Spontaneous Symmetry Breaking and Nambu-Goldstone Bosons in Quantum Many-Body Systems

1. Introduction

• The review focuses on continuous global internal symmetries without Lorentz invariance, covering intrinsically nonrelativistic systems and relativistic systems at nonzero density.The author calls both cases nonrelativistic for simplicity unless otherwise indicated.Source: §1 · p. 2
• Its progression runs from SSB and Hilbert-space subtleties through a free nonrelativistic field, Higgs-potential minimization, Goldstone theorems and counting, examples, and low-energy effective field theory.The Higgs-potential discussion includes a spin-one color superconductor; spacetime symmetry breaking is not covered.Source: §1 · p. 2
• The review omits thorough applications and detailed examples in realistic interacting quantum field theories, using general principles and model-independent effective field theory instead.The author notes that realistic theories would often require approximation schemes whose physical status must be checked; most examples are sometimes exactly solvable.Source: §1 · pp. 2–3
• The author identifies as new an exactly solvable example with a Nambu–Goldstone mode whose energy scales as a noninteger power of momentum, and an improved proof of a counting theorem.The fractional-power mode's nonanalytic long-distance behavior precludes a description by conventional effective field theory; the author also calls the free nonrelativistic particle's type-II identification novel.Source: §1 · p. 3
• The review draws chiefly on existing literature and points readers to prior reviews of Nambu–Goldstone bosons and their effective field theory.It also includes specialized reviews of relativistic chiral perturbation theory.Source: §1 · p. 3

2.1. Ground state and finite symmetry transformations

• An order parameter formalizes SSB: a translationally invariant ground state has a nonzero expectation value for its commutator with an operator that transforms under the symmetry.The construction of finite transformations of observables assumes microcausality; for nonrelativistic systems, the argument can apply when interactions are sufficiently short-ranged, typically with exponentially decaying correlations.Source: §2.1 · pp. 4–5
• In infinite volume, the integral charge and its finite symmetry transformation are not well-defined operators, and symmetry-related ground states become orthogonal.The associated excited states likewise belong to distinct Hilbert spaces, though each gives an equivalent description of the system.Source: §2.1 · p. 4

2.2. Explicit symmetry breaking and the choice of the ground state

• A weak external perturbation selects a vacuum with definite order parameter when the volume is large enough, and the infinite-volume and zero-perturbation limits cannot be reversed.At finite volume, boundary conditions can lift the degeneracy; in infinite volume the perturbation's energy difference outweighs the intrinsic level splitting.Source: §2.2 · pp. 5–6

2. Basic properties of spontaneously broken symmetries

• SSB is marked by an order parameter with nonzero vacuum expectation value, and its continuously many degenerate ground states form distinct, unitarily inequivalent Hilbert-space representations.The associated Nambu–Goldstone realization lacks the multiplet structure of the ordinary symmetry realization and has soft modes in the spectrum.Source: §2.2 · p. 6

3. Example: Free nonrelativistic particle

• Quantization gives the ISO(2) charges a central term, [Q1, Q2] = 2iΩ, although the corresponding classical translations commute.The model is quantized in finite volume to illustrate SSB and its different technical treatments.Source: §3 · pp. 6–7

3.1. Hilbert space and inequivalent ground states

• Coherent states formed by adding zero-momentum quanta have the Fock vacuum's energy, but symmetry-related ground states become orthogonal in the infinite-volume limit.The charge operators Q1 and Q2 are ill-defined on the Fock-vacuum Hilbert space in that limit.Source: §3.1 · p. 8

3.2. Spontaneous symmetry breaking and the Nambu–Goldstone boson

• The Fock vacuum breaks the two ISO(2) translations while preserving rotations, and a single one-particle state couples to the broken generators.With Ek = k^2/2m, that state is a type-II Nambu–Goldstone boson; its single mode couples to two broken generators because their commutator has nonzero vacuum expectation value from a central charge.Source: §3.2 · p. 9
• A nonzero coherent-state field expectation describes a finite-density Bose–Einstein condensate, with different choices of z corresponding to different density and/or phase.The paper selects a ground state by adding an explicit symmetry-breaking term, taking infinite volume, and then removing the term; in this example a negative chemical potential selects the Fock vacuum.Source: §3.2 · pp. 9–10
• For the same ISO(2) breaking pattern, the free relativistic massless complex scalar has two linearly dispersing Nambu–Goldstone bosons rather than the nonrelativistic theory's single quadratic mode.In the relativistic theory, particle and antiparticle contributions cancel in the shift-generator commutator, so no central charge develops.Source: §3.2 · pp. 9–10

