Abstract
Longlived dark states, in which an experimentally accessible qubit is not in thermal equilibrium with a surrounding spin bath, are pervasive in solidstate 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 nonergodic regime characterizes mesoscopic quantum dot and diamond defect systems, as we see no numerical tendency towards conventional thermalization with a finite relaxation time.
Introduction
Stateoftheart quantum technologies can control and coherently manipulate qubit systems with exquisite precision^{1,2,3,4}. The surrounding environment of the qubit however eventually decoheres the qubit and limits quantum applications^{5,6,7}. An efficient way of extending coherence times is to prepare the system in socalled dark states, in which the qubit is effectively decoupled from the bath^{8,9,10}. Dark states have been identified in several integrable central spin models^{8,10}, and are central to quantum computing^{11,12}, metrology^{13,14} and control^{15,16} applications in a variety of experimental qubit systems, including nitrogen vacancy centers in diamond^{17,18} and semiconducting quantum dots^{19,20}. A central goal of this work is to show that dark states can persist in experimentally relevant nonintegrable central spin models. At numerically accessible system sizes, they exist as exact eigenstates. In the thermodynamic limit, the qubit could eventually thermalize but only after long times.
Central spin systems are typically described by a spin1/2 Hamiltonian (\(\hbar =1\)):
where \(\omega _0\) is a local magnetic field on the central qubit, \(\omega \) is a uniform magnetic field on the bath spins, \(\alpha \) sets the anisotropy of the qubitbath interaction, and \(g_i\) sets the strength of the interaction between the central qubit and the ith bath spin for \(i = 1, 2, \ldots , L1\). Experimentally, the interaction strengths are inhomogeneous because of the randomness in the positions of the bath spins and/or the geometrical factors in dipolar interactions. For simplicity, we model the inhomogeneity as uncorrelated disorder, and take the \(g_i\) to be independently and identically distributed uniformly in the interval \([1\gamma , 1+\gamma ]\) with \(\gamma \) setting the disorder strength. Moreover, we study the model near resonance \(\omega _0 = \omega  \alpha \sum _{j=0} ^{L1} S_j^z\) where qubitbath interactions are enhanced. Since H conserves total spin magnetization \(\sum _{j=0}^{L1} S_j^z\), we set \(\omega = 0\) without loss of generality. Fig. 1 shows a schematic of the model.
The Hamiltonian H has three known integrable families. The first is the fully isotropic XXX model (\(\alpha = 1\) and arbitrary \(\gamma \)), which describes systems with contact interactions such as quantum dots in stype semiconductor bands^{19,20}. This model belongs to the class of integrable XXX RichardsonGaudin models^{21,22,23}. The second is the fully anisotropic XX model (\(\alpha = 0\) and arbitrary \(\gamma \)), which describes resonant exchange interactions in dipolar spin systems in rotating frames^{24,25,26,27}. It was only recently established that the XX model is integrable, arising as a singular limit of the class of hyperbolic XXZ RichardsonGaudin models^{10}. The third is the homogeneous XXZ model (\(\gamma = 0\) and arbitrary \(\alpha \)), which describes effective twobody interactions \(H = \omega _0 \,S_0^z + g\,(S_0^x\, S^x + S_0^y\, S^y + \alpha \,S_0^z\, S^z) \) between the qubit and the collective spin of the bath \(\mathbf {S} = \sum _{i=1}^{L1} \mathbf {S}_i\)^{28}. Fig. 2 shows the three integrable families in a broader phase diagram. The integrability of these models has enabled analytical and numerical studies of experimentally relevant systems using a variety of integrabilitybased techniques^{20,29,30,31,32,33,34,35,36,37}.
A remarkable feature of H is that it exhibits dark eigenstates of a particularly simple product form when either \( \alpha = 0 \,\) or \( \,\gamma = 0 \)^{10,28}. These product dark states \({{\mathcal {D}}\rangle }\) are states which exhibit no qubitbath entanglement. Namely, they have a product state structure \({{\mathcal {D}}\rangle }={{\downarrow }\rangle }_0 \otimes {{\mathcal {D}^}\rangle }\) or \({{\mathcal {D}}\rangle }={{\uparrow }\rangle }_0 \otimes {{\mathcal {D}^+}\rangle }\), in which the central spin is polarized along the zdirection, and the state of the bath satisfies
These states are furthermore independent of \(\omega _0\) (and \(\omega \)), making the qubit state insensitive to external axial fields, in addition to bath fluctuations. More generally, we define dark states as states in which the qubit is nearly zpolarized and is not in thermal equilibrium with the surrounding bath. Dark states allow the surrounding spin bath to be used as a robust quantum memory^{8,9,38,39}. Moreover, they pose limitations for dynamical nuclear polarization (DNP) experiments which attempt to polarize a mesoscopic bath by repeated qubit polarization and transfer^{28,40,41,42}. DNP protocols eventually prepare the system in a statistical mixture of dark states, producing effectively isolated qubits for decoherencefree quantum computation^{43}.
In this work, we first establish that dark states are robust to integrabilitybreaking perturbations that tune the anisotropy \(\alpha \) and the disorder strength \(\gamma \). Specifically, we show that dark states persist as exact eigenstates of the Hamiltonian, that only perturbatively mix with bright states (i.e. not dark states) over a broad range of values for \(\alpha \) and \(\gamma \), at system sizes amenable to numerical simulation. This perturbative mixing only slightly reduces the polarization of the qubit along the zdirection. Remarkably, while H is nonintegrable/chaotic away from its integrable lines, dark states are well protected due to the presence of quasiconserved charges.
To test the stability of dark states, we apply a recently developed exponentially sensitive probe for chaos, based on the scaling of the norm of the adiabatic gauge potential (AGP)^{44}. The AGP is defined as the operator which generates continuous adiabatic transformations between eigenstates and measures their sensitivity to perturbations of the underlying Hamiltonian^{45,46,47,48,49}. Its norm is closely related to the quantum geometric tensor and the fidelity susceptibility^{45,50,51}. The norm of the AGP was found to scale exponentially with system size for chaotic perturbations, in contrast to integrable perturbations leading to polynomial scaling^{44}.
