Spin Susceptibility of a $J=3/2$ Superconductor

Spin-orbit interaction is a source of exotic electronic states realized in topological semimetals [1, 2], topological insulators [3, 4], and topological superconductors [5, 6, 7, 8, 9, 9]. In the presence of strong spin-orbit interactions, spin $S=1/2$ and orbital angular momentum $L=1$ of an electron are inseparable degrees of freedom. Electronic properties of such materials are characterized by an electron with pseudospin $J=L+S=3/2$. Recent studies have suggested a possibility of superconductivity due to Cooper pairing between two electrons with $J=3/2$ [10, 11]. The large angular momentum of an electron enriches the symmetry of the order parameter such as pseudospin-quintet even-parity and pseudospin-septet odd-parity [12, 13] in addition to conventional spin-singlet even-parity and spin-triplet odd-parity. Such high angular-momentum pairing states would feature superconducting phenomena of $J=3/2$ superconductors [14, 15, 16, 17, 18, 19]. In particular, a large angular momentum of a Cooper pair would qualitatively change the magnetic response of a superconductor to an external magnetic field.

The spin susceptibility reflects well the internal spin structures of a Cooper pair. It is well known in spin-singlet superconductors that the spin susceptibility decreases monotonically with the decrease of temperature below $T_{c}$ and vanishes at zero temperature [20]. This phenomenon occurs independently of the direction of a Zeeman field $\bm{H}$ because a Cooper pair has no spin. In spin-triplet superconductors, on the other hand, the susceptibility can be anisotropic depending on the relative alignment between a Zeeman field and a $\bm{d}$ vector in the order parameter. For $\bm{d}\perp\bm{H}$, the spin susceptibility is constant independent of temperature. Thus, the unchanged Knight shift across $T_{c}$ in experiments could be a strong evidence of spin-triplet superconductivity. For $\bm{d}\parallel\bm{H}$, the susceptibility decreases with decreasing temperature below $T_{c}$. Such anisotropy is more remarkable when the number of components in a $\bm{d}$ vector is smaller. For $J=3/2$ superconductors, however, our knowledge of the spin susceptibility is very limited to a theoretical paper that reported vanishing the spin susceptibility at zero temperature for a singlet-quintet mixed state in a centrosymmetric superconductor [21].

In this paper, we study theoretically the response of pseudospin-quintet even-parity superconductors to an external Zeeman field. The angular momentum of a Cooper pair in such superconductors is $J=2$. Since the pairing symmetries of the pseudospin-quintet states are not well understood, we decided to calculate the spin susceptibility for the plausible pair potentials at a cubic superconductor preserving time-reversal symmetry. The spin susceptibility is analytically calculated based on the linear response formula [22]. The pair potential of pseudospin-quintet states is described by a five-component vector that couples to five $4\times 4$ matrices in pseudospin space. Such complicated internal structures of the pair potential enrich the magnetic response of $J=3/2$ superconductors. When the symmetry of the pair potential is high in pseudospin space ($A_{1g}$ state), the magnetic response is isotropic in real space and the spin susceptibility decreases monotonically with the decrease of temperature. The results are similar to those of ³He B-phase. When the pair potential is independent of wavenumber ($T_{2g}$ and $E_{g}$ states), the spin susceptibility shows unique features to pseudospin-quintet states. The magnetic response becomes anisotropic in real space in an $E_{g}$ state and the susceptibility tensor has finite off-diagonal elements in a $T_{2g}$ state.

This paper is organized as follows. In Sec. II, we described the electronic structure and the pair potential of a $J=3/2$ superconductor in terms of five $4\times 4$ matrices. The characteristic behaviors of the spin-susceptibility are discussed in Sec. III. The conclusion is given in Sec.IV. Algebras of $4\times 4$ matrices and a number of mathematical relationships used in the paper are summarized in Appendices. Throughout this paper, we use the system of units $\hbar=k_{B}=c=1$, where $k_{B}$ is the Boltzmann constant and $c$ is the speed of light.

