Nonequilibrium universality of the nonreciprocally coupled $\mathbf{O(n_1) \times O(n_2)}$ model

This paper extends the study of nonreciprocally coupled models to general $O(n_1) \times O(n_2)$ symmetries, demonstrating the emergence of nonequilibrium fixed points characterized by fluctuation-dissipation violations, underdamped oscillations, and exceptional-point-induced discrete scale invariance, while also identifying a distinct universality class for extreme nonreciprocity.

The Gist

Imagine you are watching a crowded dance floor. In a normal, calm party (what physicists call equilibrium), everyone eventually settles into a rhythm. If you push someone, they wobble a bit and then stop, slowly returning to the beat. This is "overdamped" motion: slow, steady, and predictable.

But what happens if the dance floor is chaotic, noisy, and the dancers are constantly being pushed by invisible hands from the outside? This is nonequilibrium. In this paper, the authors explore a very specific kind of chaos: nonreciprocal interactions.

The Core Idea: The One-Way Street of Influence

In a normal relationship (or a normal physical system), if Person A pushes Person B, Person B pushes back with an equal force. This is reciprocity. It's like a conversation where both people listen and speak.

In this paper, the authors study a system where the rules are broken. Imagine Person A can push Person B hard, but Person B cannot push Person A back at all. Or, even stranger, imagine Person B pushes Person A in the opposite direction of what you'd expect. This is nonreciprocity.

The authors ask: What happens to the "dance" (the phase transition) when two groups of dancers are connected by these one-way, weird rules?

The Characters: The O(n) Dancers

To study this, they use a mathematical model with two groups of dancers:

• Group 1: Has $n_1$ different ways to move (like spinning in different directions).
• Group 2: Has $n_2$ different ways to move.

In the past, they only studied the simplest case where both groups had just 1 way to move (like a simple "yes/no" switch). In this new paper, they ask: Does this weird behavior happen if the dancers have more complex moves (like spinning in 3D space)?

The Big Discoveries

Here is what they found, translated into everyday metaphors:

1. The "Hotter" Temperature

In a normal system, if you look at it from far away (large scales), the "temperature" (how much chaos there is) stays the same.
In this weird nonreciprocal system, the authors found that as you zoom out, the system gets "hotter." It's like looking at a fire: up close, it's warm, but from a distance, it feels like an inferno. The "noise" and chaos grow stronger the further you look, breaking the usual rules of physics.

2. The "Ghostly" Spiral (Discrete Scale Invariance)

Usually, when a system reaches a critical point (like water turning to ice), it looks the same no matter how much you zoom in or out. It's perfectly smooth.
But in this system, the authors found something magical: Discrete Scale Invariance.
Imagine a spiral staircase or a nautilus shell. If you zoom in, it doesn't look exactly the same; it looks the same only if you zoom in by a specific amount (like stepping up exactly one stair).
The system behaves like a fractal. The "phase boundaries" (the lines separating different states of matter) don't just meet; they spiral around the critical point. It's as if the system is whispering a secret code that repeats in a loop.

3. The "Underdamped" Dance (Oscillations)

In a normal system, if you disturb the dancers, they wobble and stop (overdamped).
In this nonreciprocal system, the dancers start swinging back and forth like a pendulum that never quite stops (underdamped). They oscillate wildly right before the system changes state. This is a completely new type of behavior that doesn't exist in normal, calm physics.

4. The "One-Way" Coupling

The authors also looked at a case where Group A influences Group B, but Group B completely ignores Group A.

• The Surprise: In the simplest version of this (the old paper), the math broke down completely. But with these more complex dancers ($n_1 \neq n_2$), the math actually works!
• The Result: Group B behaves like a normal, calm dancer (equilibrium). But Group A, which is being influenced by B, becomes chaotic and "hotter," even though B doesn't care about A. It's like a shy person (B) talking to a loud person (A); the loud person gets even louder and crazier, while the shy person stays calm.

Why Does This Matter?

