Nonequilibrium effective field theory for absorbing state phase transitions in driven open quantum spin systems

Phase transitions to absorbing states are among the simplest examples of critical phenomena out of equilibrium. The characteristic feature of these models is the presence of a fluctuationless configuration which the dynamics cannot leave, which has proved a rather stringent requirement in experiments. Recently, a proposal to seek such transitions in highly tunable systems of cold-atomic gases offers to probe this physics and, at the same time, to investigate the robustness of these transitions to quantum coherent effects. Here, we specifically focus on the interplay between classical and quantum fluctuations in a simple driven open quantum model which, in the classical limit, reproduces a contact process, which is known to undergo a continuous transition in the “directed percolation” universality class. We derive an effective long-wavelength field theory for the present class of open spin systems and show that, due to quantum fluctuations, the nature of the transition changes from second to first order, passing through a bicritical point which appears to belong instead to the “tricritical directed percolation” class.

Figure 1

Mean field phase diagram of the quantum contact process. The system can undergo a phase transition from an absorbing state towards an active, nonzero density phase, which can be either continuous (solid line) or first order (dashed line). The second- and first-order lines meet at a bicritical point. The axes represent the rescaled classical branching rate $\chi =z\kappa /\gamma$ and the quantum branching rate $\omega =z\mathrm{\Omega }/\gamma$ and correspond to the classical (A) and quantum (B) limits, respectively. The dotted diagonal line indicates the competing regime (C). On the right, the corresponding evolution of the potential as a function of the axes' parameters is shown for (i) the first-order transition and (ii) the second-order transition.

Figure 2

Scaling regimes for the second-order phase transition in dimensions $d=3,2,1$. In the white region, mean field scaling behavior according to classical directed percolation (DP) is observed, while the blue region corresponds to critical scaling of the classical DP universality class below the Ginzburg scale. In the yellow regions, the dynamics is dominated by the mean field behavior of the quantum contact process (QP). The critical behavior of the QP corresponds to the bicritical point and is found in the red region. The black (red) dashed line indicates the line of second- (first-) order transitions. Remarkably, in one dimension, the first-order transition is located partly in the critical regime and experiences strong infrared corrections.

Figure 3

Diagrammatic representation of the (a) one-loop and (b) two-loop correction of the vertex ${\mu }_3{\tilde{n}}^{2}n$. Ingoing lines represent density fields $n$, while outgoing lines represent response fields $\tilde{n}$. The retarded propagator (3.11) corresponds to a directed line, where the arrow points from earlier to later times [the propagator vanishes if this order is inverted, see (3.12)].

Figure 4

Optimal path approximation. (a) Phase space $(n,\tilde{n})$ trajectories for parameters $\omega =1.6,\chi =0.5$ [represented by the dot in parameter space in (b)]. The optimal path trajectories are displayed in red. One distinguishes between the deterministic, zero-noise trajectory (dashed line) and the noise-dominated trajectory (solid line). (b) Shift of the first-order transition line from the mean field result $\mathrm{\Gamma }\left(n_{\text{MF}}\right)=0$ (red) to the one determined by the optimal path approach $\mathcal{F}\left(n_{\text{MF}}\right)=0$ (blue) in the presence of temporal, noise-induced fluctuations.

Figure 5

First-order phase transition at a finite system size $V<\infty$. The figures (a)–(c) display the density distribution function $P\left(n\right)$ in Eq. (5.13) as in the $(\omega ,n)$ plane for fixed value of $\chi =0.4$ for different system sizes $V$ in units of the $d$-dimensional unit volume $a^d$. In the thermodynamic limit, a first-order transition occurs at ${\omega }_c\approx 1.6$. (d) The density expectation value $\langle n\rangle =\int dnnP\left(n\right)$ is plotted as a function of $\omega$ for $\chi =0.4$. The different plots correspond to the volume $V$ in (a)–(c). In both rows, the discontinuity at $\omega ={\omega }_c$ is established for increasing system size and is significant already for moderate system sizes of $V=5000a^d$.

Figure 6

First-order transition in the quantum limit for a one-dimensional chain of 12 sites. The main panels are density plots of the distribution $P\left(n\right)$ of the excitation density $n$ as a function of $\omega =\mathrm{\Omega }/\gamma$. Each inset shows an instance for fixed $\mathrm{\Omega }=8\gamma$ (indicated by the white dashed line) from the corresponding panel. The four panels correspond to (a) main model [Eq. (2.4) plus decay] in the quantum limit $\kappa =0$, (b) effective antiblockaded model corresponding to Eq. (6.2) (plus decay) with $V_{lm}$ set to 0, (c) Rydberg chain [Eq. (6.1) plus decay] with nearest-neighbor interactions only, (d) Rydberg chain with full van der Waals tails. All plots display a crossover from an (almost) absorbing state at small $\mathrm{\Omega }$ to a state with finite excitation density. For intermediate values of $\mathrm{\Omega }$ the counting statistics in the insets feature a bimodal shape which can be regarded as a signature of the anticipated first-order phase transition. All results are obtained via quantum-jump Monte Carlo calculations [114] with averages performed over 1000 runs. The simulation times are $\gamma t=4$ (a) and $\gamma t=6$ (b)–(d). For the computations for the Rydberg systems [panels (c), (d)], the remaining parameters have been set to $\mathrm{\Delta }=10\mathrm{\Omega }=-V$. Note that the color bar is bounded by 0.3, despite the peak around the absorbing state exceeding this threshold (for lower values of $\mathrm{\Omega })$. This was done in order to improve visibility of the finite-density features.

Figure 7

Optimal trajectories at $H[n,\tilde{n}]=0$ in the $\left(n,\tilde{n}\right)$ phase space for $\chi =0.05$ and $\omega =0.3671$. Black circles identify the stationary points, whereas the arrows indicate the direction the dynamics proceeds towards along each path. The dashed lines correspond to the deterministic solution $\tilde{n}=0$ (divided in paths B and D up to $n=n_{\text{MF}})$ and the irrelevant one $n=0$. The solid line is instead the escape solution $\tilde{n}={\mathrm{\Gamma }}^{\prime }\left[n\right]/\mathrm{\Xi }\left[n\right]$ and includes paths A and C. The grayed area corresponds to the action $S(n,n_U)$ which defines the rate $w(n\to n_U)=e^{-S(n,n_U)}$.

Figure 8

Stochastic process which mimics the dynamics of large fluctuations in the system. Small nodes in this graph indicate stationary $n$ field configurations, whereas larger ones denote sets of transient values, as defined in the text. Arrows indicate allowed transitions, with the letters referring to the trajectory being followed.

Figure 9

Stationary mean field phase diagram as extracted from Eqs. (F1a)–(2.1). The dark blue patch at $\chi <1$ corresponds to the absorbing phase. The solid red line highlights where the second-order transition occurs to the active phase, whereas the dashed yellow line separates the domain of real solutions (RS), corresponding to the areas labeled with “$3\text{RS}$” from where the (nonabsorbing) solutions are complex [one real solution (“$1\text{RS}$”) regions]. This line joins the second-order one at the bicritical point $(\omega ,\chi )=(1,1)$. Its upper branch denotes the appearance of a second, physically acceptable, attractive solution, whose density is displayed in the upper left portion of the diagram. The dashed-dotted, white line indicates $\chi ={\omega }^2$ and separates regions (II) and (III) defined in this section. Finally, the vertical dashed red line separates the region of stability of the absorbing solution $\left(\chi <1\right)$ from the region in which the latter is unstable $\left(\chi >1\right)$.