Quantum and classical criticality in a dimerised quantum antiferromagnet
P. Merchant,1 B. Normand,2 K. W. Kr¨amer,3 M. Boehm,4 D. F. McMorrow,1 and Ch. R¨uegg1, 5, 6 1London Centre for Nanotechnology and Department of Physics and Astronomy, University College London, London WC1E 6BT, UK 2Department of Physics, Renmin University of China, Beijing 100872, China 3Department of Chemistry and Biochemistry, University of Bern, CH–3012 Bern 9, Switzerland 4Institut Laue Langevin, BP 156, 38042 Grenoble Cedex 9, France 5Laboratory for Neutron Scattering, Paul Scherrer Institute, CH–5232 Villigen, Switzerland 6DPMC–MaNEP, University of Geneva, CH–1211 Geneva, Switzerland (Dated: January 20, 2014)
A quantum critical point (QCP) is a singular- critical (QC) regime has many special properties3. ity in the phase diagram arising due to quantum Physical systems do not often allow the free tuning of mechanical fluctuations. The exotic properties of a quantum fluctuation parameter through a QCP. The some of the most enigmatic physical systems, in- QC regime has been studied in some detail in heavy- cluding unconventional metals and superconduc- fermion metals with different dopings, where the quan- tors, quantum magnets, and ultracold atomic con- tum phase transition (QPT) is from itinerant magnetic densates, have been related to the importance phases to unusual metallic or superconducting ones4–6, of the critical quantum and thermal fluctuations in organic materials where a host of insulating magnetic near such a point. However, direct and contin- phases become (super)conducting7,8, and in cold atomic uous control of these fluctuations has been diffi- gases tuned from superfluid to Mott-insulating states9,10. cult to realise, and complete thermodynamic and However, the dimerised quantum spin system TlCuCl3 spectroscopic information is required to disentan- occupies a very special position in the experimental study gle the effects of quantum and classical physics of QPTs. The quantum disordered phase at ambient around a QCP. Here we achieve this control in pressure and zero field has a small gap to spin excita- a high-pressure, high-resolution neutron scatter- tions. An applied magnetic field closes this gap, driving ing experiment on the quantum dimer material a QPT to an ordered phase, a magnon condensate in TlCuCl3. By measuring the magnetic excitation the Bose-Einstein universality class, with a single, nearly spectrum across the entire quantum critical phase massless excitation11,12. diagram, we illustrate the similarities between Far more remarkably, an applied pressure also drives quantum and thermal melting of magnetic order. a QPT to an ordered phase13, occurring at the very low We prove the critical nature of the unconven- critical pressure p = 1.07 kbar14 and sparking detailed tional longitudinal (“Higgs”) mode of the ordered c studies15,16. This ordered phase is a different type of con- phase by damping it thermally. We demonstrate densate, whose defining feature is a massive excitation, the development of two types of criticality, quan- a “Higgs boson” or longitudinal fluctuation mode of the tum and classical, and use their static and dy- weakly ordered moment17,18. This excitation, which ex- namic scaling properties to conclude that quan- ists alongside the two transverse (Goldstone) modes of a tum and thermal fluctuations can behave largely conventional well-ordered magnet, has been characterised independently near a QCP. in detail by neutron spectroscopy with continuous pres- 19 In “classical” isotropic antiferromagnets, the exci- sure control through the QPT and subsequently by dif- 20,21 tations of the ordered phase are gapless spin waves ferent theoretical approaches . TlCuCl3 is therefore emerging on the spontaneous breaking of a continuous an excellent system for answering fundamental questions symmetry1. The classical phase transition, occurring at about the development of criticality, the nature of the QC regime, and the interplay of quantum and thermal fluc- the critical temperature TN , is driven by thermal fluc- tuations. In quantum antiferromagnets, quantum fluc- tuations by controlling both the pressure and the tem- tuations suppress long-range order, and can destroy it perature. completely even at zero temperature2. The ordered and Here we present inelastic neutron scattering (INS) re- disordered phases are separated by a quantum critical sults which map the evolution of the spin dynamics of point (QCP), where quantum fluctuations restore the TlCuCl3 throughout the quantum critical phase diagram broken symmetry and all excitations become gapped, giv- in pressure and temperature. The spin excitations we ing them characteristics fundamentally different from the measure exhibit different forms of dynamical scaling be- Goldstone modes on the other side of the QCP (Fig. 1). haviour arising from the combined effects of quantum At finite temperatures around a QCP, the combined ef- and thermal fluctuations, particularly on crossing the QC fects of quantum and thermal fluctuations bring about regime and at the line of phase transitions to magnetic a regime where the characteristic energy scale of spin order (Fig. 1). To probe these regions, we collected spec- excitations is the temperature itself, and this quantum tra up to 1.8 meV for temperatures between T = 1.8 2
