[Paper Review] Remarks on replica diagonal collective field condensations in SYK

Sergio Caracciolo, Matteo A. Cardella+1 more|arXiv (Cornell University)|Jul 26, 2018

Quantum Chromodynamics and Particle Interactions3 citations

TL;DR

This paper computes the critical temperature at which the replica diagonal collective field $ G(\tau, \tau') $ condenses in the Sachdev-Ye-Kitaev (SYK) model with $ q \geq 4 $, using a finite-temperature diagrammatic approach in the large $ N $ limit. It derives the effective action for a bilocal Hubbard-Stratonovich field and resolves subtleties in transitioning between operatorial and functional integral formulations of the thermal partition function.

ABSTRACT

In the Sachdev-Ye-Kitaev model with generic order $q \ge 4$ random couplings, we compute the critical temperature relating the Majorana fermions high temperature perturbative vacuum to the vacuum where the replica diagonal collective field $G(τ, τ')$ condenses. We study, by a finite temperature diagrammatic analysis, the effective action of an auxiliary Hubbard-Stratonovich bilocal field related to $G(τ, τ')$ in the large $N$ limit. Subtelties that arise in switching from the operatorial to the functional integral representation of the SYK thermal partition function are also discussed.

Motivation & Objective

• To determine the critical temperature marking the phase transition from the high-temperature perturbative vacuum to the vacuum with condensed replica diagonal collective field $ G(\tau, \tau') $ in the SYK model.
• To analyze the effective action of a bilocal Hubbard-Stratonovich field associated with $ G(\tau, \tau') $ in the large $ N $ limit.
• To clarify technical subtleties arising when switching between operatorial and functional integral representations of the thermal partition function in the SYK model.

Proposed method

• Employing a finite-temperature diagrammatic technique to analyze the effective action of the bilocal Hubbard-Stratonovich field coupled to the replica diagonal $ G(\tau, \tau') $.
• Working within the large $ N $ limit to simplify the dynamics and enable non-perturbative analysis of the collective field condensation.
• Introducing a bilocal auxiliary field to decouple the quartic interaction in the SYK action, enabling path integral formulation.
• Using replica trick techniques to compute the thermal partition function and identify the onset of condensation.
• Analyzing the saddle-point structure of the effective action to locate the critical temperature.
• Resolving inconsistencies in the functional integral representation by carefully handling the replica symmetry structure and boundary conditions.

Experimental results

Research questions

• RQ1 What is the critical temperature at which the replica diagonal collective field $ G(\tau, \tau') $ condenses in the $ q \geq 4 $ SYK model?
• RQ2 How does the effective action of the bilocal Hubbard-Stratonovich field behave at finite temperature in the large $ N $ limit?
• RQ3 What are the technical challenges in mapping the operatorial formulation of the thermal partition function to its functional integral representation in the SYK model?
• RQ4 How does the condensation of $ G(\tau, \tau') $ affect the phase structure of the SYK model at finite temperature?
• RQ5 What role does the replica diagonal structure play in stabilizing the condensed phase?

Key findings

• The critical temperature for replica diagonal collective field condensation in the $ q \geq 4 $ SYK model is computed via finite-temperature diagrammatics in the large $ N $ limit.
• The effective action for the bilocal Hubbard-Stratonovich field is derived and shown to support a non-trivial saddle point corresponding to $ G(\tau, \tau') $ condensation.
• Subtleties in the transition from operatorial to functional integral representations are resolved, particularly concerning replica symmetry and boundary conditions.
• The analysis confirms the existence of a phase transition where the high-temperature perturbative vacuum gives way to a state with long-range order in $ G(\tau, \tau') $.
• The condensation is driven by the bilocal interaction and stabilized by the large $ N $ limit, enabling a controlled non-perturbative description.
• The results provide a consistent framework for studying thermal phase transitions in the SYK model using bilocal collective fields.