Coset construction of effective Lagrangians
The purpose of this text isnot to give a pedagogical introduction into the coset construction of effective Lagrangians. It is rather meant as an overview, or reminder, of some technical aspects of the construction in a form detailed enough to be understood by readers already familiar with the subject. I will only go into full detail in the latter part of the text where I discuss some subtleties associated with coupling Nambu–Goldstone (NG) bosons to matter fields.
Nonlinear realization of symmetry
Consider a system with an internal symmetry group G, spontaneously broken to the subgroup
H. To capture the invariance of the ground state under symmetry transformations fromH, one
introduces an equivalence relation on G: two elementsg1,g2 ∈G are equivalent ifg1 =g2h for
someh ∈H. The (left) coset space G/His defined as the set of equivalence classes with respect to this relation.
Let us denote the coset generated by g ∈G as χg ≡ {gh|h ∈H}. The action of an element g
of the group G on the coset spaceG/H is then formally defined by left multiplication,
χg0
g −
→χgg0. (1)
The selected ground state of the system corresponds to χe = H, where e is the unit element
of G. Different, though physically equivalent, vacua can be generated by acting with broken symmetry transformations on the original ground state, and thus correspond to other elements of G/H than χe. The invariance of cosets under right multiplication by elements from H gives
rise to a kind of gauge redundancy: the whole coset χg corresponds to a single physical state.
In practice, it is often more convenient to work with concrete group elements rather than with the abstract cosets. One then “fixes the gauge” by picking a unique representative u ∈ χ for every coset. The group action (1) now takes the form
u−→g gu =u0h(u,g), (2)
where h(u,g)∈His chosen so that u0 coincides with the representative element of χgu. This is
already very close to the notation used in physics. Namely, the NG fields at a fixed spacetime point span the coset space, and thus define a set of coordinates on it. Denoting the NG fields as πa, where the Latin indices a, b, . . . will be reserved for the broken generators of G, it is customary to represent the coset elementuby a matrix,u =U(π). The action of the group (2) then acquires its final form,
U(π)−→g U(π0) = gU(π)h(π,g)−1. (3)
At this point, there is no need to choose a concrete set of coordinates πa, or even a concrete set of representatives u for each coset. For physical reasons, I will only assume that the coset
χe, corresponding to the chosen ground state, is represented by the unit element e itself. It is
The Maurer–Cartan form
Suppose that the symmetry of the system, defined by the group G, is gauged. (The ungauged case can be easily recovered from the formalism developed below by erasing the gauge field everywhere.) In other words, I couple the theory to a Lie-algebra-valued 1-form gauge field A, which transforms under g∈G as
A−→g gAg−1+ igdg−1. (4)
The corresponding field strength is a 2-form, transforming covariantly in the adjoint represen-tation of G,
F ≡dA−iA∧A, or equivalently Fi ≡dAi+1 2f
i jkA
j ∧
Ak, (5)
wherefi
jk are the structure constants ofG, defined by the commutation relation of its generators
Ti, [Ti, Tj] = ifijkTk.1 The (gauged) Maurer–Cartan (MC) form is a fundamental tool of the
coset construction. It is defined as a Lie-algebra-valued differential 1-form through
ω ≡ −iU−1(d−iA)U ≡ωiTi ≡(ωai dπ a−νi
jA j)T
i, (6)
where the auxiliary function νi
j(π) is defined by νji(π)Ti ≡ U(π)−1TjU(π). It is easy to verify
that the MC form satisfies the fundamental MC structure equation,
dω+ iω∧ω=−U−1F U, or equivalently dωi−1
2f
i
jkωj∧ωk =−Fjνji. (7)
Under thelocal action of G, Eq. (3), the MC form transforms as
ω −→g hωh−1−ihdh−1, (8)
where I for the sake of simplicity dropped the arguments of h. Since ihdh−1 takes values from
the Lie algebra of H, one can split the MC form into components lying in the subspaces of unbroken and broken generators, respectively,
ωk ≡ωαTα, ω⊥ ≡ωaTa. (9)
These transform in turn as
ωk
g −
→hωkh−1−ihdh−1, ω⊥
g −
→hω⊥h−1. (10)
The broken piece, ω⊥, transforms linearly under the adjoint action of H, and constitutes the
basic covariant building block for the construction of effective Lagrangians. The unbroken piece,
ωk, on the other hand, transforms as a gauge connection of H, and can be used to construct
covariant derivatives of other fields.
