A Near-Ideal Molecule-Based Haldane Spin-Chain

Robert C. Williams,1 William J. A. Blackmore,1 Samuel P. M. Curley,1 Martin R. Lees,1 Serena M. Birnbaum,2

John Singleton,2 Benjamin M. Huddart,3 Thomas J. Hicken,3 Tom Lancaster,3 Stephen J. Blundell,4 Fan

Xiao,5, 6 Andrew Ozarowski,7 Francis L. Pratt,8 David J. Voneshen,8 Zurab Guguchia,9 Christopher

Baines,9 John A. Schlueter,10, 11 Danielle Y. Villa,12 Jamie L. Manson,12 and Paul A. Goddard1, ∗

1Department of Physics, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK2National High Magnetic Field Laboratory, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

3Department of Physics, Durham University, South Road, Durham DH1 3LE, UK4Department of Physics, Clarendon Laboratory, Oxford University, Parks Road, Oxford OX1 3PU, UK

5Department of Chemistry and Biochemistry, University of Bern, Freiestrasse 3, CH-3012 Bern, Switzerland6Laboratory for Neutron Scattering and Imaging,

Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland7National High Magnetic Field Laboratory, Florida State University, Tallahassee, Florida 32310, USA

8ISIS Facility, STFC Rutherford Appleton Laboratory, Chilton, Oxon OX11 0QX, UK9Laboratory for Muon Spin Spectroscopy, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland

10Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA11Division of Materials Research, National Science Foundation,

2415 Eisenhower Ave, Alexandria, Virginia 22314, USA12Department of Chemistry and Biochemistry, Eastern Washington University, 226 Science, Cheney, Washington 99004, USA

The molecular coordination complex NiI2(3,5-lut)4 [where (3,5-lut) = (3,5-lutidine) = (C7H9N)]has been synthesized and characterized by several techniques including synchrotron X-ray diffraction,ESR, SQUID magnetometry, pulsed-field magnetization, inelastic neutron scattering and muon spinrelaxation. Templated by the configuration of 3,5-lut ligands the molecules pack in-registry withthe Ni–I· · · I–Ni chains aligned along the c–axis. This arrangement leads to through-space I· · · Imagnetic coupling which is directly measured for the first time in this work. The net result is anear-ideal realization of the S = 1 Haldane chain with J = 17.5 K and energy gaps of ∆‖ = 5.3 K∆⊥ = 7.7 K, split by the easy-axis single-ion anisotropy D = −1.2 K. The ratio D/J = −0.07affords one of the most isotropic Haldane systems yet discovered, while the ratio ∆0/J = 0.40(1)(where ∆0 is the average gap size) is close to its ideal theoretical value, suggesting a very highdegree of magnetic isolation of the spin chains in this material. The Haldane gap is closed by

orientation-dependent critical fields µ0H‖c = 5.3 T and µ0H

⊥c = 4.3 T, which are readily accessible

experimentally and permit investigations across the entirety of the Haldane phase, with the fully

polarized state occurring at µ0H‖s = 46.0 T and µ0H

⊥s = 50.7 T. The results are explicable within

the so-called fermion model, in contrast to other reported easy-axis Haldane systems. Zero-fieldmagnetic order is absent down to 20 mK and emergent end-chain effects are observed in the gappedstate, as evidenced by detailed low-temperature measurements.

INTRODUCTION

Background. Molecule-based magnets have proveda highly successful avenue in achieving desired low-dimensional geometries [1–3]. Several classes of materi-als which host quantum-disordered non-magnetic groundstates can undergo magnetic field and pressure drivenquantum phase transitions (QPTs), where the magneticground states undergo a dramatic reorganisation andphysical properties exhibit characteristic behavior [4, 5].

An important asset of molecule-based systems isthe modest energy scale of their exchange interactionstrengths, which makes their QPTs experimentally ac-cessible, thus providing a vital opportunity to test thepredictions of theory and simulations. Invaluable insighthas been afforded by the intensive experimental study ofprototypical realizations of models including: the idealS = 1/2 chain material Cu(C4H4N2)(NO3)2 [6]; thestrong-leg and strong-rung two-leg spin-ladder systems

(C7H10N)2CuBr4 (DIMPY) [7] and (C5H12N)2CuBr4

(BPCB) [8], respectively; the S = 1/2 dimer com-pound [Cu(pyz)(gly)](ClO4) [9]; the large-D systemNiCl24SC(NH2)2 (DTN) [10]; and the spin-liquid com-pound (C4H12N2)Cu2Cl6 (PHCC) [11]. Of note isthat several of these halide containing materials relyon through-space magnetic couplings; i.e. two-halide ex-change, rather than that offered by bridging ligands. Forthe first time, we present a novel material of this kindsupported by non-covalent I· · · I interactions.

Pioneering work by Haldane demonstrated thatHeisenberg antiferromagnetic (AFM) chains of integerspins host a disordered ground state which is topolog-ically distinct from the well-known, spin-half analogue(the Tomonaga-Luttinger liquid) [12, 13]. The signifi-cance of this was recognized via the award of the 2016Nobel Prize in Physics. The Haldane phase is protectedby its eponymous energy gap, above which excitationspropagate. Fractional excitations manifesting as S = 1/2degrees of freedom exist at the end-chains, due to the

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symmetry-protected bulk topological phase. This phe-nomenon arises naturally out of the valence bond solidmodel [14], and has been experimentally observed in sev-eral Haldane compounds [15–17]. Higher concentrationsof S = 1/2 edge-state excitations may be induced bybreaking the chains via the introduction of non-magneticsites, which has been shown to result in magnetic order-ing in PbNi2V2O8 due to the almost critical inter-chainexchange coupling [18, 19].

It has been demonstrated that extending the Hamil-tonian of the AFM S = 1 chain to include single-ionanisotropy (SIA) and inter-chain interactions gives riseto a rich phase diagram [20–22]. The resultant Hamilto-nian is given by

H = D∑

i

(Szi )2 + J

∑

〈i,j〉Si · Sj + J⊥

∑

〈i,j′〉Si · Sj′

+ µBµ0

∑

i

H · g · Si, (1)

where J and J⊥ are the dominant intra-chain and weakerinter-chain exchange interactions between S = 1 mo-ments, respectively. Here, angular brackets denote sumstaken over unique pairs of nearest-neighbor spins, primedindices refer to spins on adjacent chains, D is the axialSIA parameter and the final term is the usual Zeemaninteraction for an external magnetic field, H. The ensu-ing phase diagram has recently been extended to considerthe rhombic SIA parameter, E [23].

