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 , the absence of a specific heat anomaly, temperature- hysteresis in and divergence of the non-linear susceptibility at [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 [9, 11], attributed to singlet-like Halperin-Saslow modes [12, 13], as opposed to the linear dependence 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, , grows in both classes of materials. In GF systems, however, this trend is accompanied by a decrease of the critical temperature , in contrast with non-GF glasses as well as with common intuition. The extrapolation of the critical temperature in GF materials to the limit of zero quenched disorder gives a finite value, ∗, on the order of several Kelvin and far below the Weiss constant . 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, and at the glass transition temperature, , in conventional [(a) and (c)] and geometrically frustrated [(b) and (d)] systems. (a) and (b): The susceptibilities 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 values are plotted in the lower panels [17, 18]. (c) and (d): The susceptibilities at vs. 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 versus , 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 / ≿ 10 [28]. We note that the microscopic nature of disorder, associated with the structural parameter x, may vary in different systems. Most often, 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 changes sign from antiferromagnetic to ferromagnetic as 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 not as a function of but as a function of disorder for systems in which represents the nominal defect concentration. In non-GF systems, comes from magnetic impurities, rather than from the parent compound, and is well described by the Curie-Weiss law / for temperatures . In such conventional glasses, the Curie constant 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 . 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: 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 with such an increasing spin vacancy concentration, , 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 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 and similar to non-GF systems, we consider those GF systems in which only the spin-vacancy concentration, , is varied. As shown ∗ in Fig. 2, for such systems, increases with decreasing , and reaches a finite value, , in the limit of vanishing spin concentration for a few of the systems. This behaviour, like that of , is in stark contrast to that of non-GF systems and signals the existence of a ``hidden’’ energy scale, ∗ , on the order of several Kelvin and at least an order of magnitude below . 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 ∗ and other materials parameters for those nearly-stoichiometric GF magnets, together with the values of the Weiss constant , frustration parameters, and spin values. The Weiss constant 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 ∗ is significantly exceeded by the Weiss constant 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.