Absence of Spin Frustration in the Kagom\'e Layers of Cu2+ Ions in Volborthite Cu3V2O7(OH)2x2H2O and Observation of the Suppression and Re-entrance of Specific Heat Anomalies in Volborthite Under External Magnetic Field

We determined the spin exchanges between the Cu2+ ions in the kagome layers of volborthite, Cu3V2O7(OH)2x2H2O, by performing the energy-mapping analysis based on DFT+U calculations, to find that the kagom\'e layers of Cu2+ ions are hardly spin-frustrated, and the magnetic properties of volborthite below ~75 K should be described by very weakly interacting antiferromagnetic uniform chains made up of effective S=1/2 pseudospin units. This conclusion was verified by synthesizing single crystals of not only Cu3V2O7(OH)2x2H2O but also its deuterated analogue Cu3V2O7(OD)2x2D2O and then by investigating their magnetic susceptibilities and specific heats. Each kagome layer consists of intertwined two-leg spin ladders with rungs of linear spin trimers. With the latter acting as S=1/2 pseudospin units, each two-leg spin ladder behaves as a chain of S=1/2 pseudospins. Adjacent two-leg spin ladders in each kagome layer interact very weakly, so it is required that all nearest-neighbor spin exchange paths of every two-leg spin ladder remain antiferromagnetically coupled in all spin ladder arrangements of a kagome layer. This constraint imposes three sets of entropy spectra with which each kagome layer can exchange energy with the surrounding on lowering the temperature below ~1.5 K and on raising the external magnetic field B. We discovered that the specific heat anomalies of volborthite observed below ~1.5 K at B = 0 are suppressed by raising the magnetic field B to ~4.2 T, that a new specific heat anomaly occurs when B is increased above ~5.5 T, and that the imposed three sets of entropy spectra are responsible for the field-dependence of the specific heat anomalies.


