Towards mechanomagnetics in elastic crystals: insights from [Cu(acac) 2 ]
TTowards mechanomagnetics in elastic crystals: insights from [Cu(acac) ] E. P. Kenny, A. C. Jacko, and B. J. Powell
School of Mathematics and Physics, The University of Queensland, Brisbane, Queensland, Australia ∗ We predict that the magnetic properties of [Cu(acac) ], an elastically flexible crystal, change dra-matically when the crystal is bent. We find that unbent [Cu(acac) ] is an almost perfect Tomonaga-Luttinger liquid. Broken-symmetry density functional calculations reveal that the magnetic ex-change interactions along the chains is an order of magnitude larger than the interchain exchange.The geometrically frustrated interchain interactions cannot magnetically order the material at anyexperimentally accessible temperature. The ordering temperature ( T N ), calculated from the chainrandom phase approximation, increases by approximately 24 orders of magnitude when the materialis bent. We demonstrate that geometric frustration both suppresses T N and enhances the sensitivityof T N to bending. In [Cu(acac) ], T N is extremely sensitive to bending, but remains too low forpractical applications, even when bent. Partially frustrated materials could achieve the balance ofhigh T N and good sensitivity to bending required for practical applications of mechanomagneticelastic crystals. INTRODUCTION
Crystal adaptronics is a new and exciting field, bolsteredby the recent discovery of elastically flexible molecularcrystals.
These crystals can be bent without irreversiblychanging their structure. The mechanism by which themolecules can elastically slip past each other is beginningto be understood.
However, there are limited exam-ples of the modification of functional properties; the mostsuccessful so far being mechanochromism.
In this paper, we explore the possible changes inmagnetic properties induced by bending [Cu(acac) ](acac=acetylacetonate), a recently discovered elasticallyflexible crystal. We discuss how the geometry of thecrystal leads to these changes in the hope of motivating asearch for elastic crystals with similar geometry, but largerexchange interactions.[Cu(acac) ] is an extremely well known material. It is acommercially available reactant used in numerous organicand organometallic syntheses and is often made in under-graduate chemistry laboratories. Worthy et al. publishedatomically resolved structural information across bentsamples, providing the opportunity to use first-principlescalculations to model how its magnetic properties changeas the material is bent.We find that, apart from being elastic, [Cu(acac) ] hasexotic quantum magnetic properties – it is an almost per-fect quasi-one-dimensional magnet. The frustrated geom-etry of the crystal lattice enhances this low dimensionalityand also leads to extreme sensitivity of the magnetic prop- erties to bending. [Cu(acac) ]’s partnership of elasticityand geometrical frustration lead to it being an excellentprototype for applications for elastic crystals. We predictthat the change in geometry of [Cu(acac) ], brought on bybending, will lead to its magnetic ordering temperaturechanging by approximately 24 orders of magnitude. Thisdemonstrates the possibility of using elastic flexible crys-tals to passively sense small deformations or flexures withextremely high precision.Passive flex sensors often operate with a change in elec-trical resistivity. They are useful for measuring physicalactivity or joint movement in the human body, for facil-itating human-computer interactions, for monitoring ma-chines, and for measurement devices (for example measur-ing the curvature of a small surface). A material withdramatic magnetic changes caused by bending, such as[Cu(acac) ], could also be used for these purposes downto the micrometer scales. The magnetic ordering temper-ature, which can be detected via the concomitant diver-gence in the magnetic susceptibility, can change by manyorders of magnitude; such devices could have sensitivitiesfar exceeding those of resistive devices.The behavior of flexible quantum magnets, a new fieldopened by the discovery of elastic crystals, allows one toexamine many new questions of fundamental importance.Low dimensional magnetic crystals display fascinatingquantum phenomena. Particularly, one-dimensional mate-rials exhibit fractionalized excitations and strong quantumfluctuations that prohibit long range magnetic order. Spin-1/2 one-dimensional Heisenberg chains are described a r X i v : . [ c ond - m a t . s t r- e l ] A ug TABLE I: The ordering temperatures in units of J k of variousquasi-one-dimensional molecular crystals found in theliterature along with the value predicted for [Cu(acac) ] inthis paper. A lower value of T N /J k indicates a material closerto the 1D limit.Material T N /J k Ref.Cu (CO ) (OH) CuCl (H O) ] (PF ) ) ] 0.01 23[Cu(acac) ] 10 − This work by Tomonaga-Luttinger liquid (TLL) theory.