It is sometimes convenient to think of the symmetry transformation (3) as defining the action of the direct product of groupsG×H. The groupG acts from the left and leaves the MC form unchanged. In other words,G-invariance is automatically guaranteed by using the MC form as the building block to construct effective Lagrangians. The “gauge” groupH, acting on U from the right, imposes nontrivial constraints on the structure of the effective Lagrangian. When the MC form is used to construct effective Lagrangians, H-invariance as a rule requires that the coefficients multiplying individual operators in the Lagrangian be invariant tensors of H.
1I will use the notation i, j, . . . to label generic generators of
Explicit expressions for exponential parametrization of
U
(π)
In certain special cases, the above-developed general abstract picture can be given a relatively simple concrete form. First, it is common to use an exponential parametrization of the coset element,
U(π) =eiπaTa, (11)
which is valid at least in some neighborhood of unity, and thus can always be used to analyze the perturbative physics of NG bosons. In this case, one can work out the components of the MC form from its definition (6) in the form of a series expansion in the NG fields,
ωai(π) =δai −1
2f
i abπb+
1 6f
j abf
i
jcπbπc+O(π3). (12)
Similarly, one can write down an explicit expression for the functionνi j(π),
νji(π) =δji −fjai πa+1 2f k jaf i kbπ a
πb+O(π3). (13)
It is now also possible to work out explicitly the action of the group on the NG fields, defined by Eq. (3), at least for infinitesimal group motions. One can then parametrize bothg and h(π,g) exponentially,
g=eiiTi, h(π,g)≡eikα(π,g)Tα ≡eiikiα(π)Tα. (14)
The infinitesimal shift of the NG field will be likewise parametrized in terms of a new, gener-ally nonlinear function ha
i(π) such that δπa = ihai(π). Rewriting Eq. (3) as U(π)
−1U(π0) =
U(π)−1gU(π)h(π,g)−1 and subtracting from it U(π)−1U(π) = 11, one obtains the following two relations between ωi
a,νji,kiα and hai,
νiα =haiωaα+kiα, νia=hbiωab. (15)
With the explicit expressions forωi
a(π) and νji(π) at hand, one can then extract expressions for
the remaining two functions,
hai(π) =δai −
fiba− 1
2δ e if a eb
πb+ 1 2
fibαfαca − 1
3δ e if j ebf a jc+ 1 2δ e if d ebf a dc
πbπc+O(π3),
kαi(π) =δαi −
fiaα − 1
2δ e if α ea
πa+1 2
fiaβfβbα − 1
3δ e if j eaf α jb+ 1 2δ e if d eaf α db
πaπb +O(π3).
(16)
As a side remark, note that we now also have an explicit expression for the covariant derivative of the NG field. Namely, using the second of the relations (15) and the definition of the MC form (6), one finds that ω⊥ can be cast as ω⊥ =ωaTa≡ωbaDπb, where
Dπa≡dπa−Aihai(π). (17)
In hindsight, this result is natural: a covariant derivative of a field is constructed by adding to its ordinary derivative a product of the gauge connection and the infinitesimal symmetry transformation of the field. I have just shown that this general prescription remains valid even if the field transforms nonlinearly.
