Superconducting properties of the three-dimensional Hofstadter-Hubbard model below the critical flux for Weyl points

This paper investigates the superconducting properties of the three-dimensional Hofstadter-Hubbard model with attractive interactions, revealing a critical magnetic flux that separates two distinct regimes: one where superconductivity emerges only above a finite interaction threshold due to Weyl points, and another where it occurs for arbitrarily weak attraction with BCS-like scaling.

The Gist

Imagine a bustling city made of tiny, invisible streets (a lattice) where electrons are the citizens. Usually, these electrons move freely, but in this specific city, there's a twist: a giant, invisible magnetic wind is blowing through the streets. This wind forces the electrons to take a specific, winding path, changing how they interact with each other. This setup is called the Hofstadter model.

Now, imagine these electrons have a special personality trait: they really like to pair up and hold hands when they get close (this is called an attractive interaction). When enough of them pair up, the whole city suddenly becomes a superconductor—a state where electricity flows with zero resistance, like a perfectly smooth highway with no traffic jams.

This paper explores what happens to this electron city in three dimensions (up, down, left, right, forward, backward) when the magnetic wind changes strength. The authors discovered a fascinating "tipping point" in the wind's strength that completely changes the rules of the game.

Here is the story in simple terms:

1. The Two Worlds of the City

The researchers found that the city behaves in two completely different ways depending on how strong the magnetic wind is. They call the strength of the wind the "flux."

• The "Smooth" World (Weak Wind):
  Imagine the magnetic wind is gentle. In this world, the streets are so crowded and overlapping that the electrons can always find a partner, no matter how shy they are.

  • The Result: Even a tiny, almost invisible attraction between electrons is enough to make them pair up. The city becomes a superconductor easily. It's like a dance floor where everyone is already holding hands; you just need a tiny nudge to get the music started.

• The "Rough" World (Strong Wind):
  Now, imagine the magnetic wind gets very strong. This wind rearranges the streets so that they don't overlap anymore. Instead, the streets form distinct, separate layers with "holes" or gaps in between (these are the Weyl points mentioned in the title).

  • The Result: In this world, the electrons are isolated. They can't just pair up easily. They need a strong push (a strong attractive force) to overcome the gaps and form pairs. If the attraction is too weak, the city stays a normal metal, and no superconductivity happens. It's like a dance floor where people are stuck in separate rooms; you need a very loud, strong signal to get them to break down the walls and dance together.

2. The Critical Tipping Point

The most exciting part of the paper is the discovery of the Critical Flux ($\Phi_c$). This is the exact moment the wind changes from "Smooth" to "Rough."

• Before the Tipping Point: Superconductivity happens for free (or very cheaply).
• After the Tipping Point: You have to pay a "price" (a minimum amount of attraction, called $U_c$) to get superconductivity.

The authors mapped out exactly where this line is. They found that if you tune the magnetic wind just right, you can switch the city from a state where superconductivity is easy to a state where it's hard, right at the edge of the "Weyl points" (the gaps in the streets).

3. How the Transition Happens

The paper also looked at how the city changes as you cross this line.

• The "Gentle Slope" (Below the line): As you increase the attraction, the superconductivity grows slowly and smoothly, following a familiar pattern known from standard physics (BCS theory).
• The "Sudden Jump" (Above the line): As you approach the critical point, the superconductivity stays dead until you hit a specific threshold. Once you cross that threshold, the superconductivity "turns on" and grows according to a specific mathematical rule (like a square root). It's like a light switch that is stuck, then suddenly clicks on.

4. Why Does This Matter?

Think of this like tuning a radio.

• In the old days (standard materials), you could only get a clear signal (superconductivity) if you had a very strong antenna (strong attraction).
• This paper shows that by using the "magnetic wind" (artificial gauge fields), we can tune the radio to a frequency where the signal comes through clearly even with a tiny antenna. Or, conversely, we can tune it to a frequency where you need a massive antenna to get anything at all.

