Phase Diagram of Planar Matrix Quantum Mechanics, Tensor, and Sachdev-Ye-Kitaev Models

TL;DR

The paper analyzes the phase structure of a fermionic planar matrix quantum mechanics with $U(N)^2\times O(D)$ symmetry in the combined large-$N$, large-$D$ limit where melonic graphs dominate, showing equivalence to tensor/SYK-like models. By solving the Schwinger-Dyson equations for the Euclidean two-point function $G(t)$, the authors identify two nonperturbative branches, a high-entropy HE phase and a low-entropy LE phase, and map a mass–temperature phase diagram featuring a first-order line that terminates at a strongly coupled critical point with asymmetric, nonmean-field exponents. They report a detailed critical-point analysis, including exponents for heat capacity and susceptibilities, and demonstrate that the would-be SYK-like IR solution does not survive in the full theory. The study also analyzes purely bosonic melonic models, finding Kazakov critical points and the absence of an SYK-like phase in both unstable and stable cases, highlighting fundamental differences between fermionic and bosonic melonic theories and informing the infrared behavior of tensor/SYK-type systems.

Abstract

We compute the phase diagram of a $\text{U}(N)^{2}\times\text{O}(D)$ invariant fermionic planar matrix quantum mechanics [equivalently tensor or complex Sachdev-Ye-Kitaev (SYK) models] in the new large $D$ limit, dominated by melonic graphs. The Schwinger-Dyson equations can have two solutions describing either a high entropy, SYK black-hole-like phase, or a low entropy one with trivial IR behavior. In the strongly coupled region of the mass-temperature plane, there is a line of first order phase transitions between the high and low entropy phases. This line terminates at a new critical point which we study numerically in detail. The critical exponents are nonmean field and differ on the two sides of the transition. We also study purely bosonic unstable and stable melonic models. The former has a line of Kazakov critical points beyond which the Schwinger-Dyson equations do not have a consistent solution. Moreover, in both models the would-be SYK-like solution of the IR limit of the equations does not exist in the full theory.

Figures (5)

• Figure 1: Phase diagram of the model \ref{['Hferm']} in the $(m,T)$ plane, in the units $\lambda=1$. The plain thick line corresponds to a first order phase transition between high and low entropy phases, whereas the dashed lines delimit the region in which both the high and low entropy solutions exist. There is a nontrivial critical point at $T= T_{\text{c}} = 0.0687_2$ and $m = m_{\text{c}} = 0.345_1$.
• Figure 2: Phase diagram of the model \ref{['Hferm']} in the $(m,Q)$ plane, in the units $\lambda=1$. We have indicated several isothermal curves. For $T<T_{\text{c}}$, these curves go through the transition region delimited by the thick solid line, where both phases coexist. The area shaded in gray is a forbidden region.
• Figure 3: Entropy $S$ (black curve) and heat capacity $C_{m}$ (red curve) at $m=m_{\text{c}}$, as functions of temperature, together with the best power-law fits near $T_{\text{c}}$ (dashed curves for $T>T_{\text{c}}$ and dotted curves for $T<T_{\text{c}}$).
• Figure 4: Plot of $\alpha=1-\frac{\partial\ln |S(T,m_{\text{c}})-S(T_{\text{c}},m_{\text{c}})|}{\partial\ln |T-T_{\text{c}}|}$ as a function of ${\frac{|T - T_{\text{c}}|}{T_{\text{c}}}}$, for both $T>T_{\text{c}}$ (dashed line) and $T<T_{\text{c}}$ (dotted line). The critical exponents $\alpha_{\pm}$ for the heat capacity are defined by the limits $\alpha_{\pm}={\lim_{T\rightarrow T_{\text{c}}^{\pm}}\alpha(T)}$.
• Figure 5: Phase diagram of the model \ref{['pot1']} in units $\lambda=1$. We have plotted the quantum instability line (thick black curve), its classical limit (dashed curve) and the locus of zero entropy for the thermodynamically unfavored solution of the SD equations (dotted curve). This solution has negative entropy to the right of this curve.