Discovery of an ultra-quantum spin liquid
Yanxing Yang, Cheng Tan, Zihao Zhu, Jian Zhang, Zhaofeng Ding, Qiong Wu, Changshen Chen, Toni Shiroka, Douglas E. MacLaughlin, Chandra M. Varma, Lei Shu
DDiscovery of an ultra-quantum spin liquid
Y. X. Yang, C. Tan, Z. H. Zhu, J. Zhang, Z. F. Ding, Q. Wu, C. S. Chen T. Shiroka, D. E. MacLaughlin, C. M. Varma, ∗ L. Shu , , ∗ State Key Laboratory of Surface Physics, Department of Physics,Fudan University, Shanghai 200433, China Laboratory for Muon-Spin Spectroscopy, Paul Scherrer Institut, 5232 Villigen, Switzerland Department of Physics and Astronomy, University of California, Riverside, CA 92521, USA Department of Physics, University of California, Berkeley, CA 94704, USA Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China Shanghai Research Center for Quantum Sciences, Shanghai 201315, China ∗ Corresponding author. Email: [email protected] (C.M.V.); [email protected] (L.S.).
Quantum fluctuations are expected to lead to highly entangled spin-liquidstates in some two-dimensional spin-1/2 compounds. We have synthesizedand measured thermodynamic properties and muon relaxation rates in tworelated such triangular lattice compounds. The specific heat divided by tem-perature and muon relaxation rates are both temperature independent at lowtemperatures followed by a logarithmic decrease. Both are shown to followquantitatively from the same magnetic fluctuations which are critical in thetime dimension. Moreover ∼
64% of the magnetic entropy is missing down totemperatures of O(10 − ) the exchange energy in the compound which can bemade very pure, at zero magnetic field as well as for gµ B H up to 9 Tesla. Thismeans that quantum fluctuations lead either to a gigantic specific heat peakfrom singlet excitations below such temperatures, or to an extensively degen- a r X i v : . [ c ond - m a t . s t r- e l ] F e b rate topological singlet ground state. These results reveal an ultra-quantumstate of matter whose understanding requires a new paradigm. The study of quantum fluctuations in interacting matter is of primary interest in physics,encompassing fields as diverse as the thermodynamics of black holes [1, 2], particle physicsbeyond the standard model [3], the theory of quantum computation [4], and various phenomenain condensed matter physics. The last is often paradigmatic since it allows access and control toa wide variety of experiments, while the concepts often cut across different fields. These rangefrom quantum Hall effects [5] to the quantum criticality that governs high temperature super-conductivity [6, 7] to the spin liquid states [8, 9], all of which have been intensively studiedin the last three decades. Spin liquids, in particular, have been hard to characterize beyond thefact that quantum fluctuations prevent any conventional order in them. Despite extensive exper-iments, few precise conclusions about the nature of the ground state and low-lying excitationsare available, because the results are almost always dominated by cooperative effects, howeverinteresting, of the impurities [8, 9, 10, 11].We have synthesized the S = 1 / triangular lattice compounds Lu Cu Sb O (LCSO) andLu CuSb ZnO (LCZSO), and measured their thermodynamic properties and muon spin re-laxation ( µ SR) rate λ ( T ) down to 16 mK. Magnetic impurities are estimated to be less than apart in and other impurities about a part in in LCSO. There are no signatures of con-ventional or spin-glass order or any other cooperative effects of impurities down to the lowesttemperature in either. However a 5% density of deduced Schottky impurities in LCZSO areshown to change properties from the much purer LCSO very significantly.For T << θ W , the Curie-Weiss temperature ( ∼
20 K), the deduced specific heat C M = γT from magnetic excitations and λ ( T ) is a constant related to γ − followed by a logarithmicdecrease. These results are shown to be consistent with scale-invariant magnetic fluctuations.An even more surprising result is that the measured magnetic entropy in LCSO, obtained from2he specific heat from 100 mK to its saturation at high temperatures, is only about 36% of thetotal entropy k B ln 2 per spin 1/2. µ SR measurements effectively extend these results down to16 mK. The missing entropy does not change in a magnetic field up to 9 T at any temperatureup to θ W . The implications of these results, discussed later, shed a completely new light intothe nature of the ground and excited states of a highly pure spin-liquid.The materials synthesized include the isostructural nonmagnetic compound Lu Zn Sb O (LZSO). LCSO, LCZSO, and LZSO are variations on the R Zn Sb O series of compounds( R = rare earth) [12], with R = Lu and Zn completely (LZSO) or half (LCZSO) substituted byCu. So far only powder samples have been synthesized by solid state reaction methods. All thephysical properties shown below have been reconfirmed on independently grown samples.The crystal structure, lattice dimensions, and x-ray diffraction (XRD) pattern exhibitingnarrow Bragg peaks are shown in Supplementary Materials (SM) Sec. 2.1 for LCSO. The com-pounds form alternate parallel kagom´e planes of Lu and Sb (SM Figs. S1B and S1D). In allthree compounds, Cu and/or Zn ions sit in the slightly distorted hexagons of the Lu andSb kagom´e layers. The two layers in LCSO have Cu’s in distinct co-ordinations, octahedraof Lu and tetrahedra of Sb. In SM Sec. 2.3 we show the results of dc and ac susceptibilitymeasurements down to 0.1 K, estimate the magnetic impurity concentrations, and show evi-dence of no cooperative effects. LCSO is extraordinarily pure with less than − “orphan”spins and negligible Schottky impurities. The conclusion is extended down to ∼
16 mK by µ SRmeasurements. LCZSO has alternate planes with primarily Cu in the Lu layers and Zn in the Sblayers. We have not been able to make it with less than 5% substitutional impurities, most likelyinterchange of Cu and Zn which are in evidence from both XRD data and a low-temperatureSchottky contribution to the specific heat and which affect other properties significantly.In SM Sec. 2.2 we discuss the symmetry of the orbitals where the spins reside, which we ar-gue suggests the two-dimensional nature of magnetic interactions in both compounds. Although3he details of the microscopic Hamiltonian are not known, the fact that the magnetic suscepti-bility, the specific heat and the muon relaxation rate in LCSO can all be separated into twodistinct components, each characterized by the same two parameters, implies that the exchangeinteractions in the two layers interact dominantly only with other ions in the same layer. Theslight distortions from equilateral of the triangles will lead to slight variations in the exchangeinteractions in different directions.The results of our specific heat and µ SR measurements are central to the conclusions of thiswork. In low temperature specific heat measurements, especially in insulators, one must ensurethat thermal equilibrium is reached in the measurements. The steps we have taken to achieveequilibrium and the evidence for it are described in SM Materials and Methods.Figure 1A shows the measured zero-field specific heat C ( T ) in LCSO and LCZSO from60 mK to about 300 K, as well as that of the isostructural nonmagnetic compound LZSO; thelatter allows a very accurate subtraction of the lattice contribution. A weak bump in C ( T ) atabout 1 K and an increase below 0.2 K can be seen in both compounds. The low-temperatureincrease is the expected nuclear Schottky contribution (from quadrupolar splitting in zero ap-plied field) ∝ T − , which is possible to isolate from the bump because the latter is negligiblecompared to the former at low temperatures. We have carefully investigated the bump. Thevariation of both the rise and the bump with magnetic field are described in SM Sec. 2.4, wherewe show that the bump is a Schottky anomaly due to non-magnetic impurities [13] with an ex-cited magnetic state. All of our conclusions are affected by less than 0.5% whether or not wesubtract the Schottky contribution.The magnetic contribution C M ( T, H ) /T is deduced from C M /T ≡ [ C − ( C latt + C nuc + C imp )] /T , where C nuc and C imp are the nuclear and impurity Schottky contributions, respec-tively. C M ( T, H ) /T is shown in Fig. 1B for both compounds at various fields. For H = 0 C ( T ) /T is constant at low temperatures, below about 0.4 K for LCSO and below about 1 K4or LCZSO. There is an approximately logarithmic decrease with increasing temperature athigher temperatures in both compounds. This and other features are examined in detail in SMSec. 2.5, where the logarithmic behavior is shown to be characterized by the parameters closeto the respective Curie-Weiss temperatures.We turn next to the measurable magnetic entropy. Fig. 2A shows the normalized magneticentropy [ S ( T, H ) − S (0 . K, H )] /R ln 2 calculated by integrating C M /T from 0.1 K to T . For H = 0 , it saturates at high temperatures to 0.36 in LCSO and at 0.52 in LCZSO. From theproportionality of the µ SR relaxation rate to C M /T from 16 mK to 4 K (Fig. 3), we infer thatthe constant C M /T also continues to at least 16 mK. Therefore either about 60% of the magneticentropy resides in the ground state in LCSO, or the average C M /T below ∼
