Fermion Systems in Different Dimensions

Vincent Rivasseau

Alireza Beygi, S. P. Klevansky, Carl M. Bender

Fermionic systems differ from their bosonic counterparts, the main difference with regard to symmetry considerations being that ${\mathcal{T}}^{2}=\ensuremath{-}1$ for fermionic systems. In $\mathcal{PT}$-symmetric quantum mechanics an operator has both $\mathcal{PT}$ and $\mathcal{CPT}$ adjoints. Fermionic operators $\ensuremath{\eta}$, which are quadratically nilpotent (${\ensuremath{\eta}}^{2}=0$), and algebras with $\mathcal{PT}$ and $\mathcal{CPT}$ adjoints can be constructed. These algebras obey different anticommutation relations: $\ensuremath{\eta}{\ensuremath{\eta}}^{\mathcal{PT}}+{\ensuremath{\eta}}^{\mathcal{PT}}\ensuremath{\eta}=\ensuremath{-}\mathbb{1}$, where ${\ensuremath{\eta}}^{\mathcal{PT}}$ is the $\mathcal{PT}$ adjoint of $\ensuremath{\eta}$, and $\ensuremath{\eta}{\ensuremath{\eta}}^{\mathcal{CPT}}+{\ensuremath{\eta}}^{\mathcal{CPT}}\ensuremath{\eta}=\mathbb{1}$, where ${\ensuremath{\eta}}^{\mathcal{CPT}}$ is the $\mathcal{CPT}$ adjoint of $\ensuremath{\eta}$. This paper presents matrix representations for the operator $\ensuremath{\eta}$ and its $\mathcal{PT}$ and $\mathcal{CPT}$ adjoints in two and four dimensions. A $\mathcal{PT}$-symmetric second-quantized Hamiltonian modeled on quantum electrodynamics that describes a system of interacting fermions and bosons is constructed within this framework and is solved exactly.

Leon A. Takhtajan

Michał Papaj, Hiroki Isobe, Liang Fu

We show that disordered Dirac fermion systems in two dimensions generally exhibit bulk Fermi arc replacing the Dirac point in the clean limit, provided that the Dirac fermion arises from two distinct orbitals unrelated by symmetry. This result is explicitly demonstrated in a disordered Dirac model that we introduce, where the disorder potential acts differently on the two orbitals and drives the system into a new strongly disordered phase.

Bitan Roy, Vladimir Juričić, Igor F. Herbut

We study the renormalization group flow of the velocities in the field theory describing the coupling of the massless quasi-relativistic fermions to the bosons through the Yukawa coupling, as well as with both bosons and fermions coupled to a fluctuating U(1) gauge field in two and three spatial dimensions. Different versions of this theory describe quantum critical behavior of interacting Dirac fermions in various condensed-matter systems. We perform an analysis using one-loop ϵ-expansion about three spatial dimensions, which is the upper critical dimension in the problem. In two dimensions, we find that velocities of both charged fermions and bosons ultimately flow to the velocity of light, independently of the initial conditions, the number of fermionic and bosonic flavors, and the value of the couplings at the critical point. In three dimensions, due to the analyticity of the gauge field propagator, both the U(1) charge and the velocity of light flow, which leads to a richer behavior than in two dimensions. We show that all three velocities ultimately flow to a common terminal velocity, which is non-universal and different from the original velocity of light. Therefore, emergence of the Lorentz symmetry in the ultimate infrared regime seems to be a rather universal feature of this class of theories in both two and three dimensions.