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An attractive local pairing interaction can be decoupled with a complex Hubbard–Stratonovich transformation. Integrating the fermionic Grassmann variables gives an effective action for the pair field. Its uniform saddle is the BCS gap equation, and its slow-field expansion near the transition gives the Ginzburg-Landau theory of superconductivity. A trace over a fully doubled particle-hole basis requires the corresponding factor of one half; the displayed formula uses one independent pairing block.

The zero-field Hubbard–Stratonovich transformation of an exponentially coupled Ising model produces $U(m)=A{m}^2-\log [2\cosh (2m)]$, where $A=\tanh (\kappa /2)/J>0$. Since $U^{\prime \prime }(0)=2A-4$ and $U$ tends to positive infinity, a symmetric double well exists exactly when $A<2$. Its nonzero stationary points solve $Am=\tanh (2m)$; the ratio $\tanh (2m)/m$ decreases from 2 to 0 for $m>0$, proving that there is exactly one positive minimum. At $A=2$ the minimum is a single quartic minimum. A small conjugate field adds $-h\tanh (2m)+O(h^2)$ and tilts the wells, while a sufficiently large field can remove the metastable well. A finite barrier permits quantum tunnelling; two classical minima alone do not establish spontaneous symmetry breaking.

A pairing-channel Hubbard–Stratonovich transformation replaces an attractive fermion interaction by a complex Cooper pair field coupled to fermion pairs. A Grassmann Gaussian integral then gives a trace-log effective action, with its trace convention fixed to avoid doubling independent spin-pair blocks. The uniform stationary equation is the BCS gap equation. A small-field, slow-gradient expansion near the transition yields Ginzburg-Landau theory of superconductivity; the normalization of the phenomenological field need not equal that of the microscopic gap.

Write the finite-chain spin vector as $\sigma$ and the exchange matrix as $\mathsf{J}$. For $J>0$ and $\kappa >0$ this matrix is positive definite, as can be seen from the positive Fourier symbol in part (b), or from the explicit positive open-chain inverse. The linear-source multivariate Gaussian integral, obtained by completing the square, givesIndeed the exponent is $-(m-\mathsf{J}\sigma {)}^T{\mathsf{J}}^{-1}(m-\mathsf{J}\sigma )+{\sigma }^T\mathsf{J}\sigma$. Insert this Hubbard–Stratonovich transformation into the spin partition function. The field term remains $h{\sum }_i{\sigma }_i$, and each independent spin sum isConsequentlyThe interaction convention counts the full ordered pair sum, with no factor of one-half, so the auxiliary coupling is $2m_i{\sigma }_i$. Diagonal terms $J_{ii}{\sigma }_i^2$ only contribute a constant to the spin Hamiltonian; retaining them makes the given positive exchange matrix and normalization convenient. The auxiliary field is not itself the physical spin magnetization: the conditional spin expectation is $\tanh (2m_i+h)$, so $⟨{\sigma }_i⟩=⟨\tanh (2m_i+h)⟩$ in the auxiliary-field integral.

Write the interaction matrix as $\mathsf{J}$ to distinguish it from its scalar strength. The printed pair sum contains both orientations and has no factor $1/2$. For a positive-definite matrix $\mathsf{J}$, completion of the square gives the Hubbard–Stratonovich transformationsince $m^T{\mathsf{J}}^{-1}m-2m^T\sigma =(m-\mathsf{J}\sigma {)}^T{\mathsf{J}}^{-1}(m-\mathsf{J}\sigma )-{\sigma }^T\mathsf{J}\sigma$. For the exponential kernel with $J>0$ and $\kappa >0$, positivity follows from the positive Fourier symbol derived in part (b), or from its finite open-chain covariance matrix.

Insert this identity into the Ising model partition sum and interchange the finite spin sum with the convergent Gaussian integral. The spin variables now factor independently:It follows thatwith $C=({\pi }^N\det \mathsf{J}{)}^{-1/2}$ for the measure used here. The factor two in the spin-field coupling is required by the double pair sum; replacing it by one would decouple a statistical Hamiltonian with a different normalization. The diagonal $J_{ii}{\sigma }_i^2=J$ is a spin-independent energy. Including it as printed makes the positive Gaussian identity directly applicable; if diagonal self-couplings are removed they should first be restored as an irrelevant constant before this real auxiliary-field decoupling.

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.

A coherent state is an eigenstate of an annihilation operator. Bosonic unnormalized states have exponential overlap and the Gaussian resolution of identity used above; their overcompleteness supplies a phase-space integration measure, not an orthonormal state count. For a normal-ordered many-body Hamiltonian, coherent-state insertion gives the bosonic coherent-state path integralConstants in $H_N$ must be retained. The exact time-sliced prescription places creation and annihilation symbols on adjacent slices; discarding that prescription can change ordering corrections, as the oscillator example demonstrates. The derivative term is first order because the fields describe phase space.

Fermionic coherent states instead use Grassmann variables. Their Gaussian identity is $\int d\bar{\psi }d\psi e^{-\bar{\psi }A\psi }=\det A$, whereas complex bosons give $(\det A{)}^{-1}$. Anticommutation in the trace produces antiperiodic fields. Accordingly bosons have ${\nu }_n=2\pi n/\beta$, fermions ${\nu }_n=(2n+1)\pi /\beta$, in these inverse-energy time units.

For one mode of grand-canonical energy $\xi$, the products give $Z_B=(1-e^{-\beta \xi }{)}^{-1}$ and $Z_F=1+e^{-\beta \xi }$; convergence requires $\xi >0$ for the bosonic Gaussian. Correlation functions follow by source differentiation. Quartic interactions can be treated perturbatively or decoupled with a Hubbard–Stratonovich transformation; a saddle then describes an order parameter and quadratic fluctuations its collective modes. Such saddles require stability and an appropriate control parameter; the functional notation alone is not an exact solution of an interacting problem.

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.

Set $B(x)=\bar{\psi }(x)\psi (x)$, a Grassmann-even Dirac scalar bilinear. Completing the square givesIntroduce external Grassmann sources $\eta ,\bar{\eta }$ for the Dirac field, but no source for $\sigma$. The generating functional isAt a finite regulator the change of variable ${\sigma }^{\prime }=\sigma -gB$ has unit Jacobian: it translates a bosonic integration variable by a Grassmann-even expression independent of that variable. Its oscillatory Gaussian integral can be defined with a convergence prescription and then continued. It is a constant $\mathcal{N}$ independent of $\psi ,\eta ,\bar{\eta }$. ThusThe factor $\mathcal{N}$ cancels in $Z[\eta ,\bar{\eta }]/Z[0,0]$. Every normalized Green function with only external fermion lines is a Grassmann source derivative of this same normalized generating functional, soThis is an exact Hubbard–Stratonovich transformation in reverse, rather than just an approximation based on tree-level Feynman diagrams. At the diagrammatic level the contact auxiliary-field propagator $-i{\delta }^{(4)}(x-y)$ collapses each exchanged line into the same local four-fermion interaction. The equality uses a common regulator and does not assert perturbative renormalizability of that four-dimensional four-fermion interaction.