An im-portant prediction of TLL theory is that there will bea continuum of low-energy excitations, which are indeedobserved in neutron scattering experiments. Quasi-one-dimensional crystals contain weak interchain interactions,which become significant at low temperatures and lead toN´eel ordering below a certain temperature, T N . These ma-terials can be understood as weakly coupled chains. How-ever, at low enough temperatures, interchain interactionseventually cause long-range magnetic order.Copper II molecular crystals are well known for theirexotic magnetic properties. One of the best examplesof a quasi-one-dimensional molecular crystal is copperpyrazine dinitrate, [Cu(pz) (NO ) ] (pz=pyrazine), whichorders magnetically at 0.107 K and was recently shownto exhibit 1D quantum criticality. Its magnetic low di-mensionality has been confirmed with density functionaltheory calculations, which give an interchain coupling of J ⊥ = 0 . J k , where J k is the intrachain coupling. Theextent to which a material is 1D can be quantified with T N /J k . Table I shows some of the lowest values found todate.Below, we demonstrate that unbent [Cu(acac) ] is an al-most perfect TLL that does not order magnetically at anyexperimentally accessible temperature. We establish thisthrough a combination of first principles electronic struc-ture calculations and quantum many-body theory, reveal-ing that the presence of geometrical frustration in the lat-tice (see Fig. 1) causes two major effects: (i) [Cu(acac) ]’sextreme magnetic one-dimensionality and (ii) the signifi-cant change the N´eel temperature, T N , when the materialis bent. FIG. 1: Two examples of coupled chain geometries; (a)perpendicular interchain couplings and (b) frustratedtriangular couplings. Most of the materials in Table I havesome combination of both of these types of interactions;however, [Cu(acac) ], only has frustrated interactions (b).This is why T N /J k is much lower in [Cu(acac) ]. In thesequasi-1D materials, J k (within the spin chains) is stronglyantiferromagnetic, favoring short-range antiferromagneticcorrelation, as shown. The triangular geometry in (b)frustrates the interchain couplings, J ⊥ , as indicated by ‘?’(regardless of whether the interchain couplings areferromagnetic or antiferromagnetic). Whereas the squaregeometry in (a) is unfrustrated. We parametrize a Heisenberg Hamiltonian via broken-symmetry density functional theory (BS-DFT), whichreveals three significant exchange couplings between neigh-boring molecules, J k , J ⊥ , and J ⊥ (shown in Fig. 2). Themagnitude of the exchange coupling along the crystallo-graphic b -axis ( J k ) is much larger than the couplings in theother directions, indicating that [Cu(acac) ] can be mod-elled as weakly coupled Heisenberg spin-1/2 chains. Theinterchain couplings, J ⊥ and J ⊥ , are both geometricallyfrustrated (see Figs. 1 and 2), maintaining [Cu(acac) ] inthe 1D limit.We use the chain random phase approximation(CRPA) to predict the N´eel temperature, magnetic sus-ceptibility and dynamical structure factor of the unbentcrystal. The measured susceptibility is in good agreementwith our calculations. When the crystal is bent, the ratioof intra to interchain couplings changes significantly. Thisleads to a change in N´eel temperature of 24 orders of mag-nitude, demonstrating the dramatic potential of mechano-magnetics. COMPUTATIONAL DETAILS ANDTHEORETICAL METHODS
We use the unbent and bent [Cu(acac) ] crystal struc-tures from Worthy et al. Three nearest neighbor exchangepathways are shown in Fig. 2. In terms of the crystallo-graphic axes we label J k to be along b . The four near-est neighbour interactions in the ± ( b/ ± ( a + c )) direc-tions are equal (by symmetry) and we label them J ⊥ .Similarly, we label the four nearest neighbour interactionsalong ± ( b/ ± ( a − c )) as J ⊥ .When the crystal is bent, the lattice parameters changeapproximately linearly as a function of position across thebend. On the inside of the bend, the b -axis is compressedwhile the a and c axes are stretched. Conversely, on theoutside of the bend, the b -axis stretches while a and c arecompressed. The β angle increases approximately linearlyfrom the outside to the inside of the bend. However, theindividually measured atomic coordinates are not as pre-cise as those from bulk crystals due to the small effectivesample size. We therefore created a linearized set of latticeparameters using crystallographic data for two bends withdifferent radii of curvature, r c = 1 . r c = 3 . H Heisenberg = X ij J ij S i · S j , (1)where S i is the spin operator on the i th molecule and J ij are the exchange coupling constants. The sign of J indicates an antiferromagnetic ( J >
0) or ferromagnetic(
J <
