TY - JOUR

T1 - Symmetry Breaking in Coupled SYK or Tensor Models

AU - Kim, Jaewon

AU - Klebanov, Igor R.

AU - Tarnopolsky, Grigory

AU - Zhao, Wenli

N1 - Publisher Copyright:
© 2019 authors. Published by the American Physical Society. Published by the American Physical Society under the terms of the »https://creativecommons.org/licenses/by/4.0/» Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

PY - 2019/5/31

Y1 - 2019/5/31

N2 - 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.

AB - 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.

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U2 - 10.1103/PhysRevX.9.021043

DO - 10.1103/PhysRevX.9.021043

M3 - Article

AN - SCOPUS:85067342658

SN - 2160-3308

VL - 9

JO - Physical Review X

JF - Physical Review X

IS - 2

M1 - 021043

ER -