Quantum dynamical field theory for nonequilibrium phase transitions in driven open systems

We develop a quantum dynamical field theory for studying phase transitions in driven open systems coupled to Markovian noise, where nonlinear noise effects and fluctuations beyond semiclassical approximations influence the critical behavior. We systematically compare the diagrammatics, the properties of the renormalization group flow, and the structure of the fixed points of the quantum dynamical field theory and of its semiclassical counterpart, which is employed to characterize dynamical criticality in three-dimensional driven-dissipative condensates. As an application, we perform the Keldysh functional renormalization of a one-dimensional driven open Bose gas, where a tailored diffusion Markov noise realizes an analog of quantum criticality for driven-dissipative condensation. We find that the associated nonequilibrium quantum phase transition does not map into the critical behavior of its three-dimensional classical driven counterpart.

Figure 1

Condensate density ${\rho }_0\propto -\chi$ and number particle density $n\propto \sqrt{\left|\chi \right|}$ in the vicinity of the driven-dissipative transition.

Figure 2

Diagrammatic representation of the retarded, advanced, and Keldysh Green's functions (propagators), and of the vertices of the action (15).

Figure 3

A comparison between the Markovian nonequilibrium noise $P_{\mathrm{neq}}^K$ (b) and the equilibrium non-Markovian quantum noise $P_{\mathrm{eq}}^K={lim}_{T\to 0}coth\left(\frac{\omega }{2T}\right)=\left|\omega \right|$, which constitutes the zero-temperature limit [blue line in (a)] of the noise induced by a Caldeira-Leggett bath [red line in (a)]. In particular, in (b) the red straight line indicates that there is a region of low momenta $\left(q<{\mathrm{\Lambda }}_M\right)$ where the flat Markovian noise (red line) dominates over the diffusion Markov noise (blue line).

Figure 4

The quantum optics architecture employed to engineer the Lindblad operator $L_d={\partial }_x\varphi \left(x\right)$. Yellow curved lines stand for hopping processes among bosonic excitations inside the cavities.

Figure 5

One-loop diagram renormalizing the momentum and frequency-dependent part of the retarded propagator $G^R$. It can be seen as a two-loop diagram, where an internal Keldysh line has been cut, and the two resulting free lines have been replaced by the nonvanishing expectation value of the classical field in the ordered phase ${\overline{\varphi }}_c={{\overline{\varphi }}^{*}}_c=\sqrt{{\varphi }_0}$. Its frequency and momentum dependence for $q\to 0$ and $\omega \to 0$ generates the renormalization of $Z$ and $\overline{K}$, and allows us to compute their anomalous dimensions. In this diagram, and in others to follow, we drop the overline on the couplings.

Figure 6

The picture shows two one-loop diagrams supported by couplings present at the microscale $\left(u_q{{|}_{k\sim k_{\text{UV}}}=\lambda +i{\gamma }_t,g_1|}_{k\sim k_{\text{UV}}}=2{\gamma }_t\right)$, which contribute to effectively generate the non-Markovian noise coupling $g_2$ during RG flow.

Figure 7

One-loop corrections (tadpoles) to the spectral $\overline{\chi }$ and Keldysh mass $\overline{\gamma }$. A Keldysh Green's function $G^K$ occurs as internal line in tadpoles. In the nonequilibrium semiclassical model, the diagram contributing to $\mathrm{\Delta }\overline{\gamma }$ is, instead, a two-loop diagram where one internal Keldysh line has been cut and the two resulting external lines couple to the expectation values of the fields in the ordered phase, which is proportional to $\sqrt{{\overline{\varphi }}_0}$. Notice that the leading correction to the condensate density ${\overline{\rho }}_0$ is also determined by the tadpole occurring in $\mathrm{\Delta }\overline{\chi }$. Depending whether the truncation for ${\mathrm{\Gamma }}_k$ is centered in the ordered $\left({\overline{\rho }}_0\ne 0\right)$ or disordered phase $\left({\overline{\rho }}_0=0\right)$ [113], one has to renormalize directly $\overline{\chi }$ or the condensate density ${\overline{\rho }}_0$, through which the retarded mass is defined $\left(\overline{\chi }\sim {\overline{\rho }}_0{\overline{\kappa }}_c\right)$ (see related discussion in Secs. 2a and 3b1).

Figure 8

Sample of diagrams occurring as one-loop corrections to the quartic couplings in the quantum dynamical field theory. We distinguish three classes of one-loop diagrams, containing the combinations ${\overline{G}}^{R/A}{\overline{G}}^{A/R},{\overline{G}}^{R/A}{\overline{G}}^K$, and ${\overline{G}}^K{\overline{G}}^K$. We report a subset of the corrections to quartic couplings (specifically, the ones related to the coherent couplings of the quartic action). On the contrary, there is a single one-loop diagram, proportional to ${\overline{g}}_c^2$, in the diagrammatics of the semiclassical action since ${\overline{\varphi }}_c^{*2}{\overline{\varphi }}_c{\overline{\varphi }}_q$ is the only RG relevant operator.

Figure 9

Comparison between the fixed points of the nonequilibrium quantum action and of the semiclassical action of driven-dissipative Bose gases. The fixed point of the former displays simultaneous presence of finite dissipative and coherent couplings (absence of decoherence). The fixed point of the latter is instead entirely dissipative (decoherence at long wavelengths), and at infrared momenta the couplings flow towards the imaginary axis.

Figure 10

A renormalization group flow with all the couplings lying on a single ray is a sufficient condition for infrared thermalization at the fixed point, as occurs in the RG study of three-dimensional semiclassical (SC) criticality, in the presence of flat Markov noise. This feature is absent in the RG flow of the quantum dynamical field theory (no common rotation towards the dashed red line in the main panel), suggesting already that the quantum fixed point cannot possess thermal features.

Figure 11

Diffusion ${\gamma }_d{q}^2$ (blue line) and flat $\gamma$ (red line) Markov noises as a function of momentum. For momentum scales $q>{\mathrm{\Lambda }}_G$, the RG flow is dominated by nonuniversal effects, while below ${\mathrm{\Lambda }}_G$ it is controlled by the quantum FP, and universal effects associated to the nonequilibrium phase transitions emerge. For $q<{\mathrm{\Lambda }}_M$, the flat Markovian noise level $\gamma$ dominates over the diffusion noise, and the system departs from the universal quantum scaling regime.