## The dynamics of dipolar interactions

Well-controlled systems, such as cold atomic gases, can simulate more complicated materials. Applying this quantum simulation concept to the study of magnetism, Alvarez *et al.* add an interesting twist. Instead of cold atoms, a network of nuclear spins of hydrogen in polycrystalline adamantane serves as a simulator. Using nuclear magnetic resonance techniques, the authors could induce a transition from a state in which all spins coupled to each other to a state in which coherent spins grouped into clusters.

*Science*, this issue p. 846

## Abstract

Nonequilibrium dynamics of many-body systems are important in many scientific fields. Here, we report the experimental observation of a phase transition of the quantum coherent dynamics of a three-dimensional many-spin system with dipolar interactions. Using nuclear magnetic resonance (NMR) on a solid-state system of spins at room-temperature, we quench the interaction Hamiltonian to drive the evolution of the system. Depending on the quench strength, we then observe either localized or extended dynamics of the system coherence. We extract the critical exponents for the localized cluster size of correlated spins and diffusion coefficient around the phase transition separating the localized from the delocalized dynamical regime. These results show that NMR techniques are well suited to studying the nonequilibrium dynamics of complex many-body systems.

The complexity of many-body systems is a long-standing problem in physics (*1*–*4*). As an example, quantum states of many-body systems can be localized at well-defined positions in space or they can be delocalized, depending on parameters like disorder. In their localized regime, such systems may not reach a thermal state but retain information about their initial state on very long time scales (*5*–*10*). The role of the topology, dimension, long- and short-range interactions, and the presence of disorder is very important for the onset of these localization regimes. Much progress was achieved on the numerical and theoretical side, where these phenomena have been predicted under certain conditions. However, experimentally addressing three-dimensional (3D) many-body systems in a controlled manner poses severe experimental problems (*4*, *8*, *10*). The usual strategy to observe many-body phenomena is achieving very cold temperatures where sharp changes of the ground state as a function of a control parameter give evidence of quantum phase transitions (*11*–*13*). Alternatively, the nonequilibrium dynamics of many-body systems has been investigated to provide complementary information about a large variety of situations (*14*–*20*). This approach can even work at high or infinite temperatures; however, it is usually more challenging than monitoring static properties. The recent progress on the experimental control of cold atoms (*21*), trapped ions (*19*, *20*), Rydberg atoms (*22*), polar molecules (*23*), and nitrogen-vacancy centers in diamond (*24*) has led to promising new ways of studying the nonequilibrium dynamics and localization phenomena of many-body systems. In particular, a lot of effort is focused on studying many-spin systems with dipolar interactions of the Heisenberg type (*4*, *11**–**13*, *18**–**20*, *22*, *23*). Here, we use nuclear magnetic resonance (NMR), which provides a natural and versatile approach for coherently controlling large numbers of spins (up to ~7000) in solid-state systems, where dipolar interactions are present (*25*). NMR techniques allow to quantify the number of spins thatare coherently correlated and allow control of the interaction types and strengths of the Hamiltonians (*25*–*27*).

Our experimental system consists of the ^{1}H nuclear spins of polycrystalline adamantane (Fig. 1A, inset). All experiments were performed on a home-built solid-state NMR spectrometer in a magnetic field of 7 Tesla. The interaction of the proton spins with the static magnetic field results in a Zeeman splitting of MHz (in frequency units), which is identical for all spins. The mutual dipole-dipole interactions between the spins corresponds to a 3D spin-coupling network (Fig. 1). The dipolar interaction scales with and leads to a resonance line width of 7.9 kHz of the NMR spectrum due to the homogeneous broadening [see (*27*) for details of the sample]. The spin system is initially left to reach thermal equilibrium at room temperature. In this high-temperature limit, the two spin states are almost equally populated, with the excess of the lower energy state on the order of . Its density operator can then be described as (*26*, *28*), considering that the Zeeman interaction is much stronger than the dipolar one ( MHz kHz). We omit the unity operator since it does not evolve in time and does not contribute to the observable signal. Then is the total spin operator component in the direction of the magnetic field and that of the spin.

The Hamiltonian of the system in the interaction picture—i.e., in a reference frame rotating at the Zeeman frequency around the *z* axis—is the spin-spin interaction(1)This is the part of the dipolar interaction that commutes with the much stronger Zeeman Hamiltonian, the so-called secular term in NMR. The noncommuting terms can be neglected due to (*26*, *28*). The coupling constants are (2)with the gyromagnetic ratio, the angle between the internuclear vector and the magnetic field direction. This Heisenberg-type Hamiltonian is of growing interest in the context of quantum information and simulation science (*4*, *11*, *18**–**20*, *22*, *23*).

The initial condition corresponds to a thermal equilibrium with uncorrelated spins, and the density operator commutes with the system Hamiltonian (Fig. 1A). To generate spin clusters of correlated spins, we quench the system by suddenly changing its Hamiltonian to (3)which does not commute with the thermal equilibrium state (Fig. 1B). We use a method developed in (*29*, *30*) based on a sequence of -pulses that act equally on all spins in such a way that the time-averaged effect on a time scale much shorter than generates this effective Hamiltonian (*28*).