300 (a) (f) T = 1.81 K T = 7.71 K 250 p = 0.00 kbar p = 1.75 kbar 200
150
100 Intensity [Counts / 150 s] 50
0
300 (b) (e) T = 1.81 K T = 5.81 K 250 p = 1.05 kbar p = 1.75 kbar 200
150
100 Intensity [Counts / 150 s] 50
0
300 (c) (d) T = 1.81 K T = 3.88 K 250 p = 1.75 kbar p = 1.75 kbar 200
150
100 Intensity [Counts / 150 s] 50
0 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Energy Transfer [meV] Energy Transfer [meV] FIG. 1: Pressure-temperature phase diagram of TlCuCl3 ex- tended to finite energies, revealing quantum and thermal crit- ical dynamics. The rear panel is the bare (p, T ) phase diagram at energy E = 0 meV, in which the magenta line shows the FIG. 2: INS spectra collected at Q = (0 4 0) r.l.u., the 14 wavevector of the minimum energy gap in the QD phase, N´eeltemperature as a function of pressure, TN (p) , and the green points depict the temperature and pressure values stud- where magnetic order is induced with increasing pressure. ied. Full details of this panel are presented in Fig. 4(c). The (a-c) Evolution of triplet excitations during pressure-induced centre (E,T ) panel shows neutron intensity data collected magnetic ordering, as the lower mode (L and T1, blue) changes continuously into the longitudinal mode (L, red), from T = 1.8 K to 12.7 K at p = 1.75 kbar, where TN = 5.8 K. The rightmost (E,T ) panel shows the corresponding data while the anisotropic mode (T2, green) remains gapped. (d-f) Evolution as increasing temperature lowers the longitudinal- at p = 3.6 kbar, where TN = 9.2 K. The data in both (E,T ) panels display a clear softening of the magnetic excitations mode gap to zero at T = 5.8 K, above which all three modes are gapped in a disordered (QC) state. Error bars mark sta- at TN (p). The bottom (p, E) panel indicates the softening of the excitations, measured at T = 1.8 K, across the QPT19. tistical uncertainties in the intensity measurements. T1,T2 and L denote the three gapped triplet excitations of the quantum disordered (QD) phase. In the renormalised classical antiferromagnetic (RC-AFM) ordered phase, these scattering and a nontrivial evolution of the mode gaps become respectively the gapless Goldstone mode, which is a and spectral weights with both p and T , which is quan- transverse spin wave, a gapped (anisotropic) spin wave, and tified in Fig. 2. the longitudinal “Higgs” mode (see text). Figures 2(a-c) show respectively the measured inten- sities for pressures below, at, and above the QPT at a fixed low temperature. Fits to the lineshapes of the K and 12.7 K, and over a range of applied hydrostatic separate excitations were made by a resolution decon- pressures. Our measurements were performed primar- volution requiring both the gap and the local curva- ily at p = 1.05 kbar (' pc at the lowest temperatures), ture of the mode dispersion, which was taken from a 1.75 kbar, and 3.6 kbar, and also for all pressures at finite-temperature bond-operator description22,23. The T = 5.8 K. Most measurements were made at the order- distinct contributions from transverse and longitudinal ing wavevector, Q0 = (0 4 0) reciprocal lattice units fluctuations change position systematically as the applied (r.l.u.), and so concern triplet mode gaps. From the pressure induces magnetic order. The intensity of the INS selection rules, only one transverse mode of the or- longitudinal mode is highlighted in red in Fig. 2(c). Fig- dered phase is observable at Q = Q0, and it is gapped ures 2(d-f) show respectively the measured intensities for 19 (∆T2 = 0.38 meV) due to a 1% exchange anisotropy . temperatures below, at, and above the phase transition These features allow an unambiguous separation of the [TN (p)] at a fixed pressure p > pc. Quantitatively, the intensity contributions from modes of each transverse or intensity and the linewidth increase from the left to the longitudinal polarisation19. In the summary presented in right panels due to the temperature. Qualitatively, the Fig. 1, the contours represent scattered intensities at two thermal evolution is almost exactly analogous to a change selected pressures p > pc. Both panels show strong QC in the pressure, with the spectral weight of the longitu- 3
2 T = 1.8 K T = 5.8 K p = 1.75 kbar p = 3.6 kbar 2 dominates the low-energy scattering around the critical (a) L L (b) L L 1.8 1.8 L, T L, T L, T L, T 1 1 1 1 point, becomes soft at a N´eeltemperature TN (p) deter- T T T T 1.6 2 2 2 2 1.6 mined by the pressure. They reemerge on the right as 1.4 1.4
1.2 1.2 gapped triplets of the thermally disordered QC phase.
