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Critical fates induced by the interaction competition in three-dimensional tilted Dirac semimetals

Jing Wang, Jie-Qiong Li, Wen-Hao Bian, Qiao-Chu Zhang, Xiao-Yue Ren

TL;DR

This work develops a comprehensive one-loop renormalization-group analysis of a three-dimensional type-I tilted Dirac semimetal, incorporating Coulomb, electron-phonon, and phonon-phonon interactions on equal footing. The authors derive a coupled set of RG equations for fermion and phonon parameters, revealing three distinct fixed-point types and corresponding interaction-driven instabilities that favor chiral or triplet superconducting order. Near these fixed points, the density of states, compressibility, and specific heat exhibit pronounced, non-Fermi-liquid-like corrections and tilt-induced asymmetries, signaling rich critical behavior beyond the noninteracting case. The results emphasize how competing interactions and anisotropies jointly sculpt low-energy properties and potential superconducting phases in 3D tilted Dirac materials, offering guidance for experimental exploration of tilted semimetals.

Abstract

The interplay among Coulomb interaction, electron-phonon coupling, and phonon-phonon coupling has a significant impact on the low-energy behavior of three-dimensional type-I tilted Dirac semimetals. To investigate this phenomenon, we construct an effective theory, calculate one-loop corrections arising from all these interactions, and establish the coupled energy-dependent flows of all associated interaction parameters by adopting the renormalization-group approach. Deciphering such coupled evolutions allows us to determine a series of low-energy critical properties for these materials. At first, we present the low-energy tendencies of all interaction parameters. The tilting parameter exhibits distinct tendencies that depend heavily upon the initial anisotropy of fermion velocities. In comparison, the latter is mainly dominated by its initial value but is less sensitive to the former. Variations in these two quantities drive certain interaction parameters toward the strong anisotropy in the low-energy regime, indicating the screened interaction in specific directions, and others toward an approximate isotropy. Additionally, we observe that the tendencies of interaction parameters can be qualitatively clustered into three distinct types of fixed points, accompanied by the potential instabilities that induce an interaction-driven phase transition to a certain superconducting state. Furthermore, approaching these fixed points leads to the critical behavior of physical quantities, such as the density of states, compressibility, and specific heat, which exhibit quite different from their noninteracting counterparts and even deviate slightly from Fermi-liquid behavior. Our investigation sheds light on the intricate relationship between different types of interactions in these semimetals and provides useful insights into their fundamental properties.

Critical fates induced by the interaction competition in three-dimensional tilted Dirac semimetals

TL;DR

This work develops a comprehensive one-loop renormalization-group analysis of a three-dimensional type-I tilted Dirac semimetal, incorporating Coulomb, electron-phonon, and phonon-phonon interactions on equal footing. The authors derive a coupled set of RG equations for fermion and phonon parameters, revealing three distinct fixed-point types and corresponding interaction-driven instabilities that favor chiral or triplet superconducting order. Near these fixed points, the density of states, compressibility, and specific heat exhibit pronounced, non-Fermi-liquid-like corrections and tilt-induced asymmetries, signaling rich critical behavior beyond the noninteracting case. The results emphasize how competing interactions and anisotropies jointly sculpt low-energy properties and potential superconducting phases in 3D tilted Dirac materials, offering guidance for experimental exploration of tilted semimetals.

Abstract

The interplay among Coulomb interaction, electron-phonon coupling, and phonon-phonon coupling has a significant impact on the low-energy behavior of three-dimensional type-I tilted Dirac semimetals. To investigate this phenomenon, we construct an effective theory, calculate one-loop corrections arising from all these interactions, and establish the coupled energy-dependent flows of all associated interaction parameters by adopting the renormalization-group approach. Deciphering such coupled evolutions allows us to determine a series of low-energy critical properties for these materials. At first, we present the low-energy tendencies of all interaction parameters. The tilting parameter exhibits distinct tendencies that depend heavily upon the initial anisotropy of fermion velocities. In comparison, the latter is mainly dominated by its initial value but is less sensitive to the former. Variations in these two quantities drive certain interaction parameters toward the strong anisotropy in the low-energy regime, indicating the screened interaction in specific directions, and others toward an approximate isotropy. Additionally, we observe that the tendencies of interaction parameters can be qualitatively clustered into three distinct types of fixed points, accompanied by the potential instabilities that induce an interaction-driven phase transition to a certain superconducting state. Furthermore, approaching these fixed points leads to the critical behavior of physical quantities, such as the density of states, compressibility, and specific heat, which exhibit quite different from their noninteracting counterparts and even deviate slightly from Fermi-liquid behavior. Our investigation sheds light on the intricate relationship between different types of interactions in these semimetals and provides useful insights into their fundamental properties.
Paper Structure (16 sections, 39 equations, 25 figures, 2 tables)

This paper contains 16 sections, 39 equations, 25 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Microscopic model and effective theory
  3. RG analysis and coupled evolutions
  4. Tendencies and fates of interaction parameters
  5. Fate of $\zeta/\zeta_{0}$ and two unstable scenarios
  6. Fates of fermion velocities
  7. Fates of $\epsilon_z/\epsilon$ and $g/g_0$
  8. Fates of phonon velocities and $\lambda_z/\lambda$
  9. Fates of phonon-phonon interactions
  10. Potential instabilities
  11. Critical implications around the instabilities
  12. Density of states and compressibility
  13. Specific heat
  14. Summary
  15. Collections of one-loop corrections
  16. ...and 1 more sections

Figures (25)

  • Figure 1: (Color online) Schematic dispersion for (a) the untilted and (b) the tilted 3D DSM.
  • Figure 2: Tree-level vertexes: (a) Coulomb interaction, (b) electron-phonon interaction, and (c) phonon-phonon interaction, respectively (the solid, wavy, and dashed lines denote the free fermionic, auxiliary bosonic, and phonon propagators).
  • Figure 3: One-loop corrections caused by the Coulomb and electron-phonon as well as phonon-phonon interactions to (a)-(b) fermionic propagator, (c) auxiliary bosonic propagator, and (d)-(f) phonon propagator (the solid, wavy, and dashed lines denote the free fermionic, auxiliary bosonic, and phonon propagators, respectively).
  • Figure 4: (Color online) A schematic diagram illustrating the tendencies of 3D tDSM and potential instability as the energy scale is tuned. The label $l_c$ (or $T_c$) specifies the critical energy scale (or critical temperature) and "PT" designates the accompanying phase transition induced by the potential instability from the 3D tDSM to an $X$ phase. A further investigation into this transition will be conducted in Sec. \ref{['Sec_FP-instab']}.
  • Figure 5: (Color online) Energy-dependent evolutions of $\zeta/\zeta_{0}$ as $\zeta_{0}$ varies, starting from the (a) Anisotropy-I ($v_0/v_{z0}=0.5$) and (b) Anisotropy-II ($v_{z0}/v_0=0.5$) situations.
  • ...and 20 more figures