Vacuum structure and $\mathcal{P}\mathcal{T}$-symmetry breaking of the non-Hermetian ($i{\varphi }^3$) theory

In this work, we study the $\mathcal{P}\mathcal{T}$-symmetric ($i{\varphi }^3$) theory using the effective action formalism. To test the accuracy of the used technique, we apply it first to the $\mathcal{P}\mathcal{T}$-symmetric ($-{\varphi }^4$) theory, where we reproduce the same results obtained in the literature using the method of Dyson-Schwinger equations. In $0+1$ space-time dimensions, the one-loop effective potential prediction for the ($i{\varphi }^3$) theory ought to be more accurate than WKB results. The effective potential for the massless $\mathcal{P}\mathcal{T}$-symmetric ($i{\varphi }^3$) model is shown to be bounded from below, which is the first analytic result that advocates the vacuum stability of this theory. Our calculations show that the massless theory possesses only one stable vacuum as in the literature, but for the massive theory we find that there exist two stable vacua. For a nonzero magnetic field, we show that the $\mathcal{P}\mathcal{T}$-symmetry of the theory is broken for negative imaginary magnetic field, which agrees with the Lee-Yang theorem. We argue that $\mathcal{P}\mathcal{T}$-symmetry breaking is a manifestation of the Yang-Lee edge singularity.

Figure 1

The effective potential versus vacuum condensate ($v/i$) for the massless $\mathcal{P}\mathcal{T}$-symmetric $i{\varphi }^3$ theory. The left part for $g=1$ gives a stable vacuum for a negative imaginary condensate, while the right part represents the effective potential for $g=-1$ and is stable for a positive imaginary condensate.

Figure 2

The Stokes wedges for the massless $\mathcal{P}\mathcal{T}$-symmetric $i{\varphi }^3$ theory. There exists only one pair of noncontiguous $\mathcal{P}\mathcal{T}$-symmetric wedges, and thus according to the conjecture in Ref. [30] the theory possesses only one stable vacuum.

Figure 3

The effective potential versus vacuum condensate for the massive $\mathcal{P}\mathcal{T}$-symmetric $i{\varphi }^3$ theory. The left part for $g=1$ gives two stable vacua for a negative imaginary condensate, while the right part represents the effective potential for $g=-1$ with stable vacua for a positive imaginary condensate.

Figure 4

The Stokes wedges for the massive $\mathcal{P}\mathcal{T}$-symmetric $i{\varphi }^3$ theory. The one pair of noncontiguous $\mathcal{P}\mathcal{T}$-symmetric wedges for the massless case is now divided into two pairs. The dark gray wedges having an opening angle of $\frac{\pi }{4}$ represent the case where real parts of both terms in the potential are positive. The light gray wedges in the lower half-plane having an opening angle of $\frac{\pi }{12}$ represent the case of a positive real part of the dominant term in the potential, while the massive term has a negative real part.

Figure 5

The effective potential versus the coupling $\gamma$ for the $\mathcal{P}\mathcal{T}$-symmetric $i{\varphi }^3$ theory. For coupling values smaller than $\gamma ={\gamma }_c=-0.57435$, the vacuum energy as well as the vacuum condensate are complex and the $\mathcal{P}\mathcal{T}$-symmetry has been spontaneously broken, which resembles a Yang-Lee edge singularity.