Quantum magnetism

Quantum magnetism studies collective magnetic behavior in which quantum spin states, noncommuting observables and quantum fluctuations matter. A lattice of local spins can be modeled by a Heisenberg model, whose exchange interaction favors particular relations between neighbours. Its states need not resemble a set of classical arrows pointing in definite directions.

A two-spin spin singlet state illustrates the difference: it has zero total spin while retaining nontrivial spin correlations, rather than two independently known opposite arrows. Large ordered magnets can support spin waves, whereas strong quantum fluctuations can invalidate an expansion around a chosen ordered state. Measurements of magnetic susceptibility, heat capacity and excitation spectra probe different aspects of the same interacting system. The subject is therefore about both collective order and correlated states that resist a simple ordered reference picture.

The spin-one-half quantum XX chain has equal nearest-neighbour transverse exchange. Its continuous spin rotation symmetry conserves total longitudinal magnetization. The Jordan–Wigner transformation maps it to number-conserving free fermions. This quantum chain is distinct from the classical XY model.

For exchange $J>0$, the Jordan–Wigner transformation gives a band ${ϵ}_k=h-J\cos k$. At zero temperature, negative-energy states form a filled interval $|k|<\arccos (h/J)$. Since each fermion decreases total magnetization by one, the thermodynamic magnetization per site is $1/2-\arccos (h/J)/\pi$. It saturates at $1/2$ for $h\ge J$. A finite ring instead has discrete level crossings and parity-dependent momentum grids.

Geometrical frustration occurs when an interaction graph prevents all local preferred configurations from being satisfied simultaneously. For classical equal-length vector spins with positive antiferromagnetic exchange, a bipartite graph permits opposite vectors on its two sublattices, minimizing every bond. A triangle cannot make all three pairs antiparallel.

For three classical vectors of length $S$ on an equal-coupling triangle,The minimum has zero vector sum, hence a coplanar $120$-degree arrangement. For Ising spins on the same triangle, at least one bond remains unsatisfied. The allowed spin space matters.

The vector picture is classical. Three quantum spin-half degrees of freedom cannot have total spin zero; their frustrated states require a Quantum Hamiltonian treatment. Geometric frustration can enhance competition and low-energy ground-state degeneracy, but does not by itself prove a quantum spin liquid or a particular ordered state.

A quantum spin liquid is a phase of interacting quantum spins whose ground-state behavior avoids conventional magnetic order through quantum fluctuations and can exhibit long-range quantum entanglement and fractionalized excitations. Absence of a measured average magnetization alone is insufficient: thermal disorder or averaging over ordered states can also give a zero average.

An exactly solvable example is the honeycomb model with bond-dependent $XX$, $YY$ and $ZZ$ interactions. Its solution expresses the spin problem using fermions coupled to a discrete gauge field and yields phases whose excitations are anyons. Such a model makes fractionalization precise, beyond the informal picture of spins merely continuing to fluctuate.^[ref]

The single-band Hubbard model balances electron hopping between localized spatial orbitals against local interaction. A common form isThe first sum is over neighbouring unordered site pairs. The parameter $t$ sets hopping, while $U_H>0$ penalizes double occupancy. At $U_H=0$ electrons occupy independent energy bands; strong repulsion suppresses local charge fluctuations and can favour local-moment behaviour. This competition was the motivation for the model's treatment of narrow-band electron correlation.^[ref]

At one electron per site and $U_H\gg |t|$, virtual double occupancy yields the antiferromagnetic scale $J_{\mathrm{ex}}=4t^2/U_H$. The two-site Hubbard dimer makes this concrete: the singlet state is lowered by $(\sqrt{U_H^2+16t^2}-U_H)/2$, while triplet states have no such shift. Its small-hopping limit gives Hubbard kinetic exchange.

Interaction conventions must match. For dimensionless spin-half operators, $-2U({\mathbf{S}}^2-3n/4)=3U{n}_{\uparrow }{n}_{\downarrow }$, so that spin-square convention has $U_H=3U$, not $U_H=U$.

