Quantum Phase Transition of a Magnet in a Spin Bath

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R EPORTSThe Ho ions in LiHoF 4are placed on atetragonal Scheelite lattice with parametersa 0 5.175 ) and c 0 10.75 ). The crystal-fieldground state is a G 3,4doublet with only a ccomponent to the angular momentum andhence can be represented by the s z 0T1Isingstates. A transverse field in the a-b planemixes the higher lying states with the groundstate; this produces a splitting of the doublet,equivalent to an effective Ising model field.The phase diagram of LiHoF 4(Fig. 1A) wasdetermined earlier by susceptibility measurements(10) and displays a zero-field T cof 1.53K and a critical field of H c0 49.5 kOe in thezero temperature limit. The same measurementsconfirmed the strong Ising anisotropy,with longitudinal and transverse g factors differingby a factor of 18 (10). The suddenincrease in H cbelow 400 mK was explained byalignment of the Ho nuclear moments throughthe hyperfine coupling. Corrections to phasediagrams as a result of hyperfine couplingshave a long history (18) and were noted for theLiREF 4(RE 0 rare earth) series, of whichLiHoF 4is a member, more than 20 years ago(19). What is new here is that the applicationof a transverse field and the use of highresolutionneutron scattering spectroscopy allowus to carefully study the dynamics as wetune through the quantum critical point (QCP).We measured the magnetic excitationspectrum of LiHoF 4with the use of theTAS7 neutron spectrometer at RisL NationalLaboratory, with an energy resolution (fullwidthathalfmaximum)of0.06to0.18meV(20). The transverse field was aligned to betterthan 0.35-, and the sample was cooled in adilution refrigerator. At the base temperature of0.31 K, giving a critical field of 42.4 kOe, theexcitation spectrum was mapped out below, at,and above the critical field (Fig. 2). For allfields, a single excitation branch dispersesupward from a minimum gap at (2,0,0) toward(1,0,0). From (1,0,0) to (1,0,1), the mode showslittle dispersion but appears to broaden. Thediscontinuity on approaching (1,0,1 – e) and(1 þ e,0,1) as e Y 0 reflects the anisotropyand long-range nature of the magnetic dipolecoupling. However, the most important observationis that the (2,0,0) energy, which isalways lower than the calculated single-ionenergy (È0.39 meV at 42.4 kOe), shrinksupon increasing the field from 36 to 42.4 kOeandthenhardensagainat60kOe.Atthisqualitative level, what we see agrees with themode softening predicted for the simple Isingmodel in a transverse field. However, it appearsthat the mode softening is incomplete. Atthe critical field of 42.4 kOe, the mode retains afinite energy of 0.24 T 0.01 meV. This result isapparent in Fig. 1B, which shows the gapenergy as a function of the external field.To obtain a quantitative understanding ofour experiments, we consider the full rare-earthHamiltonian, which closely resembles that ofFig. 1. (A) Phase diagram ofLiHoF 4as a function of transversemagnetic field and temperaturefrom susceptibility (10) (circles)and neutron scattering (squares)measurements. Lines are 1/z calculationswith (solid) and without(dashed) hyperfine interaction.Horizontal dashed guide marksthe temperature 0.31 K at whichinelastic neutron measurementswere performed. (B) Field dependenceof the lowest excitationenergy in LiHoF 4measured atQ 0 (1 þ e,0,1). Lines are calculatedenergies scaled by Z 0 1.15with (solid) and without (dashed)hyperfine coupling. The dashedvertical guides show how in eithercase the minimum energy occursat the field of the transition[compare with (A)]. (C) Schematicof electronic (blue) and nuclear(red) levels as the transverse fieldis lowered toward the QCP.Neglecting the nuclear spins, the electronic transition (light blue arrow) would soften all the way tozero energy. Hyperfine coupling creates a nondegenerate multiplet around each electronic state. TheQCP now occurs when the excited-state multiplet through level repulsion squeezes the collective modeof the ground-state multiplet to zero energy, hence forestalling complete softening of the electronicmode. Of course, the true ground and excited states are collective modes of many Ho ions and shouldbe classified in momentum space. (D) Calculated ratio of the minimum excitation energy E cto thesingle-ion splitting D at