Intertwined electronic degrees of freedom: applications to superconductivity and magnetism

Daniel F. Agterberg, University of Wisconsin – Milwaukee

1- Motivation: additional electronic dofs beyond spin leads to qualitatively new physics.

2- Review of “classical” single-band superconductivity: mean-field BdG Hamiltonian, emphasis on symmetries and quasi-particle spectrum, Blount’s theorem.

3- Topological nodal classification based on superconducting symmetries

4- Examples of nodal classes

i) Bogoliubov Fermi surfaces

ii) Spin-triplet superconductivity

5- Symmetry-based construction of tight-binding Hamiltonian: space group, Wyckoff (sublattice), site symmetry irreducible representations, band representations. (Cano)

6- Applications in SG 129 – origin of FeSe Hamiltonian, sublattice degeneracy, gap functions, superconducting fitness, breakdown of Blount’s theorem, physics of CeRh₂As₂

7- Applications in SG 123 – sublattice driven topological altermagnet (Venderbos)

8- If time permits other consequences of sublattice: non-zero AHE w/o spin-splitting, odd-parity magnetism in antiferromagnets.

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Summary

• Tight binding is defined by space group, Wyckoff position (sub-lattice), IR of Wyckoff site symmetry.

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S-orbitals (A1g of D2h) on Wyckoff 2f of space group 123

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Homework Question 3

a) Find the tight-binding Hamiltonian for p_x orbitals on the 2f Wyckoff position in space group 123.

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b) Add altermagnetism to the above Hamiltonian, that is add a term

Jτ_z σ_z to the tight binding Hamiltonian above. Without SOC, find the quasi-particle energy dispersion.

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If chemical potential is in band gap, this defines a mirror Chern insulator:

Antonenko, Fernandes, Venderbos, Phys. Rev. Lett. 134, 096703 (2025)

For applications to altermagnetism for similar modes, see Roig, Phys. Rev. B 110, 144412 (2024)

Application of SG 123, with s-orbitals (so called Lieb lattice model for altermagnetism)

Tight-biding models: applications to

superconductivity

Space Group 129

Here the Fe occupy Wyckoff 2a (D_2d)

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Positions (1/4,3/4) and (3/4,1/4). Put x²-y² orbitals on these.

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Choose g₂={I|11} [note origin is at (0,0)]

Space group P4/nmm, generated by:

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{E|t₁},{E|t₂},{C₄|1/2 0},{I|0}, {m_x|1/2 0}, and {m_z|1/2 1/2}

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{E|t₁},{E|t₂} generate the group Z²

U_I=τ_x

Can show U_C4=τ_x, U_mx=τ₀, U_mz=τ₀

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Space Group 129

Can show τ₀ and τ_x are A_1g and τ_z and τ_y are B_2u in D_4h

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The resulting tight-binding Hamiltonian is:

For k=(π+k_x,π+k_y) get kp theory used for FeSe:

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Notice when λ=0, the τ_x term vanishes for any point on the BZ boundary, implying a 4-fold degeneracy. Is this general?

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Consider [T{m_x|1/2,0}]²={E|1,0}, this is -1 for k=(π,k_y) this implies a non-spin Kramer’s degeneracy all along the BZ. 4-fold degeneracy is general.

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See also, Cvetkovic and Vafek, PRB 88, 134518 (2013)

Gap functions and superconducting fitness:

Have found H_N, what about Δ?

Superconducting fitness (Ramires, PRB 98, 024501 (2018)

Fit:

Unfit:

Fit in the band basis is purely intra-band

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Unfit in the band basis is purely inter-band (this has no BCS log)

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In practice gap divides up into intra and inter-band parts

Violation of Blount’s theorem

Predicted line nodes in an odd-parity state on a mirror plane in UPt_3.(Mike Norman)

Previous argument does not explain this violation of Blount’s theorem. Note that to get line nodes, need a symmetry reason for all three of d_x, d_y, and d_z to vanish on a mirror plane. This happens on BZ boundary in non-symmorphic groups (like SG 129). Here pseusodpin for a single-band is different that usual spin.

kz= π/c

Nodes in Class DIII

(time-reversal invariant spin-singlet):

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Application to monolayer FeSe

Superconductivity in monolayer FeSe

Highest Tc in Fe superconductor family

FeSe seems “s-wave” – where have e-e interactions gone?

Zhang et al, Phys. Rev. Lett. 117, 117001

Ge et al Nature Materials14, 285 (2015).

Jian Wang’s group has data that strongly supports this state (2024 APS March Meeting)

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Gap changes signs on the two ellipses: this is a d-wave state.

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Where have the nodes gone?

Minimal kp theory

This agrees well with ARPES results with (in meV)

Note Δ=11 meV

Phys. Rev. Lett. 117, 117001

2 site and 2 spin degrees of freedom

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Superconductivity

First solve with no SOC.

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Assume d-wave pairing symmetry, leading naturally to opposite sign gaps on the two elliptical Fermi surfaces (τ_x is B1g near M)

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The superconductor is fully gapped here, where single-band limit says there should be nodes.

With SOC nodes appear

vsok0=16 meV

Nodes are protected by chiral symmetry (TC), which gives +2,-2 charges

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If the nodes move too close to each other, they can annihilate

Without spin-orbit: nodeless d-wave considered in: Nica and Si npj QM (2017), Zhu, Zhang, and Zhang PRB (2016), Chubukov, Fernandes, Vafek PRB (2016).

Reducing the SOC

vsok0=2.4 meV

Minima near nodes appear

This can explain experimentally observed anisotropy

Due to orbitally non-trivial nature of gap, the nodes annihiliate when SOC is reduced