In our present context, the AGP norm grows exponentially in accordance with quantum chaos , but interestingly, the growth rate of the logarithm of this norm is twice the rate expected for ergodic systems satisfying the eigenstate thermalization hypothesis . This rate saturates the upper bound for eigenstate sensitivity to perturbations^{44}. It reflects a very strong mixing between neighbouring eigenstates of the system and leads to ultraslow (exponentially long in system size) relaxation dynamics , reminiscent of the Arnold diffusion in classical nearintegrable systems^{52}. While a similar behavior of the AGP norm was found in previous work^{44} for spin chains with weak integrability breaking perturbations, here we find that this chaotic but nonergodic (CNE) regime extends to large perturbation strengths, even for the largest system size that we are able to simulate.
Results
Persistent dark states
Away from the integrable lines in Fig. 2, the eigenstates \({{n}\rangle }\) of H no longer admit exact product dark states. Nevertheless, we can identify persistent dark eigenstates with approximate product form using (i) the eigenstate expectation value of the central qubit zprojection \(\langle S_0^z \rangle \equiv \langle n  S_0^z  n \rangle \), or (ii) the eigenstate entanglement entropy \(\mathcal {S}_{E}^{0}\) of the qubit. The latter is defined as
where \(\rho _0\) is the reduced density matrix for the qubit obtained by tracing out the bath \(\mathcal {B}\) degrees of freedom, and \(\rho = {{n}\rangle } {\langle {n}}\) is the density matrix of eigenstate \({{n}\rangle }\) with energy \(E_n\).
Consider for reference the XX model (\(\alpha =0\)) at resonance (\(\omega _0 = 0\)) in a sector with negative net magnetization (\({\sum _{j=0}^{L1} S_j^z < 0}\))^{10}. For any product dark eigenstate \(\mathcal {D}\rangle \) of this model, \(\langle \mathcal {D}  S_0^z  \mathcal {D} \rangle = 1/2 \) and \(\mathcal {S}_E^0 = 0\). On the other hand the bright eigenstates \(\mathcal {B}\rangle \) of the model satisfy \(\langle \mathcal {B}  S_0^z  \mathcal {B} \rangle = 0\) and \(\mathcal {S}_E^0 = \ln (2)\) at resonance (far from resonance, the central spin is nearly polarized even in the bright states: \(\langle \mathcal {B}  S_0^z  \mathcal {B} \rangle \approx \pm 1/2 \) and \( \mathcal {S}_E^0 \approx 0\)).
For \(\alpha >0\), we find eigenstates \( \mathcal {D}(\alpha ) \rangle \) of H that are adiabatically connected to \( \mathcal {D}(0) \rangle \equiv  \mathcal {D} \rangle \) by a unitary transformation generated by the adiabatic gauge potential (AGP) \(\mathcal {A}_{\alpha }\):
In these eigenstates, both the zprojection and entanglement entropy of the qubit will deviate from their \(\alpha =0\) values. The question becomes whether these deviations are perturbatively small in \(\alpha \), and how this depends on the system size L. In chaotic systems, the AGP is generally a highly nonlocal manybody operator with an exponentially large norm^{44,45,53}. In the present context, the parameter \(\alpha \) breaks the integrability of the system (see section below on the chaotic but nonergodic regime). Naively, we expect qubit observables in \( \mathcal {D}(\alpha ) \rangle \) to perturbatively connect to their values in \( \mathcal {D}(0) \rangle \) only for \(\alpha \) that is exponentially small in the system size.
Remarkably, at numerically accessible system sizes, we find that \(\mathcal {A}_{0}\equiv \mathcal \,\mathcal {A}_{\alpha } (\alpha \rightarrow 0^{+})\) can be wellapproximated by fewbody operators and that perturbation theory works exceedingly well to characterize qubit observables in \(\mathcal {D}(\alpha )\rangle \). To illustrate, consider the perturbative expansion of the \(S_0^z\) expectation to leading order in \(\alpha \):
The leading term is of order \(\alpha ^2\), as the coefficient of the linear in \(\alpha \) term, \(\langle \mathcal {D}(0) i [\mathcal {A}_0,S_0^z] \mathcal {D}(0)\rangle = 0\), vanishes because \(S_0^z \mathcal {D}(0)\rangle = \pm 1/2 \mathcal {D}(0)\rangle \). Fig. 3 numerically demonstrates that the lefthand side \(\langle S_0^z \rangle + 0.5\) scales as \(\alpha ^2\) for a subset of the eigenstates over several orders of magnitude of \(\alpha \) and \(\gamma \).
In more detail, when \(\alpha >0\), the resonance condition in a given polarization sector is shifted by the mean anisotropy to \(\omega _0 =  \alpha \sum _{j=0}^{L1} S_j^z\) (see Supplemental Information). For concreteness, we focus on a single polarization sector \(\sum _{j=0}^{L1} S_j^z = 1\) such that the resonance occurs at \(\omega _0 = \alpha \); in the Supplemental Information, we generalize to other polarization sectors. Fig. 3(a) shows numerical computations of the expectation value \(\langle S_0^z \rangle + 0.5\) in every eigenstate of H at moderate disorder strength (\(\gamma =0.5\)) over several orders of magnitude in \(\alpha \). Persistent dark states (black/dark circles) are easily identifiable, as they connect smoothly to \(\langle S_0^z \rangle \rightarrow 0.5\) as \(\alpha \rightarrow 0\). The deviation from \(0.5\) scales as \(\sim \alpha ^2\), consistent with Eq. (5) (dotted lines). The bright states (red/light diamonds) are similarly perturbed around their value at resonance \(\langle S_0^z \rangle = 0\) (dashed red horizontal line). As \(\alpha \rightarrow 1\), dark and bright states attain comparable central spin projections. The inset of Fig. 3(a) shows the systemsize dependence of the averaged expectation value \([\overline{\langle S_0^z \rangle +0.5}]\), where \(\overline{\,\,.\,\,}\) denotes an average over all \(N_D\) persistent dark states, and \([\,\,.\,\,]\) denotes an average over \(N_{s}\) disorder samples. In eigenstates that satisfy the ETH, the expectation value \(\langle S_0^z\rangle _{\text {th}} = \sum _j S ^z_j/L\) approaches zero with increasing L, in a sector with fixed magnetization. However, we find that \(\langle S_0^z\rangle \) approaches its thermal value only very slowly with system size L, suggesting that dark state properties persist to system sizes much larger than we probe here. From the current analysis we cannot conclude whether or not they survive the thermodynamic limit.
