Heat capacity and its temperature and magnetic field dependence is a powerful tool to provide fundamental thermodynamical information on various solids including correlated electrons in magnetic Mott insulators. It has come to our attention that recent measurements of the heat capacity of certain quantum magnets which are candidates for an exotic state of matter called quantum spin liquid1 show a power law temperature dependence and a striking one-parameter scaling and data collapse as a function of temperature T and magnetic field H. Though the materials all appear quite different the observed scaling functions of C[H, T] as power laws in T/H suggest an unexpected hidden universality.

Power-law specific heat is a familiar consequence of the ‘random singlet’ phase seen in doped semiconductors such as Si:P, and described theoretically in 1D by Dasgupta, Ma and Fisher2,3 and in 2D and 3D by Bhatt and Lee4,5,6. In this picture the spins interact with each other via a broad distribution of antiferromagnetic exchange interactions. The spins that are most strongly coupled pair into singlets first, leaving behind spins that are further apart, eventually resulting in a power law distribution of exchanges and a power law tail of density of states. This picture follows from renormalization group analysis in 1D and has been demonstrated numerically in higher dimensions under a variety of conditions. Though the original setting for the D > 1 random-singlet phase required a dilute random network of spin–1/2 sites, without a lattice, recently a random-singlet regime has been argued to arise as a general feature of spin–1/2 lattice magnets with quantum paramagnetic ground states and random exchange energies, i.e. in highly-frustrated quantum magnets with quenched disorder7. The theory applies when the majority of spin–1/2 sites form a paramagnetic state such as a spin liquid or a valence bond crystal, and demonstrates in two different controlled limits that a small fraction of sites necessarily nucleates (as may be required by the conjectured disordered-Lieb-Schultz-Mattis restrictions7) and leads to a random network of spin–1/2 moments at low energies. This small subsystem may be captured by a random-singlet regime in its low temperature renormalization group flow, developing a power-law probability distribution of antiferromagnetic exchange energies P[J] ~ Jγ. At a given temperature T, the spins with exchange J < T behave as free spins giving rise to a heat capacity C[T] ~ T1−γ and spin susceptibility χ[T] ~ Tγ. In many cases the measurable response from this relatively small emergent subsystem may dominate over the response of the bulk phase. Such an emergent power-law energy distribution, associated with a relatively small portion of the spin–1/2 sites, serves as the a priori starting point for the present work.

The power-law distribution of entropy manifests most dramatically by varying both temperature and an external magnetic field. Consider a given singlet bond with singlet-triplet energy splitting J drawn from the distribution P[J]. Applying a magnetic field with Zeeman energy H has no effect on the singlet energy but splits the triplet manifold. At low temperatures \(T \ll H\) this bond contributes to heat capacity only when its ground state crosses over from the singlet state to the field-polarized triplet state: the width of this resonance \(\left| {J - H} \right| < k_{\mathrm{B}}T\) (Fig. 1) is set by temperature giving a heat capacity that rises linearly in T. The distribution of bond energies enters only through the H-dependent coefficient: C ~ T/Hγ at \(T \ll H\) where C ~ T1−γ at H = 0. (We have set the magnetic moment B to be unity).

Fig. 1
figure 1

Heat capacity scaling function F0[T/H] and its q = 1, 2 modification by level repulsion. The heat capacity C[H, T] for q = 0 random singlets (shown with integrated energy distribution \({\int} {P[E] = (E{\mathrm{/\Lambda }})^{1 - \gamma }}\) at γ = 0.5, blue) exhibits scaling collapse in T/H with T-linear form at \(T \ll H\). This is easily understood (bottom inset). A spin–1/2 pair with singlet-triplet splitting J acquires a resonance in magnetic fields H ≈ J, contributing to C[H, T] when |J − H| < kBT. The scaling is modified when spin-orbit coupling and lattice symmetries combine into DM interactions with singlet-triplet mixing: the resulting level repulsion changes the resonance condition ΔE < kBT to produce T-scaling with higher powers, as in the q = 1 line shown (gray)