0) interaction.We calculate the exchange couplings, J ij , within[Cu(acac) ] using broken-symmetry density functional the-ory (BS-DFT), along with the Yamaguchi spin decon- FIG. 2: The nearest neighbor exchange pathways in[Cu(acac) ]. The crystallographic axes ( a , b , and c ) areshown. The lattice is geometrically frustrated. We find that J k is strongly antiferromagnetic, favoring short-rangeantiferromagnetic correlation. Regardless of the signs of J ⊥ and J ⊥ (i.e. whether they are antiferromagnetic orferromagnetic), the triangular geometry frustrates thesecouplings. tamination procedure. In this approach, J ij = 2 E BS ij − E T ij h S i BS ij − h S i T ij , (2)where E T ij is the triplet energy of the isoloated dimercontaining molecules i and j , and E BS ij is the energy ofthe broken-symmetry state on that same dimer. h S i BS ij and h S i T ij are the corresponding expectation values ofthe spin operator, S . Calculations were perfomed inGaussian09 with the uB3LYP functional and usingthe LANL2DZ (for Cu) and 6-31+G* basis setswith an SCF convergence criterion of 10 − a.u. Bench-marking of J ij using different basis sets and functionals isdiscussed in the Supplementary Information.The dynamical magnetic susceptibility for a singleHeisenberg chain can be calculated from a combina-tion of the Bethe ansatz and quantum field theorytechniques. Within the chain random phase approxi-mation (CRPA), the full three-dimensional dynamical sus-ceptibility is a function of the interchain coupling, J ⊥ (seeEq. S.3). The CRPA susceptibility is valid abovethe N´eel temperature, T N . Generically, one expects anRPA treatment to overestimate T N . However, the geo-metrical frustration in [Cu(acac) ] enhances the range ofvalidity of this approximation; the CRPA has been com-pared with numerical methods and found to be accuratefor | J ⊥ | < . J k on a geometrically unfrustrated lattice and | J ⊥ | < . J k for a frustrated lattice. One can de-termine T N by considering the condition for a zero fre-quency pole in the CRPA expression for the dynamicalsusceptibility. Details of this calculation are given in theSupplementary Information.We use the CRPA to predict a number of experimentallymeasurable properties of [Cu(acac) ]. We fit the CRPA,using the Bonner-Fisher chain susceptibility to the ex-perimental bulk susceptibility above 2 K. We then pre-dict the low-temperature CRPA susceptibility ( T N < T < . et al . The bulk magnetic sus-ceptibility will diverge, undergoing a second order phasetransition, at T N . The dynamical structure factor (mea-sured in inelastic neutron-scattering experiments) can alsobe calculated with the CRPA susceptibility (see S.19). De-tails of the experimental predictions are also given in theSupplementary Information. RESULTS AND DISCUSSION
Unbent Crystal.
The three distinct BS-DFT near-est neighbor exchange interactions in the unbent crystal,along with their crystallographic directions, are reportedin Table II. All longer range interactions that we calculatedare smaller than the accuracy limit of our DFT results.
TABLE II: Heisenberg exchange ( J ij ) parameters for theunbent structure of [Cu(acac) ] determined with BS-DFT. J k and J ⊥ are antiferromagnetic and J ⊥ is ferromagnetic. Thedistances between Cu atoms for each dimer are also reported.Axes are shown in Fig. 2.Direction Cu ↔ Cu (˚A) J ij /k B (K) J k ± b J ⊥ ± b / ± ( a + c ) 7.818 0.04 J ⊥ ± b / ± ( a − c ) 8.133 -0.10 The exchange coupling ratios, J ⊥ /J k = 0 . , J ⊥ /J k = − . , indicate a low dimensionality in the magnetic de-grees of freedom in [Cu(acac) ]. In the limit J ⊥ = J ⊥ = 0, one has independent Heisenberg chainswhich are Tomonaga-Luttinger liquids (TLLs) at lowtemperatures. However, when there are interactions be-tween chains (i.e. J ⊥ , J ⊥ = 0), the TLL will undergo aphase transition into a N´eel ordered state below a criticaltemperature, T N .Using the CRPA susceptibility (details given in the Sup-plementary Information), we find that the N´eel tempera-ture of [Cu(acac) ] is given by T N ≈ Λ e − . R J , (3)where R J = J k / ( | J ⊥ | + | J ⊥ | ) is the ratio of the intrachaincoupling to the interchain couplings (see Table II and Fig-ure 2) and Λ = 24 . J k /k B is a non-universal parameter. Evaluating Eq. 3 for the unbent [Cu(acac) ] crystal gives T N ≈ × − K. Thus, we predict that the unbent crys-tal of [Cu(acac) ] will be magnetically disordered downto the lowest experimentally reachable temperatures – ex-perimentally [Cu(acac) ] will appear as an almost per-fect TLL. To highlight the extreme one-dimensionality of[Cu(acac) ] compared to other materials, one can makethe striking comparison of T N /J k ≈ − to the othermaterials in Table I.Given the form of Eq. 3, it is clear that T N is very sensi-tive to R J ; because T N decays exponentially as a functionof R J , a small change in R J leads to a dramatic change in T N . This extreme sensitivity is caused by the geometry of[Cu(acac) ]; the interchain interactions are geometricallyfrustrated, as illustrated in Fig. 2. It is instructive tocompare the T N calculated above with that of an unfrus-trated analogue – a cubic lattice where J ⊥ and J ⊥ arethe same magnitude as in [Cu(acac) ], but their directions CRPA with exact χ chain CRPA with Bonner-Fisher fitExperimental Data from Moreno et al.
