Evidence for a three-dimensional quantum spin liquid in PbCuTe2O6

The quantum spin liquid is a highly entangled magnetic state characterized by the absence of static magnetism in its ground state. Instead, the spins fluctuate in a highly correlated way down to the lowest temperatures. Quantum spin liquids are very rare and are confined to a few specific cases where the interactions between the magnetic ions cannot be simultaneously satisfied (known as frustration). Lattices with magnetic ions in triangular or tetrahedral arrangements, which interact via isotropic antiferromagnetic interactions, can generate such a frustration. Three-dimensional isotropic spin liquids have mostly been sought in materials where the magnetic ions form pyrochlore or hyperkagome lattices. Here we present a three-dimensional lattice called the hyper-hyperkagome that enables spin liquid behaviour and manifests in the compound PbCuTe2O6. Using a combination of experiment and theory, we show that this system exhibits signs of being a quantum spin liquid with no detectable static magnetism together with the presence of diffuse continua in the magnetic spectrum suggestive of fractional spinon excitations.

In order to verify the isotropic nature of the Cu 2+ moments in PbCuTe 2 O 6 , static (DC) susceptibility measurements on single crystal samples have been compared for three inequivalent magnetic field directions within the cubic system. For this purpose, the zero-field cooled moment was measured under a H = 1 T field applied parallel to the [1, 0, 0], [1,1,0] and [1,1,1] crystallographic directions for temperatures in the range T = 2 K to 350 K. Supplementary Figure1 shows that the susceptibility displays no indication of any phase transitions down to 2 K. Furthermore the susceptibility is almost completely independent of the field direction. Therefore, we can conclude that the Cu 2+ spin- 1 2 moments in PbCuTe 2 O 6 are isotropic in nature.

Supplementary Note 2: Muon spin relaxation spectroscopy
µSR is an invaluable technique to probe the magnetic ground state as it is extremely sensitive to the local magnetic environment. Supplementary Figure 2 shows the zero field muon depolarization spectra of PbCuTe 2 O 6 from 5 K down to 0.02 K. A classic evidence of long-range order is revealed when spontaneous oscillations appear in the muon decay spectrum due to the precession of muon around an internal Supplementary Figure 2: The zero-field µSR spectroscopy of PbCuTe 2 O 6 . The µSR depolarization spectra as a function of temperature measured in zero field along with statistical error bars. At all the temperatures, the spectra decays to reach the background level at longer time scales. At base temperature, this is clearly observed already for timescales above 2 µS pointing to the purely dynamic nature of the ground state in PbCuTe 2 O 6 .

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x/a y/b z/c Te 0.3379(7) 0.9213 (5)  magnetic field resulting from the magnetic long-range order at the time of decay. In powder samples, static magnetism also leaves nonzero polarization at long time scales due to spherical symmetry of the internal fields so that 1 3 rd of the fields are directed along the initial muon polarization. However, neither of these features are visible in case of PbCuTe 2 O 6 indicating the absence of long-range magnetic order or spin glass freezing. Nevertheless there is evidence for enhanced magnetic correlations as the relaxation rate of the muon spin polarization, directly related to the spin dynamics, slows down rapidly as the sample is cooled below 1K and becomes constant below 0.6 K. This suggests persistent spin dynamics in the ground state of PbCuTe 2 O 6 . Thus, the combination of neutron diffraction and µSR confirm the absence of static magnetic order in PbCuTe 2 O 6 down to 0.02 K and the purely dynamic nature of the ground state is established.
Supplementary Note 3: Single crystal growth and structure refinement sample measured with a photons of wavelength 0.496Å. The pattern is refined with nuclear structure of P 4 1 32 space group and lattice constant a = 12.4968(4)Åand Supplementary Table 1 lists the atomic coordinates.
The powder was then pulverized and packed into a cylindrical rubber tube and pressed hydrostatically up to 2000 bars in a cold-isostatic-pressure (CIP) machine. The pressed rod was then hung in the vacuum furnace and sintered in flowing argon for 12 hours at 550 • C to form a dense cylindrical rod with diameter about 0.6 cm and length 7 − 8 cm. Crystal growth was carried out in an optical image furnace (Crystal Systems Corp., FZ-T 10000-H-VI-VPO) equipped with four 150 W Tungsten halide lamps focused by ellipsoidal mirrors. Crystal growths were performed in under 0.1 MPa Argon atmosphere at a growth rate of 1.0 mm per hour. Single crystallinity of the as-grown crystal was checked by X-ray Laue and polarized optical microscopy. A piece of the crystal was ground and checked with x-ray powder diffraction (Brucker D8) for phase purity. We have grown several batches of PbCuTe 2 O 6 single crystals, all of which consistently reveal 5 − 10% of a non-magnetic impurity Pb 2 Te 3 O 8 . Supplementary Figure. 3b shows the diffraction pattern on a finely ground single crystal sample measured with synchrotron radiation similar to the ceramic powder where, the refinement (lattice parameter a = 12.4967(8)Åfor PbCuTe 2 O 6 ) includes the impurity phase Pb 2 Cu 3 O 8 in profile-matching mode. Although this impurity is also present in the as-grown polycrystalline sample, it is always less than 5%.

