Englert–Brout–Higgs Machanism in Nonrelativistic Systems
Haruki Watanabe1, ∗ and Hitoshi Murayama1, 2, 3, †2

1
Department of Physics, University of California, Berkeley, California 94720, USA
Theoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
3
Kavli Institute for the Physics and Mathematics of the Universe (WPI),
Todai Institutes for Advanced Study, University of Tokyo, Kashiwa 277-8583, Japan

arXiv:submit/0970502 [hep-th] 4 May 2014

We study the general theory of Englert–Brout–Higgs (EBH) mechanism without assuming Lorentz
invariance using effective Lagrangians. We clarified problems in previous studies, and present a
concrete prescription that satisfies the requirement of the charge neutrality and continuity in thelimit of vanishing gauge coupling constant for non-abelian gauge theories.

Introduction. —The discovery of the Higgs boson at
the Large Hadron Collider marks a great triumph ofunity of physics. The original idea emerged from thestudy of superconductivity and its theory by Bardeen,
Cooper, and Schrieffer (BCS) [1]. After Anderson foundthat there are collective excitations in the gap region [2],Nambu first clarified that the BCS ground state is still
consistent with gauge invariance [3], and introduced theconcept of spontaneous symmetry breaking (SSB) into
particle physics [4, 5]. Soon afterwards, Goldstone provedthat the SSB leads to massless scalar particles called
Nambu–Goldstone bosons (NGBs) [6]. Even though theoriginal theorem did not apply to non-Lorentz-invariant
systems, the present authors generalized the theorem
so that it has real-life applications in condensed matter physics, atomic physics, nuclear physics, and astrophysics [7–9]. On the other hand, Englert, Brout [10],and Higgs [11] proposed the gauged symmetry with SSB
to go around Goldstone’s theorem because Nature does
not appear to have a massless scalar boson. It is called
Englert–Brout–Higgs (EBH) mechanism. It was finallydiscovered fifty years later. At the same time, the conceptof the Higgs boson made a full circle back to condensed
matter physics, becoming a hot subject of research (see,e.g., Refs. [12, 13]).

pioneering works [21, 22], Gusynin and his collaborators
found that, in a gauged linear σ model at a finite charge
density, gauge fields Aiµ develop a finite expectation value
due to the linear term in the Lagrangian −Aµi hjµi i, generically ending up with breaking the spatial rotation. Later,
Hama and his collaborators [23] pointed out a subtletyof this result: the low-energy spectrum is discontinuous
as a function of the gauge coupling e and the mass ofthe gauge bosons does not go to zero in the limit e → 0(see the appendix of Ref. [23]). In general, there is nothing wrong with discussing non-perturbative phenomena,
but if we are interested in a perturbative, weak coupling
physics, the low-energy spectrum should be continuous
as a function of the gauge coupling. They traced the origin of the discontinuity to the non-zero charge densityof the ground state. Having in mind a situation where
the background of the system has exactly the opposite
charge densities and the net charge densities of the entire
system vanish, they subtracted the ground state charge
densities from the Lagrangian, following the prescription
called “charge neutrality” discussed in Refs. [24, 25], andfound that the mass of the gauge bosons continuously
vanishes as e → 0 this time.

Given this tremendous cross-pollination among different subareas in physics, it is natural to ask what thegeneral theory of Higgs phenomenon is without relying
on the Lorentz invariance. In particular, the non-abeliangauge theory developed in particle physics [14] is making
inroads into condensed matter physics, such as spin liquid [15], multi-layer graphene [16, 17], ultracold atoms in
optical lattices [18–20] etc. Therefore a specific questionof great importance is: what is the general theory of theEBH mechanism in non-abelian gauge theories without
Lorentz invariance?

However, as we shall see below in detail, this seemingly
reasonable prescription actually has many issues. The
crucial flaw is that the low-energy excitation spectrum isstill not continuous. In the setup they considered, there
is a NGB with a quadratic dispersion as well as its accompanying gapped mode at e = 0, but both of them areabsent in the limit e → 0. Although we will review thispathological behavior below in details, the discontinuity
in the long wavelength limit can be already seen in their
original paper; see Fig. 2 of Ref. [23]. We claim that theorigin of this sick behavior is explicit breaking of both
local and global symmetry due to the naive subtractionof non-Abelian charge densities.