Introduction
Properties of a magnetic material are described by a spin Hamiltonian defined in terms of a few spin exchange paths between magnetic ions. The repeat pattern of strong spin exchange paths forms a spin lattice, with which the spin Hamiltonian generates the energy spectrum to be utitilized in describing the magnetic properties. Consequently, use of a correct spin lattice is paramount because the nature of the energy spectrum generated by a spin Hamiltonian depends on the nature of the spin lattice chosen. Antiferromagnets possessing a kagomé arrangement of transition-metal magnetic ions M are often believed to be spin-frustrated, hence suppressing a long-range magnetic order, so they are prime candidates that can give rise to exotic magnetic ground states. 1,2 However, spin frustration is not guaranteed per se by, for example, a trigonal, a kagomé or a pyrochlore structural arrangement of magnetic transition-metal ions M because they interact strongly with the surrounding main-group ligands L to form ML n polyhedra thereby splitting its d-states. What is crucial for geometrical spin frustration is not the geometrical pattern imposed by the arrangement of magnetic ions M, but rather that of the spin exchanges these ions generate with their neighboring ions. The direction-dependence of spin exchanges between magnetic ions M is determined by their magnetic orbitals, namely, the singly-occupied d-states of their ML n polyhedra. 3,4 The d-states have the d-orbitals of M combined out of phase with the p-orbitals of the ligands L, so they are highly anisotropic in shape. Consequently, even if the magnetic ions form a kagomé arrangement, the spin exchanges between magnetic ions do not necessarily generate the same pattern. The anisotropy of spin exchange is most strongly manifested for a magnetic ion possessing only one magnetic orbital (e.g., S=1/2 ions Cu 2+ and Ti 3+ ), and least strongly for a magnetic ion with five magnetic orbitals (e.g., S=5/2 ions Mn 2+ and Fe 3+ ).
Volborthite, Cu 3 V 2 O 7 (OH) 2 2H 2 O, consisting of Cu 2+ (d 9 , S = 1/2) ions in kagomé arrangement, has received much attention over the past decade, [5][6][7][8][9][10][11] largely because it is believed to be spin-frustrated. 5,11 The kagomé layers of Cu 2+ ions are slightly distorted from an ideal kagomé shape with two different crystallographic sites for Cu 2+ in a ratio 2:1. Below room temperature, each Cu 2+ ion forms an axially-elongated CuO 6 octahedron so that, with the local z-axis taken along the elongated Cu-O bonds, the magnetic orbital of each CuO 6 octahedron is the x 2 -y 2 state contained in the CuO 4 equatorial plane, which is quite anisotropic in shape. In volborthite, the kagomé layers of composition Cu 3 O 6 (OH) 2 parallel to the ab-plane are pillared by pyrovanadates V 2 O 7 . The voids between the layers provide space for crystal water molecules. Below room temperature, volborthite undergoes two structural phase transitions, one at 292 K from a C2/c phase to a I2/a phase, and the other at about 155 K from the I2/a phase to a P2 1 /c phase. 6 The latter structural phase transition generates two slightly different kagomé layers, which are slightly different in structure. Below 1.5 K volborthite exhibits magnetic order indicated by two anomalies in the magnetic specific heat. 10 On the basis of their DFT+U calculations Janson et al. 11 described the magnetic properties of volborthite using a trigonal spin lattice made up of pseudo S=1/2 units, i.e., linear spin trimers in which adjacent Cu 2+ ions are strongly coupled antiferromagnetically. However, with this spin lattice model, it is difficult to explain why volborthite undergoes a magnetic ordering below 1.5 K. 10 Furthermore, the magnetic susceptibility Janson et al. calculated for volborthite by their exact diagonalization method for the spin Hamiltonian of the trigonal lattice show a spin gap below the susceptibility maximum. The latter is in sharp contrast to the experimental observation, 5 which reveals that the magnetic susceptibility remains rather high as the temperature is decreased toward 0 K, 5 and that the magnetic entropy removed by the long-range order is small compared to Rln2 as is typically found for low-dimensional short-range ordered (SRO) magnetic systems, e.g., for S=1/2 antiferromagnetic uniform Heisenberg (AUH) chains. Then, one might speculate if the magnetic ordering of volborthite observed below 1.5 K is associated with a magnetic ordering of such S=1/2 AUH chains, although it has been believed that volborthite is spin-frustrated. 5,11 To explore the magnetic order below 1.5 K, especially, its dependence on magnetic field as well as the origin of the apparent finite susceptibility as T 0 K, we re-analyze the spin lattice of volborthite by performing an energy-mapping analysis based on DFT+U calculations, 3,4 to find that the magnetic properties of volborthite below 75 K should be described by two-leg spin ladders with rungs of S=1/2 pseudospin units rather than by a trigonal lattice of S=1/2 pseudospin units as proposed by Janson et al. 11 We verify this conclusion by acquiring new magnetic susceptibility data and re-analyzing reported magnetization data of volborthite, to show that the kagomé layer of Cu 2+ ions is hardly spin-frustrated, and the low-temperature magnetic properties of volborthite should be described by an AUH chain composed of S=1/2 pseudospin units. We characterize the magnetic phase transitions below 1.5 K by carrying out specific heat measurements for Cu 3 V 2 O 7 (OH) 2 2H 2 O and its deuterated analogue, Cu 3 V 2 O 7 (OD) 2 2D 2 O, under magnetic field B = 0 -9 T. Our work shows that the magnetic ordering of volborthite below 1.5 K is suppressed by field B > 4.5 T while a new magnetic ordering takes place when B > 5.5 T, and that these field-dependent behaviors of the magnetic ordering originate from the three sets of magnetic entropy spectra of each kagomé layer of Cu 2+ ions created by topologically-constrained interactions between adjacent two-leg spin ladders.