1 1 The lines in this figure are best fits to power laws of γT Energy [meV] 0.8 0.8 Energy [meV] the form ∆(T ) = B|T − TN | . The similarity between 0.6 0.6 quantum and thermal melting shown in Figs. 2 and 3 is 0.4 0.4 a remarkable result. It is essential to note that the dis- 0.2 0.2 p (1.8 K) p (5.8 K) T (1.75 kbar) T (3.6 kbar) c c N N order in Fig. 3(b) is thermal, and not due to quantum 0 0 0 0.5 1 1.5 2 2.5 3 3.5 2 4 6 8 10 12 Pressure [kbar] Temperature [K] fluctuations. Thermal fluctuations in a quantum dimer
1.6 1.6 system, whose triplet excitations are hard-core bosons, (c) (d) 1.4 1.4 do not simply broaden and damp the modes of the or-
T=1.8 K p=1.75 kbar dered magnet, but cause a very specific and systematic 1.2 1.2
) ) 22 T=5.8 K p=3.6 kbar evolution of the spectral weight . On the ordered side, 1 1 the massive, longitudinal mode becomes gapless at the 0.8 0.8 ) / Energy ( ) / Energy (
classical phase transition, while on the disordered side 0.6 0.6 there is not merely a featureless paramagnet but a clear FWHM ( FWHM ( 0.4 0.4 gapped excitation. This is also the case for the pressure-
0.2 0.2 driven transition at finite temperatures [Fig. 3(a)], where
0 0 the symmetry is restored before all three excitations be- 0 0.5 1 1.5 2 2.5 3 3.5 2 4 6 8 10 12 Pressure [kbar] Temperature [K] come gapped modes of the QC phase. Figures 3(c) and (d) show the linewidths of the longi- FIG. 3: Spin dynamics at the quantum and thermal “melting” tudinal mode, measured respectively through the quan- transitions. (a) Quantum melting of magnetic order, shown tum and thermal transitions, in the form of the ratio by the triplet gaps for two different temperatures (T = 1.8 and ΓQ/Q = αL. For the pressure-induced phase transi- 5.8 K), occurs from right to left. (b) Thermal melting, shown tion in Fig. 3(c), the ratio vanishes at T = 1.8 K for for two different pressures (p = 1.75 kbar and 3.6 kbar), oc- the well-defined excitations on the disordered side and curs from left to right. Open black circles and squares give the remains constant (with αL ' 0.15) on the ordered side of 19 energies of the anisotropic transverse excitations (T2), which the QCP, demonstrating its critically damped nature . remain gapped, while filled circles and open squares show the However, this is not at all the case at T = 5.8 K, where longitudinal mode (red in the ordered phase, blue in the disor- the divergence of αL(T ) shows the longitudinal mode dered one). The lines are power-law fits, described in the text. becoming overdamped in the presence of thermal fluc- (c,d) Linewidth-to-energy ratios ΓQ/Q for the longitudinal mode across the phase transition as a function of pressure at tuations. For the thermally driven phase transition at T = 1.8 and 5.8 K (c) and of temperature at p = 1.75 and 3.6 p = 1.75 and 3.6 kbar [Fig. 3(d)], the ratio also diverges kbar (d); lines are guides to the eye and shaded regions are on approaching the critical temperature TN (p). explained in the text. Error bars in all panels mark fitting The QC regime is the area around the line p = pc uncertainties in the resolution deconvolution procedure. where the intrinsic energy scale of the system (the gap ∆ 2 in the QD phase, or TN in the ordered phase ) is lower 3 than the temperature . Near pc, the measured neutron dinal mode softening at TN (p = 1.75 kbar) = 5.8 K but intensities [Fig. 4(a)] show a broad range over which spin moving again to finite energies at temperatures above excitations are present, with a peak along a line cor- TN . responding approximately toh ¯ Q = kBT . This “ω/T ” We analyze these results in detail by extracting the scaling property3 is evident in the self-similar nature of excitation energies Q and linewidths ΓQ from the data the spectra at different temperatures. The QCP is the of Fig. 2. Figure 3(a) shows the evolution of the mode point where the intrinsic energy scale vanishes, and thus gaps (Q=Q0 ) with pressure for T = 1.8 