Hidden energy scale in geometrically frustrated systems: are quantum spin liquids achievable?

S. V. Syzranov and A. P. Ramirez Physics Department, University of California Santa Cruz, Santa Cruz, CA 95064

Despite the enormous interest in quantum spin liquids, their experimental existence still awaits broad consensus. In particular, quenched disorder may turn a specific system into a spin glass and possibly preclude the formation of a quantum spin liquid. Here, we demonstrate that the glass transition among geometrically frustrated magnets, a materials class in which spin liquids are expected, differs qualitatively from conventional spin glass. Whereas conventional systems have a glass temperature that increases with increasing disorder, geometrically frustrated systems have a glass temperature that increases with decreasing disorder, approaching, in the clean limit, a finite value. This behaviour implies the existence of a hidden energy scale (far smaller than the Weiss constant) which is independent of disorder and drives the glass transition in the presence of disorder. Motivated by these observations, we propose a scenario in which the interplay of interactions and entropy in the disorder-free system yields a temperature dependent magnetic permeability with a crossover temperature that determines the hidden energy scale. The relevance of this scale for quantum spin liquids is discussed.

1

In magnetic systems, geometrical frustration [1-3] is essential to avoid long-range order and realise quantum-spin-liquid states. As a result, such spin liquids are actively sought in materials with antiferromagnetic (AF) interactions on geometrically frustrating lattices [4], exemplified by the pyrochlore, Kagome and triangular structures. Quenched disorder (e.g. impurities, vacancies, dislocations), common in real materials, may fundamentally alter the nature of the ground state in geometrically frustrated (GF) systems, preclude the formation of a spin liquid, and induce a spin-glass (SG) state, often at extremely low defect density [3]. Understanding the role of disorder is, therefore, crucial for understanding the stability of the quantum spin liquid. Whenever a system enters an SG state, its properties are believed to be universal and similar for both conventional (non-GF) and GF spin glasses. For example, the SG transition on a pyrochlore lattice has been argued [5] to belong to the same universality class as the conventional SG transition in non-GF systems. Glass transitions in GF systems with either fully occupied lattices and bond disorder [6-8] or nearly fully occupied lattices [9] display, similar to conventional SGs, a cusp in the magnetic susceptibility 𝜒𝑇, the absence of a specific heat anomaly, temperature- hysteresis in 𝜒𝑇 and divergence of the non-linear susceptibility at 𝑇 [9, 10]. While the glass transition occurs in both GF and non-GF systems due to the freezing of the magnetic moments of impurity atoms or vacancies in a background medium (parent compound), the properties of this medium are dramatically different in these two classes of systems. In GF magnets, the parent compound may lead, for example, to a large quadratic-in-temperature specific heat 𝐶𝑇 [9, 11], attributed to singlet-like Halperin-Saslow modes [12, 13], as opposed to the linear dependence 𝐶𝑇 observed in non-GF spin glasses [14]. The properties of the background medium in GF glasses thus exhibit rich physics of non-magnetic excitations, which are sensitive to quenched disorder on their own. Thus, the interplay of disorder, frustration, and the background medium is a critical aspect of spin-liquid physics. In this paper, we compare data on spin-glass transitions that vary with the degree of disorder in both GF systems and non-GF systems. With increasing the concentration of randomly located magnetic impurities, the susceptibility, 𝜒𝑇, grows in both classes of materials. In GF systems, however, this trend is accompanied by a decrease of the critical temperature 𝑇, in contrast with non-GF glasses as well as with common intuition. The extrapolation of the critical temperature 𝑇 in GF materials to the limit of zero quenched disorder gives a finite value, 𝑇∗, on the order of several Kelvin and far below the Weiss constant 𝜃. This signals the existence of a hidden energy scale that determines the critical temperature of the disorder-driven transition in the limit of vanishing disorder. This energy scale is a property of the disorder-free system. This observation calls into question the observability of quantum spin liquids since a glass transition at a temperature independent of the amount of disorder (for achievable concentrations of impurities) may preclude the formation a quantum spin liquid. In this paper, we propose a scenario of the transition, consistent with the available experimental data, in which the interplay of interactions and the entropy of the impurity spins leads to a crossover in the effective magnetic permeability of the medium, which in turn drives the transition.

