On the Excitations of a Balian–Werthamer Superconductor

Abstract

My contribution to this collection of articles in honor of David Lee and John Reppy on their 90th birthdays is a reflection on the remarkable phenomenology of the excitation spectra of superfluid $^3$He, in particular the B-phase which was identified by NMR and acoustic spectroscopy as the Balian–Werthamer state shown in 1963 to be the ground state of a spin-triplet, p-wave superconductor within weak-coupling BCS theory. The superfluid phases of $^3$He provide paradigms for electronic superconductors with broken space-time symmetries and non-trivial ground-state topology. Indeed, broken spin- and orbital rotation symmetries lead to a rich spectrum of collective modes of the order parameter that can be detected using NMR, acoustic and microwave spectroscopies. The topology of the BW state implies its low-temperature, low-energy transport properties are dominated by gapless Majorana modes confined on boundaries or interfaces. Given the central role the BW state played I discuss the acoustic and electromagnetic signatures of the BW state, the latter being relevant if an electronic analog of superfluid $^3$He-B is realized.

Appendices

Appendix A: Vdovin’s Contribution

In 1987 I gave an invited talk at the March meeting of the American Physical Society held in New York on the “Theory of Sound Propagation and Attenuation in Superfluid $^3$He” including the Zeeman and Paschen–Back effects of the $J=2^{\pm }$ collective modes. This was the “Woodstock of Physics” meeting that highlighted the discovery of high-temperature superconductivity in the cuprates. Tony Leggett was the chair of my session, and after the talk, he told me that the collective mode spectrum had been obtained by Yu. Vdovin years before the discovery of $^3\textrm{He-B}$ and the theoretical works of Wölfle, Serene and Maki on the collective modes and their acoustic signatures. Tony kindly sent me a copy of a collection of articles published in Moscow in 1963 on “Methods of Quantum Field Theory to the Many Body Problem,” which included the article by Vdovin titled “Effects of pairing in Fermi systems in a P-state” [49]. Tony also drew my attention to a sentence at the end of the abstract stating that the work had been completed in 1961! That was two years before the publication of the work by Balian and Werthamer, and the same year as the publication of the papers by Anderson and Morel [2], Gorkov and Galitskii [84], and Vaks Galitskii and Larkin [17]. As far as I know the first reference to Vdovin’s paper in the literature on $^3$He or collective modes in superconductors was my review of collective modes and nonlinear acoustics with R. McKenzie in 1990 [62]. About the same time Vollhardt and Wölfle cited Vdovin’s paper in their treatise, “The Superfluid Phases of Helium 3,” and pointed out that Vdovin’s work “fell into oblivion.” That appears to be true, as Vaks, Galitskii and Larkin, who published work on collective excitations in higher angular momentum states in 1962 [85], appear to have been unaware of Vdovin’s work. However, the connection between Vdovin’s paper and these four early papers on the theory of pairing in higher angular momentum states is I think worth clarifying in an article reflecting on the impact of the BW ground state on both the physics of superfluid $^3$He, as well as the theory of unconventional superconductors. The existence of Vdovin’s early work, and that it appears to have been done as early as 1961, has been interpreted to imply that Vdovin should be credited equally with Balian and Werthamer for the theoretical prediction for the ground state of a spin-triplet, p-wave superconductor, i.e., what I have referred to as the BW ground state, c.f. Ref. [35]. However, that is incorrect.

Balian and Werthamer proved that the $^3$P$_0$ state with $L=1$, $S=1$ and $J=0$ was the absolute minimum of the weak-coupling BCS free energy functional within the p-wave/spin-triplet manifold. See Sect. 3, p. 1556 of the BW paper [1]. The physical reason is that within the most attractive pairing channel, the lowest free energy state(s) is the linear superposition that maximizes the pairing gap over the Fermi surface, and for the spin-triplet, p-wave manifold this is the BW state.⁵

Vdovin made no such analysis of the stability of phases within the p-wave, triplet manifold. Rather he assumed the ground state was the $^3$P$_0$ state. From paragraph 3 on p. 95 of Ref. [49], “Both single-particle and collective excitations are considered in this system. Different branches of the collective excitation spectrum, corresponding to dynamics of bound pairs with different moments J, are obtained in the assumption that the condensate is made from pairs in $^3$P$_0$ state.”