4. Achieving spontaneous symmetry breaking: Minimization of Higgs potentials

• Finding an order parameter is difficult because spontaneous symmetry breaking is nonperturbative; its calculation can use self-consistent equations of motion or an effective field.The self-consistent route is technically harder and typically requires checking an Ansatz against the equations of motion.Source: §4 · pp. 10–11

4.1. Group action on the order parameter space

• Group orbits collect order-parameter values related by symmetry, while strata collect values with conjugate little groups and therefore the same symmetry-breaking pattern.An invariant potential can be minimized as a function on the space of orbits.Source: §4.1 · pp. 11–12
• Michel’s theorem identifies critical orbits when an orbit is isolated in its stratum, while the conjecture that absolute minima have maximal little groups does not hold generally.For fixed-norm order parameters, inert states have single-orbit strata and are stationary points of any invariant function.Source: §4.1 · pp. 12–13

4.2. Minimization of Higgs potentials

• The target space comprises allowed condensate-orientation invariants, independent of couplings; minimizing a linear function over it locates the absolute potential minimum.The target-space shape reflects the symmetry group’s algebraic structure and representation.Source: §4.2 · pp. 13–14, Figure 1 · p. 14
• As quartic couplings vary, concave boundary segments yield abrupt first-order changes, whereas convex segments allow continuous evolution through an intermediate phase bounded by second-order transitions.With two quartic invariants, the target space is one-dimensional and the two phases correspond to the minimum and maximum allowed A2 values.Source: §4.2 · pp. 14–15

4.3. Example: spin-one color superconductor

• For spin-one color superconductivity, the complex matrix order parameter has eight nontrivial strata, including three inert states: A, polar, and CSL.The order parameter transforms under U(3)L × SO(3)R.Source: §4.3 · pp. 15–16
• The spin-one color-superconductor phase diagram includes a noninert ε phase, missed in an earlier analysis, and straight boundary segments permit additional coexisting states at phase transitions.The A–polar segment contains rank-one matrices, and the polar–CSL segment contains real matrices up to symmetry transformations.Source: §4.3 · p. 17
• The ε phase has a non-maximal little group despite being noninert, providing a counterexample to Michel’s conjecture; its transition to the A phase is second-order.The ε little group is a subgroup of the A-phase little group.Source: §4.3 · p. 17

5.1. Goldstone theorem

• Spontaneous breaking of a continuous global internal symmetry implies a mode whose energy vanishes as momentum approaches zero, provided the surface term vanishes in the infinite-volume limit.For long-range interactions in acausal theories, that surface-term condition must be checked case by case; the theorem guarantees a mode, not its number.Source: §5.1 · pp. 18–19
• Lorentz-invariant theories have exactly one NG boson per broken generator, while the theorem otherwise constrains low-momentum behavior, not high-momentum dispersion.In gauge theories, a gapless mode may be unphysical in covariant gauges; long-range interactions in noncovariant gauges can invalidate the theorem. Different symmetry-related Green’s functions also obey exact constraints summarized by Ward identities.Source: §5.1 · pp. 19–20

5.2. Goldstone boson counting: Dispersion relations

• Two broken generators in antiferromagnets yield two linearly dispersing NG modes, whereas ferromagnets yield one with quadratic dispersion despite breaking the same two generators.The text also gives examples with fewer NG modes than broken generators in multicomponent Bose condensates, three-component Fermi gases, and dense-matter phases.Source: §5.2 · pp. 21–22
• With translational invariance not completely broken and no long-range interactions, the Nielsen–Chadha theorem gives n_type-I + 2n_type-II ≥ n_broken; type-I and type-II energies scale as odd and even momentum powers, respectively.It establishes an inequality, not general saturation; gapless non-NG modes can obscure counting, and the dispersion need not be analytic when long-range forces are present.Source: §5.2 · p. 22

