Dyson instability for 2D nonlinear $O(N)$ sigma models

For lattice models with compact field integration (nonlinear sigma models over compact manifolds and gauge theories with compact groups) and satisfying some discrete symmetry, the change of sign of the bare coupling $g_0^2$ at zero results in a mere discontinuity in the average energy rather than the catastrophic instability occurring in theories with integration over arbitrarily large fields. This indicates that the large order of perturbative series and the nonperturbative contributions should have unexpected features. Using the large-$N$ limit of two-dimensional nonlinear $O(N)$ sigma model, we discuss the complex singularities of the average energy for complex ’t Hooft coupling ${\lambda }^t=g_0^{2}N$. A striking difference with the usual situation is the absence of the cut along the negative real axis. We show that the zeros of the partition function can only be inside a clover shape region of the complex ${\lambda }^t$ plane. We calculate the density of states and use the result to verify numerically the statement about the zeros. We propose dispersive representations of the derivatives of the average energy for an approximate expression of the discontinuity. The discontinuity is purely nonperturbative and contributions at small negative coupling in one dispersive representation are essential to guarantee that the derivatives become exponentially small when ${\lambda }^t\to 0^{+}$. We discuss the implications for gauge theories.

Figure 1

Complex values of ${\lambda }^t=1/\mathfrak{B}(M^2)$ when $M^2$ varies over horizontal (with a spacing 1) and vertical lines (with a spacing 0.5) centered about the cut in the complex $M^2$ plane.

Figure 2

Real (blue online) and imaginary (red online) part of $\mathfrak{B}(M^2)$ when $M^2$ varies over a horizontal line 0.01 above the cut in the complex $M^2$ plane. The imaginary part has a negative spike at $-4$.

Figure 3

$G(u)$ (blue online) compared to $1/(4\pi )$ (red online).

Figure 4

Complex values taken by $\mathfrak{B}(M^2)$ when $M^2$ varies over the complex plane [here on horizontal lines in the $M^2$ plane with spacing 0.1 (black, blue online) and 0.5 (gray, orange online)]. Asymptotic limits are $\pm 0.25$ in both directions.

Figure 5

Complex values taken by ${\lambda }^t$ when $M^2$ varies over lines above and below the cut with $\mathrm{Im}{M}^2=\pm 0.01$ (black, blue online); the circles are the inverses of the asymptotic lines in Fig. 4.

Figure 6

A finite number of poles and boundaries of domains described in the text for ${\lambda }_{\mathrm{simpl}}(M^2)$ in the $M^2$ plane.

Figure 7

$f(\mathcal{E})$, numerical (circle), first order strong coupling (parabola, green online), and first order weak coupling (red online).

Figure 8

Fisher’s zeros for $NV=100$: zeros of $\mathrm{Re}Z$ (small dots, blue online), zeros of $\mathrm{Im}Z$ (larger dots, red online). The solid line (blue online) is the image of a horizontal line slightly below the cut in the $M^2$ plane.