Symmetry realization via a dynamical inverse Higgs mechanism
HJE
Symmetry realization via a dynamical inverse Higgs mechanism
Ira Z. Rothstein 0 1
Prashant Shrivastava 0 1
0 Pittsburgh , PA 15213 , U.S.A
1 Department of Physics, Carnegie Mellon University
The Ward identities associated with spontaneously broken symmetries can be saturated by Goldstone bosons. However, when spacetime symmetries are broken, the number of Goldstone bosons necessary to nonlinearly realize the symmetry can be less than the number of broken generators. The loss of Goldstones may be due to a redundancy or the generation of a gap. In either case the associated Goldstone may be removed from the spectrum. This phenomena is called an Inverse Higgs Mechanism (IHM) and its appearance has a well de ned mathematical condition. However, there are cases when a Goldstone boson associated with a broken generator does not appear in the low energy theory despite the lack of the existence of an associated IHM. In this paper we will show that in such cases the relevant broken symmetry can be realized, without the aid of an associated Goldstone, if there exists a proper set of operator constraints, which we call a Dynamical Inverse Higgs Mechanism (DIHM). We consider the spontaneous breaking of boosts, rotations and conformal transformations in the context of Fermi liquids,
E ective Field Theories; SpaceTime Symmetries; Spontaneous Symmetry

nding
three possible paths to symmetry realization: pure Goldstones, no Goldstones and DIHM,
or some mixture thereof. We show that in the two dimensional degenerate electron system
the DIHM route is the only consistent way to realize spontaneously broken boosts and
dilatations, while in three dimensions these symmetries could just as well be realized via the
inclusion of nonderivatively coupled Goldstone bosons. We present the action, including
the leading order nonlinearities, for the rotational Goldstone (angulon), and discuss the
constraint associated with the possible DIHM that would need to be imposed to remove it
from the spectrum. Finally we discuss the conditions under which Goldstone bosons are
nonderivatively coupled, a necessary condition for the existence of a Dynamical Inverse
Higgs Constraint (DIHC), generalizing the results for Vishwanath and Wantanabe.
Breaking
1 Introduction
1.1
1.2
4.1
4.2
4.3
4.4
Framids
framid
4.4.1
4.4.2
4.4.3
2
3
4
5
6
7
The missing Goldstones
The paths to symmetry realization
Review coset construction Nonderivatively coupled (NDC) Goldstone bosons
Nonrelativistic framids Coset construction of Fermi liquid EFT with rotational symmetry: type I Multiple realizations of broken symmetry Power counting
Review of EFT of Fermi liquids scalings
Power counting in the coset construction
The framid as Lagrange multiplier and the Landau relation
Fermi liquid with broken rotational invariance
5.1
The stability of Goldstone boson mass under renormalization
Broken conformal symmetry: eliminating the nonrelativistic dilaton
6.1
Consequence of broken conformal symmetry via the DIHM
Conclusions A Landau relation from Galilean algebra B Landau relation from Poincare algebra 1
The canonical example of such a scenario is the Fermi liquid theory of metals where phonons
do not play a role at leading order.1 Of course, if there are no other gapless modes, or
if the Goldstones couple to sources, then they are of primary importance. An example of
such a scenario is the QCD chiral Lagrangian.
When spacetime symmetries are broken, GBs can be nonderivatively coupled. Two
canonical examples being the relativistic dilaton and the Goldstone bosons of broken
rotational invariance in Fermi liquids. Such nonderivative couplings lead to marginal or
relevant interactions which can drastically a ect the IR physics. For instance, when
rotational invariance is broken in a Fermi liquid and translations are unbroken (nematic order),
the quasiparticles decay into Goldstones [1{3] leading to a width which scale as
E
with
< 2. While relativistic dilatons can generate long range forces and, as such, their
(d 1)(p~n) h j j0X (0) j nihn j (0) j i
eiEnt
h j (0) j nihn j j0X (0) j i
e iEnt 6= 0:
We assume that the system preserves a discrete translational invariance, so that there
exists some notion of a conserved momentum. Given that X is a conserved charge, we
see that symmetry breaking implies the existence of a zero energy state when p~n ! 0.
However, we can not say anything about the associated spectral weight other than the
fact that it has to nonzero. This state may be arbitrarily wide. Thus if we are to count
Goldstone bosons when spacetime symmetries are broken we must de ne what we mean
by a Goldstone boson. For our purposes we will de ne a Goldstone mode as having to
satisfy the de nition of a quasiparticles,
E2 in the limit of vanishing energy. Also
note that (1.3) does not preclude the possibility of having multiple gapless states.
For nonrelativistic systems, there can be no symmetry breaking if the vacuum is
trivial since pair creation is disallowed. Thus a nonrelativistic system which manifests any
symmetry breaking necessarily has a ground state which breaks at least boost invariance
1Potential, o shell phonons play an indirect role in that they contribute to the attractive piece of the
four Fermi coupling once they have been integrated out.
2We assume here that there are no long range forces so that surface terms may be dropped and that
generators have canonical translational properties. See [14] for a discussion.
{ 2 {
and one can not separate spacetime from internal symmetry breaking. However, in the
literature when internal symmetries are broken, the breaking of boost symmetry is usually
ignored. We will come back to this important issue below.
Goldstone bosons may have various dispersion relations. Inequalities for counting rules
for the type I (E
p) and type II (E
p2) Goldstones3 associated with internal symmetry
breaking were rst written down by Nielsen and Chadha [11]. Since then a series of papers
ultimately led to the
nal result for the number (N ) of Goldstones [12, 13, 15{18] when
the group G is broken to H [21]
1
2
N = dim(G=H)
rank[ ]
(1.4)
ih[Xa; jba(0)]i and Xa are the broken charges of the full group G and jb(0) are
the associated charge densities. Furthermore counting rules for gapped Goldstones (with
both calculable and incalculable gaps ) have been developed [19, 20].
The analysis leading to the result (1.4) does not hold when spacetime symmetries are
broken. Consider the case of a canonical super uid. This system breaks a U(1) symmetry
corresponding to particle number and the rank of
vanishes leading to a prediction of one
Goldstone boson. However, we must ask what justi es ignoring the ersatz GB arising from
the breaking of boost invariance? The answer lies in whats known as the as the \Inverse
Higgs Mechanism" [24](IHM) (see also [25]). The counting of (gapless) Goldstones still follows once we have established the necessary criteria for the IHM. When two broken generators X; X0 obey a relation of the form
[P ; X] / X0
where P are the unbroken translations and X and X0 are not in the same H multiplet, it
may be possible to eliminate the Goldstone associated with X. As emphasized in [20] the
algebraic relation (1.5) may or may not be the signal of a redundancy. That depends upon
the nature of the order parameter. In particular, given a set of broken generators Xa a
redundancy exists when there is a nontrivial solution to the equation
a(x)XahO(x)i = 0
Ki = P it
M xi
J i = ijkxj P k
where hO(x)i is the order parameter. As an example consider the symmetry breaking
pattern for a metal. The lattice breaks rotations, translations and boosts. The boost
Goldstone i and rotation Goldstone i can be easily seen to be redundant since
and
3The dispersion relation need not be limited to these two choices. Higher order relations are possible in
in [21] are due to integrating out elds with analytic dispersion relations.
thus, assuming that the mass M is unbroken, i.e. no condensation, we have
(1.9)
so that both rotations and boosts can be compensated for by a Goldstone dependent
translation [25].