5.3. Goldstone boson counting: Charge densities

• If the vacuum expectation values of commutators of every pair of broken generators vanish, the number of NG bosons is at least the number of broken generators.This is a lower bound, not the strict equality sometimes attributed to the original theorem; nonzero commutator expectation values can arise from central charges even when all generator expectations vanish.Source: §5.3 · p. 23
• A nonzero vacuum expectation value of a broken-generator commutator implies a coupled NG mode with nonlinear low-momentum dispersion, typically quadratic, though quadratic behavior is not proven generally.The argument assumes rotational invariance; the text notes that the low-energy behavior of many solids is described to leading order by a rotationally invariant effective field theory.Source: §5.3 · pp. 24–25

5.4. Linear sigma model

• The linear-sigma-model subsection recalls a Lorentz-invariant Goldstone-theorem proof using real scalar fields in a linear symmetry representation and an invariant quantum effective potential.The fields need not form an irreducible representation, and complex fields can be decomposed into real ones.Source: §5.4 · p. 26
• With Lorentz invariance, each broken generator yields a mass-matrix zero mode, giving exactly one NG boson per broken generator.The zero modes can be made orthogonal by a change of generator basis.Source: §5.4 · p. 26
• Without Lorentz invariance, the number of mass-matrix zero modes still equals the number of broken generators but need not equal the number of NG modes.Chemical potential explicitly breaks Lorentz invariance through medium effects.Source: §5.4 · pp. 26–27
• A chemical potential lowers the mass matrix until condensation can break symmetry; in the field parametrization, NG fields Π drop out of the mass part of the bilinear Lagrangian.The modified potential includes a negative chemical-potential contribution to the mass matrix.Source: §5.4 · p. 27
• Theorem 5 asserts a basis in which only mutually commuting charges have nonzero vacuum expectation values; the text proves this for special unitary groups and their products, not generally.For those groups, the proof diagonalizes the Hermitian matrix formed from the charge expectations and generator matrices.Source: §5.4 · pp. 27–28
• Charges with nonzero expectation values are unbroken-subgroup singlets; completing them to a Cartan subalgebra and applying root decomposition pairs the remaining generators, reducing mixing to two fields.The theorem’s commuting-charge basis and the resulting pairwise mixing organize the spectrum analysis.Source: §5.4 · pp. 27–28
• When a paired sector has f(µ) ≠ 0, its spectrum contains a massive mode and a type-I NG boson with linear dispersion; when f(µ) = 0, it has a massive mode and one type-II NG boson with quadratic dispersion.The text attributes the quadratic mode to a single-time-derivative term arising from a nonzero expectation value of a commutator of broken generators.Source: §5.4 · p. 28

6.1. Nonrelativistic Boulware–Gilbert model

• For a nonrelativistic shift-symmetric field, a sufficiently fast-falling potential with finite low-momentum Fourier transform gives a gapless NG excitation associated with the conserved charge.The charge is the spatial integral of the momentum density.Source: §6.1 · p. 29
• For 1 < α ≤ 3, the long-range interaction still permits an NG boson, but its nonanalytic dispersion precludes a low-energy effective Lagrangian.For α > 3, the energy is analytic and the Nielsen–Chadha theorem applies; for α ≤ 1, the theory is ill-defined in three spatial dimensions.Source: §6.1 · p. 30
• In the screened-Coulomb limiting case, the quadratic NG mode does not violate the Nielsen–Chadha theorem because two broken conserved charges are present, one not of Noether type.At V0 = −1, both π0 and φ0 are conserved and spontaneously broken.Source: §6.1 · p. 30

6.2. Heisenberg ferromagnet

• The Heisenberg ferromagnet breaks SU(2) to U(1), and its two broken generators yield one quadratic, type-II NG boson.The two broken generators acting on the ground state produce states differing only by a phase factor.Source: §6.2 · pp. 31–32
• For α ≥ 5, the dispersion is analytic and quadratic; for 3 < α ≤ 5, the NG boson persists with nonanalytic momentum-dependent energy, exact despite the Heisenberg model remaining unsolved.Source: §6.2 · pp. 32–33

6.3. Linear sigma model

• In the SU(2) × U(1) linear sigma model, the condensate leaves τ+ unbroken and yields one type-II and one type-I NG mode.The example uses the classical approximation; φ1 represents the type-II mode and the imaginary part of φ2 the type-I mode.Source: §6.3 · p. 33
• In the linear sigma model, the type-II NG boson has low-momentum energy E−,k ≈ k2/2µ, while its opposite-polarization antiparticle is heavy, with mass 2µ.It carries the unbroken charge.Source: §6.3 · pp. 33, 35
• Unlike the linear sigma model, the ferromagnet has no spin wave with opposite circular polarization; flipping a spin already lowers the fully aligned ground-state spin.The paper relates the absence to the maximum-spin ground state and to the fixed sense of Larmor precession.Source: §6.3 · p. 35

