Tight rigorous lower bounds and long-time asymptotics for the persisting correlation of a central spin in a large spin bath

The central spin model, in which a spin-1/2 'central' spin is isotropically coupled to a large bath of spins with widely distributed coupling constants, is the canonical model of qubit decoherence by nuclear spins in quantum dots. Seifert, Bleicker, Schering, Faribault and Uhrig (Phys. Rev. B 94, 094308 (2016)) proved, via a generalized Mazur inequality, that a nonzero part of the central-spin autocorrelation $S(t)=\langle S_0^z(t)S_0^z(0)\rangle$ persists at infinite bath temperature, and established a well-defined thermodynamic limit for normalizable coupling distributions. However, at zero magnetic field their rigorous spin-spin bounds are not tight: the tight values come from an unproven 'field-field' (Overhauser) estimate based on a separation of timescales, the conserved quantities that would saturate the Mazur bound have not been identified, and the predicted long-time decay $S(t)\propto\ln(\ln t)/\ln t$ rests on the heuristic replacement of the coupling-spread parameter $x$ by $\ln t$. This record states the open problem of establishing a rigorous theory of the persisting central-spin correlation: a provable tight lower bound at zero magnetic field and a first-principles derivation of the genuine long-time asymptotics.

Background

A quantum-dot electron spin decoheres mainly through the hyperfine interaction with the $10^4$-$10^6$ surrounding nuclear spins. In the central spin model (CSM) this is idealized as a spin-1/2 'central' spin $\vec S_0$ coupled isotropically to $N$ bath spins of spin 1/2 with Hamiltonian $H_0=\sum_{k=1}^{N}J_k\,\vec S_0\cdot\vec S_k$, where the couplings $J_k$ are spread over many orders of magnitude; a standard testbed is the exponential distribution $J_k=J\exp(-kx/N)$ with spread parameter $x=N/N_0$, the ratio of the total number of bath spins to the number of spins within the localization radius of the central-spin wave function. Because the couplings correspond to temperatures far below experimental ones, the bath is taken at infinite temperature ($\rho$ proportional to the identity). The quantity of interest is the persisting correlation: the long-time (infinite-time-average) level of the central-spin autocorrelation $S(t)=\langle S_0^z(t)S_0^z(0)\rangle$, whose nonzero value quantifies the decoherence plateau that spin-noise spectroscopy, the measurement of spin-fluctuation spectra, can probe. The bath acts on the central spin through the Overhauser field $\vec B=\sum_k J_k\vec S_k$, the effective magnetic field generated by the bath. Already Khaetskii, Loss and Glazman (Phys. Rev. Lett. 88, 186802 (2002)) showed that the long-time electron-spin decay in this setting is unusually slow. The CSM is integrable in the sense of the Gaudin (Richardson-Gaudin) models, a class of exactly solvable spin models, so it possesses $N+1$ nontrivial constants of motion $H_l=\sum_{k=0,\,k\neq l}^{N}\frac{\vec S_l\cdot\vec S_k}{\epsilon_l-\epsilon_k}$ (with $\epsilon_k=-1/J_k$, $\epsilon_0=0$, assuming the couplings $J_k$ are pairwise different) commuting with $H_0$; only $N$ of the $N+1$ observables $I^zH_l$ are linearly independent. The corresponding Bethe-ansatz equations, the algebraic equations that determine the exact eigenstates, are numerically intractable for large baths, so integrability is of limited practical help. The Mazur inequality (P. Mazur, Physica 43, 533 (1969)) bounds the infinite-time average of any autocorrelation from below by the squared projection of the operator onto the space spanned by conserved quantities (operators commuting with the Hamiltonian). Uhrig, Hackmann, Stanek, Stolze and Anders (Phys. Rev. B 90, 060301(R) (2014)) introduced a generalized Mazur bound $A_{\rm low}=\vec a_C^{\,\top}\mathcal N^{-1}\vec a_C$, with $a_{C,i}=(C_i|\hat A)$ and $\mathcal N_{im}=(C_i|C_m)$ in the scalar product $(X|Y)=\mathrm{Tr}\,\rho X^\dagger Y$, which requires no orthonormalization of the conserved quantities $C_i$, and proved that persisting correlations are generic in the CSM whenever the coupling distribution is normalizable. Seifert et al. (Phys. Rev. B 94, 094308 (2016)) built on this: they proved that a well-defined thermodynamic limit $N\to\infty$ exists whenever the coupling moments $\Sigma_m=\sum_k J_k^m$ converge with $\Sigma_2=J_Q^2$ fixed ($J_Q$ is the natural energy unit); they enlarged the set of conserved operators entering the bound by including products such as $I^zH_0^{2m-1}$ and $I^zI^2H_0$ (with $I^z=\sum_{j=0}^{N}S_j^z$), whose overlaps they evaluated to leading order in $N$ by Gaussian integration; and they showed that at finite magnetic field $h\gtrsim J_Q$ a small number of conserved quantities yields bounds that are tight against numerics based on Chebyshev-polynomial expansion of the real-time evolution and on stochastic evaluation of the Bethe-ansatz formulas. At vanishing field, however, even the enlarged Mazur bounds remain clearly below the numerical values. The tight values are instead reproduced by a 'field-field' (BB) estimate: assuming the central spin precesses rapidly around a slowly evolving Overhauser field and treating the field modulus $B^2$ as approximately conserved, one factorizes the spin autocorrelation into the field autocorrelation $S^{(B)}(t)=\langle B^z(t)B^z(0)\rangle$ as $\langle S_0^z(t)S_0^z(0)\rangle\approx\frac{1}{12}\,S^{(B)}(t)/S^{(B)}(0)$, and applying the generalized Mazur construction to $B^z$ yields the field-field bound $S_{\rm low,BB}=S^{(B)}_{\rm low}/(12\,S^{(B)}(0))$, where $S^{(B)}_{\rm low}$ is the corresponding Mazur lower bound (a separation-of-timescales approximation whose rigorous status is unproven). The thermodynamic-limit data for the exponential distribution are fitted by $S_{\log}(x)=A\ln(x/B)/x$, where $A$ and $B$ are the amplitude and scale of this empirical plateau fit, distinct from the bounded operator $\hat A$, its Mazur lower bound $A_{\rm low}$, and the Overhauser field $\vec B$. Chen, Bergman and Balents (Phys. Rev. B 76, 045312 (2007)) argued heuristically that up to time $t$ only bath spins with $tJ_k\gtrsim 1$ participate, so the effective bath size grows like $\ln t$; inserting $x\mapsto\ln t$ into the fit then suggests the nested-logarithm long-time law $S(t)\propto\ln(\ln t)/\ln t$, a refinement of the previously known inverse-logarithm decay that has never been derived rigorously. Finally, the physical hyperfine case, in which infinitely many bath spins couple ever more weakly, corresponds to a non-normalizable coupling distribution, outside the moment-based machinery of the rigorous thermodynamic limit.

