Pressure-induced unconventional quantum phase transition with fractionalization in the coupled ladder antiferromagnet C9H18N2CuBr4
Tao Hong, Tao Ying, Qing Huang, Sachith E. Dissanayake, Yiming Qiu, Mark M. Turnbull, Andrey A. Podlesnyak, Yan Wu, Huibo Cao, Izuru Umehara, Jun Gouchi, Yoshiya Uwatoko, Masaaki Matsuda, David A. Tennant, Kai P. Schmidt, Stefan Wessel
aa r X i v : . [ c ond - m a t . s t r- e l ] N ov This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with theU.S. Department of Energy. The United States Government retains and the publisher, by accepting the article forpublication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United StatesGovernment purposes. The Department of Energy will provide public access to these results of federally sponsoredresearch in accordance with the DOE Public Access Plan(http://energy.gov/downloads/doe-public-access-plan). ressure-Induced Unconventional Quantum Phase Transition with Fractionalization inthe Coupled Ladder Antiferromagnet C H N CuBr Tao Hong, ∗ Tao Ying, Qing Huang, Sachith E. Dissanayake, Yiming Qiu, Mark M.Turnbull, Andrey A. Podlesnyak, Yan Wu, Huibo Cao, Izuru Umehara, Jun Gouchi, YoshiyaUwatoko, Masaaki Matsuda, David A. Tennant, Kai P. Schmidt, and Stefan Wessel Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA Department of Physics, Harbin Institute of Technology, 150001 Harbin, China Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA Department of Physics, Duke University, Durham, North Carolina 27708, USA National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA Carlson School of Chemistry and Department of Physics,Clark University, Worcester, Massachusetts 01610, USA Department of Physics, Yokohama National University, Yokohama 240-8501, Japan Institute for Solid State Physics, University of Tokyo,5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581, Japan Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA Lehrstuhl f ¨ u r Theoretische Physik I, Staudtstrasse 7,Universit ¨ a t Erlangen-N ¨ u rnberg, D-91058 Germany Theoretische Festk ¨ o rperphysik, JARA-FIT and JARA-HPC,RWTH Aachen University, 52056 Aachen, Germany (Dated: November 10, 2020)We present a comprehensive study of the effect of hydrostatic pressure on the magnetic structureand spin dynamics in the spin-1/2 coupled ladder compound C H N CuBr . The applied pressureis demonstrated as a parameter to effectively tune the exchange interactions in the spin Hamiltonianwithout inducing a structural transition. The single-crystal heat capacity and neutron diffractionmeasurements reveal that the N´eel ordered state breaks down at and above a critical pressure P c ∼ η into a quantum-disordered statecannot be described by the classic Landau’s paradigm. Using inelastic neutron scattering andquantum Monte Carlo methods, the high-pressure regime is proposed as a Z quantum spin liquidphase in terms of characteristic fully gapped vison-like and fractionalized excitations in distinctscattering channels. PACS numbers: 75.10.Jm 75.40.Gb 75.50.Ee
Landau’s symmetry-breaking theory [1, 2] has beenthe cornerstone of understanding phases of matter incondensed matter physics. Despite its remarkable suc-cess, various phenomena, including the fractional quan-tum Hall effect [3, 4] and topologically ordered quan-tum matter [5, 6] have shown the limitations of thisparadigm. Furthermore, continuous zero-temperaturequantum phase transitions (QPTs), which exhibit emer-gent gauge fields and fractionalized (deconfined) degreesof freedom, also extend beyond this conventional frame-work. In recent years, several model systems, proposedto exhibit deconfined quantum criticality, have been ex-haustively studied by analytical [7, 8] and numerical [9–12] methods. However, to the best of our knowledge,thus far there has still been no experimental realizationof such unconventional QPT with fractionalization.The S =1/2 magnetic insulator C H N CuBr (DLCBfor short) has been synthesized recently [13]. Based onthe crystal structure in Fig. 1(a), it is composed of cou-pled two-leg spin ladders with the chain direction ex-tending along the b -axis. The inter-ladder coupling inDLCB is sufficiently strong to drive the system to an an- tiferromagnetically ordered phase below 2.0 K [14]. Thestaggered moments point alternately along an easy axis( ≡ ˆ z ), the c ∗ -axis in the reciprocal space with an orderedmoment size of 0.39(5) µ B , just 40% of the saturatedmoment size due to strong quantum fluctuations.The magnetic excitations of DLCB at ambient pressurecan be described quantitatively [15] by the Hamiltonianof a two-dimensional model for the magnetic interactions: H = X γ, h i,j i J γ (cid:2) S z i S z j + λ (cid:0) S x i S x j + S y i S y j (cid:1)(cid:3) , (1)where the subscript γ reads either ‘rung’, ‘leg’, or ‘int’–for J γ being the rung, leg, or interladder exchange constant–and i and j are the nearest-neighbor lattice sites. Theparameter λ specifies an interaction anisotropy [16], with λ =0 and 1 being the limiting cases of Ising and Heisen-berg interactions, respectively. Owing to an Ising-typeanisotropy, the polarized neutron study [17] confirms thatthe gapped triplet ( S =1 and S z =0, ±
