Persistent dark states in anisotropic central spin models

Long-lived dark states, in which an experimentally accessible qubit is not in thermal equilibrium with a surrounding spin bath, are pervasive in solid-state systems. We explain the ubiquity of dark states in a large class of inhomogeneous central spin models using the proximity to integrable lines with exact dark eigenstates. At numerically accessible sizes, dark states persist as eigenstates at large deviations from integrability, and the qubit retains memory of its initial polarization at long times. Although the eigenstates of the system are chaotic, exhibiting exponential sensitivity to small perturbations, they do not satisfy the eigenstate thermalization hypothesis. Rather, we predict long relaxation times that increase exponentially with system size. We propose that this intermediate chaotic but non-ergodic regime characterizes mesoscopic quantum dot and diamond defect systems, as we see no numerical tendency towards conventional thermalization with a finite relaxation time.

where α tunes the qubit-bath interaction anisotropy and g i describes the qubit-bath interaction strengths g i = g 0 (1 + γ δ i ). The total magnetization ∑ L−1 j=0 S z j commutes with H, giving rise to polarization sectors with definite total magnetization.
Supplementary Figure S1. Spectrum and central spin entanglement on and off resonance. Left panel plots the energy E as a function of the effective central fieldω 0 , for all eigenstates of H. Right panel shows the central spin entanglement for all eigenvalues on resonance (ω 0 = 0) and off resonance (ω 0 = 4). Vertical lines in the left panel denote the field values at resonance (gray dash-dotted line) and off resonance (blue dashed) used in the right panel. On resonance, dark and bright states can be easily distinguished by E and S 0 E , while off resonance these observables become comparable. Parameters: L = 11, N s = 1 (typical sample), α = 0.5, γ = 0.5, ∑ j S z j = −0.5. This model has a natural resonance point at which the effective z-field on the central spin and the surrounding bath spins are equal. At this point, the exchange interactions between the central spin and the bath are strongly enhanced. At α = 0, resonance occurs when ω 0 = ω. At finite α > 0 and in a fixed polarization sector, the last term in H shifts the resonance point is a constant for central spin 1/2. Collecting the terms in the Hamiltonian coupled to the central spin S z 0 yields the shifted resonance condition: Without loss of generality, we set ω = 0 throughout this work, such that the resonance condition is given byω 0 = 0 =⇒ ω 0 = −α g 0 ∑ L−1 j=0 S z j . The results shown in the main text focus on the physics of the system near resonance, where the difference between bright and dark states is most pronounced. This distinction is most clearly seen in the XX limit (α = 0), where dark states are product states |↓ 0 ⊗ |D − or |↑ 0 ⊗ |D + , whereas bright states have the form c 1 (ω 0 ) |↓ 0 ⊗ |B ↓ + c 2 (ω 0 ) |↑ 0 ⊗ |B ↑ , with nonzero c 1 and c 2 dependent on ω 0 . A thorough discussion of the spectrum in the XX limit is given in Ref. 1. Dark states are insensitive to changes in ω 0 . In contrast, bright states can be tuned to equal superpositions of the central spin up and down at resonance (ω 0 = 0), or configurations where the central spin is mostly polarized along either direction (as ω 0 → ∞, c 1 → 0, c 2 → 1 and as ω 0 → −∞, c 1 → 1, c 2 → 0). Thus the central spin can be essentially decoupled from the bath in bright states with strong off-resonant fields. Figure S1 shows the energy spectrum of H (left panel) across a range of shifted central field valuesω 0 , and the central spin entanglement entropy (right panel) forω 0 = 0 (squares) andω 0 = 4 (circles) -see vertical dash-dotted and dashed lines in the left panel respectively. We have fixed the total magnetization to be ∑ L−1 j=0 S z j = −0.5 < 0, such that dark states have S z 0 ≈ −0.5. In the spectrum, bright states come in pairs exhibiting level repulsion at resonance (see bands of red curves). Dark states show up as linear bands of near degenerate states (see black lines). Far from resonance, the central spin is nearly polarized in bright eigenstates, and has low entanglement entropy. The distinction between dark and bright states (as measured by observables such as E, S z 0 , and S 0 E ) thus becomes progressively less sharp away from resonance, and must be characterized by alternative means (e.g. by their sensitivity to ω 0 ).

