Quantum Fluctuations of a Nearly Critical Heisenberg Spin Glass

TL;DR

The paper analyzes the infinite-range quantum Heisenberg spin glass with SU$(N)$ symmetry in the large-$N$ limit, solving a self-consistent, replica-structured single-site problem that reveals a spin-liquid paramagnetic phase at small $S$ and a spin-glass phase at larger $S$, with two distinct transitions $T_{sg}^{c}$ and $T_{sg}^{eq}$. In the paramagnetic regime, spin-liquid states exhibit slow, non-Fermi liquid dynamics with $G( au) o 1/ au^{1/2}$ and a gapless $ ext{Im} ext{χ}_{loc}(oldω)$; local-moment paramagnets emerge at larger $S$ with Curie-like behavior and are interpreted as mean-field artifacts. In the spin-glass phase, a one-step replica-symmetry-breaking solution yields competing marginal (replicon) and equilibrium solutions, with the replicon criterion selecting a gapless spectrum at $oldΘ_R=1/old√3$ and the equilibrium criterion yielding a gapped spectrum at $oldΘ_{eq}\napprox 0.442$. The thermodynamics shows a linear specific heat at low temperature due to gapless quantum glassy excitations, and the results have potential relevance to strongly disordered metals and heavy-fermion systems, while leaving open questions about the dynamical origin of marginal stability.

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 linear in frequency 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 dominates the properties over a significant range of temperatures and frequencies. We argue that the latter state may be relevant in understanding the properties of strongly-disordered transition metal and rare earth compounds.

Figures (8)

• Figure 1: Phase diagram of the mean field bosonic model. There is a spin glass phase below the spin glass temperature $T_{sg}^{c}$, which is determined with the marginality condition (See Section \ref{['SpinGlassPhases']}). $T^{eq}_{sg}$ is the spin glass temperature as determined with the stationarity criterion.
• Figure 2: $S$ as a function of $\theta$ for the bosonic model. The solid line is given by relation (\ref{['ValueOfTheta']}) and the points were obtained previously from a numerical solution of the saddle-point equation at zero temperature SachdevYe.
• Figure 3: Entropy as a function of the size of the spin ($q_{0}$) in the fermionic model.
• Figure 4: $\chi_{loc} (\tau )$ extracted from a numerical solution of saddle point equation (\ref{['EqBase']}) in imaginary time, for $S=1$, $J=1$. The solid curve is low temperature ($J\beta =10$), the dashed curve is high temperature ($J\beta =.1$); for intermediate temperatures, the curves interpolate between the two.
• Figure 5: Spectral densities for $G_{b}$ (left) and for the local susceptibility $\chi_{loc}$ (right), for high temperature (top) and a very low temperature (bottom). These results are extracted from a numerical solution of saddle point equation (\ref{['EqBase']}) in real frequencies.