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 Systems Biology
Fractalization drives crystalline states in a frustrated spin system

Edited by Steven M. Girvin, Yale University, New Haven, CT, and approved September 18, 2008 (received for review May 9, 2008)
This article has a Correction. Please see:
Abstract
The fractalized Hofstadter butterfly energy spectrum predicted for magnetically confined fermions diffracted by a crystal lattice has remained beyond the reach of laboratoryaccessible magnetic fields. We find the geometrically frustrated spin system SrCu_{2}(BO_{3})_{2} to provide a sterling demonstration of a system in which bosons confined by a magnetic and lattice potential mimic the behavior of fermions in the extreme quantum limit, giving rise to a sequence of plateaus at all magnetization m_{z}/m_{sat} = 1/q ratios 9 ≥ q ≥ 2 and p/q = 2/9 (m_{sat} is the saturation magnetization) in magnetic fields up to 85 T and temperatures down to 29 mK, within the sequence of previously identified plateaus at 1/8, 1/4, and 1/3 of the saturated magnetization. We identify this hierarchy of plateaus as a consequence of confined bosons in SrCu_{2}(BO_{3})_{2} mimicking the high magnetic field fractalization predicted by the Hofstadter butterfly for fermionic systems. Such an experimental realization of the Hofstadter problem for interacting fermions has not been previously achieved in real materials, given the unachievably high magnetic flux densities or large lattice periods required. By a theoretical treatment that includes shortrange repulsion in the Hofstadter treatment, stripelike spin densitymodulated phases are revealed in SrCu_{2}(BO_{3})_{2} as emergent from a fluidic fractal spectrum.
The geometrically frustrated spin gap system SrCu_{2}(BO_{3})_{2} (1–,7) is unique in its orthogonal arrangement of spin dimers, resulting in an exact direct singlet product incompressible ground state, formally described by Shastry and Sutherland (,8). Dimers comprising pairs of S = 1/2 spins on neighboring Cu^{2+} ions bound by an intradimer Heisenberg coupling, J, are coupled less strongly to orthogonal dimers by an interdimer coupling J′ (Fig. 1A Upper Inset), forming weakly coupled layers within the tetragonal crystal structure. Magnetic susceptibility and inelastic neutron scattering (INS) experiments measure a spin gap of Δ = 34 K, from which J/J′ ∼ 1.47–1.67 (J = 71.5–100 K, J′ = 43–68 K) (6, ,9–,11) is estimated, placing the system just within the predicted exact groundstate regime J/J′ ≥ 1.35 (8, ,12). In an applied magnetic field H, the ground state incorporates a finite density of spin triplets for which the groundstate configuration is no longer known.
We perform magnetization measurements on oriented single SrCu_{2}(BO_{3})_{2} crystals of ∼2 mm × 2 mm × 0.5 mm by using cantilever magnetometry and ∼0.3 mm × 0.3 mm × 2 mm by using pulsed fields. The crystals were cut from larger single crystals grown by using a self flux by floating zone image furnace techniques (13) and characterized by INS (,11) and Laue diffraction techniques. Measurements up to 35 T are made by using a 10μmthick CuBe torque cantilever rotated in situ to bring the crystalline c axis within 2° of the applied magnetic field, and repeated on another sample up to 45 T by using a 50μmthick cantilever to confirm reproducibility and facilitate comparison with previous magnetization measurements (2, ,14–,16). Torque measurements were made in a portable dilution refrigerator in continuous magnetic fields at the National High Magnetic Field Laboratory, Tallahassee—enabling a level of sensitivity exceeding that possible in pulsed magnetic fields. The measured torque is converted to absolute values of magnetization by multiplication by a constant rescaling factor and a small quadratic background subtraction obtained on comparison with measured values of pulsedfield magnetization (,14, ,15) (comparison shown in Lower Inset to Fig. 1A). In contrast to ref. 17, we find the magnetization obtained from torque measurements to correspond closely to that measured by susceptometry in pulsed magnetic fields. Measurements between 38 and 85 T are made by using a coaxially compensated wirewound magnetometer during the 10ms insert magnet pulse in the National High Magnetic Field Laboratory (Los Alamos) multishot 100T magnet. Although the short pulse duration necessarily means a relatively large noise floor for these ultrahighfield measurements, the identification of prominent features at these high fields, unachievable by other techniques, is facilitated. Pulsedfield measurements are made at T = 500 mK during 2 magnet shots on each of 2 different samples to confirm reproducibility.