In relativistic field theories, spontaneously broken
gauge symmetries do not give rise to physical NGBs —would-be NGBs are eaten by gauge fields and in turngauge fields acquire the longitudinal component and afinite mass. This is how the EBH mechanism works. It
turns out that the extension of this famous story to thegeneral non-relativistic setup is not a trivial problem. In

In this Letter, we develop the general theory of
spontaneously broken non-abelian gauge theory without
Lorentz invariance, given the puzzles above. In particular, the emergence of type-A and type-B NGBs not seenin Lorentz-invariant systems raises questions about the
number of “eaten” degrees of freedom, as there are only
half as many type-B NGBs as the broken gauge symme-

2
tries.
Main Issues. — It is clear that the basic idea of
the EBH mechanism should carry over to non-Lorentzinvariant systems. We assume rotational invariance in
d = 3 spatial dimensions throughout this Letter, andstick to the unitarity gauge to understand the physicaldegrees of freedom. In the limit of zero gauge coupling
constant, we have two massless states for each generator of the gauge group. On the other hand, a massive
spin-one boson at rest has three states of the same energy required by the rotational invariance. Their energy
should go to zero together as the gauge coupling constantis switched off. Therefore, the three states of a massive
vector boson should decompose into two massless gaugeboson states and another massless state. The last stateis said to be “eaten” by the gauge boson when it becomes
massive. Namely, a gauge boson can only eat a gaplessstate and acquires a mass as well as an additional degreeof freedom. So far it is the same as in Lorentz-invariant
case.
There are two related issues that make this familiar
story paradoxical. As the present authors showed [8],there are two types of NGBs in non-Lorentz-invariant
systems. For type-A NGBs, there is one degree of freedom per each broken symmetry. Therefore, we expectthat the EBH mechanism should work just fine. However for type-B NGBs, there is one gapless boson for two
broken symmetries. The simplest example is the Heisenberg ferromagnet. SO(3) symmetry is broken to SO(2),namely two generators are spontaneously broken. We expect two gauge bosons to acquire mass. However, there
is only one type-B NGB in the limit of zero gauge coupling, and hence there is only one state to be “eaten.” In
addition, type-B NGBs are always associated with nonzero charge density in the ground state. Namely spontaneously broken generators Q1 and Q2 excite one type-B
NGB whenZh0|[Q1 , j20 ]|0i = iǫ123 h0|j30 |0i 6= 0, Qi ≡ d3 xji0 (x). (1)Non-zero charge density coupled to the gauge field means
there is a term linear in the gauge field in the Lagrangian hj03 iA30 , namely a tadpole. Together with themass term m2A ∝ g 2 v 2 , it causes the expectation value
hA30 i ∝ hj03 i/gv → ∞ in the zero coupling limit. Therefore, we do not expect a smooth limit of zero gauge coupling constant in the presence of the finite charge density
in the ground state.
Therefore, we study these two questions together: degrees of freedom and charge density. We study a few
examples and draw a general lesson.
Gauged U(2) Model. —Let us start with discussing
the model introduced in Refs. [26, 27],1i 2) ],(2)L = (Dµ φ)2 + m2 |φ|2 − λ|φ|4 − [(Fµν )2 + (Fµν4where φ is a two-component complex scaler field andDν = ∂ν + i(e1 Aν − µδν,0 ) + ie2 Aiν (σi /2) is the covariant
derivative, with Aµ being the U(1) gauge field and Aiν
(i = 1, 2, 3) being the SU(2) gauge field. The vacuum
expectation value hφi = (0, φ0 )T [φ20 = (µ2 + m2 )/2λ]breaks the U(2) symmetry down to U′ (1) symmetrythat acts only the first component of φ. The groundstate possesses a non-Abelian charge density h[Q1 , j20 ]i =ihj30 i = −iµφ20 6= 0, in addition to U(1) charge density
hj 0 i = 2µφ20 , combining two NGBs into one type-B NGB.
When e1 = e2 = 0, the Lagrangian (2) describes fourmodes: one type-A NGB, one Higgs (amplitude) mode,
one type-B NGB, and a gapped mode accompanying thetype-B NGB. Their dispersion relations are respectively
given by (we set c = 1 in this section)p2
ωA= m2 + 3µ2 + k 2 − (m2 + 3µ2 )2 + (2µk)2 , (3)p2
ωH= m2 + 3µ2 + k 2 + (m2 + 3µ2 )2 + (2µk)2 , (4)ppωB = µ2 + k 2 − µ, ωB ′ = µ2 + k 2 + µ.
(5)
Our interest is how these modes are affected by switchingon the gauge couplings.
When we gauge only the U(1) part of the symmetry(e1 6= 0 and e2 = 0), we fix the gauge in such a way thatthe second component of φ is real, i.e., φ = (π, φ0 + h)with h ∈ R and π ∈ C. Expanding the Lagrangian L1 +L2 to the quadratic order in fields and dropping total
derivatives, we find
L = [∂µ π ∗ ∂ µ π + iµ(π ∗ π̇ − π̇ ∗ π)]i
1 h ~˙
2~
~˙
~
~ 2
~
+ A
T · AT − (∇ × AT ) − 2m1 AT · AT2+∂µ h∂ µ h − 2(µ2 + m2 )h2
1 ~
1 2
2 2
2
2 2
+ (∇A
0 ) + m1 A0 + ȦL − m1 AL
2
2~ L − 4m1 µhA0 − 2e1 µφ20 A0 ,~ 0 · ∂0 A
(6)+∇A
~ into thewhere we defined m1 ≡ e1 φ0 and decomposed A
longitudinal and transverse components with respect to~ Note that the last term is linear in A0 , corresponding∇.
to the U(1) charge density hj 0 i = 2µφ20 . Assuming theexistence of a background that neutralizes the net U(1)charge of the system, Ref. [24] suggested that one should0
add the background contribution −e1 A0 jbr= 2e1 µφ20 A0
to the Lagrangian to cancel this term.
It is then a standard exercise to derive the excitation
spectrum in this phase. The first line of Eq. (6) gives
ωB,B ′ modes in Eq. (5), unaffected by the gauge coupling.
The second line describes
the gapped transverse gaugepbosons with ωT = 2m21 + k 2 . Finally, the rest part describes the longitudinal gauge boson and the Higgs mode
with
p
2
(7)
ωL
= 2m21 + m̃2 + k 2 − m̃4 + 4µ2 k 2 ,
p
′ 2
2
2
2
4
2
2
ωH = 2m1 + m̃ + k + m̃ + 4µ k ,
(8)