FIG. 3: We predict that the experimental properties of [Cu(acac) ] will closely mimic an isolated spin-1/2 Heisenberg chain,being an almost perfect TLL. (a) A fit of the CRPA with the 1D Bonner-Fisher susceptibility (Eq. S.17 with J k = 0 .
75 K and J ⊥ + J ⊥ = 0 .
14 K) to experimental bulk susceptibility data from Moreno et al. and a low-temperature prediction with theCRPA and the exact 1D calculation from Eggert et al. (Eq. S.18). (b) Calculated plot of the dynamical structure factor of[Cu(acac) ] with Eq. S.19. The 1D model of the bulk susceptibility is very successful and the dynamical structure factorprediction shows little deviation from an isolated 1D chain. are perpendicular to J k (see Fig. 1a). The same CRPAcalculation as above then results in T cubic N ≈ . J k k B R J s log (cid:18) Λ T N (cid:19) , (4)which yields T cubic N ≈ . J k /k B ≈ .
17 K using the pa-rameters in Table II. This is 32 orders of magnitude higherthan T N for the frustrated [Cu(acac) ] lattice. Moreover,in contrast to the exponential dependence of Eq. 3, T cubic N is proportional to 1 /R J – it is larger and less sensitive tosmall changes in the value of R J . This will be importantwhen we discuss the bent crystals.The large contrast between geometrically frustrated andunfrustrated interactions is also demonstrated in previouswork on the 2D anisotropic triangular lattice Heisenbergmodel, for Cs CuCl in particular. We predict that the experimental properties of[Cu(acac) ] will closely mimic an isolated spin-1/2 Heisen-berg chain, displaying properties of an almost perfectTLL. The CRPA prediction of the bulk magnetic suscep-tibility using the exact 1D theory is limited to the low-temperature regime studied by Eggert et al . Conversely,the bulk susceptibility of [Cu(acac) ] has only been mea-sured above 2 K, with no magnetic ordering detected. Therefore, to compare our prediction with experiment, wefirst fit the CRPA using the Bonner-Fisher susceptibilityof a single spin chain, which is successful in other materials at higher temperatures.
We set J k = 0 .
75 K (our BS-DFT result) and found that the best fit corresponded to J ⊥ + J ⊥ = 0 .
14 K, in reasonable agreement our BS-DFTresults for the interchain couplings. We used this value of J ⊥ + J ⊥ to parametrize our low-temperature prediction.More details of the fit are given in the Supplementary In-formation. Fig. 3(a) shows the Bonner-Fisher fit to theexperimental data from Moreno et al. and our predic-tion of the low temperature magnetic susceptibility. Theagreement is exceptional.The dynamical structure factor for [Cu(acac) ] has notbeen measured. Our dynamical structure factor predic-tion in Fig. 3(b) (measurable via neutron scattering ex-periments) was calculated with our BS-DFT exchange pa-rameters (Table II). It shows a slight asymmetry, which isabsent for a TLL in an isolated Heisenberg chain at lowtemperatures. There are no adjustable parameters in thisprediction. Bent Crystal.
