1. Magnetism via muons

Muons are unstable elementary particles produced in particle accelerators and used to study a variety of advanced materials. A commonly used approach proceeds by shooting muons (μ⁺) at a material, thus implanting them in its crystal structure until their eventual, random decay. Once inside the material, and before they disappear (which takes a few microseconds, on average), the intrinsic magnetic moments of muons (their spins) will start to rotate away from their known initial directions under the influence of local magnetic fields at their stopping sites. By carefully measuring the time-dependent angular distribution of the muon’s decay products (positrons, antimatter partners of electrons), which depends on the average directions of muons’ spins at the time they decayed, one can extract a wealth of information on the local magnetic fields inside the studied material (Video 1).

Video 1. An animation of the muon spectroscopy technique used to study magnetic materials. Reproduced with permission from the “Physics Reimagined” team, design and direction: Dafox.

This powerful experimental technique is called muon spectroscopy [1] and has, in recent years, become crucial in the study of some of the most intriguing physical systems. These include: (i) high-temperature superconductors [2], which also underlie recent successes in quantum computing [3], (ii) topological magnets with robust, knotted vortices of magnetization called skyrmions, which could be used for low-energy information storage and processing [4,5], and (iii) peculiar disordered and dynamical, yet highly quantum-entangled, magnetic states called quantum spin liquids [6,7], which could enable robust topological quantum computers of the future based on their exotic excitations (namely, quantum waves of magnetism with dual particle properties of anyons, particles that are neither the usual bosons nor fermions) [8,9].

The main advantages of muon spectroscopy over other methods of measuring the magnetic fields in a material are that muons are sensitive to very small magnetic fields (down to ~0.1 mT) and can determine the characteristic frequencies of the dynamics of these fields over a very broad and unique window (~100 kHz to ~1 THz), which is crucial in studies of quantum magnets. Muons also sense these fields locally, at a point (their stopping site) inside the crystal structure, i.e., at the atomic scale. This is to be contrasted against bulk magnetometry methods, which can only measure the average magnetization of the whole sample. The muon spectroscopy signal of a material can thus serve as a fingerprint of its magnetic state, making such measurements highly valuable — as long as they can be reliably interpreted.

2. Quantum trouble in paradise

Despite its many successes, muon spectroscopy suffers from a seemingly fatal flaw. Namely, even though muons can sense the local magnetic fields inside a material at the atomic scale, one does not a priori know where precisely inside the crystal structure a muon will stop, and thus at exactly which position it will measure the magnetic field. While this problem has usually been tackled via large-scale ab initio (DFT) simulations of muons in crystals under the classical, point-like particle approximation of muons and nuclei [1,10], real muons are not point-like. They — like all other constituents of nature — are inherently quantum objects; and this quantumness can matter.

The main quantum effects that a muon experiences are [1]: (i) quantum uncertainty in its position, (ii) quantum tunnelling, which enables it to randomly hop to positions that would be classically unreachable [11], and (iii) quantum entanglement, which manifests in “spooky” correlations between the position the muons and those of nearby nuclei that lie beyond any classical description [12]. The interpretation of muon spectroscopy measurements as measurements of local magnetic fields at a single point (the muon stopping site) is thus thrown into question. This is the infamous quantum muon problem, which we address in our paper [13].

Qualitatively similar quantum effects of nuclei are known to be responsible for stabilizing exotic phases of water ice (ice VII, where protons continuously quantum tunnel between two possible positions) [14,15] and lithium metal [16], modification of superconducting transition temperatures [17,18] and solvation energies of Li and F ions [19], and are crucial in record-density hydrogen storage materials [20]. However, the situation is even more extreme for muons, as their low mass (they are ~9× lighter than protons, the nuclei of the lightest element, hydrogen) substantially amplifies the strength of their quantum effects.

3. An alternative proposal

Previous attempts at tackling the quantum muon problem have mostly focused on three approaches: (i) directly applying the methods used to treat quantum nuclei (which, while accurate, are very computationally expensive and do not scale to the system sizes needed for quantum muons), (ii) using crude, simplified descriptions of quantum muons (which, while scalable, often neglect crucial quantum effects like tunnelling or entanglement, often with little justification), or — by far the most commonly — (iii) simply ignoring the problem altogether.

