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Use the reduced attractive reduced BCS pairing Hamiltonian with volume V, ${\xi }_k={\epsilon }_k-\mu$ and attraction $\lambda >0$:A specified pairing shell or an ultraviolet renormalization is part of the model. The trace is grand canonical. Insert fermionic coherent states into its time-sliced trace. The resulting Grassmann variables are antiperiodic, and with inverse-energy imaginary time $0\le \tau \le \beta$ the dimensionless action isThis boundary condition is essential for Fermi-Dirac statistics; replacing it by a bosonic periodic trace changes the answer. A complex Hubbard–Stratonovich transformation uses the local identitywith time-slice factors understood. The pairing field renders the fermion action quadratic. For a uniform static field the independent Nambu spinor block $({\psi }_{k\uparrow },{\bar{\psi }}_{-k\downarrow }{)}^T$ hasIts determinant is $-({\omega }_n^2+E_k^2)$. The phase-independent determinant sign cancels under consistent reference normalization. Integrating the fermions gives the BCS pairing-field effective action, ${\mathcal{S}}_{\mathrm{eff}}={\int }_0^{\beta }V|\Delta {|}^2/\lambda d\tau -\mathrm{Tr}\ln {\mathcal{G}}^{-1}$, with the normal-ordering/reference term retained. The fermionic pairing-block determinant normalization uses one independent pairing block for every k; a fully doubled spin-particle-hole trace instead needs one half. The subtracted grand potential of the reduced BCS model is particularly unambiguous:Performing the Matsubara frequency sum yieldsup to a common constant. At zero gap this reduces exactly to the two-spin free-gas result. The Matsubara derivation of the BCS saddle follows by differentiating with respect to ${\Delta }^{\ast }$ and using ${\beta }^{-1}{\sum }_n({\omega }_n^2+E^2{)}^{-1}=\tanh (\beta E/2)/(2E)$ givesThe second equation fixes the chemical potential at specified number. This stationary action is BCS mean-field theory, not an exact integration of pairing-field fluctuations. With constant single-spin density of states ${\nu }_F$ per volume and a symmetric energy cutoff $\Lambda$, weak coupling gives $\Delta (0)\simeq 2\Lambda e^{-1/(\lambda {\nu }_F)}$ and $k_B{T}_c\simeq (2e^{{\gamma }_E}/\pi )\Lambda e^{-1/(\lambda {\nu }_F)}$. The zero-temperature condensation grand-potential density is $-{\nu }_F{\Delta }^2/2$ to leading weak-coupling order; using a spin-summed density would change these displayed factors. The BCS coherence factors are $u_k^2=(1+{\xi }_k/E_k)/2$ and $v_k^2=(1-{\xi }_k/E_k)/2$, and the associated BCS ground state is the normalized product of empty and occupied pair states. ^[ref]

Allowing a slowly varying complex pairing field generates phase stiffness and, near the transition, the Ginzburg-Landau theory of superconductivity. A neutral paired system has a gapless phase sound mode; in a charged bulk superconductor electromagnetic coupling produces the Anderson-Higgs mechanism in a superconductor and the Meissner effect. These collective fluctuations are distinct from the gapped fermionic excitations. A microscopic dynamical calculation is needed to fix collective-mode dispersion and damping; the static saddle alone does not determine them. ^[ref]

For conventional superconductivity, an attractive interaction pairs opposite-spin fermions. A complex pairing Hubbard–Stratonovich transformation introduces a Cooper pair field $\Delta$. The fermionic fields then appear quadratically, giving the pair-field effective action of a superconductorThe trace convention counts each independent spin-pair block once. The uniform saddle is the BCS gap equation; the fermionic excitations have $E_k=\sqrt{{\xi }_k^2+|\Delta {|}^2}$ in the fully gapped case. A slow-field expansion near the transition yields Ginzburg-Landau theory of superconductivity. Normalize its pair field $\Psi$ so $|\Psi {|}^2$ is an effective pair density, with pair mass $m_{\ast }$ and signed charge $q=-2e$:For $a<0$, the uniform minimum has ${\rho }_0=|\Psi {|}^2=-a/b$, with arbitrary constant phase. Writing $\Psi =(\sqrt{{\rho }_0}+h)e^{i\theta }$ separates the amplitude mode $h$ from the phase. The local potential has curvature term $2|a|h^2$ and coherence scale $\xi =\hslash /\sqrt{2m_{\ast }|a|}$. A neutral condensate has a gapless phase Goldstone mode when suitable dynamics is included.