3
′
where m̃2 ≡ m2 + 3µ2 − m21 . Observe that ωL and ωH
continuously go back to ωA and ωH in Eqs. (3), (4) inthe limit e1 → 0. It is thus clear that the type-A NGB iseaten by the longitudinal part of the gauge
p field, making
it a physical mode with a mass (gap) 2m21 .
Next let us consider gauging only the SU(2) part ofthe symmetry (e1 = 0 and e2 6= 0). This time, we fix thegauge by setting φ = (0, φ0 + h). The Lagrangian to thequadratic order in fields is then given byi
1 h ~˙ 3 ~˙ 3
~ ×A~ 3 )2 − (m2 /2)A
~3 · A~3
L=
AT · AT − (∇
T
2
T
T
2


1 1 1µν
2
− Fµν
F
+ Fµν
F 2µν + m22 (A1µ A1µ + A2µ A2µ )4+∂µ h∂ µ h − 2(µ2 + m2 )h2
1 ~ 3 2 m22 3 2 1 3 2 m22 3 2+ (∇A(A0 ) + (ȦL ) −
(AL )
0) +
2
4
2
4
~ 3 + 2m2 µhA3 + e2 µφ2 A3 ,~ 3 · ∂0 A
(9)
+∇A
L
0
0 0
0

