Fermions at finite density in the path integral approach

Abstract

We study relativistic fermionic systems in $3+1$ spacetime dimensions at finite chemical potential and zero temperature, from a path-integral point of view. We show how to properly account for the $i\varepsilon$ term that projects on the finite density ground state, and compute the path integral analytically for free fermions in homogeneous external backgrounds, using complex analysis techniques. As an application, we show that the ${\rm U}(1)$ symmetry is always linearly realized for free fermions at finite charge density, differently from scalars. We study various aspects of finite density QED in a homogeneous magnetic background. We compute the free energy density, non-perturbatively in the electromagnetic coupling and the external magnetic field, obtaining the finite density generalization of classic results of Euler--Heisenberg and Schwinger. We also obtain analytically the magnetic susceptibility of a relativistic Fermi gas at finite density, reproducing the de Haas--van Alphen effect. Finally, we consider a (generalized) Gross--Neveu model for $N$ interacting fermions at finite density. We compute its non-perturbative effective potential in the large-$N$ limit, and discuss the fate of the ${\rm U}(1)$ vector and $\mathbb{Z}_2^A$ axial symmetries.

Figures (6)

• Figure 1: Analytic structure and contour deformation for $\mu>m$ and $|\vec{p}| < p_{F}$. In the first step, we deform the integration contour from the real axis (blue) to the imaginary axis (red), see panel a). The deformation is smooth and the integrals along the two arcs in the first and third quadrants cancel each other. In the second step, we deform the contour with a shift by $-\mu$. The integral along the vertical green line reproduces the $\mu=0$ path integral, while the contributions from the discontinuity across the branch cut and the segments at infinity reproduce the finite $\mu$ free energy density of the relativistic Fermi gas, see eq. \ref{['eq:ffree']}.
• Figure 2: Magnetic susceptibility of the zero density QED vacuum in an external magnetic field, in units of $\alpha/\pi$ and as a function of $\beta\equiv 2eB/m^2$. The dashed red ($N=1$) and green ($N=2$) lines provide truncations of the small $\beta$ asymptotic expansion with $N$ terms. Increasing $N$ improves the accuracy for small $\beta$, but reduces the region in which the approximation is valid. The strong field approximation (shown as an orange dot-dashed line) provides an accurate result only for $\beta \gtrsim 100$. The intermediate region is described by the full non-perturbative result.
• Figure 3: Magnetic susceptibility of the finite density QED vacuum in a strong external magnetic field, for different values of $\tilde{\mu}= \mu/m$. We use variables that make manifest the filling of the first five Landau levels. The magnetic susceptibility has a periodic spike feature as a function of $1/\beta$, occurring for integer filling, known as the de Haas--van Alphen effect. Continuous lines: analytic form of the finite $\mu$ contribution (second term in eq. \ref{['eq:chiB_mu']}). Dots: full result, evaluated at equidistant points.
• Figure 4: Same as figure \ref{['fig:chiB_mu1']} but for a larger range of $1/\beta$ and for $\tilde{\mu}^2=\mu^2/m^2 = 3/2$. In the weak field limit (small $\beta$), the magnetic susceptibility $\chi_B$ oscillates more and more wildly. The smooth result \ref{['eq:chiB_weak']} is obtained only as an average.
• Figure 5: Finite $\mu$ contribution to the effective potential of $\sigma$ in the (generalized) Gross--Neveu model in $3+1$ dimensions, eq. \ref{['eq:Vmu']}. The result is normalized in units of $N \mu^4$ and expressed as a function of $\sigma/\mu$.