Localization-delocalization transition in the dynamics of dipolar-coupled nuclear spins

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Science  21 Aug 2015:
Vol. 349, Issue 6250, pp. 846-848
DOI: 10.1126/science.1261160

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


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 (14). 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 (510). 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 (1113). Alternatively, the nonequilibrium dynamics of many-body systems has been investigated to provide complementary information about a large variety of situations (1420). 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, 1113, 1820, 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 (2527).

Our experimental system consists of the 1H 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 Embedded Image with the static magnetic field results in a Zeeman splitting of Embedded Image 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 Embedded Image 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 Embedded Image are almost equally populated, with the excess of the lower energy state on the order of Embedded Image. Its density operator can then be described as Embedded Image (26, 28), considering that the Zeeman interaction is much stronger than the dipolar one (Embedded Image MHz Embedded Image kHz). We omit the unity operator Embedded Image since it does not evolve in time and does not contribute to the observable signal. Then Embedded Image is the total spin operator component in the direction of the magnetic field and Embedded Image that of the Embedded Image spin.

Fig. 1 Quantum evolutions and Hamiltonian characteristics.

(A) Thermal equilibrium of the proton spins in the presence of a static magnetic field at time t = 0 just before the quench. The spins are uncorrelated, the density operator is Embedded Image, where Embedded Image is the total spin magnetization operator and Embedded Image the single spin operators. The red spin in the center represents an uncorrelated spin state Embedded Image of the spin ensemble. It thus represents a cluster of correlated spins with size K = 1. (Inset) Adamantane molecule with 16 protons (small gray spheres). The large green spheres represent carbon atoms, consisting of 99% Embedded ImageC and 1% Embedded ImageC . (B) Cluster of correlated spins at time Embedded Image after the quench with Embedded Image (red spins). The cluster consisting of K > 1 correlated spins occupies a volume Embedded Image, where l is the effective coherence length. (C) Evolution of a system of K spins in the Zeeman product basis Embedded Image (Embedded Image) (black solid lines), where Embedded Image. The green arrows represent the Embedded Image interactions, which simultaneously flip two spins, causing Embedded Image to change by Embedded Image. The red arrow represents the Embedded Image interactions that conserve the quantum number Embedded Image.

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 interactionEmbedded Image(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 Embedded Image (26, 28). The coupling constants areEmbedded Image (2)with Embedded Image the gyromagnetic ratio, Embedded Image the angle between the internuclear vector Embedded Image and the magnetic field direction. This Heisenberg-type Hamiltonian is of growing interest in the context of quantum information and simulation science (4, 11, 1820, 22, 23).

The initial condition corresponds to a thermal equilibrium with uncorrelated spins, and the density operator Embedded Image commutes with the system Hamiltonian Embedded Image (Fig. 1A). To generate spin clusters of correlated spins, we quench the system by suddenly changing its Hamiltonian toEmbedded Image (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 Embedded Image-pulses that act equally on all spins in such a way that the time-averaged effect on a time scale much shorter than Embedded Image 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 HamiltoniansEmbedded Image (4)These Hamiltonians are generated by concatenating short periods during which different Hamiltonians act on the system (28): the Hamiltonian Embedded Image for a duration Embedded Image and Embedded Image for a duration Embedded Image, where Embedded Image. The total evolution operator is then equivalent to the evolution under an effective, time-independent Hamiltonian Embedded Image during a time Embedded Image. This resulting cycle is then repeated N times, giving a total evolution time Embedded Image. The control parameter Embedded Image defines a perturbation to the quench strength. If Embedded Image, there is no quench, and Embedded Image defines the strength of the quench. The two Hamiltonians Embedded Image and Embedded Image have distinct symmetries with respect to the total magnetic quantum number Embedded Image, the eigenvalue of Embedded Image. Whereas the Hamiltonian Embedded Image simultaneously flips two spins and, accordingly, changes Embedded Image by Embedded Image (green arrows in Fig. 1C), Embedded Image mixes states that conserve Embedded Image (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 Embedded Image around the z axis. The resulting signal as a function of Embedded Image consists of the terms of the density operator with the additional phase Embedded Image where Embedded Image takes only even numbers. After a Fourier transform, one obtains the distribution of coherences generated by Embedded Image (nondiagonal terms in the eigenbasis of Embedded Image) of the density matrix as a function of the difference between quantum numbers Embedded Image. 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 l3, with l the effective correlation length (Fig. 1B). Figure 2A shows the extracted cluster size K as a function of the evolution time Embedded Image for different perturbation strengths on time scales much shorter than the time required for the system to thermalize with the lattice (Embedded Image). 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 Embedded Image.

Fig. 2 Time evolution of the cluster size K for different perturbation strengths and finite-time scaling procedure.

(A) Cluster-size K as a function of the time t after the quench. The unperturbed quenched evolution (black squares) shows a cluster-size K that grows as Embedded Image at long times (dashed line is a guide to the eye). The solid symbols show the points used for the finite-time scaling analysis, while the empty symbols do not belong to the long time regime (Embedded Image ms). For the largest perturbation strengths, localization effects are clearly visible by the saturation of the cluster size. (B and C) In these two panels, we present the finite-time scaling procedure. In (B), the rescaled and squared correlation length Embedded Image as a function of the evolution time Embedded Image is plotted, where Embedded Image and Embedded Image were determined using the experimentally measured Embedded Image and the relations Embedded Image, Embedded Image that hold when Embedded Image (28). In (C), the curves of (B) are rescaled horizontally by the factor Embedded Image to obtain a universal scaling law.

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 Embedded Image. 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 Embedded Image ms and vanishing perturbation Embedded Image, where Embedded Image (25, 27). Thus, Embedded Image, 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 Embedded Image. One expects then that the cluster-size evolution will depend on Embedded Image (31, 32, 34). We use the single-parameter Ansatz for the scaling behavior at long timesEmbedded Image (5)where Embedded Image is an arbitrary function and k1 and k2 are constant parameters. We assume that Embedded Image such that the diffusion coefficient vanishes, Embedded Image, at the onset of the localized regime for Embedded Image, 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 Embedded Image for Embedded Image, 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 Embedded Image—and found the universal scaling for Embedded Image shown in Fig. 2, B and C, and in fig. S4 (28).

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

Fig. 3 Scaling factor and critical exponents.

Normalized scaling factor Embedded Image as a function of Embedded Image (blue triangles). The normalization is based on equalizing Embedded Image (28). The red solid line is a fit to the blue triangles with the expression Embedded Image, where Embedded Image, Embedded Image, the critical exponent Embedded Image, and the critical perturbation Embedded Image. From Fig. 2 we determined that Embedded Image. The two insets show the distribution of coherence orders of the density matrix as a function of the evolution time t for the perturbation strengths Embedded Image and Embedded Image, which correspond to the delocalized and localized regime, respectively. The corresponding scaling factors are indicated by the arrows.

From the power-law coefficient Embedded Image, 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 Embedded Image. This is consistent with Wegner’s scaling law Embedded Image for a 3D system (Embedded Image) (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, Embedded Image. 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

Materials and Methods

Supplementary Text

Figs. S1 to S4

References and Notes

  1. Supplementary materials are available on Science Online.
  2. 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.
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