Closed-form finite-periodic boost generators and AGP-norm asymptotics for weak integrability breaking in the Heisenberg chain

Weak integrability-breaking perturbations of the spin-1/2 XXX Heisenberg chain, generated in infinite systems by boost operators (formal sums $\sum_j j\,q_{\beta,j}$ of local conserved-charge densities), thermalize on timescales far longer than Fermi's golden rule predicts, but their generating operator has no known closed analytic form on a finite chain with periodic boundary conditions: the finite-size adiabatic gauge potential (AGP), which proxies the generator, is empirically quasi-local with exponentially suppressed weight on long Pauli strings, yet has resisted all closed-form capture. Vanovac, Surace and Motrunich constructed finite-periodic ansätze for the bilocal and step generator classes and proved exact relations reducing those AGPs to the boost AGP, but the large-$L$ norm asymptotics also remain unsettled: here $p$ denotes the exponent in the fit of the rescaled norm $\|X^{\text{AGP}}\|^2/L=a+bL^{p}$ over accessible sizes, the same convention for every class; the fits give $p\approx1.32$ and $1.546$ for two bilocal perturbations against the analytically expected dominant $L^2$ law ($p=1$), and $p\approx0.03$ or logarithmic growth for boost perturbations. The open problem is to establish the finite-periodic representation of boost-generated weak integrability breaking: a compact closed-form expression (or a rigorous nonexistence/structural characterization) for the finite-PBC boost generator, and the true asymptotic large-$L$ scaling of the AGP norms of boost- and bilocal-generated perturbations, including control of wrapping corrections.

Background

An integrable quantum spin chain, such as the spin-1/2 XXX (Heisenberg) chain with Hamiltonian $H_0=\sum_j \vec{S}_j\cdot\vec{S}_{j+1}$, possesses an extensive family of mutually commuting, translationally invariant conserved quantities $Q_\alpha^{(0)}$ (integrals of motion, IoMs), with $Q_2^{(0)}\propto H_0$. When such a model is perturbed, $H=H_0+\lambda V$, integrability is generically broken and the system is expected to thermalize on a timescale $\tau\sim\lambda^{-2}$, as predicted by Fermi's golden rule together with the eigenstate thermalization hypothesis. A special class of weak integrability-breaking perturbations thermalizes much more slowly, $\tau\sim\lambda^{-2\ell}$ with $\ell>1$, because there exist quasi-conserved quantities that commute with the full Hamiltonian only at order $\lambda^\ell$. These perturbations arise systematically as low-order truncations of long-range deformations of integrable chains — unitary rotations of the Hamiltonian and of all IoMs, introduced originally in the AdS/CFT context, that preserve integrability while giving the charges progressively longer interaction ranges. The rotation is generated by an operator $X$: to leading order the perturbation is the off-diagonal part of $i[X,H_0]$. In infinite systems, three classes of generators producing quasi-local deformations are known: extensive local operators ($X_{\text{ex}}$, a sum of local densities), boost operators $X_{\text{bo}}\sim\mathcal{B}[Q_\beta^{(0)}]=\sum_j j\,q_{\beta,j}^{(0)}$ built from a local density of an IoM (formally related to the generator of accelerated spatial translations, and ill-defined on a periodic chain because the position $j$ is ambiguous), and bilocal anticommutator sums $X_{\text{bi}}=\sum_{j,k}c_{j-k}\{q_{\beta,j}^{(0)},q_{\gamma,k}^{(0)}\}$.