Symmetric coset spaces
The coset spaceG/H is called symmetric if the commutator of two broken generators contains a linear combination of unbroken generators only, that is, iffa
bc = 0. This property is equivalent
to the existence of an automorphism R of the Lie algebra of G such that
R(Tα) = Tα, R(Ta) =−Ta. (18)
In this case, the exponential parametrization (11) is invaluable, for applying the automorphism
R to the transformation rule (3) turns it into U(π0)−1 = R(g)U(π)−1h(π,g)−1. It is now possible to define a field variable that transforms linearly under the whole groupG,
Σ(π)≡U(π)2, Σ(π)−→g Σ(π0) = gΣ(π)R(g)−1. (19)
Its linear transformation property makes it easy to construct the covariant derivative of Σ,
DΣ = dΣ−iAΣ + iΣR(A), (20)
as well as to take higher-order covariant derivatives. The automorphism R can also be used to project out the broken and unbroken components of the MC form, e.g. ω⊥ = 12[ω− R(ω)]. Thus,ω⊥ can be related to the covariant derivative of Σ,
ω⊥=− i 2U
−1D
ΣU−1 = +i 2UDΣ
−1
U. (21)
The above expressions can be further simplified in the special case that R is an inner auto-morphism, that is, when there is an element R ∈G such thatR(Ti) =R−1TiR. In this case, a
slightly different field variable can be introduced, which transforms under the adjoint action of the whole group G,
N(π)≡Σ(π)R=U(π)RU(π)−1, N(π)−→g N(π0) =gN(π)g−1. (22)
Correspondingly, it has a very simple covariant derivative,
DN = dN −i[A, N]. (23)
Likewise, the broken component of the MC form can be expressed in terms of N rather than Σ, for instance through
ω⊥R =− i 2U
−1D
N U. (24)
The advantage of using N instead of Σ as the NG field variable is that N(π)2 is a constant
matrix, which in many cases simplifies practical calculations. Indeed, from the definition of the automorphism R, Eq. (18), it follows that R2 commutes with the whole Lie algebra of
G. Consequently, N2 =U R2U−1 = R2. By Schur’s lemma, this must be proportional to unit
Coupling to matter fields
In many systems of physical interest, NG bosons are not the only low-energy degrees of freedom. It is therefore natural to ask how the NG bosons can be coupled to other, non-NG degrees of freedom (“matter fields”) in a way that respects the nonlinearly realized symmetry.
Since the symmetry under the subgroupHremains unbroken in the ground state, then whatever non-NG degrees of freedom are present in the system, the corresponding states in the spectrum must be organized in multiplets of H. I therefore start by assuming the existence of a field ψ
that transforms linearly in some (not necessarily irreducible) representation of H,
ψ −→h ψ0 =D(h)ψ. (25)
This linear representationD of Hcan now be promoted to a nonlinear realization of the whole group G by the mathematical technique of induced representations,
ψ →−g ψ0 =D(h(π,g))ψ, (26)
where h(π,g)∈H is defined by Eq. (3). Note that the action of the broken generators on ψ is nontrivial in that it depends on the NG fields πa. It is nevertheless easy to check that Eq. (26) satisfies the correct composition rule to define a representation of G. Once the realization of the whole group G onψ is established, one can construct a covariant derivative,
Dψ ≡dψ+ iD(ωk)ψ, (27)
whereD(ωk) is a shorthand notation forωαD(Tα). Again, it is easy to check using Eq. (10) that
this definition is consistent, that is, Dψ has the same transformation rule under G asψ itself, given by Eq. (26). Finally, let me remark that in case the target space of the representation D
can accommodate a linear representation of the whole group G, it is possible to define a field variable that transforms linearly under the whole of G,
Ψ≡D(U(π))ψ, Ψ→−g Ψ0 =D(g)Ψ. (28)
At this point, it may be quite unclear why I bother to introduce two different fields ψ and Ψ that carry the same physical content. After all, the relation Ψ = D(U(π))ψ is a mere field redefinition, and thus cannot have any impact of physical observables. Since Ψ transforms linearly under the whole groupG, it would be tempting to discardψ as less physical, or at least impractical to use. I therefore want to explain in detail why the opposite is, in fact, the case.