Our BS-DFT results across the bent crys-tals of [Cu(acac) ] are shown in Figure 4, where we plot J k , J ⊥ , and J ⊥ as ratios of the unbent parameters acrosseach of the bent crystals. We calculate that the interchaincoupling changes by over 20% as a consequence of the crys-tal distortion.Figure 5 shows R J and T N across the bent crystals. Thechange in geometry brought on by bending the [Cu(acac) ]crystals causes a significant change in magnetic behaviour J i / J i , unb e n t r c = 1 . r c = 3 . J i / J i , unb e n t Distance from center ( µ m) r c = 1 . r c = 3 . J k J ⊥ J ⊥ J k J ⊥ J ⊥ FIG. 4: BS-DFT calculations of the magnetic interactions inbent [Cu(acac) ] using crystal data across bent samples, withdifferent radii of curvature, r c , from Worthy et al. The intraand interchain exchange couplings change as a function of thedistance across a bent sample of [Cu(acac) ]. The center isdefined as the position where the magnitude of thecrystallographic b -axis is most similar to that of the unbentstructure (although, note that the a and c axes are quitedifferent). Lines are a guide to the eye. at different points across the bend; a small change in R J causes a very large change in the ordering temperature.In the most bent crystal, this means a change in T N of24 orders of magnitude from one side of the bend to theother.When the lattice is strained by bending, this causes asimultaneous, but opposite, change in J k and the perpen-dicular couplings, J ⊥ and J ⊥ , relative to the center ofthe bent crystal – illustrated in Fig. 6. On the inside ofthe bend, the distance between copper atoms along thechain is smaller than in the center, due to the compres-sion of the lattice along the b -axis, leading to a relativeincrease in J k . Whereas, the distance between the chains increases because the lattice is expanded along the a and c axes relative to the center, decreasing J ⊥ and J ⊥ . Bothof these processes independently decrease T N . On the out-side of the bend, the opposite effect occurs; the b -axis iselongated causing J k to decrease and the a and c axes arecompressed causing the interchain couplings to increase, leading to an increase in T N .The transition at T N is an antiferromagnetic transition,which could be detected with the divergence of the mag-netic susceptibility. We predict that, in a bent crystal,the bulk magnetic susceptibility would be a superpositionof single chain transitions resulting from the different T N values at different points across the crystal.Geometric frustration plays a vital role in this dramaticchange in T N across the bend; the extreme sensitivity of T N to the changes in the crystal described above is dueto the exponential dependence of T N on R J (Eq. 3). Ifthe lattice was cubic, the N´eel temperature would havestronger proportionality to R J (Eq. 4), and one would notobserve such a dramatic change in T N (also, the unbentcrystal would have a much larger T N ). CONCLUSIONS
In conclusion, we predict that the magnetic orderingtemperature of elastically flexible [Cu(acac) ] changes dra-matically when the material is bent. The unbent crystalwill behave, experimentally, like an almost perfect TLL(i.e. uncoupled 1D spin chains). When the sample is bentperpendicular to the chain direction, the crystal geometrychanges in such a way to maximally affect the value of theN´eel temperature, T N . A stretched crystal with a changein exchange couplings of 20% has a theoretical orderingtemperature of 0.01 mK, which is 24 orders of magnitudehigher than the unbent crystal, with a N´eel temperatureof ∼ − K. This change in T N across a bend wouldbe experimentally evidenced by measuring the bulk sus-ceptibility. The interchain interactions only weakly renor-malize the properties of [Cu(acac) ] relative to a singleHeisenberg chain. This is due in part to the weaknessof the interchain couplings, J ⊥ and J ⊥ , but mostly tothe presence of geometric frustration in the lattice; geo-metric frustration leads to the exponential suppression ofthe N´eel temperature, stabilizing the Tomonaga-Luttingerspin-liquid phase. Our results provide a powerful proof-of-principle demonstration that magnetic interactions can becontrolled via bending flexible crystals. We have demon-strated the possibility of using elastic flexible magneticcrystals to passively sense small deformations, curvatures,or flexures with extremely high precision by detecting thedivergence of the magnetic susceptibility in the sample.[Cu(acac) ] has a N´eel temperature that is highly sen- R J = J (cid:31) / ( J ⊥ + J ⊥ ) (a)(b)10 − − − T N ( K ) Distance from center ( µ m)(a)(b) r c = 1 . r c = 3 . FIG. 5: There is a significant change in the square of the ratio of intra to interchain exchange coupling R J across both bends,(a), leading to a drastic change in N´eel temperature T N , (b) and (c). Note the logarithmic scale of the ordinate in panel (b).The center is defined as the position where the magnitude of the crystallographic b -axis is most similar to that of the unbentstructure (note, however, that the a and c axes are quite different). InsideCenterOutside
