LETTERSPUBLISHED ONLINE: 7 OCTOBER 2012 | DOI: 10.1038/NMAT3432

Quantum critical state in a magnetic quasicrystalKazuhiko Deguchi1*, Shuya Matsukawa1, Noriaki K. Sato1, Taisuke Hattori2, Kenji Ishida2,Hiroyuki Takakura3 and Tsutomu Ishimasa3

Quasicrystals are metallic alloys that possess long-range,aperiodic structures with diffraction symmetries forbidden toconventional crystals. Since the discovery of quasicrystalsby Schechtman et al. in 19841, there has been considerableprogress in resolving their geometric structure. For example,it is well known that the golden ratio of mathematics and artoccurs over and over again in their crystal structure. However,the characteristic properties of the electronic states—whetherthey are extended as in periodic crystals or localized as inamorphous materials—are still unresolved. Here we report thefirst observation of quantum (T = 0) critical phenomena ofthe Au–Al–Yb quasicrystal—the magnetic susceptibility andthe electronic specific heat coefficient arising from stronglycorrelated 4f electrons of the Yb atoms diverge as T→ 0. Fur-thermore, we observe that this quantum critical phenomenonis robust against hydrostatic pressure. By contrast, there is nosuch divergence in a crystalline approximant, a phase whosecomposition is close to that of the quasicrystal and whose unitcell has atomic decorations (that is, icosahedral clusters ofatoms) that look like the quasicrystal. These results clearlyindicate that the quantum criticality is associated with theunique electronic state of the quasicrystal, that is, a spatiallyconfined critical state. Finally we discuss the possibility thatthere is a general law underlying the conventional crystalsand the quasicrystals.

The quasicrystal that we study here is a gold–aluminium–ytterbium alloy described as Au51Al34Yb15 with a six-dimensionallattice parameter a6d=0.7448 nm. In Fig. 1a we present the electrondiffraction pattern of our sample demonstrating 5-fold symmetry.This exhibits the characteristic feature of quasicrystals—long-rangetranslational but aperiodic order. Owing to this quasi-periodicity,an unusual electronic state that is neither extended nor localized isexpected.However, such an unusual state has not yet been observed.In this Letter, we show the Au–Al–Yb quasicrystal to present apeculiar quantum critical behaviour, which we propose to reflectthis unusual state expected for quasicrystals. It becomes apparentin the present system because of strong correlations induced bythe 4f electrons of Yb.

The Au–Al–Yb quasicrystal was discovered in the course ofresearch on new series of Tsai-type quasicrystals2. The Yb valencewas found to be intermediate between Yb2+ and Yb3+ by meansof X-ray absorption near edge structure (XANES) experiments,indicating the hybridization of the 4f electrons of the Yb atomswith the conduction electrons. Figure 1b shows the arrangementof Yb atoms in the structure model of the cadmium–ytterbium(Cd5.7Yb) quasicrystal3,4, which is isostructural with the presentAu–Al–Yb quasicrystal. For comparison, we illustrate the crystalstructure of the approximant Au51Al35Yb14 in Fig. 1c. The edge

1Department of Physics, Graduate School of Science, Nagoya University, Nagoya 464-8602, Japan, 2Department of Physics, Graduate School of Science,Kyoto University, Kyoto 606-8502, Japan, 3Division of Applied Physics, Graduate School of Engineering, Hokkaido University, Sapporo 060-8628, Japan.*e-mail: deguchi@edu3.phys.nagoya-u.ac.jp.

length of the icosahedron ranges from 0.545 to 0.549 nm. Inreality, this icosahedron corresponds to the third shell of theTsai-type cluster (Fig. 1d–h).