where m2 = e2 φ0 . Again, the last term is linear in
A30 , corresponding to the non-Abelian charge density
hj30 i = −µφ20 . In the same spirit as the above U(1) case,Ref. [23] subtracted this term by assuming the opposite contribution from the background −e2 A30 (jbg )03 with(jbg )03 = µφ20 . Then we notice that the last three lines ofEq. (9) is identical to those of Eq. (6) if we replace m2
by −2m1 and A3µ by Aµ . Therefore, we do not have torepeat the calculation for this sector as we already knowthat it correctly describes the type-A NGB and the Higgs
mode in the limit e2 → 0.
However, we now encounter a serious problem. The
first and second lines of Eq. (9) describe gapped
gauge bosons with the dispersion relation ωL,T =p(m22 /2) + k 2 . Although the total number of the physical modes is conserved, the ωB,B ′ modes at e2 = 0 are
missing in the limit of e2 → 0. Instead, there are twomore gapless modes with the linear dispersion ω = k inthis limit.
The Dynamics of the Background. —We attribute
this sick behavior to the incorrect treatment of the background degrees of freedom. We subtracted the background charge densities from the Lagrangian in the form
−eAiµ (jbg )µi . However, if we do not take into account the
dynamics of the background and completely freeze it, thefixed expectation value (jbg )µi can no longer transform
under the symmetry transformation acting the internal
index i. Hence this added term explicitly breaks the original symmetry. In the above example, adding −(jbr )03 A30
term explicitly breaks the U(2) symmetry down to theU(1)2 symmetry generated by σ3 and 11. This residual
symmetry explains why ωA,H modes were properly described in the limit e2 → 0 in this scheme, but because
the symmetry related to the ωB,B ′ modes were explicitly
broken, it was no wonder why they were missing in thelimit same limit.
Note that subtracting Abelian charge densities do not

explicitly break any symmetries, for Aµ changes by ∂µ χ
and the Lagrangian thus changes only by a surface termµ∂µ (−χejbg). This is why above treatment of the gauged
U(1) symmetry returned physical results.
If we fully take into account the dynamics of the background degrees of freedom, the added term −eAiµ (jbg )µi
together with other terms that are needed to describe thebackground dynamics can now perfectly respect the symmetry of the original system. However, by assumption,
the background has non-zero charge densities h(jbg )0i i,
which are supposed to cancel the charge densities of theoriginal system. These non-Abelian charge densities imply spontaneous breaking of generators Qa such thath[Qa , jb0 ]i = ifab i hji0 i 6= 0 for ∃ b. Thus the backgrounditself must support NGBs as an inevitable consequence of
the NG theorem. Then, the question is whether once takea decoupling limit, where interactions of the backgroundto other parts of the system vanish. We argue that sucha limit does not exist in the case of non-Abelian symmetries.
Decoupling Limit. —Let us first review an exampleof the proper decoupling limit in the case of the background Abelian charge densities to clarify the difficulty inthe non-Abelian cases. Our example is a superconductor,
which exhibits the EBH mechanism via the condensationof Cooper pairs of electrons. Clearly there is a charge
density of the electrons that couple to the electromagnetic gauge field. However, there is the ion backgroundwith a positive charge to ensure the charge neutrality.
For simplicity we assume an isotropic elastic medium asa model,Lion =

n0 M ˙ 2 2 ~~ u)2 ]− eAµ j µ , (10)[~u − cL (∇ ·~u)2 − c2T (∇×~2

where e, M , n are the charge, mass, and number densityof the ions, cL,T are the phonon velocities, and j µ =~ · ~u, ~u˙ ) + O(u2 ) is the number current density
n0 (1 − ∇of the ions. If we canonically normalize the displacement
field ~u so that the coefficient of ~u˙ 2 term becomes 1/2, thecoupling to the gauge field is suppressed by the factor of
M −1/2 per ~u field. Therefore, in the limit M → ∞, theion dynamics should completely decouple from the restof the system, while the charge density hj 0 i = n0 still
plays its role of electrically neutralizing the system.
In the Supplemental Material [28], we discuss a nonrelativistic superfluid as another example of the background with a proper decoupling limit.
A Ferromagnet. —Let us now move on to the caseof non-Abelian charge densities to see whether a similar decoupling limit exists. As the simplest example, we
examine the gauged version of ferromagnets. To construct an effective Lagrangian with local SO(3) symmetry, it is useful to define the gauged Maurer-Cartan
form Ω(π, A) via Ωiµ Ti = −iU † (∂µ + ieAiµ Ti )U , where Ti(i = x, y, z) is a matrix representation of SO(3) genera
ators and U (π) ≡ eiπ Ta (a = x, y). A local symmetry

4transformation g ∈ G acts NG fields π a and the gauge
field Aµ = Aiµ Ti asgU (π) = U (π ′ )h,h ≡ eiθ(g,π)Tz ∈ H,

A′µ = gAµ g −1 − ie−1 g∂µ g −1 ,

(11)
(12)

so that the Maurer-Cartan form transforms nicely:(ωµa )′ = R(θ)ab ωµb ,(ωµz )′ = ωµz − ∂µ θ,(13)

where R(θ) is the rotation matrix around the z axis byan angle θ(g, π).
The low-energy effective Lagrangian of a ferromagnet
with the local SO(3) symmetry is then given by
1 a a
ω ·~
ω
LFM = −mz ωtz − g~2

