Real critical exponents from the $\varepsilon$-expansion in an interacting $U(1)$ model with non-Hermitian $Z_4$ anisotropy

This paper demonstrates that a $U(1)$-invariant Lagrangian perturbed by a complex, $\mathcal{PT}$-symmetric $Z_4$ anisotropy yields real critical exponents in both broken and unbroken $\mathcal{PT}$ phases, with the most stable fixed point flowing to an effectively Hermitian $U(1)$ symmetric system, thereby showing that Hermiticity and $U(1)$ symmetry can emerge as intrinsic features of inherently non-Hermitian theories.

The Gist

The Big Idea: Finding Order in "Ghostly" Chaos

Imagine you are trying to understand how a crowd of people behaves at a concert. Usually, physicists assume the crowd is a "closed system"—everyone is there, no one leaves, and no one sneaks in. This is like a Hermitian system: it follows strict, predictable rules where energy is conserved, and everything stays "real" and stable.

But in the real world, things are often messier. People enter and leave (gain and loss), or the system interacts with the outside world. In physics, these are called non-Hermitian systems. They are often described as "open" systems where energy leaks out or is pumped in.

However, this paper asks a fascinating question: What if a system is "non-Hermitian" (complex and weird) by its very nature, not just because it's open? And even more surprisingly: Can such a chaotic, "ghostly" system eventually settle down and behave like a normal, stable system?

The authors say yes. They found a way for a system that starts out looking mathematically "broken" (with complex numbers and imaginary parts) to evolve into a perfectly normal, stable system with real, predictable properties.

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The Cast of Characters

To understand the story, let's meet the players:

1. The U(1) Symmetry (The Smooth Circle): Imagine a compass needle that can point in any direction. It has perfect rotational symmetry. This is the "base" of the system.
2. The $Z_4$ Anisotropy (The Four-Pointed Star): Now, imagine you put a fence around that compass that only lets it point North, East, South, or West. It breaks the smooth circle into four distinct options. This is the "anisotropy."
3. The Non-Hermitian Twist (The "Ghost" Interaction): The authors add a special, weird ingredient to the fence. It's like the fence has a "ghost" in it. Mathematically, this introduces imaginary numbers (the $i$ in $i^2 = -1$) into the rules.
   • In normal physics, imaginary numbers usually mean something is unstable or "unphysical."
   • In this paper, they use a concept called PT-Symmetry (Parity-Time symmetry). Think of this as a magical mirror. If you flip the system left-to-right (Parity) and run the movie backward (Time), the "ghost" rules look exactly the same. This keeps the system balanced, even if it looks weird.

The Experiment: The "Phase Transition"

The authors studied what happens when you heat up this system (add energy/fluctuations) and watch it cool down to a critical point (like water freezing into ice).

They used a mathematical tool called the $\epsilon$-expansion.

• The Analogy: Imagine you are trying to predict the weather on a planet where the laws of physics are slightly different from Earth. You can't calculate it exactly, so you start with Earth's laws and add a tiny "correction factor" ($\epsilon$) to see how the weather changes. The authors did this with the dimension of space (moving from 4 dimensions down to 3).

The Two Worlds: Broken vs. Unbroken

The system has two distinct modes, depending on the strength of the "ghost" interaction:

1. Unbroken PT-Symmetry (The Calm Zone):

   • Here, the "ghost" is weak. The system behaves normally. The mathematical numbers describing the system are all real numbers (like 1, 2, 3).
   • Result: Everything is stable.

2. Broken PT-Symmetry (The Chaotic Zone):

   • Here, the "ghost" is strong. The mathematical numbers describing the system become complex numbers (mixing real and imaginary parts).
   • The Surprise: In many other theories, when numbers become complex, the system falls apart. It becomes unstable, or the "critical exponents" (the numbers that tell us how the system behaves near the freezing point) become imaginary and meaningless.
   • The Paper's Discovery: Even though the internal rules (coupling constants) became complex and "ghostly," the critical exponents remained real!
   • The Metaphor: Imagine a chaotic dance floor where everyone is spinning wildly and the music is distorted (complex numbers). Yet, if you look at the average speed of the dancers or how they group together, the pattern is perfectly normal and predictable (real numbers). The chaos cancels itself out in the final result.