Given this expected scaling it came as a surprise that new experiments of ref.8 found a different scaling behavior on the spin–1/2 magnet H3LiIr2O6, a member of the family of so-called Kitaev honeycomb iridates9,10. Both magnetic frustration and disorder are likely ingredients in this compound: exchanges between Ir4+ effective spin–1/2 moments may vary randomly depending on the positions of mobile hydrogen ions, and the frustration expected from iridium’s strong spin-orbit coupling is evidently manifest in the lack of any ordering transition down to at least 0.05 K, less than a percent of the 100 K exchange energy scale. The bulk of the sites form a quantum paramagnetic phase, such as a spin liquid. The remaining fraction of sites was found to contribute a power-law heat capacity C ~ T1/2 with an appropriate small coefficient as expected from a random-singlet regime; but when a magnetic field was applied, instead of T-linear heat capacity, clear quadratic scaling C ~ T2/H3/2 was observed for \(T \ll H\). Furthermore, as shown in the inset of Fig. 4a in ref.8, the data are found to collapse to a single scaling curve of the form C[H, T]/T ~ Hγ F[T/H] where γ = 0.5.

The work of ref.8 reminds us of a system we had studied earlier, LiZn2Mo3O8. This is a layered magnet where 2/3 of the spin disappears into singlets, leaving behind 1/3 of the spins which behave almost as free spins11,12,13. The mechanism for this behavior is not well understood, since various theoretical proposals14,15,16 all require some form of short range tripling of the unit cell size, which has been searched for and not found13. However on symmetry grounds we again expect the prevalent non-magnetic Li/Zn site mixing disorder11 to generate bond randomness, whose competition with singlets requires7 low energy spin excitations to appear. Here we focus our attention on the fate of these remaining spins at low temperatures. It turns out that independent of ref.8, we had recognized that our previously unpublished heat capacity data (Fig. 2) also show clear data collapse with T2 scaling (Fig. 3). Similar data collapse, though with a smaller accessible T/H window, is seen in previous heat capacity measurements from ref.17 for synthetic herbertsmithite ZnCu3(OH)6Cl2 (Fig. 4). These materials, and related ones which we will discuss further below—1T-TaS2, YbMgGaO4, YbZnGaO4 and Ba2YMoO6—differ in their spin–1/2 lattices, magnetic Hamiltonians, spatial symmetries, and sources of randomness; but their lack of magnetic order down to the lowest temperatures measured, the presence of some randomness, and the observed power-law scaling laws consistent with a contribution from a fraction of sites, taken together call for a unified theoretical framework.

Fig. 2
figure 2

Power law heat capacity in LiZn2Mo3O8 under various magnetic fields. At zero field the heat capacity shows a non-integer power law C/TT−0.56 (purple line), changing under various magnetic fields to functional forms with quadratic CT2 behavior (orange line) at low temperature

Fig. 3
figure 3

Data collapse in LiZn2Mo3O8. Data from Fig. 2 rescaled by the scaling ansatz H0.56C/T. The data collapses to a function of the single variable T/H, asymptoting as (T/H) for \(T \ll H\) and (T/H)−0.56 for \(T \gg H\), consistent with the q = 1 scaling theory Eq. (2) and Fig. 1

Fig. 4
figure 4

Data collapse in synthetic herbertsmithite ZnCu3(OH)6Cl2. Herbertsmithite heat capacity data from ref.17, here replotted using the scaling ansatz H0.5C/T (per formula unit of ZnCu3(OH)6Cl2) to show data collapse. The phonon contribution has not been subtracted, which accounts for the upturn for T/H 1. Away from the upturn at low T/H 0.05, which is due to the nuclear Schottky contribution, the data collapses with a peak consistent with the scaling function Eq. (2) and Fig. 1

In this work we argue that the missing factor of T/H needed to explain the more recent data is captured by an extension of the random-singlet theory that includes spin-orbit coupling (SOC) and its interplay with spatial symmetries, in particular through antisymmetric Dzyaloshinskii-Moriya (DM) spin exchanges. For the sake of continuity we will denote the theory as ‘random-singlets’, even though the resulting theory describes a configuration of non-degenerate valence bonds that are no longer singlets under spin rotation. We will show how this produces three possibilities for T dependence of heat capacity in a magnetic field, associated with data collapse in T/H and a scaling exponent that can take one of three integer values.


