Scaling and data collapse from local moments in frustrated disordered quantum spin systems

Recently measurements on various spin–1/2 quantum magnets such as H3LiIr2O6, LiZn2Mo3O8, ZnCu3(OH)6Cl2 and 1T-TaS2—all described by magnetic frustration and quenched disorder but with no other common relation—nevertheless showed apparently universal scaling features at low temperature. In particular the heat capacity C[H, T] in temperature T and magnetic field H exhibits T/H data collapse reminiscent of scaling near a critical point. Here we propose a theory for this scaling collapse based on an emergent random-singlet regime extended to include spin-orbit coupling and antisymmetric Dzyaloshinskii-Moriya (DM) interactions. We derive the scaling C[H, T]/T ~ H−γFq[T/H] with Fq[x] = xq at small x, with q ∈ {0, 1, 2} an integer exponent whose value depends on spatial symmetries. The agreement with experiments indicates that a fraction of spins form random valence bonds and that these are surrounded by a quantum paramagnetic phase. We also discuss distinct scaling for magnetization with a q-dependent subdominant term enforced by Maxwell’s relations.

Scaling and data collapse from local moments in frustrated disordered quantum spin systems

SUPPLEMENTARY INFORMATION
This Supplementary contains important details for the calculations described in the main text. It is structured as follows. We begin by considering the effect of DM interactions on modifying the two-spin singlet state, as well as the competition of these effects with an applied magnetic field. We then consider the RG flow under strong disorder RG, as appropriate for random singlets, in the presence of DM interactions. Then we compute the expected scaling phenomenology and discuss the various cases for q. Finally we relate the scaling forms for various observables including heat capacity, NMR lifetimes and magnetization.
The starting point for the calculations in this Supplementary is a picture of emergent couplings with the topology of a random network, a distribution with a long power-law tail, and an associated strong-disorder RG flow. To understand the emergence of such self-similar power laws out of a microscopic lattice model with merely flat disorder distributions, we refer the interested reader to the RG flows discussed in Ref. 1. A picture for how the RG flow is initiated out of the UV lattice scale for one simple class of microscopic models is depicted in Fig. 1.

