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When spin does not split the orbital energies, a fermionic orbital can contain $2s+1$ particles with distinct spin projections. Fill these complete states in order of increasing energy to find a noninteracting many-particle ground state. Pauli exclusion does not require different spatial orbitals for particles with different available spin states. Imposed spin polarization or interactions can change the filling.

Near the first addition threshold of a repulsive single-orbital defect, the empty and singly occupied grand weights are 1 and $2e^{\beta (\mu -ϵ)}$. Equal weights occur at $\mu =ϵ-k_{B}T\ln 2$. Near the second threshold, the single/double aggregate weights are in ratio $2:e^{\beta (\mu -ϵ-U)}$, giving half-step position $ϵ+U+k_{B}T\ln 2$. These are entropic finite-temperature shifts, which vanish in energy units as $T\to 0$. They must still be retained when evaluating statistics exactly at an unshifted degeneracy point.

A spin gap is a nonzero minimum energy required to reach a specified magnetic excitation above the ground state. One must specify which spin sector or measurement couples to that excitation; not every excitation gap is a spin gap.

Two antiferromagnetically coupled spin-$1/2$ particles illustrate it. For $H=J{\mathbf{S}}_1\cdot {\mathbf{S}}_2$ with dimensionless spin operators and $J>0$, the singlet has energy $-3J/4$ and the triplet has $J/4$, giving gap $J$. Thermal magnetic response is suppressed when $k_{B}T$ is well below that scale. Extended systems require their collective excitation spectrum rather than this isolated-pair argument alone.

A spin glass has frozen, irregular local spin magnetization without the periodic magnetic order of a simple ferromagnet or antiferromagnet. Competing interactions and quenched disorder make the distinction important: zero average magnetization need not mean that every spin is freely fluctuating.^[ref]

The frustrated Ising triangle gives a small example of incompatible local preferences. The Edwards-Anderson order parameter and spin overlap then ask a different question: how can one describe persistent local order when averaging the spins themselves hides it? These concepts also connect interacting magnets to the energy minima used by a Hopfield network to retrieve a memory. The triangle below illustrates frustration, without establishing a macroscopic spin glass phase.

Spin inversion symmetry means that simultaneous reversal of every spin leaves the probability distribution or energy unchanged. For a finite spin model with this symmetry, the weights of opposite configurations pair up, so every observable that changes sign under the reversal has zero expected value. An applied magnetic field generally breaks this symmetry.

Spin isomers of hydrogen are two forms of molecular hydrogen, ${}^1{\mathrm{H}}_2$, distinguished by the combined nuclear spin of their two protons. They have the same chemical formula and electronic ground state, but different allowed molecular rotation states. Parahydrogen has total nuclear spin zero and even rotational quantum number $J$; orthohydrogen has total nuclear spin one and odd $J$. The difference makes heat capacity a probe of fermionic exchange symmetry, and makes cooling history matter for a tank of molecular hydrogen.

Assume the electronic ground state $X{}^1{\Sigma }_g^{+}$ and the vibrational ground state of a free molecule. The electronic factor is symmetric under exchanging the two protons, and the vibrational factor depends only on their separation, so it is also symmetric. These assumptions concern the exchange of protons; the electronic wavefunction separately obeys fermionic exchange symmetry under exchange of electrons. Neglect electronic excitation, vibrational excitation, external fields, and small interactions involving nuclear spin when calculating rotational energy eigenvalues.

The two protons are identical particles with spin $1/2$. Their complete wavefunction must change sign under their particle exchange operator. Interchanging the two ends reverses the internuclear direction $\hat{\mathbf{r}}$, and the spherical harmonic rotational factor satisfiesThe antisymmetric spin-one-half singlet state therefore requires symmetric, even-$J$ rotation; the symmetric spin-one-half triplet state requires antisymmetric, odd-$J$ rotation. This is exchange symmetry of two identical spin-one-half fermions applied inside a diatomic molecule, not an extra rotational force. A linear rigid rotor may omit spin-dependent splittings while still requiring this restriction on its allowed quantum states.

