Magnetic catalysis and Hall conductivity of excitonic insulators in a planar four-Fermi model

Technical Summary: Magnetic Catalysis and Hall Conductivity of Excitonic Insulators in a Planar Four-Fermi Model

Problem Statement
The excitonic insulator (EI) is an electronic phase driven by the spontaneous condensation of bound electron-hole pairs. While theoretical models exist, experimental identification remains challenging because the opening of an electronic gap does not uniquely confirm an excitonic origin. Furthermore, the interplay between external magnetic fields and excitonic ordering is not fully understood. Specifically, it is unclear how a perpendicular magnetic field modifies the excitonic instability, how it alters the phase diagram (critical temperature and chemical potential), and whether the resulting phase transitions leave identifiable signatures in transport observables like Hall conductivity. Previous studies of the extended Gross-Neveu (GN) model in (2+1) dimensions addressed EI phases in the absence of magnetic fields, but the effects of Landau quantization on the coupled scalar and excitonic condensates required further investigation.

Methodology
The authors employ an extended (2+1)-dimensional Gross-Neveu model as a low-energy effective field theory to describe the interaction channel responsible for electron-hole coherence. The model includes two interaction channels: a conventional scalar channel (chiral symmetry breaking) and an additional pseudoscalar channel representing interband coherence (the EI order parameter).

• Formalism: The study utilizes the large-$N$ approximation (where $N$ is the number of fermion flavors) and a Hubbard-Stratonovich transformation to linearize the four-fermion interactions. This introduces auxiliary scalar ($\sigma$) and excitonic ($\eta$) fields.
• Magnetic Field: A constant, perpendicular magnetic field is incorporated via minimal coupling, leading to Landau level quantization of the fermion spectrum. The orbital effect is treated exactly within the large-$N$ limit, while Zeeman splitting is neglected.
• Thermodynamics: The authors derive the finite-temperature, finite-chemical-potential thermodynamic potential, including the renormalized vacuum contribution and thermal occupation of Landau levels.
• Analysis: Coupled gap equations for the scalar and excitonic condensates are derived and solved self-consistently. The global thermodynamic minimum is identified by comparing all stationary solutions. The Hall conductivity is calculated from the equilibrium fermion number density, assuming a clean, dissipationless system.

Key Contributions and Results

1. Excitonic Magnetic Catalysis:
   The primary finding is that a perpendicular magnetic field enhances the excitonic condensate ($\bar{\eta}_c$) and increases the critical temperature ($T_c$) for the EI phase. This constitutes an "excitonic realization" of magnetic catalysis, where Landau quantization increases the low-energy spectral weight, thereby stabilizing the interaction-driven condensate against thermal fluctuations.

   • At $\mu=0$, both the zero-temperature EI gap and $T_c$ increase monotonically with the magnetic field strength ($eB$).
   • The field can induce an EI phase transition at fixed temperatures where the system would otherwise be in a normal state.

2. Behavior of the Scalar Condensate:
   A distinct feature of the model is the behavior of the scalar condensate ($\bar{\sigma}_c$). Throughout the entire EI phase (where $\bar{\eta}_c \neq 0$), the scalar condensate remains pinned to a constant value ($\sigma_0$), independent of temperature, chemical potential, or magnetic field. Magnetic catalysis of the scalar channel only becomes visible in the non-EI phase (where $\bar{\eta}_c = 0$). This indicates a transfer of the field's effect between the competing channels: the field first enhances the scalar gap in the normal phase, and once the EI phase becomes thermodynamically favored, the field dependence is carried entirely by the excitonic condensate.

3. Phase Structure and Critical Parameters:

   • Critical Temperature: $T_c$ increases monotonically with $B$.
   • Critical Chemical Potential: At $T=0$, the critical chemical potential ($\mu_c$) for the EI transition depends nonmonotonically on the magnetic field. This arises from the competition between magnetic catalysis (which favors the EI phase) and the discrete occupation of Landau levels. Weak fields show oscillations (analogous to de Haas–van Alphen effects), followed by a broad minimum, and then an increase at strong fields where magnetic catalysis dominates.
   • Tricritical Point: The magnetic field expands the EI region in the $T-\mu$ phase diagram and shifts the tricritical point (where continuous transitions meet first-order transitions) to higher temperatures and chemical potentials, thereby enlarging the region of first-order transitions.

4. Hall Conductivity Signatures:
   The Hall conductivity ($\sigma_{xy}$) serves as a complementary probe for the EI order.

   • Zero Temperature: The Hall response is zero as long as the chemical potential lies below the lowest positive-energy Landau level. The threshold for a finite Hall response is determined by the total quasiparticle gap ($\sqrt{\bar{\sigma}_c^2 + \bar{\eta}_c^2}$).
   • Correlation with EI Transition: For the parameters considered, the onset of a finite Hall conductivity coincides with the first-order transition where the EI condensate vanishes. As the EI condensate collapses, the total gap decreases, allowing the lowest Landau level to become occupied, causing a jump in Hall conductivity.
   • Continuous Transitions: In scenarios where the EI phase is induced by the magnetic field at fixed $T$ and $\mu$, the emergence of the condensate is accompanied by a characteristic change in the slope of the Hall conductivity. Near the critical field, the Hall conductivity varies linearly with the field, while the condensate follows a mean-field square-root dependence.

5. Experimental Estimates:
   The authors provide order-of-magnitude estimates for experimental systems, specifically InAs/GaSb quantum wells and monolayer WTe$_2$.

   • For InAs/GaSb, the model predicts that an EI phase can be sustained at 10 K with a magnetic field of approximately 3.5 T, and the Hall-slope feature should appear near 9.6 T. These values are within experimentally accessible ranges.
   • For monolayer WTe$_2$, the required fields are significantly higher (order of 49 T to 100 T), necessitating ultrahigh pulsed-field techniques.
   • The paper notes that materials with larger gaps, such as Ta$_2$NiSe$_5$, would require even larger fields, and their complex band structures make a direct mapping to the planar Dirac model less controlled.

Significance
The paper demonstrates that the combined analysis of phase structure and Hall response can provide complementary signatures of excitonic ordering in planar fermionic systems. The study establishes that magnetic catalysis is a robust mechanism for stabilizing the excitonic insulator phase in the semiconductor regime. Furthermore, it identifies specific transport signatures—such as the field-dependent shift of the Hall conductivity threshold and the change in slope near continuous transitions—that could help distinguish excitonic ordering from conventional insulating phases in experimental settings. The work serves as a theoretical framework for interpreting magnetic field effects in candidate EI materials, particularly those with small band gaps and Dirac-like dispersions.