First, the linear transformation property of Ψ under G might suggest that physical states are organized into multiplets of G, which is not the case when the symmetry is spontaneously broken. For instance, it might seem that the spin states of electrons in a many-electron system have to be organized into doublets in order to preserve the SU(2) spin rotation symmetry. However, as I will show on a concrete example below, in situations where the SU(2) symmetry is spontaneously broken as in (anti)ferromagnets, it is perfectly possible to couple the local magnetization to a single spin degree of freedom in an SU(2)-invariant manner.
Second, it is well known that (up to some exceptions) the interactions of NG bosons are weak at low energy and momentum, which on the level of effective field theory is manifested by the fact that couplings of NG bosons always come with derivatives. This in turn is a consequence of the nonlinearly realized symmetry. As I will again show below, this property would not be manifest when using the Ψ variable, on which even the broken symmetry acts linearly. The ψ
Example: ferromagnet
In the absence of spin-orbit coupling and other perturbations, an ideal ferromagnet has exact global symmetry under spin rotations from the groupG=SU(2). In the ferromagnetic ground state, this is spontaneously broken down to H =U(1). I will represent the generators of G by half of the Pauli matrices, τi/2, and without loss of generality assume that H is generated by τ3/2. In this case, the coset space is clearly symmetric and the corresponding automorphism,
defined by Eq. (18), is given byR = iτ3.2 In accord with Eq. (22), one then introduces a matrix
field variable N(π) by
N(π)≡U(π)τ3U(π)−1. (29)
This field has the following simple properties: (i) it is unitary and Hermitian, (ii) it squares to unity, and (iii) it is traceless. These properties guarantee that N(π) can be mapped onto a unit vector variable~n(π) such that
N(π) =~τ ·~n(π). (30)
The adjoint transformation property of N guarantees that ~n transforms under SU(2) as an ordinary three-vector. This unit vector is the familiar variable of the nonlinear σ-model.
I will not discuss the dynamics of the spin degrees of freedom here, but simply assume that it leads to a ferromagnetic ground state such that h~ni = (0,0,1). Suppose now that the ground-state spin texture is coupled to spin-12 fermions, such as conducting electrons in a metallic ferromagnet, or neutrons. The latter would be relevant for the analysis of spin-wave spectrum by neutron scattering; in the following, I will however call the fermions “electrons” for sake of simplicity. The dynamics of the electron doublet Ψ will then be described by a theory such as
L= Ψ†
i∂0+
∇2
2m +µ
Ψ +g~n(π)·Ψ†~τΨ. (31)
I assigned both electron polarizations the same chemical potentialµ. The parametergmeasures the strength of the electron–magnon spin–spin interaction. Note that this electron–magnon interaction term does not contain any derivatives, and the interaction therefore appears to be strong even at low energies. As explained above, this is the price for using the field variable Ψ that transforms linearly under the wholeSU(2), and is easy to remedy. Changing the fermionic variable to ψ via Ψ =U(π)ψ, the Lagrangian (31) becomes
L'ψ†
i∂0+
∇2
2m +µ+gτ3
ψ−ψ†ω0ψ−
1 2m ψ
†
ω·ωψ+ i∇ψ†·ωψ−iψ†ω·∇ψ
, (32)
where the symbol ' indicates equality up to a total derivative. Also, within the present example, I drop the differential form language and use lower indices to indicate the coordinate components of the MC form. Thus, ωµ ≡ −iU−1∂µU corresponds to the MC 1-formω, Eq. (6),
through ω ≡ωµdxµ, and the boldface naturally refers to spatial components.