This isn't just about abstract math. These rules describe real-world things:

• Active Matter: Flocks of birds, schools of fish, or bacteria swarms where individuals react to each other but not always symmetrically.
• Quantum Systems: Lasers and quantum computers where information flows one way.
• Engineering: Designing materials that only let sound or light travel in one direction (like a one-way mirror for sound).

The Takeaway

The authors discovered that when you break the rule of "fair exchange" (reciprocity) in a system, you don't just get a slightly messier version of normal physics. You get brand new universality classes.

It's like discovering a new element on the periodic table. These systems have their own unique "personality": they get hotter as you zoom out, they dance in spirals, and they swing wildly instead of settling down. This opens the door to understanding and designing a whole new world of materials and systems that operate far from equilibrium.

Technical Summary

1. Problem Statement

The paper addresses the fundamental understanding of nonequilibrium phase transitions, specifically focusing on systems driven by nonreciprocal interactions. While equilibrium phase transitions are well-understood via the renormalization group (RG) and universality classes, nonequilibrium systems often violate the fluctuation-dissipation theorem (FDT) and exhibit exotic critical behaviors.

Previous work by the authors (Ref. [62]) identified a new form of nonequilibrium universality in a system with two coupled Ising order parameters ($Z_2 \times Z_2$, or $O(1) \times O(1)$) exhibiting strong nonreciprocity (where the coupling between fields has opposite signs). This work extends that analysis to a generalized $O(n_1) \times O(n_2)$ model. The central questions are:

• How does the universality class change when the order parameters have continuous rotational symmetries ($n_1, n_2 > 1$)?
• What are the critical phenomena (exponents, scaling, stability) for these generalized symmetries?
• How do the results differ from the corresponding equilibrium models and the previous $Z_2 \times Z_2$ case?
• What happens in the extreme limit of one-way coupling (where one field affects the other, but not vice versa), a regime previously found to be non-perturbative for $n_1=n_2=1$?

2. Methodology

The authors employ a perturbative field-theoretic renormalization group (RG) analysis in $d = 4 - \epsilon$ dimensions.

• Model Definition: The system consists of two vector order parameters, $\Phi_1$ and $\Phi_2$, with $n_1$ and $n_2$ components, respectively. The dynamics are governed by Langevin equations with nonreciprocal coupling terms ($g_{12}\Phi_1|\Phi_2|^2$ and $g_{21}|\Phi_1|^2\Phi_2$).
• Nonreciprocity Parameter ($\sigma$): The degree of nonreciprocity is characterized by $\sigma = \text{sgn}(g_{21}/g_{12})$.
  • $\sigma = 1$: Reciprocal coupling (effectively equilibrium-like).
  • $\sigma = -1$: Strong nonreciprocity (opposite signs), the focus of the nonequilibrium study.
  • $\sigma = 0$: One-way coupling (one field is independent).
• Formalism: The authors use the Martin-Siggia-Rose-Janssen-De Dominicis (MSRJD) response function formalism. They construct an action involving the fields $\Phi_i$ and response fields $\tilde{\Phi}_i$.
• RG Analysis:
  • They calculate beta functions for the coupling constants ($u_1, u_2, u_{12}$), the temperature ratio ($v = T_2/T_1$), and the stiffness ratio ($w = D_2/D_1$) up to one-loop order for couplings and two-loop order for dynamic parameters ($\zeta, D, T$).
  • They identify Fixed Points (FPs) by solving $\beta_i = 0$.
  • They analyze the stability matrix (eigenvalues of the linearized RG flow) to determine which fixed points are stable.
  • They compute critical exponents ($\nu, z, \eta, \eta', \gamma_T$) and the maximal underdamping angle $\theta^*$.

