Quantum field theory at finite temperature for 3D periodic backgrounds

TL;DR

Problem addressed: quantum vacuum and thermal fluctuations of a scalar field in 3D periodic backgrounds formed by 1D localized potentials. Method: compute band spectra from scattering data with Floquet-Bloch conditions and derive convergent representations for E0 and the finite-temperature free energy using Cauchy integrals; apply to generalised Dirac comb and Pöschl-Teller comb and handle negative-energy bands by introducing a mass. Key findings: the PT comb has a persistent negative-energy band, while the Dirac comb's E0 can be positive, negative, or zero depending on couplings; at finite temperature ΔT F/A is negative, entropy is positive, and the Casimir pressure is repulsive. Significance: demonstrates a tractable, model-independent approach to phonon-induced quantum effects in lattices and points to extensions to Green's functions and electron-phonon coupling in realistic materials.

Abstract

The one-loop quantum corrections to the internal energy of some lattices due to the quantum fluctuations of the scalar field of phonons are studied. The band spectrum of the lattice is characterised in terms of the scattering data, allowing to compute the quantum vacuum interaction energy between nodes at zero temperature, as well as the total Helmholtz free energy, the entropy, and the Casimir pressure between nodes at finite non-zero temperature. Some examples of periodic potentials built from the repetition in one of the three spatial dimensions of the same punctual or compact supported potential are addressed: a stack of parallel plates constructed by positioning $δδ'$-functions at the lattice nodes, and an 'upside-down tiled roof' of parallel two-dimensional Pöschl-Teller wells centred at the nodes. They will be called generalised Dirac comb and Pöschl-Teller comb, respectively. Positive one-loop quantum corrections to the entropy appear for both combs at non-zero temperatures. Moreover, the Casimir force between the lattice nodes is always repulsive for both chains when non-trivial temperatures are considered, implying that the primitive cell increases its size due to the quantum interaction of the phonon field.

Figures (10)

• Figure 1: Left: Lattice of parallel Dirac $\delta \delta'$ plates (red) in three spatial dimensions and primitive cell (blue). Right: Lattice of truncated PT potentials (red) in three spatial dimensions and primitive cell (blue).
• Figure 2: First two allowed energy bands for the pure Dirac comb (solid green curve) and the generalised Dirac comb (dashed lines). On the left $w_0=-5$ for all the cases and on the right $w_0=5$. In both plots $w_1=0.3$ (purple), $w_1=0.75$ (red), $w_1=1.5$ (brown), $w_1=5$ (blue) and $w_1=-10$ (yellow). The black line in the plot on the left represents the zero energy level. In both plots $a=1$, $E=k^2$ and $q$ is the quasi-momentum in the first Brillouin zone.
• Figure 3: Left: First two allowed energy bands for the Pöschl-Teller comb for $a=1$ and different values of the compact support $\epsilon=0.2$ (blue), $\epsilon=0.6$ (red) and $\epsilon=0.9$ (green). Right: First two allowed bands for $a=3$ for compact supports $\epsilon=0.1$ (purple), $\epsilon=1$ (orange) and $\epsilon=2.5$ (yellow). In these plots $E=k^2$ and $q$ is the quasi-momentum in the first Brillouin zone.
• Figure 4: First three bands of the spectrum for the PT comb characterised by $a=1, \epsilon=0.6$. For each fixed value of the parameter $\theta \in [0, \pi]$, one has to sum over the wave vector $k$ or $\kappa$ associated to the states highlighted by green dots. Notice that in quantum mechanics $E=k^2$ when $E>0$ and $E=(i\kappa)^2$ when $E<0$. In these plots, $\theta=q a= q$ is the quasi-momentum in the first Brillouin zone.
• Figure 5: Complex contour $\Gamma$ that encloses all the zeroes of $f_\theta(k)$ as $R\to \infty$ when there are bound states in the spectrum. In the contour, $\left. k\right|_{\Gamma_3}=\{(m+R \cos \nu)+ iR \sin \nu, \nu\in[-\gamma, \gamma]\}$ and $\left. k\right|_{\Gamma_\pm}=\{m+\xi e^{\pm i \gamma}, \xi\in [0,R]\}$. Notice that $R>0$ and $0<\gamma<\pi/2$ are constants. This contour must be traversed in a counterclockwise sense when integration is carried out.