Supplementary Note 4: Low temperature characterization of single crystal
Heat capacity, µSR studies in Ref. [1,2] and our neutron diffraction ( Figure.2 of the main text) investigation on the powder samples confirm the absence of magnetic phase transitions in PbCuTe 2 O 6 down to 0.02 K. In this work, we have carried out similar investigation on the single crystals including heat capacity, thermal expansion and thermal conductivity to probe the low temperature properties. Results of these studies will be discussed in more detail in another publication.
In addition, to confirm the absence of magnetic long-range order in the single crystal of PbCuTe 2 O 6 , we have performed neutron diffraction on the same single crystal that was used for measuring the inelastic scattering. The diffraction patterns were measured on the WISH diffractometer (white neutron beam) at temperatures T < 0.1 K and at 2 K for three orientations of the crystal while focusing the flux on two types of peaks where magnetic order could be expected: 1) equivalent positions [0.3, 0.3, 1.69] where inelastic spectrum shows the intensity maxima (Fig. 3e & Fig. 4a 003) show an increment in the intensity at low temperature 0.1 K, as seen in the inset of Supplementary Fig. 4b. However, we should note that these peaks are forbidden due to the lattice symmetry (P 4 1 32) and are also present at 2 K due to the large size of the single crystal resulting in multiple scattering. Nevertheless, these reflections are weak compared to the allowed Bragg reflections. Additionally, the increased intensities do not decay as the square of the momentum transfer (in which case, the highest magnetic peak should be (0, 0, 1)) and hence these peaks cannot be attributed to magnetic order in the system. Therefore, we can exclude the possibility of magnetic order in the single crystal of PbCuTe 2 O 6 , at least, at the most-likely wave vectors. However, we cannot rule out the possibility of structural changes involving only certain symmetries of the space group P 4 1 32.

Supplementary Note 5: Low energy inelastic neutron scattering
Highly frustrated systems that undergo magnetic order at very low temperatures would result in their low-energy excitation spectra modulated to exhibit spin waves from the ordered moment. Therefore, the low energy part of the excitations in PbCuTe 2 O 6 is investigated on both the powder and single crystal samples using the LET spectrometer. The Supplementary Fig. 5a&b show the spectra measured with incident energy of E = 1.5 meV and 1.3 meV for powder and single crystal respectively where the single crystal data is integrated over the −2.35 [0, 0, l] −1 r.l.u range . It is clear that both the spectra consistently exhibit only diffuse excitations down to 0.1 meV originating at |Q| ∼0.8Å −1 and have similar intensity modulations. Hence the absence of any magnons in PbCuTe 2 O 6 is confirmed.