In what follows, we attempt to reveal the nature of thecharacteristic electronic state of quasi-periodic systems by probingthe 4f electrons and comparing physical properties between thequasi-periodic quasicrystal and the periodic approximant. Beforepresenting the low-temperature data, we briefly mention thehigh-temperature results of the electrical resistivity ρ(T ) and themagnetic susceptibility χ(T ). As shown in Supplementary Fig. S1a,both the quasicrystal and the approximant showmetallic behaviourin a wide temperature range, with a large residual resistivityρ(0), but they show different behaviour at low temperatures(inset of Supplementary Fig. S1a); whereas the approximantexhibits the conventional Fermi-liquid power-law1ρ∝T 2 (where1ρ = ρ(T )− ρ(0)), the quasicrystal rather exhibits the T lineardependence,1ρ∝T . To our knowledge, this is the first observationof the so-called non-Fermi liquid behaviour in quasicrystals. Formagnetism, both the quasicrystal and the approximant show aCurie–Weiss form above 100K (Supplementary Fig. S1b), whichyields an effective moment of µeff = 3.91µB and 3.96µB forthe quasicrystal and the approximant, respectively. These valuesindicate that the Yb-ion valence of both the quasicrystal and theapproximant is between Yb3+ and Yb2+.

Now we focus on the quantum criticality (that is, thecritical behaviour near T = 0) of the quasicrystal. The magneticsusceptibility of the quasicrystal at H = 0 shows a divergentbehaviour as T → 0 (Fig. 2a), and this quantum criticalityis characterized by a critical index n = 0.51, as defined byχ−1∝T n (Fig. 2a inset). To examine the effect of pressure onthis critical behaviour, we measured the magnetic susceptibilityunder hydrostatic pressure. The results are indicated in the insetof Fig. 2a. It is clearly seen that the divergent behaviour survives,with the novel critical exponent unchanged. Here it should benoted that the hydrostatic pressure can change the hybridizationbut not alter the crystal symmetry. In contrast, the application of amagnetic field suppresses the divergence, resulting in the saturationof χ(T ) at low temperatures. This allows us to define a crossovertemperature T ∗ into the field-induced Fermi liquid state with theenhanced Pauli susceptibility, as indicated on the figure by thearrows at which the χ(T ) curves show a maximum. In contrast, themagnetic susceptibility of the approximant increasesmonotonicallywith decreasing temperature (Fig. 2b), but does not diverge, asevidenced from the inset of Fig. 2a, χ−1 ∝ T n

+ constant with thesame exponent n as above.

Figure 3a shows the temperature–field phase diagram basedon a contour plot of the normalized uniform susceptibility,χ(T ,H )/χ(T ,0). We note that T ∗ (open squares) seems to

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a

c

d e f g h

b

0.5 nm

a = 1.450 nm

Yb Au AI Au/AI mix

Figure 1 | Structure models of quasicrystal and approximant. a, Selected-area electron diffraction pattern of the Au–Al–Yb quasicrystal. b, Atomicarrangement of Yb atoms in the isostructural Cd5.7Yb quasicrystal. The Yb atoms included in a cube with an edge length of 6 nm are projected onto theplane perpendicular to the 5-fold axis. The icosahedral aggregate is highlighted. c, Yb arrangement in the Au–Al–Yb approximant in the projection alongthe [001] direction. d–h, Concentric shell structures of Tsai-type cluster in the Au–Al–Yb approximant. Each vertex of the first cluster presented in d isoccupied by Au/Al mixed atoms with an occupancy 1/6. Chemical ordering in each shell is based on the result of structure analysis of the approximant (seeref. 2 for details). Scale bar (0.5 nm) is shown at d.

approach zero as T → 0. The crossover field H ∗ defined byχ(T ,H ∗)/χ(T ,0) = 0.95 (open circle) drops to zero as T → 0,meaning that χ(H ) diverges as H → 0 at T ∼0. These resultsindicate that there is a singular point at T = H = 0 withoutchemical doping and pressurization. In this respect, the Au–Al–Ybquasicrystal is regarded as a quantum critical matter.