1) excitation energysplits into a gapped doublet ( S =1 and S z = ±
1) as thetransverse mode (TM) and a gapped ”singlet” ( S =1 and S z =0) as the longitudinal or amplitude mode (LM) [18]reflecting spin fluctuations perpendicular and parallel tothe easy axis, respectively. Importantly, analysis of thespin Hamiltonian suggests that DLCB is close to thequantum critical point (QCP) at ambient pressure andzero field [14, 15, 17, 19] thus its magnetic propertiescould be extraordinarily responsive to an external stim-ulus such as hydrostatic pressure.In this Letter, we present the compelling AC heat ca-pacity and neutron scattering results in DLCB with hy-drostatic pressure applied up to 1.7 GPa, which directlydetect the magnetic order and dynamic structure factor(DSF) thus allowing us to address the nature of the staticand dynamic spin-spin correlations under pressure. Fig-ure 1(b) summarizes the phase diagram of DLCB as de-termined in this study as a function of temperature andpressure as well as the pressure dependence of anisotropicenergy gaps. In the following, we will demonstrate thecollapse of the AFM ordering at P c ∼ η , which is strong evidence for fractionalizeddegrees of freedom. With the aid of quantum MonteCarlo (QMC) calculations, we propose the disorderedhigh-pressure phase as a Z topological spin-liquid phasecharacterized by the spin-gapped vison-like mode andfractionalized excitation [20].Figure 1(c) shows the AC heat capacity of a deuteratedsingle crystal of DLCB as a function of temperature [21].At ambient pressure, a sharp anomaly indicates a phasetransition to the AFM state at T N =2.0 K. This peak be-comes broader and smaller at 0.48 and 0.8 GPa indicativeof suppression of the AFM order with increasing pres-sure and at or above 1.0 GPa it becomes indiscernible atleast down to the temperature of 0.15 K. To explore thiscollapse of the N´eel order in more detail, we measuredthe order parameter of DLCB using the single-crystalneutron diffraction method. Figure 2(a-b) shows thepressure-dependence of neutron diffraction θ /2 θ scans atthe AFM wavevector q =(0.5,0.5,-0.5). The scattering in-tensity of the magnetic Bragg peak becomes diminishedas hydrostatic pressure increases. At and above the hy-drostatic pressure of 1.05 GPa, there is no experimentalevidence of magnetic order down to 0.25 K. Moreover, noevidence is found of any incommensurate magnetic Braggpeak or diffuse scattering over a wide range of reciprocalspace at T =0.25 K and P =1.3 GPa in Fig. 2(c). The re-finement of additional single-crystal neutron diffractiondata at and above 1.05 GPa does not reveal any struc-tural distortion as the possible cause for the absence ofmagnetic order and the triclinic space group P¯1 is pre-served at each pressure as listed in Tab. S1 of the Sup-plementary Material [22]. The temperature-dependentorder parameter has been measured at various pressuresas well. Figure 2(d) outlines the determined pressuredependence of the ordered moment size m that is contin-uously reduced to 0.08(5) µ B at 0.95 GPa, indicating acontinuous QPT [23, 24]. The best fit to m ∝ ( P c − P ) β yields P c = 1 . β =0.65(5) that is appreciably larger than β ≃ . β is a strong signature of deconfinement at the transition, i.e., the order parameter fractionalizes into partons, andhence the critical exponents differ significantly from theconventional values within the Landau paradigm. It isnoteworthy that the determined AFM ordering temper-ature T N increases slightly upon initially increasing pres-sure and then decreases gradually with pressure untilthe abrupt change towards zero at P c . This behavioris rather distinct from the conventional phase diagram,where T N has a smooth power-law decrease towards zeroat the critical point. We consider this as an indicationthat the underlying physics is also distinct from a con-ventional order-to-quantum disorder transition scenario.As discussed in the Supplementary Material, such ap-parent sudden drop of T N near P c can be quantified by alogarithmic behavior as T N ∼ / ln( P c − P ) − [22].In order to gain more insight into the nature ofthis pressure-induced disordered phase, inelastic neutronscattering has been used to probe the evolution of thedynamic spin-spin correlation function of DLCB as afunction of pressure. Figures 3(a) and (b) show thebackground-subtracted energy scans at the same AFMwavevector for various pressures. The spectral line-shapes for P