Denoting the NG fields of the first ferromagnet by π aand those of second one by Πa (a = x, y), the linearized
Lagrangian of the whole system reads
1 ~ a~ a )2 − 1 G(∇Π
~ a − eA~ a )2− eA
L = − g(∇π
22
1 a b 1 a b
a b
a b+mz ǫabΠ Π̇ − π π̇ + eΠ At − eπ At
2
2
2
ǫ ~ i
~ i 2 ǫv (∇~ ×A~ i )2 ,+ (∇A(16)
t + ∂t A ) −22
where g, G are positive constants. ǫ is the electric permittivity v is the speed of light in the medium. Whene = 0, the Lagrangian describes two type-B NGBs with(14)

to the leading order in derivatives. Here, g is a positive constant and mz is the magnetization. Using thetransformation rule in Eq. (13), one can check that theLagrangian changes just by a surface term mz ∂t θ(g, π)under a local symmetry transformation.
Expanding the Maurer-Cartan form ωtz in NG fields
π a , one findsǫab a b1 a2 zzza b
ωt = − π π̇ + e At + ǫab π At − π At (15)22to the order of O(π 3 , π 2 A). Note that all interactions are
dictated by the SO(3) symmetry, in contrast to the caseof Abelian symmetries. Since the coupling constant e iscommon to all fields coupling to the non-Abelian gauge
field, one cannot control e arbitrarily if one wishes to use
this ferromagnet as a background of a system. One cannot tune mz either since it should be fixed by the magnetization density of the main system. This argument
proves that the coupling to the gauge field in the firstterm of Eq. (14) cannot be decoupled. Unfortunately,
this is a general property of non-Abelian gauge theories— in the presence of non-Abelian charge densities, thedynamics of the background that is supposed to neutralize the charge densities of the whole system cannot be
decoupled.
Two Copies of Ferromagnets. — This observation
motivates us to explicitly include the dynamics of the
background, at least when we are interested in the situation with non-Abelian charge densities. As a toy model,
here we discuss two copies of ferromagnets, one with themagnetization mz pointing in the positive z direction andthe other one in the opposite direction with the equalmagnitude. We assume G = SO(3) × SO(3) symmetry, which is broken down to H = SO(2) × SO(2) bymagnetizations. In this case, we can safely gauge thediagonal SO(3) symmetry thanks to the cancelation of
the magnetization. Note that, due to the non-Abeliannature of the SO(3) symmetry, gauging the vector partof SO(3) × SO(3) explicitly breaks the axial part of the
global symmetry.

ωB1 =

gk 2
,
mz

ωB2 =

Gk 2
.
mz

(17)

and (d − 1) × 3 = 6 states for transverse gauge bosons
with ωT = vk. Since Azµ stays gapless regardless of thegauge coupling, we will not discuss them below. In the
Supplemental Material [28], we discuss additional gappedmodes in the presence of terms with second power in time
derivatives so that our results can be compared to thosein Ref. [29], but we find they do not change any of the
discussions here.
For a finite coupling e 6= 0, we use the unitary gauge
to set π a + Πa = 0. Then the Lagrangian is
1 ~ a~ a )2 − 1 G(∇π
~ a + eA~ a )2
L = − g(∇π
− eA
L
L
2
2
ǫ
~ aL )2~ at + ∂t A
− 2emz ǫab π a Abt + (∇A22ǫ~ a )2 − ǫv (∇~ ×A~ a )2 − g + G (eA~ a )2 .(18)+ (∂t A
T
T
T
2
2
2
The last line describes gapped transverse gauge bosons
withqωT = m21 + (vk)2 , m21 ≡ e2 (g + G)/ǫ,(19)

and the first three lines describes two longitudinal gaugebosons withs2(g + G)k 2(g − G)k 2
2
.
(20)
±
ωL1,2 = m1 +2mz
2mz
In the limit of switching off the gauge coupling, the longitudinal modes ωL1,2 smoothly go back to two type-B
NGBs ωB1,2 in Eq. (17). Therefore, the two type-B NGBs
at e = 0 are “eaten” to become the two longitudinalgauge bosons at a finite coupling.
Therefore, we see clearly that we need two type-B
NGBs to make both A1 and A2 massive, while gappedmodes play no role, contrary to claims in Refs. [23, 29].
Given the necessity of a type-B NGB from the background, there is no decoupling limit where we can simply
replace the background system by a current density.