In any case when condition (1.5) is satis ed, it is often possible to impose a constraint
on the elds which is consistent with the symmetries. This constraint is called the Inverse
Higgs Constraints (IHC) which is associated with the IHM.
1.1
The missing Goldstones
As was pointed out in [10] there are cases for which there is no inverse Higgs constraints
and yet the Goldstones still do not appear. If particle number is spontaneously broken,
then due to the fact that [P; K] / M , there is an IHC which allows one to eliminate the
boost Goldstone. But if there is no IHM involving the boost generator, one must include
the boost Goldstone in the analysis. In [10] the authors considered two such symmetry
breaking patterns called typeI and typeII \framids". The former is a system in which the
only broken symmetry is boost invariance while the latter also breaks rotations. A cursory
check of the Galilean algebra shows that none of the broken generators satisfy (1.5) in these
cases and yet the Goldstones associated with boosts, dubbed the \framons", are nowhere
to be seen in nature.
Galilean boosts as
Another missing Goldstone boson arises in the case of nonrelativistic dilatation
invariance. The authors of [26] point out that given that the dilaton
transforms under
there is no way to write down a boost invariant kinetic term4 since the time derivative
of the dilaton transforms nontrivially. As also pointed out in [26], if the U(1) of particle
number is broken then, as a consequence of the algebraic relation,
(x; t) !
(x
vt; t)
[Pi; Kj ] = i ij M
(1.10)
(1.11)
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boost invariance is also broken (assuming translations are unbroken). As such, there is an
IHM at play and the Ward identities may be saturated without the need for a dilaton. This
begs the question, can one write down a sensible dilaton kinetic term if there is no particle
condensate? The answer, as will be discussed below, is yes as long as the framon is included
in the action. So we see that the questions of the framon and the dilaton are intimately
connected. Thus the puzzle of the nonrelativistic dilaton remains, and its resolution is
tied to the fate of the framon.
As will be discussed below a resolution of the framon puzzle is closely related to the
fact that when spacetime symmetries are broken, Goldstone bosons need not be
derivatively coupled. To explore this possibility we utilize the coset construction which will
4Matter elds transform as projective representations under boosts which allow for the canonical kinetic
energy term.
{ 4 {
allow us to generalize the criteria for nonderivatively coupled Goldstones given in [5].5
Furthermore, we will use the coset methodology to construct the theory of Fermi liquids
with the symmetry breaking pattern of type I/II framids as realized by a canonical Fermi
liquid without/with nematic order. We will see that the resolution of the framon issue
follows from the dynamics of the e ective eld theory. By treating the Goldstone boson
as a Lagrange multiplier we will generate a set of constraints, that are generalizations of
the Landau conditions in canonical Fermi liquids, which when imposed, lead to the proper
symmetry realization. We dub this the \Dynamical Inverse Higgs Mechanism" (DIHM),
because the Goldstones are absent but not for algebraic reasons. We will then go a step
further and discuss a more general symmetry breaking pattern where both boost invariance
and Schrodinger invariance are spontaneously broken, which we call a \typeIII framid".
In this case one might expect both a framon and a nonrelativistic dilaton to arise. We will
see that again, they do not, but their absence greatly constrains the form of the e ective
eld theory. This analysis was recently used to prove that degenerate electrons interacting
in the unitary limit can not behave like a Fermi liquid [6] in the unbroken phase.
1.2
The paths to symmetry realization
We see that there are three paths to spacetime symmetry realization: no inverse Higgs
constraints are applied and the system retains one Goldstone for each broken generator.
Some or all of the constraints are applied and we have a reduced number of Goldstones due
to the existence of IHCs, or a Goldstone can be eliminated via the DIHM with or without
the application of other inverse Higgs constraints. In this paper we will consider all three
scenarios in the context of degenerate fermions. We will show two examples of DIHMs, one
for boosts and the other for dilatations.
2
Review coset construction
A powerful method for generating actions with the appropriate nonlinearly realized broken
symmetries was developed for internal symmetries by CCWZ [22, 23] and later
generalized to spacetime symmetries by Volkov and Ogievetsky [27, 28]. We refer the reader to
original literature for details and here only rapidly review the salient points of this coset
construction. The method uses the fact that the Goldstones coordinatize the coset space
G=H where G is the symmetry group of the microscopic action and H is the symmetry
subgroup left unbroken by the vacuum. The vacuum manifold is parameterized by
U = ei~ X~
(2.1)
where ~ are the Goldstone elds and X~ the corresponding broken generators. The
unbroken generators will be denoted by T~ . This parameterization will be generalized when
we break spacetime symmetries. As discussed below, we may use U to write down the
5We generalize [5] in two ways. Ref. [5] states that there can be no Goldstone associated with boost
invariance due the nonvanishing commutator between boosts and the Hamiltonian. Here we show the need
for the boost Goldstone and show that it couples nonderivatively. The coset methodology also allows us
to consider relativistic generalizations.
{ 5 {
mostgeneral action consistent with the symmetry breaking pattern, including terms where
the Goldstone couples to other gapless (nonGoldstone) elds in the theory. Notice that
the coset construction seems to imply that there must be at least one Goldstone boson.
However, this need not be the case, as mentioned above. It could very well be that we can
construct an invariant action without the need for a Goldstone, even without an inverse
Higgs constraint. We will show that if this is indeed possible then the coset construction
is a useful tool in determining this nonGoldstone action.
Once spacetime symmetries are broken, the symmetry group is no longer compact.
As such the structure constants can not necessarily be fully antisymmetric6 and
consistency requires that one generalize the vacuum parameterization to include the unbroken
translations (P ),7 such that
U = eiP xei X
:
such that under a boost
rA
rA
! e 2i mv2t im~v ~x
rA :
The number of unbroken translations may be enhanced if there exist internal translational
symmetries as in the case of solids or uids [29]. In such cases the direct product of the
internal and spacetime translations are broken to the diagonal subgroup by the solid. In
this work we will not be considering such cases as we are interested in zero temperature
ground states with delocalized particles.
