# 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. doi: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

*Citation information provided by*

Web of Science

Web of Science

Works referenced in this record:

##
Multi-orientable group field theory

journal, April 2012

- Tanasa, Adrian
- Journal of Physics A: Mathematical and Theoretical, Vol. 45, Issue 16

##
Conformality lost

journal, December 2009

- Kaplan, David B.; Lee, Jong-Wan; Son, Dam T.
- Physical Review D, Vol. 80, Issue 12

##
Large *N* field theory and AdS tachyons

journal, April 2009

- Pomoni, Elli; Rastelli, Leonardo
- Journal of High Energy Physics, Vol. 2009, Issue 04

##
O(N) Random Tensor Models

journal, August 2016

- Carrozza, Sylvain; Tanasa, Adrian
- Letters in Mathematical Physics, Vol. 106, Issue 11

##
The Multi-Orientable Random Tensor Model, a Review

journal, June 2016

- Tanasa, Adrian
- Symmetry, Integrability and Geometry: Methods and Applications

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

journal, February 2018

- Azeyanagi, Tatsuo; Ferrari, Frank; Massolo, Fidel I. Schaposnik
- Physical Review Letters, Vol. 120, Issue 6

##
A universal result on central charges in the presence of double-trace deformations

journal, April 2003

- Gubser, Steven S.; Klebanov, Igor R.
- Nuclear Physics B, Vol. 656, Issue 1-2

##
Gauge theory correlators from non-critical string theory

journal, May 1998

- Gubser, S. S.; Klebanov, I. R.; Polyakov, A. M.
- Physics Letters B, Vol. 428, Issue 1-2

##
The soft mode in the Sachdev-Ye-Kitaev model and its gravity dual

journal, May 2018

- Kitaev, Alexei; Suh, S. Josephine
- Journal of High Energy Physics, Vol. 2018, Issue 5

##
The spectrum in the Sachdev-Ye-Kitaev model

journal, April 2016

- Polchinski, Joseph; Rosenhaus, Vladimir
- Journal of High Energy Physics, Vol. 2016, Issue 4

##
Traversable wormholes via a double trace deformation

journal, December 2017

- Gao, Ping; Jafferis, Daniel Louis; Wall, Aron C.
- Journal of High Energy Physics, Vol. 2017, Issue 12

##
Strongly $\gamma $ -Deformed $\mathcal{N}=4$ Supersymmetric Yang-Mills Theory as an Integrable Conformal Field Theory

journal, March 2018

- Grabner, David; Gromov, Nikolay; Kazakov, Vladimir
- Physical Review Letters, Vol. 120, Issue 11

##
Local criticality, diffusion and chaos in generalized Sachdev-Ye-Kitaev models

journal, May 2017

- Gu, Yingfei; Qi, Xiao-Liang; Stanford, Douglas
- Journal of High Energy Physics, Vol. 2017, Issue 5

##
Gapless spin-fluid ground state in a random quantum Heisenberg magnet

journal, May 1993

- Sachdev, Subir; Ye, Jinwu
- Physical Review Letters, Vol. 70, Issue 21

##
Stability in gauged extended supergravity

journal, December 1982

- Breitenlohner, Peter; Freedman, Daniel Z.
- Annals of Physics, Vol. 144, Issue 2

##
Colored Tensor Models - a Review

journal, April 2012

- Gurau, Razvan
- Symmetry, Integrability and Geometry: Methods and Applications

##
The large $N$ limit of superconformal field theories and supergravity

journal, January 1998

- Maldacena, Juan
- Advances in Theoretical and Mathematical Physics, Vol. 2, Issue 2

##
Fermion path integrals and topological phases

journal, July 2016

- Witten, Edward
- Reviews of Modern Physics, Vol. 88, Issue 3

##
Anti de Sitter space and holography

journal, January 1998

- Witten, Edward
- Advances in Theoretical and Mathematical Physics, Vol. 2, Issue 2

##
Conformal QED _{ d } , *F* -theorem and the *ϵ* expansion

journal, February 2016

- Giombi, Simone; Klebanov, Igor R.; Tarnopolsky, Grigory
- Journal of Physics A: Mathematical and Theoretical, Vol. 49, Issue 13

##
Bi-local holography in the SYK model

journal, July 2016

- Jevicki, Antal; Suzuki, Kenta; Yoon, Junggi
- Journal of High Energy Physics, Vol. 2016, Issue 7