P c ) are spin-gapped and weremodelled by superposition of two (or single) double-Lorentzian damped harmonic-oscillator (DHO) modelsconvolved with the instrumental resolution function. Atambient pressure, the best fit yields the gap energies ofTM and LM as ∆ TM =0.32(3) meV and ∆ LM =0.58(4)meV, respectively. Their values are consistent withthe previous report [17]. We find that the peak pro-files of both TM and LM for P ≤ TM and ∆ LM across the phase transition.The slightly growing gap energy of ∆ TM can delay thethermal depletion of the magnetic order to higher T N .∆ LM becomes softened with a decrease of the orderedmoment and the best fit to ∆ LM ∝ ( P c − P ) ν givesthe correlation-length exponent ν =0.32(4) that is muchsmaller than ν ≃ ν ≈ S =1/2 square-lattice J - Q model [26, 27]. At 1.06 GPa, slightly above P c , LM hasa very small gap which cannot be distinguished by thelimited instrumental resolution while ∆ TM moves furtherto 0.44(3) meV. Based on the spin Hamiltonian at am-bient pressure in Eq. (1), the ground state in the quan-tum disordered phase is expected to be a trivial dimer- Gapped QSL
Néel order (c)(a)(b)
Cu NBr DC
FIG. 1: (a) Crystal structure of deuterated C H N CuBr projected along the crystallographic c -axis to show the stacking ofdiscrete DMA + (C D N) and 35DMP + (C D N) cations. Outlined is a nuclear unit cell. (b) Phase diagram of DLCB as afunction of pressure and temperature, including the pressure dependence of anisotropic energy gaps. Circle and diamond pointsare the energy gaps of the TM and LM, respectively. The red and black solid lines are guides to the eye. The blue line wasobtained from a fit of ∆ LM ( P ) ∝ ( P c − P ) ν with P c fixed at 1.03 GPa and the fitted exponent ν =0.32(4). The olive short-dashedline is obtained from T N ( P ) ∝ / ln( P c − P ) − . (c) The AC heat capacity C P of DLCB as a function of temperature at ambientpressure, 0.48, 0.8, 1.0, 1.28 and 1.72 GPa. For clarity, the data are shifted upwards. The transition temperature is indicatedby an arrow. ized quantum paramagnet, where spins are paired intosinglet bonds arranged in a regular pattern. Such aphase has sharp excitations. Surprisingly, the spectralline shape of the TM signal becomes significantly broaderthan the instrumental resolution and the best fit to thesame double-Lorentzian DHO model gives an intrinsiclinewidth FWHM=0.42(5) meV. Such spectral broaden-ing persists at least up to 1.3 GPa beyond the QCP,where FWHM becomes 0.34(5) meV.To allow a quantitative comparison to the theoreti-cal characterization of spin dynamics, we examine theDSF using large scale QMC calculations of the same spinHamiltonian. The Hamiltonian parameters at ambientconditions were obtained to best match the experimen-tally observed magnetic dispersions as J leg =0.62 meV, J rung =0.66 meV, J int =0.20 meV and λ =0.87 [15]. Weestablish that the applied pressure is effectively tuningthe exchange coupling ratio α between the inter-ladderand intra-ladder couplings and assume no impact on theinteraction anisotropy λ and the ratio between J rung and J leg (see the Supplementary Material [22] for further de-tails). At the critical point or 1.3 GPa, α is reduced to 0.14 or 0.05 from 0.32 at ambient pressure and the param-eter set ( J leg =0.91 meV, J rung =0.97 meV and J int =0.13meV) or ( J leg =1.03 meV, J rung =1.1 meV and J int =0.05meV) [28] gives the best agreement with the experimentalvalue of ∆ TM . The calculated spectral lineshape profilesas indicated by the orange lines in Fig. 3(a) are visiblylimited by the instrumental resolution. Moreover, Fig. 4shows the comparison over the entire Brillouin zone be-tween the experimental excitation spectra at 1.06 and 1.3GPa and DSF calculated by QMC and convolved withthe instrumental resolution function. Clearly, the exper-imental data cannot be reproduced by QMC calculationsfor the Hamiltonian in Eq. (1).So what is the probable origin of the continuum-likebroad excitation of DLCB? We have carried out single-crystal neutron diffraction experiments under pressureby measuring more than 300 nuclear Bragg peaks andthe fitting outcome as listed in Tabs. S2-S5 of the Sup-plementary Material [22] does not reveal any evidence ofchemical disorder under pressure except a small H/D iso-tope effect on site occupancy. The latter has no influenceon the magnetism of non-hydrogen-bonded systems such (a) (b)(c) (d) Cytop (cid:1)(cid:2)(cid:3)(cid:4)(cid:2)(cid:5)(cid:6)(cid:3)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:8)(cid:14)(cid:2)(cid:6)(cid:3)(cid:5)(cid:15)