The MaurerCartan (MC) form decomposes into a set of well de ned geometrical
objects,
U 1
@ U = EA(PA + rA aXa + AbAT b):
The vierbein E relates the global frame to the transformed (acted upon by G=H) frame.
In this way, the covariant derivatives on the matter elds in the local frame are written as
From (2.3) we can extract the vierbein, the covariant derivative of Goldstone elds
(r ) and the Gauge elds (A) and use these objects to construct our action which will
be invariant under the full symmetry group G by forming H invariants. For a complete
discussion of the coset construction and its application to broken spacetime symmetries in
multiple contexts, we refer the reader to [32].
3
Nonderivatively coupled (NDC) Goldstone bosons
In [5] the criteria necessary to generate theories with nonderivatively coupled Goldstones
is given by
6A consequence of this fact is that it is not longer true that [T; X] = X.
7When translations are broken by localized semiclassical objects (i.e. defects) the coordinate is lifted to
the status of a dynamical variable see for instance [30{32].
[Xi; P~ ] 6= 0:
{ 6 {
(2.2)
(2.3)
(2.4)
(2.5)
(3.1)
where Xi is a broken generator and P~ are the unbroken spacetime translations. The
authors argue that the forward scattering matrix elements of broken generators X formally
diverge
~k0!~k
lim h~k j X j ~k0i ! 1:
which compensates for the explicit factor of the Goldstone momentum in the coupling. One
may be concerned with the fact that X is not a well de ned operator at in nite volume,
and that the limiting procedure is not well de ned. However, we will see below that the
coset construction supports the authors claims and allows us to, trivially, generalize their
criteria to relativistic systems. Eq. (3.1) is a necessary but not a su cient criteria for the
existence of a nonderivatively coupled Goldstones since we must also ensure that it can
not be removed via an IHM.
Within the coset formalism the search for nonderivative couplings starts with
understanding how the Goldstones couple to generic matter
elds. As such, we need to
determine under what conditions a Goldstone arises in the vierbein or connection without
any derivatives acting upon it. Thus a necessary condition for nonderivative coupling is
the generalization of (3:1), i.e.
[P ; X] 6= 0:
Note the distinction between this criteria and (3.1). First (3.3) only involves the unbroken
canonical spatial translations P which can di er from P , not only because of the zero
component, but more generally if there are internal translational symmetries.
This is
however, a distinction without a di erence because internal and spacetime symmetries
commute. But an important distinction between (3.1) and (3.3) is the fact that (3.3) allows
for the noncommutation with the Hamiltonian as being a criteria for NDC Goldstones.
As a matter of fact, this explains the NDC nature of the dilaton (both relativistic as well
as nonrelativistic8). Also we will see that whether or not G is the Poincare or Galilean
group is of no consequence as far the the criteria for nonderivative coupling is concerned.
To see that (3.3) is a su cient criteria for NDC, assuming the Goldstone boson
associated with X is not removed by the inverse Higgs mechanism, we note that the veirbein
will contribute to the measure via
S =
Z
4 p
d x E2 : : :
(3.2)
(3.3)
(3.4)
so that as long as the determinant of the vierbein contains a term linear in the Goldstone,9
there will be a NDC to matter elds. From (2.3) we can see that if [P ; X]
P then the
Goldstone associated with X will arise in E. However, the Goldstone will often be absent
from the volume factor as in the case of broken boosts or rotations. Thus the rst NDC will
come from the covariantization of the derivatives Ea ( )@ . Alternatively if [P ; X]
T ,
then the Goldstone will show up in the connection, in which case the NDC will arise from
the covariant derivative acting on the matter elds.
8The nonrelativistic case being of particular importance below.
9That E contains term linear in the Goldstone follows from the fact that the Goldstone acts as the
transformation parameter.
{ 7 {
Finally, note that if G is the Galilean group then due to relation the eq. (1.11) if the
U(1) particle number is unbroken, then the boost Goldstone will be associated with the
connection. Whereas if G is the Poincare group then the boost will be in the vierbein. But
in either case framid will be nonderivatively coupled.
broken, but all other space time symmetries are intact. The coset construction only cares
about the symmetry breaking pattern and not the de nite choice of the order parameter.
As was emphasized in [20] the choice of order parameters can a ect how the symmetry
is realized if there exist gapped Goldstones (assuming the gap size is hierarchically small
compared to the cuto ). In particular the representation of the order parameter(s) will
determine whether or not the the inverse Higgs conditions (1.5) leads to a redundancy or
a gap. However, here we are only interested in the truly gapless modes, so in this respect
the order parameter will be irrelevant. Nonetheless, we are interested in a certain class of
order parameter, i.e. those whose commutator with boost generators have a nonvanishing
vacuum expectation value (e.g. the momentum density). This class of order parameters
yield Goldstones which are collective excitations.
Whereby a \collective excitation" we
mean a quasiparticle pole (or resonance) which exists as a consequence of the fact that
the vacuum is not annihilated by some conserved charge. Put another way, the modes are
excitations of the material responsible for the breaking of boost invariance. This de nition
sets apart say the pion in QCD from the plasmon in a metal.
Cases where the framid are not collective modes correspond to speculative theories
beyond the standard model of particle physics and Relativity, such as EinsteinAether
theory [34], where a four vector gets a timelike expectation value.
hA i = n :
(4.1)
The resulting theory contains 3 Goldstone modes corresponding to the framons [35]. The
lack of the evidence for a Goldstone arising in EinsteinAether theory allows us to place
bounds on the couplings (see for instance [36]). However, we know that condensed matter
systems break boost, and if the symmetry breaking pattern is such that there are no IHC
around to eliminate the framids from the spectrum it is incumbent upon us to determine
their fate.
marginal.
It is tempting to disregard boost Goldstones since the associated generator does not
commute with the Hamiltonian and hence there is no at direction. However, the existence
of the relativistic dilaton immediately dispels this notion. Furthermore, the inclusion of the
framid into the coset parameterization is necessary for consistency. Moreover, according
to the criteria for NDC (3.3) we should expect the coupling to the framid to be at least
{ 8 {
scales as
density
Tc
?
EF
TC
To manifest framids in the laboratory we need systems which break boosts yet whose
ground state does not break any symmetry which would lead to an inverse Higgs
constraint. Thus we may eliminate electrons moving in a crystal background as well as
super( uids/conductors) from the list of possibilities. It would seem that we are relegated
to degenerate electrons in the unbroken phase. One might be concerned that the Kohn
and Luttinger [37] e ect ensures that all Fermi liquids superconduct, even if the coupling
function is repulsive in all channels in the UV. However, all we really need to manifest
a framon is for there to be a temperature window between the boost symmetry breaking
scale (EF ), and the critical temperature Tc. For a Fermi liquid the critical temperature
where
? is the strong coupling scale which is typically exponentially suppressed. Thus
there is a range of temperatures where the framid should contribute to the heat capacity.
