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With Fourier convention $c_n=N^{-1/2}{\sum }_k{e}^{-ikn}{c}_k$, the Transverse-field Ising model has $H=J{\sum }_k{\Psi }_k^{†}[(g-\cos k){\sigma }_z-\sin k{\sigma }_x]{\Psi }_k$. Expanding the spinor retains the constant $-JgN$ and pairs $k$ with $-k$. The physical quasiparticle energy is twice the positive matrix eigenvalue times $J$, because both momentum partners occur in the Nambu sum.

For nonnegative field $g$, the bulk quasiparticle gap is $2J|g-1|$. At $g=0$ the dispersion is flat, $2J$; at large $g$ it is $2Jg-2J\cos k+O(J/g)$. At $g=1$ it becomes $4J|\sin (k/2)|$, linear near the gapless point. The finite periodic spin chain has additional global excitation constraints; a single isolated domain wall is not an allowed closed-chain spin configuration.

The ideal Josephson junction has a phase-dependent energyDifferentiating and using the voltage–phase relation yieldsThus the ideal pair element stores and returns energy. At $V=0$ it carries a static electric current without electrical power loss. Under a nonzero periodic drive its instantaneous power need not vanish, but its energy returns after a full phase cycle. Real junctions also have a capacitance, quasiparticle transport and an external circuit; their total losses cannot be inferred from this conservative pair term alone.

A Josephson junction couples two regions with superconductivity through a weak link. A tunnel barrier is one implementation.

The remarkable possibility is a steady electric current across an insulating barrier at zero electric potential difference. This is the DC Josephson effect, carried by coherent transfer of Cooper pairs. It differs from single-electron superconducting tunneling, which probes the quasiparticle excitation spectrum and produces a dissipative current-voltage characteristic. Both contributions can occur in the same superconducting tunnel device. Brian D. Josephson predicted the pair contribution in 1962; his paper treats transfer that leaves the quasiparticle distribution unchanged. ^[ref]

The Josephson effect turns a difference of quantum phases into a measurable electric current, and an applied electric potential difference into a rate of phase advance. The Giaever gap experiment instead turns an electric potential difference into an excitation energy scale. These complementary measurements expose two aspects of superconductivity: its coherent order parameter and its gapped quasiparticle spectrum. The microscopic pairing calculation belongs to BCS theory and the BCS gap equation. A tunnel diode uses a different device, a semiconductor p-n junction.

Adding capacitance and an external bias makes the Gauge-invariant Josephson phase dynamical. Current-biased Josephson phase dynamics explains how its trapped energy levels and escape can be measured electrically.

A Kondo lattice develops coherent low-energy quasiparticle bands below a material-dependent coherence scale. This lattice scale need not equal the single-impurity Kondo screening scale. Coherence can produce a heavy Fermi liquid with narrow bands, large electronic heat capacity and low-temperature quadratic resistivity.

Lev Landau developed descriptions of interacting quantum fluids through their elementary excitations. In the Landau critical velocity argument, creating an excitation of energy $ϵ(\mathbf{p})$ in flowing superfluid changes the relevant energy by $ϵ(\mathbf{p})-\mathbf{v}\cdot \mathbf{p}$. Requiring this to remain positive yields the ideal bound $v_c={\min }_pϵ(p)/p$; additional processes can limit actual flow at lower speeds.^[ref] His Fermi liquid theory explains how strongly interacting fermions can retain long-lived low-energy quasiparticles, with interactions modifying their effective mass and collective response.^[ref]

A physicist.^[ref]

A Luttinger liquid is a low-energy description of many gapless one-dimensional interacting quantum fluids. Its natural excitations are collective density waves, rather than independent fermion-like quasiparticles. Long-distance correlations typically decay with interaction-dependent powers at zero temperature. The Luttinger parameter and a mode velocity encode much of that long-wavelength behavior.

A narrow channel containing interacting particles motivates the picture: in one dimension, density disturbances cannot spread around one another through transverse space, and collective motion becomes especially important. The description applies to suitable bosonic as well as fermionic fluids. Duncan Haldane's harmonic-fluid treatment explicitly identifies this shared low-energy structure in terms of long-wavelength density fluctuations and renormalized parameters.^[ref] A gap, strong lattice locking or additional low-energy modes can require an extended description, so one-dimensional geometry alone is not enough to establish a single Luttinger liquid theory.

