New Proposal for Numerical Simulations of $\theta$-Vacuum-like Systems

We propose a new approach to perform numerical simulations of $\theta$-vacuum-like systems, test it in two analytically solvable models, and apply it to $C{P}^3$. The main new ingredient in our approach is the method used to compute the probability distribution function of the topological charge at $\theta =0$. We do not get unphysical phase transitions (flattening behavior of the free energy density) and reproduce the exact analytical results for the order parameter in the whole $\theta$ range within a few percent.

Figure 1

Magnetization versus the imaginary magnetic field in the one-dimensional antiferromagnetic Ising model at $F=-1/2$ on a chain of 1000 sites: exact (dashed curve) and numerical (continuous curve) results. Statistical errors are smaller than 2%.

Figure 2

Topological charge versus $\theta$ in the two-dimensional U(1) model at $\beta =0$ and $\beta =0.6$ on a $80\times 80$ lattice. Statistical errors are not visible at this scale. The exact result (dashed curve) cannot be distinguished from the numerical result (continuous curve).

Figure 3

Topological charge versus $\theta$ in the two-dimensional U(1) model at $\beta =0$. The continuous curve is the exact result, the dashed curve is the result obtained by substituting $f(x)$ by $f[x)\mathbf{(}1+\frac{1}{2}\sin (x^2)\mathbf{]}$, and the dotted curve is the result obtained by adding a random error of the order of 0.1% to $f(x)$.

Figure 4

Topological charge versus $\theta$ in the two-dimensional $C{P}^3$ model at $\beta =0.6$ on a $100\times 100$ lattice. The continuous line (discontinuous line) reports the results obtained fitting $f^{\prime }(x)$ with a polynomial (the ratio of two polynomials). Statistical errors are at the 1% level.