Sachdev-Ye-Kitaev Model in a Quantum Glassy Landscape

Abstract

We study a generalization of `Yukawa models' in which Majorana fermions, interacting via all-to-all random couplings as in the Sachdev-Ye-Kitaev (SYK) model, are parametrically coupled to disordered bosonic degrees of freedom described by a quantum $p-$spin model. The latter has its own non-trivial dynamics leading to quantum paramagnetic (or liquid) and glassy phases. At low temperatures, this setup results in SYK behavior within each metastable state of a rugged bosonic free energy landscape, the effective fermionic couplings being different for each metastable state. We show that the boson-fermion coupling enhances the stability of the quantum spin-glass phase and strongly modifies the imaginary-time Green's functions of both sets of degrees of freedom. In particular, in the quantum spin glass phase, the imaginary-time dynamics is turned from a fast exponential decay characteristic of a gapped phase into a much slower dynamics. In the quantum paramagnetic phase, on the other hand, the fermions' imaginary-time dynamics get strongly modified and the critical SYK behavior is washed away.

Figures (13)

• Figure 1: SYK Dots in the Energy Landscape of the Spin-Glass Model: A schematic cartoon of the typical energy landscape of the $p$-spin-glass model is shown. The complex free-energy landscape consists of equilibrium states (gapped), an exponential number of metastable states, and marginal (threshold) states that are gapless. Each SYK dot is located in a different valley of this landscape, with a distinct fluctuating glassy background determined by the specific state $a, b, c, \ldots$. At very low $T$, the SYK dots are expected to occupy the low-lying minima (equilibrium states) in the semi-classical limit of the $p$-spin-glass model.
• Figure 2: Hierarchical (Parisi) structure of configurations. Left: classical. Within a state a,b,c ... almost all configurations are at the same distance from one another $q_{EA}$, a known property of high-dimensional sets. Similarly, the mutual distance between any two configurations in different states is also the same, but different from $q_{EA}$ . Right: quantum. The situation is the same, but now we have a 'collar' of configurations, the imaginary-time trajectory, whose 'beads' are labeled by $t$. The organization now extends one step inwards: two different times of a trajectory are at the same distance from any time of a trajectory in another state.
• Figure 3: Effect of fermion-boson coupling on Edwards-Anderson and break-point parameter: Panels (a) and (b) show the breakpoint parameter $m$ and the Edwards–Anderson parameter $q_{\rm EA}$, respectively, as functions of temperature $T$ for various values of the quantum parameter $\Gamma$, with the coupling strength fixed at $V/J = 5$. Solid lines correspond to the decoupled spin-glass system, while markers indicate results from the fully coupled boson–fermion system. Panels (c) and (d) display $m$ and $q_{\rm EA}$, respectively, as functions of the boson–fermion coupling strength $V$ (with $J = 1$), for the same values of $\Gamma$ as in (a), at fixed temperature $T = 0.01$. As before, solid lines represent the decoupled case, and markers denote the fully coupled system.
• Figure 4: Upper panel (a, b) -- Bosonic Green's function in thermodynamic spin-glass phase: The regular part of the bosonic Green's function, defined as $Q_{\rm reg}(\tau) = Q_{aa}(\tau) - q_{\rm EA}$, is plotted as a function of $\tau/\beta$ for various temperatures $T$. Panel (a) corresponds to quantum fluctuation parameter $\Gamma = 2.0$ and coupling strength $V/J = 1.0$, while panel (b) shows results for $V/J = 5.0$. In the legend, $Q_0$ denotes the bosonic Green's function in the absence of fermion–boson coupling ($V = 0$), and $Q$ corresponds to the fully coupled case ($V \ne 0$). Lower panel (c, d) -- Fermionic Green's function: The corresponding fermionic Green's functions are shown in panels (c) and (d), with parameters matching those in (a) and (b), respectively. In the legend, $G$ represents the fermionic Green's function of the fully coupled boson–fermion system, and $G_0$ refers to the SYK Green's function computed with a renormalized effective coupling $\mathcal{J}_{\rm eff} = V \cdot \langle Q^2(\tau) \rangle$.
• Figure 5: Crossover temperature $T^*(\Gamma,V)$ as a function of $\Gamma$ for several coupling strengths $V$, characterizing the departure from SYK behavior in the thermodynamic spin-glass phase.