Experimental and calculations
To determine the spin exchanges of the I2/a and P21/c phases of volborthite, we carried out DFT+U calculations employing the Vienna ab Initio Simulation Package (VASP) 12,13 using the projector augmented wave (PAW) 14,15 method and the PBE 16 exchange-correlation functional.
The electron correlation associated with the 3d states of Cu was taken into consideration by DFT+U calculations, i.e., by performing DFT calculation with an effective on-site repulsion U eff = U -J = 4 and 5 eV added on the magnetic ions Cu 2+ . 17 Single crystals of volborthite were grown using hydrothermal techniques as described in the literature. 6 Deuterated samples of volborthite were prepared by replacing light by heavy water (isotope enrichment 99.5 %). The magnetic susceptibilities were measured at constant field as a function of the temperature in a Magnetic Property Measurement System SQUID magnetome ter (MPMS-XL7, Quantum Design, San Diego, U.S.A.). The specific heats of a collection of oriented crystals were determined using the relaxation method of a 3-He Physical Property Measurement System (PPMS, Quantum Design, San Diego, U.S.A). In order to construct a lattice reference needed to evaluate the magnetic contribution to the specific heat of volborthite, we prepared a polycrystalline sample of the diamagnetic mineral martyite with composition Zn 3 V 2 O 7 (OH) 2 2H 2 O from an aqueous solution of NH 4 VO 3 and zinc acetate, Zn(CH 3 CO 2 ) 2 , following a recipe reported in the literature. 18

