Microscopic mechanism of high-temperature superconductivity revealed by ab initio studies on hole-doped multilayer cuprates HgBa$_2$Ca$_2$Cu$_3$O$_8$ under pressure

This study employs *ab initio* calculations with a neural network variational solver to reveal that the high-temperature superconductivity in pressurized HgBa$_2$Ca$_2$Cu$_3$O$_8$ arises from an emergent local attraction generated by reduced offsite Coulomb repulsion and the release of "false vacuum" fluctuations, offering a new microscopic mechanism for designing optimized superconductors.

The Gist

Imagine a group of electrons living in a crowded apartment building made of copper and oxygen atoms. In most materials, these electrons are like shy neighbors who avoid each other because they all carry a negative charge (repulsion). But in a special class of materials called "cuprates," something magical happens: under the right conditions, these electrons pair up and dance together without any friction, creating superconductivity (electricity that flows with zero resistance).

For decades, physicists have been trying to figure out the "secret recipe" for this dance, especially in a specific material called Hg1223, which holds the world record for the highest temperature above 130 K (above -140C) at normal pressure at which this magic (superconductivity) happens, and the magic occurs at an even higher temperature when squeezed (under pressure).

This paper is like a high-tech detective story where the authors use powerful computer simulations to peek inside the microscopic world of Hg1223 and explain why it is such a champion. Here is the story in simple terms:

1. The Building Layout: A Three-Layer Cake

Cuprate superconductors come in various numbers of stories per unit, such as single-story houses or two-story duplexes; Hg1223 is a three-story building.

• It has an Inner Layer (the middle floor) and two Outer Layers (the top and bottom floors).
• The authors found that the electrons on the middle floor and the outer floors don't behave exactly the same way. The middle floor is a bit more crowded (closer to a state where electrons stop moving entirely), while the outer floors are more free.
• Despite this difference, the layers talk to each other. The outer layers help the middle layer, and vice versa, creating a "proximity effect" where the whole building works together better than if the floors were isolated.

2. The Pressure Cooker: Squeezing the Building

When you squeeze a sponge, water flows out faster. When the scientists "squeezed" this material with high pressure (up to 30,000 times normal atmospheric pressure), the building got smaller, and the electrons got closer.

• The Result: The temperature at which superconductivity happens went up, reaching a peak.
• The Secret Sauce: The pressure didn't just push things closer; it changed the rules of the game. It reduced the "long-distance" arguments between electrons (called off-site repulsion) much more than the "in-your-face" arguments (local repulsion). This made it easier for the electrons to pair up.

3. The Paradox: Repulsion Creates Attraction

This is the most mind-bending part of the discovery.

• The Old Idea: In traditional superconductors, electrons need a "glue" (like vibrations in the building's structure) to stick together because they naturally hate each other.
• The New Discovery: In Hg1223, the authors found that the strong repulsion itself, counterintuitively, DIRECTLY creates the emergent attraction WITHOUT any 'glue'.
  • The Analogy: Imagine a room full of people who really don't want to stand next to each other (strong repulsion). If you force them to move, they might accidentally find a spot where standing next to someone else is actually less painful than standing alone.
  • In the quantum world, the strong "no-touching" rule (Coulomb repulsion) creates a situation where electrons are forced to avoid "double-occupancy" (two electrons on one spot). When they are doped (adding extra electrons), this avoidance creates an instantaneous, local attraction. It's like the electrons are saying, "I hate touching other electrons, so I prefer to stay in a sparse (low-density) area; but I find another electron feels the same way and also moves into the sparse area near me, so effectively the two of us end up attracting each other. Eventually we find a way to avoid touching by forming a pair within that sparse area - so let's pair up quickly."

4. The "False Vacuum" and the Escape

The paper uses a fascinating metaphor involving a "False Vacuum."

• Think of the electrons in the material as being stuck in a deep, uncomfortable valley (the "Mott insulator" state) where they are frozen and can't move.
• When you add carriers (doping), it's like giving them a key to escape that valley.
• The "attraction" comes from the release of tension. The electrons are no longer stuck in that uncomfortable "false vacuum" of being forced to double-up. They are freed to move into a new, smooth state (the superconducting state). This sudden release of pressure provides space/room for the electrons to come closer together in the 'released' environment, and that is what creates the pairs.

5. Why Hg1223 is the Champion

So, why does this three-layer building beat all the others?

1. Poor Shielding: The 'shielding' that normally weakens the local repulsion usually comes from the nearby (adjacent) layers; but in Hg1223 the relevant nearby layer is missing inside the three-layer unit, so the shielding is weaker. This makes the local repulsion ($U$) very strong. Paradoxically, this strong repulsion is what generates the strongest "escape attraction."
2. Pressure Sensitivity: When pressure is applied, the "long-distance" arguments ($V$) between electrons drop dramatically. Because the pair is formed by the electrons AVOIDING touching each other, the two electrons pair at OFFSITE positions (separated, not on the same site); therefore 'long-distance' (offsite) Coulomb repulsion V directly destroys/kills such a pair. So REDUCING that offsite repulsion V helps the pair survive.

The Bottom Line

The paper concludes that the secret to the highest-temperature superconductivity isn't a new type of glue, but rather a clever trick of repulsion. By squeezing the material, the scientists found a way to turn the electrons' natural hatred of each other into a powerful, instantaneous force that binds them together.

This discovery doesn't just explain Hg1223; it offers a new map for designing future materials. Instead of looking for a magical "glue," future engineers might look for ways to tune the repulsion and reduce long-distance arguments between electrons to create even better superconductors.