The nuclear spin-lattice relaxation rate divided by temper-ature, 1/T1T , deduced from 27Al nuclear magnetic resonance(NMR) measurements on the quasicrystal is plotted in Fig. 3b.(The recovery curves are shown in Supplementary Fig. S2: theyare fitted well using a single component of 1/T1, indicatingthat the Al sites around the Yb ions are not inhomogeneous.)Whereas the aforementioned uniform susceptibility χ(T ) probesmagnetic fluctuations at q = 0 (where q is the wave vectorof an applied magnetic field), 1/T1T observes the q-averagedfluctuations. The scaling observed here, 1/T1T ∝ χ(T ), togetherwith a negative Weiss temperature suggest that χ(q) is inde-pendent of q, meaning that the magnetic fluctuation associ-ated with the quantum criticality possesses a local nature in thereal space.

Let us move onto the heat capacity (Fig. 4). For the quasicrystal,logarithmic divergence at zero field is observed in the temperaturedependence of the magnetic specific heat (CM) divided bytemperature, CM/T ∝ − lnT (Fig. 4a and inset of Fig. 4b). Bycontrast, the approximant shows no divergence (Fig. 4b), althoughthe saturated value is very large, ∼0.7 J K−2 mol−1, comparedwith conventional crystals and quasicrystals. In magnetic fields,

the divergence of the quasicrystal is suppressed (Fig. 4a), butthe saturated value is still very large; CM/T ∼0.2 J K−2 mol−1 atH = 50 kOe. The approximant shows a similar field effect. Theseresults are consistent with the suppression of χ(T ) by the magneticfield, supporting the field-induced Fermi liquid state.

Combining the magnetic and thermodynamic results, in theinset of Fig. 2b we plot the ratio χ/γ (where γ = CM/T ),which is a measure of the magnetic correlation of quasiparticles.We note that the ratio is enhanced at low fields for both thequasicrystal and the approximant, suggesting the presence ofmagnetic correlations there.

The experimental results presented above are summarized asfollows. Both the periodic approximant and the quasi-periodicquasicrystal show similar transport and magnetic properties at hightemperatures. The difference becomes evident at low temperatures:whereas the approximant shows the Fermi liquid behaviour, thequasicrystal exhibits non-Fermi liquid behaviour at zero field(see also Supplementary Table S1). We interpret this difference,the presence/absence of the divergence in χ and CM/T , asthe presence/absence of the critical state unique to quasicrystalswith the quasi-periodicity5. This interpretation is supported bythe robustness of the quantum criticality against the hydrostaticpressure: for crystalline materials, a perturbation such as theapplication of external pressure gives rise to deviations fromthe critical ‘point’. As a result, we conclude that the presentquantum criticality is associated with the unique electronicstate of the quasicrystal.

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NATURE MATERIALS DOI: 10.1038/NMAT3432 LETTERS

b

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Figure 2 | Temperature dependence of the magnetic susceptibility of thequasicrystal and the approximant. a, a.c. and d.c. magnetic susceptibility ofthe quasicrystal measured in a temperature range of 0.08< T< 3.0 K(denoted by the lines) and 1.8< T< 300 K (circles), respectively. Magneticfields are described in the figure. The abscissa is plotted on a logarithmicscale. The arrows indicate a characteristic temperature T∗(H). Inset showsthe inverse susceptibility χ−1 versus T0.51 of the approximant (blue circles)and the quasicrystal (red circles) at ambient pressure, and of thequasicrystal at pressures of 0.72 GPa and 1.54 GPa. The black lines are alinear extrapolation to T=0. b, Magnetic susceptibility of the approximantmeasured in the same condition as in Fig. 2a. Insets show the fielddependence of the ratio χ/γ at T=0.1 K for the quasicrystal and theapproximant, respectively, where γ =CM/T.

