Unconventional Superconductivity

Unconventional superconductivity concerns the internal structure and symmetries of paired electrons beyond the simplest isotropic gap. Crystal representation, nodes, relative phase, spin structure and microscopic pairing interaction are related questions, but they are not interchangeable. A sign-changing $A_{1g}$ state can preserve the crystal symmetries, while a nontrivial representation need not identify the interaction that selected it.

Here we classify uniform, zero-center-of-mass-momentum pairing and connect it to measurements. Pair-density waves require additional spatial structure; topological superconductors require a separate BdG invariant. Current material and mechanism questions are discussed in Research.

Required background. BCS Theory supplies the paired-state benchmark; Symmetry of Bloch States supplies crystal representations and changes of Bloch frame. Helpful background. Ginzburg–Landau Theory explains order-parameter energetics, and spin–orbit coupling explains why pseudospin need not be physical spin.

Gap matrices and fermion exchange

Let $c_{\mathbf k}$ be a column of annihilation operators in a specified orbital, sublattice and spin basis. Define the gap coefficient by

$$H_\Delta=\frac12\sum_{\mathbf k} \left[c_{\mathbf k}^\dagger\Delta(\mathbf k) (c_{-\mathbf k}^\dagger)^T+\mathrm{h.c.}\right].$$

This sign convention defines the matrix used below. It can differ by an overall sign from another definition of the pair potential without changing physics. For even-frequency pairing, exchanging the two fermions requires

$$\Delta(\mathbf k)=-\Delta^T(-\mathbf k).$$

A frequency-dependent anomalous self-energy obeys the fuller relation $\Delta_{ab}(\mathbf k,i\varepsilon_n)=-\Delta_{ba}(-\mathbf k,-i\varepsilon_n)$, with $\varepsilon_n=(2n+1)\pi k_BT$. For factorized spin, orbital, momentum and frequency tensors, their exchange signatures satisfy $SOPT=-1$. An orbital-antisymmetric factor therefore permits even-frequency, even-parity spin-triplet pairing; “triplet means odd parity” assumes an orbital-symmetric channel.

Under a change of frame $c_{\mathbf k}=W(\mathbf k)\widetilde c_{\mathbf k}$,

$$\widetilde\Delta(\mathbf k)=W^\dagger(\mathbf k) \Delta(\mathbf k)W^*(-\mathbf k).$$

An individual matrix entry or raw band phase depends on this frame. The full BdG spectrum and consistently transformed response operators do not. Relative gap phases become physically meaningful through a specified interband coupling or probe, with its matrix elements transformed in the same frame.

Singlet, triplet and pseudospin

Any two-by-two pairing matrix can be expanded algebraically as

$$\Delta=(\psi\sigma_0+\mathbf d\cdot\boldsymbol\sigma)i\sigma_y.$$

This decomposition does not require inversion symmetry. Its interpretation does require knowing what the two basis states represent. In a physical-spin basis, $\psi$ and $\mathbf d$ are singlet and triplet amplitudes. In a well-separated centrosymmetric Kramers doublet with a suitable pseudospin frame, they describe pseudospin singlet and triplet channels. For even frequency and an orbital-symmetric channel, antisymmetry makes $\psi(\mathbf k)$ even and $\mathbf d(\mathbf k)$ odd.

Inversion permits a state to have definite parity. Without inversion, singlet and triplet components can coexist, even though each still obeys its own exchange constraint in a consistently chosen basis. Strong spin–orbit coupling also means that a pseudospin label does not directly predict the physical-spin susceptibility: magnetic matrix elements, interband contributions and orbital backgrounds matter (Anderson, 1984; Smidman et al., 2017).

For pure triplet pairing,

$$\Delta\Delta^\dagger=|\mathbf d|^2\sigma_0 +i(\mathbf d\times\mathbf d^*)\cdot\boldsymbol\sigma.$$

A unitary state has $\mathbf d\times\mathbf d^*=0$. Otherwise the two gap singular values can differ. For example, at a point where $\mathbf d=d_0(1,i,0)/\sqrt2$, their squares are $0$ and $2|d_0|^2$: a nonzero matrix can leave one branch ungapped. The momentum dependence must still satisfy the exchange constraint.