Fig. 3(b) shows \([\overline{\langle S_0^z \rangle +0.5}]\) for varying disorder strengths \(\gamma \). The markers show numerical data and the solid lines show the analytic predictions given by Eq. (5) up to \(\mathcal {O}(\alpha ^2)\). Again, we find leading order perturbation theory to be in excellent agreement with numerical simulations for \(\gamma < 1\) and the entire \(\alpha \) range between the integrable points \(\alpha = 0\) and \(\alpha =1\). As \(\gamma \rightarrow 1\) perturbation theory begins to break down (see \(\gamma = 0.9\) line in plot). When \(\gamma \gtrsim 1\), perturbation theory breaks down much faster at \(\alpha \ll 1\) (see Supplemental Information).
Persistent dark states are well captured by perturbation theory due to the quasilocality of \(\mathcal {A}_{0}\) at numerically accessible system sizes. To see this, we decompose the AGP into kbody operators:
Here \(\sigma ^{\lambda _j}_{p_i}\) with \(\lambda _j \in \{x,y,z\}\) denote the Pauli basis operators on site \(p_i\), where \(0\le p_1< p_2< \ldots < p_k \le L1\) for every \(k = 1,\ldots ,L\). Throughout this work, we define the norm of any operator \(\Theta \) by its normalized Frobenius norm:
We find \(\mathcal {A}_{0}\) has nonzero weight only for kbody operators with \(k=3,5,7,\dots \). Moreover, the total weight of kbody operators decays as \(1/k^{c}\) for \(c>0\), so that \(\mathcal {A}_{0}\) is wellapproximated by 3body operators. In the Supplemental Information, we showcase the quasilocality of \(\mathcal {A}_{0}\) and estimate \(c\approx 3\).
One can similarly find persistent dark states based on translations in the \(\gamma \)parameter space, as in Eq. (4), with a different adiabatic gauge potential \(\mathcal {A}_{\gamma }\). We find perturbatively accessible persistent dark states away from the \(\gamma = 0\) line at numerically accessible system sizes. The inset of Fig. 3(b) shows the rescaled averaged expectation value \([\overline{\langle S_0^z \rangle + 0.5}]/\gamma ^2\) vs \(\alpha \) at resonance. The data collapse for \(\gamma =0.001, 0.01, 0.1\) shows that:
at small \(\gamma , \alpha \), in perfect agreement with the perturbative result. At larger values of \(\gamma \) (\(\gamma =0.9\)), we see deviations from the perturbative result as \(\alpha \rightarrow 1\). Persistent dark states were previously found by mapping exact product dark eigenstates from the homogeneous isotropic limit \((\alpha ,\gamma ) = (1,0)\) to the inhomogeneous isotropic regime \(\alpha =1, \gamma > 0\)^{8,41}. Our results perturbatively extend dark states into a broader region of parameter space at finite size (see Fig. 2).
Persistent dark states can also be identified by their low central spin entanglement entropy (see Fig. 4). For moderate to low disorder (upper panels with \(\gamma =0.75\)), low entanglement (dark) states persist and do not fully mix with high entanglement (bright) states, even as \(\alpha \rightarrow 1\). At sufficiently large disorder (lower panels with \(\gamma =10.0\)), the persistent dark state picture breaks down as \(\alpha \rightarrow 1\), since most states acquire large central spin entanglement \(S_E^0\approx \ln (2)\).
Quasiconserved operators
The question of whether and how systems thermalize is a fundamental one in quantum statistical mechanics. Steady states of integrable systems typically have nonthermal correlations due to the presence of extensively many conserved quantities, and are described by Generalized Gibbs Ensembles (GGEs) that account for these conserved quantities^{54,55,56}. Generic integrabilitybreaking perturbations usually yield Hamiltonians which are chaotic and satisfy the Eigenstate Thermalization Hypothesis (ETH)^{57,58}. Nevertheless, the integrable Hamiltonian can control the approach to a longlived prethermal state when the strength of the integrabilitybreaking perturbation is sufficiently small^{59}.
In this section, we establish that the central spin model in Eq. (1) has longlived nonthermal states controlled by the XX and XXZ integrable lines at accessible system sizes. Specifically, we show that H has approximate conservation laws that persist away from the integrable lines, giving rise to nonthermal correlations in local qubit observables. A simple way to detect the nonthermal correlations in quench experiments is through observables, such as \(\langle S_0^z \rangle \), that differentiate between dark and bright states. As \(\langle S_0^z \rangle \) takes nonthermal values (close to \(\pm 1/2\)) in the dark state manifold, we find quenched steady states that retain memory of the initial zpolarization of the central spin.
The integrable lines of the model (\(\alpha =0\), \(\alpha =1\), and \(\gamma =0\)) constitute families of RichardsonGaudin models with extensive numbers of bilinear twobody conserved charges \(Q_i\)^{10,60}. Upon breaking integrability, there no longer exists an extensive number of exactly conserved charges. Instead we find an extensive number of quasiconserved charges, which very nearly commute with H. To find such quasiconserved charges, we numerically construct an exhaustive set of fewbody operators \(Q_k\). These operators are conserved iff \(\Vert [H,Q_k]\Vert =0\). The quasiconserved charges are those operators \(Q_k\) with very small ratio: \(\Vert \,[H,Q_k]\,\Vert /\Vert Q_k\Vert \ll \Gamma _{\text {typ}}\), where \(\Gamma _{\text {typ}} \equiv \sqrt{\,\Vert H \Vert ^2/L\,}\) sets a typical energy scale.
