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Exact Results on the ABJM Fermi Gas

Yasuyuki Hatsuda, Sanefumi Moriyama, Kazumi Okuyama

TL;DR

This work casts ABJM matrix model in the language of a Fermi gas and develops a comprehensive method to compute the grand partition function $\Xi(z)$ exactly at $k=1$. By mapping to a density matrix and employing Tracy–Widom machinery, it reduces traces $\mathrm{Tr}\,\rho^n$ to Hankel-matrix data and reveals a parity-based factorization into $\Xi_+(z)$ and $\Xi_-(z)$. A key anomaly in the coordinate-momentum commutation leads to a novel relation $G(z)=\Xi_-(z)/\Xi_+(−z)$ that fixes the grand partition function from the even sector and enables exact $Z(N)$ up to $N=9$, with all results expressed as polynomials in $\pi^{-1}$. The paper corroborates these exact results with TBA-like numerical analysis, analyzes eigenvalue distributions, and clarifies nonperturbative corrections, finding leading terms $\sim e^{-2\pi\sqrt{2N}}$ at $k=1$. These findings highlight a rich exact structure in ABJM Fermi gas and point to future explorations of hidden symmetries and extensions to Wilson loops or general $k$.

Abstract

We study the Fermi gas quantum mechanics associated to the ABJM matrix model. We develop the method to compute the grand partition function of the ABJM theory, and compute exactly the partition function Z(N) up to N=9 when the Chern-Simons level k=1. We find that the eigenvalue problem of this quantum mechanical system is reduced to the diagonalization of a certain Hankel matrix. In reducing the number of integrations by commuting coordinates and momenta, we find an exact relation concerning the grand partition function, which is interesting on its own right and very helpful for determining the partition function. We also study the TBA-type integral equations that allow us to compute the grand partition function numerically. Surprisingly, all of our exact results of the partition functions are written in terms of polynomials of 1/pi with rational coefficients.

Exact Results on the ABJM Fermi Gas

TL;DR

This work casts ABJM matrix model in the language of a Fermi gas and develops a comprehensive method to compute the grand partition function exactly at . By mapping to a density matrix and employing Tracy–Widom machinery, it reduces traces to Hankel-matrix data and reveals a parity-based factorization into and . A key anomaly in the coordinate-momentum commutation leads to a novel relation that fixes the grand partition function from the even sector and enables exact up to , with all results expressed as polynomials in . The paper corroborates these exact results with TBA-like numerical analysis, analyzes eigenvalue distributions, and clarifies nonperturbative corrections, finding leading terms at . These findings highlight a rich exact structure in ABJM Fermi gas and point to future explorations of hidden symmetries and extensions to Wilson loops or general .

Abstract

We study the Fermi gas quantum mechanics associated to the ABJM matrix model. We develop the method to compute the grand partition function of the ABJM theory, and compute exactly the partition function Z(N) up to N=9 when the Chern-Simons level k=1. We find that the eigenvalue problem of this quantum mechanical system is reduced to the diagonalization of a certain Hankel matrix. In reducing the number of integrations by commuting coordinates and momenta, we find an exact relation concerning the grand partition function, which is interesting on its own right and very helpful for determining the partition function. We also study the TBA-type integral equations that allow us to compute the grand partition function numerically. Surprisingly, all of our exact results of the partition functions are written in terms of polynomials of 1/pi with rational coefficients.

Paper Structure

This paper contains 22 sections, 165 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Grand partition function of ABJM theory
  3. Method to compute the grand partition function
  4. Eigenvalue problem and Hankel matrix representation
  5. Representation in Chebyshev basis
  6. Algebraic treatment
  7. Anomalous terms in commutation relations
  8. Implications for the grand partition function
  9. Exact results
  10. Exact partition functions
  11. Novel relation between $G(z)$ and $\Xi_\pm (z)$
  12. Numerical results
  13. Partition function from TBA-like equations
  14. Numerical results from TBA
  15. Eigenvalue distribution
  16. ...and 7 more sections

Figures (3)

  • Figure 1: The (square of) energy spectrum of the Fermi gas system against the excitation level $n$. The dots correspond to the extrapolated numerical eigenvalues in Table \ref{['tab:eigen']} while the solid line is the plot of $E^2-E_0^2=\pi^2 n/2$.
  • Figure 2: The behavior of the free energy at $k=1$. (a) The dots show our exact values while the solid line shows the free energy computed from \ref{['eq:ZMP2']}. (b) The (log of) non-perturbative correction $F_1^{\rm (np)}(N)$. The dots are the exact data, and the solid line is the fitted function \ref{['eq:fitting']} with \ref{['eq:fitting_value']}.
  • Figure 3: The behavior of the expectation value of $N$ as a function of $\mu$. The dots show the results obtained from TBA-like equations. The red solid line represents the finite-truncated expansion \ref{['eq:<N>_exp']} up to $n=9$ with the exact $\mathop{\mathrm{Tr}}\nolimits \rho^n$ while the green dashed line represents the perturbative result \ref{['eq:<N>_pert']}.