##
Universal noninteger ‘‘ground-state degeneracy’’ in critical quantum systems

journal, July 1991

- Affleck, Ian; Ludwig, Andreas W. W.
- Physical Review Letters, Vol. 67, Issue 2

##
Periodic table for topological insulators and superconductors

conference, January 2009

- Kitaev, Alexei; Lebedev, Vladimir; Feigel’man, Mikhail
- ADVANCES IN THEORETICAL PHYSICS: Landau Memorial Conference, AIP Conference Proceedings

##
Critical strange metal from fluctuating gauge fields in a solvable random model

journal, September 2018

- Patel, Aavishkar A.; Sachdev, Subir
- Physical Review B, Vol. 98, Issue 12

##
A complex fermionic tensor model in d dimensions

journal, February 2018

- Prakash, Shiroman; Sinha, Ritam
- Journal of High Energy Physics, Vol. 2018, Issue 2

##
Uncolored random tensors, melon diagrams, and the Sachdev-Ye-Kitaev models

journal, February 2017

- Klebanov, Igor R.; Tarnopolsky, Grigory
- Physical Review D, Vol. 95, Issue 4

##
Conformal symmetry and its breaking in two-dimensional nearly anti-de Sitter space

journal, November 2016

- Maldacena, Juan; Stanford, Douglas; Yang, Zhenbin
- Progress of Theoretical and Experimental Physics, Vol. 2016, Issue 12

##
Strongly Correlated Metal Built from Sachdev-Ye-Kitaev Models

journal, November 2017

- Song, Xue-Yang; Jian, Chao-Ming; Balents, Leon
- Physical Review Letters, Vol. 119, Issue 21

##
A generalization of Sachdev-Ye-Kitaev

journal, February 2017

- Gross, David J.; Rosenhaus, Vladimir
- Journal of High Energy Physics, Vol. 2017, Issue 2

##
Black holes and random matrices

journal, May 2017

- Cotler, Jordan S.; Gur-Ari, Guy; Hanada, Masanori
- Journal of High Energy Physics, Vol. 2017, Issue 5

##
Prismatic large $N$ models for bosonic tensors

journal, November 2018

- Giombi, Simone; Klebanov, Igor R.; Popov, Fedor
- Physical Review D, Vol. 98, Issue 10

##
Bosonic tensor models at large $N$ and small $\epsilon $

journal, November 2017

- Giombi, Simone; Klebanov, Igor R.; Tarnopolsky, Grigory
- Physical Review D, Vol. 96, Issue 10

##
Large $q$ expansion in the Sachdev-Ye-Kitaev model

journal, January 2019

- Tarnopolsky, Grigory
- Physical Review D, Vol. 99, Issue 2

##
Quantum criticality and duality in the $\text{Sachdev-Ye-Kitaev}/{\mathrm{AdS}}_{2}$ chain

journal, May 2018

- Jian, Shao-Kai; Xian, Zhuo-Yu; Yao, Hong
- Physical Review B, Vol. 97, Issue 20

##
Solvable model for a dynamical quantum phase transition from fast to slow scrambling

journal, April 2017

- Banerjee, Sumilan; Altman, Ehud
- Physical Review B, Vol. 95, Issue 13

##
Random tensor models in the large $N$ limit: Uncoloring the colored tensor models

journal, April 2012

- Bonzom, Valentin; Gurau, Razvan; Rivasseau, Vincent
- Physical Review D, Vol. 85, Issue 8

##
An investigation of AdS2 backreaction and holography

journal, July 2016

- Engelsöy, Julius; Mertens, Thomas G.; Verlinde, Herman
- Journal of High Energy Physics, Vol. 2016, Issue 7

##
Instability of the non-Fermi-liquid state of the Sachdev-Ye-Kitaev model

journal, May 2017

- Bi, Zhen; Jian, Chao-Ming; You, Yi-Zhuang
- Physical Review B, Vol. 95, Issue 20

##
Colored Group Field Theory

journal, March 2011

- Gurau, Razvan
- Communications in Mathematical Physics, Vol. 304, Issue 1

##
Walking, weak first-order transitions, and complex CFTs

journal, October 2018

- Gorbenko, Victor; Rychkov, Slava; Zan, Bernardo
- Journal of High Energy Physics, Vol. 2018, Issue 10

