Phases of translation-invariant systems out of equilibrium: iterative Green’s function techniques and renormalization group approaches

Abstract

We introduce a method to evaluate the steady-state non-equilibrium Keldysh–Schwinger Green’s functions for infinite systems subject to both an electric field and a coupling to reservoirs. The method we present exploits a physical quasi-translation invariance, where a shift by one unit cell leaves the physics invariant if all electronic energies are simultaneously shifted by the magnitude of the electric field. Our framework is straightaway applicable to diagrammatic many-body methods. We discuss two flagship applications, mean-field theories as well as a sophisticated second-order functional renormalization group approach. The latter allows us to push the renormalization-group characterization of phase transitions for lattice fermions into the out-of-equilibrium realm. We exemplify this by studying a model of spinless fermions, which in equilibrium exhibits a Berezinskii–Kosterlitz–Thouless phase transition.

Nonequilibrium Properties of Berezinskii-Kosterlitz-Thouless Phase Transitions

“…3), which is reassuring as quantum fluctuations should be small in these limits. In particular, the mean-field equations can be solved analytically for U → ∞ or E → ∞, and a transition between a charge ordered and a metallic state occurs at a critical value of Γ MF crit ¼ ð2=πÞU MF crit ≈ 0.64U MF crit [35]. Our RG scheme yields Γ RG crit ≈ 0.61U RG crit .…”

“…The resulting set of RG flow equations is involved, and their solution requires advanced concepts such as recursion relations for Green's functions in the presence of quasitranslation invariance. Details of this method can be found in a separate publication [35].…”

“…2. Phase diagram in the U-Γ plane for two different electric fields E. The color scale displays the susceptibility χ towards a charge density wave (white regions) obtained using our RG scheme [35]. It is known that the system is driven into a CDW phase at U crit =t ¼ 2 in the absence of reservoirs in equilibrium (Γ ¼ E ¼ 0).…”

Non-equilibrium Properties of Berezinskii-Kosterlitz-Thouless Phase Transitions

“…Phase diagram in the U -Γ-plane for two different electric fields E. The color scale displays the susceptibility χ towards a charge density wave (white regions) obtained using our RG scheme [26]. It is known that the system is driven into a CDW phase at Ucrit/t = 2 in the absence of reservoirs in equilibrium (Γ = E = 0).…”

“…The resulting set of RG flow equations is involved, and their solution requires advanced concepts such as recursion relations for Green's functions in the presence of quasi-translation invariance. Details of this method can be found in a separate publication [26].…”