In order to exploit the interplay between crystalstructure and magnetic behavior in S = 1 systems,and ultimately tune and control the parameters Jand D, we have synthesized and characterized a se-ries of Ni(II)-containing materials. Different bridgingand non-bridging ligands enable the exploration of low-dimensional phase diagrams, resulting in dramaticallydifferent magnetic behavior [24–26]. In this work wereport the discovery and characterization of a S = 1molecule-based AFM chain NiI2(3,5-lut)4 where (3,5-lut) = (3,5-lutidine) = (C7H9N). Magnetometry mea-surements reveal the AFM intra-chain coupling J =17.5(2) K, which together with the axial SIA parame-ter D = −1.2(1) K results in two triplet energy gaps∆‖ = 0.46(1) meV = 5.3(1) K and ∆⊥ = 0.66(1) meV =7.7(1) K, as observed via inelastic neutron scatter-ing (INS). The Haldane gap may be closed by the

anisotropic critical magnetic fields µ0H‖c = 5.3(1) T

and µ0H⊥c = 4.3(1) T, and the system is saturated at

µ0H‖s = 46.0(4) T and µ0H

⊥s = 50.7(8) T. The ra-

tio D/J = −0.07(1) makes this the among the mostisotropic Haldane chains reported to date, and the mod-est energy scales lead to experimentally accessible criticalfields, enabling the use of superconducting magnets foundat beamline facilities to explore the entire Haldane phase.To this end, we performed the first transverse-field muonspin relaxation (µ+SR) study of the field-driven closure of

the Haldane gap, while complementary zero-field µ+SRmeasurements confirm the absence of zero-field magneticordering down to T = 20 mK.

For the isotropic Haldane phase with D = 0, analyti-cal and numerical approaches postulate two- and three-particle continua in the zero-field magnetic excitationspectra at the Brillouin zone boundary (k = π) and cen-ter (k = 0), respectively (see Ref. [27] and referencestherein). These continua augment the well-establishedlowest-lying “single-magnon” mode defining the Haldanegap ∆. However, their existence has not yet been demon-strated experimentally [28–30]. A nearly isotropic Hal-dane chain material such as NiI2(3,5-lut)4 is thereforehighly desirable to help verify this prediction. NiI2(3,5-lut)4 also affords the possibility for controlled introduc-tion of both bond and site disorder, which makes thiscompound a model system for investigations into thephysics of the Haldane phase.Single Ion Anistropy in the Haldane Phase. In

zero-field, the presence of axial SIA serves to lift the de-generacy of the excited triplet states above the Haldaneground state, resulting in energy levels given by [31]

∆‖ = ∆0 + 1.41D, ∆⊥ = ∆0 − 0.57D, (2)

for the singlet longitudinally polarized excitation mode(denoted ‖, corresponding to the Sz = 0 triplet state) andthe transverse-polarized doublet (⊥, corresponding to theSz = ±1 triplet states). Longitudinal and transverse isdefined with respect to the unique z direction, which herecoincides with the chain axis c. The intrinsic Haldane gap∆0 = 0.41J , depends solely on the intra-chain exchangecoupling [32–36]. This energy-level arrangement may becompared to the single-ion equivalent for isolated S = 1moments subject to axial SIA, where the effective SIAparameter Deff is given by

Deff = ∆⊥ −∆‖ = −1.98D. (3)

Note Deff and D have opposite signs.There are several different quantum-field-theoretical

models which describe the evolution of these energy levelsunder the application of magnetic fields. These modelsyield anisotropic values of the critical fields at which theZeeman interaction drives a triplet state lower in energythan the Haldane ground state.

The field-dependent energy levels of the ‘fermion’ (andequivalent ‘perturbative’) model [37–40] map directlyonto the isolated S = 1 single-ion description [41], witheffective axial SIA term Deff given by Equation 3. Thefermion model predicts the following critical fields

g‖µBµ0H‖c = ∆⊥, g⊥µBµ0H

⊥c =

√∆‖∆⊥, (4)

for fields parallel and perpendicular to the unique axis.The ‘boson’ (and equivalent ‘macroscopic’) model [37,

42] differs from the fermion model for fields perpendicular

3

to the unique axis, particularly in the region close to thefield-driven closure of the Haldane gap. The boson modelpredicts the following anisotropic critical fields

g‖µBµ0H‖c = ∆⊥, g⊥µBµ0H

⊥c = ∆‖. (5)

Note that for the case of a negative SIA parameterD < 0 (easy-axis), then ∆‖ < ∆⊥ and therefore both

the fermion and boson models predict H‖c > H⊥c (and

vice-versa for the easy-plane case).

The fermion model has been shown to consistently de-scribe the behavior of the easy-plane compounds NENP[43] and NDMAP [44], as it is able to simultaneously ac-count for the anisotropic critical fields and the triplet en-ergy levels. In contrast, the boson model has been shownto successfully describe the behavior of the two easy-axisHaldane compounds PbNi2V2O8 and SrNi2V2O8 [45–47].It is believed that the success of the boson model for thesecompounds is due to their easy-axis anisotropy and crit-ical inter-chain coupling interactions (J⊥), which placethem close to the phase boundary between the Haldaneand ordered Ising AFM phases.

RESULTS

Structural Determination. Synchrotron single-crystal X-ray diffraction measurements were performedto solve the crystal structure of NiI2(3,5-lut)4 at T =100(2) K. NiI2(3,5-lut)4 crystallizes in the space groupP4/nnc, and comprises linear Ni–I· · · I–Ni chains run-ning parallel to the c-axis, as shown in Figure 1(a). Theintra-chain Ni–Ni separation is c = 9.9783(2) A, wherethe magnetic superexchange interaction J is mediatedvia the two iodine ions, and the linear configuration sug-gests AFM interactions according to the Goodenough-Kanamori rules [48–50].

( b )

( a ) CINN i

c

FIG. 1. (a) The chain structure of NiI2(3,5-lut)4, viewed alongthe [110] direction, normal to the chain axis c (H atoms areomitted for clarity). (b) The local crystal environment of eachNi(II) ion.

The local crystal environment of each Ni ion is atetragonally elongated NiI2N4 octahedron, shown in Fig-ure 1(b), where the unique axial direction coincides withthe chain axis c. The Ni–N and Ni–I bond lengths are2.123 and 2.833 A, respectively. The chains running par-allel to c lie on the axes of four-fold rotational sym-metry operators, which preclude the existence of rhom-bic SIA, and we therefore expect the magnetic behaviorof NiI2(3,5-lut)4 to be described by the Hamiltonian inEquation 1.

The non-bridging lutidine ligands serve to separate ad-jacent chains within the ab-plane, which are offset byc/2. The shortest Ni–Ni distance is actually betweenions on neighboring chains, however, there is no clearexchange pathway in this direction. The rings of the lu-tidine ligands are canted by 41.57◦ with respect to theab-plane, forming a propeller-like arrangement about thechain axes.

Not only do the methyl-substituents of the lutidine lig-ands in NiI2(3,5-lut)4 create large inter-chain separationsbut they influence the crystal packing and prompt properalignment of Ni–I· · · I–Ni chains. While NiI2(pyridine)4

is known [51], the molecules pack in such a way that po-tential I· · · I interactions are negated, making NiI2(3,5-lut)4 particularly novel. It is now possible to directlymeasure the magnetic exchange strength propagated byI· · · I couplings.