We construct \(Q_k\) using the ansatz:
and restrict \(\{\theta _i\}\) to a complete set of m traceorthogonal one and twobody spin1/2 operators with unit norm. To avoid cumbersome notation, we leave the dependence of \(q_i\) and \(\theta _i\) on k implicit. We further set \(\Vert Q_k \Vert ^2 = \sum _j q_j^2 = 1\). To determine the coefficients \(q_j\), we solve the eigenvalue problem:
where \(M_{ij} \equiv \text {Tr}([H,\theta _i][H,\theta _j])/2^{L}\) and we take \(\Gamma >0\). The eigenvectors of M then yield through Eq. (9) a set of orthogonal and bilinear operators with known decay properties. Specifically, the eigenvalue \(\Gamma ^2\) equals the norm of the commutator:
To connect \(\Gamma \) to the operator decay rate, consider the shorttime expansion of the symmetrized unequal time correlator of \(Q_k\) at infinite temperature^{61,62}:
The correlator’s decay rate is thus given by \(\Gamma = \Vert [H,Q_k]\Vert \). Hence, if \(\Gamma =0\), the unequaltime correlator equals one for all t. If instead \(0<\Gamma \ll \Gamma _{\text {typ}}\), where \(\Gamma _{\text {typ}}\) sets a typical decay rate, then the correlator is close to one for a long time \(1/\Gamma \) and \(Q_k\) is approximately conserved up to this time.
Away from the integrable lines, we generally find three kinds of eigenvalues \(\Gamma ^2\): a few \(\mathcal {O}(1)\) zero eigenvalues, \(\mathcal {O}(L^2)\) large \(\Gamma ^2 \approx \Gamma ^2_{\text {typ}}\) eigenvalues, and an extensive number \(\mathcal {O}(L)\) of eigenvalues with \(\Gamma ^2 \ll \Gamma ^2_{\text {typ}}\). Zero eigenvalues correspond to exactly conserved charges related to known conservation laws, while the extensive number of small positive eigenvalues can be identified with quasiconserved charges.
Fig. 5 shows the smallest eigenvalues \(\Gamma ^2\) (rescaled by \(\Gamma _{\text {typ}}^2\)) obtained by numerically solving Eq. (10) as a function of the index k of the corresponding operator \(Q_k\). The right panel shows the eigenvalues at several system sizes (\(L=6,8,10\)) for a fixed disorder strength \(\gamma =0.1\). We find 4 eigenvalues which are zero within numerical accuracy (shaded gray region), corresponding to exactly conserved charges; namely H, \(\sum _{j} S_j^z\), \(H^2\), \((\sum _{j} S_j^z)^2\). We also see a cluster of intermediate eigenvalues corresponding to quasiconserved charges, which are separated by a gap from a set of larger eigenvalues (only a small fraction of this set is shown). The vertical dashed lines mark the indices of the quasiconserved operators \(Q_k\) with largest eigenvalue for each system size. These maximal indices increase in precise proportion to L, showing that the number of quasiconserved charges is extensive. Furthermore, the eigenvalues themselves remain of the same order of magnitude for the different system sizes, indicating that the operator lifetimes have no significant system size dependence up to \(L=10\).
The left panel of Fig. 5 shows the \((L+1)\) smallest eigenvalues of Eq. (10) for several values of disorder strength \(\gamma \) at \(L=10\). As expected, we find exactly \((L+1)\) conserved charges as we approach the integrable line \(\gamma = 0\) (see \(\gamma = 0.001\) data within gray region). On increasing \(\gamma \), only 4 charges remain exactly conserved, while the remaining \((L3)\) charges become quasiconserved. The lifetime (\(\propto 1/\Gamma \)) of the quasiconserved charges furthermore systematically decreases with increasing \(\gamma \). Previous studies have found similar longlived quasiconserved charges in a family of near(RichardsonGaudin)integrable spin models with alltoall interactions^{63}.
The extensively many twobody quasiconserved charges \(Q_k\) control longlived nonthermal states in quench experiments. The top panel of Fig. 6 shows the relaxation of \(\langle S_0^z(t) \rangle \) to a nonthermal value in a typical sample at moderate and large disorder strength. The system is initialized in a polarization sector \(\sum _{j=0}^{L1} S_j^z = + 1\) far from resonance (\(\omega _0 = 50\)) in the mixed state
with the bath spins at infinite temperature and the central spin maximally polarized along \(+z\). The top panel plots \(\langle S_0^z(t) \rangle \equiv {{\,\text{Tr}\,}}(\rho _i S_0^z(t))/{{\,\text{Tr}\,}}(\rho _i)\) following a quench to resonance (\(\omega _0 = \alpha \)). We observe a fast decay to a positive value that is different from the thermal value \(\langle S_0^z \rangle _{\text {th}}= \sum _j S_j^z /L = 1/L\). Thus, \(\langle S_0^z(\infty ) \rangle \) retains memory of its initial condition at these system sizes. This memory is a consequence of the weight of \(\rho _i\) on the persistent dark state manifold.
Two comments are in order. First, the hybridization between the dark and bright state manifolds increases with disorder strength. Consequently, at any given \(\alpha \), \(\langle S_0^z \rangle \) in the persistent dark state manifold decreases with increasing \(\gamma \) (see Fig. 3). This explains why \(\langle S_0^z (\infty ) \rangle \) decreases with increasing \(\gamma \) in Fig. 6. Next, the initial decay in the top panel of Fig. 6 is a consequence of dephasing between the perturbed bright states. As \(\langle S_0^z \rangle \approx 0\) in each perturbed bright eigenstate, the weight of \(\rho _i\) on the perturbed bright states does not contribute to the nonzero value of \(\langle S_0^z(\infty ) \rangle \).