Supplementary Note 6: Comment on previous Density functional theory calculations in Ref. [2]
The energy mapping method applied in our density functional theory calculations employs fully charge self-consistent solutions for different spin configurations, therefore it includes at the density functional theory level the exchange contributions to arbitrary order. It is clear that this approach yields different and usually more reliable estimates than the second-order perturbation theory formula J = 4t 2 U used previously [2] which expresses only the antiferromagnetic superexchange contribution based on one virtual process. The latter approximation leaves out a large number of other exchange processes both from higher orders of perturbation theory as well as ferromagnetic contributions. The results from our DFT calculations yield interactions up to 4 th nearest neighbour with a ratio of J 1 :J 2 :J 3 :J 4 = 1.06:1:0.55:0.11 for the Curie-Weiss temperature of θ CW = −23 K. In contrast, the previous method finds only the first three neighbours to be relevant and the ratio of couplings is between the two methods is evident in the spin susceptibility maps calculated using the PFFRG method based on the Hamiltonian from these two sets of couplings as shown in Supplementary Figure6. Although the static spin susceptibility based on the Hamiltonian from the perturbation theory method [2] also produces a ring at ∼ 0.8Å −1 , as shown in Supplementary Figure6a, the maxima on the ring appear at (1, 1, 1) and equivalent positions in the [h, h, l]−plane. This is clearly different from the experimental results where the diffuse ring is broad and the maxima are at positions such as (0.3, 0.3, ±1.69) as nicely reproduced by the Hamiltonian from our DFT calculations (see Supplementary Fig. 6c). The discrepancy of the previous Hamiltonian is even clearer in the [h, k, 0]−plane (compare Supplementary Fig. 6b&d) where the double peak at (±1.69, ±0.3, 0) is missing. Supplementary Figure 6e&f show cuts through the data and the calculated spin susceptibility of the two Hamiltonians. The agreement between our Hamiltonian and the data is clearly much better than that of the previous Hamiltonian [2]. Supplementary Note 7: Ground state properties of classical Heisenberg models on the hyperkagome lattice This correlation profile can be compared with a classical Heisenberg model with J 2 hyperkagome interactions only, as it has previously been studied in Ref. [5]. The ground state of this model is macroscopically degenerate, i.e. for a system consisting of L × L × L unit cells, the dimension of the degenerate manifold scales exponentially as V = L 3 . As a consequence, the spin structure factor shows extended regions of strong intensity which fill a volume in momentum space, see Fig. 2 in Ref. [5]. Furthermore, in close similarity to classical pyrochlore systems, where an"ice rule" for each tetrahedron/triangle imposes a constraint on the local spin orientation, the structure factor exhibits pinch points.
When triangle J 1 interactions are added, further constraints are imposed on the possible ground states, which reduce the dimensionality of the degenerate ground state manifold. Particularly, it can be shown that the number of degenerate states becomes subextensive, i.e. it scales exponentially in the linear system size L [6]. As a consequence, the regions of strong intensity in the structure factor shrink Supplementary Figure 6: The static spin susceptibility profiles calculated using the PFFRG method. From the Hamiltonian produced by: a-b second-order perturbation theory that expresses only the antiferromagnetic superexchange contribution based on one virtual process (Ref. [2]): and c-d the current work that employs the full potential local orbital basis set [3] and the generalized gradient approximation functional [4]. The former results produce a sharper diffuse ring in [h, h, l]−plane with maxima at (1, 1, 1) and a diamond shaped profile in the [h, k, 0]−plane which do not agree with the experimental data. The latter results however show striking similarity with our neutron scattering maps. e-f Comparison between cuts through the experimental data and the expected magnetic intensity calculated for the two Hamiltonians using PFFRG. The data points were obtained by integrating the experimental data shown in Fig. 4a-b of the main text, over the respective shaded regions indicated in those figures. The same integration was performed over these calculations (see shaded region in a&b). The uncertainties represent statistical errors. As is clearly evident, our Hamiltonian gives much better agreement with the data than the previous Hamiltonian.
down to lines parallel to the [1, 1, 1] (and symmetry related) directions. Interestingly, the susceptibility in Fig. 4e of main text still shows remainders of the pinch points such as the point [0, 0, 2] where streaks of high intensity cross.
It is important to emphasize that even though the classical J 1 -J 2 Heisenberg model only exhibits a subextensive ground state degeneracy, there is still a macroscopic number of low-energy states through which the system can fluctuate when quantum fluctuations are considered [6]. This still remains true when the weaker J 3 and J 4 interactions are added. While this further reduces the degeneracy to a finite number of states, the full manifold of the classical J 1 -J 2 Heisenberg model still remains energetically nearby.