Spin exchanges and spin lattice of volborthite
We determine the spin exchanges J 1 -J 5 defined in Figure 1a by performing the energymapping analyses 3,4 based on DFT+U calculations (for details of calculations, see the Supplemental Material). Below room temperature down to 155 K, volborthite has a crystal structure with two equivalent kagomé layers in the unit cell (space group I2/a) phase and below 155 K a structure with two slightly nonequivalent kagomé layers of Cu 2+ ions (space group P2 1 /a). 5,8 The spin exchanges of the two phases are very similar as summarized in Table 1, where only the values obtained with U eff = 4 eV are listed. Those calculated with U eff = 5 eV are given in the Supplemental Material. Janson et al. 11 carried out DFT+U calculations using U eff = 8.5, 9.5 and 10.5 eV to determine the spin exchanges of the P2 1 /a structure by fitting the electronic band structure in terms of a tight binding approximation. In the following we first discuss the spin lattice of volborthite suggested by the spin exchanges we obtained. Subsequently, we show that the same picture is obtained from those of Janson et al. despite that their spin exchanges are considerably smaller than ours in the strengths of AFM character due to their use of very large U eff values.
As can be seen from Table 1, the strongest exchange J 2 is AFM and forms isolated linear trimers (Figure 1a). The second strongest spin exchange J 4 is also AFM but is weaker than J 2 by nearly an order of magnitude. All other spin exchanges are weaker than J 2 by more than an order of magnitude. Since J 2 is much stronger than other spin exchanges, each linear trimer at T << J 2 constitutes a pseudospin S=1/2 unit and such units form a triangular lattice, as already pointed out by Janson et al. 11 However, we note that the linear trimers interact through the AFM exchange J 4 to form two-leg spin ladders with the linear trimers as rungs (Figure 1b). This feature is hidden in a kagomé layer because nearest-neighbor spin ladders are geometrically entangled with their superposed legs.  Table 1). In each J2 path, two Cu 2+ ions make a Cu- The two-leg spin ladder model as the spin lattice of volborthite described above is also supported by Janson et al.'s spin exchanges ( Table 1), although their spin exchanges differ considerably from ours in magnitude. In general, a spin exchange J between two magnetic ions located at sites i and j can be written as the sum of the FM and AFM components (J F and J AF , respectively), namely, J = J F + J AF . 3,4,19 With the magnetic orbitals describing the spin sites i and j as  i and  j , respectively, the overlap density  ij =  i  j gives rise to the exchange repulsion K ij while the overlap integral  i | j  leads to an energy split e between the two states described by the magnetic orbitals. Then, J F and J AF are written as Since the AFM component J AF is inversely proportional to U eff , using a large U eff value in We now examine the interactions between adjacent two-leg spin ladders, under the constraint that the spin exchanges J 2 and J 4 forming the two-leg spin ladders are much stronger than J 3 and J 1 , which provide interactions between spin ladders. The latter requires that, in all possible spin ladder arrangements, all J 2 and J 4 exchange paths of each two-leg spin ladder must remain antiferromagnetically coupled (Figure 1d). In every intertwined legs, each J 2 leg has two consecutive J 3 paths. Since each J 2 leg is antiferromagnetically coupled, J 3 has no effect on the interaction between adjacent two-leg spin ladders regardless of whether it is AFM or FM. Thus , adjacent two-leg spin ladders interact only through the very weak exchanges J 1 (Figure 1a). Note that J 1 is weakly AFM for the I2/a phase, but is weakly FM for the P2 1 /a phase in our calculations.
This difference does not influence how the ordering between spin ladders in a kagomé layer is affected by J 1 (see below). Since J 1 is more than an order of magnitude weaker than J 4 , the magnetic property of each kagomé layer at low temperatures (|J 1 | < T << J 2 ) is primarily determined by those of the two-leg spin ladders with each linear trimer acting as an effective S=1/2 pseudospin rung (Figure 1c). At sufficiently low temperatures where excitations within the rungs are negligible, the dominant AFM spin exchanges form two-leg spin ladders predicting that volborthite should be regarded primarily as a system of very weakly coupled S=1/2 AUH chains. In the following we first demonstarte that the magnetic properties of volborthite below 75 K are well explained by the model of very weakly interacting S=1/2 AUH chains. Figure 2 displays the magnetic susceptibilities of volborthite, which exhibits a broad peak at 17 K and a finite susceptibility as T  0 K characteristic of a low-dimensional AFM SRO behavior. As implied by the results of the DFT+U calculations, the magnetic susceptibilities of volborthite between 3 and 75 K can indeed be very well fitted by those theoretically predicted for a S=1/2 AUH chain according to where  spin (J C , g, T) represents the magnetic susceptibility of the S=1/2 AUH chain with nearestneighbor (NN) spin exchange J C (Figure 2a). For  spin (J C , g, T) we use the Padé approximant of Quantum Monte Carlo (QMC) results for the S=1/2 AUH chain. 20 The second term refers to the temperature-independent susceptibility  0 arising from the diamagnetism of the electrons in the closed shells (-175  10 -6 cm 3 /mol per formula unit, FU) 21,22 and also from the van Vleck paramagnetic susceptibility due to excitations to the excited states of each Cu 2+ ion (100  10 -6 cm 3 /mol per Cu atom), 23 leading to  0 = +125  10 -6 cm 3 /mol per FU. Up to 75 K the experimental susceptibility is well reproduced by the susceptibility calculated for a S=1/2 AUH chain with J C = 27.8(5) K and g = 2.33 (the solid blue curve in Figure 2a). The fitted g-factor is found to be 2.33, which is at the higher end of the g-factor expected for Cu 2+ ions. 24 The S=1/2 AUH chain model readily explains the susceptibility maximum at T max = 17 K as well as the finite susceptibility below T max . The exchange J C = 27.8(5) K agrees very well with the expected value J C = T max /0.64085 = 26 K. In addition, the ratio  spin T max /g 2 = 0.0346 cm 3 K/mol concurs well with the value 0.0353 expected for the S=1/2 AUH chain. 20 The difference between the experimental data and the chain susceptibility (green solid line in Figure 1) vanishes below 75 K and gradually increases to higher temperatures. The susceptibilities above 75 K without any constraints matches very well with the susceptibility of an isolated linear spin S=1/2 trimer described by the spin Hamiltonian ) (

Experimental evidence for the spin-half AFM uniform Heisenberg chain character
represent the three spin sites. Replacing  spin (J C , g, T) in eq. (2) with the susceptibility of a spin S=1/2 trimer given, e.g., by Boukhari et al. 25 the susceptibility data above ~75 K can be well fitted without further constraints. The trimer spin exchange amounts to 197 (2) K, indicating a ratio J C /J trim = 0.14, in good agreement with the ratio J 4 /J 2 obtained from our DFT+U calculation (see Table 1) suggesting that the spin trimers should be identified as the rungs of the two-leg spin ladders running along the crystallographic b axis.
Another characteristic feature of the magnetic properties of volborthite is the lowtemperature 1/3 magnetization plateau extending over a wide magnetic field range, displayed in