K (the QPT, states become available at all energies; it is the maxi- cf. Ref. 19) and T = 5.8 K. The longitudinal mode of mum in their occupation that scales with T , and hence the ordered phase appears on the right, and softens with the temperature becomes the new characteristic energy decreasing pressure until pc(T ). The Goldstone mode is scale. The microscopic origin of this “thermal gap” in not visible due to the scattering geometry. At pressures the measured spectrum is mutual blocking of the hard- 22 below pc(T ), the effect of dimer-based quantum fluctu- core triplet excitations . As shown in Fig. 4(b), the ations is to destroy the magnetic order and gap all the linewidth ΓQ = αcQ also scales linearly with T , illus- modes. The lines are best fits to power laws of the form trating that critical damping is an essential property of γp ∆(p) = A|p − pc| , which we discuss below. QC excitations. We draw attention to the fact that these Figure 3(b) shows the evolution of the mode gaps over QC excitations are remarkably narrow, with αc ' 0.14 the temperature range 1.8 K < T < 12.7 K for pres- taking a value similar to that for the Higgs mode of the sures p = 1.75 kbar and 3.6 kbar. Here the ordered ordered phase and remaining constant to the highest tem- phase is on the left, where the longitudinal mode, which peratures measured. Narrow triplet excitations have also 4
1.8 0.18 (b) Q = (0 4 0), p = p particular the static scaling relations, of a classical criti- (a) c 1.6 L,T 1 cal (CC) regime. We show below that the scaling expo- 2 3 4 5 T 2 1.4 log. int. [cnts/150 s] 0.12 nents in the QC and CC regions have approximately the
1.2 values expected theoretically. However, for a real system
0.06 such as TlCuCl , they can differ over broad crossover re-
FWHM [meV] 3 1 gions determined by the relative size of the intrinsic and 0.8 excitation energy scales, and departures from universal Energy [meV] 0 0.6 0 2 4 6 8 10 12 Temperature [K] behaviour may also arise due to microscopic details of 0.4 120 the Hamiltonian. (d) ]
0.2 2 We begin the analysis of our results not with static Q = (0 4 0), p = p c [K 2 0 80 exponents but with the dynamical ones extracted from 0 2 4 6 8 10 12 Temperature [K] the power-law fits to the gaps in Figs. 3(a) and (b). At
15 40 the pressure-controlled transition, all of the exponents m (p)2,T=1.8 K T (p) s 2 (c) N m (T) ,p we measure fall in the range 0.46(6) ≤ γp ≤ 0.57(6),
s Bragg int. [a.u.], T T (p) T (p)2 SL N 0 which within experimental error is γp = 1/2. This is 0 1 2 3 4 5 the mean-field expectation in the generic case that the Pressure [kbar] 10 QC p-dependence of the exchange parameters Jij(p) is pre- CC (e) dominantly linear19. It applies both for the disordered 0.4 (p) phase (∆QD) and for the gap (∆L) of the longitudinal√ 0.3 max Temperature [K] 5 mode, with a multiplicative prefactor ∆L = 2∆QD at 26 0.2 [K]/T T = 0 . At higher temperatures we find no strong de- N T partures from universality in this exponent. However, fits QD RC−AFM 0.1 QCP including data points at the higher energies are generally 0 0 less reliable, suggesting that these begin to depart from 0 1 2 3 4 5 0 0.2 0.42+ 0.6 Pressure [kbar] m [µ /Cu ] s B the universal regime. For the classical (temperature-controlled) phase tran- sition, the expectation for the QC phase is ∆ ∝ ξ−1 ∝ FIG. 4: Quantum and classical criticality. (a) Scattered neu- ν (T − TN ) , where ν ' 0.67 for an XY spin symmetry tron intensity at p = pc as a function of temperature. Points and 0.70 for SU(2)27. The 1% exchange anisotropy in show the energies Q extracted from the intensity for the modes becoming gapless (L and T1, yellow) and gapped (T2, TlCuCl3 reduces the spin symmetry from SU(2) to XY black) as T → 0. (b) ΓQ as a function of T at p = pc. below energy scales of order 5 K. Our data for γT at Error bars in (a) and (b) indicate uncertainties in the reso- p = 1.75 kbar give γT = 0.67(9), in excellent agreement lution