An effective low-energy lattice model obtained from a strongly repulsive Hubbard model near one electron per site. The retained Hilbert space excludes double occupancy. Its usual nearest-neighbor Quantum Hamiltonian has projected hopping $-t{\sum }_{⟨ij⟩,\sigma }({\tilde{c}}_{i\sigma }^{†}{\tilde{c}}_{j\sigma }+h.c.)$ and antiferromagnetic superexchange $J{\sum }_{⟨ij⟩}({\mathbf{S}}_i\cdot {\mathbf{S}}_j-n_i{n}_j/4)$. Virtual intermediate states with one double occupancy cost energy $U$, giving $J=4t^2/U$ to leading order. Three-site hopping terms of the same expansion order are omitted in this simplified version.

At half filling the projected hopping is blocked. Doped holes restore charge motion coupled to spin correlations. Nearest-neighbor singlet pairing can support a d-wave bond order ansatz on a square lattice, but the Quantum Hamiltonian alone is not a proof of a unique superconducting ground state for all parameters.

An electron annihilation operator restricted to the no-double-occupancy sector of the t-J model. It removes a spin-$\sigma$ electron only when the opposite-spin site occupation vanishes. The corresponding creation operator cannot create a doubly occupied site. These projected operators do not obey the unrestricted canonical on-site fermion algebra on all states.

For spin imbalance $s=n_{\uparrow }-n_{\downarrow }$ per site, the uniform mean-field interaction energy is $U(n^2-s^2)/4$. With noninteracting density of states $D$ per spin per site, the kinetic cost is $s^2/(4D)$. The quadratic coefficient changes sign at $UD=1$, producing the Stoner mean-field instability. This does not describe every strong-coupling Hubbard phase.

Two electrons of opposite spin occupying the same spatial orbital. The mean occupation product measures its probability at a specified site. In the two-electron dimer, the sum over sites is also the probability of any doubly occupied site, since two sites cannot both be doubly occupied.

The ground-state probability of both electrons occupying one site is $P_{\mathrm{double}}=\frac{1}{2}[1-U/\sqrt{U^2+16t^2}]$. It equals the derivative of the relative ground energy with respect to $U$. It tends to zero for $|t|/U\to 0$ and to $1/2$ for $|t|/U\to \infty$.

Two localized orbitals coupled by hopping, with an on-site penalty for double occupancy. At two-electron filling the competition between hopping and local repulsion can be solved exactly within the finite basis. It gives a compact example of a molecular bonding crossover and electron correlation; it is not by itself a complete real-molecule binding curve.

At one electron per site and $U\gg |t|$, the low-energy singlet of a repulsive two-site Hubbard dimer mixes virtually with doubly occupied configurations. Its energy is $-4t^2/U$ to leading order, while the three triplets have no such shift. With dimensionless spin-half operators this yields the displayed effective antiferromagnetic Quantum Hamiltonian. Restoring angular-momentum operators with units of $\hslash$ requires the corresponding $1/{\hslash }^2$ in the spin product.

For real hopping $t$ and repulsion $U>0$, the covalent singlet and symmetric ionic state have relative Quantum Hamiltonian $\left(\begin{array}{cc}0& 2t\\ 2t& U\end{array}\right)$. The lower energy is $(U-\sqrt{U^2+16t^2})/2$, relative to the common one-electron energy. It approaches $-4t^2/U$ at small hopping, and its state continuously mixes covalent and ionic configurations.

A configuration with both electrons on one center, such as $aa$ or $bb$, in a two-center electronic basis. A coherent superposition of such configurations in a neutral molecule represents charge fluctuations; it does not imply a permanent classical ionic charge assignment.

For orthonormal orbitals $a,b$, the symmetric spatial state $C=(|ab⟩+|ba⟩)/\sqrt{2}$ multiplied by the antisymmetric spin singlet has one electron on each center and no double occupancy. The full state remains antisymmetric under fermion exchange.

The energy cost $U=\iint |a({\mathbf{r}}_1){|}^2|a({\mathbf{r}}_2){|}^2{e}^2/[4\pi {ϵ}_0|{\mathbf{r}}_1-{\mathbf{r}}_2|]{\mathrm{d}}^3{r}_1{\mathrm{d}}^3{r}_2$ for double occupancy of a localized orbital. It is positive for an ordinary nonzero orbital. A model retaining only this integral also neglects intersite and mixed Coulomb terms; orbital orthogonality alone does not make those terms vanish.

