Emergence of spin entanglement with the pseudogap onset in the Fermi-Hubbard model

By combining ultracold-atom quantum simulations with dynamical vertex approximation calculations, this study reveals that spin-singlet entanglement emerges specifically at the onset of the pseudogap regime in the two-dimensional Fermi-Hubbard model, thereby challenging purely classical theories and constraining microscopic models to those incorporating nearest-neighbor quantum correlations.

The Gist

Imagine a crowded dance floor where thousands of dancers (electrons) are trying to move around without bumping into each other. In a normal crowd, people just jostle and push; their movements are chaotic but independent. This is like a standard metal. But in certain materials, like the high-temperature superconductors scientists are trying to understand, the dancers start behaving strangely. They form a "pseudogap" phase—a mysterious state where the dance floor seems to freeze up, and the dancers start pairing up in secret, even before they actually start dancing in perfect unison (superconductivity).

For decades, physicists have been trying to figure out why this happens. The big question is: Are these dancers just reacting to each other like people in a crowd (classical correlations), or are they sharing a secret, quantum "telepathy" (entanglement) that links their minds together?

This paper, by a team of experimentalists and theorists, finally answers that question using a special "quantum simulator" made of ultra-cold atoms. Here is what they found, explained simply:

1. The Mystery of the "Ghostly Pair"

Think of entanglement as a magical link between two dancers. If you change one, the other instantly reacts, no matter how far apart they are. However, in the real world, you can't just look at the dancers to see this link because of "rules of the game" (called Superselection Rules). You can only see the link if the dancers are in specific, allowed states.

The researchers looked for this "allowed" entanglement between neighbors on the dance floor.

2. The Big Discovery: Entanglement Only Appears in the "Pseudogap"

They mapped out the dance floor at different temperatures and crowd densities.

• Outside the Pseudogap: When it's too hot or the crowd is too sparse, the dancers are just moving randomly. There is no magical link between them. They are just classical neighbors.
• Inside the Pseudogap: As the temperature drops and the crowd reaches a specific density, a "pseudogap" forms. Suddenly, the researchers detected a strong magical link (entanglement) between immediate neighbors.

The Analogy: Imagine a room full of people. At first, everyone is just chatting with anyone nearby (classical noise). But then, the lights dim (the pseudogap starts), and suddenly, every person is holding hands only with the person standing right next to them, forming a secret, invisible chain. If you look at the person two spots away, there is no hand-holding.

3. The "Hand-Holding" is Strictly Local

One of the most surprising findings is how close this link is.

• Nearest Neighbors: The "hand-holding" (entanglement) happens only between people standing right next to each other.
• Next-Door Neighbors: If you look at people standing two spots apart, the link disappears completely. They are just regular neighbors again.

This is like a rule where you can only hold hands with the person touching your elbow, but not the person touching their elbow. The paper shows that this "quantum hand-holding" is strictly confined to the very first step.

4. Why This Matters for the "Pseudogap"

For years, some scientists thought the pseudogap was caused by classical waves of people pushing and pulling (classical fluctuations).

• The Paper's Verdict: This theory is wrong. You cannot create this specific "quantum hand-holding" just by people pushing each other. You need actual quantum magic (superpositions) to create it.
• The Conclusion: The pseudogap isn't just a messy crowd; it's a state where electrons are forming tiny, quantum "singlet" pairs (like a dance couple) with their immediate neighbors. This is the first time this specific quantum link has been measured and confirmed to appear exactly when the pseudogap starts.

Summary

The paper uses a quantum simulator to prove that in the mysterious "pseudogap" phase of certain materials, electrons stop acting like a chaotic crowd and start forming quantum pairs with their immediate neighbors only. This proves that the pseudogap is driven by genuine quantum entanglement, not just classical chaos, and that this entanglement is incredibly local—it doesn't reach beyond the very next person in line.

This finding helps rule out theories that rely only on classical physics and forces scientists to focus on models that include these specific, short-range quantum connections to understand how these materials work.

Technical Summary

Technical Summary: Emergence of Spin Entanglement with the Pseudogap Onset in the Fermi-Hubbard Model

Problem Statement
The two-dimensional Fermi-Hubbard model remains a central paradigm for understanding strongly correlated electron systems, particularly high-temperature superconductivity and the enigmatic pseudogap phase. Despite decades of investigation, a complete microscopic understanding of the pseudogap regime is lacking. Conventional approaches typically rely on global observables and locally resolved correlation functions, which often fail to distinguish between classical and quantum correlations. While static magnetic susceptibility can identify magnetic ordering, it cannot differentiate between classical magnetic correlations (e.g., Ising-like) and quantum spin-singlet correlations. The authors posit that entanglement offers a complementary microscopic perspective, as it captures only quantum correlations. However, entanglement in correlated Fermi systems at finite temperatures has been difficult to characterize both experimentally and theoretically, particularly regarding its spatial structure and operational accessibility.