Yoichiro Nambu's superconductivity work led to the spontaneous symmetry breaking ideas recognized by his share of the 2008 Nobel Prize in Physics.^[ref] His 1960 treatment made the Meissner effect response consistent with gauge invariance by including collective pair contributions alongside the quasiparticles. The resulting Ward identity relates the interaction correction to the current and the self-energy; retaining the quasiparticles alone omits part of the response.^[ref]

The individual Nobel lecture explains the bridge to particle: the superconductivity gap suggested a dynamically generated fermion mass, and the accompanying collective excitation suggested a light pion. It also notes that the long-range Coulomb interaction turns the charged phase response into a plasmon. A neutral global phase mode and a charged material therefore need different low-momentum descriptions.^[ref]

In 1961, Yoichiro Nambu and Giovanni Jona-Lasinio supplied a four-fermion model with a self-consistent mass and a massless pseudoscalar bound state: an idealized pion. Their original constituents were nucleon fields. Their introduction explicitly distinguishes the zero-mass neutral collective excitation from its charged plasma counterpart and allows approximate chiral symmetry as the explanation of a nonzero pion mass. This is a model of dynamical symmetry breaking, rather than a microscopic derivation of quantum chromodynamics.^[ref]

In the modern two-light-quark account, chiral symmetry breaking leaves isospin intact and produces three Goldstone bosons in the exact non-anomalous chiral limit. Small quark masses explicitly break that chiral symmetry, so the observed pions are pseudo-Goldstone bosons. The chiral protection of a fermion mass identifies the distinct obstruction to inserting an explicit fermion mass into a chirally symmetric equation. The useful lesson is that massive individual excitations and much lighter collective excitations can coexist because they describe different motions of an interacting state.

A concrete experimental motivation for treating the electromagnetic response carefully is the Field-cooled Meissner experiment. MIT's realization cools lead in a solenoid field using helium gas. A nearby thermometer follows the cooling; a test coil feeds an integrator and oscilloscope. Expulsion gives a pulse, warming reverses it, and switching off the field while the sample is normal checks its sign. A change in alternating-field response alone can reflect normal electrical conductivity. This readout tests magnetic flux expulsion, not the spectrum or existence of a neutral Goldstone boson.^[ref]

For a global symmetry in field theory, different orientations can be physically distinct states. A local gauge symmetry instead includes redundant descriptions of the same physical state. A gauge-dependent field expectation alone is consequently insufficient as a physical criterion for spontaneous symmetry breaking. The Higgs mechanism changes the physical mode content when the gauge field is dynamical; the global Goldstone theorem alone does not prove that mechanism. In particular, the massless direction of a gauge-fixed scalar potential need not appear as a separately observable massless particle.

For four equivalent nodes, two spin quasiparticles and local dispersion $E^2={\alpha }^2{q}_1^2+{\beta }^2{q}_2^2$, one node encloses momentum area $\pi E^2/(\alpha \beta )$. Counting states with measure $d^{2}q/(2\pi {)}^2$ gives the displayed positive-excitation density. The associated clean low-temperature heat capacity is $C/\mathcal{A}=18\zeta (3)k_B^3{T}^2/(\pi \alpha \beta )$. Electron-addition spectral density alone has a different count because its coherence weights do not count all particle and hole thermal excitations.

A non-Fermi liquid is an interacting fermionic state whose low-energy behavior is not described by the long-lived quasiparticles of a Fermi liquid. In the conventional Fermi liquid picture, excitations close enough to the Fermi surface become sharp compared with their excitation energy. If that organizing principle fails, extending the same quasiparticle approximation toward zero energy can give the wrong response.

The term describes a failure of one framework, rather than one unique replacement theory. A one-dimensional Luttinger liquid provides a controlled example: its long-wavelength excitations are collective density modes, and its correlation functions need not have the free-fermion form. An anomalous finite-temperature transport curve alone does not establish a particular zero-temperature non-Fermi liquid. Spectral and thermodynamic measurements must be interpreted alongside the relevant dimensionality, interactions and accessible energy range.