An isolated spin-degenerate orbital coupled to a particle reservoir has Quantum Hamiltonian $H=ϵ(n_{\uparrow }+n_{\downarrow })+U{n}_{\uparrow }{n}_{\downarrow }$. The four occupation microstates have weights $1,z,z,q{z}^2$, where $z=e^{\beta (\mu -ϵ)}$ and $q=e^{-\beta U}$. Thus $\Xi =1+2z+q{z}^2$ and $\bar{n}=(2z+2q{z}^2)/\Xi$. The factor two counts distinct spin states. At $U=0$ the grand partition function factorizes into two independent fermionic orbitals; positive U separates the two electron-addition thresholds.

For a repulsive single orbital at temperatures small compared with U, chemical potentials strictly between the first and second addition energies select a singly occupied defect. Its mean occupation is nearly one while its charge-number variance is exponentially small away from the thermally broadened step regions. The remaining spin degeneracy does not itself produce charge-number fluctuations.

With $z=e^{\beta (\mu -ϵ)}$, $q=e^{-\beta U}$ and $\Xi =1+2z+q{z}^2$, direct four-state averaging givesFor $\beta U\gg 1$ it has two separated peaks of height $1/4$ at the spin-degeneracy-shifted charge steps. At the lower bare crossing $\mu =ϵ$, three microstates share equal limiting weight, giving $\bar{n}=2/3$ and variance $2/9$. The upper bare crossing similarly gives mean $4/3$ and variance $2/9$. Between crossings occupation is pinned near one and fluctuations are suppressed. The identity is grand-canonical; fixing total particle number would change the fluctuation question.

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.

For a single spinful fermion orbital, ${\mathbf{S}}^2=3(n-2n_{\uparrow }{n}_{\downarrow })/4$. The contraction term is $3n/4$, so $:{\mathbf{S}}^2:={\mathbf{S}}^2-3n/4=-3n_{\uparrow }{n}_{\downarrow }/2$. Empty and doubly occupied states are spinless while singly occupied states have spin one half, which also proves the identity directly.

Ferromagnetic spin polarization carried by mobile electrons rather than a separate fixed-spin lattice. A homogeneous spin field splits the electron bands and competes with a positive field cost. Mean-field instability is governed by the Stoner criterion; the ordered metal supports transverse spin waves as well as fermionic particle-hole excitations.

Spin reversal in exchange-split bands creates an electron-hole excitation with energy ${ϵ}_{k+q}-{ϵ}_k+2M$ for a majority-to-minority flip in the simple uniform model with $M>0$. Its allowed continuum depends on occupied initial and empty final states. At $q=0$ all such allowed transitions have exchange energy $2M$; finite momentum changes the thresholds. A collective ferromagnetic magnon can remain gapless below this continuum when spin rotation is exact. The continuum is not the same excitation as a longitudinal amplitude mode.

The edges of an itinerant spin-flip particle-hole continuum follow from both excitation energy and occupation constraints. Consider a zero-temperature three-dimensional parabolic-band model with majority states occupied for $|\mathbf{k}|\le k_F$ and an entirely empty minority band. For exchange splitting $2M$ and ${ϵ}_k={\hslash }^2{k}^2/(2m)$, a majority-to-minority flip carrying wavevector $\mathbf{q}$ hasFor $q>0$, the scalar product is bounded by $\pm k_{F}q$, giving continuum edgesThe extremes occur at opposite points of the occupied Fermi surface along $\mathbf{q}$. At $q=0$ the interval collapses to the single exchange energy $2M$.

The empty-minority assumption requires sufficient splitting; otherwise Pauli exclusion principle blocks some final states and the edges must be recomputed over the allowed region. These are independent particle-hole thresholds, not a calculation of the collective ferromagnetic magnon dispersion. A collective mode below the continuum can therefore have a different low-momentum energy scale.

An isotropic spin-polarized metal has a degenerate magnetization direction and a long-wavelength transverse magnon with quadratic dispersion. Nonzero spin density pairs the two transverse broken generators into one type-B Goldstone boson. Exchange-split Fermi surfaces also support fermionic low-energy excitations; the magnon is distinct from longitudinal amplitude motion and the spin-flip continuum.

