REVIEW
Nonrelativistic effective Lagrangians
High Energy Physics - Phenomenology · Submitted 10 Nov 1993 · 16 references
Pith's verdict: A clean, correct derivation of the Berry-phase term in the nonrelativistic Goldstone Lagrangian; the ferromagnet result is the real payload and it holds up.
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Referee report
The paper
The paper claims
A topological term governs ferromagnet spin waves
The reading
What carries the argument
The load-bearing device is the pair of equations that the classical action must satisfy: the pion equation of motion and the Ward identity obtained by varying the action with respect to the external gauge field. Requiring both to hold for slowly varying fields forces the leading terms of the derivative expansion into a symmetric-space form, with a metric $g_{ab}(pi)$ and Killing vectors $h_{ai}(pi)$ for $G/H$, and fixes the new couplings $e_i(pi)$ and $c_a(pi)$ up to the order parameter. The consistency conditions (a)-(e) in Eq. (13) are the actual engine: they determine the allowed vertices, show that the topological term can be gauge invariant only up to a total derivative, and reduce the ferromagnet to the Landau-Lifshitz Lagrangian. The Brouwer-degree integral in Eq. (18) is the explicit form of that term.
Extended reading notes
Core claim
The central claim is that nonrelativistic Goldstone effective theories are organized by the same Ward identities as chiral perturbation theory, but with an extra term allowed whenever the charge density orders. For $G=O(3)$ broken to $H=O(2)$, the two Goldstone fields are a unit vector $U$; the extra term is proportional to the Brouwer degree of the map $U(pi)$ and has the magnetization $Sigma$ as its coupling. Because this term is first order in time derivatives, it dominates the derivative expansion for $Sigma neq 0$, turns the wave equation into a Schrödinger-like equation, and gives $omega(k)=gamma k^2$ with $gamma=F^2/Sigma$. The resulting equation of motion is the Landau-Lifshitz equation, and the spectrum contains one magnon per momentum even though there are two real Goldstone fields. If $Sigma=0$, as in the antiferromagnet, the topological term is absent, the wave equation is second order in time, and the dispersion is linear with two magnons.
Load-bearing premise
The construction assumes that the only gapless excitations are Goldstone modes, so that no other soft degrees of freedom contaminate the low-energy correlation functions; the paper states this as a PCAC-like hypothesis.
Editorial extensions
If this is right
- For a ferromagnet, the low-energy effective theory predicts one magnon per momentum with $omega(k)=gamma k^2$, where $gamma=F^2/Sigma$ is fixed by the spin stiffness $F$ and the magnetization $Sigma$.
- The transverse charge-density matrix element remains nonzero as $k$ tends to zero, $langle 0|J^0_a|pi(k)rangle=varepsilon_a sqrt{Sigma}$, while the current matrix element is proportional to $k$; both follow from the same two constants.
- For an antiferromagnet, $Sigma=0$ removes the topological term, giving a linear dispersion $omega=v|k|$ with $v=F_2/F_1$ and two magnons, so the known chiral-perturbation results carry over with the speed of light replaced by $v$.
- For a solid, the phonons fit the same scheme as the abelian translation group, with linear sound waves and elastic moduli playing the role of the effective couplings.
- The general counting is $dim(G)-dim(H)$ effective fields, but the number of Goldstone particles can be smaller; the topological term appears only when a charge density orders.
Reading between the lines
- The topological term is a Berry-phase-like functional on the order-parameter manifold; its total-derivative gauge variation suggests that textures and finite samples may require boundary terms that the bulk Lagrangian does not display.
- The $Sigma neq 0$ versus $Sigma=0$ split gives a sharp experimental diagnostic: a nonrelativistic system with a quadratic Goldstone mode should have an ordered charge density, while a linear mode should not, so searching for exceptions tests the symmetry classification.
- Because the method fails for fluids and gases only when dissipation enters at leading order, one natural extension is a hydrodynamic effective action with noise; the Lagrangian route is the special case where dissipative terms are absent.
- The same two constants overdetermine dispersion, residue, and current matrix elements, so independent measurements can reveal extra low-energy degrees of freedom that the minimal Goldstone picture omits.
