Bose-Einstein Condensation of Triplons Close to the Quantum Critical Point in the Quasi-One-Dimensional Spin-1/2 Antiferromagnet Navopo4

Ames Laboratory Accepted Manuscripts Ames Laboratory

10-1-2019

Bose-Einstein condensation of triplons close to the quantum critical point in the quasi-one-dimensional spin-1/2 antiferromagnet NaVOPO4

Prashanta K. Mukharjee Indian Institute of Science Education and Research

K. M. Ranjith Max Planck Institute for Chemical Physics of Solids

B. Koo Max Planck Institute for Chemical Physics of Solids

J. Sichelschmidt Max Planck Institute for Chemical Physics of Solids

M. Baenitz Max Planck Institute for Chemical Physics of Solids

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Recommended Citation Mukharjee, Prashanta K.; Ranjith, K. M.; Koo, B.; Sichelschmidt, J.; Baenitz, M.; Skourski, Y.; Inagaki, Y.; Furukawa, Yuji; Tsirlin, A. A.; and Nath, R., "Bose-Einstein condensation of triplons close to the quantum critical point in the quasi-one-dimensional spin-1/2 antiferromagnet NaVOPO4" (2019). Ames Laboratory Accepted Manuscripts. 517. https://lib.dr.iastate.edu/ameslab_manuscripts/517

This Article is brought to you for free and open access by the Ames Laboratory at Iowa State University Digital Repository. It has been accepted for inclusion in Ames Laboratory Accepted Manuscripts by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Bose-Einstein condensation of triplons close to the quantum critical point in the quasi-one-dimensional spin-1/2 antiferromagnet NaVOPO4

Abstract Structural and magnetic properties of a quasi-one-dimensional spin-1/2 compound NaVOPO4 are explored by x-ray diffraction, magnetic susceptibility, high-field magnetization, specific heat, electron spin resonance, and 31P nuclear magnetic resonance measurements, as well as complementary ab initio calculations. Whereas magnetic susceptibility of NaVOPO4 may be compatible with the gapless uniform spin chain model, detailed examination of the crystal structure reveals a weak alternation of the exchange couplings with the alternation ratio α≃0.98 and the ensuing zero-field spin gap Δ0/kB≃2.4K directly probed by field-dependent magnetization measurements. No long-range order is observed down to 50 mK in zero field. However, applied fields above the critical field Hc1≃1.6T give rise to a magnetic ordering transition with the phase boundary TN∝(H−Hc1)1ϕ, where ϕ≃1.8 is close to the value expected for Bose- Einstein condensation of triplons. With its weak alternation of the exchange couplings and small spin gap, NaVOPO4 lies close to the quantum critical point.

Disciplines Condensed Matter Physics

Authors Prashanta K. Mukharjee, K. M. Ranjith, B. Koo, J. Sichelschmidt, M. Baenitz, Y. Skourski, Y. Inagaki, Yuji Furukawa, A. A. Tsirlin, and R. Nath

This article is available at Iowa State University Digital Repository: https://lib.dr.iastate.edu/ameslab_manuscripts/ 517 PHYSICAL REVIEW B 100, 144433 (2019)

Bose-Einstein condensation of triplons close to the quantum critical point in the 1 quasi-one-dimensional spin- 2 antiferromagnet NaVOPO4

Prashanta K. Mukharjee,1 K. M. Ranjith ,2 B. Koo,2 J. Sichelschmidt ,2 M. Baenitz,2 Y. Skourski,3 Y. Inagaki,4,5 Y. Furukawa,5 A. A. Tsirlin,6 and R. Nath1,* 1School of Physics, Indian Institute of Science Education and Research, Thiruvananthapuram-695551, India 2Max Planck Institute for Chemical Physics of Solids, Nöthnitzer Strasse 40, 01187 Dresden, Germany 3Dresden High Magnetic Field Laboratory (HLD-EMFL), Helmholtz-Zentrum Dresden-Rossendorf, 01314 Dresden, Germany 4Department of Applied Quantum Physics, Faculty of Engineering, Kyushu University, Fukuoka 819-0395, Japan 5Ames Laboratory, U.S. Department of Energy, Iowa State University, Ames, Iowa 50011, USA 6Experimental Physics VI, Center for Electronic Correlations and Magnetism, Institute of Physics, University of Augsburg, 86135 Augsburg, Germany

(Received 30 August 2019; revised manuscript received 9 October 2019; published 25 October 2019)

1 Structural and magnetic properties of a quasi-one-dimensional spin- 2 compound NaVOPO4 are explored by x-ray diffraction, magnetic susceptibility, high-ﬁeld magnetization, speciﬁc heat, electron spin resonance, and 31P nuclear magnetic resonance measurements, as well as complementary ab initio calculations. Whereas

magnetic susceptibility of NaVOPO4 may be compatible with the gapless uniform spin chain model, detailed examination of the crystal structure reveals a weak alternation of the exchange couplings with the alternation ratio

α 0.98 and the ensuing zero-field spin gap 0/kB 2.4 K directly probed by field-dependent magnetization measurements. No long-range order is observed down to 50 mK in zero field. However, applied fields above the 1 φ critical field Hc1 1.6 T give rise to a magnetic ordering transition with the phase boundary TN ∝ (H − Hc1 ) , where φ 1.8 is close to the value expected for Bose-Einstein condensation of triplons. With its weak alternation

of the exchange couplings and small spin gap, NaVOPO4 lies close to the quantum critical point.

