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arXiv:2602.14613

On the challenge of simulating dipolar contributions to spin relaxation with generalized cluster correlation expansion methods

Conor Ryan, Alessandro Lunghi

quant-phcond-mat.mtrl-sciSubmitted Feb 16, 2026• Announced Feb 17, 2026
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Abstract

The study of spin decoherence is often performed by assuming that spin-phonon interactions lead to relaxation at high temperatures, and spin-spin dipolar interactions instead contribute to pure dephasing at low temperatures. This has resulted in the neglect of spin relaxation due to spin-spin dipolar interactions and its influence on decoherence at low temperatures. For a complete understanding of low temperature spin dynamics, it is then imperative to focus also on the latter mechanism. One such method which has shown great promise in the efficient calculation of central spin dynamics due to spin-spin dipolar interactions with a surrounding spin bath is the Cluster-Correlation Expansion (CCE). An extension of this method through the explicit inclusion of the central spin degrees of freedom, known as the generalized Cluster-Correlation Expansion (gCCE) is capable of simulating the transfer of energy from the central spin into the bath, and thus could have the potential to investigate spin relaxation in this setting. In this work, we show that gCCE, in its standard form, is insufficient for providing even a qualitatively accurate description of spin-spin relaxation. A full mathematical deconstruction of the underlying theory of gCCE clearly points to the origin of such a breakdown and provides a starting point for its potential future resolution.