3. Key Contributions and Results

A. Generalization to $O(n_1) \times O(n_2)$ Symmetries

The study confirms that the exotic nonequilibrium fixed points (NEFPs) found in the $Z_2 \times Z_2$ model persist for general $n_1, n_2$, but with significant modifications:

• Existence of NEFPs: For $\sigma = -1$, stable NEFPs exist for a broad range of $n_1, n_2$. Unlike the $Z_2 \times Z_2$ case where a pair of stable NEFPs always exists, the number of NEFPs (0, 1, or 2) and their stability depend on the specific values of $n_1$ and $n_2$.
• Violation of FDT: At all NEFPs, the fluctuation-dissipation theorem is violated. The effective temperature becomes scale-dependent and "hotter" at longer wavelengths ($\gamma_T < 0$).
• Underdamped Oscillations: A universal feature of the NEFPs is the emergence of underdamped oscillations in the relaxation to the steady state within the doubly-ordered phase. This is characterized by a non-zero maximal angle $\theta^*$, contrasting with the overdamped dynamics of equilibrium models.

B. Discrete Scale Invariance and Exceptional Points

A major discovery is the dependence of the critical exponent $\nu$ on $n_1, n_2$:

• Complex $\nu$: For certain regions of the $(n_1, n_2)$ parameter space, $\nu$ becomes complex ($\nu^{-1} = \nu'^{-1} \pm i\nu''^{-1}$). This leads to discrete scale invariance (DSI), where correlation functions exhibit log-periodic oscillations and phase boundaries form spirals in the phase diagram.
• Real $\nu$: In other regions, $\nu$ is purely real, recovering continuous scale invariance.
• Exceptional Points: The boundary between the real and complex $\nu$ regions is an exceptional point in the RG flow. At this point, the eigenvalues of the stability matrix coalesce, leading to logarithmic corrections in scaling functions and phase boundaries that approach the critical point as $r \log r$.

C. One-Way Coupling ($\sigma = 0$)

The authors investigate the regime where one field is independent of the other ($g_{21}=0$ or $g_{12}=0$).

• Perturbative Regime ($n_1 \neq n_2$): Contrary to the $Z_2 \times Z_2$ case (where this limit was non-perturbative), the authors find a new stable perturbative fixed point when $n_2 > n_1$.
  • The independent field ($\Phi_2$) exhibits standard equilibrium criticality.
  • The dependent field ($\Phi_1$) exhibits nonequilibrium features: violation of FDT ($\gamma_{T1} \neq 0$) and scale-dependent temperature, but no discrete scale invariance or underdamped oscillations ($\theta^* = 0$).
• Marginality ($n_1 = n_2$): When $n_1 = n_2$, the fixed point becomes non-perturbative (couplings diverge). However, the divergence is slow (logarithmic), allowing for transient criticality. The system exhibits logarithmic corrections to dynamical exponents and scale-dependent anomalous dimensions before flowing to a non-perturbative regime.

D. Phase Diagram Topology

The phase diagrams of the NEFPs differ fundamentally from equilibrium bicritical/tetracritical points:

• Tetracriticality: The NEFPs generally describe tetracritical points involving four phases (disordered, two singly-ordered, and one doubly-ordered).
• Spiraling Boundaries: In the regime of complex $\nu$, the phase boundaries spiral into the multicritical point, a direct signature of discrete scale invariance.

4. Significance and Implications

• Foundational Understanding: This work establishes that nonreciprocal interactions generate a robust, intrinsically nonequilibrium universality class that is distinct from equilibrium classes, even for systems with continuous symmetries.
• New Phenomena: The identification of discrete scale invariance and underdamped critical relaxation in classical nonequilibrium systems provides new theoretical targets for experimental observation.
• Experimental Relevance: The results are applicable to a wide range of systems, including:
  • Active Matter: Flocking models and active fluids where nonreciprocal interactions are common.
  • Open Quantum Systems: Driven-dissipative condensates (e.g., exciton-polaritons) and cascaded quantum systems where measurement and feedback induce nonreciprocity.
  • Biological Systems: Models of cell signaling or predator-prey dynamics where interactions are inherently directional.
• Theoretical Tools: The paper provides a comprehensive RG framework for analyzing nonreciprocal systems, including the handling of exceptional points and transient criticality, which are crucial for understanding systems near the breakdown of perturbation theory.

In summary, the paper demonstrates that nonreciprocity is a powerful mechanism for generating new forms of criticality, characterized by complex critical exponents, discrete scale invariance, and dynamic oscillations, fundamentally altering the landscape of phase transitions beyond equilibrium physics.