This is as opposed to the bosonic case where the critical temperature is set by the number
(4.2)
(4.3)
(4.4)
(4.5)
(4.6)
(4.7)
(4.8)
HJEP05(218)4
I framid
and the boost symmetry breaking scale is of the same order.
Thus we have narrowed our search for framons to degenerate Fermi gases whose
phenomenology certainly shows no signs of nonderivatively coupled Goldstone. One might be
tempted to interpret zero sound as the boost Goldstone, however, the interaction between
electrons due to zero sound exchange vanishes in the forward scattering limit.
4.2
Coset construction of Fermi liquid EFT with rotational symmetry: type
We begin our investigation by building the coset construction for type I framids (i.e. systems
with broken boosts but unbroken rotations).
We consider the case of broken Galilean invariance, as the relativistic case will follow in a similar manner. The vacuum manifold is parameterized by
U = eiP xe iK~ ~(x)
Calculating the MCform, we can extract the vierbein
The gauge eld is given by
{ 9 {
and the covariant derivatives of the framids are (up to lowest order in elds and derivatives)
The free action for the Goldstone follows by writing down all terms which are invariant
under the linearly realized H symmetry
S =
Z
ddxdt
2
)
2
Following eq. (2.4), the coupling for the Goldstone to matter elds via the covariant
derivative is given by
Z
S0 =
1
2
where " is the unknown dispersion relation that is xed by the dynamics. Due to the
central extension of the Galilean algebra, the fermion under a boost transformation with
velocity ~v transforms as
(4.9)
(4.10)
(4.11)
HJEP05(218)4
(4.13)
while the Goldstone eld
undergoes a shift
(x; t) ! e 2i m~v2t im~v ~x (x; t):
~ ! ~ + ~v:
The 2 term will be subleading and not play role in the remainder of our discussion.
As in the standard EFT description of Fermi liquids [39{41] the quasiparticle self
interaction is most conveniently written in momentum space
ddkidt g(~ki + m~) ky1 (t) k2 (t) ky3 (t) k4 (t) d
X ki
i
(4.12)
Higher order polynomials in the matter eld
are technically irrelevant (see below). g is
the coupling function which now formally depends upon the framon. The assumption of
spherical symmetry implies g is a scalar. Notice that the
is nonderivatively coupled,
as expected from our considerations of the algebra, which can lead to non Fermi liquid
behavior. Given that He3, e.g., is well described by Fermi liquid theory, the framid must
somehow decouple, yet it must do so in such a way that the theory remains boost invariant.
4.3
Multiple realizations of broken symmetry
Before moving onto further discussion about the framids in Fermi liquids, we want to
highlight a subtle point about nonlinear realizations of broken symmetries, which is that the
same symmetry breaking pattern can lead to contrasting physical theories with very di
erent particle content. This usually happens when there are two di erent order parameters.
However, below we show that even with same order parameter we can have two di erent
realizations of the symmetry. An example of this is the case of a massive complex scalar
particle ( ) coupled to gauge elds. To power count this theory it is useful to introduce the
notion of a eld label as was introduced in Heavy Quark E ective Theory (HQET) where
one is interested in the dynamics of a massive source which interacts with light gauge elds
carrying momenta much less than the quark mass. The label is introduced by de ning a
rephased eld
such that v de nes a superselection sector [42]. Derivatives acting on hv(x) scale as
\residual momenta" (k) which obey k
m. The vacuum of the system, labeled by v, breaks
boost invariance and so we expect that framid should exist as an independent degree of
(x) =
X eimv xhv(x);
v
freedom. Typically, the Goldstone modes are associated with collective excitations of a
system which are clearly absent as the choice of vacuum is not dynamical. Nonetheless the
boost invariance must be nonlinearly realized.
Using the covariant derivatives derived in the previous section, we can write down the
most general action for
which is invariant under translations and rotations,
L
=
i
2
+
i
2
2m
im~) (t; ~x):
m~2) (t; ~x)
m~2) y(t; ~x)] (t; ~x)
i
2
(4.14)
HJEP05(218)4
If we choose c1 = 1 then
decouples from
and we get the standard nonrelativistic
kinetic term for a free particle. Had we started with a theory without the , then c1 can
be xed by requiring the theory to obey Galilean algebra, in particular by satisfying the
commutator [H; Ki] = iPi. c1 can equally well be
xed by Reparametrization Invariance
(RPI) [44], which is related to the freedom in splitting the heavy quark momentum into
a large and small piece (more on this below). However we can leave c1 to be completely
arbitrary and keep
in the spectrum and the theory will still respect all the symmetries.
The two theories (with and without ) are completely di erent and we have no reason to
believe they will lead to same physics in the IR and yet they have the same symmetry
breaking pattern and the same order parameter (local momentum density). Thus there are
multiple ways of realizing the boost symmetry. While it would seem that this is a rather
trivial example, we note that only di erence between the HQET ground state and that of
a Fermi liquid lies in the change in the number density from one to Avogadro's number.
4.4
Power counting
To determine the possible symmetry realization in a Fermi liquid, we must rst discuss the
systematics of the relevant EFT whose action is given by (4.9). The matter elds (which
we will call electrons from here on) are e ectively expanded around the Fermi surface, by
removing the large energy and momentum components via the rede nition
(x) =
X ei"(kF )te i~k( ) ~x ~k( )(x);
(4.15)
the assumption of rotational invariance implying that the magnitude of j ~k( ) j= kF . The
eld label ~k( ) is the large momenta around which we expand. As opposed to the HQET
case, here the bins are dynamical and there is no superselection rule. This case is more
akin to NRQCD [43] where the labels change due to Coulomb exchange. Notice that
there is a sum over the labels as opposed to an integral, this illustrates the fact that
we have e ectively tessellated the Fermi surfaces into \bins". The size of each bin will
scale as
E=EF . The fact that theory should not depend upon the bin size imposes
constraints on the action. That is, we should be able deform the momentum around any
xed value we wish, by an amount scaling as , and the theory should be invariant. This
reparameterization invariance (RPI) [44] implies that the action can only be a function of
~
kF + @~. In general RPI generates relations between leading order and subleading Wilson
Full theory derivatives then decompose into the RPI invariant combination P~ + i@~. In
this way we may drop the exponential factors as long as we assume label momentum
conservation at each vertex. The action becomes
S0 =
Notice that the interaction with
does not change the quasiparticle label. The reasons
for this will be discussed below once we have xed the power counting systematics. Under
a boost the labels are left invariant but the residual momentum shifts. Furthermore under
a boost the time derivative transforms as
coe cients. It is convenient for power counting purposes to introduce a label operator
P [45] such that
P~ ~k( )(x) = ~k( ) ~k( )(x):
(4.16)
(4.18)
(4.19)
(4.20)
HJEP05(218)4
4.4.1
Review of EFT of Fermi liquids scalings
We rst review the EFT of Fermi liquids and its power counting (for details see [38{41]).