A. Cu 3 V 2 O 7 (OH) 2 2H 2 O
In Figure 3a, we show the specific heat measured for an ensemble of aligned crystal of volborthite Cu 3 V 2 O 7 (OH) 2 2H 2 O in zero magnetic field with magnetic field applied along the b axis, the estimated lattice contribution to the heat capacity constructed from the specific heat of 28

B. Effect of isotope substitution
In

C. Effect of magnetic field on magnetic ordering
Given that the specific heat anomalies below 1.5 K are associated with magnetic ordering, it is reasonable to expect that it can be destroyed and the specific heat anomalies disappear under sufficiently high external magnetic field. To confirm this conjecture, we carry out specific heat measurements for volborthite under magnetic fields B = 0 -9 T (Figure 4a). As B increases, the two specific heat anomalies observed at B = 0 widen but remain separated forming two ridges until they eventually merge into one and then abruptly disappear for B > 4.2 T (see Figure 4a).
Surprisingly, however, a new specific heat anomaly occurs for fields B > 5.5 T. This new anomaly forms a single broad ridge, widening and shifting to higher temperatures with increasing B. To probe the cause for the field-dependence of the specific heat anomalies, it is necessary to find the magnetic energy spectra of volborthite available below 1.5 K with which it can exchange energy with the surrounding phonon bath. At a temperature T below 1.5 K, the free energy G i of volborthite associated with its magnetic arrangement i can be written as where H i and S i are the enthalpy and entropy of the magnetic arrangement I, respectively If volborthite has a set of different magnetic arrangements i = 1, 2, 3, , it has the corresponding energy spectrum {G 1 , G 2 , G 3 , }. Then, volborthite can absorb energy from the surrounding by using this energy spectrum reflected by the specific heat anomaly. In the next section, we show that volborthite has three distinct but different sets of energy spectra associated with the arrangements of two-leg spin ladders in each kagomé layer with respect to each other The limitation to only three energy spectra occur due to the fact that interactions between adjacent twoleg spin ladders are topologically constrained (see below).

Spin ladder arrangements
In this section we discuss the magnetic-field dependence of the magnetic ordering in volborthite below 1.5 K. Our experiments gave conclusive evidence for an effective AUH chain behavior of volborthite at low temperatures based on the two-leg spin ladders formed by J 2 and J 4 .
As already mentioned, the two-leg spin ladders (thereafter, referred to as the spin ladder, for short) interact with their neighboring spin ladders to establish two-dimensional correlations within each has one more (AA) spin ladder, while that of Group III has one more (FF) spin ladder. is weakly FM in the P2 1 /a phase (see Table 1
Consequently, S I (2m) = S I (N -2m), and S I (2m) increases as 2m increases from 0 toward at N/2 and as 2m decreases from N toward N/2.
The kagomé layers of volborthite satisfy the three conditions necessary for a crystalline solid to exhibit a purely entropy-driven phase transitions: [30][31][32] (1) It has a set of states identical in enthalpy H. (2) The members of this set are grouped into subsets i = 1, 2, 3, , with degeneracies The degeneracy  i increases steadily, e.g.,  1 <  2 <  3 < , so the associated entropy increases steadily. Thus, one may expect that already moderate magnetic fields inducing Zeeman energies comparable to J 1 are able to substantially alter the entropy balance causing dramatic effects on the low-temperature ordering behavior. In fact, as shown Figure 4a, the specific heat anomalies observed at B = 0 indeed disappears for B > 4.2 T, but a new anomaly occurs when B > 5.5 T. Below 2 K, neither electronic nor vibrational energy of volborthite is available for the energy absorption causing the specific heat anomalies. Thus, in the following, we analyze the dependence of the specific heat anomalies on T and B in terms of the entropy spectra of Groups I, II and III (Figure 4b).