The charged theory is invariant underThe gauge-invariant condensate momentum is $\hslash \nabla \theta -q\mathbf{A}$. Only $\hslash \nabla \theta -q\mathbf{A}$ and $\hslash \dot{\theta }+q\varphi$ enter physical phase dynamics. At nearly constant amplitude, varying $F$ givesFor a smooth phase, taking the curl and using Ampère's law gives ${\nabla }^2\mathbf{B}=\mathbf{B}/{\lambda }^2$: the Meissner effect screens a static bulk magnetic field over the London penetration depth $\lambda$. In a gauge with $\theta =0$, the phase stiffness is a quadratic term in $\mathbf{A}$, the superconducting version of the Higgs mechanism.

To identify the dynamical mode, a simple lossless long-wave superconducting phase-density action isHere ${\chi }_p$ is the pair compressibility coefficient. Eliminating the longitudinal electric potential givesThus the neutral sound mode becomes a gapped longitudinal plasma oscillation; transverse electromagnetic waves acquire ${\omega }^2=c_{\mathrm{vac}}^2{k}^2+c_{\mathrm{vac}}^2/{\lambda }^2$ in this simple bulk model. This redistribution of the phase degree of freedom into electromagnetic collective modes is the Anderson-Higgs mechanism in a superconductor. Local gauge symmetry is a redundancy, not an independently observable symmetry whose breaking creates another massless particle. ^[ref]

The static functional alone does not determine a frequency or damping rate. The displayed mode dispersions require the added dynamical assumptions; real screening, normal carriers, boundaries and dissipation modify them. The amplitude fluctuation is distinct from the phase mode and is not removed by the gauge choice.

Consider a conventional spin-singlet, weak-coupling, three-dimensional superconductor. Set $\hslash =k_B=1$ in the microscopic action and restore electromagnetic units below. An attraction $\lambda >0$ in an energy shell about the Fermi surface gives a Grassmann actionThe attractive sign permits pairing even when the single-particle Fermi gas is otherwise stable. A complex Hubbard–Stratonovich transformation introduces a pair field $\Delta$ proportional at the saddle to $\lambda ⟨{\psi }_{\downarrow }{\psi }_{\uparrow }⟩$. The action is then quadratic in fermions. Integrating them out gives the BCS pairing-field effective actionHere the trace uses one independent Nambu spinor pairing block; a fully doubled particle-hole representation instead carries its corresponding one-half factor. At a uniform saddle the Bogoliubov--de Gennes Hamiltonian can be written asThe negative branch describes the paired vacuum; excitations are Bogoliubov quasiparticles mixing particles and holes. Extremizing the effective action givesThe physical cutoff or a renormalized interaction is part of the microscopic model; an unregulated contact integral must not be used indiscriminately. With constant density of states $N(0)$ per spin and shell half-width ${\Omega }_c$, the zero-temperature weak-coupling solution is ${\Delta }_0\simeq 2{\Omega }_c{e}^{-1/[N(0)\lambda ]}$. The exponentially small scale follows from the logarithmic pairing susceptibility of states near the Fermi surface. For this ideal weak-coupling model, $2{\Delta }_0/(k_B{T}_c)\simeq 3.53$. The microscopic theory relates the gap, pairing and thermodynamic transition.^[ref]