Supplementary Note 1: Two spins with DM interactions
To see the effects of DM interactions on the valence bond states, consider the following Hamiltonian for two spins i, j, where in addition to a general DM term and magnetic field, for simplicity here the symmetric spin interactions are taken to consist of pure Heisenberg exchange; below we discuss generic SO(3) breaking exchanges. Defining the total spin operator S tot ≡ S i + S j , the Hamiltonian up to an overall energy shift may be written as where the DM term mixes the singlet with the member of the triplet manifold that has zero total spin moment along the DM vector, At zero field H = 0, the ground state is a mixture of the spin-singlet state and this S tot · D = 0 member of the triplet manifold, set by the Hamiltonian matrix 2H H=0 = Jσ 3 + |D|σ 2 acting on the basis of these two states. The ground state is nondegenerate; for small Supplementary Figure 1. RG flow from a class of weaklydisordered lattice models into a random network with power law distributions. Disordering short-ranged valence bonds is found, under certain conditions and even for weak randomness, to necessarily lead to defects that carry protected spin-1/2 moments. The resulting spin moments form a random network with power-law distributions, an appropriate setting for strong-disorder RG. Regardless of the ultimate fixed point of the RG, which in 2D and 3D is unknown, the system is described by emergent power laws across some low energy regime associated with the RG flow. For larger fields near resonance H ≈ J it is useful to project the Hamiltonian to the low energy space spanned by the singlet and the member of the triplet with maximal total magnetization along the magnetic field direction. In this 2-level subspace, the Hamiltonian, again up to a constant, is where σ are Pauli matrices acting on the (singlet, totalspin-up-triplet) basis, and D 1 , D 2 are the components of the DM vector perpendicular to the magnetic field axis. Indeed this equation, with modified parameters, describes the general scenario reached by adding generic spin-orbit-coupled interactions to the symmetric spin exchange matrix J µν ij S µ i S ν j . Without the DM term the singlet state remains an eigenstate and we shall assume the symmetric exchanges are sufficiently antiferromagnetic so that the singlet is the ground state of J µν ij S µ i S ν j . The triplet-manifold eigenstate brought down by the field is modified. The resulting Hamiltonian is described by Eq. 4 with J interpreted as an effective exchange parameter that may depend on the direction of H, and D 1 , D 2 interpreted as two particular components of the DM vector. The strong disorder RG (SDRG) step entails integrating out a pair of spins and considering the renormalization of interaction for every other remaining pair. We perform it analytically, within a controlled hierarchy of parameters, to establish a recursion condition that must be satisfied by any fixed-point distribution. The recursion condition ensures (1) that the Heisenberg SDRG is not modified by the presence of DM interactions, and (2) that the flow of DM interactions preserves the separation of scales discussed in the main text and used in the derivation of the specific heat scaling (Supplementary Note 3).
In one-dimensional systems SDRG is possible to perform analytically in certain cases [2]. 1D systems are special for a variety of reasons, most importantly here since integrating out two neighboring spins in a spin chain results in a system that is still exactly described as a spin chain. Numerical implementations of SDRG in higher dimensional systems, using various approximations, generally find that the SDRG assumptions become uncontrolled and the fate of the ultimate fixed point remains controversial [3][4][5][6].
Here we avoid such unresolved questions about the fixed point of d > 1 SDRG by restricting ourselves to a particular narrow question: how does the presence of DM interactions change the SDRG? We are able to answer this question rigorously by performing a single SDRG step analytically and noting two observations about the resulting recursion relation. First, we find that weak DM interaction does not enter the renormalization of the symmetric (e.g. Heisenberg) exchanges. The SDRG for Heisenberg exchange therefore proceeds identically as in the case without DM interaction. Second, we find that the renormalization of DM interactions preserves a parametric separation of scales between the DM and symmetric exchanges on each bond. The relative scale of the DM interactions therefore does not change and in particular also the control parameter for the first result is preserved under RG. In this precise sense, the DM interactions are merely spectators to the standard SDRG, which proceeds as usual.
Computation. To perform the single SDRG step it suffices to look at each cluster of four spins containing the strong pair. We assume weak DM interactions; this is the case for weak spin-orbit coupling as well as for strong spin-orbit coupling with some approximate microscopic or emergent inversion or mirror symmetries. As the starting point for an SDRG step we must postulate that the quantum state of a portion of the system, namely the fraction of spins that participate in the random-singlettype regime, is described by a power law distribution of exchange energies. Computing the SDRG step in the presence of weak DM interactions, we find that in 1D the DM and Heisenberg terms scale identically, while in 2D they scale at the same order with some distinctions in detail. The resulting random-singlet state has each spin paired with another into a nondegenerate frozen state, that differs from the spin-singlet state by the addition of a spin quadrupole (nematic) moment that preserves time-reversal, as is anyway required by the lack of spin rotation symmetry.
Consider a pair (1,2) of strongly coupled spins, with Heisenberg exchange J 12 as well as DM exchange D 12 , and consider weak Heisenberg as well as DM interactions between spins 1,2 and two additional spins 3,4. The ground state of H 12 = J 12 H Heis 12 + D 12 H DM 12 is no longer a spin-singlet under spin rotations, but nevertheless is a unique state. The low energy manifold has spins 1,2 in the ground state of H 12 , while the bare interactions H 0 34 gain a renormalized contribution by virtual excitations of the (1,2) excited states, as H 0 34 → H 0 34 + ∆H 34 . Here and below we take J 12 to be larger than all other Hamiltonian parameters i.e. we work within lowest order perturbation theory in 1/J 12 . Integrating out spins 1,2, we find the renormalized interaction ∆H 34 which can be expressed with coefficients J 34 , D 34 as ∆H 34 = J 34 H Heis 34 + D 34 H DM 34 . In writing the results below we specialize to the case where all DM vectors share the same orientation. The resulting expressions are directly applicable to the q = 1 case, e.g. a 2D layered system with effective reflection symmetry across each 2D lattice plane. This is the relevant analysis for the experimental data shown in the figures of the main text, as well as for the data of Ref. 7.
The renormalized contributions to the Heisenberg and DM interactions are given by the following expressions.
Only the emergent DM interaction is modified, Importantly, if we assume a parametric separation of scale between the DM interaction and the symmetric splitting on each bond, this small parameter appears in all three terms above, and thus the resulting DM interaction ratio D 34 /J 34 appears with precisely the same small parameter. The conclusion arising from this result, which remains true for arbitrary orientations of DM interactions, is thus as follows. A distribution of symmetric (e.g. Heisenberg) and antisymmetric (DM) interactions, characterized by a parameter denoting the ratio of strengths of antisymmetric and symmetric interactions on every bond, shows an SDRG evolution that obeys two important properties.
1. The distribution of symmetric exchanges (J's) evolves in exactly the same way regardless of the presence or absence of DM interactions. The DM interactions are spectators and do not modify the SDRG evolution of Heisenberg exchanges.
2. The distribution of DM interactions evolves in a way that does depend on the distribution of J's but nevertheless preserves the condition of separation of scales, namely the ratio of DM and J energy scales is preserved. The overall scale of the distribution of |D ij /J ij | remains fixed under SDRG.
This result allows us to take the known results on SDRG with Heisenberg terms and apply them directly to the present case with additional DM interactions.