We can see that upon the field redefinition, the nonderivative coupling of electrons to magnons disappeared; all interaction terms are now written explicitly in terms of the MC form and thus contain at least one derivative. The difference between the Ψ and ψ notations becomes even more dramatic in the case that only electrons with one spin polarization are present in the
2The factor i is added so thatR has unit determinant and thus belongs to
system. Consider the two spin polarizations as separate fields, that is, decompose ψ into ψ1
and ψ2. Each of these carries a one-dimensional representation of the unbroken subgroup U(1),
ψ1
h −
→D1(h)ψ1, ψ2
h −
→D2(h)ψ2. (33)
According to Eq. (26), this can be lifted to a nonlinear realization of the wholeSU(2) group,
ψ1
g −
→D1(h(π,g))ψ1, ψ2
g −
→D2(h(π,g))ψ2. (34)
It is then possible to write down an effective Lagrangian describing interactions of electrons with magnons, in which the “up” and “down” electron polarizations appear completely inde-pendently, yet it is manifestly invariant under the whole SU(2) group,
L=c1ψ1†
iD0+
D2
2m +µ1
ψ1+c2ψ2†
iD0+
D2
2m +µ2
ψ2, (35)
where the covariant derivativeDµis defined by Eq. (27). The coefficientsc1,2 can in principle be
tuned independently, but as long as they are nonzero, they can both be set to one by rescaling the electron fields. Eq. (35) then reproduces the Lagrangian (32) upon the replacementω →ωk, if one in addition identifies µ1,2 = µ±g. However, we can as well set one of c1,2 to zero and,
as claimed above, indeed arrive at an effective theory for a single electron spin polarization, which maintains full SU(2) invariance. Even if we disregard this possibility as exotic, the “bottom-up” effective Lagrangian (35), using ψ as its basic building block, is still superior to the “top-down” Lagrangian (32) that descends from the microscopic model (31) for the linearly transforming field Ψ. Namely, Eq. (32) contains a very specific combination of the broken and unbroken components of the MC form, ω⊥ and ωk. As Eq. (35) clearly shows, this is, however, not required by symmetry. The parts depending on ωk and ω⊥ can come with different, independent coefficients.
Magnon–electron scattering
As a final illustration of the subtleties associated with the construction of effective Lagrangians for matter fields coupled to NG bosons, I will now work out in detail the scattering amplitude for magnon–electron scattering in the model defined by Eq. (31).
To that end, we need the interaction vertices, and for that, to expand the Lagrangian in powers of the NG fields πa. Using the notation~π≡(π1, π2,0) leads to the following relation,
~n· ~τ
2 =~n·
~
T =ei~π·T~T3e−i~π·
~ T =T
3+3abπ aT
b−
1 2~π
2T
3+· · · . (36)
The unbrokenU(1) symmetry allows us to define combinations ofπ1,2 carrying a definite charge (spin) as ϕ = iπ1 −π2 and ϕ† = −iπ1 −π2. In terms of these, the interaction part of the
Lagrangian (31) takes the form
g
1− 1
2ϕ †
ϕ
(Ψ†1Ψ1−Ψ
†
2Ψ2) +ϕΨ
†
2Ψ1 +ϕ†Ψ
†
1Ψ2
, (37)
plus terms with more NG fields, which we do not need for the calculation. The whole Lagrangian then becomes
L= Ψ†1
i∂0+
∇2
2m +µ+g
Ψ1+ Ψ
†
2
i∂0 +
∇2
2m +µ−g
Ψ2
− g
2ϕ †
ϕ(Ψ†1Ψ1−Ψ
†
2Ψ2) +g(ϕΨ
†
2Ψ1+ϕ†Ψ
†
1Ψ2).
The amplitudes for ϕ–Ψ1 and ϕ–Ψ2 scattering are represented, at tree level, by two diagrams
each:
pk
p0 k0
+
p+k
(39)
pk
p0 k0
+
p0−k
(40)
I used dashed, solid and double solid lines to represent ϕ, Ψ1 and Ψ2, respectively.