The Big Picture

The authors used complex math and computer simulations to prove that magnetism and topology (the shape of the electron paths) can be used to control superconductivity in 3D materials.

• Analogy: Imagine a 3D maze.
  • In one version of the maze, the paths are all connected; you can walk anywhere easily (Weak Wind).
  • In the other version, the paths are separated by walls, and you can only move if you have a "key" (Strong Wind).
  • The paper tells us exactly how to build the walls and where to place the key so that we can control when the maze becomes a super-highway for electricity.

This research is crucial for future technologies, like designing new materials for quantum computers or ultra-efficient power grids, where we might want to turn superconductivity on or off just by changing a magnetic field.

Technical Summary

1. Problem Statement

The paper investigates the interplay between magnetic band topology and attractive interactions in a three-dimensional (3D) Hofstadter–Hubbard (HH) model.

• Context: The Hofstadter model describes particles hopping on a lattice under a uniform magnetic flux. In 3D, for a cubic lattice with an isotropic magnetic field along the body diagonal, the single-particle spectrum exhibits a critical rational flux $\Phi_c$.
• The Critical Flux: Below $\Phi_c$, magnetic bands overlap completely. Above $\Phi_c$, isolated Weyl points (band-touching points) emerge between the $m$-th and $(m+1)$-th bands, creating a 3D semimetallic phase.
• The Question: How does the presence of an attractive Hubbard interaction ($U > 0$) affect the system near this topological transition? Specifically, does the system exhibit a quantum phase transition (QPT) from a semimetal to a superconductor (SC), and how does the critical interaction strength $U_c$ depend on the magnetic flux?

2. Methodology

The authors employ a combination of mean-field theory (MFT), numerical exact diagonalization (ED), and analytical scaling analysis.

• Model Hamiltonian:
  The system is defined on a 3D cubic lattice with periodic boundary conditions. The Hamiltonian includes:

  1. Kinetic energy with Peierls phases (Hofstadter term) induced by a magnetic flux $\Phi = 2\pi m/n$ (where $m, n$ are coprime integers).
  2. On-site attractive Hubbard interaction $-U \sum c^\dagger_{r\uparrow} c^\dagger_{r\downarrow} c_{r\downarrow} c_{r\uparrow}$.
  3. Chemical potential $\mu$ to fix the filling.

• Gauge and Diagonalization:

  • The Hasegawa gauge is used to exploit translational symmetry, reducing the problem to the diagonalization of an $n \times n$ matrix for each momentum in the Magnetic Brillouin Zone (MBZ).
  • The single-particle spectrum is computed via numerical ED.

• Mean-Field Approximation:

  • The attractive interaction is treated using the Hartree-Fock-BCS approximation.
  • The system is assumed to be in a homogeneous superconducting phase with a uniform pairing field $\Delta$.
  • The problem is mapped to a Bogoliubov-de Gennes (BdG) Hamiltonian in Nambu space.

• Self-Consistent Equations:
  The authors solve coupled equations for the pairing gap $\Delta$ and the chemical potential $\mu$ (or shifted chemical potential $\tilde{\mu}$):

  1. Gap Equation: Relates $1/U$ to an integral over the Density of States (DOS) weighted by the quasiparticle energy.
  2. Number Equation: Fixes the particle density (filling $\nu = m/n$).
  • These equations are solved numerically using a Newton algorithm with backtracking (to handle convergence issues near $\Delta=0$) in both momentum and energy spaces.

• Analytical Scaling:
  Near the critical point, the authors derive scaling relations by approximating the DOS near the Weyl energy $E_w$ as quadratic: $\rho(\epsilon) \propto (\epsilon - E_w)^2$.