FIG. 2: Single-crystal neutron diffraction study under pres-sure. The representative θ/ θ scans and the temperaturedependence of the neutron scattering intensity measured atCTAX around the antiferromagnetic wavevector q =(0.5,0.5,-0.5) for (a) ambient pressure, 0.3 and 0.53 GPa and for (b)0.82, 0.88 and 0.95 GPa, respectively. For θ/ θ scans, thesolid lines are fits to the Gaussian profile. For clarity, the dataabove 1 GPa presented in the inset are shifted vertically bysuccessive increments of 400. For temperature dependence ofthe order parameter, the transition temperature at each pres-sure is indicated by the arrow and the solid lines are guidesto the eye. (c) Single-crystal neutron diffraction pattern mea-sured at CNCS at T =0.25 K and P =1.3 GPa. The ring fea-ture is caused by the cytop glue. (d) The determined orderingmoment size as a function of applied pressure. The red linebelow P c is a fit of the ordered moment m ( P ) ∝ ( P c − P ) β with the fitted exponent β =0.65(5) and P c =1.03(3) GPa. Er-ror bars represent one standard deviation. Experimental databelow and above 0.8 GPa were collected at T =1.45 K and 0.25K, respectively. as DLCB. Consequently, it can be ruled out that the ob-served broadening is due to disorder. The other possiblebroadening effect attributed to spontaneous quasiparti-cle decays [29, 30] is also discussed in the SupplementaryMaterial [22] and can be excluded mainly due to violationof the kinematic conditions [31]. Moreover, at the criticalpressure, the critical exponent η which describes the de-cay of the correlation function is related to other criticalexponents by the scaling law [25] η = 2 β/ν − ( d + z − d is the number of spatial dimensions and z isthe dynamic critical exponent. Assuming d =2 based onthe strong two-dimensional character of the magnetism inDLCB, and z =1 at the pressure-induced quantum phasetransition [23], we obtain η =3.0(5) from β =0.65(5) and ν =0.32(4) as already stated above. This is in markedcontrast to the value η ≈ η [12]. The large η canexplain the broad spectral linewidth near the transitionin DLCB and further suggests that emergence of the vison spinon (a) (b) FIG. 3: Single-crystal inelastic neutron scattering studyunder pressure. The representative background-subtractedtransferred energy scans measured at (a) MACS and (b)CNCS at the antiferromagnetic wavevector q =(0.5,0.5,-0.5)for several pressures. All data were collect at T =1.5 K ex-cept that the data at 1.3 GPa (CNCS) were measured at0.25 K. For clarity, the data and fits are shifted vertically bysuccessive increments of 150 (MACS) or 0.002 (CNCS). Thegreen and magenta dashed lines are fits to the DHO modelconvolved with the instrumental resolution for the TM andLM, respectively. The black solid lines are their sum. Thegreen and orange solid lines at 1.06 and 1.3 GPa (MACS) areconvolved calculations for the transverse mode by the DHOmodel and QMC calculations, respectively. The black dashedlines are guides to the eye. The red horizontal bars repre-sent the instrumental resolution. Error bars represent onestandard deviation. gapped phase with fractionalized excitations through thispressure-induced phase transition cannot be described byLandau’s paradigm.Consequently, we propose that the quantum disorderedphase with P >P c is a gapped quantum spin liquid (QSL)state with fractionalized degrees of freedom. In orderto stabilize such a QSL state in DLCB, a mechanismthat is not included in the original Hamiltonian is re-quired. Indeed, for unfrustrated interactions, an ordinaryquantum phase transition to a trivial quantum paramag-net emerges [23]. The possibility of diagonal frustrat-ing couplings in the ladders can be eliminated due totheir unfavorable superexchange paths in DLCB. On gen-eral grounds, higher order four-spin interactions such asthose around plaquettes are also expected to be negligi-ble for molecular ladder systems. One may also considerfurther anisotropic exchange terms, the Dzyaloshinskii-Moriya (DM) interactions. In DLCB, DM interactionsare forbidden along the rungs of the ladders and betweenthe