Technical Summary

Technical Summary: Microscopic Mechanism of High-Temperature Superconductivity in HgBa$_2$Ca$_2$Cu$_3$O$_8$

Problem Statement
The microscopic origin of high-temperature superconductivity (SC) in cuprates remains a central challenge in condensed matter physics. While ab initio studies have successfully reproduced the material dependence of the critical temperature ($T_c$) for single-, double-, and infinite-layer cuprates, the significantly higher $T_c$ observed in triple-layer compounds, specifically HgBa$_2$Ca$_2$Cu$_3$O$_8$ (Hg1223), has remained unexplained. Hg1223 holds the record for the highest ambient-pressure $T_c$ ($\sim$134 K) and exhibits a further increase under pressure to $\sim$160 K. Previous numerical analyses were hindered by the computational cost associated with the large unit cell of multi-layer systems. This study aims to elucidate the microscopic mechanism behind the high $T_c$ of Hg1223 and its unique pressure dependence using a parameter-free ab initio framework.

Methodology
The authors employed a first-principles approach based on the ab initio low-energy effective Hamiltonian derived for the antibonding orbital of Hg1223, consisting of hybridized Cu 3$d_{x^2-y^2}$ and O 2$p_\sigma$ orbitals. The Hamiltonian includes nearest-neighbor hopping ($t$), onsite Coulomb repulsion ($U$), and off-site Coulomb interactions ($V$), with parameters derived via constrained GW (cGW) calculations that incorporate self-interaction corrections and level renormalization feedback.

To solve the many-body problem, the authors utilized a refined Variational Monte Carlo (VMC) method combined with neural network techniques (Restricted Boltzmann Machine correlation factors) and the Lanczos method. The variational wave function included Gutzwiller, Jastrow, and doublon-holon correlation factors. Calculations were performed on lattices up to $28 \times 28 \times 3$ sites (representing three CuO$_2$ planes), allowing for extrapolation to the thermodynamic limit. The study covered ambient pressure and pressures up to 60 GPa, analyzing ground states to determine SC order parameters, magnetic order, and carrier densities.

Key Contributions and Results

1. Reproduction of High $T_c$ and Pressure Dependence:
   The study quantitatively reproduced the SC order parameter ($F_{SC}$) and estimated $T_c$ for Hg1223. The results show a dome-like pressure dependence of $T_c$, peaking around 30 GPa, in essential agreement with experimental indications. The calculated $T_c$ reaches $\sim$160 K at the peak, matching the experimental record.

2. Origin of High Ambient-Pressure $T_c$:
   The high $T_c$ of Hg1223 at ambient pressure compared to other cuprates is attributed to a strong local Coulomb repulsion ($U$). This strength arises from poorer screening of Coulomb interactions in multi-layer compounds, leading to a higher dimensionless ratio $U/|t_1|$.

3. Mechanism of Pressure-Induced $T_c$ Enhancement:
   The further increase in $T_c$ under pressure is ascribed to a unique interplay of three factors:

   • Increased electron hopping ($t$).
   • Decreased onsite Coulomb repulsion ($U$).
   • Crucially, a strongly reduced non-local Coulomb repulsion ($V$).
     The reduction of off-site interactions ($V$) under pressure plays a major role in enhancing SC, compensating for the reduction in $F_{SC}$ caused by the decreasing $U/|t_1|$ ratio. This finding highlights that controlling off-site interactions is critical for material design.

4. Microscopic Pairing Mechanism:
   The study identifies the origin of Cooper pairing as an emergent local attraction generated counter-intuitively from the strong bare local repulsion ($U$). This attraction arises from the "release of fluctuating doubly-occupied sites" (characterized as a "false vacuum" in the Mott insulator) to double-occupation-free $d$-wave SC states upon carrier doping. This mechanism is described as "attraction from reduced repulsion," distinct from conventional BCS SC mediated by bosonic glues where retardation is essential. The local attraction is consistent with experimental analyses supporting electron fractionalization.

5. Coexistence of SC and Antiferromagnetism (AF):
   In the underdoped region, the study demonstrates the microscopic coexistence of SC and AF orders. This is a characteristic feature of the multi-layer system driven by self-doping, which creates a difference in carrier concentration between the inner plane (IP) and outer planes (OPs). The IP, being closer to half-filling, stabilizes AF order, while the OP stabilizes SC order, with interlayer proximity enabling their coexistence.

6. Scaling and Universality:
   The estimated $T_c$ follows the scaling form $T_c \sim 0.16 |t_1| F_{SC}$. While the SC order parameter $F_{SC}$ for Hg1223 under pressure deviates from the universal trend of single- and double-layer compounds at ambient pressure (due to the significant reduction of off-site interactions), the scaling relation remains valid when accounting for the specific Hamiltonian parameters under pressure.

Significance
The paper claims to provide a quantitative and microscopic understanding of the high $T_c$ in Hg1223, resolving the long-standing challenge of explaining the record-breaking critical temperatures in multi-layer cuprates. By identifying the "emergent local attraction" arising from strong correlations and the critical role of reduced off-site interactions under pressure, the study offers a new perspective on the origin of cuprate superconductivity. The authors suggest that this understanding provides a new route for future research aimed at designing and optimizing SC materials by explicitly utilizing the channel of emergent attraction. The work validates the use of ab initio parameter-free frameworks combined with advanced many-body solvers to tackle complex, strongly correlated systems where different orders and fluctuations compete.