2 Summary of observations

Figure 1. Behaviour of susceptibilities, 𝜒𝑇 and 𝜒𝑇 at the glass transition temperature, 𝑇, in conventional [(a) and (c)] and geometrically frustrated [(b) and (d)] systems. (a) and (b): The susceptibilities 𝜒𝑇 of (Eu,Sr)S [15] and NiGa2S4 [16] as a function of temperature for various vacancy concentrations. The lines have been rendered from digitized data in the referenced publications and the 𝑇 values are plotted in the lower panels [17, 18]. (c) and (d): The susceptibilities 𝜒𝑇 at 𝑇 vs. 𝑇 for conventional and frustrated spin glasses. The reference sources for these data are provided in Table 1.

3 In order to depict the differences between GF and non-GF materials, we plot in Fig. 1 the dimensionless susceptibility 𝜒𝑇 versus 𝑇, using currently available experimental data for families of compounds with variable disorder [6, 9, 16, 19-27]. Here we consider only strongly GF systems where 𝑓𝜃/𝑇 ≿ 10 [28]. We note that the microscopic nature of disorder, associated with the structural parameter x, may vary in different systems. Most often, 𝑥 represents the concentration of spin vacancies due to the presence of non-magnetic impurity atoms [29-32], but it can also indicate substitution on a non-magnetic site [5, 22, 33]. In these cases, if the size of the impurity atoms differs significantly from that of the host atoms, exemplified by La3+ in Y2Mo2O7, chemical substitution leads to random strain, which in turn leads to random exchange interactions. Thus, the series Y2(1-x)La2xMo2O7 is not included in our survey because 𝜃 changes sign from antiferromagnetic to ferromagnetic as 𝑥 increases from 0 to 0.3, indicating that introduction of the La3+ ion for Y3+ significantly alters the character of the exchange interactions 2+ 2+ [34]. At the same time, we include Zn1-xCdxCrO4 where Cd replaces Zn because strong frustration is preserved in this series, indicating that the exchange interactions are negligibly altered [22, 35]. Notwithstanding this example, the systematics of the GF and non-GF systems as presented in Fig. 1 are faithful complements in that disorder is introduced primarily via spin vacancies (GF) or spins (non-GF). The comparison in Fig. 1 demonstrates the distinct difference in behaviour between GF and non-GF systems and, apart from confining the survey to strongly GF systems, relies on no assumptions about the nature of disorder.

We now consider the behaviour of 𝑇 not as a function of 𝜒𝑇 but as a function of disorder for systems in which 𝑥 represents the nominal defect concentration. In non-GF systems, 𝜒𝑇 comes from magnetic impurities, rather than from the parent compound, and is well described by the Curie-Weiss law 𝜒𝑇 𝐶/𝑇 𝜃 for temperatures 𝑇𝑇. In such conventional glasses, the Curie constant 𝐶 is proportional to the concentration of impurity spins. Thus increasing the concentration of magnetic impurities leads to larger magnetic susceptibilities and stronger interactions between impurity spins, resulting in higher transition temperatures 𝑇. This intuitive 2+ trend is observed in conventional spin glasses, exemplified by in EuxSr1-xS with Eu impurity ions (Figure 1a and c).

As already mentioned, GF SGs display the opposite trend: 𝑇 decreases with increasing 2+ susceptibility and defect concentration, exemplified by Ni1-xZnxGa2S4 where non-magnetic Zn 2+ ions substitute for Ni ions (Fig 1b, d). A decrease of 𝑇 with such an increasing spin vacancy concentration, 𝑥, had been commented on previously for the case of SrCr9xGa12-9xO19 [9, 20, 36]. Here in Figure 1d we show that the more robust measure of increasing 𝜒𝑇 with increasing spin vacancy concentration is a generic feature of GF SGs (excluding families which develop AF order either in addition to or instead of SG freezing).

In order to establish a constitutive relationship between 𝑇 and 𝑥 similar to non-GF systems, we consider those GF systems in which only the spin-vacancy concentration, 𝑥, is varied. As shown ∗ in Fig. 2, for such systems, 𝑇 increases with decreasing 𝑥, and reaches a finite value, 𝑇 , in the limit of vanishing spin concentration for a few of the systems. This behaviour, like that of 𝜒𝑇, is in stark contrast to that of non-GF systems and signals the existence of a ``hidden’’ energy scale, ∗ 𝑇 , on the order of several Kelvin and at least an order of magnitude below 𝜃. This energy scale determines the transition temperature in GF systems and, because it persists in the clean limit, it is a property of the parent compound rather than of impurity spins.