Finally, we discuss the implication of the robustness againsthydrostatic pressure. In general, the quasicrystal critical state can be

χ (H = 7.3 kOe)

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Figure 3 | Magnetic properties of the quasicrystal. a, Contour plot of thenormalized uniform susceptibility χ(T,H)/χ(T,0)=χ(T,H)/aT−0.51 witha=0.081 e.m.u.K0.51 mol−1. The open circles and squares denote acrossover field H∗ defined by χ(T,H∗)/χ(T,0)=0.95 and a characteristictemperature T∗, respectively. Solid lines are guides to the eye and thedashed line is an extrapolation along a contour line. b, Nuclear spin-latticerelaxation rate divided by temperature 1/T1T of 27Al NMR (left scale) andthe magnetic susceptibility χ(T) (right scale). The scaling relation1/T1T∝χ(T) is clearly observed. The abscissa is plotted on a logarithmicscale. The error bars of 1/T1T indicate the standard errors fromleast-squares fits of the recovery data of nuclear magnetization.

characterized by an extremely degenerate confined wavefunction5–7

and a singular continuous density of states5,8–12. The robustnesssuggests that the critical state is also robust against pressure; thisis naturally understood because the hydrostatic pressure does not

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LETTERS NATURE MATERIALS DOI: 10.1038/NMAT3432

6 80.1

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Figure 4 | Temperature dependence of the magnetic specific heat CM/T ofthe quasicrystal and the approximant under magnetic field. CM wasobtained by subtracting the nuclear contribution, CN=α(H)/T2, from themeasured data, where α(H) was deduced from the plot of CT2 versus T3.a, Temperature dependence of CM/T of the quasicrystal. Note that CM/Tdiverges logarithmically at H=0 whereas it tends to saturate under amagnetic field. b, CM/T of the approximant measured in the samecondition as in Fig. 4a. Inset shows a comparison of CM/T between thequasicrystal and the approximant at zero field.

change the symmetry and the quasi-periodicity. For the unusualexponents of the quantum criticality, on the other hand, thereare some models which may account for the non-Fermi liquidbehaviour. A valence criticality theory successfully accounts for ournon-Fermi liquid exponents, as seen in Supplementary Table S1

(ref. 13). Although the theory assumes crystalline materials, it mayget at the essence of the critical state (that is, the spatially localnature of the 4f electrons) by assuming the dynamical exponentz =∞. This suggests the possibility that there is a general lawunderlying the conventional crystals and the quasicrystals, howeverthe theory is unlikely to explain the robustness. Other theoriessuch as the so-called Kondo disorder and Griffiths phase seemto be also excluded: see Supplementary Information for a moredetailed discussion. To our knowledge, there is no theory to explaina quantum criticality that is robust against hydrostatic pressure butreadily destroyed bymagnetic field.We hope that the present resultswill stimulate further theoretical works.

Received 25 April 2012; accepted 24 August 2012;published online 7 October 2012

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AcknowledgementsThe authors thank Y. Tanaka and S. Yamamoto for support of the experiments. Theauthors also thank S. Kashimoto, T. Watanuki, S. Watanabe, K. Miyake andY. Takahashi for valuable discussions. This work was partially supported by agrant-in-aid for Scientific Research from JSPS, KAKENHI (S) (No. 20224015), the‘Heavy Electrons’ Grant-in-Aid for Scientific Research on Innovative Areas(No. 20102006, No. 21102510, No. 20102008, and No. 23102714) from MEXT of Japan,a Grant-in-Aid for the Global COE Program ‘The Next Generation of Physics, Spunfrom Universality and Emergence’ fromMEXT of Japan, and FIRST program from JSPS.

Author contributionsK.D., T.I., K.I. and N.K.S. wrote the paper. K.D., S.M. and N.K.S. carried outlow-temperature experiments. K.I. and T.H. carried out NMR experiments. T.I. andH.T. carried out sample preparations and structure determinations. All authorsdiscussed the results and commented on the manuscript.

Additional informationSupplementary information is available in the online version of the paper. Reprints andpermissions information is available online at www.nature.com/reprints.Correspondence and requests for materials should be addressed to K.D.

Competing financial interestsThe authors declare no competing financial interests.

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