We begin our analysis with the normal state Hamiltonian adopted in Ref. 13. The electronic states have four degrees of freedom consisting of two orbitals of equal parity and spin 1/2. In the presence of strong spin-orbit interactions, the effective Hamiltonian for a $J=3/2$ electron is given by  [23]

where $\mathrm{T}$ means the transpose of a matrix and $c_{\bm{k},j_{z}}$ is the annihilation operator of an electron at $\bm{k}$ with the $z$-component of angular momentum being $j_{z}$. The normal state Hamiltonian is represented by

with $\xi_{\bm{k}}=\epsilon_{\bm{k},0}-\mu$ and

where $\hat{k}_{j}=k_{j}/|\bm{k}|$ for $j=x,y$ and $z$ represents the direction of wavenumber on the Fermi surface. The constants $\alpha>0$ and $\beta$ determine the normal state property. The spin-orbit interactions increase with the increase of $\beta>0$. The coefficients $c_{j}$ are normalized as

where $\langle\cdots\rangle_{\hat{\bm{k}}}$ means the integral over the solid angle on the Fermi surface. The normalized five-component vector $\vec{e}=(e_{1},e_{2},e_{3},e_{4},e_{5})/|\vec{e}|$ determines the dependence of the normal state dispersions on pseudospins. The spinors for the angular momenum of $J=3/2$ are described by,

The $4\times 4$ matrices in pseudospin space are defined as

and $1_{4\times 4}$ is the identity matrix. They satisfy the following relations

where $U_{T}$ is the unitary part of the time-reversal operation $\mathcal{T}=U_{T}\,\mathcal{K}$ with $\mathcal{K}$ meaning complex conjugation. The superconducting pair potential is represented as

where a five-component vector $\vec{\eta}_{\bm{k}}$ represents an even-parity pseudospin-quintet state. Throughout this paper, we assume that all components of $\vec{\eta}_{\bm{k}}$ are real values. As a result of the Fermi-Dirac statistics of electrons, the pair potential is antisymmetric under the permutation of two pseudospins, (i.e., $\Delta^{\mathrm{T}}(\bm{k})=-\Delta(\bm{k})$). The Bogoliubov-de Gennes Hamiltonian reads,

where $\undertilde{X}(\bm{k},i\omega)\equiv X^{\ast}(-\bm{k},i\omega)$ represents the particle-hole conjugation of $X(\bm{k},i\omega)$.

The interaction with a uniform Zeeman field $\bm{H}$ is described by [23]

for $j=x,y,$ and $z$, where $\mu_{B}$ is the Bohr’s magneton, and $g_{1}$ and $g_{3}$ are the coupling constants. The matrix structures of $J_{j}$ and $J_{j}^{3}$ are displayed in Appendix A. The angular momenta in the Zeeman Hamiltonian are then given by

In the linear response theory, the spin susceptibility is calculated by using the formula [22]

The summation over the Matsubara frequency and that over the wavenumber are regularized by introducing the Green’s functions in the normal state ${G}_{\mathrm{N}}$ and the spin susceptibility $\chi_{\mathrm{N}}$ in the normal state [24].

The Green’s function for a superconducting state can be obtained by solving the Gor’kov equation

The anomalous Green’s function results in

Generally speaking, it is not easy to calculate analytically the inversion of $4\times 4$ matrices.

To proceed analytic calculation, we consider a cubic symmetric superconductor [14], in which even-parity pair potentials are classified into $A_{1g}$, $E_{g}$, and $T_{2g}$ states according to irreducible representations (irreps) of cubic symmetry. With focusing on pseudospin-quintet Cooper pairs, their pairing states are explicitly represented as

where the unit vector $\vec{h}$ determines the pseudospin structure of $A_{1g}$ state and $l_{i}\in\mathbb{C}$ ($i=1-5$). $A_{1g}$ states involve momentum-dependent coefficients, which comes from the fact that $\gamma^{4}$ and $\gamma^{5}$ $(\gamma^{1},\gamma^{2}$, and $\gamma^{3})$ themselves belong to $E_{g}$ ($T_{2g}$) irreps of cubic symmetry.

