Smeared phase transition in the dissipative random quantum Ashkin-Teller model

By combining strong-disorder renormalization group methods with adiabatic renormalization, this study demonstrates that Ohmic dissipation and quenched disorder smear two of the three quantum phase transitions in the random quantum Ashkin-Teller model, while one transition remains sharp due to a cancellation of dissipation effects on its intertwined order parameter.

The Gist

Imagine you are trying to organize a massive, chaotic dance party in a giant hall. The dancers are tiny magnets (spins), and the music is the "quantum noise" that keeps them jittery and unsure of which way to face.

In this paper, the authors are studying a specific, complex dance floor called the Ashkin-Teller Model. It's a bit more complicated than a standard dance floor because there are two groups of dancers (let's call them "Team Red" and "Team Blue") who are dancing side-by-side. They can either:

1. Face the same direction (Ferromagnetic phase).
2. Face random directions (Paramagnetic phase).
3. Do a special synchronized move where they don't face the same way individually, but their combined move is perfectly synchronized (Product phase).

The paper asks a big question: What happens if we add two things to this party?

1. Disorder: The floor is uneven, and some dancers are naturally more energetic than others (randomness).
2. Dissipation: The floor is sticky. It's like dancing in mud or thick syrup. The environment "drags" on the dancers, trying to freeze their movements.

The Main Discovery: The "Smearing" Effect

In a perfect, dry dance hall, the transition from one dance style to another is sharp and sudden. Everyone switches at the exact same moment.

However, the authors found that when you add stickiness (dissipation) to a messy floor (disorder), something strange happens. The sharp switch gets "smeared."

The Analogy of the Rare Regions:
Imagine that in your messy dance hall, there are a few lucky spots where the floor is perfectly flat and the music is just right. These are "Rare Regions."

• In a normal party, these lucky spots would eventually sync up with the rest of the crowd.
• But with the sticky floor, these lucky spots get stuck in their own little dance. They stop listening to the rest of the hall. They decide to switch dance styles on their own, at their own time, independent of everyone else.
• Because different lucky spots switch at different times, the whole hall never makes a clean, sudden switch. Instead, it slowly, gradually transitions over a long period. This is what physicists call a "smeared phase transition."

The Big Surprise: One Dance Style is Immune

Here is the twist that makes this paper special. The Ashkin-Teller model has three possible dance styles and three ways to switch between them. The authors expected the sticky floor to ruin all three transitions.

They found that it only ruins two of them.

1. The "Team Red/Blue" Switch (Ferromagnetic to Product): This transition gets smeared. The sticky floor freezes the dancers in these lucky spots, and they switch styles independently.
2. The "Random to Synchronized" Switch (Paramagnetic to Product): This transition remains sharp. It stays clean and sudden, even with the sticky floor!

Why is one immune? (The "Composite" Secret)

Why does the sticky floor affect some dancers but not others? It comes down to how the dancers hold hands.

• The "Team" Dance (Ferromagnetic): Here, the dancers are holding hands directly. If the floor is sticky, it grabs their hands and freezes them. The environment can easily "feel" what they are doing and stop them from moving.
• The "Synchronized" Dance (Product Phase): This is the weird one. In this phase, Team Red and Team Blue are not holding hands directly. They are doing a complex, invisible coordination.
  • Imagine two dancers spinning in opposite directions. If you look at just one, they look like they are spinning wildly (disordered). But if you look at them together, they are perfectly synchronized (ordered).
  • The "sticky floor" (dissipation) is like a heavy blanket that only grabs onto individual dancers. It doesn't know how to grab the invisible connection between them.
  • Because the environment can't "feel" the special synchronized move, it can't freeze the dancers in these lucky spots. They keep dancing together, and the transition remains sharp.

The Real-World Impact

This isn't just about abstract math. The authors suggest this could explain real-world materials:

• Superconductors: Materials that conduct electricity with zero resistance.
• DNA: How DNA molecules stretch and twist.
• Strange Metals: Exotic materials that don't behave like normal metals.

In all these cases, if the material has a "composite" order (like the synchronized dance), it might be surprisingly resistant to the "stickiness" of its environment. This means some quantum materials might stay stable and sharp even when they are messy and interacting with their surroundings, which is a huge relief for engineers trying to build quantum computers.

Summary

• The Setup: A messy, sticky dance floor with two groups of dancers.
• The Problem: Does the stickiness ruin the moment when the dancers change their routine?
• The Result: Yes, it ruins the routine where they hold hands directly (smearing the transition).
• The Surprise: No, it doesn't ruin the routine where they coordinate invisibly. That transition stays sharp because the "stickiness" can't grab onto the invisible connection.

The paper teaches us that sometimes, being "composite" or "hidden" is a superpower that protects a system from being frozen by its environment.

Technical Summary

1. Problem Statement

The paper investigates the interplay between quenched disorder and Ohmic dissipation in the phase diagram of the one-dimensional Random Quantum Ashkin-Teller (RQAT) model.