A pairing gap is an energy scale for fermionic excitations above a paired superfluid or superconductor ground state. In a simple isotropic paired system the quasiparticle dispersion has the form $E_{\mathbf{k}}=\sqrt{{\xi }_{\mathbf{k}}^2+|\Delta {|}^2}$, where ${\xi }_{\mathbf{k}}$ measures the unpaired single-particle energy relative to the chemical potential. A nonzero minimum gap gives exponentially small fermionic thermal occupations when $k_{B}T\ll |\Delta |$. If the gap varies with direction and has nodes, gapless directions can instead produce power-law thermodynamics. Collective excitations must be counted separately from the gapped fermionic quasiparticles.

Since every quasiparticle energy is positive, the ground state is the vacuum of all the ${\alpha }_p$ annihilation operators, including ${\alpha }_0=c_0$. Its zero-momentum mode is therefore empty. For each positive momentum, the inverse Bogoliubov transformation givesBoth annihilate the normalized pair state $(u_p+i{v}_p{c}_p^{†}{c}_{-p}^{†})|0⟩$. For example, applying ${\alpha }_p$ produces the two cancelling terms $i{u}_p{v}_p{c}_{-p}^{†}|0⟩$. ThusThe factors have even fermion parity and commute for disjoint momentum pairs. Each normalized factor is a superposition of zero and two fermions, whose pair-occupation probability is $v_p^2$. ThereforeThere is no zero-mode contribution. For fixed $L$, the sum converges since $v_p^2=O(p^{-4})$.

A normalized simple excited state is $|k⟩={\alpha }_k^{†}|\Omega ⟩$ for $k>0$. Its excitation energy is $E_k$, and its total momentum is $k$: both terms in ${\alpha }_k^{†}=u_k{c}_k^{†}+i{v}_k{c}_{-k}$ add momentum $k$. Acting on the $k,-k$ pair givesThe affected pair now contains exactly one fermion instead of mean number $2v_k^2$; all other pairs are unchanged. ConsequentlyThis is the fermion number of a Bogoliubov excitation. Its change in mean fermion number lies between zero and one for $\mu >0$, with the value one at $\Delta =0$. The Bogoliubov quasiparticle coherently mixes adding a fermion and removing the opposite-momentum fermion; it is not an eigen-excitation of the original fermion number operator. Indeed,so fermion number is not conserved when $\Delta \ne 0$, while fermion parity is conserved. The generally noninteger change is an expectation value in a coherent paired background, rather than a fractional eigenvalue of the fermion number operator. In the weak-pairing limit at fixed $k$, it tends to one, as expected for an ordinary added fermion.

Let $N_c=c^{†}c$, $N_d=d^{†}d$ and $P=cd+d^{†}{c}^{†}$. Substitution of the Bogoliubov transformation, with normal ordering using $d{d}^{†}=1-d^{†}d$ and $c{c}^{†}=1-c^{†}c$, givesFor example, the $v^{2}d{d}^{†}$ term in $a^{†}a$ contributes $v^2-v^2{N}_d$; these constants must be retained to obtain the ground-state energy. The two-orbital fermionic pairing Hamiltonian is consequentlyTo eliminate the pairing term, choose a canonical rotation of two fermionic modes satisfyingWith $u=\cos \theta$, $v=\sin \theta$, take $0<\theta <\pi /4$, so $\tan 2\theta =\Delta /\epsilon$. Setting $E=\sqrt{{\epsilon }^2+{\Delta }^2}>0$ givesThis positive choice fixes a convenient convention for the vacuum state in part (d). It yieldsBoth number operators independently have eigenvalues $0,1$. Hence the energy eigenvalues and state degeneracies areThere is one fermionic Bogoliubov quasiparticle of energy $E$ per occupied transformed orbital. The two Nambu spinor matrix eigenvalues $\pm E$ should not be mistaken for two independently occupiable levels of opposite energy; the normal ordering constant and the two positive quasiparticle excitation costs give the four physical energy eigenvalues above.