Original abstract
Chiral perturbation theory is extended to nonrelativistic systems with spontaneously broken symmetry. In the effective Lagrangian, order parameters associated with the generators of the group manifest themselves as effective coupling constants of a topological term, which is gauge invariant only up to a total derivative. In the case of the ferromagnet, a term connected with the Brouwer degree dominates the derivative expansion. The general analysis includes antiferromagnetic magnons and phonons, while the effective field theory of fluids or gases is beyond the scope of the method.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
Chiral perturbation theory is extended to nonrelativistic systems with spontaneously broken symmetry. In the effective Lagrangian, order parameters associated with the generators of the group manifest themselves as effective coupling constants of a topological term, which is gauge invariant only up to a total derivative. In the case of the ferromagnet, a term connected with the Brouwer degree dominates the derivative expansion. The general analysis includes antiferromagnetic magnons and phonons, while the effective field theory of fluids or gases is beyond the scope of the method.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Chiral perturbation theory is extended to nonrelativistic systems with spontaneously broken symmetry. | contribution 0.98 | 0.80 | 0.19 | asserted 1.00 |
| s1 | In the effective Lagrangian, order parameters associated with the generators of the group manifest themselves as effective coupling constants of a topological term, which is gauge invariant only up to a total derivative. | result 0.81 | 0.69 | 0.31 | asserted 1.00 |
| s2 | In the case of the ferromagnet, a term connected with the Brouwer degree dominates the derivative expansion. | result 0.95 | 0.72 | 0.19 | asserted 1.00 |
| s3 | The general analysis includes antiferromagnetic magnons and phonons, while the effective field theory of fluids or gases is beyond the scope of the method. | limitation 0.98 | 0.57 | 0.07 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v1
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- P abstract
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References
How later papers used it
61 later papers cite it
1 later paper by other authors is listed below: 1 used its method or tool.
Uses its method or tool 1 of 6 papers with this label
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Followed this paper for a quicker derivation.
Effective theory for deformed nuclei (2010) · 0.80 · paraphrased: the licence does not allow quoting
How the literature uses it
37 arXiv papers cite this in 83 sentences
Papers citing it · 55 sentences without a confident label
Uses its method or tool 3 papers
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To that end, we shall use the effective Lagrangian formalism, developed in Ref. [ref] , whose notation we closely follow.
Scattering amplitudes of massive Nambu-Goldstone bosons · Example of non-mNG-type pNG boson · 0.99
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We use the standard EFT methodology for describing solid materials [ref] in flat space-time.
Black hole elasticity and gapped transverse phonons in holography · Solid EFT and elasticity · 0.98
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Here, we follow Leutwyler [ref] for a quicker derivation.
Effective theory for deformed nuclei · Nambu-Goldstone modes revisited · 0.80
Compares with it 1 paper
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This is in contrast to Nielsen and Chadha [ref] as well as Leutwyler [ref] who emphasized the role of the NG boson dispersion relations.
On the number of Nambu-Goldstone bosons and its relation to charge densities · Introduction and summary · 1.00
Cites it as support 1 paper
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The argument is that a renormalisation factor $j_α^0 →Z j_α^0$ would spoil the commutation algebra $[j_α^0,j_β^0]=i f_αβ^αβγj_γ^0$, [ref] [An alternative approach based on the Ward-Takahashi identities recovers the similar result that a non-zero VEV of non-abelian charge densities induces a one-time derivative term in the effective Lagrangian [ref] .].
An introduction to Goldstone boson physics and to the coset construction · Counting rule and classification · 0.97
Cites it as background 18 papers
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As is well-known, similar statements apply to the spontaneous breaking of spacetime symmetries [ref] , albeit with a fewer number of NGBs than naively expected [ref] .
Soft Phonon Theorems · Introduction · 0.90
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Indeed, this is how the intricacies of the spectrum of Nambu-Goldstone (NG) bosons arising from spontaneous breakdown of a continuous global internal symmetry in a quantum many-body system [ref] were first understood.
Remarks on relativistic scalar models with chemical potential · Introduction · 0.80
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In the absence of the Lorentz symmetry, there is generally a mismatch between the number of NG modes and the number of the broken symmetries, and the associated (so-called type-B) NG mode often shows a quadratic dispersion relation $ω= a^2 (a ∈ℝ)$ [ref] .
Effective Lagrangian for Nambu-Goldstone modes in nonequilibrium open systems · Introduction · 0.82
Discussion
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