In our work [13], we propose a fourth approach: (iv) using carefully chosen simplified descriptions of quantum muons, adapted to the specific situation in a given material, with rigorous checks and quantification of any potential errors that might arise from using such simplifications. In this way, one can often attain almost the same accuracy as with more expensive methods of the previously mentioned approach (i) at a fraction of the computational cost, comparable to that of approach (ii). Thus, we combine the best of both worlds [ignoring approach (iii)], which should make an accurate description of quantum muon effects in many materials feasible for the first time. With such a description, a much more powerful, quantitative interpretation of muon spectroscopy data could be achieved, surpassing all previous attempts.

The practicalities of approach (iv) that we describe in our paper [13], are that we first carefully examine the expected regimes of applicability of the various simplified methods found in the literature via proxy observables that quantify the quantum regime of a muon. We identify two main observables for this purpose: (i) the degree of quantum muon entanglement, for the presence of which we establish two simple tests (so-called entanglement witnesses), and (ii) the degree to which the local potential energy of the muon differs from a simple harmonic oscillator. Secondly, we establish various ways of reliably estimating these observables at a very low computational overhead, without running full, expensive numerical simulations of quantum muon effects, both via direct approaches and through the use of new simplified toy models of quantum muons in crystals that we propose. This allows us to select the optimal, low-cost ab initio approximations of quantum muons in a given crystal, which still capture all the main quantum effects characteristic of the specific quantum regime of muons in this material (including quantum uncertainty, entanglement with nuclei, if present, etc.). Finally, we demonstrate practical ways of using the results of these optimal ab initio descriptions of quantum muons to calculate theoretical predictions of their muon spectroscopy signals and how to then use these to ultimately improve the interpretation of experimental data.

4. A proof of concept

Video 2. Crystal structure of solid nitrogen, α–N_2­­, without (left) and with an implanted muon (right) [13]. Red represents a positive induced charge on nitrogen, blue represents a negative induced charge, and the muon is shown in green.

We demonstrate the power of our approach by establishing a comprehensive ab initio description of quantum muons in solid nitrogen, α–N₂. We find that in this crystal muons form a peculiar [N₂–µ–N₂]⁺ complex with a large deformation of the crystal structure around it (Video 2), in which the muon and nearby nitrogen nuclei become highly quantum entangled (Fig. 1), and where the potential energy of the muon is highly non-harmonic [13]. To confirm these predictions, we also performed full simulations of quantum muons using more expensive numerical techniques and found that they, indeed, support these conclusions. We also calculated the expected muon spectroscopy signals both with (Fig. 2a,b) and without including quantum muon effects, and found that they differed significantly.

As the ultimate test of our new numerical predictions, we performed precise muon spectroscopy measurements on solid nitrogen crystals, which we synthesized in situ, under a range of applied magnetic fields. We found that the quantum effects of muons were indeed strong, as expected from our numerical predictions, and that they really were needed for a satisfactory fit to experimental data (Fig. 2c,d and Fig. 3). This confirmed the validity of our new theoretical approach.

While our original motivation was primarily to test our new theoretical framework, we found that the new approach worked so well, in fact, and that the fits of experimental data with quantum muon predictions were precise enough, that we could extract a new and independent estimate of an important material constant of solid nitrogen: its quadrupolar coupling constant (which is related to the strengths of electric fields inside the material) [13]. While central to solid nitrogen, the value of this constant was only known to a rather crude precision previously due to the inherent limitations of the experimental techniques used to determine it. The new results are a substantial improvement in this regard: a nearly three-fold reduction in experimental uncertainty, and the first improvement in nearly 50 years (Fig. 3) — a major achievement in precision muon spectroscopy. It serves to illustrate the quantitative potential of muon spectroscopy when supported by accurate modelling of quantum muon effects using advanced approaches like the ones we propose.