5
Conclusion. —In this Letter, we proposed a concrete prescription to realize the EBH mechanism in nonLorentz-invariant systems. The naive subtraction of
background non-Abelian charge densities, as suggested
in the literature, explicitly breaks the symmetry of thesystem and results in a discontinuity of the spectrum as
a function of the gauge coupling. When we include thedynamics of the background to properly respect the symmetry, there is no limit where the background dynamics
is decoupled from the other part of the system. We explicitly show that the gauged ferromagnet is consistentafter fully taking into account the background dynamics, with a continuous limit of zero gauge coupling andcorrect numbers of “eaten” degrees of freedom.
H. W. appreciates the financial support of the Honjo
International Scholarship Foundation. The work of H.
M. was supported by the U.S. DOE under Contract DEAC02-05CH11231, and by the NSF under grants PHY1002399 and PHY-1316783. H. M. was also supportedby the JSPS Grant-in-Aid for Scientific Research (C)(No. 26400241), Scientific Research on Innovative Areas(No. 26105507), and by WPI, MEXT, Japan.

∗

hwatanabe@berkeley.edu
hitoshi@berkeley.edu, hitoshi.murayama@ipmu.jp[1] J. Bardeen, L. Cooper,
and J. Schrieffer,Phys. Rev. 108, 1175 (1957).
[2] P. W. Anderson, Phys. Rev. 110, 827 (1958).
[3] Y. Nambu, Phys. Rev. 117, 648 (1960).
[4] Y.
Nambu
andG.
Jona-Lasinio,
Phys. Rev. 122, 345 (1961).
[5] Y.
Nambu
andG.
Jona-Lasinio,
Phys. Rev. 124, 246 (1961).
[6] J. Goldstone, Nuovo Cim. 19, 154 (1961).
[7] G. Guralnik, C. Hagen, and T. Kibble, in Advances in
Particle Physics, Vol. II (1968) p. 567.
[8] H.
WatanabeandH.
Murayama,
Phys. Rev. Lett. 108, 251602 (2012).
[9] Y. Hidaka, Phys. Rev. Lett. 110, 091601 (2013).
[10] F.
EnglertandR.
Brout,
Phys. Rev. Lett. 13, 321 (1964).
[11] P. W. Higgs, Phys. Rev. Lett. 13, 508 (1964).
[12] C. Varma, J. Low Temp. Phys. 126, 901 (2002).
[13] D.
PodolskyandS.
Sachdev,
Phys. Rev. B 86, 054508 (2012).
[14] C.-N. Yang and R. L. Mills, Phys. Rev. 96, 191 (1954).
[15] X.-G. Wen, Quantum Field Theory of Many-Body Systems (Oxford University Press, Oxford, England, 2004).
[16] P. San-Jose, J. González,and F. Guinea,
Phys. Rev. Lett. 108, 216802 (2012).
[17] S. Gopalakrishnan, P. Ghaemi,
and S. Ryu,
Phys. Rev. B 86, 081403 (2012).
[18] L. Tagliacozzo, A. Celi, P. Orland, M. Mitchell, andM. Lewenstein, Nature communications 4 (2013).
[19] D. Banerjee, M. Bögli, M. Dalmonte, E. Rico,
P. Stebler,
U.-J. Wiese,and P. Zoller,
Phys. Rev. Lett. 110, 125303 (2013).
†[20] E. Zohar,
J. I. Cirac,
and B. Reznik,
Phys. Rev. Lett. 110, 125304 (2013).
[21] V. Gusynin, V. Miransky, and I. Shovkovy, Phys. Lett.
581B, 82 (2004).
[22] V. Gusynin, V. Miransky, and I. Shovkovy, Mod. Phys.
Lett. A 19, 1341 (2004).
[23] Y. Hama,
T. Hatsuda,and S. Uchino,
Phys. Rev. D 83, 125009 (2011).
[24] J. I. Kapusta, Phys. Rev. D 24, 426 (1981).
[25] J. Kapusta and C. Gale, Finite-Temperature Field Theory, 2nd ed. (Cambridge University Press, Cambridge,
England, 2006).
[26] V.
A.
MiranskyandI.
A.
Shovkovy,
Phys. Rev. Lett. 88, 111601 (2002).
[27] T. Schäfer, D. T. Son, M. A. Stephanov, D. Toublan,
and J. J. M. Verbaarschot, Phys. Lett. 522B, 67 (2001).
[28] See Supplemental Material.
[29] S. Gongyo and S. Karasawa, arXiv:1404.1892 (2014).

