Tight Mazur bound and the Overhauser-field estimate for the non-decaying central-spin correlation in the isotropic central spin model

In the isotropic central spin model (CSM) — the paradigmatic model of hyperfine decoherence of a localized qubit spin coupled to a bath of $N$ inhomogeneously coupled spins — a fraction of the central-spin autocorrelation $S(t)=\langle S_0^z(t)S_0^z(0)\rangle$ persists forever, and its non-decaying part $S_\infty$ is bounded from below by a generalized Mazur inequality evaluated on any set of conserved operators. Uhrig, Hackmann, Stanek, Stolze and Anders [PRB 90, 060301(R) (2014)] proved this rigorously, showed that integrability plays only a limited role, and left open which conserved quantities must be included to make the bound tight, while conjecturing on numerical evidence that $S_\infty$ is quantitatively described by $S_{\mathrm{low}}^{(B)}/(12S^{(B)}(0))$, an estimate built from the Overhauser-field autocorrelation and the conserved quantities $I^z$ and $H_l^z = I^z H_l$. The problem is (i) exhibiting a sufficient set of conserved operators that saturates the Mazur bound on $S_\infty$ and (ii) deriving or refuting the factor-$1/12$ Overhauser estimate with controlled error at finite bath size; the source additionally suggests, as an outlook, investigating extensions such as external fields, anisotropic couplings and intra-bath couplings in the same manner. The direct follow-up by Seifert et al. [PRB 94, 094308 (2016)] achieved partial progress (tight bounds at moderate magnetic fields, higher-order conserved products) but explicitly left the zero-field tightness and the rigorous status of the $1/12$ estimate open; no resolution is known to date.

Background

The central spin model (CSM) is the standard microscopic model of hyperfine decoherence for a localized electronic spin, the prototypical 'qubit' in a semiconductor quantum dot or a defect center. A central spin-1/2 $\vec{S}_0$ is coupled isotropically (Heisenberg) to $N$ bath spins-1/2 $\vec{S}_k$ with inhomogeneous couplings $J_k$,

$$H_0 = \sum_{k=1}^{N} J_k\, \vec{S}_0 \cdot \vec{S}_k,$$

where the $J_k$ are the hyperfine constants set by the envelope wave function of the electron; they are generically strongly inhomogeneous, and the root-mean-square coupling $J_Q := (\sum_{k=1}^{N} J_k^2)^{1/2}$ is the natural energy scale of the model. At experimentally relevant conditions the nuclear bath is effectively at infinite temperature, $\rho \propto \mathbb{1}$, and the central-spin coherence is characterized by the autocorrelation $S(t) = \langle S_0^z(t) S_0^z(0)\rangle$ (by isotropy all components are equivalent). Its non-decaying fraction is the long-time average

$$S_\infty = \lim_{T \to \infty} \frac{1}{T} \int_0^T S(t)\, dt,$$

the zero-frequency weight of the central-spin fluctuation spectrum measured in spin-noise spectroscopy and free-induction-decay experiments. For classical bath spins, strong analytical arguments (Merkulov, Efros and Rosen, 2002, and later classical studies) show that a fraction of the central-spin correlation persists unless the bath contains a diverging number of arbitrarily weakly coupled spins.

Mazur's inequality (Mazur, 1969) states that for any set of linearly independent conserved operators $C_i$ with $[C_i, H] = 0$, the long-time average of an autocorrelation is bounded from below by the squared projection of the observable onto the space spanned by the conserved operators. Uhrig, Hackmann, Stanek, Stolze and Anders (PRB 90, 060301(R) (2014)) generalized the inequality to arbitrary linearly independent conserved sets that need not be orthogonal: with the scalar product $(X|Y) = \mathrm{Tr}(\rho X^\dagger Y)$, the norm matrix $N_{im} = (C_i|C_m)$ and the overlap vector $a_i = (C_i|A)$, the bound reads $S_{\mathrm{low}} = \mathbf{a}^\dagger N^{-1}\mathbf{a} \le S_\infty$. The bound becomes tight — an equality — only if the chosen operators span the full projection of $A$ onto the space of conserved operators; formally this is achieved by the complete set of operators $|j\rangle\langle m|$ between degenerate eigenstates, but using it requires the full diagonalization of $H$ and is useless for large baths.

For the isotropic CSM the conserved quantities are organized as follows. The $N+1$ Gaudin integrals

$$H_l = \sum_{j=0,\, j \neq l}^{N} \frac{\vec{S}_l \cdot \vec{S}_j}{\varepsilon_l - \varepsilon_j}, \qquad l = 0, 1, \dots, N, \qquad \varepsilon_0 = 0,\ \varepsilon_k = -1/J_k\ (k \geq 1),$$