Crystal representations and actual quasiparticle nodes

For a unitary crystal operation, use the sewing convention

$$g c_{\mathbf k}^\dagger g^{-1} =c_{g\mathbf k}^\dagger U_g(\mathbf k).$$

The transformed gap is

$$\Delta^g(g\mathbf k)=U_g(\mathbf k) \Delta(\mathbf k)U_g^T(-\mathbf k).$$

Classify the span of allowed matrices under this action. A channel $\Gamma$ can be written $\Delta=\sum_\mu\eta_\mu\Phi_{\Gamma\mu}$; the basis matrices describe the representation and the complex coefficients select a state within it. Nonsymmorphic operations, magnetic symmetries and antiunitary operations require their corresponding sewing or corepresentation rules.

Simple tetragonal, even-parity scalar examples are

$$\begin{aligned} A_{1g}:&\quad 1,\ \cos k_xa+\cos k_ya,\\ B_{1g}:&\quad \cos k_xa-\cos k_ya,\\ B_{2g}:&\quad \sin k_xa\sin k_ya. \end{aligned}$$

The scalar harmonics are not the complete classification when orbital matrices transform too. Nor does the irrep fix every coefficient or gap minimum.

In the isolated, unitary, intraband limit, $E_n=\sqrt{\xi_n^2+|\Delta_n|^2}$. A node requires both $\xi_n=0$ and $\Delta_n=0$. A basis function can vanish far from every Fermi surface without producing a low-energy excitation. When interband pairing or nonunitary structure matters, diagonalize

$$\mathcal H_{\rm BdG}(\mathbf k)= \begin{pmatrix} h(\mathbf k)&\Delta(\mathbf k)\\ \Delta^\dagger(\mathbf k)&-h^T(-\mathbf k) \end{pmatrix}.$$

Here $h$ includes the chemical potential. A zero of one orbital entry is not a node criterion. Band separation large compared with interband pairing allows a projection; near degeneracies, a scalar sheet approximation must be checked against the full matrix.

Enforced nodes, accidental nodes and deep minima

Consider a cylindrical Fermi surface and measure its angle $\phi$ from the crystal $x$ axis. A scalar $B_{1g}$ state $\Delta_B=\Delta_0\cos2\phi$ has diagonal nodes. A diagonal mirror leaves those momenta fixed but acts with character $-1$, forcing the gap to vanish in this model.

By contrast, the $A_{1g}$ family

$$\Delta_A=\Delta_0(r+\cos4\phi)$$

has accidental sign-changing zeros for $|r|<1$, touching zeros at $|r|=1$, and no zeros for $|r|>1$. Varying $r$ preserves the representation. In particular, $r=1.2$ gives a nonzero minimum $0.2\Delta_0$, which a coarse measurement might mistake for a node.

The upper gap changes sign at the diagonal direction. In the lower panel, all three curves belong to $A_{1g}$, while their zeros change as $r$ passes through 1. The vertical axis shows a signed scalar gap in one fixed frame, not a tunneling spectrum; a magnitude-only probe would see its absolute value.

Low-temperature power laws constrain the available low-energy phase space rather than uniquely naming a representation. For ordinary linear nodal dispersion, line nodes in three dimensions and point nodes on a two-dimensional Fermi line give $N(E)\propto|E|$ and electronic $C\propto T^2$; ordinary three-dimensional point nodes give $N(E)\propto E^2$ and $C\propto T^3$. These statements assume clean bulk quasiparticles and nonvanishing probe weights. Disorder, nonlinear nodes, extra bands and nonlocal response can change the apparent powers.

Weak coupling does not always mean the isotropic ratios

For a separable single-band weak-coupling interaction, write $\Delta(\phi)=\Delta_{\max}f(\phi)$ with $\max|f|=1$ and a normalized density-of-states-weighted Fermi-surface average. Eliminating the coupling and high cutoff between the zero-temperature and linearized gap equations gives

$$\frac{2\Delta_{\max}}{k_BT_c} =\frac{2\pi}{e^{\gamma_E}} \exp\left[-\frac{\langle f^2\ln|f|\rangle} {\langle f^2\rangle}\right].$$

The associated heat-capacity jump is

$$\frac{\Delta C}{\gamma_nT_c} =\frac{12}{7\zeta(3)} \frac{\langle f^2\rangle^2}{\langle f^4\rangle}.$$

These formulas assume a real form factor of fixed shape, constant normal density of states over the pairing shell and weak coupling. For a circular cylinder with $f=\cos2\phi$, they give $2\Delta_{\max}/(k_BT_c)\simeq4.28$ and $\Delta C/(\gamma_nT_c)\simeq0.951$. Thus a measured maximum-gap ratio above 3.528 is not by itself evidence for strong coupling. Gap anisotropy also changes the heat-capacity jump (Johnston, 2013).

