Interacting Dirac fields in an expanding universe: dynamical condensates and particle production

TL;DR

This work addresses non-perturbative particle production for interacting Dirac fermions in a curved spacetime, focusing on a $(1+1)$-D FRW background described by a scale factor $\mathsf{a}(\eta)$ and a Gross-Neveu-type four-Fermi interaction. The authors develop a lattice Wilson-Gross-Neveu-Wilson model and solve it with fermionic Gaussian states (FGS), combining imaginary-time evolution to prepare the initial state with real-time evolution to track self-consistent condensates $\Sigma(\eta)$ and $\Pi(\eta)$ that feed back into the dynamics via a self-consistent Hamiltonian $\tilde{h}(\Gamma)$. They find that, in the quench limit, interactions effectively shift the mass gap (via $\Sigma_0$ and $\Pi_0$) and modify particle production differently across the trivial, SPT, and Aoki phases; away from quench, the dynamical evolution of condensates leads to back-reaction, synchronized oscillations, and possible parity breaking in the production spectra within the Aoki phase. These results illuminate how non-perturbative mass generation and symmetry breaking interplay with real-time particle production and suggest avenues for analogue-gravity quantum simulations with cold atoms to explore beyond-perturbative regimes.

Abstract

The phenomenon of particle production for quantum field theories in curved spacetimes is crucial to understand the large-scale structure of a universe from an inflationary epoch. In contrast to the free and fixed-background case, the production of particles with strong interactions and back reaction is not completely understood, especially in situations that require going beyond perturbation theory. In this work, we present advances in this direction by focusing on a self-interacting field theory of Dirac fermions in an expanding Friedmann-Robertson-Walker universe. By using a Hamiltonian lattice regularization with continuous conformal time and rescaled fields, this model becomes amenable to either a cold-atom analogue-gravity quantum simulation, or a dynamical variational approach. Leveraging a family of variational fermionic Gaussian states, we investigate how dynamical mass generation and the formation of fermion condensates associated to certain broken symmetries modify some well-known results of the free field theory. In particular, we study how the non-perturbative condensates arise and, more importantly, how their real-time evolution has an impact on particle production. Depending on the Hubble expansion rate, we find an interesting interplay of interactions and particle production, including a non-trivial back reaction on the condensates and a parity-breaking spectrum of produced particles.

Figures (14)

• Figure 1: Scheme for the simulation of particle production in an expanding universe. First, we discretize the theory, and the expansion of the universe, here depicted in $(1+1)$-dimensional case with spatial periodic boundary conditions, is manifested in a growing separation of two lattice points, which increases in time according to the so-called scale factor $\mathsf{a}(t)$. In the middle panel, we depict how by rescaling the spinor field with the scale factor, the discretized lattice field theory in the expanding universe would correspond to a field theory in a static lattice, but endowed with time-dependent tunnelings $T_{ij}(\mathsf{a}(t))$. Finally, in the rightmost panel, by performing a coordinate transformation and introducing the so-called conformal time $\eta$, we can encode all the effects of the expanding universe in a dynamical mass term.
• Figure 2: Phase diagram for the Gross-Neveu-Wilson QFT. We depict the phase diagram of the lattice QFT as a function of the bare mass $ma$ ($a$ being the lattice spacing) and four-Fermi coupling strength $g_0^2$.The expanding universe amounts to an effective dynamical renormalization of the mass with the scale factor $\mathsf{a}(t)$, whereas the dimensionless quartic coupling $g_0^2$ remains static. Therefore, the system dynamically goes through different trajectories in parameter space as the universe expands, with a velocity determined by $\mathsf{H}$. The arrows (a), (b) and (c) depict the evolution of the parameters for an expansion within the trivial, Aoki, and SPT phases, respectively.
• Figure 3: Protocol for computing the production of particles after the expansion of the universe using FGSs. We compute the groundstate correlation matrix for both the initial and final set of parameters, $(m\mathsf{a}_0,\;g_0^2)$ and $(m\mathsf{a}_{\rm f},\;g_0^2)$, via the ITE equations, obtaining respectively the correlation matrices $\Gamma_0$ and $\Gamma_{\rm f}$. Then, we solve the RTE equations using $\Gamma_0$ as the initial condition. Finally, we diagonalize $\Gamma_{\rm f}$ via the matrix $\mathcal{U}$, that connects the fermionic operators $c_{i\alpha}$ with the correct particle/antiparticle operators defined at $\eta=\eta_{\rm f}$, $a_{\rm k}$ and $b_{-{\rm k}}^\dagger$ respectively. Acting with $\mathcal{U}$ on the real-time evolved correlation matrix $\Gamma(\eta_{\rm f})$ one can obtain the Bogoliubov coefficients $\alpha_{\rm k}$ and $\beta_{\rm k}$ that contain the information about particle production.
• Figure 4: Dynamical evolution of the $\Sigma$ condensate as a function of the scale factor for different values of ${\mathsf{H}}a$. Trivial insulator phase. The conformal time range $[\eta_0,\;\eta_{\rm f}]$ depends on the value of $\mathsf{H}$. For this reason, and in order to compare different regimes of adiabaticity, we plot the value of $\Sigma a$ as a function of the scale factor $\mathsf{a}(\eta)$ and not of the conformal time $\eta$. The grey arrow points towards increasing adiabaticity. $ma=0.1$, $\mathsf{a}_0=0.1$, $\mathsf{a}_{\rm f}=10$, $g_0^2=1$ and $N_{\rm S}=65$.
• Figure 5: Evolution of condensates within the Aoki phase. We show here the dynamical evolution of the condensates as a function of the scale factor for different values of ${\mathsf{H}}a$. The rest of the parameters are given by $ma=-1$, $\mathsf{a}_0=0.8$, $\mathsf{a}_{\rm f}=1.2$, $g_0^2=4$ and $N_{\rm S}=65$. The grey arrows points towards increasing adiabaticity. (a) Evolution of the pseudo-scalar condensate $\Pi$. We represent the absolute value of the condensate due to the ambiguity in its sign due to the $\mathbb{Z}_2$ symmetry breaking. (b) Evolution of the scalar condensate $\Sigma$.