Generally speaking, the coefficients $l_{i}$ for $i=1-5$ are determined from the steady states of the free energy [25]. For $E_{g}$ state, three distinct steady states exist: $(l_{4},l_{5})=(1,0)$, $(0,1)$, and $(1,i)/\sqrt{2}$. The first two states preserve time-reversal symmetry, while the last state breaks time-reversal symmetry. For $T_{2g}$ state, there are four distinct steady states: $(l_{1},l_{2},l_{3})=(1,1,1)/\sqrt{3}$, $(1,0,0)$, $(1,e^{i2\pi/3},e^{i4\pi/3})/\sqrt{3}$, $(1,i,0)/\sqrt{2}$. The last two states break time-reversal symmetry. Time-reversal symmetry-breaking superconducting states have a problem specific to them: the formation of Bogoliubov-Fermi surfaces [13, 14, 19]. Since the relations between the pseudospin structures of a Cooper pair and the magnetic response of a superconductor is the main issue in this paper, we focus on time-reversal symmetry respecting superconducting states. In addition to the steady states, we also consider another pseudospin states described by $(l_{4},l_{5})=(1,1)/\sqrt{2}$, $(l_{1},l_{2},l_{3})=(1,1,0)/\sqrt{2}$, and $(l_{1},l_{2},l_{3},l_{4},l_{5})=(1,1,1,1,1)/\sqrt{5}$. The last one is the admixture of $E_{g}$ and $T_{2g}$ states. These states complement possible combinations of $\gamma^{i}$ ($i=1-5$). The comparison between the calculated results for such states and those for the steady states helps us to understand the relations between the pseudospin structures of a Cooper pair and the spin susceptibility. Note that $(l_{1},l_{2},l_{3})=(1,1,0)/\sqrt{2}$ represents a possible order parameter in tetragonal symmetric superconductors with the high symmetry axis being the $y$ direction [25].

As shown in the second term in Eq. (46), the anomalous Green’s function contains the pairing correlation belonging to odd-frequency symmetry class. The stable superconducting states can be described by

which means the absence of odd-frequency pairs. Odd-frequency pairs increase the free-energy of a uniform superconducting state [26] because they indicate the paramagnetic response to a magnetic field [27, 28, 29]. A phenomenological argument on the paramagnetic response of odd-frequency Cooper pairs is given in Appendix A of Ref. [30]. Eq. (50) gives a guide that relates the stable pair potential $\vec{\eta}$ to the electronic structures $\vec{\epsilon}$. To understand this situation, we briefly summarize the case of spin-triplet superconductors in the presence of a strong Rashba spin-orbit interaction $\bm{\lambda}\cdot\bm{\sigma}$ with

where $\lambda_{\mathrm{so}}$ represents the amplitude of spin-orbit interaction, $\sigma_{j}$ and $\bm{e}_{j}$ for $j=x,y$ and $z$ are the Pauli matrix and the unit vector in spin space, respectively. The stable order parameter $i\bm{d}\cdot\bm{\sigma}\,\sigma_{y}$ is determined as

so that odd-frequency pairs are absent and the transition temperature is optimal [26, 31]. Namely, the order parameter of a spin helical state is stable in this case. The pair potentials other than the helical state would be realized when the Rashba spin-orbit interaction is sufficiently weak. The choice in Eq. (52) and that in Eq. (50) are equivalent to each other. We choose the normal state dispersion $\vec{\epsilon}_{\bm{k}}$ so that Eq. (50) is satisfied for the pair potentials in Eqs. (47)-(49). The Green’s function in the superconducting state can be expressed simply and analytically under Eq. (50). The results of the Green’s function in such a case are shown in Appedix B.

The spin susceptibility results in

with $\Omega=\sqrt{\omega_{n}^{2}+{\vec{\eta}}^{2}}$. The tensor is defined by

and its elements are calculated to be

The results in Eq. (55)-(60) describe the characteristic features of the susceptibility of $J=3/2$ superconductors.

We have studied theoretically the spin susceptibility of pseudospin-quintet pairing states in a $J=3/2$ superconductor that preserves cubic lattice symmetry and time-reversal symmetry. Within the linear response to a Zeeman field, we calculate the spin susceptibility by using the Green’s function that is obtained by solving the Gor’kov equation analytically. The pair potentials are chosen so that a superconducting state is stable under the pseudospin structures in the normal state. The calculated results indicat that the magnetic response of pseudospin-quintet states depends sensitively on the pseudospin structures of the pair potential. The susceptibility tensor in $A_{1g}$ states is isotropic in real space as that in the B-phase of superfluid ³He. For $E_{g}$ states, the susceptibility tensor becomes anisotropic in real space. We found in a $T_{2g}$ state that the susceptibility tensor has the off-diagonal elements.