Specific heat anomalies and entropy spectra
As shown above, the entropy S I (2m) of Group I is smaller than that of Group II or III by S E (2m). As the magnetic field B increases from 0, a kagomé layer starts absorbing energy by exploiting the entropy spectrum of Group I, so the anomaly below ~4.2 T arises from the entropy spectrum of Group I. Since  I (2m) becomes larger with increasing 2m from 0 to N/2 and also with decreasing 2m from N to N/2, more states are involved in the energy absorption from Zeeman energy. This allows one to understand why the intensity of the specific heat anomaly increases with increasing field B. The entropies S I (2m) from the states for 2m = 0 -N/2 as well as 2m = N -N/2 are used to absorb energy, eventually reaching the highest entropy level S I (N/2) (Figure 4b).
When all entropy levels are exploited, there is no more entropy level with which a kagomé layer can absorb energy any further even if B is increased. At this stage, the spin ladder ordering within a kagomé layer assumes a liquid-like disordered state, and the specific heat in the valley between 4.2 T and 5.5 T reaches that of the disordered state of the S=1/2 AUH chains (the green plateau in Figure 4a). It is most reasonable to assign the higher-temperature specific heat ridge to a magnetic ordering within each kagomé layer, and the lower-temperature one to a threedimensional ordering, i.e., an ordering between the magnetically-ordered kagomé layers. When the magnetic field increases further, a kagomé layer can utilize the entropy spectra of Groups II and III simultaneously, leading to the single broad specific heat anomaly above 5.5 T. However, this requires that the spin ladder arrangements of a kagomé layer must be converted from Group I to Groups II and III. Each spin ladder has two sets of J 1 exchanges, e.g., a (AF) spin ladder has one set of AFM-coupled J 1  With magnetic field, it is energetically more favorable to induce a J 1 -flip from AFM to FM rather than that from FM to AFM. In contrast, thermally driven J 1 -flips from AFM to FM or from FM to AFM are equally likely. Thus, thermal agitation will be more effective in creating J 1 -flips than Zeeman energy, approximately by a factor of 2. The positions of two ridges of the specific heat below 4.2 T are nearly parallel to the magnetic field axis, whereas the single specific heat ridge above 5.5 T is slanted with slope g B B/k B T  2. This reflects that, at strong enough magnetic field, the kagomé layers of Group II spin ladder arrangements absorb energy equally as do those of Group III spin ladder arrangements. The entropy spectra II and III are identical in the degeneracy of each entropy level, but differ in the nature of spin ladder arrangements. Whether a given kagomé layer produces the entropy spectrum II or III has no effect on the enthalpy difference.
Consequently, there is no ordering between kagomé layers, which explains the occurrence of a single specific heat anomaly at magnetic field above 5.5 T. The latter forms a single ridge with increasing B, the shape of which suggests that not all available entropy states of the entropy spectra II and III are populated at 9 T. If it is assumed that this above-5.5T specific heat anomaly ( Figure   4a) retains a symmetric shape with respect to temperature and magnetic field, one might expect that the magnetic order above 5.5 T will disappear under magnetic field higher than 13 T.
Finally, we note that in all ordered magnetic states of Groups I, II and III, there occurs a strong gain of translational entropy with respect to those states in which there is no order between neighboring spin ladders, because the ordered states have translational symmetry along the leg direction. This provides a strong driving force for the magnetic ordering. In a sense, this situation is similar to the nematic phase transition in a system of thin rods examined 6 by Onsager. 31