The "Emergent" Miracle

The most exciting part of the paper is the concept of Emergence.

• The Setup: You start with a theory that is fundamentally "non-Hermitian" (weird, complex, PT-symmetric).
• The Journey: As you look at the system from a distance (at large scales, or "long distances"), the weirdness disappears.
• The Result: The system flows toward a state that looks exactly like a standard, Hermitian system with U(1) symmetry.

The Analogy: Imagine you are looking at a pixelated image of a cat. Up close, the pixels are jagged, weird, and don't look like a cat (the non-Hermitian, complex part). But as you zoom out, the jagged pixels smooth out, and you clearly see a perfect, normal cat (the emergent Hermitian U(1) system).

The authors found that the "weirdness" of the initial rules is an illusion that washes away when you look at the big picture. The system "forgets" it was ever non-Hermitian and becomes a standard, stable system.

Why Does This Matter?

1. Beyond "Gain and Loss": Usually, when physicists talk about non-Hermitian systems, they talk about lasers (gain) and absorption (loss). This paper shows that non-Hermitian systems can exist in a deeper, more fundamental way, not just because they are open to the environment.
2. Real Physics from "Fake" Math: It proves that you can start with a theory that looks mathematically "broken" (complex numbers) and end up with a physical reality that is perfectly "real" and stable.
3. New Tools for Prediction: It suggests that even if a system looks unstable or complex, it might still have a stable, predictable "heart" (real critical exponents) that we can use to understand phase transitions in materials, from magnets to superconductors.

Summary in One Sentence

This paper shows that even if you build a physical system with "ghostly," complex rules that break standard physics, the system can naturally evolve over time to forget those rules and settle into a perfectly normal, stable, and predictable state.

Technical Summary

1. Problem Statement and Motivation

The paper addresses the critical behavior of quantum field theories that are intrinsically non-Hermitian yet possess Parity-Time (PT) symmetry. While non-Hermitian systems are often studied in the context of open systems (gain and loss), the authors investigate systems where non-Hermiticity arises from the Lagrangian itself, independent of environmental coupling.

• Context: Previous studies on non-Hermitian systems (e.g., the Abelian Higgs model with $U(n)$ symmetry) often result in complex fixed points and complex critical exponents when PT symmetry is broken, leading to runaway Renormalization Group (RG) flows and first-order phase transitions rather than critical behavior.
• Goal: The authors aim to determine if a non-Hermitian Lagrangian can yield real critical exponents and stable critical behavior, specifically investigating a $U(1)$-invariant model perturbed by a complex, PT-symmetric $Z_4$ anisotropy. They seek to understand if emergent Hermiticity and $U(1)$ symmetry can occur in the infrared (large distance) limit.

2. Methodology

The authors employ Renormalization Group (RG) analysis using the $\epsilon$-expansion technique up to two-loop order ($O(\epsilon^2)$).

• The Model: They study a $d$-dimensional complex scalar field $\psi$ with a $U(1)$ invariant Lagrangian perturbed by a non-Hermitian $Z_4$ anisotropy:
  $$\mathcal{L} = |\partial_\mu \psi|^2 + m^2|\psi|^2 + \frac{u}{2}|\psi|^4 + V_{Z_4}(\psi)$$
  where the anisotropy potential is:
  $$V_{Z_4}(\psi) = \frac{1}{2} \left[ (v + w)\psi^4 + (v - w)\psi^{*4} \right]$$
  Here, $v$ and $w$ are real coupling constants. The term with $w$ introduces non-Hermiticity.
• PT Symmetry Conditions:
  • Unbroken PT: $v^2 > w^2$ (Real spectrum).
  • Broken PT: $v^2 < w^2$ (Complex spectrum).
  • Exceptional Point: $v^2 = w^2$.
• Technical Approach:
  • The complex field is decomposed into real components ($\psi = (\phi_1 + i\phi_2)/\sqrt{2}$).
  • The potential is rewritten in terms of quartic couplings $g_{ijkl}$, revealing a structure involving $O(2)$ symmetric, cubic, and imaginary terms.
  • The authors identify an RG invariant $k$ (related to the ratio of couplings $v/w$) which allows the reduction of the system to two independent coupling constants.
  • They calculate the $\beta$-functions for the couplings and the anomalous dimensions ($\eta$ and $\eta_2$) up to two loops.
  • Stability matrices are constructed to analyze the stability of fixed points and lines.

