# Symmetry Breaking in Coupled SYK or Tensor Models

## Abstract

We study a large- $N$ tensor model with $O(N{)}^{3}$ symmetry containing two flavors of Majorana fermions, ${\psi}_{1}^{abc}$ and ${\psi}_{2}^{abc}$. We also study its random counterpart consisting of two coupled Sachdev-Ye-Kitaev (SYK) models, each containing ${N}_{\mathrm{SYK}}$ Majorana fermions. In these models, we assume tetrahedral quartic Hamiltonians which depend on a real coupling parameter $\alpha $. We find a duality relation between two Hamiltonians with different values of $\alpha $, which allows us to restrict the model to the range of $=-1\le \alpha \le 1/3$. The scaling dimension of the fermion number operator $Q=i{\psi}_{1}^{abc}{\psi}_{2}^{abc}$ is complex and of the form $1/2+if\left(\alpha \right)$ in the range $-1\le \alpha <0$, 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, which demonstrates the breaking of an antiunitary particle-hole symmetry and other discrete symmetries. We also calculate spectra of the coupled SYK models for values of ${N}_{\mathrm{SYK}}$ where exact diagonalizations are possible. For negative $\alpha $, we find a gap separating the two lowest energy states from the rest of the spectrum, leading to an exponential decay of the zero-temperature correlation functions. For ${N}_{\mathrm{SYK}}$ divisible by 4, the two lowest states have a small splitting. They become degenerate in the large- ${N}_{\mathrm{SYK}}$ limit, as expected from the spontaneous breaking of a ${\mathbb{Z}}_{2}$ symmetry.

- Authors:

- Publication Date:

- Research Org.:
- Harvard Univ., Cambridge, MA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), High Energy Physics (HEP); USDOE Office of Science (SC), Basic Energy Sciences (BES); National Science Foundation (NSF)

- OSTI Identifier:
- 1523692

- Alternate Identifier(s):
- OSTI ID: 1610986

- Grant/Contract Number:
- SC0007870; SC0019030; PHY-1620059

- Resource Type:
- Published Article

- Journal Name:
- Physical Review. X

- Additional Journal Information:
- Journal Name: Physical Review. X Journal Volume: 9 Journal Issue: 2; Journal ID: ISSN 2160-3308

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; Physics; Condensed Matter Physics; Particles and Fields; String Theory

### Citation Formats

```
Kim, Jaewon, Klebanov, Igor R., Tarnopolsky, Grigory, and Zhao, Wenli. Symmetry Breaking in Coupled SYK or Tensor Models. United States: N. p., 2019.
Web. https://doi.org/10.1103/PhysRevX.9.021043.
```

```
Kim, Jaewon, Klebanov, Igor R., Tarnopolsky, Grigory, & Zhao, Wenli. Symmetry Breaking in Coupled SYK or Tensor Models. United States. https://doi.org/10.1103/PhysRevX.9.021043
```

```
Kim, Jaewon, Klebanov, Igor R., Tarnopolsky, Grigory, and Zhao, Wenli. Fri .
"Symmetry Breaking in Coupled SYK or Tensor Models". United States. https://doi.org/10.1103/PhysRevX.9.021043.
```

```
@article{osti_1523692,
```

title = {Symmetry Breaking in Coupled SYK or Tensor Models},

author = {Kim, Jaewon and Klebanov, Igor R. and Tarnopolsky, Grigory and Zhao, Wenli},

abstractNote = {We study a large-N tensor model with O(N)3 symmetry containing two flavors of Majorana fermions, ψ1abc and ψ2abc. We also study its random counterpart consisting of two coupled Sachdev-Ye-Kitaev (SYK) models, each containing NSYK 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≤α≤1/3. The scaling dimension of the fermion number operator Q=iψ1abcψ2abc is complex and of the form 1/2+if(α) in the range -1≤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, which demonstrates the breaking of an antiunitary particle-hole symmetry and other discrete symmetries. We also calculate spectra of the coupled SYK models for values of NSYK where exact diagonalizations are possible. For negative α, we find a gap separating the two lowest energy states from the rest of the spectrum, leading to an exponential decay of the zero-temperature correlation functions. For NSYK divisible by 4, the two lowest states have a small splitting. They become degenerate in the large-NSYK limit, as expected from the spontaneous breaking of a Z2 symmetry.},

doi = {10.1103/PhysRevX.9.021043},

journal = {Physical Review. X},

number = 2,

volume = 9,

place = {United States},

year = {2019},

month = {5}

}

https://doi.org/10.1103/PhysRevX.9.021043

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