DOI: 10.1103/PhysRevB.100.144433

I. INTRODUCTION correlations at low temperatures [8–11]. Another field- induced phenomenon observed in gapped spin systems is Quantum phase transitions in low-dimensional antiferro- the Tomanaga-Lutinger liquid (TLL) phase which is mostly magnets (AFM) remain one of the most fascinating topics expected for 1D systems [12–14]. Additionally, gapped spin in condensed-matter physics [1–3]. One-dimensional (1D) systems also used to exhibit several other field-induced fea- Heisenberg spin- 1 chains are prone to quantum fluctuations 2 tures such as Wigner crystallization of magnons [15], magne- that give rise to rich ground-state properties. An individual 1 tization plataeus [16], etc. Therefore, 1D materials with 3D spin- 2 chain does not show any long-range magnetic order interchain couplings and spin ladders, which are essentially and features gapless excitations. However, in real materials coupled spin chains, can provide the opportunity to study both due to the presence of inherent interchain couplings a three- BEC and TLL physics in the same system [14,17,18]. Such dimensional (3D) long-range-order (LRO) will usually occur, phases occur between two critical fields Hc1 and Hc2, which albeit at temperatures much lower than the intrachain cou- depend on the hierarchy of coupling strengths. Experimental 1 pling [4]. Alternating spin- 2 chains, where two nonequivalent access to these fields requires appropriate design of the mate- couplings interchange along the chain, are different. Their rial with an appreciably small spin gap which will allow for energy spectrum is gapped [5,6], and this gap protects the a complete exploration of the field (H) versus temperature system from long-range ordering, unless interchain couplings (T ) phase diagram. In the past, an ample number of low- are strong enough to close the gap. dimensional magnets were studied extensively, but most of In spin-gap materials, the elementary excitations are spin-1 them feature a large spin gap. Therefore, a very high field is triplons. A magnetic field reduces the gap and causes an AFM applied to study the field-induced effects in all these materials LRO when the field exceeds the critical value of the gap [10,19–21]. closing. This magnetic order is different from the classical Herein, we have carried out a systematic investigation of magnetic ordering in many aspects and can be described as the structural and magnetic properties of the quasi-1D com- a Bose-Einstein condensation (BEC) of dilute triplons [7]. pound NaVOPO4 by performing the temperature-dependent Typically, BEC is being realized in gapped materials based x-ray diffraction, magnetization, specific heat, ESR, and on spin dimers with two-dimensional (2D) and 3D interdimer 31P NMR experiments. Our experiments are also accompanied by the density-functional band structure calculations. We compare NaVOPO4 with the isostructural compounds NaVOAsO4 and AgVOAsO4 to understand the underlying *[email protected] structure-property relationship [22–25].

other and spread in the ab plane. Figure 1(b) shows a section of the magnetic chain through the extended V–O–P–O–V path. The perfect PO4 tetrahedra also bridge the neighboring chains into a three-dimensional network [see Fig. 1(a)], similar to AgVOAsO4 and NaVOAsO4. The possible exchange couplings in this spin system are depicted in Fig. 1(d).The weak interchain couplings Jc and Ja along the V–O–V and V–O–P–O–V bonds, respectively, create frustrated interactions between the magnetic chains. Clearly, P atoms are strongly coupled to the V4+ ions, while Na atoms positioned between the chains are weakly coupled to the magnetic ions.

FIG. 1. (a) Crystal structure of NaVOPO projected in the ab 4 II. METHODS plane. The spin chains are separated by the black dashed lines. The VO6 and PO4 polyhedra are shown in green and blue colors, Polycrystalline sample of NaVOPO4 was prepared by a respectively. (b) A section of the magnetic (bond-alternating) chain conventional solid-state reaction method using the stoichio- with the intrachain couplings J and J. (c) A representative structural metric mixture of NaPO3 and VO2 (Aldrich, 99.995%). The chain extending along the c axis. (d) Sketch of the spin lattice NaPO3 precursor was obtained by heating NaH2PO4.H2O showing all relevant exchange interactions. (Aldrich, 99.995%) for 4 h at 400 ◦C in air. The reactants were ground thoroughly, pelletized, and fired at 720 ◦Cfor two days in flowing argon atmosphere with two intermediate 4+ The aforementioned V phosphates and arsenates crystal- grindings. Phase purity of the sample was confirmed by / lize in the monoclinic crystal structure (space group P21 c). powder x-ray diffraction (XRD) recorded at room temperature One peculiarity of this series is that the structural chains using a PANalytical powder diffractometer (CuKα radiation, formed by the corner-sharing VO6 octahedra do not rep- λavg 1.5418 Å). To check if any structural distortion is resent the magnetic chains [24,25]. Band structure calcula- present, temperature-dependent powder XRD was performed tions reveal that the magnetic chains run along the extended over a broad temperature range (15 K T 600 K). For V–O–As–O–V path, whereas the shorter V–O–V path along low-temperature measurements, a low-T attachment (Oxford the structural chains gives rise to a weak and ferromagnetic Phenix) while for high-temperature measurements, a high- − (FM) interaction Jc 5 K. There exists another weak AFM T oven attachment (Anton-Paar HTK 1200N) to the x-ray interaction Ja 8 K, which along with Jc constitutes a frus- diffractometer were used. Rietveld refinement of the ac- trated 3D interaction network between the chains [same as quired data was performed using the FULLPROF software Fig. 1(d)]. A quantitative comparison of the exchange cou- package [26]. plings in NaVOAsO4 with AgVOAsO4 is made in Ref. [22]. Magnetization (M) was measured as a function of tem- The Na compound is found to be more close to the 1D regime perature (0.5K T 380 K) and magnetic field (H). The due to the stronger intrachain and weaker interchain exchange measurements above 2 K were performed using the vibrat- couplings. Consequently, its spin gap is larger than in the Ag ing sample magnetometer (VSM) attachment to the Physical 5+ analog. The replacement of As (ionic radius 0.33 Å) Property Measurement System [PPMS, Quantum Design]. 5+ with P (ionic radius 0.17 Å) leaves the crystal structure For T 2 K, the measurements were done using an 3He unchanged, but the lattice parameters are reduced, thus shrink- attachment to the SQUID magnetometer [MPMS-7T, Quan- 3 ing the unit cell volume from 378.72 to 354.44 Å . Such a tum Design]. Specific heat (Cp) as a function of tempera- reduction of the cell volume may bring in a drastic change ture was measured down to 0.38 K using the thermal re- in the magnetic properties. Here we show that, compared to laxation technique in PPMS under magnetic fields up to NaVOAsO4 and AgVOAsO4,NaVOPO4 has a much lower 14 T. For T 2 K, the measurements were performed us- 3 value of Hc1 and 0/kB, thus lying very close to the quantum ing an additional He attachment to the PPMS. High-field critical point (QCP) between the 3D LRO and gapped ground magnetization was measured in pulsed magnetic field up to state. 60 T at the Dresden High Magnetic Field Laboratory. The NaVOPO4 contains one Na, V, P atom each and five details describing the measurement procedure are given in nonequivalent oxygen atoms. Each V is coordinated by six Refs. [27,28]. oxygen atoms forming a distorted VO6 octahedron. In each The ESR experiments were carried out on fine-powdered octahedron, four equatorial V–O bonds are within the range sample with a standard continuous-wave spectrometer from 1.98–2.01 Å, while the alternating short and long bond dis- 3 to 290 K. We measured the power P absorbed by the tances of V with O(1) are 1.62 and 2.13 Å, respectively. sample from a transverse magnetic microwave field (X band, Similarly, each phosphorous atom forms a nearly regular ν 9.4 GHz) as a function of the external magnetic field tetrahedron bearing the P–O distances of ∼1.54 Å. The dis- H. A lock-in technique was used to enhance the signal-to- torted VO6 octahedra are corner-shared through O(1) forming noise ratio which results the derivative of the resonance signal parallel structural chains along the crystallographic c axis dP/dH. 31 [Fig. 1(c)]. Each PO4 tetrahedron is connected to four nearest The NMR experiments on the P nuclei (nuclear spin VO6 octahedra through the O(5) and O(2) corners forming I = 1/2 and gyromagnetic ratio γ/2π = 17.235 MHz/T) bond-alternating spin chains arranged perpendicular to each were carried out using pulsed NMR technique over a wide