16 mK is about times the measured constant value above ∼
100 mK.At low temperatures a decrease of the entropy with H is as expected. Its apparent satu-ration to a smaller value with field cannot be ascertained accurately by the above subtractionprocedure at temperatures above about 25 K, where the specific heat is dominated by the latticecontribution. We determine what happens to the entropy in a magnetic field by an alternatemore accurate method, which also gives an estimate of the accuracy of the subtraction proce-dure below 25 K. In the alternate method, the magnetization M ( H, T ) of both LCSO andLCZSO is measured from 4 K to 300 K at various fields (SM Figs. S4A and S4C). We then usethe Maxwell relation ( ∂S/∂H ) T = ( ∂M/∂T ) H , and integrate ( ∂S/∂H ) T to give the change inentropy due to the magnetic field as a function of temperature. ( ∂M/∂T ) H for both compoundsare given in SM Figs. 4B and 4D. The results are displayed as red points in Figs. 2C and 2D,where the results from the direct determination by subtraction shown in Fig. 2A are shown asblack points. To get a measure of the consistency of results obtained by these quite differentmethods, we note that the standard deviation of the red and black points in LCSO is 0.02.The measurements in Figs. 2B and 2C show that the available entropy loss due to magnetic5elds at low temperatures is systematically recovered at higher temperatures to its zero-fieldvalue asymptotically. However, the missing entropy is field independent up to 9 T over thewhole temperature range. From this behavior at gµ B H ≤ k B T it follows that the missingentropy is due to purely singlet excitations. Since it is unaffected even for gµ B H >> k B T with k B T up to k B T larger than the (small) Θ W , local mutually non-interacting singlet states are alsoruled out because their population would be replaced by the doublet states favored by magneticpolarization. We have checked by measuring in a field while both warming and cooling and incooling in a field and then measuring that the behavior is unchanged. So the phenomena appearsnot to be due to metastable singlet states. µ SR is a direct probe of low-energy spin dynamics. We have carried out zero-field (ZF)and longitudinal-field (LF) µ SR measurements from 16 mK to about 20 K in both LCSO andLCZSO, the details of which are discussed in SM Sec. 2.7. Neither long-range order nor spinfreezing could be detected down to the lowest temperatures. The dynamic ZF muon spin relax-ation rate λ ZF is plotted as a function of temperature in Fig. 3. It is essentially constant belowabout 0.5 K, indicating persistent spin dynamics and a high density of magnetic fluctuations atlow temperatures [14, 15]. Also shown is the deduced C M /T . The temperature dependence of λ closely follows that of the measured C M /T . A temperature-independent relaxation rate as T → is itself extraordinary. The measured specific heat and its relation to the µ SR relaxationrate suggest a scale invariant spectral function for magnetic excitations, as discussed below andin SM Sec. 2.8.We show in the inset in Fig. 3 that the measured C M /T in LCSO can be separated intotwo parts for the two layers. The low temperature constant values for the two layers are ap-proximately inversely proportional to their respective Θ W ’s; they both decrease logarithmicallyapproximately as ln(Θ W /T ) and ln(Θ W /T ) . These forms only pertain for the ‘quantum re-gion’ below the respective Θ (cid:48) W s . The knee region between the two logarithms requires fit to6he classical region T (cid:38) Θ W . Details are given in the Supp. Sections. The integrated value,i.e. the entropy of the two layers is consistent with being equal. Similar decomposition forLCZSO is also given in the SM Sec. 2.6.1. Even with only 5% Schottky impurities which areintersubstitution of Cu and Zn in the two layers, similar separation gives that the ratio of theentropies of the two layers is no smaller than 30%. This emphasizes how important it is to havepure compounds to study spin-liquids.The theoretical results for spin liquids in relation to our experimental findings are sum-marized in SM Sec. 2.7. We have not found theoretical results on any relevant model whichcorrespond to the properties discovered here [9, 8].Both the specific heat and the µ SR relaxation rate λ ( T ) follow from a scale-invariant densityof states function A M ( ω, T ) = γ M f ( ω/T ) for magnetic fluctuations proposed in SM Sec. 2.8.Not only is the temperature dependence of λ ( T ) given by it, but its correct order of magnitude isobtained from the same coefficient γ M that reproduces the magnitude of the measured C M /T .As a function of imaginary time periodic in inverse temperature, A M ( ω, T ) is equivalent toan algebraic decay ∝ /τ . A ground-state entropy, which should more accurately be called atemperature-independent entropy, requires a more singular form A ( ω, T ) , which correspondsto a correlation function of the singlets proportional approximately to / log( τ ) . This is asquantum as one can get. Some conceptual questions related to this are briefly discussed in SMSec. 2.8. This form is chosen in the belief that the missing entropy is due to a dynamical effectand is therefore measurable by spectroscopic techniques. This form might need to be modifiedby introducing a Schottky-like term with an ultra-low energy scale with the same effect on themeasured specific heat and entropy.In summary, there are two related phenomena discovered in the very pure compound LCSO.(1) Quantitatively related constant C M /T and µ SR relaxation rates λ ( T ) below a temperaturerelated to the Curie-Weiss temperature Θ W followed by the same logarithmic cutoff in both7easurements. The excitations necessary for these are shown to be scale invariant. They areshown to carry finite spin quantum numbers because their entropy for gµ B H ≤ k B T is sys-tematically reduced due to H and leads to muon relaxation. These are shown to exhaust the measurable excitations at all temperatures up to 9 Tesla. (2) Conclusive evidence for missing entropy from colossal density of singlet excitations exist below an ultra-low energy scale com-pared to the Curie-Weiss temperature. Very interesting is also the fact that the ultra-low energyexcitations are not removed by a magnetic field as high as 9 Tesla showing that they are nottrivial local singlets but quite probably non-local and topological.In a detailed study of previous work on spin-liquids, we find that such results have not beenshown before; we think this is because LCSO can be prepared purer than any other spin-liquidinvestigated so far. The simplicity and the nature of the singularities in Eqs. (S9)–(S11), withwhich we can parametrize all the data invite important new theoretical developments. Themagnetic fluctuations suggested by A M ( ω, T ) should be accessible via neutron scattering. Thedetection of the scalar excitations A ( ω, T ) poses an interesting challenge to experimental tech-niques. Having no charge or magnetic moment, they are a form of dark matter not observableby the usual spectroscopic techniques. Acknowledgments
We are grateful to B. Hitti and D. J. Arseneau of TRIUMF and the staff of the Paul Scherrer In-stitute for their valuable help during the µ SR experiments. C.M.V. performed this work while a“Recalled Professor” at UC Berkeley, and wishes to thank the members of the condensed-mattertheory group for their hospitality.
