Nonequilibrium Fixed Points of Coupled Ising Models

Driven-dissipative systems are expected to give rise to nonequilibrium phenomena that are absent in their equilibrium counterparts. However, phase transitions in these systems generically exhibit an effectively classical equilibrium behavior in spite of their nonequilibrium origin. In this paper, we show that multicritical points in such systems lead to a rich and genuinely nonequilibrium behavior. Specifically, we investigate a driven-dissipative model of interacting bosons that possesses two distinct phase transitions: one from a high- to a low-density phase—reminiscent of a liquid-gas transition—and another to an antiferromagnetic phase. Each phase transition is described by the Ising universality class characterized by an (emergent or microscopic) ${\mathbb{Z}}_2$ symmetry. However, they coalesce at a multicritical point, giving rise to a nonequilibrium model of coupled Ising-like order parameters described by a ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ symmetry. Using a dynamical renormalization-group approach, we show that a pair of nonequilibrium fixed points (NEFPs) emerge that govern the long-distance critical behavior of the system. We elucidate various exotic features of these NEFPs. In particular, we show that a generic continuous scale invariance at criticality is reduced to a discrete scale invariance. This further results in complex-valued critical exponents and spiraling phase boundaries, and it is also accompanied by a complex Liouvillian gap even close to the phase transition. As direct evidence of the nonequilibrium nature of the NEFPs, we show that the fluctuation-dissipation relation is violated at all scales, leading to an effective temperature that becomes “hotter” and “hotter” at longer and longer wavelengths. Finally, we argue that this nonequilibrium behavior can be observed in cavity arrays with cross-Kerr nonlinearities.

Popular Summary

Nonequilibrium systems, where energy is constantly injected into and dissipated out of the system, are challenging to understand, since much of physics is tuned to describe equilibrium conditions. However, extensive theoretical and experimental investigations have made it clear that an effective equilibrium behavior typically emerges even when the system is forcefully driven away from equilibrium. A natural question then arises: When can we expect to observe genuinely nonequilibrium behavior, phases, and phase transitions? Here, we explore this question in an open quantum system that is subjected to a coherent input of energy (a “drive”) and find that an intriguing answer emerges near multicritical points, where two macroscopic tendencies of the system coalesce and compete.

The competition between the drive and dissipation takes the system to a nonequilibrium steady state after a long period of time. While the notion of temperature is not available in these nonequilibrium systems, an effective thermal behavior typically emerges near phase transitions involving a simple order parameter. Nevertheless, we show that the interplay of two order parameters at a multicritical point leads to an exotic behavior that cannot be described by any kind of effective equilibrium. Specifically, we find a fractal-like structure, spiraling relaxation and phase boundaries, and an effective temperature that becomes “hotter and hotter” across longer and longer distances.

In short, our work reveals that multicritical points provide a rich avenue for discovering new nonequilibrium universality that is qualitatively different from what can occur in equilibrium systems.

Figure 1

Schematic illustration of the physical setup. The contrast of the two checkerboard (light yellow and dark blue) sublattices defines the antiferromagnetic order parameter. On each sublattice, there is a low-population (low $n_{\mathrm{p}\mathrm{h}}$; solid curve) and a high-population (high $n_{\mathrm{p}\mathrm{h}}$; dashed curve) steady state, corresponding to the bistability order parameter. The large arrow indicates the drive, while the wavy arrows represent the dissipation. $J$ and $V$ denote the hopping and nearest-neighbor interactions, respectively.

Figure 2

A schematic RG flow diagram projected to the $g_{12}-g_{21}$ plane. (The full RG flow requires a five-dimensional space, which precludes a more complete sketch of the RG flow in this two-dimensional space; see Sec. 4.) In addition to the equilibrium fixed points where $g_{12}=g_{21}$ (green circles), a pair of stable NEFPs (orange diamonds) emerge in the sector defined by the opposite signs of $g_{12}$ and $g_{21}$. These new fixed points exhibit exotic critical behavior reflecting their truly nonequilibrium nature. Filled (black) arrows represent the stability, while partial (gray) arrows indicate the expected stability of the various fixed points in different directions. Stability is known to lowest order in $\epsilon =4-d$ only in directions that preserve the ratio $g_{12}/g_{21}$. The RG flow cannot cross the $g_{12}$, $g_{21}$ axes, which is represented by double lines. The stable equilibrium fixed point is characterized by an emergent $O(2)$ symmetry, while the unstable equilibrium fixed points include a biconical fixed point as well as various decoupled fixed points (which all lie at the origin in this diagram) corresponding to combinations of Ising and Gaussian fixed points.

