Symmetry Breaking in Coupled SYK or Tensor Models

TL;DR

The paper investigates symmetry breaking in coupled melonic fermion systems, presenting a large-$N$ tensor model with $O(N)^3$ symmetry and two Majorana flavors alongside its random SYK counterpart. A duality confines the coupling to $-1\le\alpha\le 1/3$, and the fermion-number operator $Q=i\psi_1^{abc}\psi_2^{abc}$ exhibits a complex scaling dimension for $-1\le\alpha<0$, signaling conformal instability that the system resolves through a low-temperature phase with $\langle Q\rangle\neq 0$ and spontaneous breaking of particle-hole and related discrete symmetries, producing a mass gap. The authors derive Schwinger-Dyson equations, construct bilinear scaling dimensions via Bethe-Salpeter kernels, and corroborate their predictions with exact diagonalization at finite $N_{ m SYK}$, observing ground-state structure and degeneracies consistent with $\mathbb{Z}_2$ symmetry breaking. The results illuminate how discrete symmetry breaking arises in melonic tensor/SYK models and hint at holographic interpretations in the gapped, symmetry-broken regime.

Abstract

We study a large $N$ tensor model with $O(N)^3$ symmetry containing two flavors of Majorana fermions, $ψ_1^{abc}$ and $ψ_2^{abc}$. We also study its random counterpart consisting of two coupled Sachdev-Ye-Kitaev models, each one containing $N_{\rm SYK}$ Majorana fermions. In these models we assume tetrahedral quartic Hamiltonians which depend on a real coupling parameter $α$. We find a duality relation between two Hamiltonians with different values of $α$, which allows us to restrict the model to the range of $-1\leq α\leq 1/3$. The scaling dimension of the fermion number operator $Q=iψ_1^{abc} ψ_2^{abc}$ is complex and of the form $1/2 +i f(α)$ in the range $-1\leq α<0$, indicating an instability of the conformal phase. Using Schwinger-Dyson equations to solve for the Green functions, we show that in the true low-temperature phase this operator acquires an expectation value. This demonstrates the breaking of an anti-unitary particle-hole symmetry and other discrete symmetries. We also calculate spectra of the coupled SYK models for values of $N_{\rm SYK}$ where exact diagonalizations are possible. For negative $α$ we find a gap separating the two lowest energy states from the rest of the spectrum; this leads to exponential decay of the zero-temperature correlation functions. For $N_{\rm SYK}$ divisible by $4$, the two lowest states have a small splitting. They become degenerate in the large $N_{\rm SYK}$ limit, as expected from the spontaneous breaking of a $\mathbb{Z}_2$ symmetry.

Figures (24)

• Figure 1: Pictorial representation of the antisymmetric tensor $J_{IJKL}$.
• Figure 2: Bare propagators for the Majorana tensor fields. Each thick black solid or dashed line caries three tensor indices $a,b,c$.
• Figure 3: Interaction vertices.
• Figure 4: The leading melonic correction to the full two-point function $\textbf{G}_{11}$.
• Figure 5: Schwinger-Dyson equation for the full two-point function $\textbf{G}(\tau_{12})$.

Theorems & Definitions (1)

• Conjecture