Monte Carlo calculations of the finite density Thirring model

We present results of the numerical simulation of the two-dimensional Thirring model at finite density and temperature. The severe sign problem is dealt with by deforming the domain of integration into complex field space. This is the first example where a fermionic sign problem is solved in a quantum field theory by using the holomorphic gradient flow approach, a generalization of the Lefschetz thimble method.

Figure 1

Fermion density per flavor as a function of the chemical potential $\mu$ (left) and average sign (right) for Wilson fermions at $a{m}_f=0.30(1)$ and $a{m}_b=0.44(1)$. In the left plot, the upper horizontal line is the saturation density, and the dotted curve corresponds to the free gas result; we grayed out the points with the error bars exceeding 0.3 to make the figure easier to read.

Figure 2

$⟨n⟩$ as a function of $\mu$ for several temperatures at the lattice spacing $a{m}_f=0.265$. The horizontal line is the density that corresponds to one particle in the box (per flavor.) The solid curves are splines interpolations of the data points to guide the eye. The dotted curve represents a free fermion gas in the staggered discretization on a $N_t\times N_x=40\times 10$ lattice with the mass set to the value of $a{m}_f=0.265$.

Figure 3

Particle density as a function of the chemical potential for different lattice spacings, for fixed volume ($m_{f}L\approx 3.31$) and temperature ($T/m_f\approx 0.302$). The solid line represents a spline interpolating through finest lattice spacing data points. The dotted line represents the fermion free gas result. Horizontal lines indicate the integral number of particles in the box.

Figure 4

Fermion density as a function of chemical potential on two different volumes, $10\times 10$ from Fig. 1 and $10\times 20$ using the same parameters. The dotted line is the free gas result.