Figure 3

Summary of the main features of the NEFPs contrasted with effective equilibrium fixed points. (a) Schematic correlation functions. A generic continuous scale invariance characteristic of criticality is reduced to a discrete scale invariance at the NEFP. (b) Effective temperatures $T_i$ representing the two order parameters as a function of the length scale ($q^{-1}$). The two temperatures become identical at long length scales, but while they approach a constant at the equilibrium fixed point, they diverge at large scales at the NEFPs. (c) Gap closure upon approaching the critical point. Here, ${\mathrm{\Gamma }}_{\mathcal{L}}$ denotes the Liouvillian gap with the real part describing the relaxation rate (a.k.a. the dissipative gap) and the imaginary part characterizing the “coherence gap.” For the equilibrium fixed point, the gap can close only along the real line, indicated by the arrow. In contrast, the gap for the NEFP can take complex values and close along any path lying in the shaded region, making a maximum angle of $\pi /3$ with the real line. (d) Critical exponents to lowest nontrivial order in $\epsilon =4-d$. The exponent $\nu$, typically associated with the divergence of the correlation length, becomes complex valued at the NEFPs, with its imaginary part characterizing the discrete scale invariance [cf. part (a)]. Here, $\eta$ and ${\eta }^{\prime }$ are anomalous dimensions characterizing fluctuations and dissipation with $\eta \ne {\eta }^{\prime }$ at the NEFPs indicating the violation of the fluctuation-dissipation theorem. Note that $z$ is the dynamical critical exponent.

Figure 4

Phase diagram associated with the NEFPs of the nonequilibrium Ising model of two coupled fields for $h=0$. The white region indicates the disordered phase, $\mathrm{B}$ (red vertical shading) corresponds to the phase where the bistability order parameter undergoes spontaneous symmetry breaking, AF (blue horizontal shading) denotes antiferromagnetic ordering, and $\mathrm{B}+\mathrm{AF}$ (purple square shading) corresponds to the phase where both order parameters are nonzero. The solid black lines denote second-order phase transitions. The NEFP phase diagram exhibits logarithmic spirals in the phase boundaries. The other NEFP is described by an analogous diagram upon switching the roles of the two order parameters ($\mathrm{B}\leftrightarrow \mathrm{AF}$, $r_1\leftrightarrow r_2$).

Figure 5

Mean-field dynamics near the NEFP within the doubly ordered phase with $|M_1|=|M_2|$. The arrows denote how the fields ${\varphi }_i$ evolve in time, with four possible steady states. At each steady state, there is a dissipative relaxation process as well as a “coherent” rotation, resulting in a spiraling relaxation to the steady state. Two of the steady states spiral clockwise, while two spiral counterclockwise.

Figure 6

Dynamics of gapped (fast or massive) and soft (slow or massless) modes with arrows indicating the linear (in ${\varphi }_i,{\varphi }_i^{\prime }$) relaxation of the two modes. Near the critical point, the gapped mode quickly decays to a straight line defined by the slow direction of the soft mode. (a) Relaxation of the field ${\psi }_{\mathrm{B}}$ with the soft and gapped modes lying along the angles $\pi /3$ and 0, respectively. (b) Relaxation of the field ${\psi }_{\mathrm{AF}}$ for ${\mathrm{\Delta }}_c=1/3$ with the soft and gapped modes lying along the angles $-\pi /6$ and $\pi /6$, respectively. (c) Relaxation of the field ${\psi }_{\mathrm{AF}}$ for ${\mathrm{\Delta }}_c=2/3$ with the soft and gapped modes lying along the angles 0 and $\pi /3$, respectively. (We have adopted units where $\mathrm{\Delta }+J=1$.)

Figure 7

Phase diagram for (one of the two) nonequilibrium tetracritical points as a function of effective mass terms and magnetic field. The transparent boundary indicates the location of an antiferromagnetic phase transition from a uniform phase. The horizontal (red) surface denotes a surface of first-order phase transitions. In both AF and uniform phases, the latter indicates a transition from a low- to a high-population phase; the sublattice population difference changes continuously across this surface. The (black) square boundary in the middle indicates the plane of $h=0$. The diagram for the second nonequilibrium tetracritical point will spiral in the opposite direction.

Figure 8

Diagrammatic representation of Gaussian propagators. Solid (dotted) lines correspond to classical (response) fields. The two fields are later distinguished by thickness and color. In this figure, we have shown (a) the response propagator and (b) the correlation propagator.

Figure 9

Interaction vertices. Thin black (thick cyan) lines correspond to the first (second) field, and solid (dashed) lines correspond to the classical (response) field. (a) $u_1{\varphi }_1^3{\tilde{\varphi }}_1$. (b) $u_2{\varphi }_2^3{\tilde{\varphi }}_2$. (c) $u_{12}{\varphi }_2^2{\varphi }_1{\tilde{\varphi }}_1$. (d) $\sigma u_{12}{\varphi }_1^2{\varphi }_2{\tilde{\varphi }}_2$.

Figure 10

One-loop corrections to (a,b) $r_1{\varphi }_1{\tilde{\varphi }}_1$, (c,d) $u_1{\varphi }_1^3{\tilde{\varphi }}_1$, and (e)–(h) $u_{12}{\varphi }_2^2{\varphi }_1{\tilde{\varphi }}_1$. Analogous diagrams for $r_2{\varphi }_2{\tilde{\varphi }}_2$, $u_2{\varphi }_2^3{\tilde{\varphi }}_2$ and $\sigma u_{12}{\varphi }_1^2{\varphi }_2{\tilde{\varphi }}_2$ can be obtained by switching thin black and thick cyan lines.

Figure 11

Two-loop corrections to (a)–(c) ${\zeta }_1$ and $D_1$ as well as (d,e) ${\zeta }_1{T}_1$. Analogous diagrams for ${\zeta }_2$, $D_2$, and ${\zeta }_2{T}_2$ can be obtained by switching thin black and thick cyan lines.