Effective Lagrangians for ferromagnets
In the preceding note, I explained how to construct an effective Lagrangian for Nambu– Goldstone (NG) bosons of a spontaneously broken internal symmetry in an arbitrary, not necessarily relativistic, system. In this text, I complement the general discussion by a detailed analysis of one concrete example: the ferromagnet. I will in particular show how different field parameterizations can lead to seemingly very different Lagrangians. In order that this text is self-contained, I first review the general construction of the effective Lagrangian without any proofs. Afterwards, I work out its application to ferromagnets.
Overview of the general construction
The degrees of freedom of the low-energy effective field theory (EFT) are the NG fields πa(x), which represent coordinates on the coset space of broken symmetry, denoted as G/H. In the following, I will use the notation Ti,j,... for unspecified generators ofG, Tρ,σ,... for generators of
H, and Ta,b,... for the remaining, broken generators. It is convenient to couple the EFT to a
background gauge field, Ai
µ, for each generator Ti. These allow one to determine correlation
functions of the conserved currents of the theory. At sufficiently low energies, the effective action can be organized in a covariant gradient expansion in the joint number of derivatives acting on the NG fields and of gauge fields. Owing to the lack of Lorentz invariance, spatial and temporal derivatives have to be counted separately. Denoting the part of the effective Lagrangian with s spatial and t temporal derivatives as L(s,t), the expansion takes the form
L=L(0,1)+L(0,2)+L(2,0)+· · ·, assuming in addition rotational invariance. The first terms of
the expansion can then be written as
L(0,1) =c
a(π) ˙πa+ei(π)Ai0, L
(0,2) = 1
2g¯ab(π)D0π
aD
0πb, L(2,0) =−
1
2gab(π)Drπ
aD
rπb, (1)
where Dµπa = ∂µπa −Aiµhai(π) is a covariant derivative of the NG field. Each part of the
Lagrangian is invariant, possibly up to a surface term, with respect to the simultaneous gauge transformation of the NG and gauge fields,
δπa(x) = i(x)hai(π), δAiµ(x) =∂µi(x) +fjki A j µ(x)
k
(x). (2)
Here i are the infinitesimal parameters of the transformation and fi
jk are the structure
con-stants of the symmetry algebra. The gauge invariance allows one to determine the form of the coupling functionsca(π), ei(π), gab(π) and ¯gab(π) in terms of a few unknown parameters. First,
define the element of the coset space G/H as U(π) = eiπaT
a and subsequently the Lie-algebra-valued Maurer–Cartan (MC) form as ω(π) = ωa(π) dπa = Tiωia(π) dπa = −iU(π)
−1dU(π).
Finally, define a bilinear form on the Lie algebra of G as Tr(TiTj) = Ωij. It is assumed to
be nondegenerate but not necessarily proportional to δij as is often the case, and not even
positive-definite. It can be used to raise and lower indices of generators as in Ti = ΩijT j. The
coupling functions can then be cast as
gab(π) =gcd(0)ωca(π)ω d
b(π), ei(π) = ej(0)
U(π)−1TiU(π)
j
, ca(π) = −ei(0)ωai(π),
(3) and analogously for ¯gab(π). The effective Lagrangian is now fully specified in terms of the
and ei(0), associated with charge densities in the ground state. In order to be compatible with
G-invariance, these have to satisfy further constraints,
fσac gcb(0) +fσbc gac(0) = 0, fσji ei(0) = 0, (4)
and similarly for ¯gab(0).
NG fields and transformation rules
In a ferromagnet, the symmetry-breaking pattern is given by G=O(3) and H=O(2). The generators can be chosen so that the structure constants are fully antisymmetric and equal to
fjki =εijk. The O(3) conserved charges correspond to the three components of total spin, and the external gauge field Ai
0 then has the meaning of a magnetic field, namely Ai0 =µBi, where
µis the magnetic moment of a single spin.
The O(3)/O(2) coset space is parameterized by two coordinates, π1,2, representing the NG
degrees of freedom. However, the transformation properties with respect to the O(3) rotations are much more easily expressed in terms of a unit three-vector, ~n(π). The precise mapping between these two descriptions of the NG field will be discussed later. In terms of ~n, an infinitesimal O(3) rotation with the set of parameters ~ acts as δ~n =~n×~, or δinj = εijknk.
