Phase, Amplitude, and Leggett Collective Modes

Collective modes are poles of the fluctuation propagator of a paired state, not names assigned to every maximum in a spectrum. Expanding the effective action about a self-consistent saddle separates a low-energy phase mode, an amplitude sector near the pair-breaking continuum, and—when several condensates are coherently coupled—relative-phase Leggett modes. Gauge coupling and long-range Coulomb forces qualitatively reorganize the phase sector, while damping determines whether any formal zero of a determinant is an observable excitation.

Required background. Nambu–Gor’kov propagators supplies the paired Green function. Bethe–Salpeter kernels supplies the two-particle closure.

Helpful background. BCS mean-field theory supplies the saddle and gap equation whose use is essential below.

Quadratic fluctuations about the saddle

Write the pair field as

$$\Delta(x)=[\Delta_0+h(x)]e^{i\theta(x)} \simeq\Delta_0+h(x)+i\Delta_0\theta(x).$$

After integrating out fermions and expanding to second order,

$$S^{(2)}=\frac12\sum_q \begin{pmatrix}h(-q)&\theta(-q)\end{pmatrix} \begin{pmatrix}M_{hh}(q)&M_{h\theta}(q)\\M_{\theta h}(q)&M_{\theta\theta}(q) \end{pmatrix} \begin{pmatrix}h(q)\\\theta(q)\end{pmatrix}.$$

The collective propagator is $M^{-1}$ and a mode satisfies $\det M^R(\mathbf q,\omega)=0$ after a controlled analytic continuation. The constant term in $M_{\theta\theta}(0,0)$ cancels only if the same approximation is used for the gap equation and the fluctuation kernel. This is the practical form of the broken-symmetry Ward identity; omitting it produces a spurious phase gap.

For a neutral, isotropic, weak-coupling fluid at $T=0$ and $\omega,v_Fq\ll2\Delta_0$, integrating out density fluctuations gives

$$S_\theta=\frac12\int\mathrm dt\,\mathrm d^d x \left[\kappa(\partial_t\theta)^2-\rho_s(\boldsymbol\nabla\theta)^2\right], \qquad \omega=c_sq,$$

with $c_s^2=\rho_s/\kappa$ and, in the three-dimensional weak-coupling continuum, $c_s=v_F/\sqrt3$. The coefficients depend on whether $\theta$ is the pair phase or half of it; a factor-of-four discrepancy usually reflects that convention rather than new physics. Anderson 1958, pp. 1900–1903 explains the neutral and charged distinction.

Amplitude response and damping

In an ideal particle–hole-symmetric BCS model, $h$ and $\theta$ decouple at small $q$. The amplitude kernel has structure at $\omega=2\Delta_0$, precisely where the two-quasiparticle continuum begins. The response is therefore generally a threshold or resonance, not an isolated Lorentz-invariant Higgs particle. Particle–hole asymmetry mixes amplitude and phase, while disorder and additional bands open decay channels.

A credible amplitude-mode claim states the response operator and demonstrates a pole or quantitatively modelled resonance after continuation, including its width and the continuum. Merely fitting a peak near $2\Delta$ is insufficient. Littlewood and Varma 1982, pp. 240–243 develops the amplitude response and its coupling to probes.

Relative phase in two bands

For two condensates with phases $\theta_1,\theta_2$, the lowest derivative action may be written

$$S=\frac12\int\mathrm dt\,\mathrm d^d x\, \left\{ \sum_{a=1}^2\left[\kappa_a\dot\theta_a^2-\rho_a(\boldsymbol\nabla\theta_a)^2\right] -J(\theta_1-\theta_2)^2 \right\}.$$

The in-phase combination is the ordinary Goldstone coordinate. The relative phase has, at $q=0$,

$$\omega_L^2=J\left(\frac1{\kappa_1}+\frac1{\kappa_2}\right).$$

This normalization follows directly by solving the $2\times2$ generalized eigenvalue problem. A visible Leggett mode requires $\omega_L$ below, or sufficiently weakly embedded in, the relevant two-quasiparticle continua. Long-range Coulomb interaction raises the in-phase charged oscillation toward a plasma scale but need not remove the neutral relative-phase mode. Leggett 1966, pp. 562–564 gives the microscopic two-band result.

The chapter diagram places these modes between the paired propagator and the response tests that can observe them.

Collective poles follow from a Ward-consistent quadratic kernel. Neutral phase sound, an amplitude threshold, and a multiband relative-phase mode have different gauge and damping fates. Original schematic, not to scale.

See the paired-matter claim test matrix before interpreting a response maximum as a collective pole.

Exercise

Derive the two-band frequency. Neglect gradients and find the nonzero normal-mode frequency of the two-phase action above.

Solution

The equations are $\kappa_1\ddot\theta_1+J(\theta_1-\theta_2)=0$ and $\kappa_2\ddot\theta_2-J(\theta_1-\theta_2)=0$. The determinant of the harmonic equations is $\omega^2[\kappa_1\kappa_2\omega^2-J(\kappa_1+\kappa_2)]=0$. The zero root is the common phase and the other is $\omega_L^2=J(1/\kappa_1+1/\kappa_2)$. This establishes an undamped mode only if the derivative action remains valid and the frequency avoids strong continua.

References

• Anderson, P. W. (1958). “Random-phase approximation in the theory of superconductivity.” Physical Review 112, 1900–1916. doi:10.1103/PhysRev.112.1900.
• Leggett, A. J. (1966). “Number-phase fluctuations in two-band superconductors.” Progress of Theoretical Physics 36, 901–930. doi:10.1143/PTP.36.901.
• Littlewood, P. B., and Varma, C. M. (1982). “Amplitude collective modes in superconductors and their coupling to charge-density waves.” Physical Review B 26, 4883–4893. doi:10.1103/PhysRevB.26.4883.