Inequality for Strong-Weak Spontaneous Symmetry Breaking in Fermionic Open Quantum systems

Abstract

Under decoherence, an initial Gaussian (free-fermion) state evolves into a non-Gaussian mixed state, so the resulting decohered fermionic state is not exactly solvable in general. We show through an inequality that a class of Rényi-2 correlators of the decohered fermion state are upper-bounded by the Rényi-2 correlator serving as a proximate diagnostic of strong-weak spontaneous symmetry breaking (SW-SSB) of the charge-U(1) symmetry. This inequality holds for arbitrary decoherence strength and suggests that decoherence drives fermionic quantum matter toward U(1) SW-SSB. We also make connections between our inequality and other subjects such as projected quantum spin Hall insulator and Dirac spin liquid states.

Figures (2)

• Figure 1: The Euclidean spacetime path integral (Eq. \ref{['PI0']}) representation of correlator $C(x, y)$. Action ${\cal S} _0$ is defined in the entire spacetime; the operators $O_x$ and $O_y$ as well as the effective Hamiltonian $H_{\rm eff} (g)$ arising from dephasing are inserted at $\tau = 0$.
• Figure 2: Two types of Feynman diagrams (see also Ref. vafawittenwitteninequality) potentially contributing to the correlator of $O = c^\dagger \sigma^\mu \Omega c$. Type-$a$. "Connected" diagrams, position $x$ and $y$ are connected by Fermion lines; and Type-$b$. "disconnected" diagrams. The inequality discussed in this work applies to type-$a$ diagrams. Type-$b$ diagram vanishes when $\mathrm{tr}[\Omega] = 0$.