3. Key Contributions and Results

A. Phase Diagram in the $(m, n)$ Plane

The study maps out the phase diagram parametrized by coprime pairs $(m, n)$, which define the flux $\Phi = 2\pi m/n$. The diagram reveals two distinct regimes separated by a critical line $\Phi = \Phi_c$:

1. Semimetal-Superconductor Transition ($\Phi > \Phi_c$):
   • In this regime, Weyl nodes exist, and the DOS vanishes at the Fermi level (at specific rational fillings).
   • A finite critical interaction strength $U_c \neq 0$ is required to induce superconductivity.
   • For $U < U_c$, the system is a semimetal; for $U > U_c$, it becomes a superconductor.
2. BCS-like Regime ($\Phi < \Phi_c$):
   • Magnetic bands overlap, resulting in a finite DOS at the Fermi level.
   • Superconductivity arises for arbitrarily weak attraction ($U_c = 0$).
   • The gap $\Delta$ scales exponentially with the inverse interaction strength, consistent with standard BCS theory: $\Delta \sim \exp(-b/\rho(E_F)U)$.

B. Critical Interaction Strength ($U_c$)

• The authors numerically determine $U_c$ for various coprime pairs.
• They find that $U_c$ vanishes continuously as the flux approaches the critical value from above ($\Phi \to \Phi_c^+$).
• The dependence follows a power law:
  $$U_c \propto (\Phi - \Phi_c)^\gamma$$
  where the critical exponent is estimated as $\gamma \approx 0.15$ (compatible with $3/20$).
• The critical flux converges to an asymptotic value $\Phi_c/2\pi \approx 0.1296$, consistent with previous non-interacting studies.

C. Scaling Behavior and Critical Exponents

Close to the transition ($U \approx U_c$), the authors identify universal scaling behaviors:

• Pairing Gap ($\Delta$):
  $$\Delta \propto (U - U_c)^\beta, \quad \text{with } \beta = 1/2$$
  This is the standard mean-field exponent. However, due to the quadratic vanishing of the DOS at the Weyl point, there is a marginal logarithmic correction:
  $$\Delta \sim \sqrt{\frac{U - U_c}{|\log(U - U_c)|}}$$
• Shifted Chemical Potential ($\tilde{\mu}$):
  The shifted chemical potential (relative to the Weyl energy $E_w$) scales linearly with the interaction distance:
  $$\tilde{\mu} - E_w \propto (U - U_c)^\alpha, \quad \text{with } \alpha = 1$$

D. Numerical Techniques

• The paper details the use of implicit relations for the energy spectrum for specific small $n$ (e.g., $n=2,3,4,6$) to bypass full matrix diagonalization, improving computational efficiency near the critical point.
• The implementation of backtracking and Armijo conditions in the Newton solver is highlighted as crucial for stabilizing solutions in the normal phase where $\Delta \to 0$.

4. Significance

• Topological Control of Superconductivity: The work demonstrates that magnetic band topology (specifically the presence or absence of Weyl nodes) acts as a switch for the nature of the superconducting transition. It distinguishes between a "strong-coupling" transition (finite $U_c$) driven by topological band gaps and a "weak-coupling" BCS transition driven by finite DOS.
• Universality: Despite the complex 3D band structure, the critical exponents ($\beta=1/2, \alpha=1$) remain universal mean-field values, though modified by logarithmic corrections due to the specific DOS profile of Weyl semimetals.
• Experimental Relevance: The findings are relevant for:
  • Ultracold Atomic Gases: Where synthetic gauge fields can be tuned to realize these 3D Hofstadter models.
  • High-Field Superconductors: Such as $UTe_2$, where magnetic fields are known to enhance superconductivity, potentially linked to topological band features.
• Theoretical Framework: It provides a comprehensive analytical and numerical framework for studying interacting topological semimetals, bridging the gap between non-interacting topological band theory and strongly correlated superconducting states.

In summary, the paper establishes that the 3D Hofstadter-Hubbard model hosts a unique quantum phase transition controlled by magnetic flux, where the emergence of Weyl points dictates whether superconductivity requires a finite interaction threshold or arises spontaneously, characterized by specific universal scaling laws.