deconvolution. (c) Complete experimental phase dia- with either of these exponents but certainly unable to gram, showing quantum disordered (QD), quantum critical make the subtle distinction between them; at p = 3.6 (QC), classical critical (CC), and renormalised classical (RC- kbar we do not have sufficient data for a reliable fit. At AFM) phases. The dashed lines denote energy scales marking 14 p = pc, however, we find the expected QC scaling form crossovers in behaviour. Grey symbols denote TN (p) , blue γT = 1, and the width of the crossover regime remains symbols labelled TSL(p) show the limit of classical critical scaling in the data for the staggered magnetisation, m (T ), an open question. Deducing a scaling exponent γT for s the T -dependence of the longitudinal mode gap in the and the blue bars are taken from ΓQ/Q(T ) (see text). (d) 14 Linear proportionality of the measured TN (p) and ms(p) . ordered phase remains a theoretical challenge. For this (e) Scaling of TN and ms, including one high-p data point it is important to recall that at finite temperature this taken from Ref. 25 for an absolute calibration of ms. Data Higgs mode becomes weakly overdamped, and what we for ms are normalised by Tmax = 35 K, the maximum of the show is the associated maximum in scattering intensity. 13,16 magnetic susceptibility . Red lines in panels (d) and (e) For p = 1.75 kbar we find γT = 0.54(8). represent scaling behaviour discussed in the text and error Further insight into thermal scaling exponents can be bars arise from statistical uncertainties in the intensity mea- obtained from the staggered magnetic moment, ms, mea- surements determining ms. β0 sured in Ref. 14. Fits to the form ms ∝ (TN − T ) yield values close to the classical field-theory expectation27 β0 = 0.34 (XY) or 0.37 [SU(2)]. However, the fit is fol- been found both analytically and numerically in the QC lowed only in a narrow window TSL(p) < T ≤ TN (p), 24 regime of the bilayer Heisenberg model . with TSL ' 0.8TN and points further away from the The experimental phase diagram is shown in Fig. 4(c), transition diverging clearly from classical scaling14. Thus 3,4 and contains all four regions characteristic of the QCP . ms represents very well the scaling relations expected for Between the quantum disordered and renormalised clas- a narrow CC regime in a system dominated by a QCP sically ordered regimes2, the dominant behaviour is QC [Fig. 4(c)]. The narrow nature of the CC region is also (ω/T ) scaling. On the line of classical phase transitions, evident in dynamical properties, in that the divergence the intrinsic energy scale is TN (p) but the excitation en- of the QC and longitudinal mode widths may be used to ergy is driven to zero. This results in the properties, in set dynamical criteria for the crossover from QC to CC 5
16 scaling. On the right side of Figs. 3(c), αL approaches dence may be taken as a definition of the QC regime . a constant also at finite T , but diverges on approaching Our experimental results suggest that weak departures pc(T ) (on the left side α → 0 towards the QD regime). from universality become detectable at (p, T ) values away In Fig. 3(d), quantum behaviour is evident in the con- from the QC and CC regimes, and particularly as we in- stant values of α reached far from the critical points, crease the excitation energy, presumably as microscopic a regime including the QC excitations around the line details of the fluctuation redistribution cause a mixing of p = pc (where α = αc). We define classical scaling when quantum and thermal effects. α > 1/2, and this regime is marked by the shaded regions Finally, the existence of the longitudinal “Higgs” mode in Figs. 3(c) and (d). The largely symmetric form of the has been questioned in the past. Its visibility has re- ΓQ/Q curves is the best available indicator for a classi- cently been analyzed in detail in the scaling limit21,28 cal scaling regime on the