In the homogeneous spin-vector mean-field scheme with field cost $M^2/(2U)$ and spin-summed density of states per site, the paramagnetic curvature is proportional to $U^{-1}-\int \nu (ϵ)[-n_F^{\prime }(ϵ)]dϵ$. It changes sign at $U{\chi }_0=1$. For a smooth low-temperature density of states, ${\chi }_0\simeq \nu (\mu )$, giving the familiar Fermi-level criterion. Interaction and spin-counting conventions must be specified before comparing coefficients.

For a uniform exchange field with cost $N{M}^2/(2U)$ and spin-band energies ${ϵ}_k-\sigma M$, the grand potential is even in M. Its curvature at zero field is $N[U^{-1}-{\chi }_0]$, with ${\chi }_0=\int \nu (-n_F^{\prime })dϵ$ and spin-summed density of states per site. Negative curvature proves local paramagnetic instability. It does not by itself exclude a first-order transition before that instability, nor determine the stable finite-field amplitude.

For a three-dimensional zero-temperature parabolic Fermi gas, let $\zeta =(N_{\uparrow }-N_{\downarrow })/N$ and $E_F={\hslash }^2(3{\pi }^{2}N/V{)}^{2/3}/(2m)$. Suppose the interaction energy per electron is $U{N}_{\uparrow }{N}_{\downarrow }/N^2=U(1-{\zeta }^2)/4$. The total energy per electron isIts Taylor expansion has quadratic coefficient $E_F/3-U/4$ and positive quartic coefficient $E_F/81$. Hence the unpolarized state loses stability at $U_c=4E_F/3\propto (N/V{)}^{2/3}$. This $U$ has units of energy. It differs from a contact coupling $g$ defined by interaction-energy density $g{n}_{\uparrow }{n}_{\downarrow }$, for which $g=U/n$ and $g_c=4E_F/(3n)$. Normalization must be fixed before comparing critical couplings. For $U\le 4E_F/3$, convexity of $x^{-1/3}$ implies $f^{\prime \prime }(\zeta )=(E_F/3)[(1+\zeta {)}^{-1/3}+(1-\zeta {)}^{-1/3}]-U/2\ge 0$, so the unpolarized minimum is a global minimum; a first-order polarized minimum does not pre-empt this threshold in the stated model.

Let $g_s(E_F)$ be the single-spin density of states per site per energy, and let the on-site Hubbard model repulsion have energy $U$. For a weak uniform magnetic field $H$, the half-splitting of the two spin bands is $\Delta =U\delta n/2+{\mu }_0{\mu }_{B}H$, where $\delta n=n_{\uparrow }-n_{\downarrow }$. Constant density of states gives $\delta n=2g_s\Delta$. ThereforeThis is the response per site; multiplying by the site number density gives the dimensionless SI volume magnetic susceptibility. With spin-summed density of states $g=2g_s$, the denominator is $1-Ug/2$. Its vanishing is the mean-field Stoner criterion. The unpolarized linear-response expression is stable only for $U{g}_s<1$.

For an interaction-energy density $U{n}_{\uparrow }{n}_{\downarrow }$, let $n_{\uparrow }=n/2+\delta$, $n_{\downarrow }=n/2-\delta$ at fixed total carrier density. If $g_s(E_F)$ is the unpolarized single-spin density of states per unit volume, the kinetic-energy cost of the two occupation shifts is ${\delta }^2/g_s(E_F)$, whereas the interaction changes by $-U{\delta }^2$. ThereforeFor the three-dimensional free electron model, $g_{\mathrm{tot}}(E_F)=3n/(2E_F)$ and $U_c=4E_F/(3n)$. Both the interaction normalization and the spin counting in the density of states must be specified when quoting a Stoner criterion.

Integrating free fermions in a constant auxiliary spin field gives $S_{\mathrm{eff}}=\beta N{M}^2/(2U)-{\sum }_{k\sigma }\ln [1+e^{-\beta ({ϵ}_k-\mu -\sigma M)}]$. Differentiation gives the self-consistent spin polarization. The extensive action is a mean-field restriction of the full auxiliary-field functional integral, not an exact omission of all spatial/time fluctuations.