On applying H within 2° of the crystalline c axis, a discontinuous rise in magnetization m_{z} is observed above a threshold magnetic field ∼19.5 T at 29 mK (Fig. 1A), indicating the onset of significant triplet population (2). On further increasing H, a finelyspaced sequence of plateaulike features (these are rounded because of Dzyaloshinskii–Moriya terms (9, ,18) and thermal smearing, we refer to them hereon simply as plateaus) appear in m_{z}. Maxima in the inverse differential susceptibility correspond to the midpoint of flat plateau regions, flanked by thermally rounded transitions. Magnetization plateaus values thus located are found to occur in the sequence m_{z}/m_{sat} = 1/9, 1/8, 1/7, 1/6, 1/5, 2/9, and 1/4 (within an error margin of 2%), where m_{sat} is the saturation magnetization (Fig. 1A). The corresponding magnetic fields associated with each plateau [midpoint (width)] are 27.4(0.4) T, 28.4(1.2) T, 29.7(0.8) T, 30.6(0.6) T, 32.6(0.8) T, 33.8(0.6) T, and 36(6) T, respectively—the boundaries being defined by maxima in the differential susceptibility. The 1/3 plateau is located at 58(,24) T from pulsedfield measurements (,Fig. 1A Lower Inset). Fig. 1B shows the phase boundary corresponding to melting temperatures of the cascade of ordered plateau phases located from peaks in the differential susceptibility.
The increased sensitivity of the torque measurements performed in continuous magnetic fields and significantly lower temperatures enables the observance of additional plateaus previously unobserved in magnetization measurements at elevated temperatures (T ≥ 450 mK for H ‖ c and T ≥ 80 mK for H ‖ a) using pulsed magnetic field measurements up to 70 T (2, ,14–,16), which reported only the 1/8, 1/4, and 1/3 plateaus. Subsequent measurements of NMR spectra (,19) performed down to temperatures of 0.19 K find differences between the spectral shape at field values 27.5 T, 28.7 T, and 29.9 T, providing corroboration for the fine plateaus hierarchy at 1/9, 1/8, and 1/7 of the saturation magnetization identified here by torque measurements. While the m_{z}/m_{sat} = 1/2 plateau was theoretically predicted in refs. 3, ,20, and ,21, we observe an experimental hint of its existence in the pulsedfield experiment reported here at fields exceeding 80 T. The 1/3 plateau is sufficiently stable and extended in field range to enable its unambiguous identification even against a relatively large noise floor (because of the short pulse duration in the ultrahighfield 100T magnet), whereas experimental evidence for the 1/2 plateau is more suggestive in nature.
We proceed to investigate the origin and character of the sequence of fieldtuned incompressible spin triplet configurations associated with the observed magnetization plateaus. The underlying confinement of spintriplet motion responsible for these features arises from the effect of geometrical frustration in SrCu_{2}(BO_{3})_{2}. The minimal model for describing the S = 1/2 magnetic lattice is the Shastry–Sutherland Hamiltonian: where i and j denote the lattice sites, S_{i} is the spin 1/2 operator on site i, and b_{i}^{†} and b_{i} are hardcore boson creation and annihilation operators at site i that provide an alternative description of a spin system via the Matsubara–Matsuda transformation (22): S_{i}^{+} = b_{i}^{†} and S_{i}^{−} = b_{i}. The offsite density–density interaction described by the last term (n_{i} = b_{i}^{†}b_{i} is the number operator at site i) corresponds to the Ising term, S_{i}^{z}S_{j}^{z}, in the spin representation. We note that the local magnetization along the field direction 〈S_{i}^{z}〉 = 〈n_{i}〉 − 1/2 corresponds to the particle density in the bosonic representation. We do not include in ℋ the Dzyaloshinskii–Moriya interactions (9, ,14, ,18), which are much smaller than J and J′, because we do not expect these interactions to change the number of magnetization plateaus. Their main perturbative effect is to round the plateaus, because the total magnetization along the field direction (total number of bosons) is no longer a conserved quantity.