Central spin projection: breakdown of perturbation theory.
In the main text, we established how perturbation theory captures the behavior of observables such as the central spin expectation value D(α, γ)|S z 0 |D(α, γ) , for a broad range of anisotropies α and small to moderate disorder γ. When γ 1.0, perturbation theory breaks down more rapidly as we tune α away from the α = 0 integrable line. This is shown in Fig. S2. Figure S2. Perturbation theory breaks down rapidly at large γ. Left plot shows the eigenstate expectation value of the central spin z-projection S z 0 for a typical sample of disorder with strength γ = 10.0. We see deviations from perturbation theory due to mixing between dark and bright states. The color coding used to separate dark and bright states is only nominal at sufficiently large α, as the states can no longer be precisely separated into two distinct clusters. Right plot shows the expectation [ S z 0 + 0.5] averaged over the N D eigenstates with smallest central spin projection, and N s = 500 disorder samples. The numerical data (markers) with γ = 1.0 and γ = 10.0 showcase the breakdown of perturbation theory for α 10 −4 (solid lines). Parameters L = 12, N s = 500 (right), ∑ j S z j = −1, ω = α. Locality of the adiabatic gauge potential. The adiabatic gauge potential (AGP) A α presented in the main text was used to develop a perturbation expansion (Section II), as well as establishing chaos (Section IV). The robustness of perturbation theory in our present context can be traced back to the locality of AGP; that is, A α is dominated by few-body terms at mesoscopic system sizes. In the main text, we presented the decomposition:

Supplementary
where σ λ j p i with λ j ∈ {x, y, z} denote the Pauli basis operators on site p i , where 0 ≤ p 1 < p 2 < . . . < p k ≤ L − 1 for every k = 1, . . . , L. In principle A α has contributions from operators with all possible supports. However, in Fig. S3, we show that A α for small α 1 has non-zero weight only for k-body operators with k = 3, 5, 7, . . . , and is dominated by 3-body terms.
Supplementary Figure S3. Locality of the A α . The vertical axis of the figure shows the sum of all squared-coefficients for operators with k-body terms (normalized by the trace norm squared of A α ). The horizontal axis gives the support (k). The AGP A α has contributions only from operators with odd support. It is dominated by 3-body terms, and exhibits a power law decay ∼ k −c . The exponent c ≈ 3 was found by linear regression on a log-log plot. Parameters: Inverse participation ratio for persistent dark states. In the main text, we characterized persistent dark states based on properties of the central spin in eigenstates. Persistent dark states can also be identified by their inverse participation ratio (IPR) relative to energy eigenbasis as α → 0: where {|n(0) } is the set of eigenstates (bright and dark) at α → 0, and |D(α) is any persistent dark state at α > 0. As α → 0, the persistent dark state coincides with a single unperturbed dark state; then IPR= 1 and the dark state can be thought of as being "localized" in the reference (α = 0) energy basis. We expect that the IPR decreases on increasing α from zero as the perturbed dark state has significant weight on an increasing number of unperturbed eigenstates. The IPR is bounded from below by the value 1/D, where D is the Hilbert space dimension of the appropriate polarization sector. A perturbed dark state can saturate this bound if it becomes an equal superposition of all reference states; then the dark state can be thought of as being fully "de-localized" in the reference basis. Figure S4 shows the behavior of the IPR for all persistent dark states over N s = 50 disorder realizations in a single magnetization sector ∑ j S z j = −1. The figure plots the quantity (1 − IPR) against α over two orders of magnitude. The dark unfilled circles show a distribution of IPR values for the different persistent dark states around their respective average values shown as red filled circles. Dark state IPRs collectively decrease with increasing α, with some persistent dark states approaching the bound 1 − IPR = 1 − 1/D (gold dashed line) at the highest α values. Note however that many persistent dark states are robust, showing little mixing and remaining highly localized in the reference basis even at the largest α. The average participation ratio is found to satisfy the scaling: in accordance to perturbation theory (see dotted grey lines with O(α 2 ) scaling for reference).