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FIG. 3. The lattice constants (a, b, c), monoclinic angle (β), and unit cell volume (Vcell) are plotted as a function of temperature. The solid line is the ﬁt of Vcell(T )byEq.(1).

exchange Jd = 1 eV, and around-mean-ﬁeld double-counting correction [22].

III. RESULTS AND ANALYSIS A. X-ray diffraction FIG. 2. Powder x-ray diffraction data of NaVOPO4 at 600, 300, and 17 K. The solid lines denote the Rietveld refinement of the data. Figure 2 displays the powder XRD pattern of NaVOPO4 at The Bragg peak positions are indicated by green vertical bars, and the three different temperatures (T = 600, 300, and 17 K). The bottom solid line indicates the difference between the experimental room-temperature diffraction pattern reveals that NaVOPO4 and calculated intensities. crystallizes in the primitive monoclinic unit cell with the space group P21/c and contains Z = 4 formula units per unit cell. The lattice parameters obtained from the Rietveld refinement temperature range 0.05 K T 250 K and at different mag- at room temperature are a = 6.520(1) Å, b = 8.446(1) Å, netic fields. For T 2 K, all the experiments were performed = . β = . ◦ 4 c 7 115(1) Å, 115 260(8) , and the unit cell volume using a He cryostat and a Tecmag spectrometer, while in the . 3 0.05 K T 10 K range, a 3He/4He dilution refrigerator Vcell 354 44 Å which are consistent with the previous (Kelvinox, Oxford Instruments) was used. The 31PNMR report [31]. spectra as a function of temperature were obtained either by Some of the spin-gap compounds show structural dis- performing Fourier transform (FT) of the spin echo signal at tortion upon cooling [32–35]. In our case, low-temperature a fixed field after a π/2 − π pulse sequence or by sweeping XRD measurements did not show any noticeable changes, the magnetic field keeping the transmitter frequency constant. suggesting that there is no structural distortion down to at least 15 K. The temperature variation of the lattice parameters The NMR shift, K(T ) = [ν(T ) − νref ]/νref , was determined by measuring the resonance frequency ν(T )ofthesam- obtained from the Rietveld analysis is plotted in Fig. 3.All the lattice parameters decrease in a systematic manner with ple with respect to the standard H3PO4 sample (resonance 31 no sign of any structural distortion. To estimate the average frequency νref ). The P nuclear spin-lattice relaxation rate Debye temperature (θD), we fit the temperature variation of (1/T1) was measured using a standard saturation pulse sequence. the unit cell volume [Vcell(T )] using the equation [36,37] Magnetic exchange couplings were obtained from density- V (T ) = γU (T )/K0 + V0, (1) functional theory (DFT) band-structure calculations performed in the FPLO code [29] with the generalized gradient where V0 is the unit cell volume at T = 0K, K0 is the approximation (GGA) for the exchange-correlation potential bulk modulus, and γ is the Grüneisen parameter. The in- [30]. Two complementary approaches to the evaluation of ternal energy U (T ) can be expressed within the Debye exchange couplings were used, as explained in Refs. [22,24] approximation as and in Sec. III F below. Correlation effects in the V 3d 3 θ / shell were taken into account on the mean-field DFT + U T D T x3 U T = pk T dx. = ( ) 9 B x (2) level with the on-site Coulomb repulsion Ud 4 eV, Hund’s θD 0 e − 1

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The χ(T ) data in the paramagnetic region (above 150 K) were ﬁtted by the Curie-Weiss law (bottom panel of Fig. 4), C χ(T ) = χ0 + , (3) T − θCW