Problem Statement

For the isotropic central spin model $H_0=\sum_{k=1}^{N}J_k\,\vec S_0\cdot\vec S_k$ with a spin-1/2 central spin, spin-1/2 bath, infinite-temperature bath state, and a spread of couplings (concretely the exponential distribution $J_k=J\exp(-kx/N)$ with spread parameter $x$ — the case for which the tight zero-field value and the numerical benchmarks are known, and the case addressed by components (1) and (2) below; extending the quantitative results to other normalizable coupling distributions is desirable but not required), establish a rigorous quantitative theory of the persisting correlation, i.e. of the infinite-time-average level of the central-spin autocorrelation $S(t)=\langle S_0^z(t)S_0^z(0)\rangle$. The objective comprises two linked components. (1) Tight zero-field bound. The deliverable is a rigorous tight lower bound on the persisting spin-spin correlation at vanishing magnetic field, which can be reached by either of two independent routes. (1a) Field-field (BB) estimate: decide whether the separation-of-timescales field-field (BB) bound $S_{\rm low,BB}=S^{(B)}_{\rm low}/(12\,S^{(B)}(0))$ — obtained by assuming that the central spin precesses rapidly around the slowly evolving Overhauser field $\vec B=\sum_kJ_k\vec S_k$, factorizing the spin autocorrelation as $\langle S_0^z(t)S_0^z(0)\rangle\approx\frac{1}{12}\,S^{(B)}(t)/S^{(B)}(0)$ with $S^{(B)}(t)=\langle B^z(t)B^z(0)\rangle$, and applying the generalized Mazur construction to $B^z$, with $S^{(B)}_{\rm low}$ the resulting Mazur lower bound — is a rigorous lower bound on the persisting spin-spin correlation at vanishing magnetic field. A proof of (1a) delivers a tight lower bound by this route; a disproof settles only the validity of the separation-of-timescales estimate and does not by itself deliver a tight lower bound, which then has to be supplied by (1b). (1b) Saturation by conserved operators: identify the conserved quantities or products of the known integrals of motion (beyond $I^z$, $I^zH_l$, the families $I^zH_0^{2m-1}$, and $I^zI^2H_0$) whose inclusion makes the generalized Mazur bound $A_{\rm low}=\vec a_C^{\,\top}\mathcal N^{-1}\vec a_C$ tight at zero field, and determine whether a finite or an infinite generating set of conserved operators is required to saturate the persisting correlation; a solution of (1b) delivers a tight lower bound by the conserved-operator route, independently of (1a). (2) Long-time asymptotics: derive from first principles, not via the heuristic effective-bath-size substitution $x\mapsto\ln t$, the genuine long-time behavior of $S(t)$, settling the order of the thermodynamic and long-time limits that this substitution obscures; in particular confirm or refute the nested-logarithm law $S(t)\propto\ln(\ln t)/\ln t$ suggested for exponential couplings by combining the empirical plateau fit $S_{\log}(x)=A\ln(x/B)/x$ with the growing effective spread $x(t)\propto\ln t$ implied by $tJ_{N_{\rm eff}}\approx1$. Components (1) and (2) are the two halves of one objective, the rigorous value of the persisting correlation and the rigorous law of its approach, and are linked through the same fitted function $S_{\log}(x)$.

