Symmetry Breaking in Coupled SYK or Tensor Models

Journal Article · 31 May 2019 · Physical Review. X

DOI: https://doi.org/10.1103/physrevx.9.021043 · OSTI ID:1523692

Kim, Jaewon ^[1]; Klebanov, Igor R. ^[2]; Tarnopolsky, Grigory ^[3]; Zhao, Wenli ^[2]

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1. Univ. of California, Berkeley, CA (United States)
2. Princeton Univ., NJ (United States)
3. Harvard Univ., Cambridge, MA (United States)

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.

• Research Organization: Harvard Univ., Cambridge, MA (United States)
• Sponsoring Organization: USDOE Office of Science (SC), High Energy Physics (HEP); USDOE Office of Science (SC), Basic Energy Sciences (BES); National Science Foundation (NSF)
• Grant/Contract Number: SC0007870; SC0019030; PHY-1620059
• OSTI ID: 1523692
• Alternate ID(s): OSTI ID: 1610986
• Journal Information: Physical Review. X, Vol. 9, Issue 2; ISSN 2160-3308
• Publisher: American Physical Society (APS)Copyright Statement
• Country of Publication: United States
• Language: English