FIG. 6: When [Cu(acac) ] is bent, the b -axis is stretched onthe outside of the bend relative to the center part of thecrystal (decreasing J k ), and compressed on the inside(increasing J k ). Conversely, the interchain separationdecreases on the outside (increasing J ⊥ and J ⊥ ), andincreases on the inside (decreasing J ⊥ and J ⊥ ). This leadsto a dramatic increase of T N on the outside and a dramaticdecrease of T N on the inside of the bend compared to thecentre. This happens in both planes containing the chain ((a)and (b) in Fig. 2) sitive to bending, but its extreme geometric frustrationmeans that k B T N is many orders of magnitude smallerthan the magnetic exchange interactions. Therefore, sim-ply increasing the exchange couplings would not be ex-pected to lead to experimentally accessible N´eel tempera- tures. Rather, as highlighted in Table I, the extreme geo-metrical frustration of [Cu(acac) ] is actually responsiblefor the low T N . This suggests that an incompletely frus-trated material may open the door to mechanomagneticsat experimentally accessible temperatures. However, ourresults show that there is a trade-off. Geometrical frus-tration also enhances the sensitivity of T N to bending.Unfrustrated coupling leads to a higher, measurable T N but lowers its sensitivity. Therefore, partial geometricalfrustration, e.g., imperfectly triangular couplings perpen-dicular to the chain, could provide a balance with both ahigh T N and a strong sensitivity to bending. We hope thatthis insight will play a key role in the future search anddesign of elastically flexible mechanomagnetic crystals. ACKNOWLEDGEMENT
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School of Mathematics and Physics, The University of Queensland, Brisbane, Queensland, Australia ∗ SECTION I: ORDERING TEMPERATURE CALCULATION USING THE CHAIN RANDOM PHASEAPPROXIMATION (CRPA)
The dynamical susceptibility for a single Heisenberg chain, χ chain ( ω, k k , T ), calculated from a combination of the Betheansatz and field theory techniques, is χ chain ( ω, k k , t ) = Φ( t ) Γ (cid:16) − i ω − u ( k k − π )4 πt (cid:17) Γ (cid:16) − i ω − u ( k k − π )4 πt (cid:17) Γ (cid:16) − i ω + u ( k k − π )4 πt (cid:17) Γ (cid:16) − i ω + u ( k k − π )4 πt (cid:17) , (S.1)where t = k B T /J k , Γ( x ) is the gamma function, u = π J k b is the spin velocity, andΦ( t ) = − t q ln (cid:0) Λ t (cid:1) (2 π ) / . (S.2)Here, Λ is a nonuniversal scale calculated with exact methods by Barzykin to be Λ = 24 . J k /k B . The full three-dimensional dynamical susceptibility within the CRPA is χ ( ω, k , T ) = χ chain ( ω, k k , T )1 − J ⊥ ( k ) χ chain ( ω, k k , T ) , (S.3)where ˜ J ⊥ ( k ) is the Fourier transform of the inter-chain coupling and k = (cid:0) k k , k ⊥ , k ⊥ (cid:1) is the crystal momentum alongthe respective nearest neighbor bond directions (see Table II of the main text) in units of the crystallographic constants; a = 10 . b = 4 . c = 11 . We find, for the frustrated triangularinteractions in Fig. 2 of the main text,˜ J ⊥ ( k ) = J ⊥ (cid:2) cos( k ⊥ ) + cos( k ⊥ − k k ) (cid:3) + J ⊥ (cid:2) cos( k ⊥ ) + cos( k ⊥ − k k ) (cid:3) . (S.4)One can determine T N by considering two conditions. Firstly, a zero frequency pole in χ (0 , k , T ) and secondly, that˜ J ⊥ ( k ) χ chain (0 , k k , T ) is maximised with respect to k . That is,2 ˜ J ⊥ ( k ) χ chain (0 , k k , T N ) = 1 (S.5)and ∂∂ k (cid:16) ˜ J ⊥ ( k ) χ chain (0 , k k , T N ) (cid:17) = 0 . (S.6)These two conditions give both the ordering temperature, T N , and the resulting value of k , the magnetic orderingwavevector. For a single 1D chain, the maximum in χ chain (cid:0) , k k , T (cid:1) occurs at k = π .The presence of interchain couplings will shift the ordering wavenumber to an incommensurate value, with the resultingorder occurring at k k = π + k . In [Cu(acac) ], there are four values of k ⊥ or k ⊥ (modulo 2 π ) that satisfy Eq. S.6, k ⊥ i = ± | k k | , J ⊥ i > ± | k k | ∓ π, J ⊥ i < ], we find J ⊥ > J ⊥ <
0, so the possibilities are k ⊥ = ± | k k | ,k ⊥ = ± | k k | ∓ π. (S.8) a r X i v : . [ c ond - m a t . s t r- e l ] A ug This gives four possible combinations of k ⊥ and k ⊥ . Setting k k = π + k , we find that, using Eq. S.6, all the abovepossibilities lead to the condition0 = 2 πT N u | k | + π tanh (cid:18) | k | u k B T N (cid:19) − (cid:18)
14 + i | k | u πk B T N (cid:19) , (S.9)which can be solved numerically, yielding | k | u πk B T N = J k | k | k B T N ≈ . . (S.10)Using Eq. S.5, we found that the choices in Eq. S.8 yield only two possible solutions for T N . Out of these, we takethe highest value of T N , as this is where the instability will occur. This corresponds to k = (cid:0) k k , | k k | / , | k k | / − π (cid:1) or k = (cid:0) k k , −| k k | / , −| k k | / π (cid:1) , which gives4 ( J ⊥ − J ⊥ ) J k sin (cid:18) | k | (cid:19) χ chain (0 , k k , T N ) ≈ .