4 In Table 1, we summarise the values of the hidden energies 𝑇∗ and other materials parameters for those nearly-stoichiometric GF magnets, together with the values of the Weiss constant 𝜃, frustration parameters, and spin values. The Weiss constant 𝜃 characterises the strength of the exchange interactions in the system and is usually on the order of the AF coupling between the spins. Table 1 illustrates that the hidden energy scale 𝑇∗ is significantly exceeded by the Weiss constant 𝜃 and, thus, by the exchange interaction.

ZnCr Ga O 0.8 2x 2-2x 4 BaCr Ga O 9-9x 3+9x 19 Ni Zn Ga S 1-x x 2 4 Mn Cd Te 1-x x Y Mo Ti O 2 2(1-x) 2x 7 0.6 Fe Ga TiO 3-3x 3x-1 5 Na Ir Ti O 4 3(1-x) 3x 8

0.4

0.2 - nominal vacancy concentration vacancy - nominal x T*

0.0 0.1 1 10 100

Tg (K)

Figure 2. The critical temperature 𝑇 of the glass transition in GF materials as a function of the disorder for those materials where disorder corresponds to introduction of spin vacancies. The non- vanishing of 𝑇 in the limit of vanishing disorder for NiGa2S4, Na4Ir3O8 and Y2Mo2O7 suggests the existence of a hidden energy scale 𝑇∗.

5 ∗ GF compound lattice S 𝜽𝑾(K) 𝑻𝒈 (K) f 𝑻 Ref.

ZnCr1.6Ga0.4O4 spinel 3/2 115 2.4 48 n.e. [19, 31]

Zn0.95Cd0.05Cr2O4 spinel 3/2 500 20 10 50 n.e. [22, 35]

SrCr8Ga4O19 Kagome (2D) 3/2 515 3.5 147 n.e. [9, 20]

BaCr8.1Ga3.9O19 Kagome (2D) 3/2 695 4.9 142 n.e. [37]

NiGa2S4 triangular (2D) 1 80 8.0 10 8 0.2 [11, 16]

Mn0.53Cd0.47Te fcc 5/2 292 15 3 19 n.e. [21]

Na4Ir3(1-x)Ti3xO8 hyperkagome 1/2 570 6.0 100 6.0 [26, 27]

Y2Mo2O7 pyrochlore 1 200 23.5 8.5 25 1 [6, 24]

(Fe,Ga)2TiO5 pseudobrookite 5/2 900 10 55 16 n.e. [23, 25] Table 1. Strongly geometrically frustrated magnets and their magnetic lattice, atomic spin (S), AF Weiss constant 𝜃, glass transition temperature 𝑇, frustration parameter 𝑓𝜃/𝑇 and the hidden energy scale 𝑇∗, for compounds pure enough to allow an estimate. Compounds for which T* is not estimated due to insufficient disorder range are indicated by n.e. Entries not having an error estimate can be considered accurate to one digit in the least significant place. The compounds listed are the lowest-disorder members of their respective dilution series, data for which can be found in the references.

Phenomenology of low-energy spin fluctuations in frustrated systems As is clear from the preceding discussion, the main difference between GF and non-GF spin glasses is the medium through which the defects interact. Whereas in non-GF SGs the medium is non- magnetic, in GF SGs the medium is characterised as a dense population of spins, the low-energy spin dynamics of which allow for a description in terms of the ``Coulomb phase’’ [38, 39]. In such a description, a coarse-grained classical field B (sometimes referred to as an effective electric field [4, 40]) related to the polarisations of the spins in a disorder-free frustrated system vanishes through any closed surface, leading to the condition 𝑑𝑖𝑣 𝐁0. Such a vanishing corresponds to the minimum of the AF exchange interactions on all bonds in the lattice. Spins can change their orientation under this constraint, leading to a huge degeneracy of the classical ground states in the absence of other interactions. Flipping an open chain of spins in a Coulomb-phase medium violates the condition 𝑑𝑖𝑣 𝐁0 at the end of this chain and creates, thus, two monopoles with opposite charges at the ends of this chain. Creating such monopole excitations requires large energies of order of the AF coupling and will not be central for our arguments here. Defects with spins that mismatch the spins of the parent compound also correspond to monopole charges 𝑄 at the locations of the defects.