The lower panel of Fig. 6 plots the rescaled and disorderaveraged long time value \([\langle S_0^z(\infty ) \rangle ]\) with L. The rescaling factor \(P_{D}\) is the expected polarization of the central spin due to the weight of \(\rho _i\) on the dark manifold,
where \(N^{\uparrow }_{D}\) and \(N^{\uparrow }_{B}\) are respectively the number of dark and bright states with the central spin pointing along \(+z\) in the appropriate polarization sector at large \(\omega _0\). On the integrable XX line at \(\alpha =0\), we expect that \([\langle S_0^z(\infty ) \rangle ] = P_{D}\). The blue (filled) curve in the lower panel of Fig. 6 is perturbatively accessible from the integrable line at the numerically accessible system sizes, and thus we find that \([\langle S_0^z(\infty ) \rangle ]/P_{D}\) is close to one. At larger disorder strength however, the hybridization between the dark and bright states increases with L at the accessible sizes. The long time value \([\langle S_0^z(\infty ) \rangle ]/P_D\) is thus smaller than the long time value at \(\alpha =0\), with the discrepancy growing with L (see orange curve with open markers). At large disorder strength, \([\langle S_0^z(\infty ) \rangle ]/P_D\) shows a trend toward the thermal value \(\langle S_0^z \rangle _{\text {th}}/P_D = 0.5\) with increasing L. This thermal value follows from \(\langle S_0^z \rangle _{\text {th}} = 1/L\) and \(P_D \sim 2/L\) as \(L\rightarrow \infty \) in the sector with \(\sum _j S_j^z= +1\) total magnetization; given a total magnetization density \(s\equiv \sum _j S_j^z/L\), \(\langle S_0^z \rangle _{\text {th}}/P_D \rightarrow \text {min}(0.5s, 0.5+s)\) as \(L \rightarrow \infty \). The ratio is less than one for all \(s \in (0.5,0.5)\). At the given system sizes, we cannot determine with certainty whether the qubit will saturate at or before it reaches its thermal value, even in the presence of strong disorder.
We conclude this section with two remarks. First, in addition to \(S_0^z\), generic twobody observables with significant overlap on the quasiconserved charges are expected to exhibit similar nonthermalizing behavior and nonthermal eigenstate expectation values. While it is possible that in the thermodynamic limit \(L\rightarrow \infty \) all such observables will thermalize, there is no indication that this will happen from available data at small or intermediate disorder. Even if it happens, the nonthermal state after the quench could crossover to an extremely longlived and very stable prethermal regime.
Chaotic but nonergodic regime
In previous sections, we established that dark eigenstates persist on adding putative integrabilitybreaking perturbations to Eq. (1) at numerically accessible system sizes. At these sizes, the model thus does not satisfy the ETH and fewbody observables do not thermalize in isolation. However, we expect the eigenstate and dynamical behavior to change with increasing L. In this section, we provide evidence that the model is in a chaotic nonergodic regime (CNE) characterized by an exponential sensitivity of eigenstates to small perturbations and the presence of relaxation times that are exponentially long in the system size.
Energy level statistics are a widely used tool to diagnose chaos and predict thermalization^{64,65,66}. Integrable systems generally follow Poisson level statistics, while chaotic systems exhibit WignerDyson statistics due to level repulsion in accordance with random matrix theory^{57}. The use of level statistics to diagnose chaos is limited to relatively small system sizes where exact diagonalization can be reasonably implemented. For our present model, level statistics show weaktonegligible level repulsion, thus proving insufficient to establish chaos.
Fig. 7 shows the distribution P(r) of the ratio r of consecutive energy level spacings in a sector with fixed polarization. The ratio \(r_n\) for the trio of energy levels with energies \(E_{n\pm 1}, E_n\) is defined as \(r_n = \min (s_n,s_{n1})/\max (s_n,s_{n1})\), where \(s_n = E_{n+1}  E_{n}\) and the energy levels are ordered \(E_1< E_2< E_3 < \ldots \)^{66,67}. The left panel shows that P(r) agrees with that expected for a Poisson spectrum near/on the integrable lines points with \(\alpha \approx 0\), \(\alpha = 1\), and \(\gamma \approx 0\). The center panel shows that the Poisson behavior persists in the presence of moderate anisotropy (\(\alpha =0.5\)) and disorder (\(\gamma = 0.5\)) at the largest size we access numerically. Only when the disorder strength is much larger than one (\(\gamma =10.0\)) do we see some level repulsion with a weak trend towards WignerDyson statistics with increasing L (right panel). Fig. 7 therefore shows no tendency of the model to become chaotic with increasing L at moderate values of \(\alpha \) and \(\gamma \).
Recently, Ref.^{44} proposed the norm of the AGP as a highly sensitive probe for chaos. Related measures were also proposed earlier in the context of manybody localization, see e.g. Refs.^{68,69}. Chaos manifests in the exponential scaling of the Frobenius norm of the AGP with system size, which can be interpreted as an exponential sensitivity of the eigenstates to perturbations of the Hamiltonian. In contrast, integrable perturbations show polynomial scaling^{44}.
For any Hamiltonian \(H(\alpha )\), the AGP operator can be represented as^{44,70}:
where \(\mathcal {M}_{\alpha } \equiv \sum _n n\rangle \langle n \partial _{\alpha } H n\rangle \langle n \). In the energy eigenbasis of \(H(\alpha )\), the offdiagonal matrix elements of \(\mathcal {A}_{\alpha }\) read:
where \(\Omega _{mn} = E_m  E_n\). Note that the diagonal matrix elements \(\langle m \mathcal {A}_{\alpha } m\rangle = 0\), which is a gauge choice^{45}. The (scaled) Frobenius norm is then given by:
In chaotic systems, \(\Vert \mathcal {A}_{\alpha } \Vert ^2\) fluctuates wildly with L when we take the limit \(\mu \rightarrow 0^{+}\) because the terms with the smallest energy differences \(\Omega _{mn}\) dominate the sum in the norm. This is a standard manifestation of the problem of small denominators^{71}. Instead of taking the limit \(\mu \rightarrow 0^+\), it is convenient to set \(\mu > 0\) as a regulator. This regulator provides a twofold advantage: (i) it suppresses the wild fluctuations of the norm with system size, and (ii) it allows us to retain the exponential sensitivity of the AGP norm to small perturbations if we pick \(\mu = L\, 2^{L}/c\), where c is a systemsizeindependent constant. The regulator \(\mu \) is thus parametrically larger than the level spacing, while maintaining a small deviation from the exact AGP^{44}. Physically, \(\mu ^{1}\) plays the role of a cutoff time for operator growth in Eq. (15). By picking this time to be exponentially large in L, we probe the sensitivity of eigenstates to infinitesimal perturbations.