Magnetometry. Magnetic susceptibility measure-ments were performed on a powder sample of NiI2(3,5-lut)4 using a SQUID magnetometer, with results shownin Figure 2(a). Upon cooling, a broad peak in the mo-lar susceptibility χm(T ) is visible at around 20 K whichindicates the build-up of short-range correlations and istypical for an AFM chain. As temperature is decreasedfurther, a local minimum is reached at around 2 K, dueto a paramagnetic tail which dominates the magnetic re-sponse as the sample is cooled toward zero temperature.Importantly, there is no evidence in χm(T ) for a tran-sition to magnetic long range order (LRO) down to thelowest measured temperature of 0.47 K. In fact, as will bediscussed later, our muon spin relaxation study demon-strates that in zero-field NiI2(3,5-lut)4 does not undergomagnetic ordering for temperatures down to T = 20 mK,indicating a small ratio |J⊥|/J . Taken together with thesmall ratio |D|/J revealed by the INS results, discussedbelow, this places NiI2(3,5-lut)4 in the gapped Haldaneregion of the theoretically derived phase diagram [20, 21].The paramagnetic upturn in χm(T ) is therefore ascribedto the emergence of S = 1/2 degrees of freedom at thechain-ends within the Haldane phase, as has been ob-served in susceptibility studies of related Haldane sys-tems, for example NENP [17].

Defect sites (with fraction ρ) each induce two S = 1/2end-chain moments, and are expected to arise due to lat-tice defects and crystallite boundaries within the powdersample. The temperature-dependent molar susceptibil-

4

0 1 0 2 0 3 0 4 0 5 0 6 00 . 0

0 . 5

1 . 0

3 2 3 6 4 0 4 4 4 8 5 2 5 6 6 00 . 0

0 . 4

H c

µ0 H ( T )

M/M S

M ( H )

H s | |

d M / d H

µ0 H ( T )

Arb. u

nits

T = 0 . 3 6 K

H s ⊥

( d )

0 1 2 3 4 5 6 70 . 0

0 . 5

1 . 0

1 . 5

2 . 0

M (10

−1 µ

B ion-1 )

µ0 H ( T )

1 0 K 4 K 1 . 8 K 0 . 5 K

( c )

1 1 0 1 0 05

1 01 52 02 5

χ m (1

0-8 m3 mol-1 )

T ( K )

µ0 H = 0 . 1 T

0 1 2 3 4 5 6 7T = 0 . 5 K

Arb. u

nits

d M / d H d 2 M / d H 2

µ0 H ( T )

H c

( b )

( a )

FIG. 2. Powder magnetometry results for NiI2(3,5-lut)4. (a)Magnetic susceptibility data (circles), where the solid line rep-resents a fit to the data over the entire measured temperaturerange using Equation 6. (b) Quasi-static magnetization data,where the lowest temperature (T = 0.5 K) data have beenfit to Equation 7 for the field range µ0H ≤ 2 T (solid greenline). (Inset) The first and second derivatives of the low-temperature M(H) data, with respect to magnetic field. (c)Pulsed-field magnetization data and (d) their derivative, mea-sured at T = 0.36 K.

ity data may therefore be modeled using the sum of twocomponents

χm = (1− ρ)χchain + 2ρχ1/2, (6)

where χchain is the susceptibility of an ideal isotropic Hal-dane chain [52, 53] and the end-chain contribution is cap-tured by χ1/2, which is the usual Curie-Weiss expressionfor paramagnetic S = 1/2 moments.

Despite the isotropic approximation, this model is seento account for the χm(T ) data extremely well across theentire measured temperature range (0.47 ≤ T ≤ 300 K),as shown in Figure 2(a). The extracted fit param-eters yield the fraction of defect sites ρ = 1.7(1)%,plus estimates for the intra-chain exchange interaction

strength J = 18.33(7) K and the average Haldane gap∆0 = 7.44(1) K. The resultant magnitude of the Haldanegap with respect to the intra-chain exchange strength∆0/J = 0.406(2) is found to be in excellent agreementwith the theoretically expected value of 0.41 for the idealHaldane chain [32–36]. The fit value of the powder-average g-factor is g = 2.184(2).

Magnetization measurements were performed on apowder sample of NiI2(3,5-lut)4 in quasi-static magneticfields µ0H ≤ 7 T, with results shown in Figure 2(b). Aclear kink is visible in the data close to 4 T, which be-comes more pronounced in lower temperature datasets.We ascribe this kink to the field-driven closure of theHaldane gap, as the Zeeman interaction drives the en-ergy of a triplet excited state below that of the Haldaneground state [54]. The powder-average value of the crit-ical field µ0Hc = 4.4(2) T is obtained by locating thepeak in d2M/dH2 at T = 0.5 K, shown in the insetto Figure 2(b), corresponding to the kink in the M(H)magnetization data.

Further evidence for the presence of the magneticallydisordered Haldane phase for fields below Hc, comes fromthe suppression of magnetization in this region as tem-perature is decreased from T = 10 K [Figure 2(b)]. Amagnetic response is still visible for low magnetic fieldsµ0H < 2 T for M(H) data measured at T = 0.5 K, whichis due to the paramagnetism of the end-chain S = 1/2moments already identified in the susceptibility data.These low-field magnetization data can be modeled using

M(H) = N1/2M1/2(H) +N1M1(H), (7)

where NS and MS(H) are the fractions and Brillouin re-sponses, respectively, of paramagnetic moments with spinS. The fit yields N1/2 = 2.5(3)%, comparable to the frac-tion of end-chain moments 2ρ = 3.4(2)% obtained usingthe susceptibility data, while N1 = 0.0(2)%, further con-firming the presence of fractional end-chain states arisingout of the Haldane phase.

In order to investigate the SIA-induced orientation-dependence of the critical magnetic fields, low-temperature (T = 0.5 K) magnetization measurementswere made on a small single crystal of NiI2(3,5-lut)4

with magnetic fields oriented parallel (‖) and perpen-dicular (⊥) to the unique c-axis, with results shown inFigure 3. An isotropic paramagnetic response is visi-ble at low fields, consistent with the behaviour of para-magnetic S = 1/2 moments; however, the subsequentupturn in M(H) is clearly seen to differ for the twomagnetic field orientations. By examining the secondderivative d2M/dH2 [Figure 3(b)], the critical field val-

ues µ0H‖c = 5.3(1) T and µ0H

⊥c = 4.3(1) T are deter-

mined. The observation that H‖c > H⊥c constitutes un-

ambiguous evidence for non-zero easy-axis SIA in thiscompound.

Pulsed-field magnetization measurements were per-formed on a powder sample of NiI2(3,5-lut)4, with rep-

5

0 1 2 3 4 5 6 7- 0 . 0 1

0 . 0 0

0 . 0 1

0 . 0 2

0 . 0 3

d2 M/d(µ

0H)2 (µ

B T-2 io

n-1 )

µ0 H ( T )

0 . 1

0 . 2

0 . 3

H ⊥c H | |c

T = 0 . 5 KM

(µ B io

n-1 ) H | | z H ⊥ z

( b )

( a )

FIG. 3. (a) Single-crystal magnetization measurements per-formed with magnetic fields parallel (‖) and perpendicular(⊥) to the unique local axis in NiI2(3,5-lut)4, which coincideswith the chain direction. (b) The second derivative of mag-netization with respect to field yields the anisotropic criticalfields.

resentative results shown in Figure 2(c). As for the mea-surements under quasi-static fields, a kink is clearly vis-ible where the Haldane gap is closed at Hc. As mag-netic field is increased further, the magnetization re-sponse increases approximately linearly until close tothe saturation field, where the curvature is typical of aone-dimensional system [55]. The peak in d2M/dH2 isused to determine the value of the critical field µ0Hc =4.3(3) T, which agrees closely with the value deducedfrom the SQUID magnetometry measurements. Usingthe result that D < 0 in this system then, following themean-field calculations in Ref. [56], the anisotropic satu-ration fields are found to be

g‖µBµ0H‖s = 2(2J − |D|), g⊥µBµ0H

⊥s = 2(2J + |D|),

(8)

where it has been assumed that |J⊥| � J . As shown in

Figure 2(d), µ0H‖s = 46.0(4) T and µ0H

⊥s = 50.7(8) T

are readily identifiable as the beginning and end, re-spectively, of the decrease in dM/dH accompanying theplateau in M(H) at saturation. Taking the anisotropicg-factors g‖,⊥ = 2.13(1), 2.19(1) from the ESR resultsdiscussed below, Equations 8 yield values for the SIAparameter D = −1.2(3) K and intra-chain exchange cou-pling J = 17.5(2) K, which is in reasonable agreementwith the estimate obtained from the fit to the suscepti-bility data.