Multicomponent order and time reversal

A useful two-component illustration is

$$\begin{aligned} f&=\alpha R^2+\beta_1R^4+\beta_2|Q|^2,\\ R^2&=|\eta_1|^2+|\eta_2|^2,\\ Q&=\eta_1^*\eta_2-\eta_2^*\eta_1. \end{aligned}$$

At fixed $R$, a common-phase real vector has $Q=0$, while $R(1,\pm i)/\sqrt2$ has $|Q|^2=R^4$. Positive $\beta_2$ favors the real manifold; negative $\beta_2$ favors the chiral pair. Quartic stability requires $\beta_1>0$ and, when $\beta_2<0$, also $\beta_1+\beta_2>0$. Other crystal-allowed invariants can select an orientation, and can occur at the same quartic order; this two-coefficient example is not the most general crystal functional.

In a real representation basis, time reversal complex-conjugates $\boldsymbol\eta$. The bilinear

$$\mathcal L=i(\eta_1\eta_2^*-\eta_2\eta_1^*)$$

is gauge invariant and odd under time reversal. It equals $\pm R^2$ for the two chiral partners. It is not a Chern number. A real state can break crystal rotation and be nematic, but its residual symmetry must be checked from the actual representation.

Strain can couple to $|\eta_1|^2-|\eta_2|^2$, train domains and split transitions. A split transition alone does not determine representation: gradients, competing orders and inhomogeneity can also produce multiple anomalies. Likewise, Kerr or local magnetic signals require checks for trapped flux, magnetic inclusions and domain weighting before supporting superconducting time-reversal breaking (Sigrist and Ueda, 1991; Kallin and Berlinsky, 2016).

What complementary probes can establish

Different measurements test different parts of a gap matrix:

Measurement	Primary constraint and limitation
Heat capacity, screening, transport	bulk transition and low-energy excitations; backgrounds and connectivity matter
ARPES and tunneling	momentum- or surface-weighted gap magnitudes; no direct general measurement of relative sign
Oriented Josephson junctions	relative phase with known geometry, facets and electromagnetic response
NMR and susceptibility	physical magnetic response; pseudospin, orbital and interband terms must be separated
Neutron response and QPI	conditional coherence-factor information; interaction, impurity and orbital vertices matter
Strain, ultrasound and spontaneous fields	additional symmetry breaking, subject to domains and backgrounds

For a scalar tetragonal $B_{1g}$ candidate, $\Delta(\phi+\pi/2)=-\Delta(\phi)$. Controlled corner junctions can compare the two orientations and detect the associated $\pi$ shift. The Josephson page explains how junction asymmetry, loop inductance and trapped flux enter the interpretation. Such phase tests are stronger than identifying an irrep from one power law (Tsuei and Kirtley, 2000).

In a two-sheet example, suppose gap magnitudes are $5.0$ and $7.2\,\mathrm{meV}$, with an observed neutron feature at $9.5\,\mathrm{meV}$. If the relevant momentum connects Fermi points with these locally constant gaps, the two-quasiparticle threshold is $12.2\,\mathrm{meV}$. More generally it is $\min_{\mathbf k}[E_h(\mathbf k)+E_e(\mathbf k+\mathbf Q)]$. A feature below that threshold can support a spin-exciton interpretation in a specified susceptibility model, but kinematics alone does not determine relative sign.

Even the claim “nodeless multigap $A_{1g}$” requires more than those magnitudes. They support resolved nodeless gaps only down to the measured resolution; assigning $A_{1g}$ additionally requires symmetry information or a restricted candidate model. Within two already-assumed $A_{1g}$ candidates, $s_{++}$ and $s_\pm$ have the same gap magnitudes and differ in an intersheet phase tested by a consistently transformed probe (Hirschfeld et al., 2011).