The computation of m_{z}(H) requires an approach that accurately captures the underlying hierarchy of energy levels and allows one to obtain the groundstate energy for a quasicontinuum of densities—unlike previously attempted S^{z} = 1 triplet hardcore bosonization (20, ,21) and exact diagonalization (,4, ,23) techniques. An alternative paradigm is indicated by our experimental observation of plateaus at all 1/q ratios of m_{sat} for 2 < q < 9 and p/q = 2/9—reminiscent of the quantum Hall effect (24, ,25) described by Landaulevel physics. Hence, we begin by adopting a fermionic treatment in which the density–density interactions are assumed to be irrelevant (〈n^{i}〉 = ρ is uniform), as applied to SrCu_{2}(BO_{3})_{2} by Misguich et al. in ref. 3. By using a Chern–Simons construction on the lattice (,26), we can map the hardcore bosons in ,Eq. 1 into spinless fermions: b_{i}^{†} = f_{j}^{†}e^{iΣk≠jarg(k,j)nk}, b_{i}^{†} = e^{−iΣk≠jarg(k,j)nk}f_{j}, where arg(k, j) is the angle between the relative vector, r_{k}–r_{j}, and an arbitrary direction. This transformation is equivalent to attaching a flux quantum to each fermion such that the statistical phase generated from fermionic pairexchange is cancelled by the phase generated by the flux quantum via the Aharonov–Bohm effect. After this transformation, ℋ becomes a model for a gas of spinless fermions moving on the same lattice and in the presence of a nonlocal vector potential A_{ij}(r_{i}). To simplify this problem, which is rendered intractable because of the nonlocality of the vector potential, we make the approximation of a uniform flux distribution generated by the statistical field H_{s} = 4 Φ_{0}(ρ/a^{2}) (ref. 3) (ρ = (1/N_{s}) Σi 〈n_{i}〉 is the particle density and N_{s} is the number of sites). We thus realize a gas of interacting spinless fermions in a strong magnetic field H_{s}.
The motion of the bosons on the geometrically frustrated Shastry–Sutherland lattice is therefore akin to orbital confinement in twodimensional quantum Hall systems by a (statistical) magnetic field (24, ,25). Plateaus with a finite Hall conductance arise when the chemical potential (μ = g_{c}μ_{B}H) resides in a gap of the energy spectrum of fermions with average density ρ = m_{z} + 1/2 = a^{2}H_{s}/4Φ_{0} (a^{2} is the area of the unit cell). The interplay of cyclotron (mean particle separation “a/ρ^{1/2}”)and lattice “a” lengthscales within this single particle bandfilling picture was theoretically captured by a Hofstadter butterfly of fractalized energy gaps in 1976 (27). Minimization of the groundstate energy E(m_{z}) for each value of H yields the dependence of m_{z} on H in this noninteracting limit. Following this procedure for SrCu_{2}(BO_{3})_{2} (fractal spectrum shown in Fig. 2A), theoretical m_{z} curves are obtained (shown in Fig. 2B for J = 70 K, J/J′ = 2.2 optimized to match experimental plateaus*), the shape of which agrees reasonably well with the measured magnetization and the values of observed plateaus (,Fig. 2B)—an advance provided by this fermionic treatment over previous models (4, ,6, ,20, ,21, ,23).
The importance of density–density interactions, however, is apparent from the experimental features unexplained by the uniform Chern–Simons treatment. While this treatment captures most of the measured plateau values and the overall shape of m_{z}(H) by neglecting density–density interactions, it fails to predict some experimental plateaus. In addition, the uniform incompressible liquid ground state cannot explain the broken translational symmetry observed by NMR measurements at the m_{z}/m_{sat} = 1/8 plateaus (4) or the finitetemperature phase transitions measured by using heat capacity (,29). This suggests that offsite repulsive interactions (see ,Eq. 1) play an important role in modifying groundstate character. To include this effect, we consider an unconstrained and selfconsistent mean field decoupling n_{i}n_{j} ≃ 〈n_{i}〉n_{j} + 〈n_{j}〉n_{i} − 〈n_{i}〉〈n_{j}〉, thereby allowing the mean value of the local density, 〈n_{i}〉, to relax on each site of a super unit cell. Fig. 3 shows the spin density profile corresponding to the lowestenergy solutions of each of the m_{z}/m_{sat} = p/q plateaus, revealing that stripelike density modulations of the spin (except for the 1/2 plateau) mediated solely by shortrange interactions significantly lower the energy. In addition, to facilitate the direct comparison of our results with future experimental data, supporting information (SI) Fig. S1 shows the elastic neutron scattering spectrum for each of the spindensity profiles.
