Landau and fractionalized theories of periodically driven intertwined orders

This paper investigates the phase diagrams of both conventional Landau and fractionalized theories of intertwined orders under periodic driving, utilizing the large-$N$ limit and Markovian bath coupling to reveal diverse long-time dynamical behaviors ranging from stable averages and period-doubling oscillations to quasi-periodicity and chaos.

The Gist

The Big Picture: Shaking a Jello Mold

Imagine you have a bowl of Jello. Inside this Jello, there are two different flavors trying to take over the whole bowl: Chocolate (representing Superconductivity, where electricity flows with zero resistance) and Vanilla (representing a Charge Density Wave, a static pattern of electrons).

In a normal, quiet room (equilibrium), these two flavors usually hate each other. If the Chocolate wins, the Vanilla disappears, and vice versa. They are "competing orders."

Now, imagine you start shaking the bowl rhythmically (this is the "periodic driving" or "optical driving" mentioned in the paper). The scientists in this paper asked: What happens to the Jello if we shake it really hard? Does the shaking force the two flavors to mix? Does it create a new, weird flavor? Or does it just make a mess?

They studied this using two different "recipe books" (theories) to see if they get the same answer.

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The Two Recipe Books

The authors compared two ways of describing this Jello:

1. The "Direct" Recipe (Landau Theory):

   • The Analogy: Imagine you just look at the Chocolate and Vanilla swirls directly. You treat them as two distinct, solid objects fighting for space.
   • The Science: This is the standard way physicists describe materials. It's simple and intuitive. You have two variables fighting each other.

2. The "Secret Ingredient" Recipe (Fractionalized Theory):

   • The Analogy: Imagine the Jello isn't made of Chocolate and Vanilla at all. Instead, it's made of tiny, invisible "ghost particles" (called chargons and Higgs fields) that dance around. The Chocolate and Vanilla flavors are just the shadows these ghosts cast when they dance together.
   • The Science: This is a more complex, "fractionalized" view often used for high-temperature superconductors (like cuprates). Here, the order parameters (the flavors) are actually composites of deeper, underlying fields. It's like saying a "wave" isn't a thing itself, but just the result of water molecules moving.

Why compare them? The "Direct" recipe is easier to use, but the "Secret Ingredient" recipe is thought to be more accurate for certain tricky materials (like hole-doped cuprates). The paper checks if the simple recipe gives the same results as the complex one when you start shaking the bowl.

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What Happens When You Shake the Bowl? (The Results)

When the scientists applied a rhythmic "shake" (an oscillating electric field) to their models, they found some surprising things that wouldn't happen in a quiet room:

1. The "Peace Treaty" (Coexistence)

In a quiet room, Chocolate and Vanilla usually fight to the death. But when they shook the bowl, they found a "Peace Treaty."

• The Result: In certain shaking patterns, the two flavors stopped fighting and started living together in the same spot.
• The Metaphor: It's like two rival gangs who usually fight, but when a loud DJ starts playing a specific beat, they stop fighting and start dancing together in a circle. The shaking suppressed the competition and allowed them to coexist.

2. The "Half-Time" Dance (Period Doubling)

Sometimes, the Jello didn't just move in time with the shaking. It started moving at half the speed of the shake.

• The Result: If you shake the bowl once every second, the Jello might only change its shape once every two seconds.
• The Metaphor: Imagine a drummer hitting a drum every second. The dancer (the material) decides to only spin once every two beats. This is called "period doubling." It's a sign that the system has entered a complex, rhythmic state that is locked to the beat but slower.

3. The "Metallic" Phase (The Liquid State)

In some parts of the shaking, the Jello turned into a completely different texture: a liquid metal.

• The Result: A phase appeared where the material acted like a normal metal (conducting electricity but with resistance), which shouldn't happen in that specific temperature range if the bowl wasn't being shaken.
• The Metaphor: The shaking was so effective it melted the solid structure of the Jello, turning it into a fluid soup.

4. The "Chaos" Zone

In some regions, the shaking didn't create a nice rhythm. It created a mess.

• The Result: The Jello started moving in a way that was neither a perfect circle nor a simple wave. It was chaotic or quasi-periodic (almost repeating, but never quite the same).
• The Metaphor: Imagine trying to balance a broom on your hand while someone shakes the floor. Sometimes, the broom just flails wildly in unpredictable directions. That's the chaotic zone.

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The Big Takeaway: Do the Recipes Agree?