The coordination number $z$ is the number of nearest neighbors of a lattice site. A Kac normalization divides a pair coupling by $z$ so that the interaction energy per site remains finite as $z\to \infty$.

A Néel state is a two-sublattice product state with antiparallel spin polarization on the two sublattices. It is the classical ordered reference state of a bipartite antiferromagnet.

The Néel order parameter is the slowly varying staggered magnetization of an antiferromagnet. On a bipartite lattice it changes sign between neighboring sublattices before coarse graining.

For a linear spin-wave approximation, the ground state is empty of the diagonal quasiparticles but contains the original spin bosons. If $a_k=u_k{\alpha }_k-v_k{\alpha }_{-k}^{†}$, their occupation number is $v_k^2$. The staggered magnetization is reduced from $S$ to $S-N^{-1}{\sum }_k{v}_k^2$. For the one-dimensional nearest-neighbor Heisenberg antiferromagnet, $v_k^2=(|\sin k{|}^{-1}-1)/2$ and the integral diverges logarithmically. This invalidates finite ordered magnetization within the approximation; it is not itself a determination of the exact spectral gap.

At zero temperature, linear-mode vacuum depletion scales as $\int dk{k}^{d-2}$; at nonzero temperature Bose occupation changes this to $\int dk{k}^{d-3}$. The two convergence criteria are different.

For linear spin waves, $\delta S=(2N{)}^{-1}{\sum }_k[(1-{\gamma }_k^2{)}^{-1/2}-1]$. The chain has ${\gamma }_k=\cos k$ and a logarithmic infrared divergence. This signals failure of the ordered expansion, not a physical negative staggered moment.

The staggered magnetization weights the local magnetization with opposite signs on the two sublattices of a bipartite graph. It is nonzero in a Néel state even when the uniform magnetization vanishes.

The uniform magnetization is the nonalternating component of local spin polarization. In a nearly Néel-ordered antiferromagnet it describes the small canting shared by the two sublattices.

Long-range order means that an appropriate order-sensitive correlation function remains nonzero as the separation becomes arbitrarily large in the thermodynamic limit. In a ferromagnet, a spin-spin correlation can approach a positive constant, indicating a persistent relation between distant spins. An antiferromagnet requires an alternating sign convention, captured by staggered magnetization, to reveal its corresponding order.

The choice of average matters. A finite zero-field equilibrium sample can preserve symmetry and have zero mean magnetization, while distant spin correlations still diagnose the approach to an ordered thermodynamic phase. Within a selected symmetry-broken state, the disconnected contribution can carry the order and the connected correlation can instead decay. One must therefore state the observable, state and order of limits before applying a nonzero-correlation criterion.

This also explains a locality constraint: a circuit with bounded depth and bounded-range gates cannot generate arbitrary-distance connected correlations from a product state. Its finite propagation range limits which separated regions can share correlations.

For a two-species uniform boson number state, set $S_i^{+}=a_i^{†}{b}_i$. A single spin ladder operator changes the species occupation numbers, so its expectation value vanishes. At distinct sites, $⟨S_i^{+}{S}_j^{-}⟩=n_A{n}_B/L^2$, proving the displayed transverse two-point correlation function. For $n_A/L,n_B/L$ tending to nonzero constants, this correlation function has a nonzero large-distance limit despite the global U(1) symmetry. This diagnoses long-range order in the auxiliary state; its interpretation as a state of fixed physical spins still requires the Schwinger boson constraint.

An antiferromagnetic spin wave is a small collective variation of the ordered spin pattern in an antiferromagnet. In a bipartite Néel state, neighbouring spins point oppositely; the slowly varying Néel order parameter describes their staggered orientation. A long-wavelength disturbance changes that orientation smoothly rather than independently flipping each site.

For an isotropic ordered antiferromagnet in the usual long-wavelength regime, the low-energy dispersion relation is linear, $\omega \simeq c|\mathbf{k}|$, with spin-wave velocity $c$. This differs from the quadratic low-wavenumber ferromagnetic magnon in a conventional isotropic ferromagnet. Anisotropy or an applied magnetic field can alter the branches or introduce gaps. The starting ordered state also needs justification: strong quantum fluctuations, especially in low dimension, can invalidate a naive linear spin-wave approximation. A computed divergence is then a warning about the approximation, not a negative physical magnetization.

