Condensation of Lee-Yang zeros in scalar field theory

We show that, at the critical temperature, there is a class of Lee-Yang zeros of the partition function in a general scalar field theory, which location scales with the size of the system with a characteristic exponent expressed in terms of the isothermal critical exponent $\delta$. In the thermodynamic limit the zeros belonging to this class condense to the critical point $\zeta =1$ on the real axis in the complex fugacity plane while the complementary set of zeros (with $\text{Re}\zeta <1)$ covers the unit circle. Although the aforementioned class degenerates to a single point for an infinite system, when the size is finite it contributes significantly to the partition function and reflects the self-similar structure (fractal geometry, scaling laws) of the critical system. This property opens up the perspective to formulate finite-size scaling theory in effective QCD, near the chiral critical point, in terms of the location of Lee-Yang zeros.

Figure 1

The zeros of the partition function ${\mathcal{Z}}_M$ given by Eq. (19) in the complex fugacity plane for $M=80,{\alpha }_c=1$, and $\delta =5$. The red and green lines are the analytical results of the saddle point approximation to the integral form of ${\mathcal{Z}}_M$ given by Eq. (20). The unit circle is presented with the black line. The exact zeros lie on the intersection of the lines obeying $\text{Re}{\mathcal{Z}}_M=0$ (blue), with the lines obeying $\text{Im}{\mathcal{Z}}_M=0$ (orange).

Figure 2

The zeros (red and green lines) of the partition function saddle-point approximation ${\mathcal{Z}}_{\text{sp}}$ given in Eq. (31) in the complex fugacity plane. We use $M=80,1000,10000,{\alpha }_c=1$ and $\delta =5$. The red arcs define the critical set of zeros, which shrinks to the single point ${\zeta }_c=(1,0)$ for $M\to \infty$.

Figure 3

The zeros (colored points) of Eq. (12) in the square $\text{Re}\zeta \in [1,1.12],\text{Im}\zeta \in [0,0.3]$ for ${\alpha }_c=1,\delta =5$, and $M=100,200,..,1000$. The solid lines are drawn to guide the eye.

Figure 4

(a) The real and (b) the imaginary part of ${\rho }_k\left(M\right)-1$ as a function of $M$ for $M=100,200,..,1000,2000,..,10000$ and $k=1,..,6$. In both plots the black line indicates the result of a linear fit in the large $M$ $\left(M>4000\right)$ range for the zero with $k=1$. As in Fig. 3, we use ${\alpha }_c=1$ and $\delta =5$.