The most important question was: Do the "Direct" recipe and the "Secret Ingredient" recipe give the same map of what happens?

• The Good News: For the most part, yes! Both recipes predicted the same types of phases: the peace treaties (coexistence), the half-time dances, and the chaos. This suggests that for many experiments, the simpler "Direct" recipe is good enough.
• The Bad News: There are subtle differences. The "Secret Ingredient" recipe showed that the "Peace Treaty" (coexistence) is much harder to achieve starting from a Vanilla state than the "Direct" recipe suggests.
• The Real-World Implication: This matters for real experiments on superconductors. If scientists see a material turn superconductive when they shine light on it, the "Direct" recipe might say, "Oh, that's easy, they just coexist!" But the "Secret Ingredient" recipe says, "Wait, that's actually really hard to do; maybe we are missing some hidden physics (like the SU(2) gauge field) that we need to account for."

Summary

This paper is like a physics lab where scientists took two different maps of a territory (Superconductors) and drove a car through it while shaking the steering wheel. They found that the shaking creates new, weird landscapes where enemies become friends, rhythms get weird, and chaos reigns. While both maps mostly agree on the geography, the "Secret Ingredient" map warns us that some paths are trickier than they look, reminding us that the deep, hidden physics of these materials is still full of surprises.

Technical Summary

1. Problem Statement

The paper investigates the non-equilibrium dynamics of intertwined orders (specifically Superconductivity (SC) and Charge Density Wave (CDW)) in strongly correlated electron systems, such as cuprate materials, under periodic external driving (Floquet engineering).

• Context: Recent experiments show interplay between SC and CDW under optical driving. While equilibrium theories exist, the behavior of these competing orders under strong periodic driving is less understood.
• Core Question: Can periodic driving suppress the competition between SC and CDW to stabilize coexistence, or select phases unfavorable in equilibrium? How do different theoretical frameworks (conventional Landau vs. fractionalized) respond to driving?
• Challenge: Standard equilibrium phase diagrams often show mutual exclusion (competition) between SC and CDW. The authors aim to determine if driving can unlock a "coexistence" regime or induce exotic non-equilibrium states like chaos or quasiperiodicity.

2. Methodology

The authors employ two distinct theoretical frameworks, both analyzed in the large-$N$ limit and coupled to a Markovian thermal bath to model dissipation and prevent unphysical heating.

A. Theoretical Models

1. Fractionalized Theory (Section III):

   • Framework: Based on a multi-component Higgs field $B_n$ transforming as a doublet under an emergent $SU(2)$ gauge field.
   • Order Parameters: SC and CDW are defined as distinct gauge-invariant composites of the Higgs field: $\phi_{SC} = 2b_+b_-$ and $\phi_{CDW} = b_+^2 - b_-^2$.
   • Motivation: This approach is motivated by hole-doped cuprates, where small hole pockets (observed in transport) are naturally explained by fractionalization.
   • Approximation: The $SU(2)$ gauge field is neglected in the bulk dynamics (decoupled in large-$N$) but included for electromagnetic response calculations.

2. Conventional Landau Theory (Section IV):

   • Framework: A standard $O(N) \times O(M)$ Landau-Ginzburg functional with two vector order parameters $\phi_1$ (CDW) and $\phi_2$ (SC).
   • Interaction: Coupled via a quartic term $v \phi_1^2 \phi_2^2$.
   • Motivation: A phenomenological approach applicable to various materials (e.g., pnictides) where fractionalization is not assumed.

B. Computational Approach

• Keldysh Formalism: The system is treated using the Keldysh path integral to handle non-equilibrium dynamics.
• Large-$N$ Saddle Point: The authors perform Hubbard-Stratonovich transformations to decouple interactions, deriving self-consistent equations of motion for the order parameters and correlation functions.
• Driving Protocol: The coupling constants (mass $r$ and interaction $v$, or relative mass $\Delta r$) are modulated periodically: $X(t) = X_0 + 2X_1 \cos(\Omega t)$.
• Numerical Solution: The resulting coupled differential equations are solved numerically to extract steady-state behaviors, phase diagrams, and time-evolution profiles.