where χ0 accounts for the temperature-independent contribution consisting of the core diamagnetic susceptibility (χcore) of the core electron shells and the Van-Vleck paramagnetic 4+ susceptibility (χVV) of the open shells of the V ions in the compound. The second term in Eq. (3) is the Curie-Weiss (CW) law with the CW temperature θCW and Curie constant C. The fitted data in the temperature range 150–380 K −5 3 3 yields χ0 7 × 10 cm /mol, C 0.373 cm K/mol, and θCW −14.14 K. From the value of C, the effective moment 4+ was calculated to be μeff 1.72 μB/V which is in close agreement with the expected spin-only value of 1.73 μB for S = 1/2. This value of μeff also corresponds to a g value of g 1.98 which is nearly equal to the obtained g value from the ESR experiment (discussed later). The negative value of θCW indicates that the dominant exchange couplings 4+ between V ions are antiferromagnetic in nature. The χcore −5 3 of NaVOPO4 was estimated to be −7.3 × 10 cm /mol by adding the core diamagnetic susceptibility of the individual 1+ 4+ 5+ 2− ions Na ,V ,P , and O [39]. The χVV was calculated −5 3 to be ∼14.3 × 10 cm /mol by subtracting χcore from the experimental χ0 value. To shed light on the nature of magnetic interactions in NaVOPO4 and to estimate the corresponding exchange cou- FIG. 4. Upper panel: χ(T ) measured at H = 1 T. The solid line plings, χ(T ) was fitted by the following expression: represents the fit using Eq. (4). Inset: χ(T ) in the low temperature Cimp regime measured in different applied fields to focus on the low χ(T ) = χ0 + + χspin, (4) temperature broad maximum. Lower panel: 1/χ vs T and the solid T line is the CW fit using Eq. (3). where the second term is the generic Curie law that accounts for the impurity contribution. Cimp stands for the Curie constant of the impurities and the third term χspin is the expression 1 for the spin susceptibility of a 1D spin- 2 Heisenberg AFM In the above, p stands for the total number of atoms inside the with uniform or alternating exchange couplings. Such an k expression is available in the whole range of alternation ratios unit cell and B is the Boltzmann constant. The best fit of the datadowntoT = 15 K (lower panel of Fig. 3) was obtained α,0 α(=J /J) 1, and in a large temperature window γ −5 −1 / . with the parameters: θD 530 K, 6.73 × 10 Pa , (kBT J 0 01) [40]. K0 3 The fit of the χ(T ) data down to 8 K by Eq. (4) taking and V0 352.3Å. χspin for the alternating spin chain is shown in the upper panel of Fig. 4 (solid line). The best fit of the data was −4 3 obtained with the parameters: χ0 1.47 × 10 cm /mol, 3 Cimp 0.012 cm K/mol, g 1.95 (fixed from ESR), alterna- B. Magnetization tion parameter α 0.98, and the dominant exchange coupling Magnetic susceptibility χ(T )[≡ M(T )/H] measured in J/kB 36.5K.TheCimp value obtained above is equivalent = ∼ . 1 the applied field of H 1 T is shown in the upper panel to 3 2% impurity spins, assuming that they are of spin- 2 of Fig. 4. The inverse magnetic susceptibility 1/χ(T )for nature. From the values of J/kB and α, the magnitude of the same field is shown in the lower panel of the figure. the zero-field spin gap is estimated to be 0/kB = J/kB[(1 − As the temperature is lowered, χ(T ) increases in a Curie- α)3/4(1 + α)1/4] 2.4K[5,40]. This spin gap corresponds Weiss manner as expected in the paramagnetic regime and to the critical field of Hc1 = 0/(gμB) 1.74 T [24]. It is to max then shows a broad maximum around Tχ 20 K, indicative be noted that this formula overestimates the actual value of of short-range magnetic order and revealing magnetic one- the spin gap since it neglects interchain couplings that tend to dimensionality of NaVOPO4. Although no clear indication reduce the gap size. of any magnetic LRO can be seen down to 0.5 K, another We have also fitted the bulk susceptibility data down to broad feature is visible at around 4 K. As shown in the inset 8 K taking χspin for the uniform spin chain (not shown here). of Fig. 4, this broad feature is field-dependent. Below 1.5 K, The fitting procedure was completed by fixing the g value to χ(T ) shows an upturn likely caused by extrinsic paramagnetic 1.95 (from ESR) and the obtained parameters are χ0 1.59 × −4 3 3 impurities or defect spins present in the powder sample [38]. 10 cm /mol, Cimp 0.013 cm K/mol, and J/kB 36.4K.

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FIG. 6. χ(T ) measured in the low temperature region and in different applied fields. The downward arrows point to the field FIG. 5. Magnetization M vs field H measured at T = 1.5K, induced magnetic transition. using pulsed magnetic field. Lower inset: dM/dH vs H in the high field regime and the arrow points to Hc2. Upper inset: M vs = . H measured at T 0 5 K using MPMS. The solid line is the fit where χ is the intrinsic susceptibility which was kept fixed, using Brillouin function [Eq. (5)]. M is the magnetization after cor f is the molar fraction of impurities, S is the impurity subtraction of the impurity contribution. imp imp spin, gimp is the impurity g-factor, NA is the Avogadro’s

number, BSimp (x) is the Brillouin function with the modified argument x = SimpgimpμBH/[kB(T − θimp)] [43]. The Since both the alternating-chain and uniform-chain models fit obtained fitting parameters are fimp 0.013(3) mol%, the χ(T ) data nicely and yield similar values of the exchange Simp 0.50(3), and gimp 2.06(1). The corrected coupling, it is apparent that the system lies close to the magnetization (Mcor) after the subtraction of this extrinsic boundary between the uniform and alternating-chain models. contribution from the raw data is also presented in Fig. 5. Figure 5 presents the magnetization (M) versus field (H) Clearly, Mcor remainszerouptoHc1 1.6 T that corresponds curve at T = 1.5 K measured using pulsed magnetic field. M to the zero-field spin gap of 0/kB 2 K between the singlet increases almost in a linear fashion up to 40 T. Above 40 T, it ground state and the triplet excited states. exhibits a pronounced upward curvature and then approaches To check for the field-induced effects, if any, χ(T )was the saturation value. Such a curvature in the magnetization measured at low temperatures (0.5 K T 2 K) and in data is a clear signature of strong quantum fluctuations, as different applied fields. As shown in Fig. 6, it shows a change 1 ∼ . anticipated for a spin- 2 1D system. From the derivative of in slope at 0 7 K in the 2.5 T data. As the field increases, M with respect to H versus H plot (lower inset of Fig. 5)the the transition shifts toward high temperatures marked by the saturation or upper critical field is found to be Hc2 57 T. The downward arrows in Fig. 6. This cusplike anomaly above value of Hc2 corresponds to an exchange coupling of J/kB = H 2.5 T and its field variation are robust signatures of the gμBHc2/2kB 37.3 K, assuming the 1D uniform spin chain field-induced magnetic LRO [9]. The variation of TN with model [41]. This value of J/kB matches well with the one respect to H is presented in Fig. 15. obtained from the χ(T ) analysis. Since the system lies at the phase boundary between the uniform-chain and alternating-chain models, it is expected C. ESR that either the system should have a magnetic LRO or a spin ESR experiment was performed on the powder sample gap at very low temperatures. To probe the ground state, we and the results are displayed in Fig. 7. The inset in the measured the magnetization isotherm at T = 0.5Kupto7T top panel of Fig. 7 shows a typical ESR powder spectrum (inset of Fig. 5). Below 2 T, it exhibits a dome-shaped feature at room temperature. The spectra at different temperatures above which the variation of M is linear with H.Thisisa could be fitted well using the powder-averaged Lorentzian clear indication of the existence of a spin gap with the critical line for the uniaxial g-factor anisotropy. The fit to the spec- field of gap closing Hc1 2 T. Typically, in the gapped spin trum at room temperature reveals anisotropic g-tensor com- systems, the magnetization remains zero up to Hc1. However, ponents: parallel component g 1.931 and perpendicular a nonzero value of M below Hc1 is likely due to the saturation component g⊥ 1.965. From these values the average g / of some extrinsic paramagnetic contributions and or defects factor is calculated to beg ¯ 1/3(g + 2g⊥) 1.951. A in our powder sample. Therefore, the data below 1.5 T were slight deviation of the estimated g factor (g/g 0.02) from fittedby[42] = 4+ 1 the free-electron value (g 2) is typical for V (spin- 2 )- based compounds having the VO6 octahedral coordination = χ + μ , M H NA fimpSimpgimp BBSimp (x) (5) [24,44].

