Crossover to Sachdev-Ye-Kitaev criticality in an infinite-range quantum Heisenberg spin glass

Abstract

We study the equilibrium dynamics of an infinite-range quantum Heisenberg model with random couplings, in which local magnetic moments arise from $\mathcal{N}_f$ flavors of spinful fermions. We employ an expansion in $\mathcal{N}_f$, which controls the strength of quantum fluctuations, and self-consistently include $1/\mathcal{N}_f$ corrections to the Luttinger-Ward functional. In the large-$\mathcal{N}_f$ limit, where quantum fluctuations are weak, the high- and low-temperature phases are respectively paramagnetic and spin glass ordered, with a transition temperature independent of $\mathcal{N}_f$. For small numbers of fermionic flavors, however, quantum fluctuations substantially suppress the ordering temperature. We show that this behavior reflects the proximity of the system to a Sachdev-Ye-Kitaev (SYK) phase, where both fermionic and spin spectral densities display critical behavior over a broad range of finite frequencies, with the latter exhibiting the scale-invariant form $χ''(ω)\sim \operatorname{sgn}(ω)$. At the lowest energies and temperatures, spin-glass dynamics ultimately take over, producing a universal sub-Ohmic dynamical spin susceptibility $χ''(ω)\sim \operatorname{sgn}(ω)\sqrt{|ω|}$. Our results establish a minimal framework for understanding dynamical crossovers between SYK criticality and spin-glass ordering.

Figures (4)

• Figure 1: The phase diagram of the system in terms temperature ($T$) and number of fermionic flavors ($\mathcal{N}_f$). The latter controls the strength of quantum fluctuations.
• Figure 2: Diagrammatic contributions to the fermionic self-energy, at leading non-vanishing order in the $1/\mathcal{N}_f$ expansion, include an infinite series which is resummed in this work.
• Figure 3: Numerical solutions of the self-consistent equations. (a) Fermionic spectral density for $\mathcal{N}_f=5$ in the spin glass ($T/J=0.21$) and paramagnetic (rest of the curves) phases. (b) Spin spectral density for $\mathcal{N}_f=5$ limit, showing Ohmic (black dashed) and sub-Ohmic (red dash-dotted) low-energy behavior in the paramagnetic and spin-glass phases, respectively. (c) Temperature dependence of the static magnetic susceptibility for different values of $\mathcal{N}_f$. The red curve shows the analytical result obtained for $\mathcal{N}_f\to\infty$. For large values of $\mathcal{N}_f$, the results overlap, with a plateau at $J|\chi(0)|=1$ appearing below the critical temperature $T_c=J/4$ (dotted line). For small values of $\mathcal{N}_f$, the transition is shifted to lower temperatures due to quantum fluctuations. (d) Spin-glass order parameter as a function of temperature for different values of $\mathcal{N}_f$. We observe the saturation and suppression of $q$ at large and small values of $\mathcal{N}_f$, respectively.
• Figure 4: Emergence of SYK physics in the small-$\mathcal{N}_f$ limit. (a) Fermionic spectral density, showing the emergence of power-law behavior at low temperatures. (b) Spin spectral density, exhibiting a crossover from an SYK-like plateau at finite $\omega$ to linear and square-root behavior at low frequencies in the paramagnetic and spin-glass phases, respectively. In both panels, $\mathcal{N}_f=1/10$.