Topological anisotropic non-Fermi liquid from a Berry-dipole semimetal

This paper investigates the effects of long-range Coulomb interactions on a three-dimensional Berry-dipole semimetal using large-$N_f$ and $\epsilon$-expansion renormalization-group analyses, revealing the emergence of a spatially anisotropic non-Fermi liquid with an enhanced Berry-dipole moment and providing scaling relations and observational criteria to distinguish this state from the original semimetal.

The Gist

The Big Picture: A Dance of Electrons and Topology

Imagine a ballroom filled with dancers (electrons). In most materials, these dancers move in a very predictable, polite way. They form neat lines, follow a strict rhythm, and if you push one, the others react in a standard, orderly fashion. Physicists call this a Fermi liquid. It's the "normal" state of matter we see in most metals.

However, this paper explores a special, exotic ballroom where the rules are different. Here, the dancers are on a Berry-dipole semimetal. Think of this as a stage with a hidden, swirling vortex in the center (the "Berry dipole"). The dancers aren't just moving; they are swirling around this invisible whirlpool, creating a unique "topological" pattern.

The researchers asked a big question: What happens if we turn up the music and make the dancers interact more strongly? Specifically, they looked at what happens when the electrons start pushing and pulling on each other over long distances (like long-range Coulomb interactions).

The Discovery: The "Anisotropic" Chaos

The answer they found is surprising. When these swirling electrons interact strongly, they don't just get louder or faster; they completely change their dance style.

1. The Breakdown of Order (Non-Fermi Liquid):
   The polite, predictable lines of the Fermi liquid dissolve. The dancers stop moving in a coordinated rhythm. Instead, they enter a chaotic state called a Non-Fermi Liquid. In this state, you can't describe the dancers as individual people anymore; they act like a single, messy, collective fluid. It's like the difference between a marching band playing in perfect sync and a mosh pit where everyone is moving wildly but still connected.

2. The "One-Way Street" Effect (Anisotropy):
   Here is the most creative part of the discovery. Usually, when things get chaotic, they get messy in all directions equally. But in this specific ballroom, the chaos is anisotropic.

   • The Analogy: Imagine the dancers are on a trampoline. If you jump in the middle, the fabric stretches equally in all directions. But in this new state, the trampoline behaves like a slippery slide.
   • If the dancers try to move sideways (in the $x$ and $y$ directions), they get stuck or move very slowly.
   • If they try to move up and down (along the $z$ axis), they zoom freely.
   • The interaction forces the electrons to behave very differently depending on which way they are facing. The "slippery slide" direction becomes the dominant path, while the other directions become "sticky."

The "Berry Dipole" and the Magnetic Compass

The paper focuses on a specific type of material called a Hopf Insulator boundary.

• The Analogy: Imagine a magnetic compass. In a normal material, the needle points North. In this "Berry-dipole" material, the magnetic field lines don't just point North; they form a dipole (a North and South pole) right at the center of the stage, creating a vortex.
• The researchers found that when the electrons start interacting, this vortex gets supercharged. The "flux" (the strength of the magnetic swirl) becomes so intense that it effectively blows up to infinity.
• The Result: This massive surge in magnetic swirl is the smoking gun. It proves the material has shifted from a simple semimetal to this new, chaotic, one-way "Non-Fermi Liquid" state.

How Did They Figure This Out?

The authors used two powerful mathematical tools to predict this behavior, which are too complex to explain in full detail, but here is the gist:

1. The "Large Crowd" Trick (Large-$N_f$):
   Imagine trying to predict the behavior of a single person in a crowd is hard. But if you imagine a crowd of infinite people, the math becomes easier because the average behavior smooths out the chaos. They used this trick to see the big picture of how the electrons interact.

2. The "Zoom Lens" (Renormalization Group / $\epsilon$-expansion):
   Imagine looking at a forest. From far away, it looks like a green blob. As you zoom in, you see individual trees. As you zoom in closer, you see leaves. As you zoom in even closer, you see veins in the leaves.
   The researchers used a "zoom lens" to look at the electrons at different energy scales. They found that as they zoomed in (looking at lower and lower energies), the "stickiness" of the sideways movement and the "slipperiness" of the up/down movement became more and more extreme.

