Superradiance enhances and suppresses fermionic pairing based on universal critical scaling rate in two order parameters systems

This paper proposes a universal critical scaling rate framework based on Landau's theory to analyze how superradiant phase transitions in two-order-parameter systems can either enhance or suppress fermionic pairing, as demonstrated through the two-mode Rabi and 1D Fermi Dicke models.

The Gist

The Big Idea: The "Domino Effect" in Physics

Imagine you are trying to bake the perfect cake. You have two main ingredients you can control: Sugar (which makes it sweet) and Flour (which makes it sturdy). Usually, scientists study how changing the sugar affects the taste, or how changing the flour affects the texture. They treat these ingredients as if they don't really talk to each other.

But in the real world, ingredients do interact. If you add too much sugar, the cake might collapse, changing how much flour you need. If you change the oven temperature, it might affect both the rise and the sweetness at the same time.

This paper is about a new way to understand systems where two different "ingredients" (called Order Parameters) are fighting or helping each other. Specifically, the authors look at what happens when a system suddenly changes its state (a "phase transition")—like water turning to ice—and how that sudden change instantly boosts or crushes a second effect happening at the same time.

The Two Main Characters

To understand the paper, we need to meet the two "characters" (Order Parameters) the authors are studying:

1. The Super-Flash (Superradiance): Imagine a group of people in a dark room. If they all whisper at different times, it's just noise. But if they suddenly decide to shout in perfect unison, the sound becomes incredibly loud. In physics, this is Superradiance. It's when light (photons) and atoms suddenly sync up and start acting like a single, giant laser beam.
2. The Handshake (Fermionic Pairing): Imagine a dance floor. Usually, people dance alone. But in a superconductor (a material that conducts electricity with zero resistance), electrons pair up and dance together in perfect steps. This "handshake" or pairing is what creates the Superconducting Band Gap. It's the glue that holds the super-conducting state together.

The Problem: How Do They Influence Each Other?

The authors asked a simple question: If the "Super-Flash" suddenly turns on, does it help the "Handshake" get stronger, or does it break the dancers apart?

In the past, to figure this out, scientists had to do incredibly difficult math to simulate the whole system from scratch. It was like trying to predict the weather by calculating the movement of every single air molecule.

The Solution: The "Cheat Code"

The authors developed a universal rule (a mathematical shortcut) based on an old theory called Landau's Theory.

Think of it like this: Instead of watching the whole storm develop, you just look at the exact moment the wind speed hits a critical point. By looking at how the wind starts to change at that exact tipping point, you can instantly predict whether the rain will start pouring or if the clouds will clear up.

They found a formula that tells you:

• If the number is positive: The Super-Flash will boost the Handshake (making superconductivity stronger).
• If the number is negative: The Super-Flash will suppress the Handshake (breaking the superconductivity).

This is a huge deal because it lets scientists predict the outcome just by looking at the "tipping point," without needing to simulate the entire complex system.

The Two Experiments (The Proof)

To prove their rule works, they tested it on two different "playgrounds":

1. The Two-Mode Rabi Model (The Tunable Light Bulb)

• The Setup: Imagine two light beams hitting a pair of spinning tops (atoms). One beam is steady, and the other is a "floppy" beam that can be tuned.
• The Result: They found that by carefully tuning the "floppy" beam, they could make the Super-Flash turn on.
• The Surprise: Depending on the settings, turning on the Super-Flash could either strengthen the electron pairing (helping the dance) or weaken it (stopping the dance). It was like having a remote control that could switch the effect from "Help" to "Hinder" just by turning a dial.

2. The 1D Fermi Dicke Model (The One-Dimensional Dance Line)

• The Setup: Imagine a line of dancers (electrons) in a narrow hallway, with a giant spotlight (the cavity) shining on them.
• The Result: When the spotlight got bright enough to trigger the Super-Flash, the dancers' "handshake" (the superconducting bond) got weaker.
• The Lesson: In this specific scenario, the Super-Flash acted like a bully, pushing the dancers apart and breaking the superconducting bond.