6 The free energy of the coarse-grained field B of the spins may be assumed to be quadratic in the limit of small B and includes the temperature-independent energy contribution 𝑩𝑑𝒓 (measured in appropriate units) as well as the ``entropic’’ contribution [41] 𝑩𝑑𝒓 linear in the temperature T, scaled by a constant prefactor 1/𝑇. The temperature-independent part may come, for example, from the magnetic dipole-dipole interactions between the spins or from the interplay of dipolar and exchange interactions [42]. Another possible mechanism for generating this energy is quantum tunnelling [43-45] between classical ground states of the regions with 𝑑𝑖𝑣 𝐁0. Such a tunnelling may lift the degeneracy of the classical ground states. While it does not require the existence of an energy scale other than the exchange energy, the amplitude of the tunnelling is in general exponentially suppressed by the number of spins [43-45] that need to be rotated in order to arrive from one classical ground state to another. The exact nature of the energy of the field B is not important for our consideration and cannot be uniquely inferred from the available data, which is why we leave the investigation of the origin of this energy for future studies. The entropic contribution 𝑩𝑑𝒓 accounts for the decrease of the entropy with increasing B. Combining all the discussed contributions gives the free energy of low-energy excitations in a GF material in the form 1 𝑇 𝑇 1 𝐹𝜙, 𝑩 𝑩𝑑𝒓 𝜙 𝑑𝑖𝑣 𝑩 𝑑𝒓 8𝜋 𝑇 4𝜋 (1) ∑ 𝜙𝒓 𝐸 𝑄 𝒉⋅ 𝑩 𝑑𝒓 ℎ 𝑩 𝑑𝒓,

where 𝑄 and 𝒓 are the charges and the locations of the monopoles introduced by the defects; 𝐸 is the energy of the defect, which may depend on the state of the defect spin; 𝒉𝒓 and ℎ𝒓 are weak quenched random fields that account for disorder other than monopole charges created by magnetic vacancies, e.g. random strain. In Eq. (1), we have introduced the field 𝜙, minimisation with respect to which enforces the Gauss theorem in the form 𝑑𝑖𝑣 𝑩4𝜋 ∑ 𝑄𝛿𝒓 𝒓 . For states that minimise the free energy (1), the scalar field 𝜙 plays the role of the potential of the Coulomb field, with 𝛁𝜙 𝑩4𝜋𝒉 ℎ 𝑩, where the random fields 4𝜋𝒉 and ℎ 𝑩 come from the fluctuations of the effective magnetisation and magnetic permeability. The free energy (1) describes a system of randomly located magnetic charges (defects) in a disordered medium with the average magnetic permeability determined by the properties of the disorder-free system and given by 𝑇 (2) 𝜇𝑇 . 𝑇𝑇 The permeability (2) displays a crossover from 𝜇𝑇 1 at low temperatures 𝑇≪𝑇 to strongly suppressed permeability 𝜇𝑇 ∝ 1/𝑇 at high temperatures 𝑇≫𝑇. We argue below that this crossover may drive the glass transition in sufficiently disordered samples. We emphasise that the permeability 𝜇𝑇 given by Eq. (2) characterises the interaction between excitations in the system and is in general unrelated to the magnetic susceptibility 𝜒𝑇, which

7 characterises response to the external magnetic field. The energy 𝐸𝐸 ∑, of a |𝒓𝒓| system of monopoles in the material consists of a temperature-independent contribution 𝐸 determined by the exchange coupling between spins and the interaction contribution ∝ 𝜇𝑇. The temperature dependence of the permeability 𝜇𝑇 can thus be independently verified via spin resonance spectroscopy of multi-charge, e.g, dipole, excitations in the system. The free energy (1), which we used to derive the permeability (2), describes the system in terms of the classical coarse-grained field B which varies continuously. However, in the presence of magnetic dipolar interactions, the system may have excitations with discrete energies and substantially suppressed concentrations at temperatures smaller than the characteristic value of those interactions. The temperature dependence of the permeability 𝜇𝑇 may then deviate from Eq. (2). This dependency, however, will still display a crossover between 𝜇1 at low temperatures and strongly suppressed 𝜇 at high temperatures and our results will hold qualitatively.