Inelastic Neutron Scattering. In order to examinethe magnetic excitations of NiI2(3,5-lut)4 and resolve the

SIA-split Haldane energy gaps in zero-field, INS measure-ments were performed on a powder sample using the LETinstrument at ISIS, UK. Representative data collected atT = 1.8 K for incident neutron energy Ei = 2.2 meVare shown as a (|Q|, E) spectrum in Figure 4(a), where|Q| and E denote momentum and energy transfers, re-spectively. Corresponding T = 12 K data are treated asa non-magnetic background, and have been subtractedfrom the data. The two components of the triplet ex-citations are visible as bands of excitations for energiesabove E ≈ 0.4 and 0.6 meV. Intensity for energy trans-fers below E ≈ 0.2 meV is attributed to the elastic andquasi-elastic contribution from nuclear scattering.

The Haldane gaps may be clearly quantified by inspect-ing an energy cut through the data: Figure 4(b) showscuts obtained by integrating over the full measured rangeof |Q| for the background-subtracted data for T = 1.8, 3and 7 K. The SIA-split Haldane gaps are visible as thelow-energy onsets of the bands of spin excitations, wherethe large density of states at each excitation band mini-

0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 1 . 4- 101234567

∆ | |

Norm

alised

intens

ity (a

rb. un

its)

E ( m e V )

1 . 8 K 3 K 7 K

∆⊥

I n s t r u m e n t r e s .

Intensity (arb. units)

| Q | ( Å - 1 )

E (me

V)

( b )

( a ) 1 . 8 K

FIG. 4. (a) Time-of-flight INS data, presented as a (|Q|, E)heat-map for T = 1.8 K and incident neutron energy Ei =2.2 meV, after background subtraction. (b) Representativeenergy cuts, obtained by integrating over the full measuredrange of |Q|, for the background-subtracted data at T = 1.8,3 and 7 K.

6

mum leads to a rapid increase in scattering intensity atenergies centered around each gap energy [57, 58]. Theexcitation bands become more pronounced as tempera-ture is lowered, confirming that these features originatein the magnetic contribution to the scattering.

Taking the midpoint of the low-energy onset of eachpeak yields values for the SIA-split Haldane gaps of∆‖ = 0.46(1) meV = 5.3(1) K and ∆⊥ = 0.66(1) meV =7.7(1) K, where labels ‖ and ⊥ are assigned using theconclusion that SIA in this system is easy-axis in na-ture, based on the magnetization results discussed above.Using Equation 2, these gap values correspond to aneasy-axis value of the SIA parameter D = −1.2(1) K,in excellent agreement with the estimate based on the

anisotropic saturation fields H‖,⊥s . The intrinsic Hal-

dane gap of ∆0 = 0.60(1) meV = 7.0(1) K predictsan intra-chain AFM exchange interaction J = 17.0(2),where both of these values are in reasonable agreementwith the result of the powder magnetometry measure-ments described above. The ratio of SIA to intra-chainexchange is therefore determined to be D/J = −0.07(1)which, together with the absence of LRO for temper-atures down to T = 20 mK revealed using muon spinrelaxation, places NiI2(3,5-lut)4 firmly in the Haldanephase at low temperatures, as per the phase diagram inRef. [21].

Muon Spin Relaxation. Zero-field muon spin relax-ation (µ+SR) measurements were made on a powder sam-ple of NiI2(3,5-lut)4 to search for LRO in the absence ofany applied magnetic field. No oscillations were resolvedin the muon spin polarization for temperatures down toT = 20 mK, consistent with the absence of magneticorder. Instead, we observe exponential relaxation withvery little temperature-dependence, implying dynamicsin a disordered magnetic field distribution [59] and nophase transition to an ordered state. This observation isconsistent with the realization of a Haldane state.

To investigate the closure of the Haldane gap,transverse-field µ+SR spectra were measured for a pow-der sample in applied fields 2.0 ≤ µ0H0 ≤ 7.0 T at fixedtemperatures T = 0.02 K and T = 0.75 K. The distri-bution of local magnetic field strength p(B) across thesample volume may be obtained by taking the Fouriertransform of the measured time-dependent muon spinpolarization. Example Fourier spectra for T = 0.02 K,obtained using the maximum entropy method [60], areshown in Figure 5. The spectra consist of a narrow mainpeak (common to all) and a low-field shoulder whosecharacter changes with applied field µ0H0. The mainpeak is due to muons experiencing fields close to µ0H0,in sites in the sample where there are not significantstatic internal local magnetic fields. This main peakshifts to slightly higher fields as the external field µ0H0

is increased, which could reflect a small (motionally nar-rowed) paramagnetic contribution to the total local fieldat these sites due to dilute end-chain spins. The broader

0.0

0.2

0.4

0.6

0.8

1.02.0 T (a)

0.0

0.2

0.4

0.6

0.84.0 T (b)

0.0

0.2

0.4

0.6

0.8

Nor

mal

ized

Spec

tral

Den

sity

4.3 T (c)

0.0

0.2

0.4

0.6

0.85.0 T (d)

0.0

0.2

0.4

0.6

0.8

-6 -4 -2 0 2 4 6

B 0H0 (mT)

6.0 T, T = 0.02 K6.0 T, T = 0.75 K

(e)

−

FIG. 5. Fourier transform transverse-field µ+SR spectra mea-sured at T = 0.02 K. In (e) the corresponding spectrum atT = 0.75 K is shown as the dashed line.

low-field feature is well coupled to the magnetic behavior,and reflects the contribution from muons experiencing aninternal local field in addition to the applied field. A sig-nificant local field at the muon site below µ0Hc is unex-pected, but we note that a high-field feature was observedin the µ+SR spectra of the molecular spin-ladder BPCB[61] which was attributed to a muon-induced distortionat the muon site. While both the main peak and the low-field shoulder (and therefore the average spectral weight)shift to higher fields with increasing µ0H0, the shift islarger for the main peak, resulting in increasing separa-tion of the peaks as µ0H0 increases. At µ0H0 ≥ 6.0 T[Figure 5(e)] the shoulder broadens significantly, which

7

1.5

2.0

2.5

3.0

3.5

4.0(a) T = 0.02 K

1.5

2.0

2.5

2 3 4 5 6 7

0H0 (T)

(b) T = 0.75 KPeak

FWH

M(m

T)

FIG. 6. Full width half maximum (FWHM) for the low-fieldpeak at (a) T = 0.02 K and (b) T = 0.75 K. The solid linesare guides to the eye, and are the same for both temperatures.

reflects the widening of the local magnetic field distribu-tion as the field at the muon site increases. The spectrameasured at T = 0.75 K are similar to those measuredat T = 0.02 K [see Figure 5(e)].