Scaling and data collapse

The derivation of the T-linear scaling shown in Fig. 1 assumed that the downshifted triplet state crosses the singlet state freely. This assumption no longer holds in the presence of spin orbit coupling. The triplet will in general be split but in a magnetic field one level will still move down to approach the ground state with a modified g factor. Any off-diagonal matrix element D will mix these states, leading to level repulsion and resulting in an additional, independent, constraint D < T supplementing the diagonal constraint \(\left| {J - \left| H \right|} \right| < T\). It is important to note spin-orbit coupling is a necessary but not sufficient ingredient for the level repulsion: spin-orbit coupling modifies the magnetic exchanges in two distinct ways depending on spatial symmetries. With an inversion center at the bond midpoint, or certain other combinations of symmetries, the matrix \(J_{ij}^{\alpha \beta }\) of the spin exchange \(J_{ij}^{\alpha \beta }S_i^\alpha S_j^\beta\) is required to be symmetric. The two-spin singlet state is odd under the inversion symmetry and does not mix with the triplet manifold. However without these symmetries, the spin exchange matrix gains an antisymmetric contribution to \(J_{ij}^{\alpha \beta }\) which is conventionally described by the DM term D · (Si × Sj). The magnitude of the DM vector D is linear in the strength of SOC for weak SOC. By breaking inversion symmetry i ↔ j it mixes the singlet and the triplet and produces the desired level repulsion.

The preceding argument may be considered in more detail. While the zero-field splitting is given simply by \(\left( {1{\mathrm{/}}2} \right)\sqrt {J^2 + \left| D \right|^2}\), in a magnetic field this splitting becomes (Supplementary Note 1)

$${\mathrm{\Delta }}E = \frac{1}{{2\sqrt 2 }}\sqrt {2\left( {\left| H \right| - J} \right)^2 + D_1^2 + D_2^2}$$

where D1, D2 are the components of the DM vector D that lie perpendicular to the magnetic field H. It is clear that each component of the DM vector D that enters into the resonance condition Eq. (1) contributes a factor of T/D to the scaling of specific heat at low T (Supplementary Note 3). Furthermore, as long as the ratio of DM interactions to symmetric exchange interactions remains roughly fixed in the RG flow, these factors can be replaced by T/H. We show that this is indeed the case within the strong-disorder-RG renormalization step (Supplementary Note 2). The details of the scaling functions can depend on crystal symmetry as well as the dimensionality of the magnetic lattice (Supplementary Note 4). We thus find three possibilities: the linear scaling C[T] ~ T/Hγ may stay the same or gain a factor linear or quadratic in T/H. This result can be captured by a general scaling function Fq for the density of states as measured by heat capacity,

$$\begin{array}{l}\frac{{C[H,T]}}{T}\sim \frac{1}{{H^\gamma }}F_q\left[ {\frac{T}{H}} \right]\\ F_q[X]\sim \left\{ {\begin{array}{*{20}{c}} {X^q} & {X \ll 1} \\ {X^{ - \gamma }\left( {1 + \frac{{c_0}}{{X^2}}} \right)} & {X \gg 1} \end{array}} \right.\end{array}$$

The sub-dominant scaling term at large T/H must be added in order to satisfy the Maxwell relation between entropy and magnetization. A related subdominant term must be added to the scaling of magnetization at \(T \ll H\) yielding M/H ~ H−γ(1 − m1(T/H)q+2) (for m1 and further discussion see Supplementary Note 5). Here the coefficient c0 is given by c0 = ((1 + γ)γ/2)/(CH=0/TχH=0) when the zero-field susceptibility is χH=0 ~ T−γ. This scaling form shows the familiar non-universal exponent 0 γ 1 that characterizes the random-singlet distribution, but in addition a new integer index q appears that takes one of the three values {0, 1, 2} for the three cases described above; its choice depends on the interplay of SOC with spatial symmetries as we will discuss below.