In the EFT, the power counting is such that the momenta perpendicular to the Fermi
surface (k?) scale as
E= , where the theories' breakdown scale is
EF . With
this scaling, the most relevant terms in the action come from expanding the energy and
coupling function around the Fermi surface and keeping the leading term in k .
?
The scaling of the electron eld
(~k; t)
1=2;
10We are ignoring spin as it will not play a role in our discussion.
follows from the equal time commutator
f (~k; t = 0); y(p~; t = 0)g
(k?
p?) d 1(kk
p )
k
1=
since kk does not scale. Thus ignoring the framon for the moment, the leading order action
is given by10
SFL =
X Z
~
k
+ X Z
~
ki
ddldt ~y (t; l)(i@0 + ~l? ~vF )
k
~k(t; l)
2
ddlidt
g(~ki) ky1 (t; l1) k2 (t; l2) ky3 (t; l3) k4 (t; l4) d
X li :
i
(4.21)
back to back con guration which leads to Cooper pair condensation. (b) Forward scattering, in
which the nal state momenta lie on top of the initial state momenta.
sTuhrefaFcee.rmIni vtheleolcaitsyt tdeermnetdhearse ~visFa=Kr@@ok"n?iejkcFkerisdceoltnastfaonrtthone laabseplhmeroicmalelnytasytmhamteitsriicmFpeliremd.i
The residual momenta scale as l?
and lk
1. The latter scaling might seem odd given
that it is a residual momentum. However, lk scales as the bin size, which does not play
a role for Fermi surfaces which are featureless Another way of saying this is that the lk
integral can be absorbed into the label sum.
Naively the interaction terms looks irrelevant because the delta function scale as 0
for generic kinematic con gurations so that, once the scaling of the measure is taken into
account (
3
), the operator will scale like . However, there are two con gurations
for which one of the delta function will scale as 1= : the BCS con guration (back to
back incoming momenta) and forward scattering. These two con gurations are shown in
gure 1, where it can be seen that these are the only two possible con gurations that allow
for momentum conservation that keep all momentum within
of the Fermi surface.
It is convenient to decompose the BCS coupling into partial waves gl. A one loop
calculation (which is exact) of the beta function shows that gl are either marginally
relevant/irrelevant for attractive/repulsive UV initial data. The forward scattering coupling
does not run, but plays an important role in the IR nonetheless. Interestingly, below we
will show that Galilean invariance is su cient to prove that the forward scattering and
BCS kinematics are the only possible marginal/relevant interactions. This result follows
without the need to consider the e ects of the special kinematics on the power counting of
the four Fermi operator.
4.4.2
Power counting in the coset construction
Let us now derive the power counting from the coset construction. We begin with the
kinematics of the framid interactions. The two allowed scattering con gurations are shown
in
gure 2. Figure (a) shows the interaction of a quasiparticle with a framid that is far
o its mass shell in the sense that E
k, in the EFT language this would be called a
\potential framid" and can be integrated out. Thus these interactions are swept, along
with those of the phonon and screened electromagnetic interactions, into a nonlocal
coupling. Note that the potential is e ectively local because the labels on the incoming and
outgoing quasiparticles can not be the same and hence it is analytic in (the small) residual
S0 = X Z
~k( )
= X Z
~k( )
Sd=2 =
0
X Z
~k( )
volves an o shell framid, which can be integrated out. (b) shows the interaction with a soft framid
leading to near forward scattering.
momenta.11 Figure (b) shows the interaction with an onshell framon whose momentum
is necessarily soft k
E
then we only know that k
EF : If we de ne our power counting parameter as
E=EF ,
n, where n is yet to be determined. However, symmetries
x n as the covariant derivative must scale homogeneously in
for the theory to be boost
invariant. That is,
must scale in the same way as the residual momentum of order , so
that
n we can x n by considering the canonical commutator
[ i(x); _j (0)]
n+2
d(x) ij
dn
(4.22)
thus n = d 1
2 . Thus we see that in two spatial dimensions k
2 and the framons can
not change the (residual) momentum of the quasiparticles and only their zero mode is
relevant. This however is not the case in three dimensions where the framon carries o
residual momentum k
.
Expanding the action (4.17)
~(x) ~vF ( )(m
(4.23)
~k( )(x) + : : : (4.24)
multipole expand the framon eld to preserve manifest power counting [46], which leaves
only the coupling to the framon zero mode. The leading order action is given by
~(0) ~vF ( )(m
i
~k( )(x) + : : :
(4.25)
From here on to simplify the notation we will be dropping the label sum and the bold
font for labels as all momenta unless stated otherwise will be labels.
11In this sense it is better to think of the 1=k2 in the interaction as a Wilson coe cient.
Before we move on to determine the consequences of the multipole expansion let us
pause to clarify this unusual scaling. Typically in an EFT the scaling of the elds follows
from the scaling of the momenta not the other way around as in this case. Indeed, it
would be useful to understand what happens to loops with momenta scaling as
and
not 2
. However, symmetries forbid such contributions and it must be that if we do not
multipole expand the framon interaction, that power counting and boost invariance are
incompatible. Thus we see that in two spatial dimensions, the symmetries can not be
realized via a Goldstone as the framon equations of motion allow us to eliminate it from
the theory, as will be discussed below. In three spatial dimensions this conclusion does
not follow.
The framid as Lagrange multiplier and the Landau relation
Let us consider the rami cations of the multipole expansion of the framon in two
dimensions.12 Since the kinetic piece of the Framon action vanishes for the constant zero mode
plays the role of a Lagrange multiplier.
Expanding the action for the fourFermi interaction term leads to the coupling
@g(kj ) ky4 (t) ky3 (t) k2 (t) k1 (t) d
X ki :
i
(4.26)
Using the equations of motion for
gives the operator constraint OiB = 0 where
OiB =
i
Z
ddp
(2 )d
m Z
2
py(t) pi
4
Y ddpa (d)
a=1 (2 )d
m
X pi
i
p(t)
i
py4 (t) py3 (t) p2 (t) p1 (t) : (4.27)
This is a strong operator constraint both technically and colloquially. Notice that the
constraint is nonlocal in the sense that it is integrated. This is crucial, as the constraint
is a function of the Noether charges. Indeed, current algebra imposes this same constraint
OiB = 0 as shown in the appendix where we also derive the relativistic generalization of
this constraint.
The power counting of the terms in this constraint deserve attention. The rst two
terms scale as 0 while the last term naively scales as 2 as the measure scales as 4. Recall
that at this point we have not made any assumption about special kinematics so the delta
function does not scale. We are trying to derive the fact that the only relevant couplings
have these special kinematics. Thus we might naively think that we can drop the quartic
term in the constraint. In general this is true, but there is an exception as we now explain.