To parametrize the shapes of the field spectra we fitthem to the sum of three Gaussian peaks, one to modelthe main peak, a second to capture the additional weighton the low-field side and a low amplitude, broad contri-bution to capture the tails of the main peak:

A(B) =

3∑

i=1

Ai exp

[− (B −Bi)

2

2σ2i

]. (9)

The spectral weight (given by the product Aiσi) of thelow-field component is approximately constant, with thepeak amplitude decreasing as the feature broadens. Thelow-field shoulder accounts for ≈ 25% of the total spec-tral weight, with the two other components contributingapproximately equally to the remaining spectral weight.As seen in Figure 6, the width of the shoulder is ap-proximately constant up to µ0H0 ≈ 4.0 T beyond whichthe full width half maximum (FHWM) increases signif-icantly with increasing µ0H0 for both T = 0.02 K andT = 0.75 K. Similar increases in the width of the fielddistribution have been seen in µ+SR experiments on themolecular magnet [Cu(pyz)(gly)](ClO4) [9] and in thecandidate Bose-Einstein condensation material Pb2V3O9

[62], where rapid rises in the relaxation rate were ob-served at field-induced transitions. This suggests thatthe system undergoes a crossover in behavior or phasetransition for fields µ0H0

>∼ 4.0 T.The solid line in Figure 6(a) is a guide to the eye, cap-

turing the field-dependence of the low-field peak FHWMfor T = 0.02 K. We estimate µ0Hc ≈ 4.4(2) T for the

critical field, in agreement with the powder magnetiza-tion results. The same solid line is overlaid with the datafor T = 0.75 K in Figure 6(b), showing that the behaviorof the peak FHWM is similar at both temperatures.Electron Spin Resonance. In order to further elu-

cidate the nature of the axial SIA induced by the Ni(II)tetragonal local environment, ESR measurements wereperformed on a powder sample of NiI2(3,5-lut)4. Trans-mission spectra collected in the first-derivative mode athigh temperature are presented in Figure 7(a), whereT = 30 K is sufficiently high, relative to the energy scalesof the intra-chain exchange coupling J and resultant Hal-dane gap ∆0, that the system is within the thermallydisordered phase. The spectra display a single exchange-narrowed transition and are therefore not indicative ofthe value of D. This has been seen elsewhere, for exam-ple in the related compound PbNi2V2O8 [45]. A linearfit to the transition positions in the field–frequency planeis shown as a dashed line in Figure 7(a), which yields thepowder-average g-factor g = 2.19(1), in close agreementwith the value obtained using magnetic susceptibility.

At low temperatures (T = 3 K), the Haldane singletphase is the ground state for applied magnetic fields be-

low the anisotropic critical values µ0H‖c = 5.3 T and

µ0H⊥c = 4.3 T, whereupon a triplet state is driven lower

in energy. ESR transitions between the singlet Haldanephase and triplet states are forbidden by momentum con-servation, and are not observed in our data. The centralfeature which dominates at low frequencies [indicated bygreen arrows and circles in Figures 7(b) and (c), respec-tively] is ascribed to fractional S = 1/2 end-chain degreesof freedom, as observed in the magnetometry measure-ments, and is therefore considered separately to all otherobserved resonances. This resonance is still observed forfields above the critical values, as seen in the Haldanecompound NENB, for example [63], albeit with a sup-pressed amplitude as the Haldane state depopulates forH > Hc. The resonance positions are fit to an isotropicmodel for S = 1/2 moments [green line in Figure 7(c)],which yields the g-factor g = 2.191(4), consistent withthe high-temperature result for the bulk S = 1 moments.

The remaining resonances observed at T = 3 K are in-dicated by red and blue arrows and circles in Figures 7(b)and (c), and comprise satellite features at fields on ei-ther side of the central resonance, plus the ‘half-field’resonances corresponding to transitions with ∆mS = ±2[64]. At low frequencies and fields (H <∼ Hc) there arefewer transitions between excited triplet states, resultingin small amplitudes for the satellite peaks relative to thecentral end-chain resonance. However, for fields abovethe critical value a low-energy triplet state becomes theground state, and these resonances rapidly gain intensity.

Based on their lineshapes [64] (peak versus peak-derivative), the non-central transitions are ascribed tofields parallel to the z-direction (color-coded blue) andthe x-direction (red). These positions may be fitted to

8

0

1 0 0

2 0 0

3 0 0

4 0 0

0 2 4 6 8 1 0 1 2 1 4

H | | x H | | z E n d - c h a i n

µ0 H ( T )

( c ) T = 3 K0

1 0 0

2 0 0

3 0 0

4 0 0

5 0 0

S i m u l a t i o n

( b ) T = 3 K

Frequ

ency (

GHz)

2 0 3 G H z

3 2 4 G H z4 0 3 G H z

1 0 2 G H z

1 0 02 0 03 0 04 0 05 0 0

g = 2 . 1 9∆m S = 1

( a ) T = 3 0 K

2 0 3 G H z2 9 5 G H z

4 0 6 G H z

1 0 2 G H z

Transm

ission

dI/dH

(Arb.

units)

FIG. 7. The frequency-dependence of ESR transmission spec-tra for a powder sample of NiI2(3,5-lut)4 at (a) T = 30 K and(b) T = 3 K. (c) The positions of transitions observed atT = 3 K. The dashed line in (a) and solid lines in (c) arefits described in the text and the simulated spectrum in (b)is obtained with ν = 324 GHz.

the fermion model, which is equivalent to the descrip-tion of isolated moments possessing axial SIA [41]. Thisprocedure captures the positions of the resonances, asshown by the blue and red lines in Figure 7(c), andyields the the anisotropic g-factors g‖ = gz = 2.13(1) andg⊥ = gx = 2.19(1), and an easy-plane effective SIA pa-rameter Deff = +1.11(6) K, and hence an easy-axis D inthe Hamiltonian. A simulated spectrum created by sum-ming the two contributions outlined above is comparedto the data measured for ν = 324 GHz in Figure 7(b),where it may be seen that the lineshapes and positions ofthe main resonances (and the relative intensities of thetwo satellite peaks) are well described by the simulation.

DISCUSSION

The INS and single-crystal magnetization studies pro-vide direct measurements of the SIA-split Haldane gapsand anisotropic critical fields, respectively, for NiI2(3,5-lut)4. These quantities can be related via the model-dependent expressions given for the fermion and bosontheories in Equations 4 and 5, respectively. In order totest the validity of these theories for this system, theanisotropic g-factors that each model predicts, given the

experimentally observed values of ∆‖,⊥ and H‖,⊥c , are

shown in Table I. The two models yield an identical valueof g‖ = 2.15(5) which is physically reasonable for Ni(II)ions, and in close agreement with the value determinedusing the low-temperature ESR measurements. Thefermion value of g⊥ = 2.21(6), and the resultant pow-der average gpowder = (g‖ + 2g⊥)/3 = 2.19(4) are also inexcellent agreement with the high- and low-temperatureESR and powder susceptibility results. However, the bo-son value of g⊥ = 1.85(6) is smaller than is plausible forthis ion, and the powder average value gpowder = 1.95(4)is not in good agreement with the aforementioned tech-niques.

