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.

​

​

Key Results

• Key symmetries of superconductivity and H_BdG: I,T, C
• In single band limit, find symmetry required nodes.
• Single Band (with T and I symmetry): Excitation spectra are either fully gapped, have point nodes, or have line nodes.
• Experiments show differences from single-band limit: robust nodes without symmetry, gaps where nodes should exist, violation of Blount’s theorem.

​

​

Homotopic Classification of Nodes

In 3D

Space of Hamiltonians M

Homotopic Classification of Nodes

1- Identify relevant symmetries and symmetry classes

2- In each class find the dimensionality of nodes (co-dimension arguments)

3- Identify topological invariants associated with nodes:

Superconducting Nodal Symmetries

These symmetries are: TI and CI and S= (CI)(TI)=CT.

S 

AU

AU

U

Same symmetry conditions as Altland-Zirnbauer classes: ten-fold way

Many topological nodal classifications based on symmetries: Beri, Bzdusek, Fischer, Ryu, Sato, Yanase, Samokhin, Sato, Schynder, Shiozaki, Sigrist, Sumita, Ryu, Volovik, and Yanase

​

1- Include key superconducting symmetries T and I

2- Also include C since all superconductors have this.

3- For nodes wants symmetries that take k to k

​

Here: T. Bzdušek and M. Sigrist, Phys. Rev. B 96, 155105 (2017)

​

Mark H Fischer

UZH

2018-12-12

Nodal Classes

TI

CI

S

Node Dimension: Consider class DIII

Minimal model has pseudospin (σ) and particle-hole (τ) symmetry:

Imply: only c_0z(k)=ε(k) and c_yy(k)=ψ(k) are non-zero

This has codimension δ=2, allowing line nodes in 3D

TI=iσ_yK

CI=τ_xK

S=iτ_xσ_y

Breaks Inversion symmetry

Homework Question 2

1. Use codimension arguments to show that class CII has no nodes (spin-triplet with time-reversal symmetry).
2. Use codimension arguments to find what nodes are allowed in class AIII (broken inversion symmetry but with time-reversal). Here care must be taken to choose the minimal model so that it does not include too many degrees of freedom (ask what happens to the spectrum of the single band H_N when inversion symmetry is broken).

​

​

​

​

​

Nodal “AZ+I” SC Classes

Point nodes are classified by Chern number (surface Weyl arcs), line nodes by winding number (flat band –Majorana surface states) – found using S symmetry.

Surprise: Bogoliubov Fermi surfaces

Point Nodes

Line Nodes

Surface Nodes

Nodes in Class D

(broken time-reversal spin-singlet):

​

Topologically Protected

Bogoliubov Fermi Surfaces

Key Result:

In clean even parity, multiband superconductors with spontaneous time-reversal symmetry breaking, the excitation spectrum is either

1. Fully gapped
2. Has topologically protected Bogoliubov Fermi surfaces

Materials Motivation

Key is multiband materials: many examples:

Kerr Effect : Schemm et al PRB RC (2015)

URu₂Si₂ j=5/2 bands: Ikeda et al Nature Physics (2012).

URu₂Si₂ likely pairing state: ψ(k)=kz(kx+iky)

Also spontaneous broken time-reversal superconductivity: examples are UPt₃, UBe₁₃, PrOs₄Sb₁₂, SrPtAs, URu₂Si₂, YPtBi, Cu-doped BiSe₂.

Will later consider this state and j=3/2 fermions

j=3/2 fermions

Two-band model: consider j=3/2 fermions with spherical symmetry.

Two bands, leading to one or two spherical Fermi surfaces:

Kohn-Luttinger Hamiltonian:

4x4, j=3/2 matrices

On-site j=3/2 pairing

on-site interactions: can make more than “s-wave” local Cooper pairs

Pauli exclusion: only J=0 and J=2 on-site Cooper pairs, 6 possibilities

​

Four species to make Cooper pairs from: 3/2,1/2,-1/2,-3/2

J=0

Here we use

Same symmetry as ψ(k)=kz(kx+iky)

Bogoliubov Fermi Surfaces

Why was missing in single-band theory?

TI

CI

S

Consider class D

Consider a minimal model that has pseudospin (σ) and particle-hole (τ) symmetry:

CI=τ_xK

Physical Origin of Bogoliubov Fermi surfaces

The superconductor has created an internal pseudospin magnetic field!

Perturbation Theory

If h(k)=0:

Pseudospin magnetic field

Topological Protection

Kobayshi et al PRB (2014), Zhao et al PRL (2016),

Bzdusek and Sigrist PRB (2017). DFA, Brydon, Timm, PRL (2017)

​

If (CI)²=1,then: Fermi surfaces can be topologically protected with a Z₂ invariant.

​

We find Z2 invariant is defined through the Pfaffian.

If δ=1, then topologically non-trivial

Fermi surface is stable to any perturbation that preserves CP symmetry.

​

No surface/edge states since edge/surface breaks I

Oh and Moon, PRB 102, 020501(R) (2020): parity breaking BCS-like instability

Nodes in Class CII

(no nodes in odd-parity):

​

Application in UTe₂

Bount’s Theorem

Class CII has no nodes, to get nodes we need to add a symmetry to classification.

​

Mirror symmetry Mz does the job, we can assign symmetry to spin-matrices:

​

For spin ½, Mz=iσ_z , since σ_zσ_xσ_z=-σ_x we have σ_x (and σ_y) are Mz odd, while σ_z is Mz even.

​

​

Hence for an odd-parity gap function that is even under Mz, near a mirror plane at k_z=0.

​

​

Now co-dimension arguments allow point nodes on the mirror plane.

Results on UTe₂

Gu et al Science 2025

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.

kz= π/c

Tight Binding Hamiltonians

​

Tight-binding Hamiltonians and band representations

The τ−matrices above describe Fe-site degree of freedom – suggests a link between real space tight-binding theories and kp theories

​

Formally, this corresponds to building band representations for local degrees of freedom.

​

Exploited in topological quantum chemistry module in the Bilbao crystallographic server. Linked to the notion of irreducible band representations (Joshua Zak). Here will use J. Cano, PRB 97, 035139 (2018) and J. Cano San Sebastian Lecture notes as references.

​

​

For FeSe, local in k-space theory (kp theory). Formally, this theory is derived from band-representations (for example in Bilbao Crystallographic server).

​

Here k=M=(π,π)