Singular spin fluctuations in the strange-metal phase of La2-xSrxCuO4

By employing high magnetic fields and an NMR protocol tailored to electronic inhomogeneity, this study reveals that singular, quantum-critical spin fluctuations persist in the overdoped strange-metal phase of La2-xSrxCuO4 (x=0.25) due to spatially inhomogeneous nanoscale electronic states, challenging the conventional view that such fluctuations are confined to the spin-stripe critical doping.

The Gist

Imagine a group of dancers on a floor. In a normal metal (like copper wire), these dancers move in a predictable, orderly way, bumping into each other occasionally. But in a "strange metal," a mysterious state found in certain superconductors, the dancers move in a chaotic, perfectly synchronized chaos where their resistance to moving (electrical resistivity) changes in a very strange, linear way as the room gets colder. Scientists have been trying to figure out why they dance this way.

For a long time, many suspected that the dancers were reacting to invisible "spin fluctuations"—tiny, rhythmic wiggles in their magnetic orientation. However, in the specific region of the dance floor where the "strange metal" behavior happens (called the overdoped regime), previous measurements suggested these magnetic wiggles were too weak to cause the chaos. It was like trying to explain a hurricane by looking at a gentle breeze.

The New Discovery: Turning Up the Volume

This paper reports a breakthrough in observing these wiggles in a specific material called La2−xSrxCuO4 (LSCO). The researchers faced two main problems:

1. Superconductivity: At low temperatures, the dancers usually stop moving chaotically and start gliding perfectly without friction (superconductivity). This hides the "strange metal" behavior.
2. The Wrong Lens: Previous tools used to measure the magnetic wiggles (looking at Copper atoms) were getting "blinded" by the chaos, missing the signal entirely.

To solve this, the team used a massive magnetic field (26 Tesla, about 500,000 times stronger than a fridge magnet). Think of this as a giant "pause button" that forces the dancers to stop gliding and start moving chaotically again, revealing the underlying strange metal state.

They also switched their "camera lens." Instead of looking at the Copper atoms (which were too jittery and lost the signal), they looked at the Lanthanum atoms. These atoms act like a more stable, wide-angle lens that can see the whole dance floor without getting confused.

What They Found

When they looked through this new lens under the giant magnetic field, they saw something surprising:

• The Wiggles Explode: As the temperature dropped toward absolute zero, the low-energy magnetic wiggles didn't fade away; they grew stronger and stronger, almost infinitely so.
• The Paradox: This explosion of magnetic activity was happening in a part of the dance floor where scientists thought the "stripe" patterns (ordered magnetic lines) had already disappeared. It's like hearing a massive orchestra playing a crescendo in a room where you thought the musicians had left.

The Hidden Clue: A Patchwork Floor

The data also revealed that the dance floor isn't uniform.

• The Stretching Effect: The way the dancers relaxed back to order wasn't the same everywhere. Some parts of the floor were very active, while others were calmer.
• The Explanation: The researchers propose that the floor is actually a patchwork of tiny puddles. In some small puddles (about 25-30% of the floor), the local conditions are still just right for the magnetic stripes to exist and wiggle wildly. In the rest of the floor, the stripes are gone.
• The Analogy: Imagine a large crowd where most people are just walking randomly, but there are small, hidden pockets where a riot is happening. If you look at the whole crowd from far away, you might miss the riots. But if you have a special camera that can see the "heat" of the riots, you realize the whole crowd's chaotic behavior is actually being driven by these hidden pockets of intense activity.

Why It Matters

This study suggests that the "strange metal" behavior isn't a mystery of the whole material, but rather a result of these tiny, hidden pockets of intense magnetic activity (quantum critical fluctuations) that persist even when the material is heavily doped. It provides a new, concrete piece of evidence that these magnetic wiggles are indeed the engine driving the strange electrical properties of these materials, solving a puzzle that has confused physicists for decades.

In short: By using a giant magnet to stop the superconductivity and a better "camera" to see the details, the scientists found that the strange metal behavior is driven by intense, hidden magnetic wiggles that were previously invisible.

Technical Summary

Technical Summary: Singular Spin Fluctuations in the Strange-Metal Phase of La$_{2-x}$Sr$_x$CuO$_4$

Problem Statement
The electronic ground state of hole-doped cuprates in the overdoped regime ($p > p^* \approx 0.2$) remains a central open problem. This regime hosts an extended "strange-metal" phase characterized by electrical resistivity linear in temperature ($T$) down to the lowest temperatures. While spin fluctuations are a leading candidate mechanism for this behavior, compelling evidence has been lacking. Previous Nuclear Magnetic Resonance (NMR) studies suggested that the low-energy dynamical spin susceptibility, $\chi''(q, \omega)$, was too weak to account for linear resistivity. Furthermore, the temperature, energy, and doping dependence of these fluctuations remained incompletely characterized, particularly above the pseudogap critical point $p^*$. A specific challenge has been the difficulty in accessing the intrinsic normal state at low temperatures due to the persistence of superconductivity and the potential loss of NMR signal sensitivity (signal "wipeout") in the presence of strong spin correlations.