disordered side of TN (p). We for systems in two and three dimensions. Our results have used these parallel static and dynamic approaches confirm that it is a genuine example of quantum critical to estimate the width of the CC regime in both pressure dynamics in three dimensions. Its critical nature makes and temperature, as represented in Fig. 4(c). it infinitely susceptible to thermal fluctuations, so that Our precise level of control over quantum criticality it becomes overdamped as soon as these become notice- in TlCuCl3 has inspired recent numerical and analytical able. While still possessing a significant spectral weight, studies of the finite-T properties of dimer systems at the the longitudinal mode of the pressure-ordered phase is coupling-induced QPT. The authors of Ref. 16 argue that overdamped at finite temperatures, although its critical ms(p) and TN (p) should have the same behaviour, and nature is restored on passing into the finite-T disordered demonstrate good scaling of ms with TN by Quantum phase. There are several reasons for the ready visibility – Monte Carlo simulations, independent of the functional in the longitudinal rather than the scalar susceptibility21 form in p; this technique cannot address the longitudinal – of the longitudinal mode in TlCuCl3, meaning for 20 mode. In an effective√ quantum field theory approach , the anomalously low value of αL, despite its critically ms(p) ∝ TN (p) ∝ p − pc for a linear p-dependence damped nature. These include the high dimensionality of the exchange couplings, and ∆L(p) also has this de- of the system, its low phase space for magnon scattering, pendence. This analysis is predicated on proximity to the collinearity of the ordered moments, and the fact that a QCP, but in neglecting the classical critical regime, one of the spin waves contributing to decay processes is the field theory does not return the correct behaviour massive. It is reasonable to assume that the same fac- for static quantities around TN , although its dynamical tors also control the anomalously low value of αc, allow- predictions remain valid. The exchange anisotropy in ing ready observation of the QC excitations at p = pc 20 TlCuCl3 is found to have small quantitative effects on [Figs. 4(a) and (b)], and also the very narrow regime of the calculated quantities, but no detectable qualitative logarithmic corrections to scaling, which are indiscernible ones (e.g. on exponents). From our measurements, the in our data. While logarithmic corrections are expected best fits to the pressure exponents for ms and TN lie to be relevant in a system at the upper critical dimension 14 17 30 close to the classical value of 0.35 , although the quan- (dc = 4, as here ) as pc is approached , the width of tum value of 0.5 is not beyond the error bars very close to the “log-correction” regime is a non-universal quantity. the QCP. From experiment, the two quantities scale well Because our data provide no signs of such corrections in together near the QCP, as shown in Figs. 4(d) and (e), either the thermal or the pressure exponents, we conclude 16 but depart from universal scaling around an ordered that this regime is unusually small in TlCuCl3. Both the moment of 0.4µB/Cu [Fig. 4(e)]. visibility of the longitudinal mode and our level of control We have shown that the effects on the spectrum of over both quantum and thermal fluctuations in TlCuCl3 17,18 quantum and thermal melting are qualitatively very sim- remain significantly superiour to any other magnetic , 31 32 ilar. Both result in the systematic evolution of excita- charge-density-wave , or cold-atom systems displaying tions whose gap increases away from the classical phase this “Higgs boson.” transition line, rather than simply a loss of coherence due In summary, high-resolution neutron spectroscopy ex- to thermal fluctuations. Microscopically, quantum fluc- periments on the quantum antiferromagnet TlCuCl3 al- tuations in a