We begin by noticing that the quartic term is time dependent while the quadratic terms
(being conserved charges) are not. Thus it would seem that the last term must vanish (to
the order we are working). However, if we insert the quartic term in a two point function
the time dependence will cancel.
generalize it latter to d=3.
12Even though our arguments in this section are strictly valid only for d=2, we keep d arbitrary to
Consider taking matrix of element of (4.27) in a 1particle state with momentum ~k
(j~kj = kF )
ki = m
ddp
+
(4.28)
We can now see why that the interaction term is enhanced because the radial integral,
naively scaling as
is actuality scaling as order one. This is a consequence of the power
divergence of the integral. Such mixing of orders is commonplace in e ective eld theories
when a cuto regulator is used. However, here the cuto (the radius of the Fermi surface)
is physical.13
ddp
(2 )d
(pF
p)g(p; k) + 2m
Z
ddp
(2 )d g(p; k) ("F
")
: (4.29)
The second term on the r.h.s. vanishes by spherical symmetry.
tion in Legendre polynomials, g( ) = Pl glPl(cos ),14 we get
Next using the assumption of rotational invariance, and expanding the coupling
funcUsing this result we get the famous Landau relation [47] for a Fermi Liquid
m
kF = vF +
2pF Z
(2 )2
d cos
X glPl(cos ):
l
m?
m
= 1 +
1 2m?
3 (2 )2 g1
(4.30)
(4.31)
Notice that at this point it is not clear that this result will hold to all orders in perturbation
theory.
its scaling.
It is interesting to ask whether or not more information can be extracted from the
constraint by considering a two body state. However, as is seen by inspection the insertion
of the constraint operator on external lines will will automatically be satis ed once the
Landau condition is imposed, and furthermore the insertion of the quartic function into a
four point amplitude will be suppressed since there is no power divergence that can enhance
We can glean more information from the Landau criteria by utilizing the fact that the
equation is RG invariant. Di erentiating it with respect to the RG scale implies that the
beta function vanishes. For generic momenta the four point one loop interaction diverges
logarithmically. To avoid this conclusion we impose a kinematic constraint to suppress the
one loop result. If we consider the forward scattering interaction,
SF =
Z
ddxdt X g( 1; 2) ky( 1)(x; t) k( 1)(x; t) ky( 2)(x; t) k( 2)(x; t)
(4.32)
k( i)
13In canonical EFT's one uses dimensional regularization exactly to avoid this mixing issue which
complicates the power counting.
14We take d=2 for sake of simplicity but the results are valid for arbitrary d.
then the one loop result vanishes since the constraints imply that the loop involves no sum
over the large label bins leading to a power suppressed result.
It would seem that we have ruled out the possibility of a BCS interaction which has a
nonvanishing beta function at one loop. However, this is not the case as such an interaction
would not contribute to the Landau relation since the tadpole diagram vanishes for the
BCS interaction.
Thus we have reached the conclusion that the only allowed interactions are BCS and
forward scattering. We are not claiming that this is a rigorous proof since we have
assumed that the only sensible coupling with vanishing beta function is forward scattering.
Furthermore, our argument regarding the acceptability of the BCS coupling is based on
the fact that our arguments allow for any coupling which leads to a vanishing tadpole
(with no associated counterterm). It is possible that there are other allowed kinematic
con gurations, however, assuming a featureless Fermi surface15 we have not been able to
nd any sensible examples.
Recall at this point the result in the section only hold at one loop. However, now
that we have restricted our interactions to BCS and forward scattering we know that that
the Landau relation holds to all orders. This well known result follows from the fact
that tadpole corrections to the one loop insertion of the constraint are pure counterterm
and vanish.
Finally recall that this result assumed that the framon acts as a Lagrange multiplier.
However, this was only forced upon us in two dimensions. In three dimensions, there is
the logical possibility that the framon remains in the spectrum and there is no DIHM at
play. This will be discussed below when we list the possible paths to symmetry realization
in three dimensions.
5
Fermi liquid with broken rotational invariance
Let us now consider the case where the rotational symmetry is broken by the Fermi surface
(the typeII framid). We work in two spatial dimensions for the sake of simplicity. Again,
to avoid an algebraic inverse Higgs constraints, we assume that the U(1) particle number
is unbroken. The vacuum will be parameterized by
U (~; ; x) = eiP xe iK~ ~(x)e iL (x)
The rotational Goldstone boson ( ) is called the \angulon" has been studied in the context
of electronic systems [1] as well as in neutron stars [48], although to our knowledge its
nonlinear self interactions have not been previously derived.
Calculating the MCform we may extract the vierbein
where R( ) is the two dimensional rotation matrix. The gauge elds are
Ai =
Rij ( ) j ;
A0 =
15In cases where the Fermi surface is singular there are other relevant interactions whose self contractions
would vanish [49] algebraically.
(5.1)
(5.2)
(5.3)
r0
Z
the covariant derivatives of the angulons are
The quadratic piece of the quasiparticle action is given by
ri
(5.4)
S
m~2 + "(R( )ij (i@j + m j ))
:
(5.5)
1
2
The kinetic piece of the angulon Lagrangian consistent with time reversal and parity
invariance is given by
LKE = ( _ )2 + Dij (ri )(rj );
(5.6)
so the angulon is a \type I" Goldstone, i.e. E
p. Unlike the framid, the angulon
scaling is not xed by symmetry and its momentum scaling is determined by the maximum
momentum transfer consistent with the e ective theory, i.e. the scattering of an electron
with an angulon should leave the electron near the Fermi surface to within
thus the
angulon momentum scales as
and following the same arguments as above the eld (x)
1=2 in two spatial dimensions and as
in three.
Expanding the action (5.5) and keeping on the leading order piece we have
S
i
:
(5.7)
We see that for d = 2, the interaction with the angulon is relevant and thus destroys Fermi
liquid behavior. In d = 3 it is classically marginal and the fate of Fermi liquid behavior is
determined by the sign of the beta function for this coupling.
Notice that the breaking of rotational symmetry does not e ect the operator
relation (4.31) imposed by the nonlinearly realization of boost invariance. However, at least
in two spatial dimensions, the Landau relation (4.31) is no longer justi ed, as the angulon
coupling becomes strong in the IR and quasiparticle picture breaks down. In three
dimensions it is possible that a perturbative result for the Landau relation could follow if the
theory remains weakly coupled. In any case the operator constraint (4.27) must hold for
the system to be boost invariant. However, in strong coupling it is not easy to deduce the
physical rami cations. It would be interesting to utilize this constraint to generate new
prediction in systems with broken rotational symmetry. In particular it is interesting to
ask whether or not one can impose a DIHC to eliminate the angulon from the spectrum.
