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 models, each one 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 $1/2+if(\alpha )$ 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. 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 $N_{\mathrm{SYK}}$ where exact diagonalizations are possible. For negative $\alpha$ we find a gap separating the two lowest energy states from the rest of the spectrum; this leads to 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.

arXiv comment37 pages, v2: some improvements, references added. v3: added a discussion of separation of spectra into sectors and an Appendix on zero-energy states. The version to appear in Physical Review X

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• Jackiw-Teitelboim Gravity and Rotating Black Holes
• Lindbladian dissipation of strongly-correlated quantum matter
• Euclidean wormhole in the SYK model
• Interplay between superconductivity and non-Fermi liquid at a quantum-critical point in a metal. I: The $\gamma$-model and its phase diagram at $T=0$. The case $0<\gamma <1$
• Lindbladian dynamics of the Sachdev-Ye-Kitaev model
• Quantum Phase Transition in the Yukawa-SYK Model
• Phases Of Melonic Quantum Mechanics
• Quantum criticality of granular SYK matter
• Traversable wormhole and Hawking-Page transition in coupled complex SYK models
• Excitation spectra of quantum matter without quasiparticles I: Sachdev-Ye-Kitaev models
• Intrinsic superconducting instabilities of a solvable model for an incoherent metal
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• Line of fixed points in a bosonic tensor model
• Supersymmetric Tensor Model at Large $N$ and Small $\epsilon$
• Superconducting instabilities in a spinful Sachdev-Ye-Kitaev model
• Quenched vs Annealed: Glassiness from SK to SYK
• Rotating Black Holes in AdS, Extremality and Chaos
• The chiral SYK model
• Phases of $\mathcal{N}=2$ Sachdev-Ye-Kitaev models
• Melonic Dominance in Subchromatic Sextic Tensor Models
• Non-Perturbative Defects in Tensor Models from Melonic Trees
• Phase diagram of a two-site coupled complex Sachdev-Ye-Kitaev model
• Reliability of the local truncations for the random tensor models renormalization group flow
• Chaos in Three-dimensional Higher Spin Gravity
• Disconnecting a Traversable Wormhole: Universal Quench Dynamics in Random Spin Models
• Spectrum of a Gross-Neveu Yukawa model with flavor disorder in $d=3$
• Hagedorn Temperature in Large $N$ Majorana Quantum Mechanics
• One SYK SET
• Spontaneous symmetry breaking in fermionic random matrix model
• Chiral Sachdev-Ye model: Integrability and chaos of anyons in 1+1d
• Thermalization of Randomly Coupled SYK Models
• Interacting Majorana modes at surfaces of noncentrosymmetric superconductors
• Emergent Topology in Many-Body Dissipative Quantum Matter
• Charge fluctuation entropy of Hawking radiation: a replica-free way to find large entropy
• Precise Low-Temperature Expansions for the Sachdev-Ye-Kitaev model
• Sixfold Way of Traversable Wormholes in the Sachdev-Ye-Kitaev Model
• Coupling renormalization flow in the strongly interacting regime of an asymptotically free quantum field theory in four dimensions
• Attenuating Dynamics of Strongly Interacting Fermionic Superfluids in SYK Solvable Models
• Statistic behaviors of gauge-invariance-dominated 1D chiral current random model
• Scale invariance beyond criticality within the mean-field analysis of tensorial field theories
• Lyapunov exponents in a Sachdev-Ye-Kitaev-type model with population imbalance in the conformal limit and beyond
• Towards Entanglement Entropy of Random Large-N Theories
• Instabilities of a Generalized Gross-Neveu Quantum Criticality
• A bipartite Sachdev-Ye-Kitaev model: Conformal limit and level statistics