Quantum critical behavior of cuprate superconductors observed by inelastic X-ray scattering

Using high-resolution resonant inelastic X-ray scattering, researchers uncovered conclusive evidence for a quantum critical point in La$_{2-x}$Sr$_x$CuO$_4$ cuprate superconductors by observing universal scaling of charge-charge correlations with a critical exponent consistent with O(4) symmetry, despite the point being buried beneath the superconducting dome.

The Gist

Imagine a cup of coffee that, instead of just getting cold, suddenly decides to conduct electricity perfectly without any resistance. This is the world of cuprate superconductors, materials that have baffled scientists for decades. They are like a complex dance floor where electrons (the dancers) suddenly stop bumping into each other and glide in perfect unison.

But there's a mystery: Why do they do this? And what happens right before they start dancing?

This paper is like a detective story where scientists finally found a hidden clue to solve the mystery of these materials. Here is the story, broken down into simple concepts.

The Mystery: The "Buried" Treasure

Scientists have long suspected that cuprates have a special "switch" hidden inside them called a Quantum Critical Point (QCP). Think of a QCP as a specific spot on a map where the rules of physics change completely. If you could stand right on this spot, the material would be in a state of perfect chaos and order at the same time, creating the conditions for superconductivity.

However, there was a problem: The treasure was buried.
In these materials, the superconducting state (the perfect dance) forms a "dome" that covers the QCP. It's like trying to find a specific tree in a forest, but a thick fog (the superconductivity) hides the tree from view. Every time scientists tried to look for the QCP, the fog just got in the way, making it look like the tree didn't exist at all.

The New Tool: X-Ray "Slow Motion"

To see through the fog, the researchers used a high-tech camera called Resonant Inelastic X-ray Scattering (RIXS).

• The Analogy: Imagine you are in a dark room and you want to see how a crowd of people is moving. You can't see them directly, so you throw a handful of glowing confetti (X-rays) at them. By watching how the confetti bounces off and changes color (energy), you can figure out how the people are moving, even if you can't see them clearly.
• They used this "confetti" to watch the Charge Density Waves (CDW). Think of CDW as a ripple or a wave pattern that electrons make as they move through the material, like ripples in a pond.

The Discovery: The "Universal Collapse"

The team looked at these ripples in different types of cuprates (with different amounts of "doping," which is like adding salt to the coffee) and at different temperatures.

They found something magical:
When they measured how far these ripples could travel before fading away (the correlation length), they noticed that for all the different samples and temperatures, the data points collapsed onto a single, perfect curve.

• The Metaphor: Imagine you have a bunch of different-sized rubber bands. If you stretch them, they behave differently. But if you stretch them all to a specific "critical point," they all snap back with the exact same force and pattern. That is what the scientists saw. All the messy, different data points lined up perfectly on one universal line.

This "collapse" is the smoking gun. It proves that the hidden Quantum Critical Point is real. The fact that the data fits a specific mathematical pattern (with a number called an exponent, $\nu$, of roughly 0.74) confirms that the material is undergoing a fundamental phase transition, even though it's hidden under the superconducting fog.

The Twist: A "Team Sport" Symmetry

Here is the most exciting part. The researchers expected the QCP to be driven by just one thing: the Charge Density Waves (the ripples).

But the math told a different story. The number they found (0.74) didn't match the pattern for just ripples. It matched a pattern for a team of four players working together.

• The Analogy: Imagine a sports team. You thought the game was just about the quarterback (the Charge Wave). But the data showed that the quarterback was actually passing the ball to three other players: Pair-Density Waves (a different kind of electron pairing), Superconductivity itself, and the Charge Wave.
• These four "players" were intertwined, holding hands, and acting as a single unit. The researchers call this O(4) symmetry. It's like the material isn't just doing one thing; it's doing four things simultaneously, and that complex teamwork is what creates the superconductivity.

The Conclusion: Why It Matters

For years, scientists were frustrated because they couldn't see the "engine" driving these superconductors. This paper says: "The engine is there, but it's running so hot and fast that it looks like static noise."

The "noise" they saw wasn't a failure of the experiment; it was the sound of the material fluctuating wildly between different states (superconducting, charge waves, etc.) right at the critical point.

In simple terms:

1. The Problem: We couldn't find the "magic switch" (QCP) because superconductivity was hiding it.
2. The Solution: We used X-rays to watch the "ripples" in the electrons.
3. The Proof: The ripples from all different samples lined up perfectly on one curve, proving the switch exists.
4. The Surprise: The switch isn't controlled by one force, but by a complex team of four different quantum orders working together.

This discovery gives us a new map for understanding how superconductors work, potentially helping us design materials that can superconduct at room temperature, which would revolutionize our power grids, transportation, and electronics.

Technical Summary

1. Problem Statement

The physics of high-temperature cuprate superconductors remains one of the most significant unsolved problems in condensed matter physics. A central hypothesis is that the complex phase diagram, particularly the "strange metal" behavior and the pseudogap phase, is driven by a Quantum Critical Point (QCP) buried beneath the superconducting dome.

• The Challenge: Direct evidence for this QCP has been elusive because the superconducting state (SC) masks the critical fluctuations. The presumed QCP is associated with Charge-Density Waves (CDW), but the interplay between CDW and SC orders makes it difficult to distinguish whether the observed anomalies are due to a true QCP or merely the competition between orders.
• Specific Question: Does a QCP exist in the hole-doped cuprate $La_{2-x}Sr_xCuO_4$ (LSCO) associated with CDW fluctuations, and if so, what is its universality class and critical behavior?

