JHEP05(2018)014 Springer May 3, 2018 : April 17, 2018 : January 8, 2018 : February 21, 2018 : Published Accepted Received Revised Published for SISSA by https://doi.org/10.1007/JHEP05(2018)014 [email protected] , . 3 1712.07795 The Authors. Effective Field Theories, Space-Time Symmetries, Spontaneous Symmetry c

The Ward identities associated with spontaneously broken symmetries can , [email protected] Department of Physics, CarnegiePittsburgh, Mellon PA University, 15213, U.S.A. E-mail: Open Access Article funded by SCOAP Higgs Constraint (DIHC), generalizing the results forKeywords: Vishwanath and Wantanabe. Breaking ArXiv ePrint: inclusion of non-derivatively coupledthe Goldstone bosons. leading order We non-linearities, presentconstraint for the associated action, the with including the rotational possible Goldstonefrom DIHM (angulon), the that and spectrum. would need discussnon-derivatively Finally to the coupled, be we imposed a discuss to the necessary remove conditions condition it under for which the Goldstone existence bosons of are a Dynamical Inverse a Dynamical Inverse Higgs Mechanismboosts, (DIHM). rotations We consider and the conformalthree spontaneous possible transformations breaking paths of in to the symmetryor realization: context some mixture of pure thereof. Goldstones, Fermi no Wethe liquids, show Goldstones DIHM finding that and in route DIHM, the isdilatations, two while dimensional the in degenerate three only electron dimensions system these consistent symmetries way could just to as well realize be spontaneously realized via broken the boosts and and its appearance haswhen a a well Goldstone defined bosonenergy mathematical associated theory condition. despite with the However, a lack thereshow broken of that are the generator in cases existence does such of not casesan an the associated appear associated relevant Goldstone, IHM. in broken if In the symmetry there this can low exists paper be a we will realized, proper without set the of operator aid constraints, of which we call be saturated bythe Goldstone number bosons. of Goldstoneless However, bosons when than necessary space-time the toredundancy symmetries number non-linearly or are of realize the broken, broken the generationremoved generators. from of symmetry the a can spectrum. gap. The be This phenomena loss In is of called either an case Goldstones Inverse Higgs the may Mechanism associated (IHM) be Goldstone due may to be a Abstract: Symmetry realization via amechanism dynamical inverse Higgs Ira Z. Rothstein and Prashant Shrivastava JHEP05(2018)014 8 19 15 18 21 6 13 17 12 10 24 23 5 – 1 – 5 4 8 11 1 23 9 4.4.1 Review4.4.2 of EFT of Fermi Power liquids counting4.4.3 scalings in the coset The construction framid as Lagrange multiplier and the Landau relation framid 5.1 The stability of Goldstone boson mass under renormalization 6.1 Consequence of broken conformal symmetry via the DIHM 4.1 Non-relativistic framids 4.2 Coset construction of Fermi liquid4.3 EFT with rotational Multiple symmetry: realizations4.4 of type broken I symmetry Power counting 1.1 The missing1.2 Goldstones The paths to symmetry realization The realization of the broken(GBs). symmetry, in general, When willcoupled space-time include and symmetries gapless are Goldstone irrelevant are in bosons associated the unbroken, with IR. symmetry Goldstone If breaking, there bosons the areThe Goldstones are canonical other may example be gapless of derivatively ignored modes such a to in scenario first the is approximation. the spectrum, Fermi not liquid theory of metals where phonons 1 Introduction When symmetries are broken spontaneously they are manifested non-linearly in the IR. A Landau relation from Galilean algebra B Landau relation from Poincar´ealgebra 6 Broken conformal symmetry: eliminating the non-relativistic dilaton 7 Conclusions 5 Fermi liquid with broken rotational invariance 2 Review coset construction 3 Non-derivatively coupled (NDC) Goldstone bosons 4 Framids Contents 1 Introduction JHEP05(2018)014 . α ]. 0. 5 E any 6= 0 (1.2) (1.3) (1.1) → ∼  t n n ]. More- ~p iE 8 , − 7 e i Ω | (0) X 0 j | n is a conserved charge, we ih n X such that | φ (0) φ . , | ) x in the limit of vanishing energy. Also Ω ( = 0 X 0 2 ] for a discussion. i 6 − h E 14 Ω xj t ] leading to a width which scale as Γ 1 | n 3 ≤ − ] – d iE e 1 d – 2 – i X, φ Ω Z [ | Of course, if there are no other gapless modes, or | ] (this includes the aforementioned non-relativistic = 1 Ω 14 (0) h X ]. The necessary conditions for non-derivatively coupled φ 4 | n ih n | (0) X 0 j . Suppose we have a order parameter | X Ω h  2 ) does not preclude the possibility of having multiple gapless states. ) n ~p 1.3 ( 2. While relativistic dilatons can generate long range forces and, as such, their 1) − d ( α < δ For non-relativistic systems, there can be no symmetry breaking if the vacuum is For relativistic theories the breaking of internal symmetries leads to a one to one corre- When space-time symmetries are broken, GBs can be non-derivatively coupled. Two Potential, off-shell phonons play an indirect role inWe that assume they here contribute that to there the are attractive no piece long of the range forces so that surface terms may be dropped and that 1 2 n X trivial since pair creation issymmetry disallowed. breaking Thus a necessarily non-relativistic has system a which manifests ground state whichfour breaks Fermi at coupling least once they boost have invariance been integrated out. generators have canonical translational properties. See [ Goldstone bosons when space-timeby symmetries a are Goldstone broken we boson.satisfy must the define For definition what our of we purposesnote mean a that we ( quasi-particles, will Γ define a Goldstone mode as having to We assume that theexists system some notion preserves of asee a discrete that conserved symmetry translational momentum. breaking invariance,However, implies we so Given can the that that not existence there fact say