The crossover in the permeability (2) leads to a crossover in the ratio 𝛿𝐸/𝑇∝𝜇𝑇 𝑇 of the fluctuation of the interaction energy of impurity spins to the temperature T. In weakly disordered samples, this ratio is small at high temperatures 𝑇≫𝑇 and grows rapidly for 𝑇≪𝑇, favouring the glass state. The crossover in the magnetic permeability (2) suggests that the glass transition, for sufficiently strong disorder, occurs at a temperature 𝑇∗ on the order of 𝑇. Similarly, the crossover will drive the glass transition if the disorder is dominated by the quenched parameters 𝒉 and ℎ in Eq. (1) rather than randomly located impurities. The existence of a glass state requires sufficiently strong disorder that may come from randomly located impurities creating monopole charges as well as other sources of randomness represented by the quenched parameters 𝒉 and ℎ in Eq. (1). We emphasise that the hidden energy scale 𝑇∗, the glass transition temperature at zero impurity concentration, cannot be explained solely by the residual disorder distinct from impurities because the glass transition temperature grows with decreasing disorder. This trend, the exact Figure 3. Conjectured dependence of the critical microscopic mechanism of which we temperature 𝑇 of the glass transition as a function of leave for future studies, suggests also a the amount of quenched disorder, which may be significant role of the disorder-free represented by impurity spins as well as other sources, medium for the ``hidden’’ energy scale in a GF magnetic system. The transition temperature 𝑇∗. vanishes in the limit of zero disorder, but the extrapolation from the range of experimentally We conjecture that GF spin-glass states accessible disorder strength suggests a finite glass- observed to date have been observed in transition temperature 𝑇∗ in a disorder-free sample. sufficiently disordered materials, with the glass-transition temperature

8 determined by the crossover in the effective magnetic permeability of the medium. While the glass transition temperature vanishes with vanishing disorder, as shown in Figure 3, the extrapolation of the transition temperature from the range of disorder accessed to date gives a finite value 𝑇∗. A careful verification of the scenario of the glass transition proposed here will require a thorough experimental investigation of the details of the transition or an advance in the synthesis of clean frustrated magnetic systems. As mentioned, quantum spin liquids are expected to be found in geometrically frustrated (GF) systems with s=½ spins. Indeed, much experimental effort on s=½ GF systems has resulted in observations consistent with a spin liquid interpretation [4, 46-50], but questions remain. Some s=½ GF systems, such as Herbertsmithite [51], H3LiIr2O6 [52, 53] and Cu3Zn(OH)6FBr [54], show no signs of spin freezing at low temperatures. Other systems, such as Na4Ir3O8 [26, 27] and YbMgGaO4 [55], exhibit spin glass freezing. Some of the materials not exhibiting spin glass behaviour are found to have field-dependent specific heat that scales in a manner suggestive of a collection of singlets with random pairing energies [53, 56, 57]. While such states would seem to be inimical to quantum spin liquid, they illustrate the possibility of other types of ground states of disordered GF systems. These results call for a thorough investigation of the role of quenched disorder in GF systems. While weak disorder is perturbatively irrelevant in a GF material [58], non-perturbative fluctuations of the locations of the impurities and inter-spin couplings may still result in a glass state for arbitrarily weak average strength of disorder. Importantly for future studies, however, knowledge of the existence of 𝑇∗ may enable a systematic approach to making GF materials purer specifically by pursuing synthetic routes that yield progressively smaller 𝜒𝑇 anomalies. Similar to the use of mobility as a proxy for purity in 2DEG systems [59], the inverse magnitude of 𝜒𝑇 may be used to measure the approach towards a quantum-spin-liquid-bearing material. Conclusion In this paper, we have analysed known SG transitions in GF magnets and showed that they differ qualitatively from those in conventional glasses. The freezing temperature of such SG transitions grows with decreasing the amount of quenched disorder in the system and reaches a finite value in the limit of vanishing disorder. This observation calls into question the achievability of quantum spin liquids. Indeed, if a glass transition occurs at a temperature determined by the properties of the medium and not by the amount of disorder, a system may always be entering the spin-glass state instead of becoming a spin liquid. Here, we have also proposed a phenomenological model for glass transitions in geometrically frustrated systems. In this model, the interactions between impurity spins in frustrated systems capable of supporting a Coulomb phase are strong below a characteristic temperature 𝑇 and suppressed above this temperature. This leads to a crossover in the magnetic permeability of the frustrated medium which drives the glass transition in GF systems for experimentally achievable impurity concentrations. We conjecture that the SG transition temperature vanishes at strictly zero disorder. However, for realistically available impurity concentrations, glass freezing is driven by the crossover in the magnetic permeability rather than by the amount disorder. As a result, an extrapolation of the SG transition temperature to the limit of zero disorder from currently achievable impurity

9 concentrations leads to a finite value. The phenomenological picture of the SG transition developed here calls for a thorough investigation of the critical features of the transition and for synthesis of cleaner GF materials. Acknowledgements. We have benefited from useful discussion with S. Brown, J.T. Chalker, S.-H. Lee, R. Moessner, A.W. Sandvik and A.P. Young. APR would like to acknowledge support by the U.S. Department of Energy Office of Basic Energy Science, Division of Condensed Matter Physics grant DE-SC0017862.

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