On a finite chain of length $L$ with periodic boundary conditions (PBC), only the extensive-local class has an obvious well-defined counterpart. The finite-size proxy for the other two is the adiabatic gauge potential (AGP), $X^{\text{AGP}}$, defined through its matrix elements $\langle n|V|m\rangle=i(E_m-E_n)\langle n|X^{\text{AGP}}|m\rangle$ between eigenstates of $H_0$ with $E_n\neq E_m$, and set to zero within each degenerate subspace; the AGP was originally introduced as a very sensitive measure of quantum chaos. Its norm distinguishes strong from weak breaking: $\|X^{\text{AGP}}\|^2$ grows exponentially with $L$ for generic (strong) perturbations but only polynomially or logarithmically for weak ones. For the Heisenberg chain, Orlov et al. (Phys. Rev. B 107, 184312) found $\|X^{\text{AGP}}\|^2/L\sim\log L$ for the boost-generated next-nearest-neighbor perturbation, and Pozsgay et al. (arXiv:2402.12979) showed in perturbed free-fermion models that polynomial AGP scaling can even occur for some strong perturbations. Vanovac, Surace and Motrunich (Phys. Rev. B 110, 144309) systematized the finite-size picture: they constructed finite-PBC ansätze $\widetilde{X}^{\beta\gamma}$ (bilocals) and $\widetilde{X}^{\beta}$ (generators of strictly local step-type perturbations) that reproduce the corresponding AGPs up to controlled $1/L$ corrections, and proved exact relations expressing the bilocal and step AGPs through the boost AGPs. Formulating long-range deformations on finite PBC chains is a recognized open problem in its own right; the known technical route is the wrapping-corrections program for long-range spin chains (Gombor, Phys. Rev. Lett. 129, 270201; de Leeuw and Retore, SciPost Phys. 15, 241), which tames the finite-size 'wrapping' effects that arise when a long-range density wraps around the periodic chain. What remains missing is the boost generator itself: the authors could not capture the boost AGP — which is numerically quasi-local, with Pauli-string weights decaying exponentially in both support (number of sites touched) and geometric range — with any closed analytic expression, and the true asymptotic large-$L$ growth of the AGP norms for the boost and bilocal classes is not settled.

Problem Statement

Establish the finite-periodic representation of boost-generated weak integrability breaking in the spin-1/2 XXX Heisenberg chain, in two linked parts.

(1) Closed form of the finite-PBC boost generator. For boost-type weak perturbations $V_{\text{bo}}$ generated in infinite systems by $X_{\text{bo}}\sim\mathcal{B}[Q_\beta^{(0)}]$, construct for general system size $L$ a compact closed-form analytic operator $X_L$ on the periodic chain that satisfies the generator relation $i[X_L,H_0]=V_{\text{bo}}-(V_{\text{bo}})_{\text{diag}}$ (where $(V_{\text{bo}})_{\text{diag}}$ is the part of $V_{\text{bo}}$ diagonal in the eigenbasis of $H_0$) up to corrections controlled in the large-$L$ limit (for the known bilocal ansatz the analogous corrections are $O(1/L)$), reproduces the numerically observed quasi-locality (exponentially decaying Pauli-string weights in support and range), and reduces to the infinite-system boost generator as $L\to\infty$; or else prove rigorously that no such closed form exists within a natural class of expressions that the solver states explicitly and fixes independently of the conclusion (a class tailored to make the nonexistence statement trivial does not count); or provide the structural characterization (for example via wrapping corrections or generalized transfer matrices) that explains the observed quasi-locality by deriving the exponentially decaying Pauli-string weights with error bounds controlled in the large-$L$ limit. Any one of these three outcomes resolves part (1).

(2) Asymptotic AGP-norm scaling. Determine the true large-$L$ scaling of the adiabatic gauge potential norms for boost- and bilocal-generated weak perturbations of the Heisenberg chain: whether $\|X^{\text{AGP}}[V_{\text{bo}}]\|^2$ grows as $L\log L$ or as $L^{1+p}$ (and what fixes $p$; current fits over accessible sizes give $p\approx0.03$), and whether the bilocal AGP norm realizes the analytically expected dominant $\sim L^2$ law (the proven bound $\|\widetilde{X}^{\beta\gamma}\|^2=bL^2+O(L)$ for the constructed ansatz, $p=1$), resolving the discrepancy with the effective exponents $p\approx1.32$ and $1.546$ obtained from the same fits over accessible system sizes.

These two parts constitute one problem, and both are mandatory deliverables: the exact relations proved by Vanovac et al. express the bilocal and step AGPs through the boost AGP, so a finite-size construction or structural characterization of the boost generator (part 1) is the route to settling the norm asymptotics of the whole boost/bilocal family (part 2); part (2) is satisfied either by an independent asymptotic analysis or by making explicit how the part (1) result fixes the scaling.

References

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