A spin ladder is a system of coupled chains of spins, with interactions along the chains and between neighboring chains. The chains are called legs and their transverse couplings are called rungs; the model provides a controlled step between one-dimensional and two-dimensional magnetism.

In a two-leg spin-$1/2$ ladder with strongly antiferromagnetic rung coupling, pairs on each rung approximately form singlets. Exciting a rung to a triplet costs finite energy, illustrating a possible spin gap. Different couplings and leg numbers need not share that behavior. Magnetic response and excitation spectra test the model rather than the ladder geometry alone fixing the ground state.

Exchange of energy between magnetic spin degrees of freedom and the surrounding lattice, allowing their temperatures to approach thermal equilibrium. It permits adiabatic demagnetization to cool a solid and its attached load.

Each allowed wavevector occupies area $(2\pi {)}^2/A$ under periodic boundary conditions, for each independent spin state. For parabolic dispersion, the number below energy $ϵ$ is $g_{s}Amϵ/(2\pi {\hslash }^2)$. Thus the often-quoted coefficient $Am/(2\pi {\hslash }^2)$ is per spin. Unpolarized electrons have $g_s=2$; the same convention must be used in particle counting and the Fermi energy. A linear Dirac dispersion has a different energy dependence.

Spinodal decomposition is spontaneous growth of composition fluctuations in a homogeneous mixture whose local mixing free energy has negative composition curvature. For a mean $\varphi$, equal regions at $\varphi \pm \delta \varphi$ change the mean free energy by $f^{\prime \prime }(\varphi )(\delta \varphi {)}^2/2+O((\delta \varphi {)}^4)$, which is negative inside the spinodal. Interfacial costs suppress sufficiently short wavelengths, while conservation of composition constrains the dynamics. Outside the spinodal but inside the binodal, the uniform state is instead metastable and requires nucleation to begin separation.

A spinodal point is the limit of local metastability of a phase. At a mean-field spinodal, a local minimum merges with a neighboring maximum and the corresponding second derivative of the free energy vanishes.

Within the $l=1$ multiplet, $|1,0⟩$ has energy ${\hslash }^2(a_x+a_y)$. The symmetric and antisymmetric combinations of $|1,1⟩$ and $|1,-1⟩$ have the other two pairwise coefficient sums. Principal-axis permutation relates these energies; equality requires corresponding coefficient degeneracies rather than angular-momentum conservation alone.

For a spin-one-half ensemble, spin polarization measures unequal occupation or coherent alignment of spin states. A spin-dependent energy splitting does not alone determine the ensemble's response: occupation, temperature and spin relaxation matter. A spin polarization is distinct from a net change of charge density and from the optical polarization of waves.

For locally linear electron confinement potential energy $U(y)$, two Zeeman splitting branches cross the same Fermi energy at positions differing by the energy splitting divided by $|U^{\prime }|$. The expression neglects self-consistent electrostatic screening and exchange interaction enhancement. If the quoted slope is an electrostatic voltage gradient instead of an electron-energy gradient, its magnitude must be multiplied by the elementary charge. A screened compressible quantum Hall edge strip is not a single sharp crossing with an imposed bare slope.

For the static energy $E=\int [{\varphi }^{\prime 2}+W^{\prime }(\varphi {)}^2]dx/2$, completing the square givesThe Bogomolny bound is the absolute boundary difference. Equality requires the Bogomolny equation ${\varphi }^{\prime }=\pm W^{\prime }(\varphi )$ with the sign selected by that difference. Differentiating it gives the Euler-Lagrange field equation ${\varphi }^{\prime \prime }=W^{\prime }{W}^{\prime \prime }$. A stationary solution of that second-order equation need not saturate the bound in a general field theory.