Methodology
To address these limitations, the authors investigated La$_{1.75}$Sr$_{0.25}$CuO$_4$ ($x = 0.25$) using a combination of high magnetic fields and a tailored NMR protocol:

• High Magnetic Fields: Experiments were conducted at magnetic fields up to 26 T, applied perpendicular to the CuO$_2$ planes. This field strength is comparable to the upper critical field ($B_{c2}$) at this doping, effectively suppressing superconductivity to access the low-temperature normal state where strange-metal behavior occurs.
• Isotope Selection ($^{139}$La vs. $^{63}$Cu): The study utilized $^{139}$La NMR rather than the conventional $^{63}$Cu NMR. The authors demonstrate that $^{63}$Cu NMR suffers from severe signal loss and "wipeout" effects at low temperatures due to short spin-spin relaxation times ($T_2$) and inhomogeneous spin dynamics, rendering it unreliable for quantitative probing of low-temperature spin dynamics in this regime. In contrast, $^{139}$La, with its longer $T_2$ and weaker hyperfine coupling, provides a robust probe.
• Relaxation Measurements: The spin-lattice relaxation rate ($1/T_1$) was measured for $^{139}$La. The quantity $1/T_1T$ is proportional to the low-frequency slope of the dynamical spin susceptibility, $\chi''(q, \omega)/\omega$, in the limit $\omega \to 0$.
• Analysis of Inhomogeneity: The recovery of nuclear magnetization was analyzed using a stretched exponential function to extract a stretching exponent $\beta$, which phenomenologically accounts for a distribution of $T_1$ values, indicating spatial inhomogeneity.

Key Results

1. Singular Divergence of Spin Susceptibility: At 26 T, the $^{139}$La $1/T_1T$ increases monotonically upon cooling down to the base temperature of 1.34 K, with no indication of saturation. The data are well-described by both a Curie–Weiss form (with a very small Weiss temperature $\theta \approx -4.6$ K) and a power law ($1/T_1T \propto T^{-(1+\alpha)}$ with $\alpha \approx -0.67$). Both fits imply a divergent or nearly divergent $\chi''(q, \omega \to 0)$ as $T \to 0$.
2. Field Dependence: At 20 T, the behavior is indistinguishable from that at 26 T. At 15 T, the low-temperature increase is weaker, yielding a larger Weiss temperature ($\theta \approx -20$ K), suggesting that residual superconducting correlations suppress the intrinsic divergence at lower fields.
3. Spatial Inhomogeneity: The magnetization recovery curves deviate from a single exponential, requiring a stretching exponent $\beta < 1$. This deviation begins below $\sim 100$ K and becomes field-independent at low temperatures. A two-component fit reveals two distinct relaxation rates differing by a factor of five, with the faster (more magnetic) component accounting for roughly 25–30% of the sample volume. This suggests that the spin dynamics are spatially inhomogeneous, likely arising from nanoscale electronic inhomogeneity where local doping variations create "puddles" with doping levels near or below $p^*$.
4. Exclusion of Density of States Effects: The observed temperature dependence cannot be explained solely by a Van Hove singularity in the density of states (DOS), as the inferred DOS variation corresponds to a doping level ($p \approx 0.22$) distinct from the sample ($p=0.25$), and the static susceptibility does not show the expected Van Hove signature.

Significance and Claims
The paper claims to provide direct evidence for "hitherto hidden" singular low-energy spin fluctuations in the overdoped strange-metal phase of LSCO. The significance is fourfold:

1. Methodological Breakthrough: The strong enhancement of $\chi''$ was previously undetected because prior studies did not use sufficiently high magnetic fields and relied on $^{63}$Cu NMR, which is insensitive to these fluctuations in this regime due to signal wipeout.
2. Direct Correlation: The divergence is established in the field-induced normal state under the exact same conditions (doping, temperature, field) where linear resistivity is observed, removing the need for extrapolations from zero-field data.
3. Quantum Criticality: The singular behavior of $\chi''(q, \omega \to 0)$ is suggestive of quantum-critical dynamics. While the authors do not claim to prove that strange metallicity is quantum critical, they posit that these fluctuations are a plausible microscopic origin for the linear-in-$T$ resistivity, as they are observed under the same conditions.
4. Role of Inhomogeneity: The observation of spatial inhomogeneity suggests that nanoscale electronic variations may underlie the apparent paradox of observing quantum-critical-like fluctuations well beyond the nominal spin-stripe critical doping ($x \approx 0.19$). The authors propose that persistent, quantum-critical-like dynamics may arise from the inability of spin stripes to freeze in local regions with $p_{loc} \lesssim p^*$, even when the bulk doping is higher.

The authors conclude that these observations provide quantitative benchmarks for microscopic descriptions of the strange-metal state and highlight the necessity of characterizing spatial inhomogeneity in future studies across different doping levels and cuprate families.