[Paper Review] Effective Interacting Hamiltonian and Pairing Symmetry of LaOFeAs

Junren Shi|ArXiv.org|Jun 2, 2008

Iron-based superconductors research3 citations

TL;DR

This paper derives the general effective electron-electron interaction Hamiltonian for LaOFeAs based on its $D_{2d}$ point group symmetry, revealing a unique form of interaction due to the multi-orbital nature and symmetry-protected degeneracy at the M-point. It systematically identifies ten possible pairing states, with symmetry and physical constraints narrowing the candidates to $s$-wave, $d_{x^2-y^2}$, and $p$-wave states, particularly favoring mixed-symmetry pairing due to strong on-site Coulomb repulsion and band splitting.

ABSTRACT

We establish the general form of effective interacting Hamiltonian for LaOFeAs system based on the symmetry consideration. The peculiar symmetry property of the electron states yields unusual form of electron-electron interaction. Based on the general effective Hamiltonian, we determine all the ten possible pairing states. More physical considerations would further reduce the list of the candidates for the pairing state.

Motivation & Objective

• To establish the general form of the effective interacting Hamiltonian for LaOFeAs based on its crystal and electronic symmetry.
• To determine all possible pairing states allowed by the system's symmetry, particularly near the M-point where $d_{xz}$, $d_{yz}$, and $d_{xy}$ orbitals hybridize.
• To analyze the stability of these pairing states against band energy splitting and on-site Coulomb repulsion.
• To provide a general framework applicable to other iron-based superconductors with similar electronic structure.

Proposed method

• Construct the effective electron-electron interaction Hamiltonian using the symmetry group $D_{2d}$ and its irreducible representations, particularly focusing on the two-dimensional $E$ representation at the M-point.
• Apply group theory to classify all possible pairing states based on orbital, spin, and spatial symmetries, using irreducible representations of $D_{2d}$.
• Use the generalized Bloch theorem and reduced $1\times1$ primitive cell to define continuous Bloch wavefunctions across the Brillouin zone, enabling consistent symmetry transformation rules.
• Analyze the transformation properties of the two degenerate bands at the M-point under $D_{2d}$ operations, leading to the form of the Hamiltonian in Eq. (1).
• Derive the ten possible pairing states by classifying the pairing matrix $\Delta_{ij}$ according to orbital parity ($P_{\text{orbit}}$), spin parity ($P_{\text{spin}}$), and irreducible representation (I.R.).
• Assess the stability of each pairing state under band splitting and on-site Coulomb repulsion, using the relative strength hierarchy $d_{x^2-y^2} > s \approx p \approx g > d_{xy}$.

Experimental results

Research questions

• RQ1 What is the general form of the effective electron-electron interaction Hamiltonian in LaOFeAs, consistent with its $D_{2d}$ symmetry and multi-orbital electronic structure?
• RQ2 How many distinct pairing states are allowed by the symmetry of the system, particularly near the M-point where the Fermi surface consists of two elliptical pockets?
• RQ3 Which pairing states are most robust against band energy splitting and on-site Coulomb repulsion, and how do these effects influence the pairing symmetry?
• RQ4 Can the symmetry-protected degeneracy at the M-point lead to unconventional pairing channels, such as $p$-wave or mixed-symmetry states, despite the dominance of $d$-orbital character?

Key findings

• The effective electron-electron interaction Hamiltonian is uniquely determined by the $D_{2d}$ symmetry and the generalized translational symmetry involving $T_xP_z$ and $T_yP_z$, which enables the $1\times1$ primitive cell construction.
• Ten distinct pairing states are identified, classified by orbital and spin parity and irreducible representation, including $s$-wave, $d_{x^2-y^2}$, $p$-wave, and mixed-symmetry states.
• The pairing states (1)–(4) and (6) are less sensitive to band splitting due to dominant intraband pairing components, while pure interband states (5) and (7)–(10) are suppressed.
• On-site Coulomb repulsion strongly suppresses $s$-wave components in states (1)–(3), favoring $d$-wave symmetry, and disfavors spatially localized pairing due to small Fermi pockets.
• The $p$-wave pairing state (6) is viable and supported by symmetry, contrary to some two-orbital models, and is stabilized by Hund's coupling and three-orbital hybridization.
• The analysis provides a general framework that constrains possible superconducting pairing symmetries and serves as a consistency check for microscopic models of iron-based superconductors.