Funding:
This research was funded by the National Researchand Development Program of China, No. 2016YFA0300503 and No. 2017YFA0303104, theNational Natural Science Foundations of China, No. 11774061, and the Shanghai MunicipalScience and Technology (Major Project Grant No. 2019SHZDZX01 and No. 20ZR1405300).8 uthor contributions:
Y.X.Y., C.M.V., and L.S. designed the experiments. Y.X.Y. and J.Z.grew the samples. Y.X.Y. carried out X-ray structure refinement and magnetic characterizations.Y.X.Y., Z.H.Z., and Q.W. performed the specific heat measurements. Y.X.Y., C.T., D.E.M., andL.S. carried out the µ SR experiments, with site assistance from T.S. Y.X.Y., L.S. and C.M.V. an-alyzed the data. C.M.V. provided the theoretical framework. All authors participated in discus-sion. The manuscript was written by Y.X.Y., L.S., C.M.V., and D.E.M.
Competing interests:
The authors declare that they have no competing interests.
Data and materials availability:
All data needed to evaluate the conclusions in the paper are present in the main text or thesupplementary materials.
Supplementary Materials
Materials and MethodsSupplementary TextFigs. S1 to S7Table S1References 9 .1 1 10 1000.0010.010.1110100 A LCSOLCZSOLZSO C ( J m o l - K - ) T (K) Zero Field 0.1 1 100.00.20.40.6 B H = 0 T 1 T 3 T 6 T 9 T C M / T ( J K - m o l - C u - ) T (K)LCSO
Fig. 1. Specific heat of LCSO, LCZSO, and LZSO. (A)
Measured specific heats in zerofield. (B)
Intrinsic magnetic contribution C M ( T, H ) /T to the specific heat coefficient at variousmagnetic fields for LCSO (filled symbols) and LCZSO (open symbols), after subtraction of thelattice, nuclear-Schottky, and impurity-Schottky contributions (See text and SM Sec. 2.4).10
100 200-0.4-0.3-0.2-0.10.0 0 10 20-0.4-0.20.00 5 10 15 200.00.10.20.30.4 B [ S M ( T , H ) - S M ( . K , H ) ] / ( R l n2 ) T (K) H = 0 T 1 T 3 T 6 T 9 TLCSO S M ( J K - m o l - C u - ) T (K)LCSO A Fig. 2. Entropy of LCSO. (A)
Change S M ( T, H ) − S M (1 K , in magnetic entropy, normal-ized to R ln 2 per spin, . ≤ T ≤ K, ≤ H ≤ T. (B) Red symbols: change ∆ S M inmagnetic entropy in an applied field of 9 Tesla as a function of temperature from 2 K to 300 Kfrom measurements of magnetization in LCSO (see text and SM Sec. 2.4). Black symbols: thesame quantity up to 20 K from the direct determination of magnetic entropy given in Panel A.The lost magnetic entropy is fully recovered at high temperatures, proving that the missingentropy is independent of the applied field. 11 ig. 3. Muon spin relaxation rate and specific heat in LCSO. A. Temperature dependenciesof zero-field muon spin relaxation rate λ ( T ) (red dots: data taken at PSI; red squares: datataken at TRIUMF) and C M ( T ) /T at zero field. It is remarkable that the relaxation rate tendsto a constant value at low temperatures, and that it follows the temperature dependence of C M /T over the entire temperature range. In the inset we show that the specific heat can beseparated into two parts for the two layers. The constant values are approximately inverselyas their respective Θ (cid:48) W s as determined by the fit to the magnetic susceptibility measurements.The characteristic temperature of the two logarithmic terms are also similar to the respective12 (cid:48) W s . Detailed numbers are given in a SM Sec. 2.5. The knee between the two logarithms, for T (cid:38) Θ W requires a classical form, which we fit to the expression mandated for T >> Θ W .With this fit, the measured entropy of the two layers is consistent with being the same.13 upplementary MaterialsThis PDF file includes: • Materials and Methods• Supplementary Text• Figs. S1 to S7• Table S1• References 14 Materials and Methods
We have synthesized the compound Lu Cu Sb O (LCSO) by the solid state reaction method.Stoichiometric amounts of Lu O , CuO and Sb O were thoroughly mixed using an agate mor-tar, and heated to 1030 ◦ C for 60 hours with intermediate regrinding and reheating. So faronly powder samples could be synthesized. The crystal structure was determined from pow-der X-ray diffraction (XRD) data taken at room temperature using a Bruker D8 advance XRDspectrometer ( λ = 1 . ˚A). Rietveld refinement of the X-ray data was made using the GSASprogram [16].LCSO belongs to the rhombohedral pyrochlore family [12, 17], in which kagom´e latticesare formed by alternating layers of filled-shell ( S = 0 ) Sb and Lu ions. The spin-1/2 Cu ions sit at the centers of the kagom´e hexagons (Cu1-Sb and Cu2-Lu). To determine whetherthe observed properties are specific to the 2D layers, we have also synthesized the related com-pound Lu CuZnSb O (LCZSO), in which nonmagnetic layers, where S = 0 Zn ions replaceCu1, alternate with Cu2 layers. We have been unable to synthesize this compound with lessthan about ∼ substitutional disorder, despite efforts with different growth protocols. DC magnetic susceptibility measurements above 2 K were made using a Magnetic PropertyMeasurement System (MPMS, Quantum Design). The AC magnetic susceptibility was mea-sured over the temperature range 0.1 K–4 K in a PPMS equipped with AC Susceptibility anddilution refrigerator options. The AC Susceptibility measurements covered the frequency rangefrom 631 Hz to 10000 Hz. 15 .3 Specific heat measurements
Specific heats were measured by the adiabatic relaxation method, using a Physical PropertyMeasurement System (PPMS, Quantum Design) equipped with a dilution refrigerator. Datawere taken at temperatures between 50 mK and 300 K for LCSO and LCZSO, and 0.2 K–300 Kfor the isostructural nonmagnetic compound Lu Zn Sb O (LZSO). We took special care toensure that thermal equilibrium was achieved during the low-temperature measurements. As anexample, the lowest temperature ( ∼
50 mK), the measurement took 70 minutes. The thermalcoupling factor between the sample and sample platform reported by the PPMS was 95% at100 mK and 99% for temperatures above 0.6 K. Measurements were made during coolingdown to the lowest temperature as well as warming up. Similarly when measuring the specificheat in a magnetic field, the sample was both cooled in a magnetic field and then measuredon warming, as well as with the field applied at low temperatures and then measured duringwarming. The measurements were always consistent with each other.