the critical field as a function of temperature. This measures how far theelectronic system is from the coherent limit, for which E c/D 0 0.HoF 3(21, 22). Each Ho ion is subject to thecrystal field, the Zeeman coupling, and thehyperfine coupling. The interaction betweenmoments is dominated by the long-rangedipole coupling, with a small nearest neighborexchange interaction J 12:H 0 X EH CFðJ i ÞþAJ i I I i j gm B J i I H^ij 1 X XJ2D D ab ðijÞJ ia J jbijj 1 2abX n:n:ijJ 12 J i I J jð2Þwhere J and I are the electronic and nuclearmoments, respectively, and for 165 Ho 3þ J 0 8and I 0 7/2. Hyperfine resonance (23)andheatcapacity measurements (24) show the hyperfinecoupling parameter A 0 3.36 meV as forthe isolated ion, with negligible nuclearquadrupolecoupling. The Zeeman term isreduced by the demagnetization field. Thenormalized dipole tensor D ab(ij) is directly calculable,and the dipole coupling strength J Dissimply fixed by lattice constants and the magneticmoments of the ions at J D0 (gm B) 2 N 01.1654 meV, where m Bis the Bohr magneton.This leaves as free parameters various numbersappearing in the crystal-field HamiltonianH CFand the exchange constant J 12. The formerare determined (25) largely from electron spinresonance for dilute Ho atoms substituted forYinLiYF 4, whereas the latter is constrainedby the phase diagram determined earlier (10)(Fig. 1A). We have used an effective mediumtheory (9) previously applied to HoF 3(26) tofit the phase diagram, and we conclude that agood overall description—except for a modest(14%) overestimate of the zero-field transitiontemperature—is obtained for J 120 –0.1 meV.On the basis of quantum Monte Carlo simulationdata, others (27) have also concluded thatJ 12is substantially smaller than J D.Having established a good parameterizationof the Hamiltonian, we model the dynamics,where expansion to order 1/z (where z is thenumber of nearest neighbors of an ion in thelattice) leads to an energy-dependent renormalizationE1 þ S(w)^–1 (on the order of10%) of the dynamic susceptibility calculatedin the random phase approximation, with theself energy S(w) evaluated as described in(26). For the three fields investigated in detail,the dispersion measured by neutron scatteringis closely reproduced throughout the Brillouinzone. As indicated by the solid lines in Fig. 2,the agreement becomes excellent if the calculatedexcitation energies are multiplied by a renormalizationfactor Z 0 1.15. The point is notthat the calculation is imperfect but rather thatit matches the data as closely as it does. Indeed,it also predicts a weak mode splitting of about0.08 meV at (1,0,1 – e), consistent with theincreased width in the measurements. Theagreement for the discontinuous jump between(1,0,1 – e) and(1þ e,0,1) as a result of thelong-range nature of the dipole coupling showsthat this is indeed the dominant coupling.39015 APRIL 2005 VOL 308 SCIENCE www.sciencemag.org
R EPORTSFig. 2. Pseudocolor representationof the inelastic neutron scatteringintensity for LiHoF 4at T 00.31 K observed along the reciprocalspace trace (2,0,0) Y (1,0,0)Y (1,0,1) Y (1.15,0,1). Whitelines show the 1/z calculation forthe excitation energies asdescribed in the text. White ellipsesaround the (2,0,0) Bragg peakindicate 5 times the resolution tail(full width at half maximum).Fig. 3. Measured intensities of the excitationsalong Q 0 (h,0,0) at the same values of thefield as in Fig. 2. Lines are calculated withgeometric and resolution corrections appliedto allow comparison to the neutron data.The simple origin of the incomplete softeningand enhanced critical field (Fig. 1, B andC) is easiest to understand if we start fromthe polarized paramagnetic state above H c,where the experiment, the purely electroniccalculation, and the theory including the hyperfinecoupling all coincide. At high fields,the only effect of the hyperfine term is tosplit both the ground state and the electronicexcitation modes into multiplets that aresimply the direct products of the electronicand nuclear levels, with a total span of2AbJÀI , 0.1meV(Fig.1C).Uponloweringthe field, the electronic mode softens andwould reach zero energy at H c00 36 kOe inthe absence of hyperfine coupling. The hyperfinecoupling, however, already mixesthe original ground and excited (soft mode)states above H c. As this happens, the