The fermion model is therefore in excellent agreementwith the experimental results for the Haldane phase ofNiI2(3,5-lut)4, demonstrating that its applicability is notlimited to easy-plane Haldane compounds, as been re-cently suggested [46, 47]. The resultant energy levelsare shown in Figure 8, where the values of g‖,⊥ fromTable I reproduce the experimentally observed energy

gaps ∆‖,⊥ and critical fields H‖,⊥c . The boson model has

previously been successful in describing compounds withcritical inter-chain interactions [36, 46, 47], and its fail-ure to consistently describe the experimental findings forNiI2(3,5-lut)4 may constitute further evidence for small|J⊥| in this compound.

INS provides a direct, zero-field measurement of theSIA-split triplet energy levels, and therefore unequivo-cally yields D = −1.2(1) K (and Deff = +2.4(2) K),in excellent agreement with the estimate based on theanisotropic saturation behavior probed using pulsed-fieldmagnetometry. Interestingly, the fermion model appearsto describe powder ESR data for transitions betweentriplet states under the application of magnetic field, butwith Deff = +1.11(6) K, which is close to half of the INSvalue. This discrepancy could be due to the fact that

Theory g‖ = gz g⊥ = gx gpowder

Fermion 2.15(5) 2.21(6) 2.19(4)

Boson 2.15(5) 1.85(6) 1.95(4)

TABLE I. The anisotropic and powder g-factors predicted bythe fermion (Equations 4) and boson (Equations 5) models,using the values of ∆i and Hi

c directly observed using INSand magnetization, respectively.

9

0 1 2 3 4 5 60

2

4

6

8

1 0

1 2

1 4

1 6

1 8

| D e f f |

∆ | | = 0 . 4 6 m e V = 5 . 3 K

∆⊥ = 0 . 6 6 m e V = 7 . 7 K

� 0 H ⊥c = 4 . 3 T

Energ

y (K)

� 0 H ( T )

F i e l d | | z F i e l d ⊥ z

� 0 H | |c = 5 . 3 T

g z = 2 . 1 5g x = 2 . 2 1

FIG. 8. The anisotropic field dependence of energy levelswithin the Haldane phase, according to the fermion model.The anisotropic critical fields and zero-field energy gap valueshave been set to the values determined by magnetization (Fig-ure 3) and INS (Figure 4), respectively. The g-factors havebeen fixed to the values for the fermion model from Table I,in order to satisfy the relations in Equation 4. The effectiveSIA term Deff = +2.4(2) K defined by Equation 3 is labelled.

the energy levels of the Haldane chain are not well un-derstood for fields above the critical field Hc [65]. Thefermion and boson models have been shown to fail in thedescription of energy levels above Hc, albeit in systemswhich undergo 3D magnetic ordering above the criticalfield. In the related Haldane compounds SrNi2V2O8 andNDMAP, INS [44, 47, 66] and ESR [65, 67] have demon-strated that the triplet energy levels change gradient asmagnetic field is swept through Hc, in contrast to theprediction of the fermion model.

The intra-chain exchange coupling J = 17.5(2) K isdeduced from the saturation behavior observed in thepulsed-field magnetization measurements, while the INSmeasurement yields ∆0 = 7.0(1) K. The resulting ra-tio ∆0/J = 0.40(1) is very close to the value of 0.41expected for an ideal isolated chain, implying that theinterchain exchange should not play a significant role inour material, at least at the temperatures explored here.In contrast, the SIA-split Haldane gaps in PbNi2V2O8

and SrNi2V2O8 are driven lower in energy by substantialinter-chain dispersion due to J⊥ [19].

Hence NiI2(3,5-lut)4 is remarkable as a highly isolatedHaldane chain with a single crystallographic Ni(II) site,experimentally accessible energy scales (J = 17.5(2) K,

∆‖,⊥ = 5.3, 7.7 K, H‖,⊥c = 5.3, 4.3 T) and an

anisotropy ratio D/J = −0.07(1). Systems with com-parable degrees of anisotropy include Y2BaNiO5 [69],PbNi2V2O8 and SrNi2V2O8 [19], while AgVP2S6 is virtu-ally isotropic [70]. The parameters of all five compoundsare compared in Table II. The large energy scales ex-

hibited by both Y2BaNiO5 and AgVP2S6 preclude field-tuning through the quantum critical region associatedwith the closure of their Haldane gaps using DC mag-netic fields. While H⊥c for PbNi2V2O8 and SrNi2V2O8

are within the range of superconducting magnets, bothsystems have near-critical values of |J⊥|/J , lying on thevery edge of the Haldane phase as evidenced by the de-parture of ∆0/J from the ideal value of 0.41 and the ob-servation of long-range magnetic order at liquid heliumtemperatures in applied fields. NiI2(3,5-lut)4 is thereforea uniquely ideal Haldane system that provides easy ac-cess to the entire Haldane phase for all field directions.In particular, the entire quantum critical region can beprobed with techniques that may have hitherto not beenviable for other compounds. This includes instrumentsat central beamtime facilities, as our preliminary µ+SRinvestigations demonstrate.

Furthermore, the molecular building blocks that makeup NiI2(3,5-lut)4 permit a higher degree of controlledstructural modification than is possible for the inorganicsystems. Of particular interest is the controlled introduc-tion of quenched bond disorder. Randomly distributednon-magnetic defects on the transition metal sites andsubstitutions which locally modify superexchange path-ways have proven to be a pathway to exciting new physics[71], yet still represent an unsolved problem. Controlledchemical substitution in related molecule-based materialshas been demonstrated to give rise to entirely new phases,such as the elusive Bose-glass phase in the large-D systemDTN [72] and the spin-ladder IPA-CuCl3 [73]. Bond dis-order in the Haldane phase has been considered throughnumerical simulations, which predict localized impuritybound states [74, 75]. However, to date, the paucity ofphysically realized compounds where controlled introduc-tion of bond disorder can be achieved has led to a lack ofexperimental verification of these theoretical predictions.