Although the predicted triplet superstructures in Fig. 3 have some features in common with spin configurations predicted in refs. ,4, ,20, and ,21, a qualitative difference is seen in the alternation of onedimensional superstructures with arrays of spirallike clusters to accommodate coexisting microscopic length scales of the lattice “a” and the mean particle separation (analogous to the cyclotron radius) “a/ρ^{1/2}”. This commensuration of length scales is characteristic to the Hofstadter fractal spectrum, the gap hierarchy of which is inextricably linked to the distinctive pattern of spin density modulation at each incompressible plateau state. Starting with the fluidic spectrum (Fig. 2A), the gap structure is rearranged because of a change in lattice potential as each p/q incompressible plateau state stabilizes triplet stripes separated by an average distance of q lattice parameters (where p/q is the density of S^{z} = 1 triplets).^{‡} The role of interactions is further amplified by the quasidegenerate energy level subspectrum at lower m_{z}/m_{sat} plateaus that determine the density–density susceptibility. For the associated low p/q ratios with q ≫ 1 (where q determines the number of subbands), interactions stabilize a unique hierarchy of gaps from the fine gap substructure within a bounded spectrum (seen in Fig. 2A).
Spectacularly, the nonuniform solution corresponding to plateaus spindensity profiles in Fig. 3 captures the entire sequence of measured magnetization plateaus, including those observed at p/q = 1/7 and 2/9, which were not predicted by the uniform model; comparison with experiment is shown in Fig. 2B for a modified interactiondriven spectrum computed for values of J = 75.5 K and an optimized “effective” ratio J/J′ = 2.2 that best fit experimental data. The remarkable correspondence of predicted and experimental plateaus as well as the abrupt step in magnetization preceding the onset of the plateaus at p/q = 1/9, and the background shape of the experimentally measured magnetization, reveals a marked improvement over the uniform treatment.
We thus interpret the observed plateau states in SrCu_{2}(BO_{3})_{2} to represent the emergence of a previously unrecognized type of stripelike crystalline state resulting from the instability of the incompressible Hall fluid to interactions. Unlike Wigner crystalline states, the structure of these densitymodulated states is seen to be determined by the density–density susceptibility of the uniform solution, as inferred from the p/q fractions that are stabilized as a function of field— similar to the sequence that appears in the fluid Hofstadter butterfly. We are now provided with a new perspective from which to question the continued stability of ground states in metallic systems (composed of charged fermions) in the formidably high equivalent experimental magnetic field limit. The intermediate interaction regime in which the geometrically frustrated SrCu_{2}(BO_{3})_{2} system lies raises the intriguing possibility of a dual ground state in which crystalline density modulation retains topological properties of a Hall fluid, enabling analogies to be made with “Hall crystals” (29) and fieldinduced chargedensity wave states (,30).
Acknowledgments
We acknowledge useful discussions with S. M. Girvin, S. A. Kivelson, and S. Todadri. Experiments performed at the National High Magnetic Field Laboratory were supported by the National Science Foundation Division of Materials Research, the Department of Energy, and the State of Florida. B.D.G. is supported by the Natural Sciences and Engineering Research Council and the Canadian Institute for Advanced Research. N.M. is supported by a postdoctoral fellowship from the Spanish Ministerio de Educación y Ciencia. S.E.S. received support from Trinity College (Cambridge) and the Institute for Complex Adaptive Matter.
Footnotes
 ^{1}To whom correspondence should be addressed. Email: suchitra{at}phy.cam.ac.uk

Author contributions: S.E.S. and N.H. designed research; S.E.S., N.H., S.F., E.P., T.M., N.M., H.A.D., and B.D.G. performed research; S.E.S., N.H., P.S., and C.D.B. analyzed data; and S.E.S. and C.D.B. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/cgi/content/full/0804320105/DCSupplemental.

↵* This “effective” ratio J/J′ is higher than that obtained from fits to low magnetic field susceptibility and INS data (6, ,9, ,10, ,11).

↵‡ The system prefers onedimensional superstructures over twodimensional ones because of the associated commensuration of triplets with the lattice.
 © 2008 by The National Academy of Sciences of the USA
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