Without knowing the precise form of the function πa(~n), we can now easily show that this
transformation rule has the correct Lie-algebraic structure,
δiδjπa−δjδiπa=δi
∂πa ∂nkεjk`n
`
−(i↔j) = ∂π
a
∂nkεjk`εi`sn s+
XX XX
XX
XXX
X
∂2πa ∂nk∂nsεjk`n
`ε
istnt−(i↔j)
= ∂π
a
∂nkn s
(δjsδik−
H
H H
δijδks)−(i↔j) =
∂πa
∂nin
j− ∂πa
∂njn i
(5)
= ∂π
a
∂nsn tε
ijkεkst =εijkδkπa.
This is equivalent to the differential equation for the Killing vectorsha
i(π), derived by Leutwyler,
dihaj −djhai = fijkhak, where di = hai ∂
∂πa. Our transformation rule for ~n(π) is thus compatible with the general expression (2).
In order to be able to use the solution (3) for the coupling functions, we next have to find the relation between the matrix variableU(π) and the vector~n(π). To that end, we first define the matrix variable N(π) =Tini(π). Under a transformation by the rotation g =ei
iT
i, we expect this to change naturally as N → gN g−1. To check that this agrees with the above-defined
transformation rule for~n(π), it is sufficient to consider an infinitesimal rotation,
Tjδnj =δN = i[iTi, N] =−injεijkTk =i(εijknk)Tj, (6)
which reproduces the rule δinj = εijknk. Let us now choose without lack of generality the
ground state as~n0 = (0,0,1), that is, N0 =T3. The relation between πa and ~n can be fixed, at
least in the vicinity of the ground state, by setting
N(π) = U(π)N0U(π)−1. (7)
The broken generators are then Ta with a = 1,2. The whole argument could be reversed
Effective coupling functions from the coset construction
All the expressions in Eq. (3) are independent of the choice of representation for the generators
Ti. However, from now on I will use the two-dimensional representation of O(3) so that I can
take advantage of some special properties of the Pauli matrices. Since the normalization is fixed by the structure constants, we have to set Ti =σi/2 so that Tr(TiTj) = 12δij, and hence
Ti =σ i.
Let us first determine the metric gab(π). From the condition (4) one easily finds that gab(0)
must be proportional to the unit matrix. (This is true by Schur’s lemma whenever the broken generators span an irreducible representation of the unbroken subgroup H.) We can thus set
gab(0) =F2δab. Given the normalization of the generators and Eq. (3), the kinetic term for the
NG fields can be expressed, in the absence of gauge fields, as
gab(π) dπadπb =gcd(0)ωca(π)ω d b(π) dπ
adπb =F2δ
cdωc(π)ωd(π) = 2F2Tr[ω⊥(π)ω⊥(π)], (8)
where ω⊥(π) is the component of the MC form in the subspace of broken generators. This can be extracted using the fact that the Pauli matrices anticommute with each other, namely as
ω⊥= 12(ω−σ3ωσ3). We thus obtain
gab(π) dπadπb =F2Tr
ω(π)ω(π)−ω(π)σ3ω(π)σ3
=F2TrdU(π) dU−1(π)−ω(π)σ3ω(π)σ3
.
(9) With Eq. (7), the trace can be cast as 2 TrdN(π) dN(π) = d~n(π)·d~n(π). The spatial part of the kinetic term in absence of external gauge fields thus takes the form in accord with Eq. (1),
L(2,0) =−1
2F
2∂
r~n(π)·∂r~n(π). (10)
If needed, background gauge fields can be reintroduced by replacing ordinary derivatives with covariant ones. This form of the kinetic term could of course have been guessed from the linear transformation rule for~n(π). Using instead the general solution (3) shows that it is unique.
The temporal kinetic termL(0,2) takes the same form. However, it is not needed at the leading
order of the derivative expansion, since in the case of ferromagnets, the termL(0,1) is nontrivial.