The time-differential µ SR technique [18] was used, in which the evolution of the ensemblemuon-spin polarization after implantation into the sample is monitored via measurements of thedecay positron count-rate asymmetry A ( t ) . µ SR experiments were performed down to 16 mKusing the DR spectrometer on the M15 beam line at TRIUMF, Vancouver, Canada, and theDolly spectrometer at the Paul Scherrer Institute, Villigen, Switzerland. Samples were attachedto a silver cold-finger sample holder in the DR spectrometer, to ensure good thermal contactwith the mixing chamber. Appropriate functional forms of A ( t ) were fit to the asymmetry datausing the MUSRFIT µ SR analysis program [19].16
SUPPLEMENTARY TEXT
Fig. S1 shows the XRD powder pattern for LCSO, together with details of the crystal structure.Table S1 lists the fitted values of the structure parameters.
Fig. S1. Rietveld refinement of XRD data for LCSO. (A)
Powder XRD pattern. Red points,black line, and blue line: experimental data, calculated patterns, and residuals, respectively.Black bars: Bragg reflections. (B)
Unit cell. (C)
Inequivalent Cu1 and Cu2 environments. (D)
Inequivalent triangular lattices formed by Cu the spin-1/2 Cu ions in the ab plane.In LCSO, inequivalent Cu1 and Cu2 sites form two sets of triangular lattices surrounded by Sband Lu atoms, respectively. According to the XRD refinement results, Cu1 ions are displacedfrom the center 3 b sites (0, 0, 0.5) and located at 18 g sites ( x , 0, 0.5) with a partial site occupationof 1/6. Such a slight distortion of the Cu1 position (the refined value of x is 0.0505) is consistentwith previous work [12, 17].In LCZSO, ∼
5% Cu2 sites on the Cu-Lu layer are occupied by Zn ions, and ∼
5% Zn sites onZn-Sb layer are occupied by Cu1 ions. We show below that a ∼
5% impurity level is consistentwith the analysis of the impurity Schottky specific heat.17 able SI. Rietveld XRD fitting results. Lu Cu Sb O a = b = 7 . ˚A; c = 16 . ˚A α = β = 90 ◦ ; γ = 120 ◦ Space Group: R - m Atom Wyckoff positions x y z
Occ.Sb 9d 0.5 0 0.5 1Cu1 18g 0.0505(4) 0 0.5 0.1667Cu2 3a 0 0 0 1Cu2 (disorder) 9e 0.5 0 0 0Lu 9e 0.5 0 0 1Lu (disorder) 3a 0 0 0 0O1 6c 0 0 0.3922(7) 1O2 18h 0.4838(2) -0.4838(2) 0.1308(4) 1O3 18h 0.0299(5) -0.0299(5) -0.0274(1) 1Lu CuZnSb O a = b = 7 . ˚A; c = 16 . ˚A α = β = 90 ◦ ; γ = 120 ◦ Space Group: R - m Atom Wyckoff positions x y z
Occ.Sb 9d 0.5 0 0.5 1Cu1 18g 0.0320(1) 0 0.5 0.0083Zn1 18g 0.0401(4) 0 0.5 0.1584Cu2 3a 0 0 0 0.95Zn2 3a 0 0 0 0.05Lu 9e 0.5 0 0 1Lu (disorder) 3a 0 0 0 0O1 6c 0 0 0.5359(4) 1O2 18h 0.5016(1) -0.5016(1) 0.1231(8) 1O3 18h 0.3087(2) -0.3087(2) -0.3267(1) 1 ions and the symmetry of the orbitals From the detailed structure analysis and the Rietveld refinement, one finds that the Cu in theLu layers are tetrahedrally coordinated and those in the Sb layers are octahedrally coordinatedby the O − ions (see structure in Fig. S1B–D). The tetrahedra do not appear to be distorted, but18he octahedral axis perpendicular to the Cu layers is elongated.In an ideal tetrahedral crystal field, the xy , yz and zx orbitals are higher in energy anddegenerate. Their linear combinations form bonds with the oxygen. The Cu d holes then sitin these linear combinations. In an ideal octahedral crystal field, the levels are reversed andthe higher energy states are the degenerate d x − y and d z − r orbitals. Due to the elongationof the octahedra, there is a further splitting so that d z − r is the half occupied highest orbital.Moreover, in LCSO the Cu ions in the Sb layers sit in the center of the triangles formed by theCu in the Lu layers, each of which has orbitals with four-fold phase variation. The exchangeintegral between the Cu ions in different layers would then be zero.This suggests that in LCSO, where the intralayer distance between Cu sites is shorter thanthe interlayer distance, the interaction between layers is weak. Therefore LCSO has similarproperties per Cu ion as LCZSO, where Cu ions only occupy the Lu layer. We expect thatboth are two-dimensional as far as the magnetic interactions between the spin-1/2 Cu ionsare concerned. These arguments are only suggestive, however, and more evidence, e.g, frommeasurements of crystal-field levels in single crystals, would be required to substantiate themas well as to learn about effects on the levels due to other small distortions of the octahedra. Fig. S2A shows the measured inverse DC susceptibility from 2 K upwards for both LCSO andLCZSO. The former shows a low field Weiss temperature θ W of 4.4K and µ eff = 1 . µ B ,consistent with a weakly perturbed free Cu ion. In Sec. 2.5 below we show that in LCSO thesusceptibility is best fit by two Curie-Weiss laws of equal amplitude, with µ eff ≈ . µ B andWeiss temperatures 4.37 K and 26.9 K. 19
20 40 60 80 1000246 7.07.58.08.59.02 3 4024 H = 0.5 T LCSO / d c ( e m u - O e m o l - C u ) T (K)
A B ’ a c ( a r b . un i t ) C LCZSOLCSOStatic field: 0 OeDriving field: 1 Oe Frequency: 631 Hz0.1 1 4.55.05.56.06.5T (K) LCSO H = 5 T d c ( - e m u O e - m o l C u - ) T (K) 0246 LCZSO / ( d c - ) ( e m u - O e m o l - C u ) Fig. S2. Magnetic susceptibility. (A)
Temperature dependence of inverse magnetic sus-ceptibility /χ dc of LCSO and LCZSO, Inset: χ dc of LCSO, ≤ T ≤ K. Tempera-ture dependence of the real part of the AC susceptibility χ (cid:48) ac in LCSO (B) and LCZSO (C) , . ≤ T ≤ . K. Similar temperature dependencies were found for frequencies up to10 kHz.We also tried to estimate the impurity density by adding a Curie law ( θ W ≈ , appropriate tonearly-free impurity spins) and readjusting the Weiss temperatures. Equally good fits are foundwith an impurity concentration of O (5 × − ) .Much stronger estimates of impurity concentrations are obtained from the AC susceptibilitymeasured over the temperature range 0.1 K–3 K, shown in Figs. S2B and S2C. By ascribing allthe temperature dependence of the AC susceptibility in this temperature range to a Curie law,we find accurate upper limits to the concentrations of free-spin magnetic impurities to be lessthan about − in LCSO and about twice that value in LCZSO. No frequency dependence isfound up to 10 kHz (only the lowest frequency results are reported in Figs. S2 B and C). Thusno spin freezing behavior is observed down to 0.1 K. The AC susceptibility, as distinct fromthe DC susceptibility, is not given in absolute units. The region where no transition of