Methodology
The study employs a dual approach combining experimental quantum simulation and advanced theoretical calculations:

1. Experimental Quantum Simulation: The authors utilize a 2D ultracold-atom quantum simulator using $^6$Li atoms in an optical lattice to realize the Fermi-Hubbard model. They employ a spin-resolved quantum gas microscope to measure site-resolved number and spin correlations. Crucially, they reconstruct the two-site reduced density matrix ($\rho_{ij}$) for nearest-neighbor (NN) and next-nearest-neighbor (NNN) sites.
2. Superselection Rules (SSR) and Entanglement Measures: To address the operational constraints of local measurements (which cannot change local charge or fermionic parity), the authors apply Superselection Rules (SSR) to the density matrix. This projects out matrix elements connecting distinct local superselection sectors, isolating the "operationally accessible" entanglement. They calculate the SSR logarithmic negativity ($N$), a measure of entanglement that detects non-separability under these constraints. They also compute the SSR mutual information ($I$) to quantify total correlations (classical and quantum).
3. Theoretical Framework: Theoretical results are generated using the Dynamical Vertex Approximation (D$\Gamma$A), a diagrammatic extension of Dynamical Mean-Field Theory (DMFT) that includes non-local spin and charge fluctuations via ladder diagrams. The two-site reduced density matrix is computed from two- and four-point Green's functions obtained via D$\Gamma$A.

Key Contributions and Results

• Entanglement Onset Coincides with Pseudogap: The primary finding is that operationally accessible spin-singlet entanglement emerges precisely at the onset of the pseudogap regime. In the phase diagram of temperature ($T$) versus doping ($\delta$), the SSR negativity is vanishingly small (indicating no entanglement) in the metallic regime. It becomes finite only when the system enters the pseudogap phase (identified by the suppression of antinodal spectral weight).
• Spatial Confinement to Nearest Neighbors: The entanglement is strictly confined to nearest-neighbor sites.
  • For nearest neighbors, the ratio of spin-singlet probability to spin-triplet probability ($p_{\text{singlet}}/p_{\text{triplet}}$) exceeds unity within the pseudogap, satisfying the condition for SSR entanglement.
  • For next-nearest neighbors, this ratio remains well below the entanglement threshold ($<1$) even deep within the pseudogap regime. Consequently, no SSR entanglement is detected beyond nearest neighbors.
• Distinction from Classical Correlations: While SSR entanglement is short-ranged (NN only), the SSR mutual information (total correlations) extends beyond nearest neighbors, exhibiting a correlation length of approximately 2.5 lattice sites. This indicates that while long-range classical correlations persist, the quantum entanglement is a short-range, nearest-neighbor phenomenon.
• Agreement Between Experiment and Theory: There is excellent quantitative agreement between the ultracold-atom experimental data and the D$\Gamma$A calculations across the sampled temperature and doping range.

Significance and Claims
The paper claims that these results provide a sharp, falsifiable constraint on microscopic theories of the pseudogap:

1. Exclusion of Purely Classical Theories: The observation of finite SSR negativity (requiring non-zero off-diagonal coherence $\rho_{8,9}$ and a singlet-to-triplet ratio $>1$) disfavors theories of the pseudogap based entirely on classical spin fluctuations. A purely classical thermal mixture of singlet and triplet states cannot exceed the entanglement threshold.
2. Requirement for Quantum Singlet Structure: Any viable microscopic model for the pseudogap must develop nearest-neighbor spin-singlet entanglement at the onset of the regime. This supports theories involving quantum superpositions, such as Anderson's resonating valence bond (RVB) state, though the specific "resonating" character cannot be distinguished from a static singlet structure using the measured static density matrix.
3. Microscopic Explanation for Quantum Fisher Information: The findings offer a microscopic explanation for the previously observed increase in Quantum Fisher Information in cuprate pseudogap regimes, linking it directly to the emergence of nearest-neighbor spin-singlet entanglement.
4. Operational Accessibility: The work demonstrates that SSR-constrained entanglement is a robust, measurable quantity in quantum simulators that distinguishes quantum from classical correlations in finite-temperature Fermi systems, providing a new tool to test candidate models for high-temperature superconductivity.

The authors modestly note that while their results rule out purely classical fluctuation theories, they do not exclude additional entanglement mechanisms (e.g., from quantum criticality or superconducting fluctuations) that might emerge at lower temperatures or longer ranges, which remain subjects for future investigation.