Disorder can lift accidental nodes, broaden deep minima or break pairs. Rapid suppression of $T_c$ is not a unique representation measurement. Surfaces can reconstruct, and different probes can weight different domains or sheets. Comparing the same thermodynamic state across measurements is therefore essential. Proximity Effect explains induced interface correlations, which need not constitute a separate bulk phase.

Exercises

1. Apply the complete exchange constraint

For even frequency, find the momentum parity of spin-singlet and spin-triplet states with either orbital-symmetric or orbital-antisymmetric tensors.

Solution

From $SOPT=-1$, an orbital-symmetric singlet is even and an orbital-symmetric triplet is odd. Orbital antisymmetry reverses both answers: the singlet is odd and the triplet even. Odd frequency reverses them again. This classification assumes the indicated factors have definite exchange parity; a general mixed tensor should be tested with the full matrix relation.

2. Check a change of Bloch frame

Show that the stated frame transformation preserves fermionic antisymmetry and the BdG eigenvalues.

Solution

Transposing the transformed matrix at $-\mathbf k$ gives

$$\begin{aligned} \widetilde\Delta^T(-\mathbf k) &=W^\dagger(\mathbf k)\Delta^T(-\mathbf k)W^*(-\mathbf k)\\ &=-\widetilde\Delta(\mathbf k). \end{aligned}$$

The normal block transforms as $\widetilde h=W^\dagger hW$. The full Nambu transformation is unitary with particle block $W(\mathbf k)$ and hole block $W^*(-\mathbf k)$, so it preserves all eigenvalues. The sewing matrices and probe vertices must be transformed too. This is the same unitary equivalence used for ordinary operators, now applied to paired Bloch states.

3. Evaluate the anisotropic weak-coupling benchmark

For $f(\phi)=\cos2\phi$ on a uniform circular cylinder, evaluate the averages entering the maximum-gap ratio and heat-capacity jump.

Solution

The averages are

$$\begin{aligned} \langle f^2\rangle&=\tfrac12,\qquad \langle f^4\rangle=\tfrac38,\\ \langle f^2\ln|f|\rangle&=\tfrac14-\tfrac12\ln2. \end{aligned}$$

The logarithmic average follows by differentiating the integral of $|\cos2\phi|^p$ with respect to $p$ at 2. Substitution gives

$$\begin{aligned} \frac{2\Delta_{\max}}{k_BT_c} &=\frac{4\pi}{e^{\gamma_E+1/2}}\simeq4.279,\\ \frac{\Delta C}{\gamma_nT_c} &=\frac{8}{7\zeta(3)}\simeq0.951. \end{aligned}$$

The different numbers arise within weak coupling. Comparing an experiment requires matching its Fermi-surface weights and gap definition to the model.

4. Minimize the two-component example

Compare $R(1,0)$, $R(1,1)/\sqrt2$ and $R(1,i)/\sqrt2$. Determine their quartic energies and whether time reversal can be restored by a global phase.

Solution

The first two have $f_4=\beta_1R^4$ and $\mathcal L=0$. The third has $f_4=(\beta_1+\beta_2)R^4$ and $\mathcal L=R^2$. Its conjugate has the same energy and opposite $\mathcal L$. No common phase maps $(1,-i)$ to $(1,i)$: the first component requires phase $1$, the second phase $-1$. The chiral partners break time reversal; the real states preserve it in this basis. The displayed functional leaves their real orientations degenerate.

5. Resolve an accidental node

For $\Delta_A/\Delta_0=r+\cos4\phi$, find the zeros in $0\leq\phi\leq\pi/2$ at $r=0.6$. What changes at $r=1$ and 1.2?

Solution

The zeros are $\phi=\tfrac14\arccos(-0.6)$ and $\pi/2-\tfrac14\arccos(-0.6)$, about 31.72° and 58.28°. They merge at 45° when $r=1$, where the touching zero is quadratic in angular displacement. At 1.2 the minimum is $0.2\Delta_0$ and no zero remains. All three states have the same $A_{1g}$ symmetry. Their low-energy phase space differs, and a measurement with resolution wider than the minimum may not distinguish the last state from a nodal one.

References

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