3. Key Contributions

1. Discovery of Real Critical Exponents in Broken PT Regime: Unlike previous models where broken PT symmetry leads to complex exponents and non-critical behavior, this model yields real, $k$-independent critical exponents even when the coupling constants become complex (in the broken PT region).
2. Emergent Hermiticity and Symmetry: The most stable fixed point corresponds to the flow towards an effectively Hermitian $U(1)$ symmetric system. This demonstrates that both $U(1)$ symmetry and Hermiticity can be emergent features of a theory that is fundamentally non-Hermitian and anisotropic at the microscopic level.
3. Novel Fixed Point Transition Mechanism: The authors describe a unique transition mechanism for fixed points as the system crosses the exceptional point ($k=1$). Instead of the standard "collision and annihilation" of fixed points in the real plane, the fixed points diverge to infinity in the real domain and then reappear in the complex plane as a "scattering" process.
4. Fixed Lines: The analysis reveals the existence of fixed lines (continuous families of fixed points parameterized by $k$) for the Ising and Cubic universality classes, rather than isolated fixed points.

4. Key Results

• Fixed Points and Lines:
  • Gaussian Fixed Point: Unstable.
  • Heisenberg Fixed Point: Completely stable. This point corresponds to the $U(1)$ symmetric limit ($w=0$).
  • Ising and Cubic Fixed Lines: These exist for $k \neq 0$.
    • In the Unbroken PT region ($k^2 < 1$), these lines are real.
    • In the Broken PT region ($k^2 > 1$), the coupling constants on these lines become complex, but the lines persist.
• Critical Exponents:
  • The critical exponents $\eta$ (anomalous dimension) and $\nu$ (correlation length exponent) are calculated for all fixed points.
  • Crucial Finding: The values of $\eta$ and $\nu$ are independent of the parameter $k$.
  • Specifically, for the stable Heisenberg point: $\eta_H = \epsilon^2/50$ and $\nu_H = 1/2 + \epsilon/10 + 11\epsilon^2/200$.
  • These values match those of the standard Hermitian $O(2)$ model with cubic anisotropy, despite the underlying non-Hermitian nature of the Lagrangian.
• Stability Analysis:
  • The Heisenberg fixed point is the only completely stable attractor.
  • The Ising and Cubic fixed lines are stable only along specific directions (one relevant direction), making the model more predictive than its Hermitian counterparts (which typically have two relevant directions).
• Unitarity and IR Bounds: The paper notes that the anomalous dimension $\eta$ is positive. This is significant because, in some non-Hermitian theories (like the $\phi^3$ model with imaginary coupling), a negative $\eta$ violates infrared bounds and unitarity. The positive $\eta$ here suggests the theory remains physically meaningful and stable.

5. Significance

• Beyond Gain and Loss: The work challenges the standard interpretation of non-Hermitian systems as merely open systems with gain and loss. It provides a concrete example of an intrinsically non-Hermitian system that flows to a physically stable, Hermitian critical point.
• Universality: It establishes a new universality class where non-Hermitian perturbations do not destroy criticality but instead lead to emergent symmetries.
• Theoretical Implications: The result suggests that non-unitary quantum field theories, when analytically continued to Euclidean space, can yield real critical exponents under specific conditions (positive $\eta$), offering a potential pathway to understanding complex systems like QCD at finite density or effective field theories of valence-bond solids.
• Experimental Relevance: The findings are relevant for systems exhibiting $Z_4$ anisotropy (like 4-state clock models) in condensed matter and quantum optics, suggesting that even if the microscopic description is non-Hermitian, the macroscopic critical behavior may be indistinguishable from a standard Hermitian system.

In summary, the paper demonstrates that non-Hermiticity does not necessarily preclude real critical behavior. Through a sophisticated RG analysis, the authors show that a non-Hermitian $U(1)$ model with $Z_4$ anisotropy flows to a stable, Hermitian $U(1)$ fixed point with real critical exponents, regardless of whether the PT symmetry is broken or unbroken.