##
Perturbative search for fixed lines in large- *N* gauge theories

journal, August 2005

- Dymarsky, Anatoly; Klebanov, Igor R.; Roiban, Radu
- Journal of High Energy Physics, Vol. 2005, Issue 08

##
Remarks on the Sachdev-Ye-Kitaev model

journal, November 2016

- Maldacena, Juan; Stanford, Douglas
- Physical Review D, Vol. 94, Issue 10

##
Chaos in ${\mathrm{AdS}}_{2}$ Holography

journal, September 2016

- Jensen, Kristan
- Physical Review Letters, Vol. 117, Issue 11

##
The 1/N expansion of colored tensor models in arbitrary dimension

journal, August 2011

- Gurau, R.; Rivasseau, V.
- EPL (Europhysics Letters), Vol. 95, Issue 5

##
Critical behavior of colored tensor models in the large N limit

journal, December 2011

- Bonzom, Valentin; Gurau, Razvan; Riello, Aldo
- Nuclear Physics B, Vol. 853, Issue 1

##
The Complete 1/N Expansion of Colored Tensor Models in Arbitrary Dimension

journal, July 2011

- Gurau, Razvan
- Annales Henri Poincaré, Vol. 13, Issue 3

##
Dynamical thermalization in isolated quantum dots and black holes

journal, January 2017

- Kolovsky, Andrey R.; Shepelyansky, Dima L.
- EPL (Europhysics Letters), Vol. 117, Issue 1

##
${\mathbb{Z}}_{2}$ fractionalized phases of a solvable disordered $t\text{\u2212}J$ model

journal, August 2018

- Fu, Wenbo; Gu, Yingfei; Sachdev, Subir
- Physical Review B, Vol. 98, Issue 7

##
Sachdev-Ye-Kitaev model and thermalization on the boundary of many-body localized fermionic symmetry-protected topological states

journal, March 2017

- You, Yi-Zhuang; Ludwig, Andreas W. W.; Xu, Cenke
- Physical Review B, Vol. 95, Issue 11

##
Diving into traversable wormholes: Diving into traversable wormholes

journal, May 2017

- Maldacena, Juan; Stanford, Douglas; Yang, Zhenbin
- Fortschritte der Physik, Vol. 65, Issue 5

##
Strange half-metals and Mott insulators in Sachdev-Ye-Kitaev models

journal, October 2018

- Haldar, Arijit; Shenoy, Vijay B.
- Physical Review B, Vol. 98, Issue 16

##
Models of AdS2 backreaction and holography

journal, November 2015

- Almheiri, Ahmed; Polchinski, Joseph
- Journal of High Energy Physics, Vol. 2015, Issue 11

##
Numerical study of fermion and boson models with infinite-range random interactions

journal, July 2016

- Fu, Wenbo; Sachdev, Subir
- Physical Review B, Vol. 94, Issue 3

##
Ads/CFT correspondence and symmetry breaking

journal, September 1999

- Klebanov, Igor R.; Witten, Edward
- Nuclear Physics B, Vol. 556, Issue 1-2

##
Bekenstein-Hawking Entropy and Strange Metals

journal, November 2015

- Sachdev, Subir
- Physical Review X, Vol. 5, Issue 4

##
Effects of interactions on the topological classification of free fermion systems

journal, April 2010

- Fidkowski, Lukasz; Kitaev, Alexei
- Physical Review B, Vol. 81, Issue 13

Works referencing / citing this record:

##
Line of fixed points in a bosonic tensor model

journal, June 2019

- Benedetti, Dario; Gurau, Razvan; Harribey, Sabine
- Journal of High Energy Physics, Vol. 2019, Issue 6

##
Interacting Majorana modes at surfaces of noncentrosymmetric superconductors

journal, January 2020

- Rückert, Janna E.; Roósz, Gergő; Timm, Carsten
- Physical Review B, Vol. 101, Issue 2

##
Supersymmetric tensor model at large $N$ and small $\epsilon $

journal, January 2020

- Popov, Fedor K.
- Physical Review D, Vol. 101, Issue 2

##
Solvable Strong-Coupling Quantum-Dot Model with a Non-Fermi-Liquid Pairing Transition

journal, January 2020

- Wang, Yuxuan
- Physical Review Letters, Vol. 124, Issue 1