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FIG. 8. Upper Panel: Specific heat Cp vs T of NaVOPO4 in zero applied field. The dashed line is the phonon contribution to the specific heat (Cph) obtained using a Debye fit [Eq. (6)]. The solid line indicates the magnetic contribution to the specific heat Cmag. Lower FIG. 7. Upper Panel: Integrated ESR intensity vs temperature. Panel: Left y axis shows Cmag/T and right y axis shows the magnetic Inset: ESR spectrum at room temperature measured at a microwave entropy Smag vs T , respectively. frequency of 9.4 GHz together with the powder-averaged Lorentzian fit (solid line). Lower Panel: Temperature variation of g values (both perpendicular and parallel components) obtained from Lorentzian fit. we subtracted the phonon contribution (Cph) from the total χ Inset: IESR vs with temperature as an implicit parameter. measured specific heat Cp. For this purpose, the experimental data at high temperatures (T 60 K) were fitted by a linear combination of four Debye functions [45]: The temperature-dependent integrated ESR intensity (IESR) θ is presented in the upper panel of Fig. 7. It resembles the 4 3 Dn 4 x T T x e bulk χ(T ) behavior with a pronounced broad maximum at C (T ) = 9R c dx. (6) ph n x 2 max χ θDn 0 (e − 1) around TESR 20 K. When IESR is plotted as a function of , n=1 it indeed follows a straight-line behavior (inset of the lower panel of Fig. 7) down to 18 K. Similar to the χ(T ) data, Here, R is the universal gas constant, the coefficients cn account for the number of individual atoms, and θDn are IESR (T ) also exhibits another weak but broad feature at around 4 K. As shown in the lower panel of Fig. 7, both g-components the corresponding Debye temperatures. Since we have four are constant over a wide temperature range down to ∼15 K. different types of atoms with varying atomic masses and The anomalous behavior below 15 K might be related to the Debye temperature is inversely proportional to the atomic effect of defects/impurities. mass, we have used four Debye functions corresponding to the Na, V, P, and O atoms. Cmag(T ) extracted by subtracting Cph(T )fromCp(T )is D. Specific heat also shown in Fig. 8. The above procedure was verified by Specific heat [Cp(T )] data measured under zero field are calculating the magnetic entropy Smag through the integration shown in the upper panel of Fig. 8. In the high-temperature re- of Cmag(T )/T which gives Smag 5.6J/mol K at 100 K gion, Cp is entirely dominated by phonon excitations, whereas (lower panel of Fig. 8). This value is very close to the expected = = . / 1 the magnetic part dominates only at low temperatures. It entropy Smag R ln 2 5 76 J mol K for a spin- 2 system. shows a broad feature below about 20 K. There is no clear As shown in the upper panel of Fig. 8, Cmag has a broad = . max max anomaly in the data down to T 0 38 K, which excludes maximum at TC 20 K with an absolute value Cmag . / max the possibility of any magnetic LRO in zero field. To quan- 2 80 J mol K. This value of Cmag is indeed very close to tify the magnetic contribution to the specific heat (Cmag), the expected value ∼0.35R = 2.9J/mol K for a uniform

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FIG. 9. Cp/T vs T of NaVOPO4 in the low-temperature regime in different magnetic ﬁelds up to 14 T.

1 max spin- 2 chain [46]. Similarly, the value of TC 20 K, which reflects the energy scale of the leading exchange interaction, / = max/ . corresponds to J kB TC 0 48 41 K in the framework of the uniform spin chain model [40]. It is indeed very close to the values obtained from the other experiments. Furthermore, FIG. 10. Temperature-dependent FT 31P NMR spectra of = . an anomaly in Cmag is visible at around ∼5 K, similar to NaVOPO4 measured at H 2 6 T down to 1.8 K. The dash-dotted the χ(T ) data suggesting that it is an intrinsic feature of the line represents the reference field of the nonmagnetic H3PO4 compound. C T In addition, we have also measured p( )inthelow- 1. NMR spectra temperature region (down to 0.38 K) in magnetic fields up 31 to 14 T. Figure 9 shows the data at few selected fields. In P NMR spectra were measured on the powder sample zero field, no anomaly associated with the magnetic LRO is either by doing Fourier transform (FT) of the echo signal at found down to 0.38 K, but it shows a rapid decrease below 1 K a constant magnetic field or by sweeping the magnetic field 31 toward zero which could be due to the opening of a spin gap. at a fixed frequency. Since P has the nuclear spin I = 1/2, a single 31P NMR line is expected corresponding to one allowed An exponential fit below 1 K yields 0/kB 2.6K which is consistent with the value obtained from the χ(T ) analysis. transition. Figure 10 displays representative NMR spectra When the field is increased, no extra features are seen up to taken for 1.8 K T 238 K at H = 2.6 T. Indeed, a single 2 T. However, for H 2 T, it displays a λ-type peak which spectral line is observed down to 1.8 K and the line width is moves toward high temperatures with increasing field. This found to increase systematically with decreasing temperature. can be ascribed to the crossover from a spin-gap state to a 3D This confirms that NaVOPO4 has a single P site, which is commensurate with the crystal structure. Typically, in a LRO state under external magnetic field. TN as a function of H is plotted in Fig. 15. magnetically ordered state, the NMR nucleus senses the static internal field and hence the NMR line broadens drastically.