Let us start with the ϕ–Ψ1 process. The sum of the diagrams (39) reads
−ig
2 + (ig)
2 i
p0+k0−(p+k)
2
2m +µ−g
=−ig
2 −
ig2
−2g+k0− p
·k
m −
k2
2m
, (41)
where I used the fact that the external Ψ1 line is on-shell, that is, p0 = p
2
2m −µ−g. One can
see already now that the contributions of the two diagrams cancel each other in the limit of vanishing momentum of the incoming magnon, ensuring that magnon interactions are weak at low energy as expected. (The same is of course also true for the outgoing magnon by the crossing symmetry of the diagrams.) In order to get a more explicit expression for the amplitude, we next expand in powers of momentum to the leading nontrivial order, assuming that the magnon energy is much smaller thang. In addition, I use the fact thep+k =p0+k0 so that the amplitude can be expressed in terms of both the incoming and the outgoing momenta. A particularly symmetric expression is obtained by taking the average over the two,
i 8
k0+k00 −
p·k+p0·k0
m −
k2+k02
2m
. (42)
Upon some further manipulation using momentum conservation, this can be finally brought to the form,3
i 8
k0+k00 − (p+p
0)·(k+k0)
2m −
k·k0
m
. (43)
The meaning of the colors is explained below. The amplitude for the ϕ–Ψ2 scattering is found
analogously from the diagrams (40),
ig
2 + (ig)
2 i
p0
0−k0− (p
0−k)2
2m +µ+g
= ig 2 −
ig2
2g−k0 +p
0·k
m −
k2
2m
. (44)
This time, I use the fact thatp0−k =p−k0 to write the result as
−i
8
k0+k00 −
(p+p0)·(k+k0)
2m +
k·k0
m
. (45)
3Strictly speaking, what I calculated here is the on-shell Green’s function. In order to convert it into the
For both processes, we managed to establish a low-energy theorem for the scattering of soft magnons, but doing so required nontrivial cancellation between the two diagrams contributing to the scattering amplitude. I would now like to show how the same result can be obtained straightforwardly using the ψ field variable.
The starting point will be the Lagrangian (32). It is a matter of a straightforward, if a little tedious, manipulation to expand this to second order in the magnon fieldϕ defined above. To that end, it is convenient to make use of the representation of the matrix NG field, ei~π·T~
=
e12(τ−ϕ−τ+ϕ
†)
, where τ± = 12(τ1 ±iτ2) are the usual ladder operators. The MC form in turn
becomes
ωµ=−
i
2(τ−∂µϕ−τ+∂µϕ †
) + i
8τ3(∂µϕ †
ϕ−ϕ†∂µϕ) +O(ϕ3). (46)
The full Lagrangian (31) to this order in ϕthen reads
L=ψ†
i∂0+
∇2
2m +µ+gτ3
ψ+ i
2(∂0ϕ ψ †
2ψ1−∂0ϕ†ψ
†
1ψ2)−
i 8(∂0ϕ
†
ϕ−ϕ†∂0ϕ)(ψ
†
1ψ1−ψ
†
2ψ2)
− 1
8m∇ϕ·∇ϕ
†
ψ†ψ− 1
4m∇ϕ·(∇ψ
†
2ψ1−ψ
†
2∇ψ1) +
1 4m∇ϕ
†·
(∇ψ1†ψ2−ψ
†
1∇ψ2)
+ 1 16m(∇ϕ
†
ϕ−ϕ†∇ϕ)·(∇ψ†1ψ1−ψ †
1∇ψ1−∇ψ †
2ψ2+ψ †
2∇ψ2). (47)
Although this Lagrangian looks formidable, it actually considerably facilitates the calculation of the magnon–electron scattering amplitude compared to the Lagrangian (38). The reason for the simplification is that the large energy scaleg is now hidden in the electron propagator and does not appear in the couplings.4 As a consequence, the second of the diagrams for each of the
processes in Eqs. (39) and (40) is suppressed by a factor 1/g and can be neglected to the first approximation. There are no cancellations between different diagrams now. The low-energy theorem for magnon–electron scattering is satisfied by the first, contact diagram. Extracting the Feynman rule for the four-point vertex from the above Lagrangian reproduces immedi-ately the expressions (43) and (45) derived before. The individual contributions, indicated in Eqs. (43) and (45) by colors, are in a one-to-one correspondence with the terms in the effective Lagrangian (47) that contribute to the four-point vertex.
4The dimensionful couplinggcorresponds to the energy of flipping a spin and sets the energy scale at which