Concluding remarks
The spin exchanges of volborthite show that each kagomé layer of Cu 2+ ions is hardly spinfrustrated, but rather consists of very weakly interacting two-leg spin ladders with linear trimers as rungs. Below 75 K, these rungs act as S=1/2 pseudospin units, making each two-leg spin ladder behave as a S=1/2 antiferromagnetic uniform Heisenberg chain. This conclusion was confirmed by synthesizing single crystal samples of volborthite Cu 3 V 2 O 7 (OH) 2 2H 2 O and its deuterated analogue Cu 3 V 2 O 7 (OD) 2 2D 2 O and subsequently measuring their magnetic susceptibilities and specific heat anomalies. Under magnetic field B higher than 4.2 T, the specific heat anomalies below 1.5 K are suppressed. However, a new specific heat anomaly appears when B is raised above 5.5 T. The dependence of the specific heat anomalies on magnetic field are governed by the fact that the magnetic properties of each kagomé has three sets of entropy spectra with which it can exchange energy with the surrounding. These entropy spectra arise from the topologicallyconstrained interactions between adjacent two-leg spin ladders in each kagomé layer that all spin exchange paths forming each two-leg spin ladder should remain antiferromagnetically coupled.
The present work makes it clear that use of a correct spin lattice is critical in describing the properties of a magnetic material. As a criterion for finding a proper spin lattice, the geometrical pattern of magnetic ion arrangement can be misleading because a spin lattice is decided by the geometrical pattern of the strong spin exchange paths. 3,4,19 If volborthite were to be treated as a spin-frustrated kagomé lattice model, the S=1/2 antiferromagnetic uniform Heisenberg chain behaviors of its magnetic susceptibility and magnetization are exotic and novel, and so are its specific heat anomalies below 1.5 K. By the same token, all other magnetic properties of volborthite not explained by a spin-frustrated kagomé lattice model would be novel and surprising.
Conversely, each kagomé layer of Cu 2+ ions consists of very-weakly interacting two-leg spin ladders, so the seemingly exotic magnetic properties previously attributed to magnetic frustration are simply explained by a well-studied S=1/2 antiferromagnetic uniform Heisenberg chain model.
The most unbiased, straightforward way to find a correct spin lattice for any given magnetic material is to evaluate the relative strengths of various possible spin exchanges of a given magnetic system by using the energy-mapping analysis based on first principles DFT calculations. 3

Spin exchanges of the I2/a and P2 1 /a phases of volborthite by energy mapping analysis
To extract the values of the spin exchanges for the I2/a and P2 1 /a phases of volborthite, we carry out DFT calculations encoded in the VASP 1,2 using the augmented plane wave method 3,4 and the PBE exchange-correlation functional. 5 To take into account the effect of electron correlation of the Cu 3d states, we use the DFT plus on site repulsion (DFT+U) method 6 with U eff = U -J = 4 and 5 eV.

A. I2/a phase
To determine the five spin exchanges J 1 -J 5 of the I2/a phase, we consider six ordered spin states FM, AF(i) (i = 1 to 5) shown in Figure S1. Then, the total spin exchange energies of these states can be written as = ∑ n i J i S 2

=1
(S1) where S refers to the spin of the Cu 2+ ion (i.e., S = 1/2). The values of n i ( i = 1 to 5) found for the six spin states are listed in Table S1. The relative energies (meV/FU) obtained for the FM, and AFi (i = 1 -5) states by DFT+U calculations are listed in Table S2. By mapping the relative energies of the ordered magnetic states determined by DFT+U calculations to those determined by the spin exchange energies, we obtain the values of the spin exchanges J 1 -J 5 listed in Table S3 and Table 1 of the text.

B. P2 1 /a phase
To determine the 10 spin exchanges of the P2 1 /a phase, namely, J 1 -J 5 for the layer 1 and J 1 -J 5 for the layer 2, we consider 11 ordered spin states FM, AF(i) (i = 1 to 10) shown in Figure   S2. Then, the total spin exchange energies of these states can be written as  Table S4. The relative energies (meV/FU) obtained for the FM, and AFi (i = 1 -10) states by DFT+U calculations are listed in Table S5. By mapping the relative energies of the ordered magnetic states determined by DFT+U calculations to those determined by the spin exchange energies, we obtain the values of the spin exchanges listed in Table S6 and Table 1 of the text.