2. Methodology

The authors employed high-resolution Resonant Inelastic X-ray Scattering (RIXS) to probe the dynamical charge-charge correlations in LSCO.

• Samples: Single crystals of $La_{2-x}Sr_xCuO_4$ with hole concentrations $x = 0.12, 0.15, 0.17,$ and $0.18$.
• Experimental Setup: Measurements were conducted at the National Synchrotron Radiation Research Center (Taiwan) using the O K-edge RIXS technique.
  • Energy Resolution: 16–25 meV.
  • Conditions: Measurements were taken at various temperatures (down to 24 K) and momentum transfers ($q$) along the anti-nodal direction.
• Data Analysis Strategy:
  1. Spectral Decomposition: The raw RIXS spectra were decomposed into elastic scattering, acoustic phonons, optical phonons (buckling, apical oxygen, breathing modes), and the dynamical CDW fluctuation component.
  2. Phenomenological Modeling: The CDW fluctuations were modeled using a Ginzburg-Landau approach. The charge susceptibility $\chi_{CDW}(q, \omega)$ was described by a relativistic form:
     $$\chi_{CDW}(q_\parallel, \omega) = \frac{1}{m^2 + c^2q^2 - (\omega + i\Gamma)^2}$$
     Where $m$ is the characteristic energy (mass term), $c$ is the excitation speed, and $\Gamma$ is the inverse lifetime (relaxation rate).
  3. Scaling Analysis: The characteristic energy $m$ (inversely proportional to the correlation length $\xi$) was analyzed against doping ($x$) and temperature ($T$) to test for quantum critical scaling: $m(x, T) \propto T^\nu F(|x-x_c|/T)$.

3. Key Contributions

• Unmasking the QCP: The authors demonstrated that the apparent absence of a diverging susceptibility peak (a standard QCP signature) is a "disguise" caused by the intertwining of CDW and Superconducting orders. This intertwining enhances the relaxation rate ($\Gamma$), causing the resonance peak to broaden and the intensity to decrease, rather than diverge.
• Universal Scaling: They successfully collapsed data from various dopings and temperatures onto a single universal scaling curve, providing definitive evidence for a QCP.
• Universality Class Identification: The study identified the critical exponent $\nu$ and determined that the QCP belongs to the O(4) universality class, suggesting a symmetry enlargement where CDW order couples with Pair-Density Wave (PDW) order.
• Dissipative Nature: The paper characterizes the QCP as highly dissipative, with a breakdown of the quasiparticle picture near the critical point.

4. Key Results

• Critical Exponent ($\nu$): The analysis yielded a critical exponent of $\nu = 0.74 \pm 0.08$.
  • This value satisfies the Harris criterion ($\nu \ge 2/D$), indicating the QCP is governed by a clean model unaffected by disorder.
  • The value is inconsistent with a mean-field theory or a simple O(2) model (typical for CDW alone, $\nu \approx 0.67$).
  • It closely matches the theoretical prediction for the O(4) model ($\nu \approx 0.74$).
• Scaling Collapse: When plotting the inverse correlation length ($m/T^\nu$) versus the scaled doping distance ($|x-x_c|/T$), all data points for different dopings and temperatures collapsed onto a single curve defined by $F(y) = Ay^\nu + B$. This confirms the existence of a QCP at a critical doping $x_c \approx 0.195$.
• Dynamics near QCP:
  • As the system approaches $x_c$, the characteristic energy $m$ (gap) decreases.
  • Simultaneously, the relaxation rate $\Gamma$ increases significantly.
  • Near the QCP, $\Gamma$ exceeds $m$, leading to the breakdown of the bosonic quasiparticle picture. The CDW fluctuations become overdamped, explaining the lack of a sharp peak in the RIXS intensity.
• Symmetry Enlargement: The O(4) symmetry implies that the order parameter is a vector $(\rho_Q, \Delta_Q)$, where $\rho_Q$ is the CDW order and $\Delta_Q$ is the Pair-Density Wave (PDW) order. This suggests the QCP is rooted in the microscopic SO(4) symmetry found in the Hubbard model at half-filling.
• Dissipation and Superfluid Density: The inverse lifetime of the CDW quasiparticles scales with the superfluid density ($\rho_s$). This indicates that the quantum fluctuations are driven by the coupling between the phase gradients of the CDW and the superconducting order.

5. Significance

• Resolution of a Long-standing Debate: This work provides the first conclusive experimental evidence of a QCP in cuprates buried within the superconducting dome, resolving skepticism regarding the "ideal QCP" scenario.
• New Perspective on Phase Transitions: It redefines the quantum phase transition in cuprates not as a simple transition between two phases, but as a transition between a state of intertwined orders (CDW, PDW, SC) and a pure d-wave superconducting state.
• Theoretical Implications: The identification of the O(4) universality class links the macroscopic phenomenology of cuprates directly to the microscopic SO(4) symmetry of the Hubbard model, offering a unified framework for understanding charge, spin, and pairing orders.
• Methodological Advance: The study demonstrates that high-resolution RIXS, combined with rigorous scaling analysis, can extract critical exponents even when the system is strongly dissipative and the critical point is obscured by competing orders.

In summary, the paper establishes that the strange-metal behavior in cuprates arises from a quantum critical point characterized by O(4) symmetry, where charge density waves and pair-density waves fluctuate collectively, leading to non-Fermi liquid behavior and Planckian dissipation.