What we need to do now is therefore to find the functions ei(π) andca(π). Let us start with
the former. First, the condition (4) implies that e1(0) = e2(0) = 0. Let us denote e3(0) = Σ;
this represents the magnetization in the ground state. From Eq. (3), we then obtain
ei(π) = Σ Tr
T3U(π)−1TiU(π)
= Σ TrTiU(π)T3U(π)−1
= Σ TrTiN(π) = Σni(π). (11)
This establishes the expected result that the background magnetic field couples to the total spin of the system though a simple scalar product. The remaining unknown coupling, containing the time derivative of the NG field, also follows from Eq. (3), ca(π) dπa =−ei(0)ωi(π) =−Σω3(π).
From the definition of the MC form, this becomes after some manipulations
ca(π) ˙πa = iΣ Tr
T3U(π)−1∂0U(π)
= iΣ
Z 1
0
dλTrT3U(λπ)−1(i ˙πaTa)U(λπ)
=−Σ
Z 1
0
dλTrπ˙aTaU(λπ)T3U(λπ)−1
'Σ
Z 1
0
dλ ~π·~n˙(λπ).
(12)
dependence on the choice of the orientation of ~n0 has now dropped. Since we know how ~n
depends on πa and thus on λ, we can now in principle integrate over λ directly. However, to
keep the index-free notation, I use a different approach. Since ~n is normalized to unity, ˙~n is perpendicular to it, so we can write~π·~n˙ = (~n×~π)·(~n×~n˙). Next, we express using Eq. (7)
Ti[~n(λπ)×~π]i = [iπaTa, N(λπ)] =∂λN(λπ), (13)
whence ~n(λπ)×~π = ∂λ~n(λπ), which allows us to eliminate explicit π-dependence from the
action. Altogether, the L(0,1) part of the Lagrangian thus acquires the form
L(0,1) = Σ
Z 1
0
dλ ∂λ~n(λπ)·
~
n(λπ)×~n˙(λπ)+µΣB~ ·~n(π). (14)
Together with Eq. (10), this defines the low-energy effective Lagrangian of ferromagnets in external magnetic fields. Its structure is determined by symmetry up to two unknown low-energy coupling constants, Σ and F. The former is given directly by the magnetization of the ground state, while the latter can be extracted for instance from the dispersion relation of spin waves, discussed below.
Topological nature of the
c
a(
π
) ˙
π
aterm
The Lagrangian L(0,1) was derived before by Leutwyler by a straightforward integration of the
differential equation for ca(π), that is, hbi(∂bca−∂acb) = ∂aei. The auxiliary variable λ there
appears as an additional spacetime dimension, similarly to the construction of the Wess–Zumino term in chiral Lagrangians. Although the derivation based on Eq. (3) is more straightforward as well as completely general, it is illuminating to use this analogy to gain a deeper insight in the structure of this term. Let us rewrite Eq. (14), for simplicity in a vanishing magnetic field, formally as
L(0,1)
B=0 = Σ
Z 1
0
dλ ∂λm~(λ, π)·
~
m(λ, π)×m~˙ (λ, π). (15)
Herem~(λ, π) with 0 ≤λ≤1 defines a smooth path on the unit sphere such thatm~(0, π) =~n0
and m~(1, π) = ~n(π) for all values of πa. Eq. (14) corresponds to m~(λ, π) = ~n(λπ). However,
I will now show that the action of the theory is independent of the choice of the path. In other words, by choosing any other path, the Lagrangian will at most change by a total time derivative.
