Interaction-driven quantum criticality in two-dimensional quadratic band crossing semimetals with time-reversal symmetry breaking

A systematic investigation into all sixteen marginally relevant fermion-fermion interactions in two-dimensional time-reversal symmetry-breaking kagom´e semimetals hosting a quadratic band crossing point reveals how these interactions drive quantum critical behavior and determine the dominant low-energy phases.

The gist

The RG analysis of fermion-fermion interactions reveals coupling divergence at a critical low-energy scale, signaling quantum criticality governed by fixed points whose character depends on structural parameters, leading to ten distinct fixed points across three structural cases.

Microscopic Model and Effective Theory

The study begins with the microscopic noninteracting model for a two-dimensional quadratic band crossing point (QBCP) semimetal on a kagom´e lattice, characterized by the Hamiltonian density involving structure parameters d0, d1, d2, and d3. The energy dispersion relations are given by Eq. (3), where rotational symmetry is restored when d2 = 2d1. A stable QBCP requires the stability criterion of Eq. (4): d0 < r / [d21 cos2 2θ + (1/4)d22 sin2 2θ + d23]. The effective action, Seff (Eq. 6), incorporates these noninteracting dynamics alongside all sixteen marginally relevant short-range four-fermion interactions, Sint (Eq. 5).

Renormalization Group Analysis and Fixed Points

The momentum-shell renormalization group (RG) method is employed to derive coupled flow equations for the interaction parameters, dλµν dl = Fµν, with µ, ν = 0, 1, 2, 3 (Eq. 12). The analysis identifies fixed points (FP) at a critical energy scale lc where couplings diverge. These fixed points are classified into three cases based on structural parameters:

1. Case I: Rotational symmetry is restored (d2 = 2d1). This case yields two stable FPs: FP1 and FP2.

2. Case II and Case III: Rotational symmetry is explicitly broken, allowing for a richer structure with nine additional FPs (FP3 through FP10).

Classification of Fixed Points and Boundary Conditions

The ten fixed points are cataloged in Table I, which summarizes their critical values for the rescaled couplings at the critical scale lc. The analysis establishes boundary conditions (BCs) in the structural parameter space (u = log d1, v = log d2, w = log d3) required to flow into these FPs. For Case I, BC-FP1-Case I is bounded by four lines (Eqs. 19). For Case II and III, the boundary conditions are more complex, involving multiple planes (e.g., BC-FP3-case II is bounded by eight planes in Eq. 25).

Instabilities and Dominant Phases

To determine the physical consequence of each FP, external symmetry-breaking perturbations (source terms) are introduced into the effective action (Eq. 30), and their susceptibilities are computed. The analysis clusters the ten FPs into four groups:

(Group-A: FP1)

The leading instability is s-wave SC and chiral SC2, with CDW as a subleading instability.

(Group-B: FP5, FP6, FP9, and FP10)

FP5 and FP6 share the CDW as their common leading instability. FP9 and FP10 both display a degenerate leading instability between chiral SC1 and SC2.

(Group-C: FP2, FP4, and FP8)

The leading instability is consistently the CDW for all three FPs.

(Group-D: FP3 and FP7)

FP3 is dominated by the x-current state, while FP7 is dominated by bond density.

This analysis establishes that the basic results are presented in Table III, which summarizes the leading and subleading instabilities near each fixed point, elucidating rich low-energy phenomena induced by marginally relevant fermion-fermion interactions.

Conclusion

The study successfully derives a theoretical framework for understanding the low-energy critical behavior in 2D kagom´e QBCP systems. The results show that the system's low-energy physics is dictated by the ten distinct fixed points, with CDW and chiral SC2 being primary competing orders, while other states like x-current and bond density emerge as significant subleading instabilities depending on the specific fixed point. This provides useful insights into interaction-driven quantum criticality and phase transitions in 2D kagom´e QBCP materials.

How it works

The core methodology relies on the momentum-shell RG approach to derive energy-dependent flow equations for all sixteen fermion-fermion interactions, treating them "on equal footing.