To study the effect of the quench and monitor the generation of clusters of correlated spins, we compare its evolution under a parametric set of Hamiltonians (4)These Hamiltonians are generated by concatenating short periods during which different Hamiltonians act on the system (*28*): the Hamiltonian for a duration and for a duration , where . The total evolution operator is then equivalent to the evolution under an effective, time-independent Hamiltonian during a time . This resulting cycle is then repeated *N* times, giving a total evolution time . The control parameter defines a perturbation to the quench strength. If , there is no quench, and defines the strength of the quench. The two Hamiltonians and have distinct symmetries with respect to the total magnetic quantum number , the eigenvalue of . Whereas the Hamiltonian simultaneously flips two spins and, accordingly, changes by (green arrows in Fig. 1C), mixes states that conserve (red arrows in Fig. 1C).

After the quench, the Hamiltonians (Eq. 4) generate correlations between the different spins. We measure the average number of correlated spins in the system (the cluster size) by decomposing the corresponding density operator according to its symmetry under rotations around the *z* axis, adapting a method from (*30*). The method is based on an interferometric detection of the overlap of the density operator with and without a rotation by an angle around the *z* axis. The resulting signal as a function of consists of the terms of the density operator with the additional phase where takes only even numbers. After a Fourier transform, one obtains the distribution of coherences generated by (nondiagonal terms in the eigenbasis of ) of the density matrix as a function of the difference between quantum numbers . We determine the average number of correlated spins *K* in the generated clusters from the width of these distributions (*25*, *27*, *28*). We associate them to an effective volume *l*^{3}, with *l* the effective correlation length (Fig. 1B). Figure 2A shows the extracted cluster size *K* as a function of the evolution time for different perturbation strengths on time scales much shorter than the time required for the system to thermalize with the lattice (). For the unperturbed evolution (black squares), the cluster size grows indefinitely within the time range measured before the experimental signal disappears due to decoherence processes (*25*, *27*, *28*). This changes qualitatively when the perturbation is turned on: The growth of the clusters generated by the perturbed Hamiltonian (colored symbols in Fig. 2A) does not continue indefinitely but saturates at a certain level, which we call the localization size. This localization size decreases with increasing perturbation strength .

To quantitatively analyze the transition from the delocalized to the localized dynamical regimes, we exploit the powerful finite-time scaling technique (*31*, *32*). This procedure allows one to extract from finite-time experimental data the localization length *l* at . Without perturbation, the cluster size is expected to grow with apower law in agreement with several experimental observations in solid-state spin networks (*33*). In our system, this growth law is observed for times ms and vanishing perturbation , where (*25*, *27*). Thus, , where *D* is a generalized diffusion coefficient and α is the exponent of the “diffusion” process (*33*, *34*). This time evolution behaves in a qualitatively different way if the system remains localized. In our case, the system undergoes a phase transition to a localized evolution if the perturbation strength exceeds the critical value . One expects then that the cluster-size evolution will depend on (*31*, *32*, *34*). We use the single-parameter Ansatz for the scaling behavior at long times (5)where is an arbitrary function and *k*_{1} and *k*_{2} are constant parameters. We assume that such that the diffusion coefficient vanishes, , at the onset of the localized regime for , with *s* as a critical exponent of the delocalized phase.

In the localized regime, we found experimentally that the localization cluster size is a power-law function of the perturbation strength *p* (*25*, *27*). Therefore, we assume that at long times for , as is typically assumed for localization phenomena; *v* is the critical exponent for the localized phase (*31*, *32*, *34*). We performed the finite-time scaling analysis for different relations between the two critical exponents—i.e., varying β in the relation —and found the universal scaling for shown in Fig. 2, B and C, and in fig. S4 (*28*).

The scaling factor that leads to the universal scaling behavior , with an arbitrary function, is shown in Fig. 3 as the blue triangles. The solid red line is a fit with the expression where *B* accounts for decoherence processes that smooth out the critical transition (*31*, *32*). We thus obtain a critical perturbation strength of and the critical exponents . This result is consistent with the scaling law Ansatz of Eq. 5. The insets in Fig. 3 show the probability distribution of coherences (nondiagonal terms in the eigenbasis of ) in the density matrix (*28*) as a function of the coherence order and the evolution time in both regimes. Although for a perturbation strength , the coherence distribution spreads indefinitely (delocalized regime), for the coherence distribution becomes localized after a given time.

From the power-law coefficient , experimentally determined in the unperturbed free diffusion regime, our analysis shows a critical behavior on the transition from the localized to the delocalized dynamical regime with critical exponents . This is consistent with Wegner’s scaling law for a 3D system () (*35*), in agreement with the assumption that the cluster-size *K* determines an effective volume occupied by the correlated spins and its respective effective correlation length, . Although a microscopic theory should be developed to confirm our findings, the present results represent strong evidence for a critical transition of the dynamic behavior of the coherence length of our system after the quench. This critical behavior is induced by competing dipole-dipole interactions in the many-body dynamics of the cluster of correlated spins.

## Supplementary Materials

www.sciencemag.org/content/349/6250/846/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S4

## References and Notes

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**Acknowledgments:**We thank J. Chabé, F. Hebert, and E. Altman for fruitful discussions. This work was supported by the Deutsche Forschungsgemeinschaft through Su 192/24-1. G.A.A. acknowledges the support of the Alexander von Humboldt Foundation and of the European Commission under the Marie Curie Intra-European Fellowship for Career Development grant PIEF-GA-2012-328605.