Let us calculate the variation of the Lagrangian under an infinitesimal shiftδ ~m(λ, π). From the boundary conditions at λ= 0,1, this variation must satisfy δ ~m(0, π) = 0 andδ ~m(1, π) = δ~n(π) for all πa. We thus obtain
δL(0B=0,1) = Σ
Z 1
0
dλ∂λδ ~m·(m~ ×m~˙) +∂λm~ ·(δ ~m×m~˙) +∂λm~ ·(m~ ×δm~˙)
. (16)
The middle term vanishes since due to the constraint|m~|= 1, all vectors∂λm, δ ~~ m,m~˙ lie in the
two-dimensional plane perpendicular to m~. In the last term, we “integrate by parts in time” and in the first term with respect to λ,
δL(0B=0,1) = Σ[δ ~m·(m~ ×m~˙)]10+ Σ
Z 1
0
dλ
((((
((((
((((
((((
(((
hhhhhh
hhhhhh
hhhhhh
h
−δ ~m·(m~ ×∂λm~˙)−∂λm~˙ ·(m~ ×δ ~m) +∂0[∂λm~ ·(m~ ×δ ~m)]
=δ~n·(Σ~n×~n˙) + Σ∂0
Z 1
0
When we merely vary the pathm~(λ, π) but keep the end points fixed,δ~n =~0 and the action does not change. Since the unit sphere is simply connected, any such path can be smoothly deformed into any other, which implies the above claim that the action is completely independent of the choice of the path. The integration over λ can be carried out explicitly by choosing that path simply as m~(λ, π) = λn1(π), λn2(π),p
1−λ2[(n1(π))2+ (n2(π))2]
. Writing the integrand in Eq. (15) as ˙m~ ·(∂λm~ ×m~), we obtain straightforwardly
L(0,1)
B=0= Σ(n
2n˙1−n1n˙2)
Z 1
0
dλp λ
1−λ2+λ2(n3)2. (18)
Upon integration over λ the full L(0,1) becomes
L(0,1) = Σ
1 +n3(π)
n2(π) ˙n1(π)−n1(π) ˙n2(π)+µΣB~ ·~n(π). (19)
This representation of the Lagrangian still does not depend explicitly on the coordinates πa. On the other hand, as emphasized above, it is somewhat ambiguous; it can be shifted by a total time derivative upon a different choice of the path m~(λ, π).
Equation of motion
In order to derive the equation of motion for the NG mode in the ferromagnet, the magnon, directly using the variable~n(π), we have to further add to the Lagrangian the term 12α(1−~n2), where α is the Lagrange multiplier for the constraint |~n|= 1. The variation of δL(0,1) can be
read off Eq. (17) as the coefficient at δ~n. The equation of motion then acquires a neat form
Σ~n×~n˙ +µΣB~ +F2∆~n=α~n, (20)
known as the Landau–Lifshitz equation. Upon expanding in small fluctuations around the ground state~n0, this gives a dispersion relation for the magnon as E(k) =µB+F
2
Σk 2.
The relation between the two forms for the effective Lagrangian, Eqs. (14) and (19), is nontrivial. As the last point of this text, I therefore wish to demonstrate explicitly that despite different appearance, they lead to the same equation of motion. I do so by showing that the variation of the Lagrangian (19) reproduces Eq. (17) for n1 and n3; the case of n2 is obtained from n1
by a simple exchange of the labels 1 ↔ 2. Since we use a different parameterization of the Lagrangian, we have to modify the Lagrange multiplier as well to, say, β. The variation of
L(0,1) in Eq. (19) with respect to n3 then reads
δ3L(0,1) =δn3
− Σ (1 +n3)2(n
2
˙
n1−n1n˙2) +µΣB3−βn3
. (21)
Note that if we redefine the Lagrange multiplier as
β =α+ Σ 2 +n
3
(1 +n3)2(n
2n˙1−n1n˙2), (22)
the variation of the Lagrangian takes the formδ3L(0,1) =δn3[−Σ(n2n˙1−n1n˙2) +µΣB3−αn3],
consistent with Eq. (17). The equation of motion for n1, on the other hand, follows from the
variation (up to a total time derivative)
δ1L(0,1) =δn1
−2Σ n˙
2
1 +n3 + Σ
n2n˙3
(1 +n3)2 +µΣB
1−βn1
Finally, we have to use the same redefinition of the Lagrange multiplier as in Eq. (22). After some manipulations, we obtain δ1L(0,1) =δn1(Ξ +µΣB1−αn1), where
Ξ = −Σn
1(2 +n3)
(1 +n3)2 (n 2
˙
n1−n1n˙2)−2Σ n˙
2
1 +n3 + Σ
n2n˙3 (1 +n3)2
= Σ
(1 +n3)2
n
(2 +n3)−n2 n1n˙1
| {z }
−n2n˙2−n3n˙3
+(n1)2n˙2−2(1 +n3) ˙n2+n2n˙3o
= Σ
(1 +n3)2
n
˙
n2(2 +n3)(n1)2+ (2 +n3)(n2)2−2(1 +n3)
| {z }
=(2+n3)[1−(n3)2]−2(1+n3)=−n3(1+n3)2
+ ˙n3(2 +n3)n2n3+n2
| {z }
=n2(1+n3)2
o
= Σ(n2n˙3−n3n˙2),
(24)
