Phase transition and chaos in charged SYK model
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Springer
Received: April 1, 2020 Accepted: June 19, 2020 Published: July 8, 2020
Nilakash Sorokhaibam National Institute of Science Education and Research, HBNI, Bhubaneswar 752050, Odisha, India
E-mail: [email protected] Abstract: We study chaotic-integrable transition and the nature of quantum chaos in SYK model with chemical potential. We use a novel numerical technique to calculate the partition function explicitly. We show the phase transition in the presence of large chemical potential. We also show that a mass-like term consisting of two fermion random interaction (q = 2 SYK term) does not give rise to a sharp transition. We find that turning on the chemical potential suppresses the Lyapunov exponent in the chaotic phase exponentially. Keywords: AdS-CFT Correspondence, Field Theories in Lower Dimensions, Holography and condensed matter physics (AdS/CMT) ArXiv ePrint: 1912.04326
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP07(2020)055
JHEP07(2020)055
Phase transition and chaos in charged SYK model
Contents 1 Introduction and summary
1
2 Complex SYK model in imaginary time formalism 2.1 Numerical method
6 9 11
4 No phase transition in (q = 2, 4) model
13
5 Complex SYK model in real time formalism
15
6 Calculation of Lyapunov exponent 6.1 Comparison with large q result
16 21
7 Conclusions
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A Conventions
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B Mass versus chemical potential
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C Fluctuation-dissipation theorem with chemical potential
25
D Fermion partition function
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1
Introduction and summary
The Sachdev-Ye-Kitaev (SYK) model is a quantum system of many fermions (N in number, N is large) with random all-to-all interaction [1, 2]. It has been a subject of great interest in the last few years [3–8]. The model has many remarkable properties. It does not have any quasi-particle excitations. The gaps in the spectrum are exponentially suppressed in N . It flows to a conformal theory in deep infrared. It also saturates the quantum chaos bound of [9]. All these properties point to the existence of a bulk dual of the theory. There has been many proposals and other related works on the gravity side [10–18]. The original SYK model is a model with Majorana fermions. If one considers SYK model with complex fermions, one can turn on the mass term in the Hamiltonian or consider a thermal state with chemical potential turned on. With chemical potential turned on, a first order phase transition has been observed [19]. The high temperature phase is chaotic while the low temperature phase is integrable (non-chaotic). Henceforth, the two phases will be called chaotic phase and integrable phase. The integrable phase is effectively described by a weakly interacting massive theory. In this phase, the Lyapunov exponent is
–1–
JHEP07(2020)055
3 Phase transition in (q = 4) model with chemical potential
also practically zero. This phase transition is like Hawking-Page transition between black hole phase and thermal AdS phase [20]. In this paper we study the phase transition in more details. We
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