Close to a continuous transition, for fields varying slowly compared with microscopic scales, expand the effective action in powers and gradients of $\Delta$. Its static part is the Ginzburg-Landau theory of superconductivity:The normalization of $\Delta$ fixes the coefficients; here $a(T)$ changes sign at $T_c$. The uniform minimum has $|{\Delta }_0{|}^2=-a/b$ for $a<0$, condensation free-energy density $-a^2/(2b)$, and characteristic coherence-length scale $\xi =\sqrt{K/|a|}$. Above the transition the minimum is at zero. The quartic stabilization and gradient cost are precisely the ingredients of Landau theory for a complex order parameter. This controlled near-transition expansion supplies a bridge between microscopic BCS theory and macroscopic magnetic response; its coefficients and validity depend on the material and fluctuation regime.^[ref]

Write $\Delta =(|{\Delta }_0|+\eta )e^{i\varphi }$. The radial fluctuation has positive local restoring curvature, whereas the phase has no local potential in a neutral theory with global $U(1)$ symmetry. Freezing the amplitude givesThe neutral case supports a gapless collective phase mode. In the charged case the electromagnetic field couples to this phase rigidity. Under $\mathbf{A}\mapsto \mathbf{A}+\nabla \chi$ and $\varphi \mapsto \varphi +q\chi /\hslash$, the displayed combination is invariant. Locally choosing a gauge with constant phase leaves a quadratic term in $\mathbf{A}$. Varying the free energy gives the transverse London current and magnetic screening scale,Combining this relation with static Maxwell equations gives exponential field penetration, the Meissner effect. Single-valuedness of the pair field around a loop with negligible current gives flux in units ${\Phi }_0=2\pi \hslash /|q|=\pi \hslash /e$. These are gauge-invariant physical consequences of the condensate stiffness.

For the dynamics, the neutral phase and density mode is reorganized by the Coulomb interaction and the dynamical electromagnetic field. In a bulk three-dimensional system the long-wave longitudinal mode has a plasma gap, rather than remaining the neutral sound mode. This is the Anderson-Higgs mechanism in a superconductor: a phase coordinate becomes part of the electromagnetic collective response instead of surviving as an independent massless physical particle. Local gauge transformations are redundancies of the description, so a literal breaking of a local redundancy is not an observable. Phase coherence, current response and magnetic screening provide the physical statements.^[ref]

A static Landau functional alone cannot determine that full dynamical spectrum. A two-dimensional film, for example, need not have the same long-wave plasma dispersion as a bulk charged system. Nor does a positive radial curvature automatically guarantee a sharp Higgs amplitude resonance: coupling to quasiparticles and other modes can damp it. The chosen mean-field phase is also a thermodynamic symmetry-breaking description; an isolated finite system can be formulated with definite particle number. The microscopic pairing saddle, the near-transition Landau expansion and the electromagnetic Anderson-Higgs response describe successive scales of the same superconducting state, with distinct assumptions at each step.

A neutral superfluid has a complex order parameter whose phase describes coherent particle transport. For a Bose condensate it is represented by a field amplitude; for a paired Fermi superfluid it is a pair amplitude. In a symmetry-breaking description, selecting its phase breaks the global particle-number $U(1)$ symmetry. A common phase shift leaves the energy unchanged, while a phase gradient costs energy and produces superfluid velocity. Long-wavelength phase oscillations give a gapless Goldstone mode, generally a sound wave when density fluctuations provide the conjugate dynamics. A finite isolated state with exact particle number need not have a nonzero field expectation: the phase-selected language describes the ordered limit and relative-phase coherence.

In a superconductor, the paired order parameter carries charge of magnitude $2e$ and couples to the electromagnetic field. The meaningful spatial phase combination is $\hslash \nabla \theta -q\mathbf{A}$, with signed pair charge $q$. Gauge symmetry is a redundancy, so “breaking local gauge symmetry” is not a literal breaking of a measurable physical symmetry. The gauge-invariant electromagnetic response includes the Meissner effect and a finite London penetration depth. Long-range Coulomb forces move the would-be neutral phase mode to a plasma scale through the Anderson-Higgs mechanism in a superconductor.

