Sachdev-Ye-Kitaev Model as Liouville Quantum Mechanics (1607.00694v1)

Abstract: We show that the proper inclusion of soft reparameterization modes in the Sachdev-Ye-Kitaev model of $N$ randomly interacting Majorana fermions reduces its long-time behavior to that of Liouville quantum mechanics. As a result, all zero temperature correlation functions decay with the universal exponent $\propto \tau^{-3/2}$ for times larger than the inverse single particle level spacing $\tau\gg N\ln N$. In the particular case of the single particle Green function this behavior is manifestation of the zero-bias anomaly, or scaling in energy as $\epsilon^{1/2}$. We also present exact diagonalization study supporting our conclusions.

• The paper demonstrates the reduction of the SYK model to Liouville quantum mechanics using soft reparameterization modes to uncover its low-temperature dynamics.
• It applies exact diagonalization techniques to validate analytical predictions, showing a universal τ^-3/2 decay in zero-temperature correlation functions.
• The study highlights the role of Goldstone modes from broken conformal symmetry, offering insights relevant to holographic dualities and disordered quantum systems.

Sachdev-Ye-Kitaev Model as Liouville Quantum Mechanics

The paper presents an intricate paper of the Sachdev-Ye-Kitaev (SYK) model, illustrating its representation as Liouville quantum mechanics by incorporating soft reparameterization modes. The SYK model, influential in both quantum gravity and condensed matter theory, consists of $N$ randomly interacting Majorana fermions. The couplings are described by Gaussian random variables, and remarkable implications have linked this model to holographic dualities and strongly correlated quantum systems.

Core Findings and Methodology

The authors explore the low-temperature behavior of the SYK model, focusing on reparameterization invariance in the time domain, where a slight modification manifests as an emergent conformal symmetry. Here, the sophisticated dynamics of the Majorana fermions simplify into a Liouville quantum mechanics framework. The work highlights the importance of Goldstone modes, stemming from the spontaneously broken conformal symmetry, which are crucial in the infrared regime of the model.

A prominent conclusion of this paper is the universal decay of zero-temperature correlation functions at long time scales, described by an exponent $\tau^{-3/2}$. The paper provides a deeper understanding of how this behavior affects the single-particle Green function, indicative of the zero-bias anomaly. Such manifestations are crucial for understanding low-frequency behavior in systems governed by the SYK model.

The research employs exact diagonalization techniques to corroborate analytical results, presenting compelling evidence that supports the theoretical framework proposed. The adaptive mapping to Liouville quantum mechanics facilitates a non-perturbative examination of correlation functions in the SYK model. This approach could promote further exploration into the model's universality class and its applications.

Implications and Future Directions

The implications of the paper are manifold both theoretically and practically. From a theoretical standpoint, the reduction of the SYK model to Liouville quantum mechanics provides a new lens through which to interpret low-dimensional holographic correspondences. This work recognizably adds to the dialogue between quantum mechanics and statistical physics, suggesting new avenues for studying nearly conformally invariant theories.

Practically, the findings about the correlation functions could substantially influence how low-temperature electronic phenomena in disordered systems are understood. Since the zero-bias anomaly reflects a universal behavior across various parameter regimes, its applications transcend the specific boundaries of the holographic context and extend into experimental observations.

The paper raises potential for subsequently disentangling complex problems in high-energy physics, condensed matter, and quantum information science. Future work could explore in greater depth the $NAdS_2/NCFT_1$ correspondence and pursue a comprehensive understanding of symmetry breaking and energy scaling in chaotic quantum systems.

This research not only extends the mathematical apparatus available for studying complex systems but also presents a foundation for ongoing and prospective investigations, encouraging a reevaluation of existing paradigms within the realms of holography and many-body physics. Such continued exploration will undoubtedly yield further insights into the enigmatic and universally significant behavior of the SYK model.