CONCLUSIONS

In summary, we have shown that NiI2(3,5-lut)4 is arealization of a S = 1 Heisenberg AFM 1D chain, withintra-chain exchange coupling J = 17.5(2) K manifestedby unique Ni–I· · · I–Ni magnetic couplings. Muon spin re-laxation measurements demonstrate the absence of zero-field magnetic LRO down to T = 20 mK, indicatinga high degree of one dimensionality (i.e. a small ratio|J⊥|/J). The results of INS measurements prove thiscompound resides within the Haldane region of the the-oretical phase diagram from Ref. [21], and quantify theHaldane energy gaps for the SIA-split triplet excitations∆‖ = 0.46(1) meV and ∆⊥ = 0.66(1) meV. Theseenergy gaps correspond to an easy-axis SIA parame-ter D = −1.2(1) K, so that D/J = −0.07(1), makingthis one of the most isotropic Haldane systems reportedto date. Single-crystal magnetization measurements re-

10

Compound J (K) D (K) D/J ∆‖ (K) ∆⊥ (K) ∆0/J H‖c (T) H⊥c (T) References

NiI2(3,5-lut)4 17.5(2) -1.2(1) -0.07(1) 5.3(1) 7.7(1) 0.40(1) 5.3(1) 4.3(1) This study

PbNi2V2O8 104 -5.2 -0.05 21 28 0.25 19.0 14 [18, 19, 45]

SrNi2V2O8 101 -3.7 -0.04 18.2 29.9 0.26 20.8 12.0 [46, 53]

Y2BaNiO5 280 -9.4 -0.03 87 100, 110 0.35 ∼ 70 [68, 69]

AgVP2S6 780 4.5 0.006 300 0.38 ∼ 200 [57]

TABLE II. A comparison of parameter values for selected Haldane compounds.

veal the anisotropic critical fields µ0H‖c = 5.3(1) T and

µ0H⊥c = 4.3(1) T, which are readily accessible in com-

mercially available superconducting magnets, and pow-der pulsed-field measurements demonstrate the system is

saturated at µ0H‖s = 46.0(4) T and µ0H

⊥s = 50.7(8) T.

These energy gaps and critical fields are explicable usingthe fermion model, in contrast to other reported easy-axis Haldane systems where J⊥ is critically large and theboson model is more consistent with experiment [45–47].

NiI2(3,5-lut)4 is also ideally suited to the introductionof quenched disorder via substitution of the Ni(II) ionand the halide ions which mediate the superexchange in-teraction. This model system therefore provides a uniqueopportunity to explore this relatively uncharted territoryand gain fresh insight into the physics of the Haldanephase using chemistry to control and tune the exchangeinteractions and SIA.

EXPERIMENTAL SECTION

Synthesis. All chemical reagents were purchased fromcommercial sources and used as received. A typicalsynthesis involves slow mixing of aqueous solutions ofNi(NO3)2 ·6H2O (0.4002 g, 1.38 mmol) with excess NH4I(0.9971 g, 6.88 mmol). In 5-mL of acetonitrile, 2-mL of3,5-lutidine was dissolved and slowly added by pipette tothe aqueous Ni solution. The resulting solution is mostlyblue with a slightly green hue. Slow solvent evaporationover the period of a few days resulted in yellow-greenmicrocrystals. Larger single crystals can be grown us-ing a vapor diffusion method wherein 3,5-lutidine is al-lowed to slowly diffuse into an aqueous solution contain-ing Ni(NO3)2 ·6H2O and NH4I. In either case, crystals ofthe product are obtained in better than 90% yield. Thecrystals were found to be largely air-stable although theywere generally stored in a refrigerator.

Structural Determination. Experiments were con-ducted on the ChemMatCARS 15-ID-B beamline of theAdvanced Photon Source at Argonne National Labora-tory. A microcrystal of NiI2(3,5-lut)4 measuring 10 ×10 × 2 µm3 was selected from a bulk sample using acryo-loop and mounted on a Huber 3-circle X-ray diffrac-tometer equipped with an APEX II CCD area detector.The sample was cooled to 100(2) K using a LN2 cryojet.Synchrotron radiation with a beam energy of 32.2 keV

(λ = 0.41328 A) was selected and the beam size at thesample position was 0.1 × 0.1 mm2. The distance be-tween sample and detector was set at 60 mm. A totalof 51, 036 reflections were collected of which 1, 272 wereunique [I > 2σ(I)].

Data collection and integration were performed us-ing the APEX II software suite. Data reduction em-ployed SAINT.[76] Resulting intensities were correctedfor absorption by Gaussian integration (SADABS).[77]The structural solution (XT)[78] and refinement (XL)[79]were carried out with SHELX software using the XPREPutility for the space-group determination. Consideringsystematic absences, the crystal structure was solved inthe tetragonal space group P4/nnc (#126).[80] LutidineH-atoms were placed in idealized positions and allowed toride on the carbon atom to which they are attached. Allnon-hydrogen atoms were refined with anisotropic ther-mal displacement parameters.

Magnetometry. Quasi-static magnetometry mea-surements were performed using a Quantum DesignSQUID magnetometer in fields up to 7 T, where aniQuantum 3He insert enabled temperatures T >∼ 0.5 K tobe reached. The powder and crystal samples were loadedinto a gelatin capsule, which was then attached to a non-magnetic sample holder at the end of a rigid rod. Thetemperature-dependent molar magnetic susceptibility isobtained in the linear limit via χm = M/(nH) where thepowder contains n moles of the compound.

Isothermal pulsed-field magnetisation measurementswere performed at the National High Magnetic Field Lab-oratory in Los Alamos, USA. Fields of up to 65 T withtypical rise times ≈ 10 ms were used, and a 3He cryostatprovides temperature control. A powdered sample wasmounted in a 1.3 mm diameter PCTFE ampoule whichwas attached to a probe containing a 1500-turn, 1.5 mmbore, 1.5 mm long compensated-coil susceptometer, con-structed from 50 gauge high-purity copper wire. Whenthe sample is centered within the coil and the field ispulsed, the voltage induced in the coil is proportional tothe rate of change of magnetization with time (dM/dt).The total magnetization is obtained by numerical inte-gration of the signal with respect to time. A subtractionof the integrated signal recorded using an empty coil un-der the same conditions is used to calculate the magne-tization of the sample. The magnetic field is measuredvia the signal induced within a coaxial 10-turn coil and

11

calibrated using de Haas-van Alphen oscillations withinthe copper coils of the susceptometer.

Inelastic Neutron Scattering. Zero-field inelasticneutron scattering measurements were performed on apowder sample of NiI2(3,5-lut)4 (mass 3.5 g) using thehigh-flux chopper setting on the direct geometry time-of-flight spectrometer LET, at the ISIS facility, UK. A pri-mary incident energy Ei = 3.7 meV was used, providinga number of incident energies with various flux and reso-lution combinations in the energy window of interest forthis compound, of which we focus on the Ei = 2.2 meVdata. The sample was mounted in a 4He cryostat anddatasets were collected at temperatures of T = 1.7, 3, 7and 12 K, where the latter was treated as a non-magneticbackground.

Muon Spin Relaxation. Zero-field µ+SR measure-ments on NiI2(3,5-lut)4 were made on the LTF and GPSinstruments at the Swiss Muon Source (SµS), Paul Scher-rer Institut, Switzerland. Transverse-field µ+SR mea-surements were carried out on a polycrystalline sampleusing the HAL-9500 spectrometer at SµS. For the zero-field measurements, a powder sample was packed in Agfoil envelopes (foil thickness 12.5 µm), attached to a sil-ver plate and mounted on the cold finger of a dilutionrefrigerator. For transverse-field measurements, a pow-der sample was packed in foil and glued to a silver holderthat was mounted on the cold finger of a dilution refrig-erator. Data analysis was carried out using the WiMDAanalysis program.[81]

Electron Spin Resonance. High-field, high-frequency ESR spectra of powdered samples wererecorded on a home-built spectrometer at the EMR fa-cility, National High Magnetic Field Laboratory, Talla-hassee, Florida, USA. Microwave frequencies in the range52 ≤ ν ≤ 422 GHz and temperatures T = 3 and 30 Kwere used in the measurement. The instrument is atransmission-type device and uses no resonant cavity. Apowdered sample was loaded into thin teflon vessel andlowered into a 4He cryostat. The microwaves were gen-erated by a phase-locked Virginia Diodes source, withgenerating frequency (13± 1) GHz, and equipped with acascade of frequency multipliers to generate higher har-monic frequencies. The resultant signal was detected us-ing a cold bolometer, and a superconducting magnet gen-erated fields up to 15 T.