References

1. U. Seifert, P. Bleicker, P. Schering, A. Faribault, and G. S. Uhrig, Persisting correlations of a central spin coupled to large spin baths, Phys. Rev. B 94, 094308 (2016), DOI: 10.1103/PhysRevB.94.094308, arXiv:1603.08894, https://arxiv.org/abs/1603.08894
2. G. S. Uhrig, J. Hackmann, D. Stanek, J. Stolze, and F. B. Anders, Conservation laws protect dynamic spin correlations from decay: Limited role of integrability in the central spin model, Phys. Rev. B 90, 060301(R) (2014), DOI: 10.1103/PhysRevB.90.060301, arXiv:1402.1277, https://arxiv.org/abs/1402.1277
3. G. Chen, D. L. Bergman, and L. Balents, Semiclassical dynamics and long-time asymptotics of the central-spin problem in a quantum dot, Phys. Rev. B 76, 045312 (2007), DOI: 10.1103/PhysRevB.76.045312, https://doi.org/10.1103/PhysRevB.76.045312
4. P. Mazur, Non-ergodicity of phase functions in certain systems, Physica 43, 533 (1969), DOI: 10.1016/0031-8914(69)90185-2, https://doi.org/10.1016/0031-8914(69)90185-2
5. A. V. Khaetskii, D. Loss, and L. Glazman, Electron Spin Decoherence in Quantum Dots due to Interaction with Nuclei, Phys. Rev. Lett. 88, 186802 (2002), DOI: 10.1103/PhysRevLett.88.186802, https://doi.org/10.1103/PhysRevLett.88.186802
6. R. Röhrig, P. Schering, L. B. Gravert, B. Fauseweh, and G. S. Uhrig, Quantum mechanical treatment of large spin baths, Phys. Rev. B 97, 165431 (2018), DOI: 10.1103/PhysRevB.97.165431, arXiv:1711.08919, https://arxiv.org/abs/1711.08919
7. B. Fauseweh, P. Schering, J. Hüdepohl, and G. S. Uhrig, Efficient algorithms for the dynamics of large and infinite classical central spin models, Phys. Rev. B 96, 054415 (2017), DOI: 10.1103/PhysRevB.96.054415, https://doi.org/10.1103/PhysRevB.96.054415
8. M. Yarmohammadi, K. Bolsmann, Y. Ribbeheger, T. Gräßer, and G. S. Uhrig, Truncated Wigner approximation for the bosonic model of large spin baths, Phys. Rev. B 107, 125421 (2023), DOI: 10.1103/PhysRevB.107.125421, arXiv:2301.06346, https://arxiv.org/abs/2301.06346
9. C. Bengs, Centralizer theory for long-lived spin states, J. Chem. Phys. 154, 244107 (2021), DOI: 10.1063/5.0050419, https://doi.org/10.1063/5.0050419
10. P. Schering, P. W. Scherer, and G. S. Uhrig, Interplay of spin mode locking and nuclei-induced frequency focusing in quantum dots, Phys. Rev. B 102, 115301 (2020), DOI: 10.1103/PhysRevB.102.115301, arXiv:2004.02623, https://arxiv.org/abs/2004.02623
11. M. Krishna, P. Solanki, M. Hajdušek, and S. Vinjanampathy, Measurement-Induced Continuous Time Crystals, Phys. Rev. Lett. 130, 150401 (2023), DOI: 10.1103/PhysRevLett.130.150401, https://doi.org/10.1103/PhysRevLett.130.150401
12. A. Faribault and C. Dimo, 'Bethe-ansatz-free' eigenstates for spin-1/2 Richardson-Gaudin integrable models, J. Phys. A 55, 415205 (2022), DOI: 10.1088/1751-8121/ac92ac, https://doi.org/10.1088/1751-8121/ac92ac
13. W. M. Witzel, J. J. Lutz, M. D. Grace, N. D. Foster, R. M. Jock, and D. R. Luhman, Predicting the Slow Drift of Nuclear Spin Noise in Semiconductor Spin Qubits, arXiv:2607.26019 (2026), https://arxiv.org/abs/2607.26019