The basis for Vdovin’s assumption of a $^3$P$_0$ ground state is the paper of Gorkov and Galitskii (GG) [84]. However, the paper by GG contains fundamental errors and is not a proof that the $^3$P$_0$ state is the ground state. GG start from an ansatz for the two-particle density matrix, $\rho _{\alpha \beta ;\gamma \rho } (p,-p;p',-p')\equiv \langle \psi _{\alpha }(p)\psi _{\beta }(-p) \psi ^{\dag }_{\gamma }(p') \psi ^{\dag }_{\delta }(-p')\rangle$, which is not a BCS condensate, but rather a fragmented condensate [87], i.e. $(2l+1)$ condensates with macroscopic eigenvalues of the form (1st equation on p. 793 of Ref. [84]),

$$\begin{aligned} \rho _{\alpha \beta ;\gamma \rho }(p,-p;p',-p') \rightarrow \sum _{m=-l}^{+l} F_{m,\alpha \beta }(p)\,F^{\dag }_{m,\gamma \delta }(p'). \end{aligned}$$

(51)

GG posit an equation for each m of the form,

$$\begin{aligned} \hat{\Delta }_m(\mathbf{p}) = \int d\mathbf{p}'\,V(|\mathbf{p}-\mathbf{p}'|) \,\int d\omega \,\hat{F}_m(\mathbf{p}',\omega ) , \end{aligned}$$

(52)

then assert that “since the angular momentum is zero,” the diagonal (quasiparticle) propagator is isotropic with

$$\begin{aligned} G_{\alpha \beta }(p) = G(|\mathbf{p}|,\omega )\,\delta _{\alpha \beta }. \end{aligned}$$

(53)

With this assumption, GG eliminate all pairing states that do not have an isotropic excitation gap. Specifically, GG argue that since the diagonal propagator is isotropic then $\hat{F}_m(p)\propto Y_{lm}(\mathbf{p})$ as is $\hat{\Delta }_m(p)$, and thus based on Eq. (2), each $\hat{\Delta }_m(p)$ has the same amplitude, in which case the addition theorem for the spherical harmonics generates an isotropic excitation gap given by,

$$\begin{aligned} |\Delta |^2 = \frac{1}{2}|\Delta _m|^2\,(2L+1)(2S+1)\,P_{L}(\theta =0). \end{aligned}$$

(54)

It is a circular argument disconnected from the BCS free energy functional and the BCS gap equation, which is the stationarity condition of the former  [86].

By contrast BCS condensation corresponds to macroscopic occupation of a single two-particle state

$$\begin{aligned} \rho _{\alpha \beta ;\gamma \rho }(p,-p;p',-p') \rightarrow F_{\alpha \beta }(p)\,F^{\dag }_{\gamma \delta }(p'), \end{aligned}$$

(55)

where the spin- and orbital structure of the Cooper pair amplitude, $F_{\alpha \beta }(p)=\langle \psi _{\alpha }(p)\psi _{\beta }(-p)\rangle$ is determined self-consistently by the BCS mean field gap equation,

$$\begin{aligned} \hat{\Delta }(\mathbf{p}) = \int d\mathbf{p}'\,V(|\mathbf{p}-\mathbf{p}'|)\,\int d\omega \,\hat{F}(\mathbf{p}',\omega ) . \end{aligned}$$

(56)

The linearized form of the gap equation separates into a set of eigenvalue equations determined by pairing interactions, $V_l$, for each of the irreducible representations of the symmetry group of the normal state, which in this case is $\texttt {SO(3)}_{\mathrm{L}}$. The superconducting transition is then driven by the most attractive pairing interaction, e.g., $V_1$, resulting in an anomalous self energy of the form

$$\begin{aligned} \hat{\Delta }(\mathbf{p}) = \sum _{m_s=-1}^{+1}\sum _{m_L=-1}^{+1} \Delta _{m_s,m_L}\,\hat{S}_{1,m_s}Y_{1,m}(\mathbf{p}), \end{aligned}$$

(57)

where $\hat{S}_{1,m_s}$ are the $2\times 2$ matrix representation of spin states $\displaystyle \vert \,{1,m_s}\,\rangle$ and $Y_{1,m_L}(\mathbf{p})$ are the p-wave orbital basis states, i.e., the $L=1$ spherical harmonics. The amplitudes $\Delta _{m_s,m_L}$ are determined by solutions to the full nonlinear BCS gap equation, which is the stationarity condition for the weak coupling BCS free energy functional. The lowest energy state among the solutions to the gap equation is the ground state, which for $L=1$, $S=1$, is the BW state.