Two different values of q are necessary to capture the observed experimental scaling we are aware of so far. We have already discussed the quadratic scaling (q = 1) observed for H3LiIr2O6 and LiZn2Mo3O8. The q = 1 scaling in these layered magnets may be understood as arising from a DM vector that is forced to point perpendicular to the lattice plane by an approximate reflection symmetry across the plane that emerges during the course of RG flow: in each strong-disorder RG step the new spin exchanges are generated based on the pattern of valence bonds that have been integrated out at higher energies, and for a 2D lattice any such configuration of valence bonds preserves the mirror reflection across the plane, leading (Supplementary Note 4) to a single-component DM vector and q = 1.

In Fig. 4 we show the data collapse for the well studied spin–1/2 kagome lattice material ZnCu3(OH)6Cl2. It is known that approximately 15% of the Zn sites which are off the kagome planes are replaced by Cu, forming S = 1/2 local moments18. Furthermore, there is significant DM coupling. We should mention two caveats. First the nuclear contribution makes the low temperature spin contribution inaccessible and second, the bulk spin gap is estimated to be 0.7 meV, so that the high field data may have some bulk contribution due to the closing of the spin gap. With these reservations, the data collapse shown in Fig. 4 may also be consistent with the q = 1 case.

Next we consider the layered material 1T-TaS2, where a charge density wave (CDW) transition at intermediate temperatures leads to a cluster Mott insulator with one spin–1/2 per 13-site unit cell arranged into a triangular lattice, with phenomenology consistent with a spin liquid state19. A small fraction of the spins (less than than a few % per cluster) were recently observed to produce a low temperature T-linear term in the heat capacity with a coefficient that decreases monotonically with a magnetic field (ref.20 Fig. 4) roughly as C ~ H−2/3T. This is consistent with the argument above for linear scaling (q = 0) for 1T-TaS2 due to its high crystal symmetry, preserved even in the CDW state, that forbids DM interactions from being generated for not only nearest neighbor and second neighbor bonds but further neighbor bonds as well. A replot of the C[H, T] two-parameter data (Y. Dagan and I. Silber, personal communication) indeed shows data collapse consistent with Eq. (2) with q = 0.

It will be interesting to test the data collapse scaling form for other frustrated spin–1/2 quantum magnets. Other layered compounds such as YbMgGaO4 and YbZnGaO4, both with effective spin–1/2 moments from Yb arranged on a triangular lattice and with intrinsic Mg/Ga and Zn/Ga charge disorder21,22 show clear C ~ T1−γ scaling in zero field measurements21,23,24; but the \(T \ll H \ll J_0\) scaling limit, given the small lattice magnetic exchange energy J0 ~ few K, is not yet clear. The case q = 2 is expected to arise most naturally in magnets with fully 3D magnetic lattices such as Ba2YMoO625, which we leave for future work.

Finally, we note that the appearance of scaling and data collapse sheds light on two interrelated subsystems of the quantum magnet. The first subsystem is formed by the local moments that contribute to the heat capacity scaling. When the material disorder does not directly change the sites of its spin–1/2 lattice but rather impacts it (weakly or strongly) only through bond randomness in the magnetic exchanges—likely the case for all the materials discussed above except synthetic herbertsmithite—then the local moment subsystem is emergent via an unusual RG flow7. The quantum critical scaling functions which can be exhibited by this subsystem are interesting both in their own right and as a signature of a coexisting quantum paramagnet state for the second subsystem, consisting of the remaining spins. This coexisting quantum paramagnetic phase must involve valence bonds which may be frozen, possibly with a relic of valence-bond-solid order, or resonating, as in a quantum spin liquid and associated topological order. The interplay of the quantum paramagnet with the local moments that produce the T/H scaling merits further exploration.