formationof a composite spin from mixednuclear and electronic contributions immediatelystabilizes ordering along the c axisof the crystal. In other words, the hyperfinecoupling shunts the electronic mode, raisingthe critical field to the observed H c0 42.4kOe, where the mode reaches a nonzero minimum.This process is accompanied by transferof intensity from the magnetic excitation ofelectronic origin to soft modes of much lowerenergy (in the 10-meV range) that have anentangled nuclear/electronic character. Coolingto very low temperatures would revealthese modes as propagating and softeningto zero at the QCP, but at the temperaturesreachable in our measurements there is thermalization,dephasing the composite modes toyield the strong quasi-elastic scattering appearingaround Q 0 (2,0,0) and zero energyat the critical field, as in Fig. 2.The intensities of the excitations are simplyproportional to the matrix elements kbf k P jexp(iQ I R j)J jþk0Àk 2 , and therefore provide adirect measure of the wave functions via theinterference effects implicit in the spatialFourier transform of J j. Figure 3 showsintensities recorded along (h,0,0) for the threefields 36, 42.4, and 60 kOe. They follow amomentum dependence characterized by abroad peak near (2,0,0), which is welldescribed by our theory. In the absence ofhyperfine interactions, the intensity at H c0would diverge as q approaches (2,0,0),reflecting that the real-space dynamical coherencelength x cof the excited state grows toinfinity. The finite width of the peak observedat H ccorresponds in real space to a distanceon the order of the interholmium spacing;because the hyperfine interactions forestall thesoftening of the electronic mode, the implicationis that these interactions also limit thedistance over which the electronic wavefunctions can be entangled (4).Thus,Fig.3is a direct demonstration of the limitation ofquantum coherence in space via coupling to anuclear spin bath. x cis obtained from a sumover matrix elements connecting the groundstate to a particular set of excited states,whereas the thermodynamic correlation lengthx tis derived from the equal time correlationfunction S(r), which is the sum over all finalstates. x tdiverges at second-order transitionssuch as those in LiHoF 4, where the quasielasticcomponent seen in our data dominates thelong-distance behavior of S(r) atT c(H). It isthe electronic mode, and hence x c, that dictatesto what extent LiHoF 4can be characterizedand potentially exploited as a realization of theideal transverse-field Ising model.Beyond providing a quantitative understandingof the excitations near the QCP of a modelexperimental system, we obtain new insight bybringing together the older knowledge from rareearthmagnetism and the contemporary ideas ofentanglement, qubits, and decoherence. Althoughthe notion of the spin bath was developed toaddress decoherence in localized magneticclusters and molecules (1), our work disclosesits importance for QPTs. In particular, weestablish that the spin bath is a generic featurethat will limit our ability to observe intrinsicelectronic quantum criticality. This may notmatter much for transition metal oxides withvery large exchange constants, but it couldmatter for rare earth and actinide intermetalliccompounds, which show currently unexplainedcrossovers to novel behaviors at low (G1 K)temperatures Esee, e.g., (28)^.For magnetic clusters, decoherence can beminimized in a window between the oscillatorbath–dominated high-temperature regionsand the spin bath–dominated low-temperatureregions (29). Our calculations suggest thatthe dense quantum critical magnet shows analogousbehavior. Here the interacting electronspins themselves constitute the oscillator bath,and the extent to which the magnetic excitationsoftens at T c(H), as measured by the ratioof the zone center energy E cto the fieldinducedsingle-ion splitting D (Fig.1D),gaugesthe electronic decoherence. E c/D achieves itsminimum not at T 0 0 but rather at an intermediatetemperature T , 1 K, exactly wherethe phase boundary in Fig. 1A begins to beaffected by the nuclear hyperfine interactions.www.sciencemag.org SCIENCE VOL 308 15 APRIL 2005 391
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Quantum Phase Transition of a Magnet in a Spin Bath

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