31 As one can see in Fig. 10, no significant line broadening E. PNMR was observed around 5 K, ruling out the possibility of any NMR experiments were performed over a wide tempera- magnetic LRO taking place at this temperature. The line ture range down to 40 mK to address several key questions, position is found to shift with temperature. namely: (i) Is the anomaly at 4 K an intrinsic feature? (ii) As we have discussed earlier, specific heat in zero field What is the ground state in zero field? (iii) Does the system does not show any magnetic LRO down to 0.38 K. To ensure undergo field-induced magnetic LRO? In magnetic insulators, that there is no LRO in zero field, NMR spectra were mea- the Hamiltonian is relatively simple and well-defined, which sured down to 40 mK. As one can see in Fig. 11,theNMR allows the powerful local tools like NMR to access the struc- line shape remains the same and there is no significant line ture, static and dynamic properties of a spin system. Since broadening down to 0.04 K at 17.235 MHz (1 T). However, 31P nuclei are strongly coupled to the V4+ ions in the crystal for 68.9424 MHz (4 T), the line broadens abruptly below structure, one can investigate the properties of the spin chains ∼1 K. This can be taken as an evidence that the system by probing at the 31P site. does not undergo magnetic LRO below 1 T and shows a

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31 FIG. 11. Field-sweep P NMR spectra of NaVOPO4 measured 31 at ν = 17.235 MHz (left panel) and 68.9424 MHz (right panel) in FIG. 12. Temperature-dependent PNMRshift[K(T ))] as a the low-temperature region. function of temperature for two different magnetic fields. The solid line is the fit of K(T )byEq.(7). The downward arrows point to the field-induced magnetic transitions. Inset: K vs χ (measured at field-induced magnetic transition at TN 1Kin4Twhich 4 T) with temperature as an implicit parameter. The solid line is the is consistent with the χ(T ) and Cp(T ) measurements. The linear fit. absence of line broadening and any extra feature in the NMR spectra at 1 T also exclude the possibility of any structural or lattice distortions, consistent with our temperature-dependent powder XRD. This is in contrast with that reported for the spin-Peierls compounds CuGeO3 where spin dimerization is is usually very small compared to the transferred hyperfine accompanied by the lattice distortion and α -NaV2O5 where coupling, and therefore neglected. From Eq. (7), Ahf can be the singlet ground state is driven by the charge ordering calculated from the slope of the linear K versus χ plot with at low temperatures [47,48]. 63Cu-NMR experiments in the temperature as an implicit parameter. The inset of Fig. 12 χ well-known BEC compound TlCuCl3 revealed that the field- presents the K versus plot, which obeys a straight line induced magnetic transition is accompanied by a simultane- behavior down to 50 K. The data for T 50 K were fitted ous lattice deformation [49]. This also implies strong spin- well to a linear function and the slope of the fit yields − . /μ phonon coupling at the critical field that drives the BEC K0 1219 65 ppm and Ahf 6445 Oe B. A large value 31 of Ahf indicates that the P nucleus is very strongly coupled phenomenon in TlCuCl3. However, in our compound for + 31 4 H > Hc1 the P NMR line only broadens without any splitting to the V ions. This value is almost twice the hyperfine 4+ or extra features, ruling out the possibility of significant lattice coupling of As with V ions in AgVOAsO4 and NaVOAsO4 deformations in the field-induced state. [22,23]. This also illustrates the fact that the spin chains run along the extended V4+–O–P–O–V4+ path rather than the 4+ 4+ 2. NMR shift short V –O–V pathway. One advantage of the NMR experiment is that the NMR Figure 12 shows the variation of the NMR shift K(T ) with shift directly probes χ and is inert to impurity contri- = . spin temperature measured at H 2 6 T and 4 T. Both the data butions. Therefore, in low-dimensional spin systems K(T ) sets resemble each other. With decrease in temperature, K(T ) data are often used for a reliable estimation of the mag- passes through a broad maximum around 25 K in a similar netic parameters instead of the bulk χ(T ). For a tentative way as that of the bulk susceptibility. With further decrease estimation of the magnetic parameters, we have fitted the in temperature it starts to decrease. A weak but broad hump K(T )usingEq.(7) down to 6 K, taking χ for both the spin appears at T 4 K, reflecting that it is an intrinsic feature alternating-chain and uniform-chain models [40]. To mini- of the compound. At very low temperatures, the small peaks mize the number of fitting parameters, the value of g was . (TN 0 89 K at 2.6 T and 1.05 K at 4 T) are likely due to fixed to 1.95 obtained from ESR experiment. The resultant field-induced magnetic transitions. fitting parameters using the alternating-chain model are K0 Since K(T ) is a direct and intrinsic measure of the spin −1182 ppm, Ahf 6873 Oe/μB, α 0.98, and J/kB 39 K. susceptibility χspin(T ), one can write Taking these values of α and J/kB,wearriveatthespingapof 0/kB 2.37 K [5,40]. However, the fit using the uniform- NAK(T ) = K0 + Ahf χspin, (7) chain model yields K0 −1198 ppm, Ahf 6938 Oe/μB, where K0 is the temperature-independent NMR shift and Ahf and J/kB 39 K. Thus, approximately the same values of 31 is the total hyperfine coupling between the P nuclei and J/kB are obtained from both fits, suggesting that the system 4+ V spins. Ahf is a combination of the transferred hyper- approaches the uniform-chain regime but reveals a small spin fine coupling and the nuclear dipolar coupling contributions, gap. These findings are quite consistent with our analysis of both temperature-independent. The nuclear dipolar coupling the χ(T ) data.

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shows a change in slope similar to that observed in χ(T ), Cp(T ), IESR(T ), and K(T ). At very low temperatures, it exhibits a peak at TN 0.62 K due to the critical slowing down of the ﬂuctuating moments and reﬂects the transition to a magnetic LRO. Similarly, the 1/T1 data for H = 4 T develops an upward trend below 10 K due to the growth of strong magnetic correlations and then approaches the magnetic LRO at TN 1K. The AFM correlations in a spin system can be assessed by analyzing 1/T1(T ). The general expression connecting 1 with the dynamic susceptibility χM (q ,ω0 ) can be written T1T as [53]

1 2γ 2k χ (q ,ω ) = N B | |2 M 0 , 2 A(q) (9) T1T N ω0 A q

where γN is the nuclear gyromagnetic ratio, A(q )isthe form factor of the hyperﬁne interactions between nuclear and χ ,ω electronic spins, and M (q 0 ) is the imaginary part of the dy- χ ,ω = χ ,ω + χ ,ω namic susceptibility [ M (q 0 ) M (q 0 ) i M (q 0 )] at the nuclear Larmor frequency ω0.Forq and ω0 = 0, the χ ,ω real component M (q 0 ) corresponds to the uniform static susceptibility χ(T ). As K(T ) is a direct measure of the static susceptibility χ, χ(T ) in the above expression can be replaced by K(T ). To see the persistent spin correlations, we have plotted 1 against temperature. If the dynamic susceptibility is KT1T dominated by the uniform component (q = 0), then 1/KT1T should be constant over the whole temperature range. As one can see in the bottom panel of Fig. 13, 1 remains constant KT1T down to T ∼ 30 K as expected, demonstrating that 1/T1 at high temperatures is dominated by the q = 0 contribution.