5.1
The stability of Goldstone boson mass under renormalization
As can be seen from the actions (4.8) and (5.6), a Goldstone boson mass is forbidden
despite the fact that the Goldstone boson need not be derivatively coupled. There are
no gapped Goldstones as a consequence of the fact that there is no inverse Higgs
mechanism for our chosen symmetry breaking pattern. If there is no anomaly then we should
expect that this masslessness should persist to all orders in perturbation theory, indeed
it should hold nonperturbatively. Vishwanath and Watanabe showed the cancellation of
angulon mass correction at one loop [5] but they did not consider the framon. Given that
could contribute to a mass. At zero external momentum the two diagrams cancel as dictated by
we have constructed the full action, the all orders proof follows from the Ward identity.
Nonetheless is it instructive to study the one loop case in order to distinguish the framon
from the angulon. The Goldstone mass can be read o
by considering the quadratic piece
of the e ective action generated by integrating out the electrons in a constant Goldstone
background. For the angulon we nd
(5.8)
(6.1)
(6.2)
R d!ddpLog [!
R d!ddpLog [!
"(R( )p~)]
"(p~)]
which is independent of as a consequence rotational invariance of the measure. This result
tells us that, at the level of the integrals, there must be an algebraic cancellation between the
two diagrams which contribute to the mass at one loop shown in gure 3. Note this should
NOT be expected for the framon, since boost symmetry breaking is sensitive to the UV
scale EF , whereas the angulon only knows about the shape of the Fermi surface and not its
depth. Of course, boost invariance dictates the framon mass must vanish if we use a boost
invariant regulator, i.e. not a cuto . The situation is analogous to the case of the dilaton
whose mass corrections vanish in dimensional regularization but necessitates counterterms
when using a cuto . Such counterterms should not be considered
ne tuning.
6
Broken conformal symmetry: eliminating the nonrelativistic dilaton
As mentioned in the introduction, consequences of spontaneous breaking of conformal
invariance in nonrelativistic systems is unique as the nonrelativistic kinetic term for the
dilaton appears to be in tension with boost invariance [26]. As such, we will study systems
for which the broken symmetries are dilatations (D), special conformal transformations (C)
and boosts (Ki). The relevant commutators of the Schrodinger group (the nonrelativistic
conformal group) are
as these relations imply a reduction in the naive number of Goldstones. Furthermore,
note that (6.1) implies that if dilatations are broken then so are the special conformal
transformations. The vacuum is parameterized via
U = eiP xe iK~ ~e iC e iD :
The algebra implies that both and are redundant degrees of freedom. The ensuing
HJEP05(218)4
vierbein is given by
The gauge elds are and the covariant derivatives are given by
E00 = e 2
E0i = i
e
i =
)
=
~
0
0
0
~x + t ~
t
2 t
(6.3)
(6.4)
(6.5)
(6.6)
(6.7)
(6.8)
(6.9)
(6.10)
(6.11)
(6.12)
(6.13)
The invariance of these objects under boosts, dilatations and special conformal
transformations follows by
rst determining the nonlinear transformation properties of the
Goldstones via the relation
gU ( ) = U ( 0; g)h(g; );
where g 2 G and h 2 H. Table 1 gives the resulting transformation properties of the
Goldstones.
We see that there are two possible inverse Higgs constraints coming from setting the
covariant derivatives in (6:6) and (6:8) to zero. Linearizing yields the two possible inverse
Higgs relations from (6:6) and (6:8)
Let us now address the question of the possible symmetry realizations. We will see that
no matter what path is chosen, the systems will not behave like a canonical Fermi liquid [6].
=
=
1
3
_ + : : :
We may choose not to eliminate any Goldstones, however note that in this case, the
gets
gapped as (6.10) is time reversal invariant and thus an allowed term in the action without
squaring it. This realization includes two nonderivatively coupled Goldstones which would
invalidate a Fermi liquid description [5]. If we use one IHC then again we will have the
same spectrum and the same conclusion is reached. Finally we may consider using both
constraints such that we equate
which would lead to a theory which appears nonlocal.16 Thus, although we have two
possible constraints we can only impose one while maintaining locality. This is a consequence
so that the two constraints are linked establishing the fact that the criteria for Goldstone
elimination stated in (1.5) must be amended. If two of the relation involve the same
generator on the l.h.s. then there is one fewer allowable constraint. We know of no other
cases where this happens. The
nal possibility is that we eliminate both
and
using
DIHMs as discussed in the next section.
Consequence of broken conformal symmetry via the DIHM
To derive the relevant DIHCs we will again build the coset and treat both the dilaton and
the framon as Lagrange multipliers. As in the previous cases, in two spatial dimensions
this is not a choice as a consequence of power counting and symmetry. Notice that
will
not play a role as it shows up neither in the vierbein nor the connection. We have already
written down the most general boost invariant interaction in (4.17) and (4.26) which we
now amend using the new version of the vierbein and gauge eld (6.4), (6.5). The invariant
action for the quasiparticle is given by
S0 =
Z ddpdt 2
(2 )d
e
h
e 2 ~ ~p + "~(e
i
(p~ + m~)) + F
~p(t);
(6.16)
Here the energy functional "~(p) is the energy of the quasiparticle measured from the
Fermi surface since we have explicitly included the chemical potential F in the action.
For notational convience we will drop the explicit factor of
F and rede ne the energy
functional as "(p) =
F + "~(p). As far as the interactions are concerned we have
Sint =
4
aY=1 (d2dp)ad dt (d)(p1 + p2
p3
p4)e(2 d)
g(e
~pi + e
m~i; e
) ~py1 (t) ~p2 (t) ~py3 (t) ~p4 (t):
(6.17)
16This nonlocality in EFT arises due to a poor choice of variables and is not in any sense fundamental
since the underlying theory is local.
~
k
ddpdt
~py(t) 2"(p)
(2
d)g(~pi; )
~pi
p
~p(t) +
The constraint follows from imposing S
= O
= 0.
O
~
k
ddpdt
~py(t) 2"(p)
(2
d)g(~pi; )
~pi
p
~p(t) +
ddpadt
ddpadt
~py1 (t) ~p2 (t) ~py3 (t) ~p4 (t) = 0
and
Let us now see if a Fermi liquid description is consistent with these constraints. Given
our assumption of rotational invariance and the notion of a well de ned Fermi surface,
the marginal coupling is only a function of the angles which are scale invariant. Thus the
second term in the last line of (6.19) vanishes, and, as such, if we take the one particle
matrix element we see that the quadratic and quartic terms must vanish separately since
the quadratic term will depend upon the amplitude of the incoming external momentum
and the quartic will not. In three dimensions we see that the coupling has power law
running which is inconsistent with Fermi liquid theory, and in two dimensions the theory
is free. Thus we conclude that: fermions at unitarity are not properly described by Fermi
liquid theory.