For systems satisfying the ETH^{57} the states at energy density corresponding to infinite temperature satisfy
where \(R_{mn}\) is a random variable with zero mean and unit variance, and \(f(\Omega )\) is a smooth function that is proportional to the Fourier transform of the correlation function \({{\,\text{Tr}\,}}(\partial _{\alpha }H(t)\partial _{\alpha }H(0)+\partial _{\alpha }H(0)\partial _{\alpha }H(t))\) at the frequency \(\Omega \). In general, the function f also depends on the average energy \((E_m+E_n)/2\). However, as the summation in Eq. (17) is dominated by the eigenstates corresponding to infinite temperature, we suppress this additional dependence. Typically, \(f(\Omega )^2\) increases as \(\Omega \) decreases until \(1/\Omega \) becomes comparable to the slowest relaxation time scale in the system (such as the Thouless time) and saturates for smaller values of \(\Omega \).
Interestingly, it was recently observed that the function \(f(\Omega )^2\) can be defined and remains smooth even in generic integrable systems^{72}. Then \(f(\Omega )^2\) vanishes as \(\Omega \rightarrow 0\) for deformations of the Hamiltonian along integrable directions^{44,73,74,75}.
Combining Eq. (17) at finite \(\mu \) and Eq. (18) at exponentially small scales \(\Omega = \mu \sim L\,2^{L}\), we get the following estimate for the AGP norm:
Thus if our system were to satisfy ETH,
up to polynomial corrections. In contrast, we would expect only a polynomial scaling of \(\Vert \mathcal A_\alpha \Vert ^2\) with the system size in an (interacting) integrable system for deformations of the Hamiltonian along an integrable direction^{44}.
Figure 8 shows the exponential divergence of the disorderaveraged norm \([\Vert \mathcal {A}_{\alpha }\Vert ^2]\) of the AGP corresponding to perturbations of the anisotropy parameter \(\alpha \) at moderate disorder strength \(\gamma =0.5\) in the \(\sum _j S_j^z = 1\) polarization sector. At \(\alpha =0\), \([\Vert \mathcal {A}_{\alpha } \Vert ^2]\) scales polynomially with system size L. Away from the integrable point, the scaling of \([\Vert \mathcal {A}_{\alpha } \Vert ^2] \) is polynomial until a critical length \(L^*(\alpha )\), which marks the onset of exponential growth and thus chaos (see vertical dashdotted lines). The critical length increases with decreasing \(\alpha \) as:
such that \(L^{*}\) becomes infinite in the integrable limit \(\alpha \rightarrow 0\). The power \(\nu \approx 1.25\) is found using linear regression (see inset).
For L larger than the critical length \(L^*(\alpha )\), the norm of the AGP in Fig. 8 grows exponentially at twice the rate predicted by the ETH:
From Eq. (19), we obtain \(f(\mu )^2\sim 1/\mu \sim 2^L\). It follows from Eq. (18) that the offdiagonal matrix elements \(\langle m \partial _{\alpha } H n \rangle \) are not exponentially suppressed with system size in the narrow energy interval \(\Omega \sim \mu \sim 2^{L}\), in contrast with the ETH prediction. The absence of an exponential suppression in the offdiagonal matrix elements is shown in Fig. 9. The figure shows nearest neighbor (in energy) matrix elements \(\langle n \partial _{\alpha } H n + k \rangle  ^2\) for \(k\in \{1,2,3\}\), averaged over disorder samples and eigenstates \(n\rangle \); we denote this averaging by the doubleoverline \(\overline{\overline{[\cdots ]}}\). These matrix elements differ by several orders of magnitude between low (\(\gamma =0.5\)) and high (\(\gamma =10.0\)) disorder. However, no exponential decay with L is observed at either disorder strength at numerically accessible systems sizes. This absence of exponential suppression can only persist when polynomially many nearby eigenstates mix. In contrast, ETH would require that a given eigenstate mix equally with exponentially many nearby eigenstates upon perturbing the system.
The behavior of the chaotic nonergodic regime is manifest separately within the dark and bright manifolds, and jointly in the interactions between these manifolds. We plot the various contributions to the disorderaveraged AGP norm (AGP) in Eq. (17) from dark and bright classes of eigenstates in Fig. 10. The DD (BB) contribution comes from terms involving matrix elements between dark eigenstates (bright eigenstates in the same band; the bright states in the XX model come in LandauZener (LZ) pairs that can be continuously followed as a function of \(\omega _0\) in each magnetization sector. These bright states form two bands, consisting of the positive and negative energy states of each LZ pair, respectively. See the Supplemental Information). The matrix elements between the dark and bright states contribute the DB piece, while the matrix elements between the two bright state bands contribute the LandauZener (LZ) piece. At \(\alpha =0.1\) and \(\alpha =0.5\), the sum in the AGP norm is dominated by the intraband brightbright (BB) matrix elements. Consequently, the offdiagonal matrix elements between neighboring bright states are not exponentially suppressed (see the discussion below Eq. (22)). Darkdark contributions also exponentially increase with L; however their total value is many orders of magnitude smaller than the BB contribution at these sizes. The DB contributions show a striking difference between the left and right panels of Fig. 10 at the accessible sizes. The DB contributions only grow exponentially with L in the right panel with \(\alpha =0.5\), reflecting the strong hybridization between neighboring dark and bright states in the spectrum on perturbing \(\alpha \) (see Supplemental Information). For \(\alpha =0.1\), on the other hand, the dark and bright state manifolds are separated in energy at the accessible sizes. This limits the hybridization between the two manifolds. However, we expect that the DB contribution diverges exponentially with L at sufficiently large sizes at any \(\alpha >0\). We remind the reader that perturbation theory worked exceedingly well to characterize qubit observables in dark states, even in parameter regimes where the DB contribution exponentially increases with L. This suggests that the strong DB mixing should primarily affect bath observables in persistent dark states at these sizes. In the next section, we discuss the potential implications of this behavior in the thermodynamic limit.