FIG. 13. Upper panel: 1/T1 plotted as a function of temperature As the temperature decreases and becomes comparable to the 1 for two different magnetic fields. The solid line guides through the energy scale of J/kB, increases abruptly. This reflects KT1T linear regime in the H = 2.6 T data. Lower panel: 1 plotted as a the dominant q =±π contribution to 1/T at low tempera- KT1T 1 function of T for H = 2.6T. tures due to the growth of AFM correlations, as the system approaches the low-dimensional short-range order [54]. 3. Spin-lattice relaxation rate Another crucial quantity determined in the NMR experi- F. Microscopic magnetic model ment is the spin-lattice relaxation rate (1/T1). At the central To assess individual magnetic couplings in NaVOPO4,we resonance frequency of each temperature, the sample is irra- use two complementary approaches. First, we extract hopping diated with a single π/2 saturation pulse and the growth of parameters ti between the V 3dxy states in the uncorrelated longitudinal magnetization is monitored for variable delays. band structure, and introduce them into an effective one- These recovery curves are fitted well by a single exponential orbital Hubbard model that yields antiferromagnetic part of function AFM = 2/ = . the superexchange as Ji 4ti Ueff , where Ueff 4 5eV M(t ) / is the on-site Coulomb repulsion in the V 3d bands [55,56]. 1 − = Aet T1 , (8) M(0) Second, we use the mapping procedure [57,58] and obtain exchange couplings Ji from total energies of collinear spin where M(t ) stands for the nuclear magnetization at a time t configurations calculated within DFT + U. after the saturation pulse, and M(0) is the equilibrium mag- The results from both methods are summarized in Table I. netization. Figure 13 (upper panel) displays the variation of Similar to AgVOAsO4 and NaVOAsO4, we find leading 1/T1 with temperature for H = 2.6 T and 4 T extracted from exchange couplings J and J along the 110 and 110¯ crystal- the above fit. For both the fields, 1/T1 is almost temperature- lographic directions oblique to the structural chains of the independent down to T 100 K due to random fluctuation VO6 octahedra. Only a minor difference between the two of paramagnetic moments at high temperatures [50]. For couplings is observed, with (t /t )2 0.93 and J/J 0.93, H = 2.6T,1/T1 decreases linearly down to ∼4 K. Such a in agreement with the experimental alternation ratio α close 1 behavior is typically observed in 1D spin- 2 systems in the to 1.0. The interchain couplings are represented by the weakly temperature range of T ∼ J/kB where 1/T1 is dominated FM Jc that runs along the structural chains via the V–O–V by the uniform (q = 0) contribution [51,52]. At T 4K,it bridge and by the AFM Ja through the single V–O–P–O–V

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TABLE I. The V–V distances (in Å), hopping parameters ti TABLE II. Comparison of magnetic parameters for the / α (in meV) of the uncorrelated (GGA) band structure, and exchange NaVOPO4-like spin-chain compounds. The J kB and values are χ / couplings Ji (in K) obtained via the DFT + U mapping procedure in taken from the (T ) analysis, while 0 kB and Hc1 are taken from NaVOPO4 (this work) and NaVOAsO4 (Ref. [22]). the high-ﬁeld magnetization data. / α = / / NaVOPO4 NaVOAsO4 Compounds J kB (K) ( J J) 0 kB (K) Hc1 (T)

dV−V ti Ji dV−V ti Ji AgVOAsO4 40 0.62 13 10 NaVOAsO4 52 0.65 21.4 16 J 5.385 89 54 5.519 99 57 NaVOPO4 39 0.98 2 1.6 J 5.337 86 50 5.489 81 54 Ja 5.952 42 13 6.073 25 8 Jc 3.565 0 −6 3.617 9 −5 and AFM regions. We have placed some of the rigorously studied gapped and antiferromagnetically ordered compounds bridge. These couplings are similar in nature and magnitude at their respective positions according to the magnitude of to those in the isostructural V4+ arsenates [22,24]. their spin gap and TN. From the phase diagram we infer The hopping parameters in Table I confirm that the su- that NaVOPO4 lies in the proximity of the QCP but in the perexchange pathways for J and J are quite similar, resulting gapped regime of the phase diagram. This renders NaVOPO4 α an ideal candidate for probing field-induced effects in gapped in the alternation ratio close 1.0. Microscopically, these χ / quantum magnets. Further, (T ), Cp(T ), IESR(T ), K(T ), and couplings are mediated by double bridges of the PO4 AsO4 / ∼ tetrahedra and may depend on three geometrical parame- 1 T1 all show a second broad maximum at 4K whichis ters [59]: (i) the in-plane offset of the VO octahedra; (ii) found to be intrinsic to the sample. The energy scale of this 6 feature may be comparable to the size of the spin gap in the out-of-plane offset of the VO6 octahedra, and (iii) the tilt of the tetrahedra with respect to the VO octahedra. In zero field. However, the feature survives in magnetic fields 6 of 4–5 T, well above the gap closing. Therefore, we attribute NaVOPO4-like structures, the in-plane offset plays the crucial / role. Indeed, we find similar offsets of d 0.534 Å (J) and this feature to the effects of disorder and or interchain frustration. A similar broad feature at low temperatures has been 0.515 Å (J )inNaVOPO4 (see Fig. 1), where α approaches 1.0, as opposed to the largely different offsets of 0.593 Å (J) reported in the frustrated spin chain compound LiCuVO4, which is ascribed to the effect of magnetic frustration of the and 0.826 Å (J )inNaVOAsO4 (α 0.65), with the stronger coupling J corresponding to the smaller offset. We conclude interchain interactions that tends to establish an additional that the in-plane offsets control the alternation ratio of the spin short-range order between the chains [60,61]. In NaVOPO4, chains in this family of compounds. we indeed have significant interchain couplings that form a frustrated network. Therefore, interchain frustration may be a more plausible explanation for the second broad anomaly IV. DISCUSSION around ∼4 K. To rule out the impurity effect, we repeated The above assessments convincingly demonstrate that the experiments on different sample batches and obtained 1 identical results. Even our ESR experiments do not detect any NaVOPO4 reveals the magnetism of a spin- 2 alternating chain with a weak alternation of the exchange couplings. This extra signal at this temperature, thus confirming that the signal material does not undergo any magnetic LRO down to at is intrinsic to the sample and not related to an impurity phase. least 50 mK, and rather shows a tiny spin gap of 0/kB We have also thoroughly checked the sample quality, which is 2 K in zero field. In Fig. 14, we constructed a conceptual found to be perfect according to XRD. phase diagram showing the QCP separating the spin-gap For a better understanding of the spin lattice, we compare NaVOPO4 with its structural analogs AgVOAsO4 and NaVOAsO4. Magnetic parameters for all the three compounds are summarized in Table II. The most interesting aspect of NaVOPO4 is arguably the field-induced magnetic LRO. The field evolution TN obtained from the χ(T ), Cp(T ), M(H ), and 1/T1(T ) measurements is shown in Fig. 15. It traces a single dome-shaped curve in contrast to the double-dome reported for the isostructural compound AgVOAsO4 [25]. The shape of the H-T phase diagram is quite similar to other spin-gap systems undergoing BEC of triplons under external magnetic field [10,19,20,62]. The applicability of the BEC scenario is usually verified by the critical exponent obtained from fitting the H-T boundary with the following power-law [63,64]:

1 φ TN ∝ (H − Hc1) , (10) FIG. 14. Illustration of a conceptual phase diagram showing the spin-gap and AFM regions separated by the QCP. Some reported where φ = d/2 is the critical exponent, which reﬂects the compounds are placed at their respective positions. universality class of the transition. Here, H − Hc1 controls

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ﬁt returns Hc1 1.3 T and φ 1.8. In the inset of Fig. 15, 1/1.8 we have plotted TN versus (H − Hc1) to highlight the linear regime. The obtained value of Hc1 1.3 T leads to 0/kB 1.6 K. This value of the spin gap differs slightly from the one obtained from the magnetic isotherm data. Simi- larly, φ 1.8 is more close to (though slightly larger than) 1.5 obtained theoretically [9,63,64]. Nevertheless, such discrep- ancies are not uncommon in other quantum magnets, including Ba3Cr2O8,Pb2V3O9,IPA-CuCl3, etc., undergoing triplon BEC [20,72–74].

V. CONCLUSION 1 In conclusion, we have discovered a new spin- 2 quasi- 1D compound NaVOPO4, which is well described by an alternating spin chain model with a dominant AFM exchange coupling of J/kB 39 K and a tiny spin gap of 0/kB 2K. External magnetic field of Hc1 1.6 T closes the spin gap and triggers magnetic LRO. Such a small spin gap and the onset of magnetic LRO already in low magnetic fields place FIG. 15. H-T phase diagram of NaVOPO4 obtained using the NaVOPO in the vicinity of the QCP separating the spin- data points from the χ(T ), Cp(T ), and 1/T1(T ) measurements. Two 4 gap and AFM LRO states in the phase diagram. The field- critical fields (Hc1 and Hc2) extracted from the M vs H data are also placed in the phase diagram. The dashed line traces the anticipated induced magnetic LRO is indeed confirmed from the χ(T ), / phase boundary between the two critical fields. The lower (Hc1)and Cp(T ), K(T ), and 1 T1(T ) measurements and found to move upper (Hc2) critical fields separate the quantum disordered state from toward high temperatures with the field. A power-law fit to the the 3D-XY AFM state and fully saturated state from the the 3D-XY H-T phase boundary yields an exponent φ 1.8, a possible 1/1.8 AFM state, respectively. Inset: TN vs (H − Hc1 ) and the solid line signature of triplon BEC. This compound appears to be an is the linear fit. ideal system for high-field studies. Moreover, it also demands further experimental studies including neutron scattering and the boson density at a given temperature. This power law NMR on single crystals to elucidate microscopic nature of the < . max field-induced ordered phase. is typically fitted in the critical regime T 0 4Tc in the vicinity of the QCP for a meaningful estimation of φ, where T max is the maximum temperature of the magnetically ordered c ACKNOWLEDGMENTS regime [65].Ina3Dsystem(d = 3), the value of the critical exponent is predicted to be φ = 1.5[8,9,62] which has been P.K.M. and R.N. acknowledge BRNS, India, for finan- experimentally verified in quantum magnets TlCuCl3 [66,67], cial support bearing Sanction No. 37(3)/14/26/2017-BRNS. DTN [68,69], (CH3)2CHNH3CuCl3 [70], etc. However, for Work at the Ames Laboratory was supported by the U.S. De- a 2D system, the theory predicts φ 1 over a wide tem- partment of Energy, Office of Science, Basic Energy Sciences, perature range except at very low temperatures [71]. Indeed, Materials Sciences and Engineering Division. The Ames Lab- this has been experimentally verified in K2CuF4 [11]. At oratory is operated for the U.S. Department of Energy by Iowa sufficiently low temperatures (i.e., temperatures lower than State University under Contract No. DEAC02-07CH11358. the interlayer coupling), because of the interlayer interaction, Y.I. thanks JSPS Program for Fostering Globally Talented one may expect a crossover from 2D BEC to 3D BEC with Researchers which provided an opportunity to be a visiting φ = 3/2. Of course, exchange anisotropy, which is inherent scholar at Ames Laboratory. We also thank C. Klausnitzer to real materials, will often alter the universality class of the (MPI-CPfS) for technical support. A.A.T. was funded by transition. the Federal Ministry for Education and Research through To corroborate the above BEC scheme here, the H-T the Sofja Kovalevskaya Award of Alexander von Humboldt phase boundary of NaVOPO4 wasfittedbyEq.(10). Given Foundation. We also acknowledge the support of the HLD the limited temperature range available experimentally, we at HZDR, member of European Magnetic Field Laboratory varied the fitting range between 0.75 K and 1.25 K. Our (EMFL).

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