6
Supplemental Material
for “Higgs Phenomena in Nonrelativistic Systems”
1. Gauged U(2) Model

Here we present the full derivation of Eqs. (7) and (8).
The (h, A0 , AL ) sector of the Lagrangian (6) reads


 


Z


1 2
1 ~
2 2
2
2 2
~
~
A
−
4m
µhA
+
∇A
·
∂
+
L = d3 xdt ∂µ h∂ µ h − 2(µ2 + m2 )h2 + (∇A
Ȧ
−
m
A
)
+
m
A
1
0
0
0 L
0
1 L
1 0
2
2 L





Z 3

d kdω  2
1 2
1 2=
ω − k 2 − 2(µ2 + m2 ) h∗ h +k + m21 A∗0 A0 +
ω − m21 A∗L AL
4
(2π)
2
2

1
− kω(A∗0 AL + A∗L A0 ) − 2m1 µ(h∗ A0 + A∗0 h)
2
 
Z 3
h
d kdω ∗ ∗ ∗   
h
,
A
,
A
A
=
M
,
(21)
0
0
L
(2π)4
AL

where

 2

ω − k 2 − 2(µ2 + m2 ) −2m1 µ0
1 22−2m1 µ− 21 kω  ,M ≡
2 k + m1
1
1 2
2
0
− 2 kω
2 ω − m1
m2 
detM = 1 ω 4 − 2(m2 + 3µ2 + m21 + k 2 )ω 2 + k 4 + 2(m2 + µ2 + m21 )k 2 + 4m21 (m2 + 3µ2 ) .
2Dispersion relations, Eqs. (7) and (8), can be obtained by solving detM = 0:q
ω 2 = (m2 + 3µ2 + m21 + k 2 ) ± (m2 + 3µ2 + m21 + k 2 )2 − [k 4 + 2(m2 + µ2 + m21 )k 2 + 4m21 (m2 + 3µ2 )]q= 2m21 + (m2 + 3µ2 − m21 ) + k 2 ± (m2 + 3µ2 − m21 )2 + (2µk)2 .

(22)
(23)

(24)

2. Two Copies of Ferromagnets

In this section, we discuss the two copies of ferromagnets, including additional terms with second power in time
derivatives. The linearized Lagrangian reads
L=

1 ~ a
1
~ a )2 + 1 Ḡ(Π̇a + eAa )2 − 1 G(∇Π
~ a − eA~ a )2ḡ(π̇ a + eAat )2 − g(∇π
− eA
t
2
2
2
2


2
1 a b 1 a b
ǫ ~ i
a b
a b
~ i 2 ǫv (∇~ ×A~ i )2 ,+mz ǫabΠ Π̇ − π π̇ + eΠ At − eπ At + (∇At + ∂t A ) −
2
2
2
2

(25)

where g, ḡ, G, Ḡ are positive constants. for the sake of brevity we set G = g and Ḡ = ḡ, but one can work out without
this simplification if one wishes.
When e = 0, the Lagrangianiih1h~ a )2 + 1 ḡ(Π̇a )2 + mz ǫab Πa Π̇b − g(∇Π~ a )2ḡ(π̇ a )2 − mz ǫab π a π̇ b − g(∇π
L=
2 h
i 2
ǫ ~ i
i
2
2
i
2
~ ) − v (∇
~ ×A
~ ) ,
(26)
+ (∇At + ∂t A
2
describes two type-B NGBs with


r
g2 k4
gḡ 2
m2z
gḡ 2
2
+ O(k 6 ),(27)
ωB = 2 1 + 2 2 k − 1 + 4 2 k ≃
2ḡ
mz
mz
m2z
two gapped modes with


r
m2
gḡ 2
gḡ 2
2g
≃ 2z + k 2 + O(k 4 ),1 + 2 2k + 1 + 4 2k
2ḡ
mz
mz
ḡ
ḡ

m2
2
ωG
= z2

(28)

7
and (d − 1) × 3 transverse gauge bosons with
ωT = vk.