Why Should We Care? (The Real-World Test)

You might ask, "This sounds cool, but how do we know it's real?" The paper suggests a way to test this in a lab.

• The Test: They propose measuring something called the Non-Linear Hall Effect.
• The Analogy: Imagine you are pushing a shopping cart. In a normal store, if you push it forward, it goes forward. If you push it sideways, it goes sideways.
  In this new "Berry-dipole" state, if you push the electrons with a specific type of electric field, they don't just go straight; they generate a massive, unexpected sideways current that grows giant as the interaction gets stronger.
• The Prediction: If scientists build these materials (using acoustic lattices or special circuits) and see this "giant" sideways current, they will know they have successfully created this new "Topological Anisotropic Non-Fermi Liquid."

Summary

In short, this paper predicts that if you take a special 3D material with a swirling magnetic core and let the electrons interact strongly, the material will undergo a transformation. It will stop being a normal metal and become a chaotic, one-way fluid. The electrons will find it easy to move up and down but hard to move sideways, and the internal magnetic swirl will become infinitely strong. It's a new state of matter where the rules of physics get rewritten by the dance of the electrons.

Technical Summary

1. Problem Statement

The paper investigates the interplay between topology and electron-electron interactions in condensed matter physics, specifically focusing on a three-dimensional Berry-dipole semimetal. This system exists at a Topological Quantum Critical Point (TQCP) separating a Hopf insulator from a trivial insulator.

• The Gap: While non-interacting topological phases are well-classified, the fate of these phases under long-range Coulomb interactions remains an open question. In systems with vanishing density of states (like quadratic band-touching points), Coulomb interactions are often relevant perturbations that can destroy the Fermi-liquid description.
• The Specific Question: What happens to the Berry-dipole semimetal (characterized by a finite Berry-dipole moment and quadratic band touching) when subjected to long-range Coulomb interactions? Does it remain a semimetal, transition to an insulator, or evolve into a novel non-Fermi liquid phase?

2. Methodology

The author employs two complementary theoretical frameworks to analyze the renormalization of the system's parameters under Coulomb interactions:

1. Large-$N_f$ Expansion:

   • The fermionic flavor number $N_f$ is extended from 2 to $2N_f$ components.
   • A frequency-dependent bosonic field is used to avoid infrared divergences associated with the zero-frequency mode.
   • The one-loop bosonic self-energy is calculated to determine the renormalized bosonic propagator.
   • The fermionic self-energy is then derived using the renormalized boson propagator to study the flow of anisotropy parameters.

2. $\epsilon$-Expansion within the Renormalization Group (RG) Scheme:

   • The 3D system is generalized to a $(2+d)$-dimensional manifold by extending the rotational $z$-axis to $d$ dimensions, where $\epsilon = 2-d$ acts as the expansion parameter (with the physical case being $d=1, \epsilon=1$).
   • The Clifford algebra is extended from 2 to 4 dimensions to accommodate the higher-dimensional form factors.
   • Wilsonian RG equations are constructed to track the flow of coupling constants and anisotropy parameters ($\gamma_{21}, \gamma_{31}, \alpha$) towards the infrared limit.

Key Physical Inputs:

• Hamiltonian: A tight-binding model for Hopf insulators is expanded near the TQCP ($m=3$) to yield an effective low-energy Hamiltonian $H_3(k)$ with quadratic band touching and $C_4^z$ rotational symmetry.
• Interactions: Long-range Coulomb interactions are introduced via a Hubbard-Stratonovich field. Due to the vanishing density of states at the Fermi level, screening is weak, making the interaction a relevant perturbation.

3. Key Contributions and Results

A. Emergence of Spatial Anisotropy

Both the large-$N_f$ and $\epsilon$-expansion methods reveal that long-range Coulomb interactions induce a strong spatial anisotropy in the system.