Two sublattices give two bosonic modes for each of N/2 wavevectors in a magnetic Brillouin zone. For a symmetric quadratic pairing block with number coefficient A and anomalous coefficient B, a bosonic Bogoliubov transformation gives two equal positive energies $E_k=\sqrt{A^2-B_k^2}$ and the displayed vacuum shift. There are N modes overall. The quantum depletion of Néel order per site is $(2/N){\sum }_{\mathrm{MBZ}}{v}_k^2$, where $v_k^2=(A/E_k-1)/2$. Equivalently one can use one branch over a full zone with the corresponding factors of one half. Mixing these conventions doubles the vacuum energy or depletion.

The nearest-neighbour isotropic chain has linear-spin-wave energy ${\omega }_k=2JS|\sin k|$ in units $\hslash =a=1$. Gapless points occur at zero and the zone boundary. Integer-spin quantum gaps and strongly fluctuating finite-spin spectra require physics beyond this approximation.

For $H=J{\sum }_{⟨i,j⟩}{\mathbf{S}}_i\cdot {\mathbf{S}}_j$, with $J>0$, dimensionless spin length $S$ and lattice spacing $a$ on a $d$-dimensional hypercubic lattice, the linear spin-wave approximation givesThe angular frequency is ${\omega }_{\mathbf{k}}=E_{\mathbf{k}}/\hslash$. Near a gapless point, $E_{\mathbf{k}}\simeq 2JSa\sqrt{d}|\mathbf{k}|$, so the physical spin-wave velocity is $2JSa\sqrt{d}/\hslash$. Writing the energy formula directly as an angular frequency assumes $\hslash =1$.

The zero-temperature spin-boson occupation is $[1/\sqrt{1-{\gamma }_{\mathbf{k}}^2}-1]/2$, producing quantum depletion of Néel order. Exact zero modes need an infrared cutoff or separate global-rotation treatment. The energy, angular frequency and spin normalization must remain consistent when comparing different conventions.

The spin-wave velocity is the slope $c={\lim }_{k\to 0}\omega (k)/k$ of a linearly dispersing spin wave. For the leading square-lattice Heisenberg antiferromagnet continuum theory, $c=2\sqrt{2}JSa$.

Zeeman splitting is the separation of energy levels or mode frequencies by a background magnetic field. For an antiferromagnetic spin wave, opposite circular spin polarizations receive opposite frequency shifts.

A ferromagnetic spin chain is a one-dimensional array whose exchange interactions favor aligned neighbouring spins. For $J>0$, the nearest-neighbour Heisenberg model conventionhas fully aligned ground states. A collective small deviation from alignment produces a ferromagnetic magnon rather than a set of unrelated local spin flips.

For spin length $S$, lattice spacing $a$ and this exchange convention, the one-magnon energy is $E(k)=2JS[1-\cos (ka)]$, giving $E(k)\simeq JS{a}^2{k}^2$ at long wavelength. The gapless point reflects the freedom to rotate a uniform aligned state. This ground-state and excitation statement does not establish finite-temperature spontaneous order in an infinite one-dimensional short-range model. Finite length, anisotropy and coupling between chains can change what an experiment observes.

Easy-plane anisotropy makes spin orientations within one plane energetically preferable to orientations normal to it. For example, a classical single-site term $D(S^z{)}^2$ with $D>0$ penalizes out-of-plane components and leaves the $xy$ plane degenerate. At sufficiently low energies, an in-plane angle can then be the main orientational degree of freedom.

The resulting approximate angular description connects a magnet to the XY model, but the microscopic form matters. For a quantum spin one-half, $(S^z{)}^2$ is proportional to the identity, so that single-site term alone does not generate orientational anisotropy. Anisotropic exchange interactions provide another route. Stating the spin size and Quantum Hamiltonian prevents transferring a classical mechanism to a quantum model where it is ineffective. Residual in-plane anisotropy can further select a finite set of directions rather than a continuous circle.