any kind20including a spin-glass transition) could be evidenced through the AC susceptibility is extendeddown to 20 mK by muon spin relaxation experiments, shown below in Fig. S6. The lattice contribution to the specific heat of LCSO and LCZSO can be obtained very accu-rately from measurement of the specific heat of the nonmagnetic isostructural compound LZSO(main article, Fig. 1A). As is evident from this figure, after subtracting the lattice contribution,the specific heat in both LCSO and LCZSO exhibits a weak bump at about 1 K and an increasebelow about 0.2 K.The low-temperature rise is due to the nuclear Schottky contribution C nuc ( T, H ) , whichvaries as C nuc ( T, H ) = A ( H ) T − (S1)for temperatures high compared to the nuclear spin splitting (quadrupolar in zero and low fields).By fitting Eq. (S1) to the rise at the lowest temperatures (where contributions from the lattice andthe bump are negligible), the coefficient A ( H ) could be determined. The nuclear contributionat various H and the field dependence of A ( H ) are shown in Fig. S3A.The weak bump is expected to be a Schottky contribution due to impurities with nonmag-netic ground states. If so, the integrated entropy under the bump should remain constant as themagnetic field is varied, even though the bump may move linearly to higher temperature withincreasing field due to excited magnetic states. This new hypothesis was tested by using thewell-known contribution due to Schottky defects with a density of impurities n : C imp ( T ) = nR (cid:18) ∆ T (cid:19) g g exp(∆ /T )[1 + ( g /g ) exp(∆ /T )] , (S2)where R is the molar gas constant, ∆ is the energy level splitting, and g and g are the degen-eracies of the lower and upper levels. This expression provides a very good fit to the bump with g /g set to 1, ∆ = 1 K, and n = 0 . ± . at zero applied field for LCSO.21 ig. S3. Nuclear and impurity Schottky specific heats. (A) Nuclear contribution to specificheat C nuc at different magnetic fields. Curves: fits of Eq. (S1) to the data. Inset: fitting parameter A using C nuc = A ( H ) T for both LCZSO and LCSO. (B) Impurity Schottky contribution C imp to specific heat at different applied magnetic fields for LCSO Curves: fits of Eq. (S2) to the data. (C) Field dependence of the parameters n and ∆ for LCZSO and LCSO from the fits of Panel B. (D) Specific heat divided by temperature in LCSO at zero magnetic field. Red: measured total.Black: lattice contribution. Green: nuclear Schottky contribution. Yellow: impurity Schottkycontribution. Blue: intrinsic magnetic contribution C M /T .As shown in Fig. S3C, n remains constant as H is varied and ∆ increases linearly with H : ∆ = ∆ + gµ B H , where µ B is the Bohr magneton and the Land´e g factor is found to be22.3. The area under the bump remains constant as shown in Fig. S3B. In LCZSO, the valueof n = 0 . ± . , which is the same value given for impurities in LCZSO by the Rietveldrefinement of the structure. This consistency is strong evidence for the hypothesis of an impuritySchottky contribution. For LCSO n is too small to be determined from the XRD powder pattern. (d (cid:1) / d T ) H (emu mol-Cu-1 K-1) (cid:1) H ( T )5 K7 K1 5 K2 5 K5 0 K2 0 0 K C DBA
L C Z S O (d (cid:1) / d T ) H (emu mol-Cu-1 K-1) (cid:1) H ( T )5 K7 K1 5 K2 5 K5 0 K2 0 0 K1 1 0 1 0 005 0 01 0 0 01 5 0 02 0 0 0 (cid:1) H = 1 T 3 T 5 T 6 T 7 TL C S O (cid:1) (emu mol-Cu-1) T ( K )1 1 0 1 0 005 0 01 0 0 01 5 0 02 0 0 02 5 0 03 0 0 0 (cid:1) H = 1 T 3 T 5 T 7 TL C Z S O (cid:1) (emu mol-Cu-1) T ( K ) Fig. S4. Magnetization and entropy. (A)
Measured magnetization from 2 K to 300 K forvarious magnetic fields for LCSO. (B) ( ∂M/∂T ) H for LCSO at a few selected temperatures asa function of magnetic field. (C, D) As above, for LCZSO. These results are used to obtain thechange in entropy with field at various temperatures as shown in the main article, Figs. 2B and2C.The intrinsic magnetic contribution C M ( T ) to the specific heat is then obtained by sub-23racting the nuclear contribution C nuc ( T ) and the nonmagnetic impurity Schottky contribution C imp ( T ) as determined above from the difference of the total specific heat and the lattice con-tribution. The various contributions at H = 0 are separately shown in Fig. S3 D. C M ( T, H ) forboth LCSO and LCZSO at various fields is shown in the main article, Fig. 1B.A further test of the consistency of the procedure is to determine the variation of entropy S ( T, H ) through the measurement of magnetization M ( H, T ) at various H and T . We use theMaxwell relation (cid:18) ∂S∂H (cid:19) T = (cid:18) ∂M∂T (cid:19) H , (S3)and compare S ( T, H ) obtained by this method with that determined directly from C M ( T, H ) . M ( H, T ) and ( ∂M/∂T ) H are shown in Fig S4, and the deduced change in entropy for µ H =9 T is shown in Figs. 2B and 2C of the main article. The agreement also provides a quantitativemeasure of the consistency between the results of these quite different techniques. As discussedin the main article, S ( T, H ) determined through the magnetization serves to determine the slowcrossover at high temperatures of the measured entropy at finite H to its value at H = 0 . d c ( - e m u O e - m o l - ) T (K) 05101520 B LCSO: Cu1 Cu2 / d c ( e m u - O e m o l ) A T (K) / d c ( e m u - O e m o l ) d c ( - e m u O e - m o l - ) T (K) 010203040LCZSO: Cu1 Cu2 / d c ( e m u - O e m o l ) T (K) / d c ( e m u - O e m o l ) Fig. S5. Two components in susceptibility of LCSO and LCZSO. (A)
DC magnetic suscep-tibility χ dc of LCSO measured at µ H = 0 . T (black circles). Curves: two Curie-Weiss24ependencies, with the same effective moment and different Weiss temperatures Θ CW , = − . K, Θ CW , = − . K. Inset: inverse /χ dc vs T , showing downward curvature atlow temperatures. (B) DC magnetic susceptibility χ dc of LCZSO measured at µ H = 0 . T(black circles).Same as in LCSO, two Cu sublattices contribute to the susceptibility of LCZSO,but only ∼ of total susceptibility is from Cu1 sublatice.For LCZSO, while χ dc can also be fit by a sum of two Curie-Weiss terms, the majoritycontribution comes from Cu1 sublattice, χ = 0 . N A µ k B ( T − Θ CW , ) + 0 . N A µ k B ( T − Θ CW , ) , with a common value of µ eff = 1 . µ B and two Weiss temperatures Θ CW , = − K and Θ CW , = − . K. This is consistent with the observation from XRD that ∼ Zn sites onZn-Sb layer are occupied by Cu1 ions.