To summarize, Vdovin’s contribution was the original prediction of the Bosonic collective modes based on the assumed BW ground state using the field theory method developed by Vaks, Galitskii and Larkin [17]. However, neither Vdovin, nor Gorkov and Galitskii proved that the ground state of a spin-triplet, p-wave superconductor is the $^3$P$_0$ state. That was the work of Balian and Werthamer.

Appendix B: Evaluation of the Response Function

Equation 23 for the static condensate response is obtained by evaluating Eq. 8 with $\omega =0$ and changing the integration variable to $\xi =\mathrm{sgn}(\varepsilon )\sqrt{\varepsilon ^2-|\Delta |^2}$. The symbol implies principal part integration in the neighborhood of the singularities on the real $\xi$ axis at $\pm \eta /2$. This integral is most easily evaluated by using the Matsubara representation for the hyperbolic tangent function,

$$\begin{aligned} \frac{\tanh (\sqrt{\xi ^2+|\Delta |^2}/2T)}{2\sqrt{\xi ^2+|\Delta |^2}} =T\sum _{\varepsilon _n}\frac{1}{\xi ^2 + \varepsilon _n^2 + |\Delta |^2}, \end{aligned}$$

(58)

where $\varepsilon _n=(2n+1)\pi T$ are the Fermion Matsubara frequencies with $n\in {\mathbb {Z}}$. Thus, Eq. 23 becomes

(59)

The principal part integral on the real axis is a component of the integral over the closed contour shown in Fig. 5, i.e., where is the path of the principal part integral on the real $\xi$-axis, ${\mathscr {C}}_{\pm }$ is an infinitesimal half circle in the upper half $\xi$-plane of radius $\delta \rightarrow 0^+$ centered at $\xi _{\pm }=\pm \eta /2$, and ${\mathscr {C}}_{\infty }$ is a half circle in the upper half plane of radius $R\rightarrow \infty$. The integrand

$$\begin{aligned} I(\xi )=\frac{1}{\xi ^2+\varepsilon _n^2+|\Delta |^2} \times \frac{1}{\frac{1}{4}\eta ^2 - \xi ^2}, \end{aligned}$$

(60)

is analytic on contour ${\mathscr {C}}_{\infty }$, except at isolated points on the imaginary axis that can be avoided and vanishes faster than $1/|\xi |$ for $|\xi |\rightarrow \infty$ which implies that the corresponding integral of the integrand in Eq. 59 vanishes. For the small semi-circles $\xi =\pm \eta /2+\delta e^{i\theta }$ for $\theta \in \{0,\pi \}$. Integration around the small semi-circles yields,

$$\begin{aligned} \int _{{\mathcal {C}}_{\pm }}d\xi \,I(\xi )= \mp \frac{i\pi }{\eta } \times \frac{1}{\varepsilon _n^2+|\Delta |^2+\frac{1}{4}\eta ^2}. \end{aligned}$$

(61)

Thus, $\int _{{\mathcal {C}}_{+}+{\mathcal {C}}_{-}}d\xi \,I(\xi ) \equiv 0$, yielding a regular response function for $\eta \rightarrow 0$ and . Contour ${\mathcal {C}}$ encloses a meromorphic integrand with a simple pole at $\xi =+i\sqrt{\varepsilon _n^2 + |\Delta |^2}$. Evaluating Eq. 59 with the residue of the integrand yields Eq. 24.

Fig. 5

Integration contours for evaluating the principal part integral in Eq. 59 for the static condensate response function $\lambda (\eta )$ (Color figure online)