Why Does This Matter?

This paper gives scientists a new toolkit.

In the past, if you wanted to make a better superconductor (a material that carries electricity perfectly), you had to search for new materials with different chemical properties. It was like trying to find a new cake recipe from scratch.

Now, thanks to this paper, scientists can take an existing material and use "phase transitions" (like turning on a laser or changing a magnetic field) to manipulate the material's properties. They can essentially "tune" the strength of the electron pairing without changing the material itself.

The Takeaway

Think of the universe as a giant, complex machine with many levers. This paper tells us that if you pull one lever (trigger a phase transition in one part of the system), you can predict exactly how it will jiggle the other levers (change the pairing strength).

It turns a chaotic, complex problem into a predictable game of cause-and-effect, opening the door to designing smarter quantum computers and more efficient energy materials by simply "tweaking" the transitions rather than rebuilding the whole machine.

Technical Summary

1. Problem Statement

The paper addresses the complex interplay between two or more order parameters in quantum many-body systems. While Landau's theory of phase transitions is well-established for systems with a single order parameter (e.g., magnetism in Ising models), systems with multiple coupled order parameters exhibit richer phase diagrams and non-trivial interactions where the transition of one parameter influences the magnitude of another.

Specifically, the authors aim to:

• Derive a universal analytical framework to predict how a continuous phase transition in one order parameter (e.g., superradiance) affects the scaling behavior of a second order parameter (e.g., fermionic pairing/superconductivity) immediately beyond the critical point.
• Determine whether the superradiant phase transition enhances or suppresses fermionic pairing (Cooper pairs) in specific quantum models.
• Provide a method to manipulate physical quantities (like band gaps) by inducing phase transitions in coupled observables, offering a new paradigm for quantum simulation and material design.

2. Methodology

Theoretical Framework: Universal Critical Scaling Rate

The authors extend Landau's mean-field theory to systems with two macroscopic order parameters, $O_1$ and $O_2$, governed by a free energy functional $F(O_1, O_2; \vec{\lambda})$, where $\vec{\lambda}$ represents environmental parameters.

• Assumptions: The system possesses $Z_2$ symmetry for the second order parameter ($O_2$). The transition occurs at a critical point $\vec{\lambda}_c$ where $O_2$ transitions from zero (normal phase) to non-zero (ordered phase).
• Derivation: By expanding the free energy around the critical point and minimizing it, the authors derive a universal scaling relation (Eq. 5) describing the change in the first order parameter ($\delta O_1$) as a function of the second ($\delta O_2$) just beyond the critical boundary:
  $$\delta O_1 \approx -\frac{\partial_1 \partial_{v_{m1}}^2 F(\bar{O}_1^c, 0; \vec{\lambda}_c)}{v_{m1}! \partial_1^2 F(\bar{O}_1^c, 0; \vec{\lambda}_c)} (\delta O_2)^{v_{m1}}$$
• Key Insight: The sign and magnitude of the mixed partial derivative of the free energy determine whether the first order parameter increases (enhancement) or decreases (suppression) when the second order parameter emerges. This allows prediction of the ordered phase's properties using only critical point data, saving significant computational resources.

Model Systems

To validate the theory, the authors analyze two specific models:

1. Floquet-Engineered Two-Mode Rabi Model:
   • Construction: A system of two spin-1/2 particles coupled to two optical modes ($b$ and $c$). Mode $b$ has a linear coupling to spins, while mode $c$ has a higher-order coupling designed via Floquet engineering (using periodic driving) or circuit QED techniques.
   • Goal: To manipulate the two-spin pairing strength ($\Delta_c$) via the superradiance of mode $b$.
2. 1D Fermi-Dicke Model:
   • Construction: A 1D gas of fermions interacting with a single bosonic cavity mode, including attractive fermion-fermion interactions (BCS pairing).
   • Goal: To study the competition between the superradiant phase transition and the superfluid (pairing) order parameter ($\Delta$).