dimer-based system cause enhanced singlet low us to probe the spin excitations of all phases in and formation and loss of interdimer magnetic correlations, around the QC regime by varying the pressure and tem- while thermal fluctuations act to suppress the spin corre- perature. We demonstrate a number of remarkable prop- lation function hSi ·Sji on both the dimer and interdimer erties arising at the interface between quantum and clas- bonds. These correlation functions may be estimated sical physics. Quantum and thermal fluctuations have from neutron-scattering intensities23 and also measured remarkably similar effects in melting the magnetically in dimerised optical lattices of ultracold fermions29. In ordered phase and in opening excitation gaps, but op- TlCuCl3, both methods of destroying interdimer coher- erate quite independently close to the QCP. In the QC ence cause the triplet modes to evolve in the same way. A regime there is robust ω/T scaling of the energies and key question in the understanding of quantum criticality widths of critically damped excitations. This scaling is whether quantum and thermal fluctuations can be con- crosses over to a classical critical form in a narrow re- sidered as truly independent, and whether this indepen- gion around the phase transition line TN (p). The criti- 6 cally damped longitudinal, or Higgs, mode of the ordered measurements for each mode were fitted with a thermal phase is exquisitely sensitive to thermal fluctuations and damping Ansatz33, which has been used for the accurate becomes overdamped in the classical regime. modelling of phonon damping at finite temperatures34 and shown in Ref. 22 to be reliable for triplet spin ex- Materials and Methods citations. The magnon is modelled as a damped har- High-quality single crystals of TlCuCl3 were grown by monic oscillator (DHO), whose scattering intensity has the Bridgman method. INS studies were performed on the double-Lorentzian lineshape the cold-neutron triple-axis spectrometer IN14 at the In- A[n(ω) + 1]4Γ ω stitut Laue Langevin (ILL). This was operated at con- S(Q, ω) = Q Q , (1) ˚−1 2 2 2 2 2 stant final wavevector kf = 1.15 A , with a focusing [ω − DHO(Q) ] + 4ΓQω pyrolytic graphite analyser and monochromator, collima- tion open-60’-open-open and a cooled Be filter positioned where n(ω) expresses the thermal magnon population. 2 2 2 between sample and analyser. The temperature and the Here DHO(Q) = Q + ΓQ is a renormalised energy ex- applied hydrostatic pressure were controlled with a He pressed in terms of the real excitation energy, Q, and the cryostat and a He gas pressure cell (precision ±50 bar). linewidth of the scattered intensity, taken as the FWHM, Spin excitations with longitudinal or transverse polari- ΓQ. The fits presented in Fig. 2 are based on a four- sation were distinguished unambiguously by working at dimensional convolution in momentum and energy of the the wavevector Q = (0 4 0), where there are no contribu- model cross-section [Eq. (1)] with the instrument resolu- tions from mode T1, while mode T2 is gapped and evolves tion, which causes the asymmetric peak shapes. Excita- with pressure in a different way from mode L. This pro- tions measured throughout the (p, T ) phase diagram were cedure is described in Ref. 19, where mode T1 was also characterised in this way by their energies Q, linewidths measured independently at Q = (0 0 1). The intensity ΓQ, polarisation and intensities [Eq. (1)].
