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2PI effective action for the SYK model and tensor field theories

Dario Benedetti, Razvan Gurau

TL;DR

The paper develops and deploys the $2PI$ effective action framework to the SYK model and to tensor field theories, showing it can reproduce the bilocal reformulation of SYK without replicas and serve as the bilocal backbone for tensor models where replicas are problematic. In SYK, the leading and next-to-leading $1/N$ terms coincide with the leading and one-loop corrections of the bilocal action, respectively, while higher orders reveal limitations due to non-commutativity of disorder averaging and on-shell procedures. For tensor models (CTKT and GW), the $2PI$ formalism provides the natural route to a bilocal description, with melonic dominance guiding the large-$N$ expansion and subleading terms appearing as trace-log determinants that can be interpreted as one-loop bilocal fluctuations; in the GW case the expansion extends up to NNNLO and exhibits a universal log-det structure. Overall, the $2PI$ perspective clarifies soft modes and potential holographic interpretations in tensor theories, and offers a robust platform for investigating subleading $1/N$ effects and possible NNLO analyses in CTKT and related models.

Abstract

We discuss the two-particle irreducible (2PI) effective action for the SYK model and for tensor field theories. For the SYK model the 2PI effective action reproduces the bilocal reformulation of the model without using replicas. In general tensor field theories the 2PI formalism is the only way to obtain a bilocal reformulation of the theory, and as such is a precious instrument for the identification of soft modes and for possible holographic interpretations. We compute the 2PI action for several models, and push it up to fourth order in the $1/N$ expansion for the model proposed by Witten in arXiv:1610.09758, uncovering a one-loop structure in terms of an auxiliary bilocal action.

2PI effective action for the SYK model and tensor field theories

TL;DR

The paper develops and deploys the effective action framework to the SYK model and to tensor field theories, showing it can reproduce the bilocal reformulation of SYK without replicas and serve as the bilocal backbone for tensor models where replicas are problematic. In SYK, the leading and next-to-leading terms coincide with the leading and one-loop corrections of the bilocal action, respectively, while higher orders reveal limitations due to non-commutativity of disorder averaging and on-shell procedures. For tensor models (CTKT and GW), the formalism provides the natural route to a bilocal description, with melonic dominance guiding the large- expansion and subleading terms appearing as trace-log determinants that can be interpreted as one-loop bilocal fluctuations; in the GW case the expansion extends up to NNNLO and exhibits a universal log-det structure. Overall, the perspective clarifies soft modes and potential holographic interpretations in tensor theories, and offers a robust platform for investigating subleading effects and possible NNLO analyses in CTKT and related models.

Abstract

We discuss the two-particle irreducible (2PI) effective action for the SYK model and for tensor field theories. For the SYK model the 2PI effective action reproduces the bilocal reformulation of the model without using replicas. In general tensor field theories the 2PI formalism is the only way to obtain a bilocal reformulation of the theory, and as such is a precious instrument for the identification of soft modes and for possible holographic interpretations. We compute the 2PI action for several models, and push it up to fourth order in the expansion for the model proposed by Witten in arXiv:1610.09758, uncovering a one-loop structure in terms of an auxiliary bilocal action.

Paper Structure

This paper contains 12 sections, 131 equations, 11 figures.

Table of Contents

  1. Introduction
  2. 2PI effective action for vector models
  3. Large-$N$ expansion
  4. Large-$N$ expansion as the loop expansion of an auxiliary theory
  5. 2PI effective action for the SYK model
  6. Next-to-leading order action
  7. 2PI effective action for tensor field theories
  8. The bosonic CTKT model in $d=0$
  9. The fermionic CTKT model in $d=1$
  10. The fermionic GW model in $d=1$
  11. Summary and outlook
  12. Orthogonal decomposition of the fluctuations

Figures (11)

  • Figure 1: The two vertices from Eq. \ref{['eq:vec-int']}.
  • Figure 2: An example of a graph with one tetravalent and two trivalent vertices.
  • Figure 3: The leading order contribution of the large-$N$ expansion in the $O(N)$ model.
  • Figure 4: The next-to-leading order contribution of the large-$N$ expansion in the $O(N)$ model.
  • Figure 5: The fundamental melon for $q=4$. The dashed line represents the Wick contraction associated to the quenched average.
  • ...and 6 more figures