• Context: In disordered quantum systems, "rare regions" (RRs)—spatially large regions with locally ordered couplings—can dominate low-energy physics, leading to Griffiths singularities. When coupled to a dissipative bath (e.g., a metallic environment), the dynamics of these RRs can be frozen, potentially "smearing" sharp quantum phase transitions (QPTs) into inhomogeneous crossovers.
• The Model: The RQAT model consists of two coupled transverse-field Ising chains. Unlike the standard Ising model, it exhibits three distinct phases:
  1. Paramagnetic (PM): Disordered spins.
  2. Ferromagnetic (FM): Ordered spins ($\langle S^z \rangle \neq 0$).
  3. Product (PROD): A phase with composite order where individual spins are disordered ($\langle S^z_1 \rangle = \langle S^z_2 \rangle = 0$), but the product of spins on the same site is ordered ($\langle S^z_1 S^z_2 \rangle \neq 0$).
• The Question: Do the transitions between these phases (PM-PROD, PROD-FM, and PM-FM) remain sharp or become smeared when the system is coupled to a dissipative bath?

2. Methodology

The authors employ a combination of two powerful analytical techniques:

1. Adiabatic Renormalization: Used to integrate out high-frequency bath oscillators. This procedure renormalizes the tunneling parameters (transverse fields $h$ and couplings $g$) of the system, effectively reducing the quantum fluctuations due to dissipation.
2. Strong-Disorder Renormalization Group (SDRG): A real-space RG method adapted for dissipative systems. It iteratively decimates the highest energy scales in the system (either local couplings or bath oscillators) to determine the flow of parameters and the nature of the ground state.

Key Technical Steps:

• Variable Transformation: The Hamiltonian is reformulated using "product variables" ($\sigma$) and "original variables" ($\eta$).
  • $\sigma^z_i = S^z_{1,i} S^z_{2,i}$ (Product order parameter).
  • $\eta^z_i = S^z_{1,i}$ (Original spin order parameter).
• Coupling Analysis: The system-bath coupling is analyzed in these new variables. The authors show that the bath couples differently to the original spins ($\eta$) versus the product spins ($\sigma$).
• RG Flow: The authors track the flow of the dissipation strength $\alpha$ under the SDRG decimation steps. They identify specific decimation cases where the effective dissipation strength increases as the volume of a rare region grows.

3. Key Contributions and Results

A. Asymmetric Smearing of Phase Transitions

The central finding is that dissipation does not affect all phase transitions equally:

1. PROD-FM Transition (Smeared): The transition between the Product phase and the Ferromagnetic phase is smeared.

   • Mechanism: In the Griffiths region near this transition, rare regions are locally Ferromagnetic. These regions involve the ordering of the original spin variables ($\eta$). Since the dissipation couples directly to $\eta$, the effective dissipation strength $\tilde{\alpha}$ grows with the size of the rare region.
   • Outcome: For sufficiently large RRs, $\tilde{\alpha}$ exceeds a critical threshold ($\alpha_c = 1/2$), freezing the quantum tunneling. These regions order independently of the bulk at different control parameter values, turning the sharp transition into a smeared crossover. The Griffiths Product phase transforms into an Inhomogeneous Ferromagnetic (IFM) phase.

2. PM-PROD Transition (Sharp): The transition between the Paramagnetic and Product phases remains sharp.

   • Mechanism: In the Griffiths region near this transition, rare regions are locally in the Product phase. The order parameter here is the composite variable $\sigma^z = S^z_1 S^z_2$.
   • Crucial Insight: The dissipation couples to the individual spins ($S^z_1, S^z_2$), not directly to their product. In the Product phase, the individual spins fluctuate rapidly and independently, but their product remains ordered. The authors demonstrate that the dissipation effects on the two constituent spins cancel out regarding the dynamics of the composite order parameter.
   • Outcome: The quantum fluctuations of the Product rare regions are not overdamped. They continue to tunnel coherently, preserving the sharpness of the phase transition.

B. Robustness

The authors verify that these conclusions hold for:

• Ohmic and Sub-Ohmic dissipation: Both regimes lead to smearing of the FM-related transitions.
• Super-Ohmic dissipation: Transitions remain sharp as tunneling is not suppressed.
• Alternative Coupling Schemes: The results persist even if the two spin colors couple to different baths (color-dependent coupling) or share a single bath.

4. Physical Interpretation

The paper highlights a fundamental distinction between local order and composite (intertwined) order in the presence of dissipation:

• Local Order (FM): Dissipation couples directly to the order parameter. Large RRs freeze, leading to smearing.
• Composite Order (PROD): The order parameter is a product of two fields. Dissipation couples to the constituents, but the specific symmetry of the composite state leads to a cancellation of dissipative effects on the collective mode. Consequently, the rare regions remain dynamic, and the transition remains a true quantum critical point belonging to the infinite-randomness universality class.

5. Significance

• Theoretical Advancement: This work extends the theory of smeared phase transitions (previously established for the Random Transverse-Field Ising Model) to models with composite order parameters. It reveals that the "smearing" phenomenon is not universal to all disordered quantum transitions but depends critically on the symmetry of the order parameter relative to the bath coupling.
• Experimental Relevance: The findings provide a theoretical framework for interpreting experiments in materials like $Sr_{1-x}Ca_xRuO_3$ and $CePd_{1-x}Rh_x$, where smeared transitions are observed. It suggests that in systems with composite orders (e.g., nematic order in iron pnictides or quadrupolar order in frustrated magnets), dissipation might not smear the transition, even if the system is coupled to a metallic bath.
• Methodological Impact: The successful application of the combined Adiabatic-SDRG method to a multi-component spin chain with dissipation sets a precedent for studying complex dissipative quantum many-body systems.

In summary, the paper demonstrates that while dissipation and disorder generally conspire to smear quantum phase transitions involving local order, intertwined composite orders can be robust against this smearing, preserving sharp quantum criticality.