Additional symmetries can be broken by particular pairing states. Conventional isotropic spin-singlet pairing need not break spatial rotations or time reversal. A d-wave superconductor has a gap transforming nontrivially under crystal symmetries, and some triplet or multicomponent states can also break spin-rotation or time-reversal symmetries. These additional choices are not required by superconductivity itself. A nonzero pairing amplitude, phase coherence and electromagnetic screening describe related but distinct aspects of the ordered state.

A neutral superfluid can select a phase of its complex order parameter, breaking a global $U(1)$ symmetry. A spatially uniform change of that phase costs no energy. Slowly varying changes therefore produce a gapless Goldstone mode. For a dilute weakly repulsive Bose gas, the Bogoliubov quasiparticle dispersion isso $E_k\simeq \hslash c_{s}k$ at long wavelengths, with $c_s=\sqrt{g_B{n}_0/m_B}$. Density and phase fluctuations are canonically coupled components of this sound branch, rather than two independent positive-frequency modes. Superfluid helium and dilute atomic condensates supply neutral examples, although the dilute-gas formula is not quantitatively valid for strongly interacting liquid helium.

In a superconductor, the pair field is charged and its phase couples to the electromagnetic gauge field. Only combinations such as $\nabla \theta -q\mathbf{A}/\hslash$ are physically meaningful. The Anderson-Higgs mechanism in a superconductor produces electromagnetic screening and removes the neutral sound mode as an independent gapless charged-density oscillation. In a homogeneous three-dimensional continuum, long-range Coulomb forces give a longitudinal plasma gap, approximately ${\omega }^2={\omega }_p^2+c_s^2{k}^2$ at small $k$. The static transverse response gives the Meissner effect andwhen $n_s$ counts superconducting electrons of effective mass $m$. Counting pairs instead uses density $n_s/2$, charge $-2e$ and mass $2m$, giving the same stiffness. The condensate amplitude may also have a gapped Higgs mode, but that amplitude oscillation is distinct from the electromagnetic Higgs mechanism. Local gauge transformations remain a redundancy; this is not spontaneous breaking of an observable local symmetry in the same sense as the neutral global phase symmetry.

The superconducting order is a coherent pair amplitude, schematically $\Delta \propto ⟨c_{\downarrow }{c}_{\uparrow }⟩=|\Delta |e^{i\varphi }$. Under a global fermion phase change $c\mapsto e^{i\alpha }c$, it acquires phase $2\alpha$. A symmetry-broken thermodynamic description selects a pair phase. This must be distinguished from electromagnetic gauge freedom: a local gauge transformation changes the phase and potentials together and is a redundancy of the description, not an observable symmetry that the material literally violates.

A conventional centrosymmetric s-wave superconductor usually has an even-parity orbital pair state and a spin singlet state, consistent with the antisymmetry of two fermions. Its ordinary zero-field equilibrium state preserves time reversal. The electromagnetic coupling makes phase dynamics and screening part of the Anderson-Higgs mechanism in a superconductor; the Meissner effect is a measurable consequence. The condensate phase description does not imply that electric charge is no longer conserved.

In an unconventional superconductor, the gap can transform nontrivially under the crystal point group or have other nonstandard pairing structure. A common example is d-wave superconductivity, with sign changes and possible nodes in the gap. This does not automatically imply an additional observable fourfold-rotation breaking: a rotation that reverses the gap's sign can be accompanied by a gauge phase change. A nematic superconducting state instead breaks an actual crystal rotation symmetry in its gauge-invariant responses.

A chiral combination such as $p_x+i{p}_y$ can break time reversal, while a spin-triplet order parameter can also distinguish spin directions when the applicable spin-rotation and spin-orbit constraints permit it. Helium-3 superfluidity illustrates paired-fermion order with richer spin and orbital structure. These possibilities must be tested rather than equated: unconventional pairing does not by itself mean spin-triplet pairing, time-reversal breaking, or nematicity. Gap angular dependence, phase-sensitive interference, spin response and spontaneous magnetic signatures probe different symmetry questions. The phase-sensitive test of d-wave pairing is more specific than inferring a gap sign merely from its magnitude.