A ferromagnetic magnon is a quantum of a collective spin wave about ferromagnetic order. In a uniform isotropic exchange ferromagnet, slowly varying misalignment costs less energy than rapid variation. Its long-wavelength energy is typically quadratic, $E(\mathbf{k})\simeq D{k}^2$, and its angular frequency is $E/\hslash$.

A one-magnon state in a fully polarized magnet reduces total spin projection by one quantum while spreading that change coherently across the sample. It is therefore different from assigning each site an independent classical tilt. The ferromagnetic spin chain gives an explicit cosine dispersion relation whose small-wavenumber limit is quadratic. An applied magnetic field or symmetry-breaking anisotropy can introduce a nonzero excitation gap, while interactions between magnons matter when their population is no longer dilute. The mode picture is most useful when the ordered reference and low-excitation approximation are justified.

For $H=-J{\sum }_n{\mathbf{S}}_n\cdot {\mathbf{S}}_{n+1}$ and unit spacing, the one-magnon energy is $2JS(1-\cos k)=4JS{\sin }^2(k/2)$. The exact one-magnon hopping matrix has diagonal $2JS$ and neighbour entries $-JS$. The same result follows from the linear spin-wave approximation. Its long-wavelength stiffness is $JS$, with energy proportional to $k^2$.

For a periodic spin-one-half Heisenberg model $H=-(J/{\hslash }^2){\sum }_{n=1}^N{\mathbf{S}}_n\cdot {\mathbf{S}}_{n+1}$, $J>0$, the fully aligned state has $E_0=-NJ/4$. In its single-excitation subspace, let $|n⟩$ denote a down spin at site $n$ in an otherwise up background. Direct action of each bond givesThe normalized translation eigenstates are $|q⟩=N^{-1/2}{\sum }_n{e}^{iqna}|n⟩$, with $q=2\pi \ell /(Na)$ modulo $2\pi /a$. Their energy eigenvalues are $E_0+J[1-\cos (qa)]$. This is an exact one-magnon calculation, not a many-magnon harmonic approximation. Near $q=0$ the energy above $E_0$ is $J{a}^2{q}^2/2$.

The wavenumber restriction comes from requiring a periodic plane-wave coefficient pattern, equivalently requiring this vector to be an eigenvector of a one-site translation with eigenvalue $e^{-iqa}$. The identity $T^N=I$ alone imposes no restriction on a general state: it already holds throughout the ring Hilbert space. An arbitrary finite sum with coefficients $e^{iqna}$ still returns after $N$ translations, but need not be a one-site translation eigenstate unless $e^{iqNa}=1$.

Positive nearest- and next-nearest-neighbour exchanges give one-magnon energy $2S[J_1(1-\cos k)+J_2(1-\cos 2k)]$. At small momentum this is $S(J_1+4J_2)k^2$. The quadratic-boson result describes the harmonic many-magnon spectrum; higher-order spin-wave interactions remain in the full Quantum Hamiltonian.

A damped ferromagnetic spin wave has a complex pole $\omega =\pm {\omega }_k-i{\Gamma }_k$. Both poles of its retarded Green function must lie in the lower half of the complex frequency plane so that the response respects causality.

The spin coherent-state path integral contains a Berry phase equal, modulo gauge choices, to the spin $S$ times the solid angle swept on the Bloch sphere. Single-valued path-integral amplitudes require $2S\in \mathbb{Z}$.

For a spin coherent-state history, a Wess--Zumino term records the oriented target-sphere area bounded by the spin path. Its variation is local even though a single gauge-patch expression for the area need not be global.

In a bipartite antiferromagnet, the smooth staggered contributions to the spin Wess--Zumino terms cancel between sublattices. Their term linear in uniform canting is proportional to $\mathbf{m}\cdot (\mathbf{n}\times {\partial }_t\mathbf{n})$.

The CP1 spinor representation writes a unit vector as $n_i=z^{†}{\sigma }_{i}z$ with $z^{†}z=1$. The phase redundancy $z\mapsto e^{i\alpha }z$ identifies normalized spinors that represent the same point of the two-sphere.

The ferromagnetic Wess–Zumino term is the first-order kinetic term $i\kappa z^{†}\dot{z}$ in the CP1 description. A local phase change of $z$ changes it only by the total derivative $-\kappa \dot{\alpha }$.