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Supplementary Figure S4. Dependence of the dark state inverse participation (IPR) ratio on the anisotropy parameter. Adiabatic gauge potential norm for variations in disorder strength. The adiabatic gauge potential which generates translations in γ-space is denoted by A γ . The behavior of A γ is analogous to A α , and can be used to study integrability-breaking perturbations, as well as the onset of chaos by tuning γ. Figure S5 shows the exponential divergence of the Frobenius norm of A γ as a function of system size L.
Supplementary Figure S5. Exponential divergence of the of disorder averaged norm A γ with system size. Close to the integrable point (γ = 0.01), the norm scales sub-exponentially. The curves for larger γ break off from the γ = 0.01 line at a critical size L * and subsequently grow exponentially with slope 2 ln(2), reflecting slow relaxation. Parameters: N s = 200, α = 0.5, ω = α, ∑ j S z j = −1, c ≈ 1.

Persistent dark states and adiabatic gauge potential norm scaling in different magnetization sectors.
In the main text, our numerical results focused on the magnetization sector ∑ j S z j = −1 with zero magnetization density lim L→∞ ∑ j S z j /L → 0. This sector is the largest one containing both dark and bright states (note the ∑ j S z j = 0 sector contains only bright states). In this section, we present analogous results (specifically analogs of Figures 3 and 8 in the main text) in a magnetization sector with non-zero density ∑ j S z j /L = −1/4. As the Hilbert space dimension of this sector is smaller than that of the | ∑ j S z j | = 1 sector, our numerical simulations probe larger system sizes L ≈ 20.  Figure S7 extends the results of Figure 8 of the main text to the ∑ j S z j = −L/4 sector. The figure shows the adiabatic gauge potential norm as a function of system size L for several values of the anisotropy parameter α. As in the main text, the gauge potential norm is a polynomial of L in accordance with an integrable perturbation of an integrable system up to a critical system size L * . For L > L * , the AGP norm grows exponentially with L:

Supplementary
The inset shows that the critical size L * is linearly dependent on log 2 (α), just as in the inset of Figure 8 in the main text. The slope ν ≈ 2.5 is however twice as large as the value in the ∑ j S z j = −1 sector. These results are consistent with an exponentially diverging relaxation time τ r ∼ C|α| 2ν 2 L . Therefore, the system size scaling of the gauge norm and the associated relaxation time continues to hold in magnetization sectors with non-zero density.

Resonant energy gap and dark-bright hybridization.
In the main text, we discussed various contributions to the AGP norm in the chaotic non-ergodic regime ( Figure 10). In particular, we found that the contribution from dark-bright mixing (DB) increased exponentially with L only at large α (α = 0.5). Here, we correlate this rise with a closing of a finite-size energy gap between the dark and bright manifolds in the spectrum. Figure S8 shows the distribution of dark and bright energies as a function of system size L, for α = 0.1 and 0.5. At α = 0.1 (left panel), the dark and bright manifolds are separated by an energy gap at the accessible system sizes, so that the dark and bright states only weakly hybridize. This explains the lack of exponential growth with L of the DB component of the AGP norm in the left panel of Figure 10 of the main text. On the other hand, at α = 0.5 (right panel), the dark and bright manifolds overlap in energy for L ≥ 14 and can strongly hybridize. This strong hybridization results in the exponential rise of the DB component of the AGP norm for L ≈ 12 in the right panel of Figure 10 in the main text.