611 ( J ⊥ − J ⊥ ) J k s ln (cid:18) Λ T N (cid:19) = 1 (S.11)Where we have made a small angle approximation in the second line (this will be strongly vindicated post hoc ). In unitsof J k , Eq. S.5 then becomes T N ≈ Λ exp " − . J k ( J ⊥ + J ⊥ ) ≈ . × − J k ≈ . × − K (S.12)Finally, Eq. S.9 now yields | k | = 2 . × − /b , where b is the lattice spacing along the chain. Since the value of k is so small (vindicating our small angle approximation), the magnetic ordering wavevector along the chain, k k , isapproximately π .More generally, regardless of the signs of J ⊥ and J ⊥ , two frustrated interchain couplings will result in ordering at T N ≈ Λ exp " − . J k ( | J ⊥ | + | J ⊥ | ) . (S.13)When we remove geometric frustration from the lattice (see Fig. 1b of the main text), setting all interchain couplingsto be unfrustrated, Eq. S.4 becomes ˜ J cubic ⊥ ( k ) = J ⊥ cos( k ⊥ ) + J ⊥ cos( k ⊥ ) (S.14)The same CRPA calculation as above then results in T cubic N ≈ .
56 ( | J ⊥ | + | J ⊥ | ) k B s log (cid:18) Λ J k k B T N (cid:19) , (S.15) Experimental Predictions
We use the CRPA to predict a number of experimentally measurable properties of [Cu(acac) ]. We calculate the 3Dsusceptibility via Eq. S.3 with two different methods; first with a fit of the CRPA using the Bonner-Fisher expressionfor the 1D bulk susceptibility and, secondly, using the exact temperature dependent bulk susceptibility of a singleantiferromagnetic Heisenberg chain, χ chain (0 , , T ), calculated numerically by Eggert et al. The Bonner-Fisher susceptibility is χ BF = 1 k B T (cid:18) .
25 + 0 . x + 0 . x . x + 0 . x + 6 . x (cid:19) (S.16)where x = | J k | / (2 k B T ). We fit the expression χ expt ( T ) = N g µ B (cid:18) χ BF ( T )1 − J ∗⊥ χ BF ( T ) (cid:19) (S.17)where the Land´e g-factor g and the interchain coupling J ∗⊥ , are free parameters and J k is fixed to our BS-DFT result,0.75 K. Our fit to the experimental data between 2 K and 10 K from Moreno et al. results in g = 2 . ,J ∗⊥ = 0 .
28 K . The result for g is a typical value for other cuprates. The value J ∗⊥ corresponds to k = ( ) in Eq. S.4, so our resultpredicts ˜ J ⊥ ( ) = 2 ( J ⊥ + J ⊥ ) = 0 .