5. The road ahead

This work represents the first step towards a comprehensive resolution of the quantum muon problem in materials. While covering many cases of practical importance, it still has certain limitations that need to be resolved in the future. Primary among these is that it still neglects quantum tunnelling in many cases and instead mostly focuses on the situation where the muon has only one stable stopping site. In the limit of low quantum entanglement, it also neglects nuclear quantum effects, which could be important. Resolving these issues could finally realise the full quantitative potential of the muon spectroscopy technique in an even wider range of materials. This might also lead to novel insights into the problem of quantum nuclei in pristine systems, with wide-ranging implications.

The application of these quantum muon methods would be helped immensely by their implementation into user-friendly programs, as this would significantly lower the barrier-to-entry for muon spectroscopy users interested in quantitative interpretation of their experimental data. One candidate for such an implementation is our MuFinder program [10], which was originally designed to help with ab initio determination of classical muon stopping sites, and in which we are actively working on implementing the main methods presented in our paper [13] and beyond.

Our vision is nothing less than a new era of material science, where quantum effects of muons and nuclei are not neglected, but rather embraced, for their rich complexity and where they form an integral part of our understanding of condensed matter. Even though the road ahead is long, the present study represents a major, quantum leap ahead.

References

[1] Blundell, S. J., De Renzi, R., Lancaster, T. & Pratt, F. L. Muon Spectroscopy: An Introduction. (Oxford University Press, Oxford, 2021).

[2] Ghosh, S. K. et al. Recent progress on superconductors with time-reversal symmetry breaking. J. Phys.: Condens. Matter 33, 033001 (2021).

[3] Kjaergaard, M. et al. Superconducting Qubits: Current State of Play. Annu. Rev. Condens. Matter Phys. 11, 369–395 (2020).

[4] Lancaster, T. Skyrmions in magnetic materials. Contemp. Phys. 60, 246–261 (2019).

[5] Hicken, T. J. et al. Megahertz dynamics in skyrmion systems probed with muon-spin relaxation. Phys. Rev. B 103, 024428 (2021).

[6] Broholm, C. et al. Quantum spin liquids. Science 367, eaay0668 (2020).

[7] Gomilšek, M. et al. Kondo screening in a charge-insulating spinon metal. Nat. Phys. 15, 754–758 (2019).

[8] Kitaev, A. Anyons in an exactly solved model and beyond. Ann. Phys. 321, 2–111 (2006).

[9] Janša, N. et al. Observation of two types of fractional excitation in the Kitaev honeycomb magnet. Nat. Phys. 14, 786–790 (2018).

[10] Huddart, B. et al. Mufinder: A program to determine and analyse muon stopping sites. Comput. Phys. Commun. 280, 108488 (2022). You can download the program from: https://gitlab.com/BenHuddart/mufinder

[11] Storchak, V. G. & Prokof’ev, N. V. Quantum diffusion of muons and muonium atoms in solids. Rev. Mod. Phys. 70, 929 (1998).

[12] Work on quantum entanglement has recently been awarded the Nobel Prize in Physics 2022.

[13] Gomilšek, M. et al. Many-body quantum muon effects and quadrupolar coupling in solids. Commun. Phys. 6, 142 (2023).

[14] Benoit, M., Marx, D. & Parrinello, M. Tunnelling and zero-point motion in high-pressure ice. Nature 392, 258–261 (1998).

[15] Ceriotti, M. et al. Nuclear quantum effects in water and aqueous systems: Experiment, theory, and current challenges. Chem. Rev. 116, 7529–7550 (2016).

[16] Ackland, G. J. et al. Quantum and isotope effects in lithium metal. Science 356, 1254–1259 (2017).

[17] Errea, I. et al. Quantum hydrogen-bond symmetrization in the superconducting hydrogen sulfide system. Nature 532, 81–84 (2016).

[18] Errea, I. et al. Quantum crystal structure in the 250-kelvin superconducting lanthanum hydride. Nature 578, 66–69 (2020).

[19] Duignan, T. T., Baer, M. D., Schenter, G. K. & Mundy, C. J. Real single ion solvation free energies with quantum mechanical simulation. Chem. Sci. 8, 6131–6140 (2017).

[20] Ranieri, U. et al. Formation and stability of dense methane-hydrogen compounds. Phys. Rev. Lett. 128, 215702 (2022).