1 Goldstone, J., Salam, A. & Weinberg, S. Broken Symme- Kr¨amer, K. W., G¨udel, H. U. & M´el´esi, L. Pressure- tries. Phys. Rev. 127, 965-970 (1962). Induced Quantum Phase Transition in the Spin-Liquid 2 Chakravarty, S., Halperin, B. I. & Nelson, D. R. Two- TlCuCl3. Phys. Rev. Lett. 93, 257201 (2004). dimensional quantum Heisenberg antiferromagnet at low 15 Matsumoto, M., Normand, B., Rice, T. M., & Sigrist, M. temperatures. Phys. Rev. B 39, 2344-2371 (1989). Field- and pressure-induced magnetic quantum phase tran- 3 Sachdev, S. Quantum Phase Transitions (Cambridge Uni- sitions in TlCuCl3. Phys. Rev. B 69, 054423 (2004). versity Press, Cambridge, 2011). 16 Jin, S. & Sandvik, A. W. Universal N´eeltemperature in 4 von L¨ohneysen,H., Rosch, A., Vojta, M. & W¨olfle,P. three-dimensional quantum antiferromagnets. Phys. Rev. Fermi-liquid instabilities at magnetic quantum phase tran- B 85, 020409(R) (2012). sitions. Rev. Mod. Phys. 79, 1015-1075 (2007). 17 Affleck, I. & Wellman, G. F. Longitudinal modes in quasi- 5 Knafo, W., Raymond, S., Lejay, P. & Flouquet, J. Antifer- one-dimensional antiferromagnets. Phys. Rev. B 46, 8934- romagnetic criticality at a heavy-fermion quantum phase 8953 (1992). transition. Nature Phys. 5, 753-757 (2009). 18 Lake, B., Tennant, D. A. & Nagler, S. E. Novel Longi- 6 Stockert, O. et al. Magnetically driven superconductivity tudinal Mode in the Coupled Quantum Chain Compound in CeCu2Si2. Nature Phys. 7, 119-124 (2011). KCuF3. Phys. Rev. Lett. 85, 832-835 (2000). 7 Bourbonnais, C. & J´erome,D. in The Physics of Organic 19 R¨uegg,Ch., Normand, B., Matsumoto, M., Furrer, A., Mc- Superconductors and Conductors, ed. Lebed, A. (Springer, Morrow, D. F., Kr¨amer,K. W., G¨udel,H. U., Gvasaliya, Heidelberg, 2008). S., Mutka, H. & Boehm, M. Quantum Magnets under 8 Kanoda, K. & Kato, R. Mott Physics in Organic Conduc- Pressure: Controlling Elementary Excitations in TlCuCl3. tors with Triangular Lattices. Annu. Rev. Condens. Matter Phys. Rev. Lett. 100, 205701 (2008). Phys. 2, 167-188 (2011). 20 Oitmaa, J., Kulik, Y. & Sushkov, O. P. Universal finite- 9 Bloch, I., Dalibard, J. & Zwerger, W. Many-body physics temperature properties of a three-dimensional quantum with ultracold gases. Rev. Mod. Phys. 80, 885-964 (2008). antiferromagnet in the vicinity of a quantum critical point. 10 Zhang, X.-B., Hung, C.-L., Tung, S.-K. & Chin, C. Ob- Phys. Rev. B 85, 144431 (2012). servation of Quantum Criticality with Ultracold Atoms in 21 Podolsky, D., Auerbach, A. & Arovas, D. P. Visibility of Optical Lattices. Science 335, 1070-1072 (2012). the amplitude (Higgs) mode in condensed matter. Phys. 11 Nikuni, T., Oshikawa, M., Oosawa, A. & Tanaka, H. Bose- Rev. B 84, 174522 (2011). 22 Einstein Condensation of Dilute Magnons in TlCuCl3. R¨uegg,Ch., Normand, B., Matsumoto, M., Niedermayer, Phys. Rev. Lett. 84, 5868-5871 (2000). Ch., Furrer, A., Kr¨amer,K. W., G¨udel,H. U., Bourges, 12 Giamarchi, T., R¨uegg,Ch. & Tchernyshyov, O. Nature Ph., Sidis, Y. & Mutka, H. Quantum Statistics of Inter- Phys. 4, 198-204 (2008), and references therein. acting Dimer Spin Systems. Phys. Rev. Lett. 95, 267201 13 Tanaka, H., Goto, K., Fujisawa, M., Ono, T. & Uwatoko, (2005). Y. Magnetic ordering under high pressure in the quantum 23 Normand, B. & R¨uegg,Ch. Complete bond-operator the- spin system TlCuCl3. Physica B 329-333, 697-698 (2003). ory of the two-chain spin ladder. Phys. Rev. B 83, 054415 14 R¨uegg, Ch., Furrer, A., Sheptyakov, D., Str¨assle, T., (2011), and references therein. 7
24 Shevchenko, P. V., Sandvik, A. W. & Sushkov, O. P. E. C. Temperature dependence of S(Q, ω) in liquid 4He Double-layer Heisenberg antiferromagnet at finite temper- under pressure. Phys. Rev. B 38, 11229-11244 (1988). ature: Brueckner Theory and Quantum Monte Carlo sim- 34 F˚ak,B. & Dorner, B. Phonon line shapes and excitation ulations. Phys. Rev. B 61, 3475-3487 (2000). energies. Physica B 234-236, 1107-1108 (1997). 25 Oosawa, A., Kakurai, K., Osakabe, T., Nakamura, M., Takeda, M. & Tanaka, H. Pressure-Induced Successive Acknowledgements Magnetic Phase Transitions in the Spin Gap System We are grateful to S. Sachdev, A. Sandvik and especially TlCuCl3. J. Phys. Soc. Jpn. 73, 1446-1449 (2004). 26 Sachdev, S. Quantum Theory of Condensed Matter, pre- M. Vojta for helpful comments. We thank the sample en- sented at 24th Solvay Conference on Physics, Brus- vironment team at the ILL, where these measurements sels, Oct 2008. Available at