3. Key Contributions

1. Universal Scaling Law: The derivation of a general formula (Eq. 5) that predicts the critical scaling rate of one order parameter based on the emergence of another in a two-order-parameter system. This provides a "shortcut" to understanding complex phase diagrams without solving the full non-linear equations for the entire ordered phase.
2. Mechanism for Control: The paper demonstrates that superradiance can be used as a "knob" to tune fermionic pairing. Depending on the specific coupling terms (e.g., Kerr non-linearities or interaction strengths), the superradiant transition can either enhance or suppress the pairing gap.
3. Analytical vs. Numerical Validation: The authors provide strict analytical derivations for the scaling rates and verify them against numerical simulations (mean-field and finite-difference methods), showing high precision agreement near critical points.

4. Results

Case 1: Two-Mode Rabi Model (Enhancement vs. Suppression)

• Setup: The system involves a pairing order parameter $\Delta_c$ (related to mode $c$) and a superradiant order parameter $B$ (related to mode $b$).
• Findings:
  • The effect of superradiance on pairing depends critically on the Kerr non-linearity ($\chi_3$).
  • Weak Kerr ($\chi_3$ small): The emergence of superradiance ($B \neq 0$) initially enhances the pairing strength $\Delta_c$.
  • Strong Kerr ($\chi_3$ large): The cross-Kerr term ($\chi_3 B^2 C^2$) dominates, causing the superradiance to suppress the pairing order $\Delta_c$ immediately after the transition.
  • Validation: Numerical plots of $\Delta_c$ vs. coupling strength $\lambda_b$ match the linear fitting predicted by the derived scaling rate (Eq. 10).

Case 2: 1D Fermi-Dicke Model (Suppression)

• Setup: A 1D fermionic system with attractive interactions ($U < 0$) coupled to a cavity.
• Findings:
  • The superradiant phase transition always suppresses the superconducting band gap ($\Delta$).
  • The scaling rate is determined by the mixed derivative $\partial^2 F / \partial \Delta \partial |\alpha|^2$, which is found to be positive, leading to a negative slope for $\Delta$ as the superradiant order $\alpha$ grows.
  • Numerical Results: Simulations show that as the dimensionless coupling strength $\tilde{g}$ exceeds the critical value, $\Delta$ decreases linearly with the emergence of $|\alpha|^2$. The analytical scaling rates ($-0.186$ and $-1.268$ for different filling factors) match the numerical data perfectly.
• Note on 1D Limitations: The authors acknowledge that in 1D, quantum fluctuations (Mermin-Wagner theorem) destroy long-range order at finite temperatures, but the zero-temperature mean-field results remain valid for understanding the fundamental interplay principles.

First-Order Transitions

• The paper also briefly discusses scenarios where the transition becomes discontinuous (first-order), particularly in the Fermi-Dicke model at high filling factors ($k_F \approx 0.96$). In these cases, both order parameters jump simultaneously. While the universal scaling law (derived for continuous transitions) does not apply directly, the authors note that similar manipulation of pairing is possible via these discontinuous jumps.

5. Significance

• New Paradigm for Quantum Control: The work proposes a novel strategy for quantum simulation: instead of searching for new materials with intrinsic large band gaps, one can manipulate existing systems by inducing phase transitions in coupled order parameters to tune the desired physical effect (e.g., enhancing superconductivity).
• Theoretical Efficiency: The derived universal scaling rate allows researchers to predict the behavior of complex ordered phases using only critical point data, significantly reducing the computational cost of simulating large quantum systems.
• Experimental Relevance: The proposed models (Rabi and Fermi-Dicke) are realizable in circuit QED and ultracold atom experiments. The ability to engineer specific coupling terms (like the higher-order Rabi coupling via Floquet engineering) makes the theoretical predictions experimentally testable.
• Broader Impact: The framework extends beyond the specific models studied, offering a general tool for understanding complex systems in condensed matter physics, quantum optics, and quantum materials where multiple order parameters coexist and interact.