3. Key Contributions and Results

A. Phase Diagrams and Coexistence

• Drive-Induced Coexistence: In both models, strong periodic driving ($v_1$ or $\Delta r_1$) can suppress the competition between SC and CDW.
  • In the fractionalized model, coexistence (pink regions in Fig. 3) emerges when the drive amplitude is large enough to make the coupling $v(t)$ oscillate between positive and negative values.
  • In the Landau model, similar coexistence regions appear when the relative mass detuning $\Delta r(t)$ oscillates across zero.
• Arnold Tongues: The phase diagrams exhibit "Arnold tongue" structures (stability regions of Mathieu-Hill equations). Nested tongues appear due to the distinct self-consistent equations for the two order parameters.
• Metallic Phase: A driven metallic phase emerges in the deep interior of the Arnold tongues, a state not present in the equilibrium phase diagram for these parameters.

B. Dynamical States of Order Parameters

The authors identify rich dynamical behaviors beyond simple locking to the drive frequency $\Omega$:

1. $\Omega$-locked: Order parameters oscillate at the drive frequency with a non-zero mean.
2. Period-Doubling ($\Omega/2$): Order parameters oscillate at half the drive frequency.
   • Crucial Distinction: Unlike single-order-parameter theories where period-doubling must occur around zero mean, the intertwined nature allows period-doubling to occur around a finite non-zero mean. This is due to the coupling of fluctuations between the two orders.
3. Mixed States: Four distinct coexistence phases are identified based on the frequency locking of $\phi_{SC}$ and $\phi_{CDW}$:
   • $(\Omega, \Omega)$, $(\Omega, \Omega/2)$, $(\Omega/2, \Omega)$, and $(\Omega/2, \Omega/2)$.
4. Quasiperiodicity and Chaos:
   • Hatched Regions: The phase diagrams contain regions where the dynamics do not settle into simple periodic states.
   • Quasiperiodicity: Characterized by incommensurate frequency peaks in Fourier spectra and closed continuous curves in stroboscopic plots.
   • Chaos: Characterized by broad frequency continua and diffuse clouds of points in stroboscopic plots (chaotic attractors).
   • Significance: These chaotic/quasiperiodic regimes were not found in previous studies of single-order-parameter driven systems, highlighting the complexity introduced by intertwined orders.

C. Comparison of Models

• Symmetry: The Landau model phase diagram (Fig. 8) is symmetric under the exchange of SC and CDW (due to fine-tuned parameters), whereas the fractionalized model (Fig. 3) lacks this symmetry, reflecting the underlying physics of hole-doped cuprates.
• Coexistence from CDW: Experiments show light-induced SC from a CDW state. The fractionalized model predicts only small coexistence regions starting from CDW, suggesting that neglecting the full $SU(2)$ gauge field fluctuations might be insufficient to fully capture experimental observations.

4. Significance and Implications

1. Floquet Engineering of Intertwined Orders: The paper demonstrates that periodic driving is a powerful tool to stabilize coexistence of orders that compete in equilibrium, potentially offering a route to control superconductivity in correlated materials.
2. New Non-Equilibrium Phenomena: The discovery of chaotic and quasiperiodic phases in the large-$N$ limit of driven intertwined orders is a major theoretical finding. It suggests that complex many-body systems under driving can exhibit rich dynamical complexity not seen in simpler models.
3. Period-Doubling Mechanism: The identification of period-doubling around a finite mean value challenges standard Floquet symmetry expectations for single-order systems, providing a new signature for intertwined orders in time-resolved experiments.
4. Theoretical Validation: By comparing the fractionalized (microscopic) and Landau (phenomenological) approaches, the authors validate the utility of Landau theory for order-parameter dynamics while highlighting where fractionalization is essential (e.g., for transport properties and specific asymmetries in cuprates).
5. Experimental Relevance: The results provide a theoretical framework for interpreting recent optical pump-probe experiments in cuprates, suggesting that the observation of SC-like signals in CDW states might require careful consideration of gauge fluctuations and the specific nature of the drive.

5. Limitations

• Large-$N$ Approximation: The analysis neglects fluctuations that could lead to long-time thermalization or modify the stability of phases.
• Spatial Homogeneity: The study assumes spatially homogeneous condensates, ignoring spatially inhomogeneous states (e.g., stripe phases) which are relevant in cuprates.
• Gauge Field Treatment: In the fractionalized model, the $SU(2)$ gauge field is largely neglected in the dynamics (though included for response), which may limit the quantitative accuracy for specific transport predictions.

In summary, this work establishes a comprehensive theoretical framework for understanding how periodic driving alters the phase competition and dynamics of intertwined orders, revealing a landscape of coexistence, period-doubling, and chaos that extends significantly beyond equilibrium physics.