# Symmetry Breaking in Coupled SYK or Tensor Models

**Authors:** Jaewon Kim, Igor R. Klebanov, Grigory Tarnopolsky, Wenli Zhao  
**arXiv:** [1902.02287](https://arxiv.org/abs/1902.02287)  
**Date:** 2019-02-06  
**Summary page:** https://www.summarizepaper.com/en/paper/1902.02287v3/

## Key points

- Investigated large $N$ tensor model with $O(N)^3$ symmetry
- Explored random counterpart consisting of two coupled Sachdev-Ye-Kitaev (SYK) models
- Discovered duality relation between Hamiltonians with different values of $\alpha$
- Restricted model to range of $-1\leq \alpha\leq 1/3$
- Scaling dimension of fermion number operator is complex in range $-1\leq \alpha<0$, indicating instability of conformal phase
- Used Schwinger-Dyson equations to solve for Green functions
- Fermion number operator acquires expectation value in low-temperature phase, breaking anti-unitary particle-hole symmetry and other discrete symmetries
- Spectra calculations performed for coupled SYK models at certain values of $N_{\rm SYK}$
- Negative $\alpha$ results in gap between lowest energy states and rest of spectrum, exponential decay of zero-temperature correlation functions observed
- Small splitting between lowest states when $N_{\rm SYK}$ divisible by 4, but become degenerate in large $N_{\rm SYK}$ limit aligning with expected spontaneous breaking of a $\mathbb{Z}_2$ symmetry
- Provides insights into behavior of coupled SYK or tensor models and sheds light on symmetry breaking phenomena within these systems

## Summary

In this study, the authors investigate a large $N$ tensor model with $O(N)^3$ symmetry that includes two flavors of Majorana fermions, denoted as $\psi_1^{abc}$ and $\psi_2^{abc}$. They also explore its random counterpart, which consists of two coupled Sachdev-Ye-Kitaev (SYK) models. Each SYK model contains $N_{\rm SYK}$ Majorana fermions. The Hamiltonians in these models are assumed to be tetrahedral quartic and depend on a real coupling parameter $\alpha$. The authors discover a duality relation between two Hamiltonians with different values of $\alpha$, leading them to restrict the model to the range of $-1\leq \alpha\leq 1/3$. They find that the scaling dimension of the fermion number operator $Q=i\psi_1^{abc} \psi_2^{abc}$ is complex and takes the form $1/2 +i f(\alpha)$ in the range $-1\leq \alpha<0$, indicating an instability of the conformal phase. To further analyze the system, Schwinger-Dyson equations are employed to solve for the Green functions. The results demonstrate that in the true low-temperature phase, the fermion number operator acquires an expectation value, signifying a breaking of anti-unitary particle-hole symmetry as well as other discrete symmetries. Additionally, spectra calculations are performed for coupled SYK models at values of $N_{\rm SYK}$ where exact diagonalizations are possible. For negative values of $\alpha$, a gap is observed between the two lowest energy states and rest of spectrum, resulting in exponential decay of zero-temperature correlation functions. In cases where $N_{\rm SYK}$ is divisible by 4, there is a small splitting between these two lowest states; however they become degenerate in large $N_{\rm SYK}$ limit which aligns with expected spontaneous breaking of a $\mathbb{Z}_2$ symmetry. Overall, this study provides insights into behavior of coupled SYK or tensor models and sheds light on symmetry breaking phenomena within these systems. The findings contribute to our understanding complex quantum systems and their properties.