An independent diagonalization of a matrix checks both the pairing sign and the additive constant. In the ordered orthonormal basis $|0⟩,|11⟩$, the even-fermion parity block isbecause $ab|11⟩=-|0⟩$ and $b^{†}{a}^{†}|0⟩=-|11⟩$. Its characteristic polynomial is $x^2-2\epsilon x-{\Delta }^2$, giving $x=\epsilon \pm E$. The odd-fermion parity Fock states $a^{†}|0⟩,b^{†}|0⟩$ are unaffected by pairing and both have energy $\epsilon$.

Equivalently, with the Nambu spinor $\Psi =(a,b^{†}{)}^{\mathsf{T}}$,Replacing $\Psi$ by $R(c,d^{†}{)}^{\mathsf{T}}$ gives $E{c}^{†}c-Ed{d}^{†}+\epsilon =E(N_c+N_d)+\epsilon -E$, reproducing the result by normal ordering.

The canonical anticommutation relations make $(a^{†}{b}^{†}{)}^2=0$, so the proposed exponential function terminates exactly:Use the inverse Bogoliubov transformation $c=ua-v{b}^{†}$, $d=v{a}^{†}+ub$, and the actions $a|11⟩=b^{†}|0⟩$ and $b|11⟩=-a^{†}|0⟩$. The two vacuum state conditions becomeBoth therefore require $B=v/u$. Normalizing this quantum state gives $1=A^2(1+B^2)$. Choosing positive $A$, which fixes an irrelevant overall global phase, yields $A=u$ because $u^2+v^2=1$. Thus the two-mode fermionic paired vacuum isThe other sign for $A$ describes the same physical quantum state. The positive relative coefficient is fixed by the paper's pairing order and the convention $|11⟩=a^{†}{b}^{†}|0⟩$.

As a direct eigenvector check, $\Delta v=(E-\epsilon )u$ and $\Delta u=(E+\epsilon )v$ implyIt is therefore the unique ground state, as also follows from positive quasiparticle excitation energy $E$. It has definite even fermion parity but is not an eigenstate of the original total number operator: the empty-pair and occupied-pair probabilities are $u^2$ and $v^2$. In particular $⟨N_a⟩=⟨N_b⟩=v^2$. The pair amplitude is $⟨ab⟩=-uv$, consistently with $ab|11⟩=-|0⟩$.

For the limiting check $\Delta \to 0^{+}$ at fixed positive $\epsilon$, $A\to 1$, $B\to 0$, so $|\tilde{0}⟩\to |0⟩$. For $\epsilon \to 0^{+}$ at fixed positive $\Delta$, $A\to 1/\sqrt{2}$, $B\to 1$, giving equal amplitudes for the empty and occupied pair. These checks also fix which branch of the Bogoliubov transformation produces the ground state.

In a conventional superconductor, an attractive pairing interaction forms Cooper pairs and a phase-coherent condensate. The elementary fermionic excitations have a superconducting energy gap $\Delta$; breaking a pair into two excitations costs at least $2\Delta$. The density of thermally excited quasiparticles is exponentially small for $k_{B}T\ll \Delta$. Nevertheless the vanishing DC electrical resistivity below the critical temperature is a property of the coherent supercurrent, not a claim that all thermal quasiparticles disappear. Below the relevant critical-current and critical-field limits, a homogeneous bulk sample carries dissipationless current; above the superconducting transition the normal resistivity returns.

In a semiconductor, the band gap separates filled valence states from empty conduction states because of the periodic crystal potential and orbital hybridization. Thermally exciting carriers across the gap produces both conduction electrons and valence holes. For an intrinsic sample,Thus its electrical resistivity generally rises rapidly on cooling, with an activated factor $e^{E_g/(2k_{B}T)}$ up to mobility and density-of-states prefactors. This differs from the zero-resistivity condensate response. Dopant freeze-out and extrinsic-carrier regimes require their own activation scales; the intrinsic formula is not universal for every doped sample.