• Anisotropy Flow: The ratio of fermionic anisotropy parameters $\beta_{21} = t_2/t_1$ flows to zero ($\beta_{21}^* \ll 1$), while $\beta_{31} = t_3/t_1$ flows to infinity ($\beta_{31}^* \gg 1$).
• Physical Consequence: The term $(k_x^2 + k_y^2)\sigma_3$ in the Hamiltonian becomes irrelevant, while the $k_z$-dependent terms dominate. The system effectively decouples in the $xy$-plane compared to the $z$-axis.

B. Non-Fermi Liquid Behavior

The system transitions from a Berry-dipole semimetal to a Topological Anisotropic Non-Fermi Liquid.

• Breakdown of Quasiparticles: The fermionic anomalous dimension $\eta_\psi$ becomes non-zero ($\approx 0.197$ at the fixed point). Consequently, the fermionic spectral function loses its sharp Dirac delta peaks and becomes a Lorentzian distribution, signaling the invalidation of the single-particle quasiparticle picture.
• Anomalous Dimensions:
  • Fermionic: $\eta_\psi \approx 0.197$
  • Bosonic: $\eta_\phi \approx 0.5$ (indicating strong renormalization of the bosonic spectrum).
  • Dynamic Critical Exponent: $\eta_\omega \approx 1.803$, indicating anisotropic scaling between energy and momentum ($E \sim k^{\eta_\omega}$).

C. Enhancement of Topological Flux

A striking result is the fate of the Berry curvature flux ($C_{z<0}, C_{z>0}$) through the upper and lower half-$z$ planes.

• Non-Interacting Case: The flux is quantized and finite.
• Interacting Case: Due to the scale dependence of the anisotropy parameters, the flux diverges ($C \to \pm \infty$) at the interacting fixed point.
• Implication: While the quantization is destroyed, the magnitude of the topological flux is significantly enhanced, leading to the term "Topological" in the phase name.

D. Scaling Relations of Physical Observables

The paper derives modified power-law scaling relations for various physical observables, deviating from standard Fermi-liquid predictions. The scaling depends on the probed energy/temperature ($\nu, T$) and the anisotropy parameters:

• Density of States (DOS): $\rho(\nu) \propto |\nu|^{1/2 + \eta_1}$
• Specific Heat: $C_v(T) \propto T^{3/2 + \eta_1}$
• Compressibility: $\kappa(T) \propto T^{1/2 + \eta_1}$
• Diamagnetic Susceptibility: Anisotropic scaling between $xy$-plane and $z$-axis ($\chi_{xy} \propto T^{1/2-\eta_2}$, $\chi_z \propto T^{1/2+\eta_3}$).
• Optical Conductivity: Distinct power laws for in-plane ($xx, yy$) and out-of-plane ($zz$) components.

4. Observational Criteria

The author proposes a concrete experimental criterion to distinguish this phase from the non-interacting Berry-dipole semimetal:

• Non-linear Hall Effect: The intrinsic and extrinsic non-linear Hall conductivities ($\sigma_{xxz}, \sigma_{zxx}$) are linearly related to the Berry curvature flux.
• Prediction: As the system approaches the topological anisotropic non-Fermi liquid phase, these conductivities should exhibit a giant enhancement (divergence) due to the unbounded growth of the Berry flux.

5. Significance

• Novel Phase of Matter: The work identifies a new class of quantum matter: a Topological Anisotropic Non-Fermi Liquid. It demonstrates that interactions can not only destroy Fermi-liquid behavior but also fundamentally alter the topological character of a system (enhancing flux while destroying quantization).
• Experimental Relevance: The paper connects theoretical predictions to realizable platforms, specifically acoustic lattices (where Berry-dipole semimetals have been realized) and nitrogen-vacancy centers or topoelectric circuits.
• Theoretical Framework: It provides a robust theoretical toolkit (combining large-$N_f$ and $\epsilon$-expansion) for studying interacting topological critical points, showing that spatial anisotropy is a generic feature of such systems under Coulomb interactions.

In summary, the paper establishes that long-range Coulomb interactions drive a 3D Berry-dipole semimetal into a strongly anisotropic, non-Fermi liquid state characterized by divergent topological fluxes and distinct power-law scaling in transport and thermodynamic properties.