Inequivalent Cu1 and Cu2 sites in LCSO form two triangular sublattices coordinated by Sbatoms and Lu atoms, respectively. Here we present evidence that these two sublattices makeadditive contributions to the specific heat and the susceptibility. C M /T is constant below about 0.6 K in LCSO (main article Fig. 1B) and similar temperaturein LCZSO (See below). A fit of two components to the data for LCSO is shown in Fig. 2B,where each component is constant at low temperatures followed by a logarithmic decrease athigher temperatures [Eqs. (S5) and (S6)]. The ratio of the constants at low temperature isapproximately as the inverse of the ratio of their respective Θ W ’s.Quantitatively, C mag /T = C /T + C /T , (S4)25here C /T ≈ . , T < . K , . . /T ) , . K < T < K , . /T, T > K (S5)and C /T ≈ (cid:26) . , T < K , .
096 ln(20 /T ) , T > K . (S6)In the fitting of the specific heat in Fig. 3 as well as in Fig. S7, the knee between the twoapproximately logarithmic decrease is observed. We fit this with the contribution mandatoryin the ‘classical regime’, i.e.
T >> Θ W where C M /T = α (1 /T )( Θ W T ) and α is a constantwhich depends on lattice and number of nearest neighbors.We note that the characteristic temperatures in the logarithms in the specific heat are alsoclose to the Weiss temperatures or characteristic exchange energies measured in the magneticsusceptibility (Fig. S5A). As shown in Fig. S5A, the temperature dependence of the DC mag-netic susceptibility χ dc can be fit by a sum of two Curie-Weiss terms with equal proportion χ = N A µ k B ( T − Θ CW , ) + N A µ k B ( T − Θ CW , ) , with a common value of µ eff = 1 . µ B and two Weiss temperatures Θ CW , = − . Kand Θ CW , = − . K. The inset shows the downward curvature of /χ dc vs. T that is thesignature of a second contribution with a smaller value of Θ CW . Zero-field (ZF) and longitudinal-field (LF) µ SR experiments were performed over the temper-ature range 16 mK–20 K. Positive muons implanted in the sample are highly sensitive to thelocal magnetic fields, with a resolution about 0.1 mT [20].26 ig. S6. Zero- and Longitudinal-field µ SR data.
Temperature and field dependencies of µ SRasymmetry spectra. (A)
LCSO. (B)
LCZSO. Curves: fits discussed in the text. A constantsignal from muons that miss the sample and stop in the sample holder has been subtracted fromthe data.Representative ZF- µ SR asymmetry spectra are shown in Fig. S6 for both LCSO and LCZSO.A constant background signal from muons that stop in the sample holder has been subtracted.No long-range order or spin freezing is observed. This is evidenced by the lack of any spon-taneous coherent oscillation or initial asymmetry loss in the spectra [21, 22]. Static spin-glassbehavior is also excluded due to the absence of a / recovery tail of the muon depolarizationdue to the random distribution of the static fields [23].The ZF spectra are best described by the functional form A ( t ) = A exp( − λ ZF t ) G KTZF ( σ, t ) , (S7)where A is the initial count-rate asymmetry, which is found to be temperature independent,and G KTZF ( σ, t ) = 13 + 23 (1 − σ t ) exp (cid:0) − σ t (cid:1) (S8)27s the ZF Kubo-Toyabe (KT) function, expected [24] from a Gaussian distribution of randomly-oriented static or quasistatic nuclear dipolar fields at muon sites. In Eqs. (S7) and (S8) σ/γ µ isthe the rms width of this distribution, γ µ = 2 π × . MHz/T is the muon gyromagnetic ratio,and λ ZF is the rate of exponential damping due to dynamic fluctuations of the local Cu +2 spins. σ is found to be temperature independent at low temperatures, with σ = µ s − for LCSOand σ = µ s − for LCZSO, typical values of the nuclear dipolar field distribution fromthe host. With decreasing temperature λ gradually increases, saturates below 1K, and remainsessentially constant down to the lowest measured temperature of 16 mK.A stretched exponential function A ( t ) = A exp − ( λt ) β has also been used to fit the ZF data.The fits are fine but with relatively less goodness of fit. Similar temperature dependence of λ is observed, and β is temperature independent for all the temperature measured, with β ∼ . .This justifies the validity of Eq. (S7).By applying longitudinal magnetic fields (i.e., along the muon initial spin polarization), theKubo-Toyabe form G KTZF ( σ, t ) in Eq. (S7) changes to G KTLF ( H L , σ, t ) [24]. Fig. S6 also showsasymmetry spectra for representative values of H L . When the applied fields are small, thefield dependence is dominated by decoupling of the static relaxation. At higher fields, thestatic contribution is completely decoupled, and the µ SR asymmetry spectra show a singleexponential decay. Although the dynamic relaxation becomes slower with increasing magneticfield, it is not completely suppressed even at 2.5 T. This is evidence that the observed relaxationarises from persistent dynamic spin fluctuations.28 .6.1 Specific heat, Entropy and Muon spin relaxation in LCZSOFig. S7. Data for LCZSO A : Magnetic specific heat divided by temperature C M /T , and zero-field µ SR relaxation rate λ ZF ( T ) . The inset shows the partition of two components of thespecific heat for the two layers in LCZSO deduced in the same way as Fig. (3B) for LCSO. B : C M /T at the specified magnetic fields. C : The magnetic entropy obtained by integrating Fig. S7 (B). D : Black points give the change in entropy in a magnetic field by direct measurement of the spe-cific heat, while the red points are deduced from the magnetization measurements as describedin the textThe µ SR relaxation rate λ ( T ) and C M ( T ) /T for LCZSO in Fig. S7 show the same generalbehavior as those for the purer LCSO shown in Fig. 3 of the main text. In both compounds thetwo quantities are nearly constant at low temperatures. In LCZSO the fall-off of λ ( T ) occurs29t a higher temperature compared to that in C M /T while in LCSO they occur at about the sametemperature. The crossover from the constant C M /T to its decreasing value