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The study of spin decoherence is often performed by assuming that spin-phonon interactions lead to relaxation at high temperatures, and spin-spin dipolar interactions instead contribute to pure dephasing at low temperatures. This has resulted in the neglect of spin relaxation due to spin-spin dipolar interactions and its influence on decoherence at low temperatures. For a complete understanding of low temperature spin dynamics, it is then imperative to focus also on the latter mechanism. One such method which has shown great promise in the efficient calculation of central spin dynamics due to spin-spin dipolar interactions with a surrounding spin bath is the Cluster-Correlation Expansion (CCE). An extension of this method through the explicit inclusion of the central spin degrees of freedom, known as the generalized Cluster-Correlation Expansion (gCCE) is capable of simulating the transfer of energy from the central spin into the bath, and thus could have the potential to investigate spin relaxation in this setting. In this work, we show that gCCE, in its standard form, is insufficient for providing even a qualitatively accurate description of spin-spin relaxation. A full mathematical deconstruction of the underlying theory of gCCE clearly points to the origin of such a breakdown and provides a starting point for its potential future resolution.
Decoherence of a center spin or qubit in a spin bath is essentially determined by the many-body bath evolution. We develop a cluster-correlation expansion (CCE) theory for the spin-bath dynamics relevant to the qubit decoherence problem. A cluster-correlation term is recursively defined as the evolution of a group of bath spins divided by the cluster correlations of all the subgroups. This correlation accounts for the authentic (nonfactorizable) collective excitations within a given group. The bath propagator is the product of all possible cluster correlation terms. For a finite-time evolution as in the qubit decoherence problem, a convergent result can be obtained by truncating the expansion up to a certain cluster size. The two-spin cluster truncation of the CCE corresponds to the pair-correlation approximation developed previously [W. Yao et al., Phys. Rev. B 74, 195301 (2006)]. In terms of the standard linked cluster expansion, a cluster-correlation term is the infinite summation of all the connected diagrams with all and only the spins in the group flip-flopped, and thus the expansion is exact whenever convergence occurs. When the individual contribution of each higher-order correlation term to the decoherence is small (while all the terms combined in product could still contribute substantially), as the usual case for relatively large baths, where the decoherence could complete well within the bath spin flip-flop time, the CCE coincides with the cluster expansion [W. M. Witzel and S. Das Sarma, Phys. Rev. B 74, 035322 (2006)]. For small baths, however, the qubit decoherence may not complete within the bath spin flip-flop time scale and thus individual higher-order cluster correlations could become significant. In such cases, only the CCE converges to the exact coherent dynamics of multispin clusters. We check the accuracy of the CCE in an exactly solvable spin-chain model.
The clock transitions (CTs) or optimal working points (OWPs) of central qubits have long coherence times because their frequency fluctuations vanish in the linear order of external field noise (such as Overhauser fields from nuclear spin baths). Therefore, CTs (or OWPs) are useful for quantum technologies. Also, the quadratic dependence of frequencies on noises makes the CT decoherence an interesting physics problem. Thus we are motivated to study the decoherence of CTs. We consider noise from spin baths, which is one of the most relevant mechanisms of qubit decoherence. Various quantum many-body methods have been developed to study the decoherence of a central spin in spin baths. In particular, the cluster-correlation expansion (CCE) systematically accounts for the many-body correlations that cause the central spin decoherence. However, the CCE cannot be straightforwardly applied to CTs in spin baths, for the expansion may fail to converge due to the effective long-range interactions resulting from the quadratic term of the noise (e.g., the second-order interaction mediated by hyperfine interactions for a nuclear spin bath). In this work, we develop a modified CCE method to tackle this class of decoherence problems. By diagonalizing the central spin Hamiltonian for each bath eigenstate of the hyperfine interaction, we find that the effects of long-range interactions are absorbed as fluctuations of central spin eigenenergies in the form of single-spin correlations. We apply the method to two important systems, namely, nitrogen vacancy center electron spins in near zero magnetic field and the singlet-triplet transition of two electrons in a double quantum dot. The numerical simulation shows that the modified CCE converges rapidly for the CTs. This modified cluster correlation expansion provides a nonperturbative approach to treating the central spin mediated long-range interactions in central spin decoherence problems.
Decoherence of a center spin or qubit in a spin bath is essentially determined by the many-body bath evolution. The bath dynamics can start either from a pure state or, more generally, from a statistical ensemble. In the preceding article [W. Yang and R. B. Liu, Phys. Rev. B 78, 085315 (2008)], we developed the cluster-correlation expansion (CCE) theory for the so-called single-sample bath dynamics initiated from a factorizable pure state. Here we present the ensemble CCE theory, which is based on similar ideas of the single-sample CCE. The bath evolution is factorized into the product of all possible cluster correlations, each of which accounts for the authentic (nonfactorizable) collective excitation of a group of bath spins and for the finite-time evolution in the qubit decoherence problem. Convergent results can be obtained by truncating the ensemble CCE by keeping cluster correlations up to a certain size. A difference between the ensemble CCE and single-sample CCE is that the mean-field treatment in the latter formalism of the diagonal part of the spin-spin interaction in the bath is not possible in the former case. The ensemble CCE can be applied to nonfactorizable initial states. The ensemble CCE is checked against the exact solution of an XYXYXY spin bath model. For small spin baths, it is shown that single-sample dynamics is sensitive to the sampling of the initial state from a thermal ensemble and hence very different from the ensemble average.
We propose a new approach to simulate the decoherence of a central spin coupled to an interacting dissipative spin bath with cluster-correlation expansion techniques. We benchmark the approach on generic 1D and 2D spin baths and find excellent agreement with numerically exact simulations. Our calculations show a complex interplay between dissipation and coherent spin exchange, leading to increased central spin coherence in the presence of fast dissipation. Finally, we model near-surface nitrogen-vacancy centers in diamond and show that accounting for bath dissipation is crucial to understanding their decoherence. Our method can be applied to a variety of systems and provides a powerful tool to investigate spin dynamics in dissipative environments.
We present a cluster expansion method for approximating quantum spin-bath dynamics in terms of a classical Gaussian stochastic process. The cluster expansion produces the two-point correlation function of the approximate classical bath, permitting rapid evaluation of noise-mitigating quantum control strategies without resorting to computationally intensive dynamical decoupling models. Our approximation is valid for the wide class of models possessing negligible back-action and nearly-Gaussian noise. We study several instances of the central spin decoherence problem in which the central spin and randomly-located bath spins are alike and dipolarly coupled. For various pulse sequences, we compare the coherence echo decay computed explicitly quantum mechanically versus those computed using our approximate classical model, and obtain agreement in most, but not all, cases. We demonstrate the utility of these classical noise models by efficiently searching for the 4-pulse sequences that maximally mitigate decoherence in each of these cases, a computationally expensive task in the explicit quantum model.
We simulate the coherence of two coupled electron spins interacting with a bath of nuclei using the generalized cluster correlation expansion method. An exchange interaction between the electrons facilitates a family of entangling gates that can be spoiled by nuclear-induced dephasing. Consequently, we study the dephasing of the coherent two-electron system by characterizing the T2 and T2* of the two-electron reduced density matrix for various system parameters in the range mimicking magnetic molecules, including magnetic field strength and orientation, exchange interaction strength, distance between the two spins, minimum distance between electron and nuclei and between nuclei, and nuclei density. We find the optimal regime for each parameter in which the coherence time is maximized and provide a physical understanding of it.