ACKNOWLEDGEMENTS

We thank J. Liu for useful discussions. This projecthas received funding from the European Research Coun-cil (ERC) under the European Union’s Horizon 2020research and innovation programme (Grant AgreementNo. 681260). WJAB thanks the EPSRC for additionalsupport. A portion of this work was performed at theNational High Magnetic Field Laboratory, which is sup-

ported by National Science Foundation (NSF) Coopera-tive Agreement No. DMR-1157490, the State of Floridaand the U.S. Department of Energy (DOE) through theBasic Energy Science Field Work Proposal “Science in100 T”. Work at EWU was supported by NSF grant No.DMR-1703003. NSF’s ChemMatCARS Sector 15 is prin-cipally supported by the Divisions of Chemistry (CHE)and Materials Research (DMR), NSF, under grant num-ber NSF/CHE-1834750. Use of the Advanced PhotonSource, an Office of Science User Facility operated for theU.S. DOE Office of Science by Argonne National Labora-tory, was supported by the U.S. DOE under Contract No.DE-AC02-06CH11357. JAS acknowledges support fromthe Independent Research/Development (IRD) programwhile serving at the National Science Foundation. FX ac-knowledges funding from the European Union’s Horizon2020 research and innovation program under the MarieSkodowska-Curie grant agreement No 701647. A portionof this work was performed at the Swiss Muon Source,Paul Scherrer Institute, Switzerland and we acknowl-edge support from EPSRC under grants EP/N023803/1,EP/N024028/1 and EP/N032128/1. BMH thanks STFCfor support via a studentship. Data presented in this pa-per resulting from the UK effort will be made availableat (URL here).

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Supporting Information accompanyingA Near-Ideal Molecule-Based Haldane Spin-Chain

In a muon spin relaxation (µ+SR) experiment, a beamof spin-polarized positive muons is incident upon a ma-terial [1, 2]. Muons stop within the sample at particu-lar sites, where their spins precess in the local magneticfield, before decaying after, on average, 2.2 µs. The ob-served property of the experiment is the average spin po-larization of muon ensemble. In a zero-field (ZF) µ+SRexperiment the muon spins precess entirely in the inter-nal magnetic field of the sample. If a material showsquasistatic LRO, we expect spontaneous coherent oscil-lations in Pz(t), the longitudinal spin polarization alongthe initial muon spin direction. In the TF µ+SR exper-imental geometry [1] the externally applied field µ0H0

is directed perpendicular to the initial muon spin direc-tion. Muons precess about the total magnetic field B atthe muon site, which is a vector sum of µ0H0 and fieldsdue to the Ni(II) moments. The observed property ofthe experiment is the time evolution of the muon spinpolarization Px(t). The Larmor relation between muonprecession frequency and local magnetic field strengthω = γµB (where γµ/(2π) = 135.5 MHz T−1 is the muongyromagnetic ratio) allows the determination of the dis-tribution p(B) of local magnetic fields across the samplevolume by means of a Fourier transform of Px(t).

Zero-field µ+SR measurements. Example ZFspectra spanning the measured temperature range for apowder sample of NiI2(3,5-lut)4 are shown in Figure 1.The asymmetry data were fitted in the time domain withtwo exponentially decaying components and a constant

25

30

35

40

45

Asy

mm

etry

(arb

.%

)

0.0 0.5 1.0 1.5 2.0

Time ( s)

20 K

0.75 K

0.02 K

FIG. 1. Asymmetry from ZF µ+SR measurements onNiI2(3,5-lut)4 at three different temperatures. (The spectraare offset for clarity.) The solid lines are fits to Equation 1.

0.0

0.2

0.4

0.6

0.8

1.0

1(M

Hz)

0.02 0.05 0.1 0.2 0.5 1 2 5 10 20

Temperature (K)

FIG. 2. Values of λ1 obtained by fitting ZF µ+SR data toEquation 1. Red dashed line shows λ1 ∝ exp (−∆/kBT ) with∆ = 7 K.

background Abkg,

A (t) = A1e−λ1t +A2e

−λ2t +Abkg, (1)

where λi are relaxation rates. The two separate ex-ponentials capture both fast-relaxing and slow-relaxingcomponents of the data. The fast relaxing componenttypically arises from muons in bound states [3] whereasthe slow relaxing component likely arises from fluctua-tions of magnetic fields detected by diamagnetic muonstates, and is therefore sensitive to the magnetism of thesystem. The relaxation due to muon bound states is notexpected to change with temperature, and as the value ofλ2 ' 38 MHz remained roughly constant over the wholetemperature range, it was fixed at this value for the fits.Similarly, the ratio of muons in bound states to those indiamagnetic states should remain constant with temper-ature (as the muon site is determined by the structureof the material). Approximately constant values of A1

and A2 were found over the whole temperature range, asexpected, and hence A1 and A2 were also fixed.

The value of λ1 as a function of temperature is shownin Figure 2. At a phase transition, one would expecta dramatic increase in the value of λ1, indicating thatthere is a large variation in the values of internal mag-netic fields, or a slowing of dynamics. As the value of λ1stays approximately constant throughout the whole tem-perature range this suggests there is no phase transition,indicating that there is no transition to long-range ordereven down to the lowest measured temperatures.

Previous work [4] has shown that the relaxation rateλ1 in a Haldane chain should vary as ∝ exp (−γ∆/kBT ),where ∆ is the gap magnitude and the value of the con-

arX

iv:1

909.

0790

0v1

[co

nd-m

at.s

tr-e

l] 1

7 Se

p 20

19

2

stant γ depends on the temperature regime and details ofthe model. Figure 2 shows this scaling law for ∆ = 7 Kand γ = 1 (expected for low temperatures), suggestingthat the slight increase in λ1 above the gap temperatureis consistent with the predicted behaviour. Althoughthere is insufficient data density to test which value ofγ best describes this system, the location of the upturnis supportive of the value ∆ ≈ 7K found through othertechniques.

There are several other possible causes of the occur-rence of the relaxation parametrized by λ1, all consistentwith the realisation of a Haldane phase in this material.One possibility is that chain ends give rise to magneticmoments, which introduce some magnetic field dynam-ics that cause the relaxation. Another possibility is thatdefects in the sample affect the local field configuration,again leading to a small, fluctuating magnetic field. It isalso possible that the muon itself distorts the local envi-ronment at the implantation site inducing a small local

moment. This has been observed previously in the mag-netically disordered state of molecular spin-ladder sys-tems [5], where it was shown that it did not prevent themuon being a faithful probe of the intrinsic physics of thesystem.

[1] S. J. Blundell, Contemporary Physics 40, 175 (1999).[2] A. Yaouanc and P. Dalmas de Reotier, Muon Spin Rota-

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