[Paper Review] Two indices Sachdev-Ye-Kitaev model

Jinwu Ye|arXiv (Cornell University)|Sep 18, 2018

Advanced Condensed Matter Physics16 references3 citations

TL;DR

This paper introduces the two-index Sachdev-Ye-Kitaev (SYK) model using Majorana fermions, avoiding local constraints present in the original SU(M) SY model and reducing complexity compared to the four-index SYK model. Through a 1/M expansion at N=∞, it shows the quantum spin liquid (QSL) state remains conformally invariant and stable at finite M, with 1/M corrections being exactly marginal. Crucially, the 4-point out-of-time-ordered correlation function (OTOC) shows no quantum chaos at N=∞ or M=∞, suggesting chaos emerges only at finite N and M, potentially via 1/N quantum fluctuations.

ABSTRACT

We study the original Sachdev-Ye (SY) model in its Majorana fermion representation which can be called the two indices Sachdev-Ye-Kitaev (SYK) model. Its advantage over the original SY model in the $ SU(M) $ complex fermion representation is that it need no local constraints, so a $1/M $ expansion can be more easily performed. Its advantage over the 4 indices SYK model is that it has only two site indices $ J_{ij} $ instead of four indices $ J_{ijkl} $, so it may fit the bulk string theory better. By performing a $1/M $ expansion at $ N=\infty $, we show that a quantum spin liquid (QSL) state remains stable at a finite $ M $. The $ 1/M $ corrections are exactly marginal, so the system remains conformably invariant at any finite $ M $. The 4-point out of time correlation ( OTOC ) shows quantum chaos neither at $ N=\infty $ at any finite $ M $, nor at $ M=\infty $ at any finite $ N $. By looking at the replica off-diagonal channel, we find there is a quantum spin glass (QSG) instability at an exponentially suppressed temperature in $ M $. We work out a criterion for the two large numbers $ N $ and $ M $ to satisfy so that the QSG instability may be avoided. We speculate that at any finite $ N $, the quantum chaos appears at the order of $ 1/M^{0} $, which is the subleading order in the $ 1/M $ expansion. When the $ 1/N $ quantum fluctuations at any finite $ M $ are considered, from a general reparametrization symmetry breaking point of view, we argue that the eThis work may motivate future works to study the possible new gravity dual of the 2 indices SYK model.ffective action should still be described by the Schwarzian one, the OTOC shows maximal quantum chaos.

Motivation & Objective

• To propose a two-index SYK model in Majorana fermion representation to avoid local constraints present in the original SU(M) SY model.
• To compare the two-index model favorably with the four-index SYK model by reducing tensor rank from four to two indices, potentially improving compatibility with bulk string theory.
• To analyze the stability of the quantum spin liquid (QSL) state at finite M using a 1/M expansion at N=∞.
• To investigate the presence or absence of quantum chaos via the 4-point out-of-time-ordered correlation function (OTOC) across different limits of N and M.
• To derive a criterion for avoiding quantum spin glass (QSG) instability at exponentially suppressed temperatures in M.

Proposed method

• Formulate the two-index SYK model using Majorana fermions, with Hamiltonian H = ∑_{i,j} J_{ij} χ_i χ_j χ_k χ_l, where J_{ij} is Gaussian-distributed with variance ⟨J²_{ij}⟩ ∝ 1/N.
• Perform a 1/M expansion at N=∞ to study corrections to the QSL ground state, leveraging the absence of local constraints.
• Use the replica trick and path integral formalism with two auxiliary fields Q^{ab} and P^{ab} to decouple interactions via Hubbard-Stratanovich transformations.
• Derive the effective action for Q^{ab} by integrating out fermionic fluctuations δP^{ab}, including quadratic corrections in δP^{ab} to capture 1/M quantum fluctuations.
• Analyze the saddle-point solution and self-energy structure, showing P^{0}_{ab} depends on Q^{ab} through the relation Σ^{0}_{ab} = 8Q^{ab}P^{0}_{ab} in the conformal limit.
• Use the Schwarzian action derived from time reparametrization symmetry breaking to argue that quantum chaos may emerge at finite N and M via 1/N corrections.

Experimental results

Research questions

• RQ1 Does the two-index SYK model in Majorana fermion representation preserve conformal invariance and quantum spin liquid order at finite M?
• RQ2 Why does the 4-point out-of-time-ordered correlation function (OTOC) show no quantum chaos at N=∞ or M=∞, despite the presence of random interactions?
• RQ3 What is the role of 1/M corrections in the stability of the QSL state, and why are they exactly marginal?
• RQ4 At what scale does quantum chaos appear in the two-index SYK model, and how does it depend on the interplay between finite N and finite M?
• RQ5 Can a criterion be derived to avoid the quantum spin glass (QSG) instability at exponentially low temperatures in M?

Key findings

• The quantum spin liquid (QSL) state remains stable and conformally invariant at any finite M due to 1/M corrections being exactly marginal, which only rescale parameters like entropy and specific heat without breaking conformal symmetry.
• The 4-point out-of-time-ordered correlation function (OTOC) shows no quantum chaos at N=∞ for any finite M, nor at M=∞ for any finite N, indicating chaos is absent in these limits.
• Quantum chaos may only emerge at finite N and finite M, likely at the subleading order in the 1/M expansion, suggesting a non-trivial interplay between 1/N and 1/M corrections.
• A quantum spin glass (QSG) instability appears at an exponentially suppressed temperature in M, and a criterion is derived to avoid it by tuning the relative sizes of N and M.
• The effective action for the system remains governed by the Schwarzian action when 1/N quantum fluctuations are included at finite M, implying maximal quantum chaos via a Lyapunov exponent λ_L = 2π/β.
• The inclusion of the log Q^{ab} term in the effective action—previously neglected—may lead to a zero mode at order 1 in the 1/M expansion, suggesting non-trivial structure in the quantum fluctuation spectrum.