In the Supplemental Information, we show that the analysis in this section extends to a polarization sector with nonextremal density \(0<\sum _j S_j^z/L < 1/2\),. In particular, we find no deviation from the exponential scaling in Eq. (22).
In sum, the exponential divergence of \([\Vert \mathcal {A}_{\alpha } \Vert ^2 ]\) provides strong evidence that arbitrarily small in \(\alpha \) perturbations are integrabilitybreaking, with a growth rate that is twice that predicted by the ETH at numerically accessible system sizes. In this chaotic nonergodic regime, eigenstates in exponentially small shells of order \(\mu \) hybridize, with interaction matrix elements that show no exponential suppression with system size.
Possible fates of the CNE regime in the thermodynamic limit
We have provided evidence that the family of central spin models in Eq. (1) is chaotic, but nonergodic at moderate disorder strengths and numerically accessible system sizes. Eigenstate chaos manifests in the exponential scaling of the AGP norm with L, while the nonergodicity is manifest in both the nonthermal value of the central spin polarization in persistent dark states and the nonETH scaling of the AGP norm.
Ref.^{44} argued that the nonETH scaling of the AGP norm in Eq. (22) indicates slow relaxation with exponentially long in L relaxation times. We repeat the argument for completeness. From Eq. (19) we find:
As \(\Omega \rightarrow 0\), \(f(\Omega )^2\) is proportional to the relaxation time \(\tau _r\) of the operator \(\partial _\alpha H=\sum _i g_i S_0^z S_i^z\). Using that \(\mu \propto L \,2^{L}\) and Eq. (21), we find that for \(L>L^*(\alpha )\):
where \(2\nu \approx 2.5\) and the constant C can have a weak (power law) dependence on L. As the DD, DB and BB components of the AGP norm exhibit nonETH scaling in the right panel of Fig. 10, we expect that Eq. (24) characterizes certain relaxation processes in both the dark and bright sectors.
In the dark eigensector, the exponentially long relaxation times largely arise from the darkbright mixing (cf. Fig. 10) and coexist with a robust nonthermal value of the central spin magnetization. This suggests the following ‘cartoon’ for the decomposition of the persistent dark states in the eigenbasis of the XX integrable model,
with a nonzero central spin residue \(Z \in (0,1]\). Above, we use the eigenbasis of the XX model at resonance (\(\omega _0 =0\)) in a magnetization sector with fixed positive value, \(\mathcal {D}^+\rangle \) is a dark states satisfying Eq. (2), and \(\tilde{\mathcal {B}}\rangle \) is a normalized superposition of bright states with the property \(\langle \mathcal {B}_iS_0^z\mathcal {B}_j\rangle = 0\). Note: In the XX model, \(S_0^z\) only connects pairs of bright states with equal and opposite energy^{10}. As the bright states hybridize in (exponentially small in L) energy shells due to the \(\alpha \)perturbation, it is plausible that \({{\tilde{B}}\rangle }\) statistically does not involve these pairs. From this property, we obtain:
In contrast, the central spin polarization equals \(\sum _j S_j^z/L\) in an infinite temperature eigenstate of the thermalizing system, which vanishes as \(L^{1}\) in a fixed magnetization sector.
Fig. 3 indicates that at intermediate disorder \(\gamma =0.5\), the residue Z is wellcaptured by perturbation theory with no noticeable system size dependence. For stronger disorder, we however find that Z slightly decreases with L (this can be inferred from the bottom panel of Fig. 6). This suggests that the (exponentially) long relaxation times are associated with slow dynamics of the bath spins adjusting to long time (likely nonthermal) configurations. Likewise, the exponential increase of the AGP norm in the dark sector is mostly due to their mixing with bright states (cf. Fig. 10), in turn implying that the \(\tilde{\mathcal B}\rangle \) part of the wave function in Eq. (25) is chaotic, i.e. exponentially sensitive to infinitesimal perturbations.
From the presented data, it is not possible to predict what happens as the system size L increases beyond \(L\approx 16\). We propose two possible distinct possibilities:

1.
KAMtype: the residue Z remains finite, the bath remains nonergodic, and \(\mathcal A_\alpha ^2\propto \exp [2\log (2) L]\) persists as \(L \rightarrow \infty \).

2.
Ergodic: the residue Z ultimately vanishes as \(L\rightarrow \infty \) and the AGP norm crossovers to the ETH scaling, \(\mathcal A_\alpha ^2\propto \exp [\log (2) L]\).
One can also imagine other more exotic scenarios, where, for example, the residue Z remains finite at \(L\rightarrow \infty \) while the bath becomes ergodic, or conversely, \(Z\rightarrow 0\) but the whole system remains nonergodic. We do not discuss these further as we see no indications that they could be realized.
Figure 11 schematically depicts the possible scenarios of (i) KAM and (ii) Ergodic behavior. The figure shows the system size dependence of the (log) relaxation time \(\tau _r\) (upper panel) and the disorder averaged central spin magnetization in the dark manifold (lower panel). At system sizes smaller than the critical size \(L^*(\alpha )\), we have a region where the dynamics of the system are dominated by the integrable lines and the system quickly relaxes to a nonthermal steady state, with at most polynomial dependence of \(\tau _r\) on L. In the chaotic nonergodic (CNE) regime, eigenstate mixing gives rise to an exponentially increasing relaxation time for the bath (cf. Eq. (24)), while persistent dark states maintain a nonthermal qubit polarization. As we approach the thermodynamic limit \(L\rightarrow \infty \), scenario (i) would result in a continuation of the CNE regime (see dotted curves), while (ii) would show a second critical size \(L_e(\alpha )\) marking the onset of ergodic dynamics. For \(L \gg L_e(\alpha )\), \(\tau _r\) saturates and the system always reaches local thermal equilibrium under its own isolated dynamics.