shown in A. issmoother than in LCSO. The inset in Fig. S7 - A shows the partitioning of the specific heat intothose for two layers. The ratio of the entropy in the two layers is close to 3:1 although in theideal case, i.e. without Cu and Zn substitution in the two layers, there would be no magneticentropy in one of the layers. The substitution determined by x-ray diffraction and susceptibilitymeasurements, as described earlier is 5% but it has a much larger effect in disrupting the system. Detailed numerical calculations [25] on the S = 1 / Heisenberg model on a triangular latticegives an ordered three-sublattice state with reduction of the order parameter by zero-point fluc-tuations of about 36% for the nearest-neighbor interaction model. We do not know how theslight distortion of the triangular lattice in these compounds might affect the numerical resultsfor the nearest neighbor Heisenberg model. Numerical calculations on models with substantialnext-nearest-neighbor interactions on a triangular lattice [26] have given a quantum-disorderedstate with gapless excitations conjectured to be spinons , but no ground state entropy or indica-tions of a gigantic peak in exponentially low energy singlet excitations. Spinons have a Fermisurface and therefore a linear-in- T specific heat, but the magnetic fluctuations associated withthe Fermi surface lead to a relaxation rate proportional to the density of thermal excitations. Itis therefore proportional to T at low temperatures, similar to the Korringa rate in metals andunlike the constant rate found here.Calculations on models for ice [27] or spin ice [28] [which agree with experiments[29, 30]],and glass or spin-glass models, possess ground state entropy, but they are are obviously inappli-cable here. Very careful calculations [31] and analysis of a high-temperature series expansionup to 17th order for S = 1 / Heisenberg spins on a kagom´e lattice have found a missing entropy30f about 1/2 the total value down to temperatures of O ( J/ , the lowest to which the calcu-lations are reliable. The most studied kagom´e compound, herbertsmithite [10] has been foundexperimentally to have scale-invariant magnetic excitations and a specific heat closely related tothe form suggested below, but the compound suffers from substantial disorder. No determina-tion of magnetic entropy is available for herbertsmithite, because its nonmagnetic counterparthas not been found.The only quantum-mechanical models known definitively to give ground-state entropy aremodels of impurity spins in a metal with parameters tuned to give singularities at T → : the2-channel Kondo model [32, 33, 34], the two-interacting Kondo-impurity model [35, 36, 37],and mixed-valence impurity models [38, 39]. All these models are supersymmetric at criticality,and have Majorana excitations proportional to the density of dilute noninteracting impurities.Holographic field theory models [40, 41, 42] do have ground state entropy as well as observablespecific heat with various power laws including linear. (0+1)-dimensional disordered effective-impurity models such as the SYK model [43, 44] also have extensive ground-state entropy aswell as gapless fermion excitations giving a linear-in- T specific heat. The mapping of the SYKmodel to AdS theory of black holes has also been discussed [44]. Black holes are conjecturedto be quantum-mechanical and their physics is fashioned parallel to the thermodynamic laws[45]. They are believed to have an observable linear-in- T entropy [1, 2]. The properties reported here can be used to specify some features of the frequency-dependentcorrelation functions that a fundamental theory might provide. Let us consider only the purelimit of the experimental results for H = 0 . We show that the measured specific heat and themuon relaxation rate follow if there are magnetic fluctuations with local density of states ofa specific scale-invariant form, A M ( ω, T ) . The ground state entropy requires a more singular31orm A ( ω, T ) . A loc ( ω, T ) ≡ (cid:88) q A ( q , ω, T ) δ ( ω − ω q ) = A ( ω, T ) + A M ( ω, T ) , (S9) A ( ω, T ) = S ωω + T e − ω/T , (S10) A M ( ω, T ) = γ M ωT , for ωT (cid:28) , = γ M ln (cid:18) ωT + T x (cid:19) , for T (cid:28) ω (cid:46) T x . (S11)Here q specifies the quantum numbers of the fluctuations, which are different for the two contri-butions. A M ( ω, T ) , as shown below, provides the measured specific heat C M /T ≈ ( γ M k B ) for T (cid:28) T x , and the µ SR relaxation rate, which requires magnetic field fluctuations. A ( ω, T ) , thelocal density of states of singlet excitations, can easily be modified if later experiments reveala gigantic peak in C M /T at very low temperatures. S and γ M T with a cutoff T x of order theWeiss temperatures, are related through the sum rule that the total entropy is k B ln 2 per spinat high temperatures. The fluctuations are assumed to obey Bose-Einstein statistics with zerochemical potential. Were it to turn out that they are hard core bosons or neutral fermions, obvi-ous modifications in the hypotheses above would be required. The possibility that the spectrumrepresents unfamiliar particles with unfamiliar statistics should also be entertained.The functional form of A M ( ω, T ) is derived at criticality in all the impurity models men-tioned above, the 2D dissipative quantum xy model [46, 47, 48] and the SYK impurity model[43, 44]. In every case, the low energy excitations in these toy models are topological and soare the T → states.The free energy F is given by F = − kT ln Tr Z, Tr Z = (cid:88) λ e − βE λ , (S12)where λ contains the quantum numbers q as well as their occupation number. Summing over32he occupation numbers gives, as usual, ln Tr Z = (cid:88) q ln (cid:18)
12 sinh( βω q / (cid:19) . (S13)Let us first calculate the entropy due to A ( ω, T ) . One finds the entropy from the free energy[ S = − ( ∂F/∂T ) V ]: S = k B ddT T (cid:90) dω A ( ω, T ) ln (cid:18)