Based on this research paper, here are several specific improvements that could be implemented in AI systems, along with what those improved systems could achieve:

)AI System Improvement 1: Quantum Criticality Prediction Engine (Q-CPE)

The core of the paper is mapping interaction parameters to fixed points (FPs) and predicting resulting phase transitions. An AI system trained on this data could revolutionize materials discovery.

• [Specific Improvement]: Develop a machine learning model (e.g., a neural network or a reinforcement learning agent) that takes the structural parameters of a candidate material (d0, d1, d2, d3) as input and predicts the most likely Quantum Critical Fixed Point (FP1 through FP10). The model should be trained on the derived boundary conditions (BCs) from Table I.

• [Specific Improvement]: Implement a secondary predictive layer that uses the predicted FP and the structural parameters to predict the dominant low-energy phase transition (e.g., CDW, superconductivity, x-current). This would involve classifying inputs based on which group (A, B, C, or D) the FP belongs to.

• [What it can do]: Automatically screen vast chemical spaces for novel 2D kagom´e semimetals. Instead of slow experimental synthesis and measurement to find QCPs, the system could rapidly filter thousands of hypothetical materials based on their structural parameters and predict exactly which quantum phase (e.g., s-wave SC vs. CDW) will emerge under interaction effects, significantly accelerating the discovery of exotic quantum materials.

)AI System Improvement 2: Interaction Flow Simulator (IFS) for RG Dynamics

The paper meticulously derives coupled flow equations (Eq. 12) and source term evolutions (C1-C16). An AI system could simulate the evolution of these flows dynamically.

• [Specific Improvement]: Create a simulation environment that numerically integrates the coupled RG flow equations (Eq. 12) over energy scales to visualize the trajectory of all sixteen fermion-fermion couplings. The system should be able to track when and where couplings diverge, signaling quantum criticality at scale lc.

• [Specific Improvement]: Integrate the source terms (C1-C16) into this flow simulation to predict how external symmetry-breaking perturbations will drive the system toward specific FPs or instabilities. For example, simulating a perturbation that favors a chiral SC state near FP9/FP10.

• [What it can do]: Provide high-fidelity theoretical verification for experimentalists. If an experiment observes a material exhibiting unusual transport properties at low temperatures, the IFS could simulate the expected interaction flow trajectory and suggest which fixed point (and thus which quantum phase) is most consistent with those observations, guiding targeted experimental probes.

)AI System Improvement 3: Instability Classifier & Phase Transition Predictor (IC-PTP)

The paper concludes by analyzing candidate instabilities using susceptibilities (Table II). An AI system can automate the interpretation of these complex results.

• [Specific Improvement]: Train a classifier on the input features derived from the susceptibility calculations (e.g., the relative magnitudes and signs of divergences for charge, spin, and particle-particle channels) to determine if a specific candidate phase (from Table II) is leading or subleading near any given FP.

• [Specific Improvement]: Develop an inference engine that maps the combination of structural parameters (d0-d3) and FP identity onto a definitive list of leading and subleading instabilities, effectively automating the results summarized in Table III.

• [What it can do]: Perform rapid hypothesis testing on theoretical predictions. Researchers could input a specific material structure and instantly receive a ranked list of predicted quantum ground states (e.g., 90% chance of CDW dominance, 5% chance of chiral SC2, based on the derived susceptibility landscape).

)AI System Improvement 4: Boundary Condition Solver (BCS) for Parameter Space Exploration

The paper provides explicit linear and plane-fitting boundary conditions (Eqs. 19, 20, B1-B9) that define the parameter space boundaries for each FP.

• [Specific Improvement]: Implement a geometric solver that takes the structural parameters (d1, d2, d3) as coordinates and uses the derived BC equations to perform rapid region testing. This solver would instantly determine if a given set of material parameters falls within the stable region of FP1 or FP2.

• [What it can do]: Provide precise guidance for experimental tuning. If an experiment needs to tune a material parameter (e.g., doping or pressure) to achieve a specific quantum critical point, the BCS solver could map out the exact required range in structural space (d1, d2, d3) needed to hit that target FP. This removes the guesswork from parameter optimization.