mutually commute, $[H_l, H_p] = 0$, which underlies the Bethe-ansatz (Gaudin) solvability of the model; the $l = 0$ integral coincides with the Hamiltonian $H_0$ itself. They also commute with the $z$ component of the total spin, $I^z = \sum_{j=0}^{N} S_j^z$. However, the infinite-temperature trace of a product in which some site carries exactly one spin operator vanishes, so the bilinear integrals $H_l$ have zero overlap with the linear observables $S_0^z$ and the Overhauser field $\vec{B} = \sum_k J_k \vec{S}_k$ (the effective bath field acting on the central spin); the conserved quantities actually usable in the Mazur bound are the products $H_l^z := I^z H_l$ (the source's notation), together with $I^z$ itself. These are the 'N+1 integrals $I^z$ and $H_l^z$' of the source: the set $\{I^z\} \cup \{H_l^z : 1 \leq l \leq N\}$ contains (and therefore reproduces the bound of) the three low-order conserved quantities $I^z$, $H_0^z = I^z H_0$ and $I_Q^z$ used in the simplest bounds, because $\sum_{l=0}^{N} H_l^z = 0$ and $I_Q^z = \sum_{l=1}^{N} \varepsilon_l H_l^z$. Uhrig et al. used these generic isotropic conservation laws to prove rigorously that a finite fraction of the central-spin correlation persists for finite $N$ unless the average coupling vanishes, showed that integrability plays only a limited role (adding an intra-bath coupling $J_{\mathrm{ex}}$ between two bath spins, which spoils integrability completely, degrades the bound only smoothly), and observed that the Mazur bound on $S_0^z$ obtained from the $N+1$ quantities $I^z, H_l^z$ comes close to but does not saturate the numerically computed $S_\infty$.

On the quantitative side, Merkulov, Efros and Rosen (2002) derived, for a classical and large Overhauser field, the estimate $S_\infty = S_\infty^{(B)}/(12 S^{(B)}(0))$, where $S^{(B)}(t) = \langle B^z(t) B^z(0)\rangle$ is the autocorrelation of the Overhauser field. Combining this with the Mazur bound for $B^z$ computed from the $N+1$ quantities $I^z$ and $H_l^z$, Uhrig et al. found excellent agreement with the available numerical data (Chebyshev-expansion calculations and published stochastic Bethe-ansatz results for baths of up to $N = 48$ spins; Hackmann and Anders, 2014; Faribault and Schuricht, 2013), and conjectured that the non-decaying fraction is quantitatively described by $S_{\mathrm{low}}^{(B)}/(12 S^{(B)}(0))$.

The open questions posed by the source are then: which additional conserved operators — presumably higher powers of the Hamiltonian combined with the integrals, such as $I^z H^2$, or non-commuting conserved operators — must be included for the Mazur bound to be tight, and on what rigorous footing the factor-$1/12$ Overhauser estimate stands in the quantum model at finite bath size. As an outlook, the source further notes that various extensions of the model — by magnetic fields, anisotropies, or more intra-bath couplings — suggest themselves to be investigated in the same manner.

Problem Statement

Determine rigorously the non-decaying fraction $S_\infty$ of the central-spin autocorrelation in the isotropic central spin model with generic inhomogeneous couplings at infinite temperature, by resolving the following linked questions as posed by Uhrig, Hackmann, Stanek, Stolze and Anders [PRB 90, 060301(R) (2014)]:

1. Tight set of conserved quantities. Exhibit an explicit, computable set of conserved operators — presumably including products of the known integrals with powers of the Hamiltonian, such as $I^zH^2$, or non-commuting conserved operators beyond the commuting $N+1$ quantities $\{I^z, H_l^z\}$ — such that the generalized Mazur bound computed from this set equals $S_\infty$. The set must be defined without reference to the eigenbasis of $H$ — constructing it must not require the full diagonalization of the Hamiltonian — and the equality must hold uniformly in the bath size, either for all $N$ or in the thermodynamic limit under stated conditions on the coupling distribution.

2. Overhauser-field estimate. Prove, with a controlled error estimate in the bath size and coupling distribution, or refute by an explicit instance, the conjecture that $S_\infty$ is quantitatively described by $S_{\mathrm{low}}^{(B)}/(12S^{(B)}(0))$, where $S_{\mathrm{low}}^{(B)}$ is the generalized Mazur bound of the Overhauser-field autocorrelation $S^{(B)}(t) = \langle B^z(t)B^z(0)\rangle$ computed from the $N+1$ conserved quantities $I^z$ and $H_l^z = I^z H_l$ ($l = 1, \dots, N$). This includes settling the status, at finite $N$, of the two semiclassical assumptions in the derivation of the underlying relation $S_\infty = S_\infty^{(B)}/(12S^{(B)}(0))$: the separation between the fast central-spin precession and the slow Overhauser-field dynamics, and the (asymptotic-only) conservation of $|\vec{B}|^2$.

These two components jointly support one objective: replacing the numerically supported picture of persistent central-spin coherence by a rigorous quantitative theory of $S_\infty$. The source's outlooks — investigating extensions of the model (a finite magnetic field on the central spin, anisotropic hyperfine couplings, additional intra-bath couplings) in the same manner, and applying the generalized Mazur inequality in other contexts (for example, estimating Drude weights of other models) — lie outside this problem's completion boundary.

References

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