactions, whose low-order truncations are weak perturbations. Three generator classes are known in the infinite chain: extensive local operators $X_{\rm ex}=\sum_j O_j$; boost operators $X_{\rm bo}=-B[Q_\beta]$ with $B[Q_\beta]=\sum_j j\,q_{\beta,j}$, an IoM density $q_{\beta,j}$ weighted by the site position $j$; and bilocal operators combining IoM densities at two independently positioned centers. Boosts and bilocals are ill-defined on finite PBC chains (a position weighting is incompatible with periodicity), so no closed-form finite-size counterpart of these generators was known.

The adiabatic gauge potential (AGP) provides a finite-size proxy. For the Hamiltonian $H_0+\lambda V$ the AGP $X_{\rm AGP}$ has, between non-degenerate eigenstates of $H_0$, matrix elements equal to the perturbation matrix element divided by the energy difference (times $i$), and vanishes inside degenerate subspaces; it satisfies $i[X_{\rm AGP},H_0]=V-V_{\rm diag}$ exactly on any finite system. Its norm diagnoses chaos: for generic strong perturbations it grows exponentially with system size $L$ (Pandey et al. 2020), while weak perturbations show polynomial or logarithmic growth (Orlov et al. 2023). In practice one uses a regularized AGP, replacing energy denominators $\omega^{-1}$ by $\omega/(\omega^2+\mu^2)$ with a cutoff $\mu$ of order the many-body level spacing, and computes Frobenius norms in the zero-magnetization ($S^z_{\rm tot}=0$) sector.

Vanovac, Surace, and Motrunich [Phys. Rev. B 110, 144309 (2024)] computed regularized AGPs for representatives of all weak-perturbation classes of the Heisenberg chain. For the boost-generated perturbations $V_{{\rm bo},1}=\sum_j \vec\sigma_j\cdot\vec\sigma_{j+2}$ and $V_{{\rm bo},2}=\sum_j(\vec\sigma_j\times\vec\sigma_{j+3})\cdot(\vec\sigma_{j+1}+\vec\sigma_{j+2})$, the rescaled norm $\|X_{\rm AGP}\|^2/L$ fits $a+bL^p$ with $p=0.03$ but is also consistent with $c+d\log L$ (Orlov et al. earlier proposed $\|X_{\rm AGP}[V_{{\rm bo},1}]\|^2\sim L\log L$). Expanding the AGPs in Pauli strings — tensor products of single-site $\sigma^x,\sigma^y,\sigma^z$ operators with coefficients $c_S$ and norm fractions $w_S=|c_S|^2/\|X_{\rm AGP}\|^2$, organized by string support size and spatial range — they found that for all weak perturbations the weight on long-support, long-range strings is strongly and apparently exponentially suppressed, an emergent quasi-locality that is statistically 'rare' in operator space (the number of strings grows exponentially with support). They also constructed finite-PBC ansätze $\widetilde X^{\beta\gamma}=\sum_{j,k} t((k-j)\,{\rm mod}\,L)\,\{q_{\beta,j},q_{\gamma,k}\}$ and $\widetilde X^{\beta}$ that reproduce the AGPs of bilocal- and step-generated perturbations exactly in terms of boost-type ones, and showed that recovering the block-diagonal parts of generators requires the generalized AGP: augmenting the construction with quasi-IoM corrections $Q_\alpha^{(1)}$ to lift the degeneracies of $H_0$ (for example, including $Q_3^{(1)}$ recovers $\widetilde X^{\beta=2}$ exactly at the studied sizes). What the paper does not supply is the mathematics behind these observations: the true $L\to\infty$ norm asymptotics, a mechanism guaranteeing quasi-locality, and a general characterization of the recovery algorithm.

Problem Statement

For the spin-1/2 Heisenberg chain $H_0=\sum_j \vec\sigma_j\cdot\vec\sigma_{j+1}$ with periodic boundary conditions and its boost-generated weak integrability-breaking perturbations — with $V_{{\rm bo},1}=\sum_j \vec\sigma_j\cdot\vec\sigma_{j+2}$ and $V_{{\rm bo},2}=\sum_j(\vec\sigma_j\times\vec\sigma_{j+3})\cdot(\vec\sigma_{j+1}+\vec\sigma_{j+2})$ as representative cases — establish the mathematical characterization of the finite-size generators, as proxied by the regularized adiabatic gauge potential, in three jointly required parts:

1. Asymptotic scaling. Determine the exact $L\to\infty$ asymptotic behavior of the regularized AGP norm in the $S^z_{\rm tot}=0$ sector (Frobenius norm; regulator $\mu$ of the order of the many-body level spacing, for instance the source paper's choice $\mu=L/D$ with $D$ the Hilbert-space dimension of the $S^z_{\rm tot}=0$ sector): whether the squared norm $\|X_{\rm AGP}[V_{{\rm bo},i}]\|^2$ grows as $L\log L$, as $L^{1+p}$ with $p>0$, or otherwise. Current exact-diagonalization data at $L\leq 18$ fit $\|X_{\rm AGP}\|^2/L=a+bL^{0.03}$ — a squared norm near-linear in $L$ up to this weak correction — and are equally consistent with $\|X_{\rm AGP}\|^2\sim c\,L\log L$; the asymptotic form is not established.

2. Quasi-locality. Prove or disprove that the finite-PBC boost AGPs are quasi-local: either exhibit the underlying mathematical mechanism, with explicit bounds showing that the Pauli-string weights $w_S=|c_S|^2/\|X_{\rm AGP}\|^2$ are exponentially suppressed in string support size and spatial range as $L$ grows, or provide a counterexample (a boost-generated weak perturbation whose finite-PBC AGP has non-negligible weight on long strings at large $L$). Any proof should make contact with the finite-PBC ansätze $\widetilde X^{\beta\gamma}$ and $\widetilde X^{\beta}$ constructed in the source paper.

3. Exact recovery. Provide an algorithmic characterization of the exact recovery of finite-size generators, including their block-diagonal parts, via the generalized AGP: determine which and how many quasi-IoM corrections $Q_\alpha^{(1)}$ must be included, for arbitrary system sizes and symmetry sectors, so that the recovered generator coincides with the exact generators (for example, so that $\widetilde X^{\beta=2}$ is recovered for the step-generated perturbation $V_{{\rm step},0}=(\vec\sigma_j\times\vec\sigma_{j+1})\cdot\vec\sigma_{j+2}$ acting at a single fixed position $j$ — the local energy current, i.e., the local density of the charge $Q_3$).

Results for bilocal- and step-generated perturbations may be obtained through the exact relations derived in the source paper linking their AGPs to boost AGPs. A complete solution turns the source paper's numerical and heuristic picture into established mathematics.

References

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