We can also consider how these symmetry constraints can be utilized if we assume that
the microscopic theory is de ned via the action (4.21) (i.e. its is not an e ective theory)
as done in simulations. In this case since there is no restriction to forward scattering there
is no mechanism by which the quadratic term can cancel with the quartic for all choices of
states. Then taking the one particle matrix element of (6.19) we have the constraints
" =
p
2
2m?
0 = (2
d)g(~pi; )
~pi
Here we have also introduced the renormalization scale
in the coupling. The Landau
relation (4.27) which ensures boost invariance remains unchanged but we generate a new
constraint by setting
to zero and varying the action (6.16) and (6.17) with respect to .
Expanding (6.16) and (6.17) to leading order in ,
For Swave scattering (g(p; )=g( )), m = m? due to the Landau condition in eq. (4.31)
and (2
d)g( ) =
(g). For higher angular momentum channels, (6.21) gives us the beta
function to all orders.
~py1 (t) ~p2 (t) ~py3 (t) ~p4 (t)
HJEP05(218)4
(6.18)
(6.19)
(6.20)
(6.21)
The predicitve power of symmetries is not lost when the ground state is not invariant. The
symmetries are simply realized in a nonlinear fashion. For internal symmetries Goldstone
boson appears which saturate the relevant Ward identities. When spacetime symmetries
are broken, new pathways to symmetry realization arise. We have shown in particular,
that the inverse Higgs mechanism which leads to a reduction in the number of Goldstones
can be generalized. We introduced the notion of a dynamical inverse Higgs mechanism
(DIHM) whereby a strong operator constraint (DIHC) is imposed which enforces symmetry
realization. A simple example of a DIHC was presented here in the context of Fermi liquid
theory. The missing boost Goldstone is seen to be absent as it is unnecessary once the
DIHC is imposed. In this case the constraint leads to the Landau relation which relates
the coupling to the e ective mass. This in itself does not explain why we know of no
systems which manifest a boost Goldstone, as in three dimensions the framon (the boost
Goldstone) path to symmetry realization seems to be perfectly consistent. Thus there is
no a priori reason why we would expect no such systems to arise in nature. However, as
we have shown, the framon is nonderivatively coupled, thus it is natural to expect that in
the IR, strong dynamics will set in and change the relevant degrees of freedom. As such,
the framon could be hiding under the shroud of strong coupling.
In general we do not know a priori if a given DIHC can be satis ed. In the case
of the breaking of boost invariance we showed that the constraints force all low energy
interactions to be relegated to particular kinematic con gurations, i.e. backtoback (BCS)
and forward scattering. At the same time, canonical Fermi liquid theory tells us that these
are the only possible marginal interactions based on power counting. Clearly this is no
coincidence.
We then presented an example of a DIHC which can not be satis ed by studying the
realization of the Schrodinger group broken by a Fermi sea. We showed how the broken
symmetries can be realized by the inclusion of a dilaton and a framon. Using the boost
DIHC (Landau relation) to eliminate the associated Goldstone leads to another DIHC
which allows for the elimination of the dilaton as well. In two dimensions the DIHC can be
satis ed and is the only path to symmetry realization while in three dimension the DIHC
leads to a constraint that contradicts the Lagrangian dynamics. The full Schrodinger
symmetry can be consistently realized by including a dilaton and a framon, while the
Goldstone of special conformal transformation gets gapped. However, the ensuing theory
is not consistent with a Fermi liquid description as the nonderivative coupling of the
Goldstone would leads to (at least) marginal Fermi liquid behavior [6].
A
Landau relation from Galilean algebra
The Landau relation can also be derived (similar to Landau's original derivation) by
demanding that the Fermi Liquid action, without including the boost Goldstone, should be
Galilean boost invariant. This is equivalent to satisfying the Galilean algebra by using
the Noether charges constructed from the Fermi Liquid action. The only commutator
H =
Gi = t
Pi =
Z
Z
Z
ddp pypi p
ddp pypi p
i
im
Z
of the Galilean algebra we need to satisfy is [H; Gi] = iPi where Gi is the generator of
Galilean boost, H is the Hamiltonian and Pi is the momentum operator. In terms of the
quasiparticle elds, these operators are given by
ddp py"(p) p + Y Z
ddki
g(ki) y y
2
k1 k2 k3 k4
(d)(k1 + k2
k3
k4)
Using anticommutation relation f p
; py0 g = d(p
p0) and satisfying [Gi; H] = iPi,
we get back the operator relation in (4.27).
B
Landau relation from Poincare algebra
Here we derive the Landau relation for a relativistic Fermi liquid from current algebra. The
same result can be reached by using the relativistic coset construction. The derivation of
Landau relation for the relativistic Fermi liquids [51] is a little more involved then compared
to the Galilean case. The commutator we need to satisfy is still [H; Ki] = iPi where Ki
is the generator of the Lorentz boost's but the Noether charges are di erent from their
Galilean counterparts. Denoting H0 as the free Hamiltonian and V as the interaction
H0 =
Z
ddp py"(p) p
V = Y Z
ddki
ki = t
Wi = Y Z
i
Z
i
Z
Pi =
ddp pypi p
(A.1)
(B.1)
(B.2)
(B.3)
g(ki) y y
2
k1 k2 k3 k4
i
Z
ddp pypi p
(d)(k1 + k2
k3
k4)
(d)(k1 + k2
k3
k4)
where Wi is the correction to the boost operator due to presence of interactions. This is
because Ki = tPi
R xiT 00 where T
is the energy momentum tensor, and T 00 is the
sum of free and the interacting Hamiltonian density. So we de ned Ki = ki + Wi where ki
is from free part of the Hamiltonian density and Wi is due to interactions. The Poincare
algebra condition now becomes
[H0; ki] + [H0; Wi] + [V; ki] + [V; Wi] = iPi
Assuming weak interactions between quasiparticles and neglecting terms of O(g2), (taking
one particle matrix elements for a state with external momentum, k)
"(k) + "(k)
Z
ddph0j py pj0ig(p; k)
Z
("(p)g(p; k)) = ki
For forward scattering g(p; k) = g(cos ) where is angle between p~ and ~k and h0j py pj0i =
(pF
p) to leading order in g. Using "(kF ) = , where
is the chemical potential and
the de nitions of e ective mass, m and the density of states at the Fermi surface, D( ),
m
1 +
G1
1
3
where we assumed d = 2 and expanded the coupling function g( ) in Legendre polynomials,
g( ) = Pl glPl(cos ) and de ned Gl = D( )gl.
Acknowledgments
This work supported by the DOE contracts DOE DEFG0204ER41338 and
FG0206ER41449. The authors thank Riccardo Penco for comments on the manuscript.
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
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