Quantum and Classical Potts Criticality in Driven-Dissipative Bosonic Lattices

Technical Summary: Quantum and Classical Potts Criticality in Driven-Dissipative Bosonic Lattices

Problem Statement
The emergence of critical phenomena in the nonequilibrium steady states (NESS) of open quantum systems has been a focal point of research, particularly regarding whether these systems can exhibit universal behaviors analogous to equilibrium statistical mechanics. While driven-dissipative photonic lattices have successfully demonstrated transitions belonging to the $Z_2$ Ising universality class (both classical and quantum, depending on the presence of two-photon losses), a significant gap remains regarding higher discrete symmetries. Specifically, the archetypal $q$-state Potts model ($q > 2$), which governs rich critical phenomena in equilibrium physics (such as continuous transitions for $q \leq 4$ in 2D and non-Ising critical exponents for $q=3$), has no established counterpart in the universality classes of open quantum matter. Furthermore, characterizing symmetry breaking beyond $Z_2$ is computationally challenging; three-photon driving and losses generate strongly correlated, non-Gaussian many-body states with Wigner function negativities that render standard stochastic phase-space methods (like truncated-Wigner or positive-P) intractable due to the breakdown of the Fokker-Planck structure.

Methodology
To address these challenges, the authors employ the Variational Multi-Gaussian (VMG) approach. This method represents the many-body Wigner function as a superposition of complex-centered Gaussian components, allowing for systematically controllable accuracy.

• Model: The study focuses on a lattice of coupled quantum nonlinear optical resonators subject to three-photon parametric driving, described by a cubically-driven Bose-Hubbard model. The system is coupled to a Markovian bath featuring both single-photon losses (rate $\gamma$) and three-photon losses (rate $\eta_3$).
• Symmetry: The Lindbladian admits a $Z_3$ symmetry ($\hat{a}_j \to e^{i2\pi k/3}\hat{a}_j$). The order parameter is defined as the $Z_3$ parity operator, $\hat{Z}_3 = \exp(i\frac{2\pi}{3}\sum_j \hat{a}^\dagger_j \hat{a}_j)$.
• Simulation: The VMG ansatz is evolved via the Dirac–Frenkel variational principle using automatic differentiation. Crucially, the variational components are organized into $Z_3$-symmetric triples to respect the underlying symmetry of the Liouvillian. The order parameter is computed analytically from the Weyl symbol of the ansatz.
• Validation: The method is benchmarked against exact numerical simulations of a single-mode cubically driven Kerr oscillator, demonstrating high accuracy even in regimes with strong three-photon losses and pronounced Wigner negativities.

Key Results
The authors perform finite-size scaling analyses in both 1D and 2D to uncover the nature of the phase transitions:

1. 2D Lattices (Single-Photon Losses Only): In the absence of three-photon losses ($\eta_3 = 0$), the system undergoes a continuous symmetry-breaking transition. Finite-size scaling collapse using the critical exponents of the 2D classical three-state Potts model ($\beta = 1/9, \nu = 5/6$) confirms that the nonequilibrium transition belongs to this classical universality class.
2. 1D Chains (Single- and Three-Photon Losses): When three-photon losses are present ($\eta_3 \neq 0$), the system in 1D exhibits a continuous transition governed by the 1D quantum three-state Potts universality class. The scaling collapse utilizes the same critical exponents ($\beta = 1/9, \nu = 5/6$), consistent with the standard quantum-to-classical correspondence where the 1D quantum Potts model maps to the 2D classical Potts model.
3. Effective Hamiltonian: In the strong-driving limit, the effective energy of the lattice maps exactly to the ferromagnetic $q=3$ Potts Hamiltonian (or its equivalent three-state clock representation), providing a theoretical basis for the observed universality.

Significance and Claims
The paper claims to provide the first identification of Potts criticality (specifically the three-state case) in driven-dissipative bosonic lattices, extending the known landscape of emergent universality classes beyond the $Z_2$ Ising paradigm.

The authors establish a general correspondence in multiphoton driven-dissipative bosonic systems:

• The order of the multiphoton drive ($n$-photon driving) determines the underlying discrete symmetry ($Z_n$) and the associated universality class.
• The presence or absence of corresponding multiphoton losses determines whether the emergent critical behavior is classical (absence of $n$-photon losses) or quantum (presence of $n$-photon losses).

This work demonstrates that equilibrium universality classes, such as the Potts model, can emerge from genuinely nonequilibrium many-body dynamics. The authors suggest this framework may extend to higher discrete symmetries (e.g., $n=4$ potentially realizing the Ashkin–Teller universality class), though this remains an open question not addressed in the current study.