ladders by the inversion symmetry, but are allowedalong the ladder legs. DM anisotropy has been investi-gated by ESR measurements in other metal-organic lad-der compounds including(C H N) CuBr (BPCB) [32]and (C H N) CuBr (DIMPY) [33, 34], and was foundto be about 5% of the dominant exchange interactions.In general, DM interactions prefer non-collinear magneticstructures, and in DLCB the effects of the DM interac-tions on the collinear ordered state within the N´eel phaseshould thus be very weak. Indeed, the field dependenceof the anisotropic energy gaps at ambient pressure [17] FIG. 4: Comparison between the experimental data and QMCcalculations. False-color maps of the excitation spectra as afunction of energy and wavevector transfer along two high-symmetry directions (H,H,-0.5) and (0.5,0.5,-L) in the recip-rocal space, respectively. Experimental data were collected atMACS at T =1.5 K and (a) P =1.06 GPa and (c) P =1.3 GPa.Data for H less than 0.4 r.l.u. are not shown due to a contam-ination by the direct neutron beam. (b) and (d) are dynamicstructure factors of the TM calculated by QMC using the pa-rameter sets at the critical point and 1.3 GPa, respectively, asdescribed in the text. Simulations were convolved with the in-strumental resolution function where the neutron polarizationfactor and the magnetic form factor for Cu were included.(e) High-resolution inelastic neutron scattering measurementsat CNCS at T =0.25 K and P =1.3 GPa. The excitation spec-tra at T =15 K are shown in Fig. S4(c) of the SupplementaryMaterial [22]. No smoothing or symmetrization was appliedto all experimental data. agrees well with the Zeeman spectral splitting, which sug-gests that S z is a good quantum number. It is possible,that the effects of the DM interactions become enhancedwithin the QSL phase. Although we note that a recentstudy finds that QSL phases can be stable at weak DMinteractions [35], further theoretical work is needed toestablish the proper lattice Hamiltonian for the entirephase diagram.One simple example of gapped QSL with fractionaliza-tion is the Z spin liquid, which has been shown to existin quantum dimer model on the triangular lattice [36] andalso XXZ spin model on the Kagom´e lattice [37]. Sucha Z QSL state exhibits non-trivial topological order, which is characterized by, e.g., the topological degener-acy of the ground states. It furthermore exhibits two pri-mary types of excitations: spinons that carry a Z gaugecharge and visons that carry Z gauge flux [38, 39]. Theseexcitations are bosonic, with a semionic mutual statis-tics. In the easy-axis limit, the visons transform as S z =0(singlet) excitations under the U(1) symmetry about theeasy-axis [37], i.e., they reside within the LM channelfor DLCB. High-resolution inelastic neutron scatteringwas used in order to nail down the possible vison excita-tions deep in the QSL phase. The excitation spectra at P =1.3 GPa in Fig. 4(e) clearly show two well-separatedgapped modes including one broad continuum-like andanother almost dispersionless excitation. At the AFMwavevector, cf. Fig. 3(b), we clearly identify two dis-tinct excitations: (i) a broad peak at about 0.56(4) meVwith a FWHM of 0.36(4) meV, which we identify as aris-ing from fractionalized excitations as spinons, and (ii) anadditional, resolution-limited excitation at 0.25(3) meV,identified as the vison excitation.In summary, we have performed neutron scatteringexperiments under pressure on DLCB at low temper-ature and continuously tuned the ground state of thecompound C H N CuBr from the AFM N´ e el state,through an unconventional QPT with a large anomalousexponent η , to an exotic quantum disordered state. Bycontrasting with QMC calculations for the conventionalspin model for DLCB, the unique ability of neutron scat-tering to probe spin pair correlation functions allows oneto experimentally identify fully gapped vison-like andfractionalized excitations consistent with a possible Z QSL state. Our study offers a much-needed experimen-tal platform to search for QSL physics along with theassociated quantum criticality within a single compound.We gratefully acknowledge the helpful discussions withCollin Broholm, Masashige Matsumoto and GuangyongXu. We are indebted to Cenke Xu for the unique com-ments and insights. We also thank Matthew Collins,Michael Cox, Saad Elorfi, Cory Fletcher, Juscelino Le˜ao,Christopher Redmon, Christopher Schmitt, Randall Sex-ton, Erik Stringfellow and Tyler White for the technicalsupport in neutron scattering experiments. 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