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Asymptotics of spin-spin correlators weighted by fermion number measurements with low rapidity threshold in the 2D Ising free-fermion QFT

Yizhuang Liu

TL;DR

This work studies the averaged fermion-number observable above a low rapidity threshold $Y$ in the spin-spin correlators of the 2D Ising QFT at the free-fermion point. By embedding the observable into the Sinh-Gordon/Painlevé-III integrable framework, the authors derive two-variable linear differential equations in $(r,Y)$ and identify two scaling functions $f_ obreak ext{_ obreak}(t)$ and $g(t)$ that determine the small-distance behavior in the scaling limit $r o0$, $t= rac{r}{2}e^{Y}$ fixed. The scaling functions admit Barnes representations, reduce to explicit Bessel-$K$ forms at the physical point $oldsymbol{ u} o1$, and relate to an integrated four-point function in the Ising CFT, thereby connecting massive form-factor data to massless CFT structures. The authors show singularity cancellations at the physical value and provide a resummation of collapsed power corrections, achieving small-$r$ expansions of the observables up to ${ m O}(r^3)$ in the scaling limit. Overall, the work demonstrates a precise, integrable-control mechanism for non-Gaussian observables in a massive theory, with clear links to Ising-CFT correlators and potential applications to similar measurements in other integrable QFTs.

Abstract

In the work, we study the averaged number of massive fermions above a low rapidity threshold $Y$, underlying the form-factor expansions of the spin-spin two-point correlators at an Euclidean distance $r$, in the 2D Ising QFT at the free massive fermion point. Despite the on-shell freeness, the spin operators are still far away from being Gaussian, and create particles in the asymptotic states with complicated correlations. We show how the number observables can still be incorporated into the integrable Sinh-Gordon/Painleve-III framework and controlled by linear differential equations with two variables $(r,Y)$. We show how the differential equations and the information of two crucial scaling functions arising in the $r\rightarrow 0$, $e^{Y}r={\cal O}(1)$ scaling limit, can be combined to fully determine the small-$r$ asymptotics of the observables, in the $λ$-extended form. The scaling functions, on the other hand, are analyzed by summing the exponential form-factor expansions directly, generalizing the traditional Ising connecting computations. We show carefully, how the singularities cancel in the physical value limit $λπ\rightarrow 1$ and how the power-corrections that collapse at this value can be resummed. In particular, we show for the physical $λ$-value, the scaling functions are related to an integrated four-point function in the Ising CFT and continue to control the asymptotics of the number-observables in the scaling limit up to ${\cal O}(r^3)$.

Asymptotics of spin-spin correlators weighted by fermion number measurements with low rapidity threshold in the 2D Ising free-fermion QFT

TL;DR

This work studies the averaged fermion-number observable above a low rapidity threshold in the spin-spin correlators of the 2D Ising QFT at the free-fermion point. By embedding the observable into the Sinh-Gordon/Painlevé-III integrable framework, the authors derive two-variable linear differential equations in and identify two scaling functions and that determine the small-distance behavior in the scaling limit , fixed. The scaling functions admit Barnes representations, reduce to explicit Bessel- forms at the physical point , and relate to an integrated four-point function in the Ising CFT, thereby connecting massive form-factor data to massless CFT structures. The authors show singularity cancellations at the physical value and provide a resummation of collapsed power corrections, achieving small- expansions of the observables up to in the scaling limit. Overall, the work demonstrates a precise, integrable-control mechanism for non-Gaussian observables in a massive theory, with clear links to Ising-CFT correlators and potential applications to similar measurements in other integrable QFTs.

Abstract

In the work, we study the averaged number of massive fermions above a low rapidity threshold , underlying the form-factor expansions of the spin-spin two-point correlators at an Euclidean distance , in the 2D Ising QFT at the free massive fermion point. Despite the on-shell freeness, the spin operators are still far away from being Gaussian, and create particles in the asymptotic states with complicated correlations. We show how the number observables can still be incorporated into the integrable Sinh-Gordon/Painleve-III framework and controlled by linear differential equations with two variables . We show how the differential equations and the information of two crucial scaling functions arising in the , scaling limit, can be combined to fully determine the small- asymptotics of the observables, in the -extended form. The scaling functions, on the other hand, are analyzed by summing the exponential form-factor expansions directly, generalizing the traditional Ising connecting computations. We show carefully, how the singularities cancel in the physical value limit and how the power-corrections that collapse at this value can be resummed. In particular, we show for the physical -value, the scaling functions are related to an integrated four-point function in the Ising CFT and continue to control the asymptotics of the number-observables in the scaling limit up to .

Paper Structure

This paper contains 15 sections, 185 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Fredholm determinants and the fermion number observables
  3. The fermion number observables
  4. Conventions of the expansions
  5. Summary of main results
  6. Scaling functions and asymptotics of the observables
  7. The scaling functions $f_{\pm}(t)$
  8. The scaling functions at $\sigma=1$
  9. The small-$r$ expansion of $\Psi_{\pm}(r,Y)$
  10. The scaling limit at $\sigma=1$
  11. Conclusion and comments
  12. The number observables of the Ising CFT on a cylinder
  13. Resumming the collapsed power corrections
  14. Frequently used linear equations
  15. Numerical evidences for the asymptotic formula

Figures (1)

  • Figure 1: The comparison at $r=0.01$ (upper left), $r=0.05$ (upper right) and $r=0.1$ (lower), between the $N_{3}(t,r)$ (black solid), $N_{\rm asym}(t,r)$ (dotted) and the "wrong" asymptotic formula with $\mathfrak{g}(t)$ removed (orange-dashed). The horizontal-axis is the scaling variable $t$. Each of the figure are composed of $20$ or $21$ data points at $t \in 0.1\mathbb{Z}$ connected by straight lines. The ranges of $t$ are: $0.1\le t\le2$ ($r=0.01$), $0.2\le t\le 2.1$ ($r=0.05$) and $0.5\le t \le 2.5$ ($r=0.1$). The lower values of $t$ are to guarantee that $2t/r =e^Y \ge 8$. In all these plots, the three particle truncation $N_3(t,r)$ and the asymptotic formula $N_{\rm asym}(t,r)$ are sufficiently close to each other and hard to distinguish. At $r=0.05$ and $r=0.1$, the discrepancies are below $5\times 10^{-4}$, while at $r=0.01$ the discrepancies increase to $5\times 10^{-3}$. On the other hand, the plots with $\mathfrak{g}(t)$ removed deviate from the other two curves as $t$ decrease, and the discrepancies are quite significant at $r=0.05$ and $r=0.1$.