Quantum fluctuations of a nearly critical Heisenberg spin glass

Abstract

We describe the interplay of quantum and thermal fluctuations in the infinite-range Heisenberg spin glass. This model is generalized to SU (N ) symmetry, and we describe the phase diagram as a function of the spin S and the temperature T . The model is solved in the large N limit and certain universal critical properties are shown to hold to all orders in 1/N . For large S, the ground state is a spin glass, but quantum effects are crucial in determining the low T thermodynamics: we find a specific heat linear in T and a local spectral density of spin excitations, χ ′′ loc (ω) ∼ ω for a spin glass state which is marginally stable to fluctuations in the replicon modes. For small S, the spin-glass order is fragile, and a spin-liquid state with χ ′′ loc ∼ tanh(ω/2T ) dominates the properties over a significant range of T and ω. We argue that the latter state may be relevant in understanding the properties of strongly-disordered transition metal and rare earth compounds.The study of intermetallic compounds of the transition metals and rare earths has been a subject at the forefront of condensed matter physics for some time now [1,2]. A rich and complex variety of behaviors is observed in low temperature electrical and magnetic measurements, much of which lacks a comprehensive theoretical description. The complexity arises from the dominant role played by the local magnetic moments on the d and f orbitals and their interactions with each other and the itinerant charge carriers.It is convenient to begin our discussion in a phase with well-established magnetic order, in which each magnetic moment is effectively static. This static moment could be polarized in a regular manner (as in a commensurate antiferromagnet or an incommensurate spin density wave), or point in random directions (as in a spin glass state). In most realistic systems, the magnetic moment is either quite small, or has averaged to zero by dynamic quantum fluctuations: so it is useful to consider mechanisms which reduce the magnetic moment, and eventually cause it to vanish at a quantum phase transition to some paramagnetic state. Two distinct routes to such a quantum phase transition can be envisaged, and, we believe, the interplay between them is at the heart of the complexity of the problem. In the first route, originally discussed by Doniach [3], the moment is quenched by Kondo screening by the itinerant electrons: theories of such quantum critical points have been proposed [4,5,6,7] in which the predominant role of the itinerancy is to overdamp the collective magnetic excitations. In the second route, the exchange interactions between the moments play a more fundamental role: a pair of spins interacting with an antiferromagnetic exchange prefers to form a singlet valence bond, and the proliferation of such singlets can destroy the magnetic order. Analytic theories for such transitions has been made mainly for systems without quenched disorder [8]. Simple models of crossovers between these two routes have also been presented [9,10,11].This paper will...

Remarks on the Sachdev-Ye-Kitaev model

“…The temperature independent piece can be compared with the result obtained in [1] for general q (see the earlier [31] for the q = 4 case using the Sachdev-Ye model)…”

“…We assume that the system does not have a spin glass transition [31] and we work to leading order in the 1/N expansion. Though the model generically has a unique ground state, we work at temperatures which are fixed in the large N expansion, implying that we access an exponentially large number of low energy states, of order O(e αN ), α > 0.…”

Infinite-range quantum random Heisenberg magnet

“…The exact numerical solution of these equations was later obtained with quantum Monte Carlo techniques in the paramagnetic phase [13], and the solution was found to become unstable towards spin-glass order at a critical temperature T g ≈ 0.14J. The study of SU(M) extensions of this model has been treated in the limit of M→ ∞ [14][15][16], and spin-glass and spin-liquid phases where found at sufficiently low temperatures. We have recently investigated the dynamical response of the hamiltonian (1) at zero temperature with exact diagonalization techniques [10].…”

“…Moreover, its functional form is very similar to that of the quantum spin-liquid discussed in Refs. [14][15][16] for the SU(M) generalization of the Heisenberg model in the regime of large M and small S. We also investigate the behavior of the specific heat C v as a function of temperature. We find that this quantity displays a smooth maximum at a temperature T M well above the freezing temperature T g , and we link this fact to the presence of strong quantum fluctuations.…”

Strongly correlated Fermi–Bose mixtures in disordered optical lattices

“…In particular, this provides bond frustration, which is essential for the appearance of the spin glass phase and turns out to introduce severe difficulties for analytical and numerical analyses. We thus think that experiments with ultracold atoms can provide a useful quantum simulator to address challenging questions related to spin glasses such as the nature of the ordering of its ground-and possibly metastable states [4,39,40], broken symmetry and dynamics of spin glasses [5,41]. In the following two sections, we outline some general properties of spin glasses and then we apply the replica method under the constraint of a fixed magnetization and argue that this preserves the occurrence of a symmetry breaking characteristics of spin glasses in the Mézard-Parisi theory [4].…”