12 sinh( βω/ (cid:19) , (S14) = k B S (cid:16) (cid:90) ∞ dx x e − x x f ( x ) (cid:17) , f ( x ) = ln( 12 csc( x/ . (S15)The value of the integral is approximately 0.275.We briefly comment on the use of the Bose-Einstein distribution with zero chemical poten-tial. The reason this works is that even with zero chemical potential, as can be easily calculated,the number of excitations remains independent of temperature with the choice of the singu-lar density of states of excitations. The divergent damping of the excitations implied by thedensity of states also obviates a Bose-Einstein condensation. The colossal degenerate fluctuat-ing singlet state is however likely to be unstable to other states by perturbations, for examplesuperconductivity on promoting itinerant charge states by doping.The local density of states function A M ( ω, T ) gives a free energy proportional to T withlogarithmic corrections at high temperatures, an entropy of the deduced form, and a measurablespecific heat C M ( T ) /T ≈ γ M k B . (S16)at low temperatures, with a logarithmic cutoff for temperatures above T x . From the measuredvalue of C M ( T ) /T ≈ mole at low temperatures, γ M ≈ × − s. It is noteworthythat its inverse is close to the measured Weiss temperature.In zero magnetic field, the muon relaxation rate due to magnetic field (dipolar) fluctuations33t the muon site given by λ ( T ) = γ µ lim( ω → Tω (cid:88) q | B loc ( q ) | Im χ ( q , ω ) . (S17)where γ µ = 2 π × . MHz/T is the muon gyromagnetic ratio and Im χ ( q , ω ) is the spectrumof magnetic fluctuations, which summed over q is identified as A M ( ω, T ) . This is often writtenas λ ( T ) = γ µ (cid:104) B (cid:105) τ M ( T ) , (S18)where (cid:104) B (cid:105) is the mean-square fluctuation of the local magnetic fields at muon sites, and τ M ( T ) is the characteristic correlation time of the local field fluctuations [24]. For A M ( ω, T ) given byEq. (S11), the temperature dependence of λ ( T ) is seen to be the same as that of C M /T , as in theexperimental results shown in Fig. 3 of the main article. Since A ( q , ω ) is the absorptive part ofthe magnetic fluctuation spectra, it follows that at low temperatures τ M ≈ γ M . Quantitatively,from the measured λ ( T ) and the deduced γ M , we can deduce (cid:104) B (cid:105) using Eq. (S18). Themeasured λ ≈ . µ s − and τ M ≈ γ M ≈ × − s, which gives a local rms fluctuatingfield (cid:104) B (cid:105) / of about 0.1 T. This is roughly the field from ∼ µ B moments at a distanceof about 4 ˚A, which is what is to be expected for S = 1 / Cu dipolar fields at a typicalmuon location. Thus the same fluctuations which contribute to the specific heat account semi-quantitatively for the muon relaxation rate. References [1] S. Carlip,
Int. J. Mod. Phys. D , 1430023 (2014).[2] J. Maldacena, arXiv e-prints p. arXiv:1810.11492 (2018).[3] J. Ellis, Nucl. Phys. A , 187c (2009). PANIC08.[4] Y. Li,
The Theory of Quantum Computation (2016).345] R. Prange, S. E. Girvin,
The Quantum Hall effect (Springer Verlag, Third Edition, 1990).[6] C. M. Varma,
Rep. Prog. Phys. , 082501 (2016).[7] C. M. Varma, Rev. Mod. Phys. , 031001 (2020).[8] C. Broholm, et al. , Science (2020).[9] L. Savary, L. Balents,
Rep. Prog. Phys. , 016502 (2016).[10] T.-H. Han, et al. , arXiv e-prints p. arXiv:1402.2693 (2014).[11] K. Kitagawa, et al. , Nature , 341 (2018).[12] M. B. Sanders, J. W. Krizan, R. J. Cava,
J. Mater. Chem. C , 541 (2016).[13] C. He, et al. , Appl. Phys. Lett. , 102514 (2009).[14] Y. J. Uemura, et al. , Phys. Rev. Lett. , 3306 (1994).[15] Z.-F. Ding, et al. , Phys. Rev. B , 174404 (2018).[16] B. H. Toby, R. B. V. Dreele, J. Appl. Crystallogr. , 544 (2013).[17] K. Li, et al. , J. Solid State Chem. , 80 (2014).[18] A. Yaouanc, P. Dalmas de R´eotier,
Muon Spin Rotation, Relaxation, and Resonance: Ap-plications to Condensed Matter , International Series of Monographs on Physics (OxfordUniversity Press, New York, 2011).[19] A. Suter, B. M. Wojek,
Phys. Procedia , 69 (2012).[20] A. Amato, Rev. Mod. Phys. , 1119 (1997).[21] A. Keren, et al. , Phys. Rev. B , 6451 (1996).3522] X. G. Zheng, et al. , Phys. Rev. Lett. , 057201 (2005).[23] Y. J. Uemura, T. Yamazaki, D. R. Harshman, M. Senba, E. J. Ansaldo, Phys. Rev. B ,546 (1985).[24] R. S. Hayano, et al. , Phys. Rev. B , 850 (1979).[25] F. Mezzacapo, J. I. Cirac, New J. Phys. , 103039 (2010).[26] S. Hu, W. Zhu, S. Eggert, Y.-C. He, Phys. Rev. Lett. , 207203 (2019).[27] L. Pauling,
The Nature of the Chemical Bond (Cornell University Press, Third Edition,1960).[28] P. W. Anderson,
Phys. Rev. , 1008 (1956).[29] W. F. Giauque, J. W. Stout,
J. Am. Chem. Soc. , 1144 (1936).[30] A. P. Ramirez, Ann. Rev. Mat. Sci. , 453 (1994).[31] G. Misguich, B. Bernu, Phys. Rev. B , 014417 (2005).[32] P. Nozi`eres, A. Blandin, J. Phys. France , 193 (1980).[33] I. Affleck, A. W. W. Ludwig, Phys. Rev. Lett. , 161 (1991).[34] V. J. Emery, S. Kivelson, Phys. Rev. B , 10812 (1992).[35] B. A. Jones, C. M. Varma, J. W. Wilkins, Phys. Rev. Lett. , 2819 (1988).[36] I. Affleck, A. W. W. Ludwig, Phys. Rev. Lett. , 1046 (1992).[37] C. Sire, C. M. Varma, H. R. Krishnamurthy, Phys. Rev. B , 13833 (1993).[38] I. E. Perakis, C. M. Varma, A. E. Ruckenstein, Phys. Rev. Lett. , 3467 (1993).3639] C. Sire, C. M. Varma, A. E. Ruckenstein, T. Giamarchi, Phys. Rev. Lett. , 2478 (1994).[40] T. Faulkner, N. Iqbal, H. Liu, J. McGreevy, D. Vegh, arXiv e-prints p. arXiv:1003.1728(2010).[41] K. Jensen, S. Kachru, A. Karch, J. Polchinski, E. Silverstein, Phys. Rev. D , 126002(2011).[42] J. Zaanen, K. Schalm, Y.-W. Sun, Y. Liu, Holographic Duality in Condensed MatterPhysics (Cambridge University Press, 2015).[43] S. Sachdev, J. Ye,
Phys. Rev. Lett. , 3339 (1993).[44] A. Kitaev, S. J. Suh, J. High Energy Phys. , 183 (2018).[45] J. M. Bardeen, B. Carter, S. W. Hawking,
Comm. Math. Phys. , 161 (1973).[46] V. Aji, C. M. Varma, Phys. Rev. Lett. , 067003 (2007).[47] L. Zhu, Y. Chen, C. M. Varma, Phys. Rev. B , 205129 (2015).[48] C. Hou, C. M. Varma, Phys. Rev. B94