(29)

Note that ωG modes in general is beyond the scope of an effective Lagrangian, as they balance O(∂t ) term withO(∂t2 ) term; hence their gap may be beyond the cutoff scale and not belong to the effective Lagrangian. Especially,
they should disappear in the limit ḡ → 0. Nevertheless, here we assume the existence of the ωG modes to see theirfate after coupling to gauge fields. Since Azµ is decoupled from the other part of the system, regardless of the gauge
coupling, we will not discuss them below.
For a finite coupling e 6= 0, we use the unitary gauge to set π a + Πa = 0. The Lagrangian is then decomposed into
i
i h
i h
h
~ a )2 − ge2 (A~ a )2~ a )2 + ḡe2 (Aa )2 + ǫ (∂t A~ a )2 + ǫ (∇A
L = ḡ(π̇ a )2 − g(∇π
L
L
t
t
2
2
a
a b
a
~L
~ t · ∂t A
(30)
−2emz ǫab π At + ǫ∇A2ǫ~ aT )2 − v 2 (∇~ ×A~ aT )2 − 2e g (A~ aT )2 .
(31)+ (∂t A2ǫ
The last line describes (d − 1) × 2 gapped transverse gauge bosons withp
ωT′ = (2e2 g/ǫ)2 + (vk)2 .

(32)

The first two lines describes 1 × 2 gapped longitudinal gauge bosons. In the basis (π 1 , A1t , A1L , π 2 , A2t , A2L )T , thequadratic Lagrangian in the Fourier space can be represented by the matrix 2ḡω − gk 2
0
0
0
−emz
0
ǫ 2
2


− 2ǫ kωemz
000
2 k + ḡe


ǫ 2
ǫ
2


0
0
0
0
− 2 kω
2 ω − ge
.
(33)
M =
2
2


0
emz
0
ḡω − gk
0
0


ǫ 2
2
 −emz− 2ǫ kω 
000
2 k + ḡe
ǫ 2
ǫ
2
0
0
0
0
− 2 kω
2 ω − ge
By solving detM = 0, one gets



s


2
2
2
2
2
e gḡ
gḡ
gḡ
e gḡ
1+
+ 2 2 k2 −+ 4 2 k2 
1−
ǫm2z
mz
ǫm2z
mz


2
2 4

m2
2
ωL
= z2 2ḡ

≃ (2e2 g/ǫ)2 1 −

gḡk
g k
m2z
m2z
6

+

3 + O(k ),2e2 gḡ2
2 gḡ 2e
1 − ǫm2
1 − ǫm2
z
z



s

2

2
2
2
2
gḡ
e gḡ
gḡ
e gḡ
+ 2 2 k2 +1+
+ 4 2 k2 
1−
ǫm2z
mz
ǫm2z
mz
!
1
g
m2z
1+
k 2 + O(k 4 ).
≃ 2 +2e2 gḡ2
ḡ
ḡ
1−
2

m2
′ 2
ωG
= z2 
2ḡ

(34)

(35)

ǫmz

p
One can see that ωL has the same gap 2e2 g/ǫ as the transverse components, as expected by the spatial rotational′
symmetry. This mode is continuously connected to the type-B NGB ωB in the limit e → 0. Also, ωG
continuously
goes back to ωG in this limit.
3. Another Example of Background With a U(1) Charge

As an example of a background that does break an internal symmetry, let us take a non-relativistic superfluid of
charged bosons, described by the Galilean invariant Lagrangian#2"~ − eA)
~ 2
(∇ϕ
n0
2
M cs − (ϕ̇ + eAt ) −
.
(36)
LSF =
2M c2s2M

8Here M is the mass of the bosons and cs is the speed of sound. One may worry that the superfluid phonon (NGB)
messes up the system or thepbroken symmetry gives a mass to the U(1) gauge field by the standard EBH mechanism.
However, redefining ϕ′ = ϕ n0 /M c2s , we find
LSF =

1 ′ 2
~ ′ )2 ] − en0 At + O(M −1/2 ).
[(ϕ̇ ) − c2s (∇ϕ2(37)

Therefore
the superfluid dynamics is completely decoupled in the limit of M → ∞. Especially, the Higgs masspn0 e2 /M vanishes in this limit, while the U(1) charge n0 remains finite.