Both possibilities outlined above are very interesting and have nontrivial implications. If the KAM scenario (i) is realized, then there is a true nonergodic phase in the thermodynamic limit. In this case, both the dark and the bright sectors behave nonergodically at all system sizes, violating the ETH. They are characterized by bathspin relaxation times that are exponentially long in L. These violations will necessarily lead to a breakdown of various thermodynamic relations such as the fluctuationdissipation theorem, which heavily rely on ETH^{57}. If the Ergodic scenario (ii) is realized, then the system will eventually relax in a finite time to a thermal steady state with a thermal value of the centralspin magnetization. Even if this scenario is realized, according to our numerical results, this can only happen at extremely large system sizes (cf. the lower panel in Fig. 6). As the relaxation time \(\tau _r\) scales exponentially with L, it could be astronomically large at \(L \approx L_e\) before saturation. This suggests that the dark states, while not exact eigenstates in scenario (ii), will be extremely stable and long lived.
Our numerical results do not predict which of the two possibilities is realized. Based on the available data the KAM scenario (i) seems to be the most likely, at least for moderate \(\alpha \) and \(\gamma \), as there are no visible deviations from \(Z(L)=\text {const.}\) (Fig. 3) and \(\mathcal A_\alpha ^2\propto \exp [2\log (2) L]\) (Fig. 8). The absence of deviations from these scalings is especially remarkable since there are no small parameters in the system, so there is no obvious estimate for the length scale \(L_e\). Nevertheless, a more careful analysis is needed to reach a definite conclusion.
Discussion
Physically, dark states can be realized in several qubit systems with mesoscopic environments. For example, in diamond systems, a nitrogen vacancy (NV) center serves as a qubit and the electronic spins on the surface act as a bath. In a suitable rotating frame, the Hamiltonian is well approximated by the XX central spin model. Furthermore, as the qubitbath interactions are dipolar, they decay sufficiently rapidly as the distance between the NV and a surface spin grows that only a handful of surface spins can be experimentally accessed^{17,18}. Our results imply that such small NVsurface spin systems exhibit dark states that are robust to the presence of moderate anisotropy and disorder.
One potential avenue for applications involves quantum information processing in the manifold of persistent dark states. To initialize the system in the persistent dark state manifold, one can implement dynamic nuclear polarization (DNP) to repeatedly polarize the central spin and transfer its polarization to the bath. DNP works by harnessing the flipflop interactions (\(S_0^{+}S_j ^{}+S_0^{}S_j ^{+}\)) already present in the Hamiltonian H. This transfer can be achieved with several methods, e.g. tuning external fields to resonant HartmannHahn conditions where flipflop interactions dominate^{24,25}. By Eq. (2), dark states are unaffected by flipflop interactions, and therefore only bright state populations will be continually transferred to dark states and other bright states. Repeating the process, the system tends to a statistical mixture of persistent dark states^{41,76}. The low qubitbath entanglement of the resulting manifold ensures robust qubit states with large decoherence times for highfidelity quantum computing.
A closely related application of DNP is to fully polarize a mesoscopic bath. It has long been known that DNP protocols populate dark states in which the bath is only partially polarized, preventing complete bath polarization and severely limiting this goal^{77}. Methods to overcome these limitations have been proposed^{28,40}. Our results extend these limitations to mesoscopic central spin systems with moderate anisotropy and disorder. A promising avenue for future research would be to characterize experimentally relevant integrabilitybreaking perturbations which destroy mesoscopic persistent dark states.
Our results also extend the class of mesoscopic systems relevant for applications to quantum memory. Dark states have been proposed for the storage and retrieval of qubit states^{8,38}. In one scheme, the qubit is initialized in an arbitrary state, which can be expressed as a superposition of bright and dark states. By controlling the external field which does not couple to dark states, the information about the qubit state can be completely transferred to the surrounding bath state and retrieved at a later time^{8}. The scheme immediately generalizes to persistent dark states with moderate anisotropy and disorder, opening avenues for quantum memory in new systems.
To conclude, we investigated the robustness of dark states in a family of central spin models with anisotropic and inhomogenous qubitbath interactions. The model is integrable along three lines in parameter space, two of which exhibit exact product dark eigenstates in which the central spin is unentangled with its environment. At moderate deviations away from these exact lines, we found persistent dark states whose central spin polarization and entanglement entropy are welldescribed by perturbation theory at numerically accessible system sizes. We furthermore showed that the extensive set of conserved operators at the integrable lines morph into an extensive set of quasiconserved operators away from the integrable lines. In quench experiments, these quasiconserved operators result in nonthermal correlations in a longlived nonthermal state. To address the possibility of chaotic behavior at larger system sizes than numerically accessible, we investigated the scaling behavior of the norm of the generator of adiabatic deformations of eigenstates with system size. Although the scaling predicts the onset of chaos at any nonzero strength of the (integrabilitybreaking) perturbation, the ETH is not obeyed and the relaxation time of the system diverges exponentially with system size. While these effects may disappear in thermodynamically large systems, we see no evidence for this at the numericallyaccessible system sizes.
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Acknowledgements
The authors thank D. Sels, C.R. Laumann and A. Sushkov for insightful discussions and collaborations on related topics, and M. Rigol for useful feedback on the manuscript. The authors acknowledge support from the Sloan Foundation through a Sloan Research Fellowship (A.C.), from the Belgian American Educational Foundation (BAEF) through the Francqui Foundation Fellowship (P.W.C.), from the Banco Santander Boston UniversityNational University of Singapore grant (M.P.), and from the BU CMT Visitor Program (P.W.C.). Numerics were performed on the BU Shared Computing Cluster with the support of the BU Research Computing Services. This work was supported by EPSRC Grant No. EP/P034616/1 (P.W.C.), NSF DMR1813499 (T.V. and A.P.) and NSF DMR1752759 (T.V. and A.C.), and AFOSR FA955016 10334 (A.P.).
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T.V. and P.W.C. led the project and conducted numerical simulations and analyses. All authors discussed the results and cowrote the manuscript.
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Villazon, T., Claeys, P.W., Pandey, M. et al. Persistent dark states in anisotropic central spin models. Sci Rep 10, 16080 (2020). https://doi.org/10.1038/s41598020730151
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DOI: https://doi.org/10.1038/s41598020730151
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