Supplementary Note 3: Derivation of scaling in a magnetic field
Here we discuss in detail the impact of DM interactions on the scaling relation for the density of states, as observable by heat capacity, in a magnetic field.
The entropy distribution at low energies is set by the distribution of couplings that emerges under RG flow. Consider then the distribution of bonds picked by the strong-disorder RG. It defines the (highly correlated) distributions of three energy parameters: (1) the symmetric splitting J; (2) the antisymmetric splitting set by the components D i of the DM vector D; (3) the resulting true splitting between the lowest state and the second state. Let us denote the distributions of these parameters by P 1 , P 2 , P 3 respectively. The Hamiltonian matrix elements for the lowest two states are given above, and the resulting splittings are as follows. With no magnetic field, this splitting is In a magnetic field, this splitting (see Eq. 4) becomes where D 1 , D 2 are the components of the DM vector that lie perpendicular to the magnetic field H. The entropy distribution seen directly by C[T ] at zero field is the distribution of the splittings P 3 [E]. Self consistency within the strong-disorder RG framework requires a broad tail for P 3 [E], generally power law P 3 [E] = E −γ , with γ defined by the specific heat exponent at zero field, Now let us add a magnetic field and consider the scaling of specific heat at temperatures far below the Zeeman energy, T H. First recall the case of no DM interactions, D = 0.
Here the field picks out the distribution P 3 [H], ie P 3 at energy ∆E = H. Now we may linearize P 3 for energy splittings E near ∆E = H. It has an analytic polynomial expansion, which to first order may be written as Due to quantum mechanical level repulsion, a low excitation energy ∆E < T entails conditions on the amplitude of three parameters: ||H| − J| < T , |D 1 | < T , and |D 2 | < T . We must now consider three distinct physical scenarios, depending on the crystal symmetries or the effective approximate symmetries that emerge in SDRG, as well as potentially depending on the direction of the magnetic field. Recall that the scaling relation is C[H, T ] ∼ T 1−γ for H T , and is for T H. Case (1): D × H = 0. This is the case if symmetries constrain the DM vector to lie along a particular axis, and the magnetic field is taken to lie along the same axis. Then the specific heat scaling reduces to the case with no DM terms, and we find q = 0.
Case (2): only one component of D perpendicular to H is nonzero. This is the case if symmetries constrain the DM vector to lie along a particular axis, for any magnetic field direction away from that special axis (or for a powder average). 1 For example, this can occur when there is a mirror symmetry consisting of reflection across the 2D plane: this symmetry constrains D to lie perpendicular to the plane. Even if the crystal does not have such a symmetry microscopically, it may be reasonable to expect that, within a 2D magnetic layer, the magnetic exchanges that develop under RG will depend only on the single layer, and that such an mirror symmetry may emerge as an approximate symmetry under RG. In particular, within an approximation where each magnetic layer is treated independently as a purely 2D system, a given SDRG step will involve only the configuration of valence bonds within the 2D layer that was generated in previous RG steps, and reflection across the plane becomes a symmetry. In this case, only one component D i is relevant, and we find q = 1.
Case (3): both D 1 , D 2 are nonzero. This is the case if no symmetries constrain the DM vector. (This is also the case if symmetries constrain the DM vector to lie in a plane, and the magnetic field is oriented perpendicular to this plane.) Then both D 1 , D 2 are relevant, and we find q = 2.
Thus we find a different scaling form, with different powers of T /H, depending on the crystal or emergent symmetries as well as the direction of the magnetic field. The linear scaling of 1T-TaS 2 (q = 0) may be understood by noting that the inversion centers in the 1T-TaS 2 crystal structure forbid DM interactions from being generated not only for nearest-neighbor bonds but also for 2nd, 3rd, 4th, etc neighbors, so it may be reasonable that no sizable DM interactions arise during RG. Quadratic scaling (q = 1), as in H 3 LiIr 2 O 6 and LiZn 2 Mo 3 O 8 , can arise for generic or powder-averaged field directions through an emergent approximate mirror symmetry consisting of reflections across the 2D layer, a reasonable scenario for layered magnets. Cubic scaling (q = 2) may thus be expected to arise in valence bond materials with 3D lattices, such as Ba 2 YMoO 6 [8]. Experimental observation of the q = 2 case is left for future work.
in the density of states, 1/T 1 ∼ C/Γ with some lifetime Γ which is a priori not related to the density of states, and not known.) In general there may be two contributions to the spin-lattice relaxation: one from the minority of spins that participate in the random-singlet scaling, and another from the phase of the bulk of the sites. However generally the fluctuations will be dominated by the few sites in the tail of the energy distribution, i.e. the minority of spins in the random-singlet regime, producing a continuum of lifetimes for the spin-lattice relaxation and rendering exponential decay fits with a single lifetime T 1 difficult to interpret.