28 K. This is comparable with our BS-DFT results.We then calculated the low-temperature prediction using these parameters as χ pred ( T ) = N g µ B (cid:18) χ chain (0 , , T )1 − J ∗⊥ χ chain (0 , , T ) (cid:19) (S.18)with χ chain (0 , , T ) from Eggert et al. The magnetic susceptibility is related to the dynamical structure factor, which is measured in inelastic neutron-scattering experiments, by S ( ω, k , T ) = − − exp ( − ω/T ) Im χ ( ω, k , T ) . (S.19) SECTION II: GENERATION OF LINEARIZED CRYSTALS
Figure 1 shows the lattice parameters of the bent crystals, from Worthy et al. , along with our linear regressions. Forradius of curvature r c = 1 . a a = − . d + 0 . b b = 0 . d − . c c = − . d + 0 . β β = − . d − . , (S.20)where d is the distance from the center of the crystal (defined to be where the data for b is closest to b ) in µ m. For r c = 3 . a a = − . d + 0 . b b = 0 . d − . c c = − . d + 0 . β β = − . d − . . (S.21)Using these regressions, we produced a linear set of lattice parameters for points across the bent crystals. In orderto minimise noise in our predictions, we then produced a set of nearest neighbor dimers to use as input for our BS-DFT calculations. These structures are attached as ‘bent dimers.zip’. To correctly simulate the change of the dimercoordinates across the bends, we kept the intramolecular bond lengths constant by producing new Wyckoff coordinates, -2.0-1.5-1.0-0.50.00.51.01.52.0 % C h a n g e i n l a tt i ce p a r a m e t e r s r c = 1 . r c = 3 . % C h a n g e i n l a tt i ce p a r a m e t e r s Distance from center ( µ m) r c = 1 . r c = 3 . abcβabcβ FIG. 1: Linear regressions (Eqs. S.20 and S.21) of the change in lattice parameters across both crystals. The data points arefrom Worthy et al. The change in lattice parameters is relative to the unbent crystal. The center is defined as the point in theeach bent crystal where b is closest to the unbent b . The data are labelled by the radius of curvature, r c . ( x , y , z ), for each dimer using the Wyckoff coordinates from the unbent crystal structure, ( x , y , z ), measured byWorthy et al. ; x x . . .y y . . .z z . . . = a − a cot( β )0 b
00 0 c csc( β ) a c cos( β )0 b
00 0 c sin( β ) x x . . .y y . . .z z . . . , (S.22)where a , b , c , and β are the new, linearized crystallographic paramaeters. The Wyckoff coordinates include thecoordinates for one copper atom and one acetylacetonate (acac) unit. After transforming the Wyckoff coordinatesas above, we created the other coordinates (one more copper atom and three more acac units) using the symmetrytransformations of the crystal space group, P /c . For all dimers, the first molecule is the Wyckoff coordinates, ( x, y, z )+(0 , , − x, − y, − z ) + (1 , , x, y, z ) + (0 , ,
0) and( − x, − y, − z ) + (1 , , ⊥ − x + 0 . , y + 0 . , − z + 0 .
5) + (0 , ,
0) and( x + 0 . , − y + 0 . , z + 0 .
5) + ( − , , − ⊥ − x + 0 . , y + 0 . , − z + 0 .
5) + (1 , ,
0) and( x + 0 . , − y + 0 . , z + 0 .
5) + (0 , , − . TABLE I: Heisenberg exchange ( J ij ) parameters for the unbent structure of [Cu(acac) ] determined with BS-DFT usingdifferent basis sets. All calculations were performed in Gaussian09 with the uB3LYP functional and an SCF convergencecriterion of 10 − a.u.Basis Set J k (K) J ⊥ (K) J ⊥ (K)6-31+G* and LANL2DZ (Cu) 0.75 0.04 -0.1TZVP and LANL2DZ (Cu) 0.82 0.03 -0.09aug-cc-pVTZ J ij ) parameters for the unbent structure of [Cu(acac) ] determined with BS-DFT usingdifferent exchange-correlation functionals. All calculations were performed in Gaussian09 using the LANL2DZ (for Cu)and 6-31+G* basis sets with an SCF convergence criterion of 10 − a.u.Functional J k (K) J ⊥ (K) J ⊥ (K)uPBEPBE We chose this method to make the bond distances in our linearized crystals the same as the unbent structure, whilethe distances and angles between molecules changed; this is how the coordinates change in the original crystollographicdata. SECTION III: BS-DFT BENCHMARKING
Table I shows BS-DFT results for all three nearest neighbour couplings with different basis sets. The magnitudes ofthe interchain couplings are all very similar and the range of J k values is 0.15 K, which is 20% of our reported value J k = 0 .
75 K.As shown in Table II, changing the functional between a pure functional, PBE, a hybrid, B3LYP, and a hybrid meta-GGA, uTPSSh, results in a larger range of couplings, which is expected. However, these three functionals still result ina large change in couplings across a bend (Table III).Importantly, all results show that the magnetic exchange model of [Cu(acac) ] is highly one-dimensional. There aretwo cases, in Tables I and II, where J ⊥ changes sign. This does not change our result, as T N depends only on themagnitudes of the interchain couplings (see Eq. S.13). TABLE III: Intrachain Heisenberg exchange ( J k ) for the inside and outside of the bent crystal with r c = 1 . with an SCF convergence criterion of 10 − a.u.Functional Basis Set In. J k (K) Out. J k (K) % ChangeuPBEPBE , LANL2DZ (Cu) 1.25 0.67 46TZVP , LANL2DZ (Cu) 1.36 0.78 44uB3LYP , LANL2DZ (Cu) 0.69 0.46 33TZVP , LANL2DZ (Cu) 0.75 0.49 34uTPSSh , LANL2DZ (Cu) 0.53 0.25 52TZVP , LANL2DZ (Cu) 0.58 0.31 47 ∗ [email protected] H. Z. Bethe,
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