Write $Q=c^{†}c+d^{†}d$ and $P=cd+d^{†}{c}^{†}$. Expanding the number operators and restoring the displayed order with the canonical anticommutation relations givesThe minus sign in the first line follows from $c^{†}{d}^{†}+dc=-(cd+d^{†}{c}^{†})$. Substitution into the Hamiltonian operator yieldsLet $E=\sqrt{{\epsilon }^2+{\Delta }^2}>0$. A convenient real Bogoliubov transformation hasThese obey $u^2+v^2=1$, $u^2-v^2=\epsilon /E$ and $2uv=\Delta /E$. Equivalently $u=\cos \theta$, $v=\sin \theta$ with $0<\theta <\pi /4$ and $\tan (2\theta )=\Delta /\epsilon$. The coefficient of $P$ vanishes, while the coefficient of $Q$ is $E$. The constant isThus the diagonalized two-mode fermion pairing Hamiltonian isThe two transformed fermionic occupation numbers independently take values zero or one. The energy eigenvalues and their eigenspace dimensions are thereforeThe positive quasiparticle energy $E$ makes the transformed vacuum state the unique ground state. The constant $\mu$ is essential for the correct ground-state energy.

As a separate sign and spectrum check, use the original fermionic Fock space basis. Both singly occupied states are energy eigenstates with energy $\epsilon$, since the pairing terms annihilate them. In the ordered even-fermion parity basis $(|0⟩,a^{†}{b}^{†}|0⟩)$,The off-diagonal signs follow from $b^{†}{a}^{†}=-a^{†}{b}^{†}$ and $ab{a}^{†}{b}^{†}|0⟩=-|0⟩$. Its characteristic polynomial is $z^2-2\epsilon z-{\Delta }^2$, giving the remaining two eigenvalues $\epsilon \pm E$, in agreement with the Bogoliubov transformation.

At the critical field,The quasiparticle gap closes and the low-energy branch is linear, unlike a massive excitation. This is the Ising-chain quantum critical point of the Transverse-field Ising model; the cusp in the nonnegative dispersion represents the two opposite propagation directions.

Expanding $r_k$ at large positive $g$ gives $r_k=g-\cos k+{\sin }^{2}k/(2g)+O(g^{-2})$, soThe branch is strongly gapped and relatively flat, with width approaching $4J$ against a mean energy of order $2Jg$. These are field-polarized spin-flip quasiparticles with a small relative exchange modulation. Averaging the expansion over momentum also gives $E_0/N=-J[g+1/(4g)+O(g^{-3})]$.

At zero field, $r_k=1$ and ${ϵ}_k=2J$ for every $k$. The Ising-chain quasiparticle dispersion is flat: the bulk quadratic excitation costs no momentum-dependent energy. In spin language these are domain-wall excitations. A periodic finite spin configuration has an even number of domain walls, so the single bulk quasiparticle energy $2J$ does not imply an allowed isolated one-wall spin state; the first such closed-chain spin excitation costs $4J$.

Write $a_k=g-\cos k$, $s_k=\sin k$ and $r_k=\sqrt{a_k^2+s_k^2}$. The Ising-chain Bogoliubov diagonalization uses the eigenvalues $\pm r_k$ of the two-by-two Nambu matrix. For distinct paired modes $k\ne -k$, choose $\cos {\theta }_k=a_k/r_k$, $\sin {\theta }_k=-s_k/r_k$ and ${\theta }_{-k}=-{\theta }_k$. The rotation $U_k=e^{-i{\theta }_k{\sigma }_y/2}$ satisfies $U_k^{†}{M}_k{U}_k=r_k{\sigma }_z$, and its first transformed component isThe opposite signs of the paired angles preserve the canonical anticommutation relations. Self-paired spinless fermion modes have zero pairing and are treated directly as occupied or empty number levels; they do not require this paired-angle formula.

Both $k$ and $-k$ occur in the sum, so the physical quasiparticle coefficient is $2J{r}_k$, not $J{r}_k$. ConsequentlyThis is the bulk quadratic result under the stipulated boundary simplification. Restoring the finite-chain parity sectors adjusts allowed modes and global excitation constraints. In the thermodynamic limit, the ground-state energy per site isFor $g\ge 0$, the bulk quasiparticle gap is $2J|g-1|$. The requested three regimes are shown in the original sketch and discussed separately below.