An elementary excitation is an approximately independent low-energy normal mode above a many-body equilibrium state. A Bogoliubov transformation mixes creation and annihilation operators to diagonalize a quadratic fluctuation Hamiltonian. For weakly interacting bosons,interpolating between sound and particle-like motion. Bosonic canonical commutators require $|u_k{|}^2-|v_k{|}^2=1$. In BCS theory, fermionic particle–hole mixtures instead have $|u_k{|}^2+|v_k{|}^2=1$ and excitation energies $E_k=\sqrt{{\xi }_k^2+|{\Delta }_k{|}^2}$. These fermionic quasiparticles must be distinguished from collective amplitude and phase oscillations of the pair field.

For the stated dimensionless first-order equation, a nonzero uniform static solution obeys $(|{\psi }_0{|}^2-1){\psi }_0=0$. Thus ${\psi }_0=e^{i{\theta }_0}$. The zero field is also stationary, but is not the condensed minimum of the local potential $\beta (|\psi {|}^2-1{)}^2/2$.

Choose ${\theta }_0=0$. To linear order, $\psi =1+\rho /2+i\theta$, so $|\psi {|}^2-1=\rho$. Substitution and separation of real and imaginary parts giveDifferentiating the first relation and using the second yieldsThe same elimination holds for $\theta$. A plane wave $e^{i(kx-\omega t)}$ therefore givesThe density and phase components share this frequency because of density-phase conjugacy. They are coupled components of one positive-frequency Bogoliubov mode, not two independent excitations that happen to have equal energies. At $k=0$, a constant phase change is a zero mode; the nonzero-$k$ elimination should not be used to assign an independent oscillation to a uniform density change.

Replacing the first-order time derivative by $-\gamma \ddot{\psi }$ changes the number and structure of dynamical degrees of freedom. The linearized real and imaginary equations now decouple:The amplitude and phase modes of a second-order complex field consequently haveThe amplitude mode has a nonzero gap $\sqrt{2\beta /\gamma }$; the phase mode is gapless with speed $1/\sqrt{\gamma }$. Frequencies are nonnegative, so the printed expression $k/\sqrt{\gamma }$ is understood for $k\ge 0$ or replaced by $|k|/\sqrt{\gamma }$.

In the complex plane, an amplitude oscillation moves radially relative to the chosen equilibrium $\psi =1$, and a phase oscillation moves tangentially, or along a short unit-circle arc beyond linear order. A first-order coupled mode combines them in quadrature. For a fixed spatial factor and $k\ne 0$, one possible parametrization isso $\delta \psi =\rho /2+i\theta$ traces a small ellipse about $1$. Choose $R$ small enough that both components remain within the linear approximation.

A weak neutral Bose condensate realizes the first-order structure and its sound-like Bogoliubov spectrum. A particle–hole-symmetric effective complex field can realize the second-order structure, with a gapless Goldstone mode and a distinct amplitude mode. A neutral BCS superfluid likewise has collective phase and amplitude responses, although the amplitude response near pair-breaking energy need not be a sharp undamped particle; its fermionic quasiparticles are separate excitations.

In a charged bulk superconductor, long-range Coulomb interactions and electromagnetic dynamics remove the neutral sound-like phase branch from the low-frequency spectrum. The Anderson-Higgs mechanism in a superconductor produces a longitudinal plasma mode, schematically ${\omega }^2\simeq {\omega }_p^2+c_s^2{k}^2$ in an appropriate long-wavelength model. The amplitude scale is typically associated with $2\Delta /\hslash$ in a simple BCS regime, with damping and mixing dependent on material and symmetry. One should not simply retain the neutral Goldstone dispersion after charging the field, nor identify the static Ginzburg-Landau functional alone with a complete time-dependent microscopic theory.