In modern theoretical physics, analyzing strongly coupled quantum systems remains a central challenge, as conventional perturbation methods fail outside the weak-coupling regime. Traditional techniques expand around a non-interacting (or Gaussian) vacuum, which becomes unstable or inappropriate in strongly interacting contexts, such as quantum field theory in four dimensions or spin-imbalanced fermionic systems. These limitations hinder our understanding of spontaneous symmetry breaking, phase transitions, and the emergence of non-perturbative phenomena in high-energy physics and condensed matter systems.
I have focused on extending and applying the adaptive perturbation method, a non-standard perturbative approach originally introduced in quantum mechanics, to address these obstacles. This method avoids direct expansion in the coupling constant by using a variational principle to adapt the unperturbed Hamiltonian to better reflect the system's physical vacuum. Through this approach, I have demonstrated new capabilities for tackling non-trivial vacua and providing analytical insights into regimes traditionally dominated by numerical methods such as lattice simulations or Monte Carlo calculations.
Spontaneous symmetry breaking occurs when the underlying laws of a physical system are symmetric, but the vacuum state chosen by the system is not. The (3+1)d $\phi^4$ theory is relatively simple compared to other more complex theories, making it a good starting point for investigating the origin of non-trivial vacua. The adaptive perturbation method is a technique used to handle strongly coupled systems. The study of strongly correlated systems is useful in testing holography. It has been successful in strongly coupled QM and is being generalized to scalar field theory to analyze the system in the strong-coupling regime. The unperturbed Hamiltonian does not commute with the usual number operator. However, the quantized scalar field admits a plane-wave expansion when acting on the vacuum. While quantizing the scalar field theory, the field can be expanded into plane-wave modes, making the calculations more tractable. However, the Lorentz symmetry, which describes how physical laws remain the same under certain spacetime transformations, might not be manifest in this approach. The proposed elegant resummation of Feynman diagrams aims to restore the Lorentz symmetry in the calculations. The results obtained using this method are compared with numerical solutions for specific values of the coupling constant $\lambda = 1, 2, 4, 8, 16$. Finally, we find evidence for quantum triviality, where self-consistency of the theory in the UV requires $\lambda = 0$. This result implies that the $\phi^4$ theory alone does not experience SSB, and the $\langle \phi\rangle = 0$ phase is protected under the RG-flow by a boundary of Gaussian fixed-points.
From ultracold atoms to quantum chromodynamics, reliable ab initio studies of strongly interacting fermions require numerical methods, typically in some form of quantum Monte Carlo calculation. Unfortunately, the nonrelativistic systems at finite density generally have a sign problem. In the relativistic case, imaginary chemical potentials solve this problem. Is this feasible for nonrelativistic systems? We introduce a complex chemical potential to avoid the sign problem in the nonrelativistic case. To give a first answer to the above questions, we perform a mean-field study of the finite-temperature phase diagram of spin-1/2 fermions with imaginary polarization.