For $V(\varphi )=m^2{\varphi }^2/2-\mu {\varphi }^3/6+\lambda {\varphi }^4/24$, positive $\lambda$ makes the classical potential bounded below. For $m^2\ge 0$, completing the square givesThus zero is a global minimum precisely when ${\mu }^2\le 3\lambda m^2$. At a nontrivial equality there are degenerate minima at zero and $2\mu /\lambda$. If $m^2>0$ and the inequality fails, zero is a false vacuum: its Hessian is positive but another minimum has lower energy. A negative quartic coupling, or a nonzero cubic term without a quartic term, is unbounded below. These are classical or tree-level criteria.

ConsiderThe perturbation connects the oscillator ground state only to the number state $|r⟩$. Its squared matrix element is $(\hslash \omega {)}^{2}r!$, and the energy denominator is $-r\hslash \omega$. Thus second-order nondegenerate perturbation theory gives the formal shift $-{\lambda }^2\hslash \omega (r-1)!$. A finite formal coefficient does not establish that the full Hamiltonian has a ground state.

For $r>2$, a normalized coherent state with label $R{e}^{i\theta }$ hasFor nonzero $\lambda$, choose the phase so that the last term is negative. It dominates as $R\to \infty$, proving that the Hamiltonian is unbounded below on these trial states. The perturbative vacuum branch cannot be an exact lowest-energy state.

For $r=2$, put $Q=(a+a^{†})/\sqrt{2}$ and $P=(a-a^{†})/(i\sqrt{2})$, so $[Q,P]=i$. ThenIt is bounded below for $|\lambda |\le 1/2$. A positive-frequency oscillator and a normalizable ground state require the strict inequality $|\lambda |<1/2$, giving frequency $\omega \sqrt{1-4{\lambda }^2}$. At $\lambda =1/2$ only $Q^2$ remains; at $\lambda =-1/2$ only $P^2$ remains. Their zero infimum is not attained by a normalizable vector. This is the zero-mode obstruction to bosonic Bogoliubov diagonalization, rather than unboundedness below.

For $r=1$, completing the displacement gives $H_1=\hslash \omega [(a^{†}+\lambda )(a+\lambda )+1/2-{\lambda }^2]$, which has a normalizable displaced oscillator ground state for every real $\lambda$.

In static coordinates on anti-de Sitter spacetime, write $f=1+r^2/L^2$, conserved energy per unit rest mass $E=f\dot{t}$, and conserved specific angular momentum $h=r^2\dot{\varphi }$. Timelike normalization gives ${\dot{r}}^2=E^2-V_{\mathrm{eff}}$ with $V_{\mathrm{eff}}=f(1+h^2/r^2)$. A circular geodesic at $r_0>0$ obeys $V_{\mathrm{eff}}^{\prime }=0$, so $h^2=r_0^4/L^2$ and $E=f(r_0)$. Thus $\dot{t}=1$ and $\dot{\varphi }=\pm 1/L$. Both its static-coordinate period and its own proper time period are $2\pi L$, independent of radius. Stability follows from $V_{\mathrm{eff}}^{\prime \prime }(r_0)=8/L^2>0$. A static clock at the orbital radius measures the different period $2\pi L\sqrt{f(r_0)}$, which does depend on radius.

A pair potential is stable when its total interaction energy obeys this lower bound for every allowed configuration, with a constant $B$ independent of $N$. It prevents collapse with an arbitrarily negative energy per particle. Stability alone does not control the long-distance contribution needed for the ordinary thermodynamic limit.

An unpaired electron does not set a universal reaction rate. A stable radical can retain radical character during isolation and storage under specified conditions; a persistent radical survives long enough under specified conditions to be observed. These descriptions concern lifetime and handling, rather than immunity to every possible chemical reaction. DPPH is a practical counterexample to the claim that all free radicals react immediately.

Two mechanisms help explain persistence. Spreading electronic character over several chemical atoms, represented by chemical resonance, can lower the energy relative to a localized description. Bulky surrounding groups can impede approach to a reactive site and slow recombination. Lower energy and slower chemical kinetics are different advantages: neither alone guarantees a long lifetime. A primary computational study explicitly separates these effects, using electronic and geometric descriptors; it does not directly measure every predicted lifetime.^[ref]