7. Low-energy effective field theory for NG bosons

• EFT retains low-energy degrees of freedom and encodes short-distance dynamics in couplings; for SSB, NG bosons alone typically describe the low-energy theory when no other gapless modes exist.When long-range interactions are absent, low-energy observables admit a systematic expansion in powers of momentum or derivatives.Source: §7 · pp. 35–36

7.1. Coset construction of effective Lagrangians

• NG fields parameterize the coset G/H, and the Maurer–Cartan form’s transverse component supplies building blocks for the most general G-invariant Lagrangian.The kinetic normalization may be chosen independently for each irreducible representation of the unbroken subgroup H.Source: §7.1 · pp. 36–38

7.2. Geometric interpretation

• The two-derivative NG kinetic term defines a G-invariant metric on G/H; broken transformations shift NG fields at leading order, while the unbroken symmetry acts linearly.In the presence of a gauge field, the shift induced by group motion determines the gauge-covariant derivative of the NG fields.Source: §7.2 · pp. 38–39

7.3. Nonrelativistic e

• Nonrelativistic Ward identities require action—not necessarily Lagrangian—invariance and constrain EFT coefficients; Ci is determined up to a gradient, which changes the Lagrangian only by a total time derivative.The gradient ambiguity has no observable consequences because it leaves the action unchanged.Source: §7.3 · pp. 40–41
• With nonzero non-Abelian charge density, a single-time-derivative term yields quadratic-dispersion type-II NG bosons; at zero density, NG modes have linear dispersion.For linear dispersion, space and time derivatives have the same power-counting order; for quadratic dispersion, a time derivative counts as two spatial derivatives. The charge density is given by ϱa(0).Source: §7.3 · pp. 41–42
• The antiferromagnet has two linearly dispersing NG bosons, and its leading EFT uses a unit vector on SU(2)/U(1) with a two-derivative kinetic term.Its vanishing charge densities exclude the single-time-derivative term.Source: §7.3 · p. 42
• The ferromagnet has one quadratically dispersing NG boson; its nonzero magnetization permits a single-time-derivative term whose coefficient has topological character.The coefficient is related to the Brouwer degree of the map n(π), and the defining integral is path-independent.Source: §7.3 · pp. 42–43
• The ferromagnetic EFT retains time-reversal invariance: the ground state breaks it, while the symmetry’s implementation on the coset permits a single-time-derivative term [67].In the antiferromagnet, time reversal acts on the NG fields so as to forbid that term.Source: §7.3 · p. 43

7.4. Applications of effective field theory

• At leading derivative order, a relativistic superfluid’s NG-boson effective action is determined by its equation of state and can yield transport coefficients and other low-energy observables.Treating the chemical potential as a background gauge field restores Lorentz invariance; after integrating out the Higgs mode, gauge invariance restricts dependence to D_μ π = ∂_μ π − A_μ, and the pressure fixes the action.Source: §7.4 · pp. 43–44

• The review presents a self-contained introduction to spontaneous symmetry breaking and Nambu–Goldstone bosons, demonstrating its general results with mostly exactly solvable examples.Source: §8 · p. 44
• Nonrelativistic-specific features include unconventional Nambu–Goldstone boson dispersion relations and improved counting rules.Source: §8 · p. 44
• Weak explicit breaking yields pseudo-Nambu–Goldstone bosons with small nonzero zero-momentum energy; counting blurs when approximately linear and quadratic dispersions cannot be distinguished.Source: §8 · p. 44
• When part of the Lagrangian has higher symmetry, an approximation can produce extra gapless states that gain a gap from quantum corrections, potentially yielding naturally light states.Source: §8 · p. 45
• The review covers continuous internal symmetries; spacetime-symmetry breaking can yield fewer Nambu–Goldstone bosons than broken generators, with helical-ferromagnet modes type-I axially and type-II transversely.Source: §8 · p. 45
• The stated aim is to bridge high-energy and condensed-matter or atomic-physics communities with a framework covering relativistic many-body and intrinsically nonrelativistic systems.Source: §8 · p. 45