of that anything it a has about zero to the energy non-zero. associated state This when spectral state weight may other be than arbitrarily wide. the Thus if we are to count It follows that broken generator where spondence between Goldstone bosons andover, generators the which Goldstone shift boson the isspace-time vacuum manifested [ symmetries as are a broken deltacase) function [ we in are no the longer spectral assured density. about When the existence of a Goldstone boson associated with a relevant interactions which can drasticallytional affect invariance is the broken IR in physics. athe Fermi For liquid quasi-particles instance, and decay translations when into are rota- with Goldstones unbroken (nematic [ order), couplings are highly constrained [ Goldstones in non-relativistic theories were discussed by Vishwanath and Watanabe in [ if the Goldstones couplesuch to a sources, scenario then is they the are QCD of chiral primary Lagrangian. importance.canonical An examples example being of thetational relativistic invariance dilaton in and Fermi the liquids. Goldstone bosons Such of non-derivative broken couplings ro- lead to marginal or do not play a role at leading order. JHEP05(2018)014 a a ] the (1.5) (1.7) (1.8) X (0) are ] when 20 b j 18 – 15 , 13 , 12 ]. 20 , . As emphasized in [ obey a relation of the form 19 0 X ]. Since then a series of papers ] (1.4) 11 ρ X,X associated with internal symmetry are not in the same H multiplet, it 3 0 ) of Goldstones [ i vanishes leading to a prediction of one X rank[ = 0 (1.6) k N 0 1 2 ρ i P ) Mx X j − x and x ( − ∝ t O ] i H) ijk X h  / a P – 3 – ]). ) does not hold when space-time symmetries are ) Goldstones = X 2 can be easily seen to be redundant since ¯ = 25 ) P,X i p 1.4 [ i i x J ( θ ∼ K a are the broken charges of the full group G and π E = dim(G a X N ] 21 and i (0)] ](IHM) (see also [ ) may or may not be the signal of a redundancy. That depends upon a b ) and type II ( 24 , j p 1.5 a ∼ X [ h is the order parameter. As an example consider the symmetry breaking E i and rotation Goldstone i − i ) η x = ( are the unbroken translations and O ab ¯ h P ρ be possible to eliminate the Goldstone associated with ] are due to integrating out fields with analytic dispersion relations. The counting of (gapless) Goldstones still follows once we have established the neces- The analysis leading to the result ( Goldstone bosons may have various dispersion relations. Inequalities for counting rules The dispersion relation need not be limited to these two choices. Higher order relations are possible in 21 3 and some systems, e.g. thein vibrations [ of a stiff rod. Non-analytic dispersion relations such as those discussed where pattern for aGoldstone metal. The lattice breaks rotations, translations and boosts. The boost the nature of theredundancy order exists parameter. when there In is a particular, non-trivial given solution a to set the of equation broken generators where may algebraic relation ( Goldstone boson. However, we mustthe ask breaking what of justifies ignoring boost theHiggs invariance? ersatz Mechanism” GB The [ arising answer from lies in whats known assary the criteria for as the the IHM. “Inverse When two broken generators the associated charge densities.both Furthermore calculable counting and rules incalculable for gaps gapped ) have Goldstones been (with developedbroken. [ Consider the casecorresponding of to a particle canonical number superfluid. and This the system rank breaks of a U(1) symmetry ultimately led to thethe final group result G for is broken the to number H ( [ where literature when internal symmetries areignored. broken, the We will breaking of come boost back to symmetry is this usually important issuefor below. the type I ( breaking were first written down by Nielsen and Chadha [ and one can not separate space-time from internal symmetry breaking. However, in the JHEP05(2018)014 (1.11) ) in these 1.5 transforms under σ = 0 (1.9) i ) ], if the U(1) of particle x since the time derivative ( 26 4 O h )) a i∂ )( ) (1.10) x ( a vt, t M π ij − iδ x ) + ( i ] the authors considered two such symmetry σ ] = it∂ j – 4 – 10 → )( ) ,K x i ( i , there is an IHC which allows one to eliminate the P [ η x, t ( ) is satisfied, it is often possible to impose a constraint M σ is unbroken, i.e. no condensation, we have ) + ∝ 1.5 j ] M i∂ ] point out that given that the dilaton ( i 26 x P,K ] there are cases for which there is no inverse Higgs constraints ijk 10  ) x ( k θ ( ]. 25 As will be discussed below a resolution of the framon puzzle is closely related to the Another missing Goldstone boson arises in the case of non-relativistic dilatation in- In any case when condition ( Matter fields transform as projective representations under boosts which allow for the canonical kinetic 4 fact that when space-timetively symmetries coupled. are broken, To Goldstone explore bosons need this not possibility beenergy we term. deriva- utilize the coset construction which will begs the question, can onecondensate? write The down answer, a as sensible will dilatonin be kinetic discussed the term below, action. if is yes there Soconnected. as is long we no as Thus see particle the framon the that istied the puzzle included to questions of the of the fate the of non-relativistic framon the dilaton and framon. remains, the and dilaton its are resolution intimately is number is broken then, as a consequence of the algebraic relation, boost invariance is also brokenIHM (assuming at translations play and are the unbroken). Ward identities As may such, be there saturated is without the an need for a dilaton. This variance. The authorsGalilean of boosts [ as there is no wayof to the write dilaton transforms down non-trivially. a As boost also invariant pointed kinetic out term in [ only broken symmetry is boostcheck invariance of while the the Galilean latter algebra also showscases that breaks and none rotations. of yet A the the broken cursory to Goldstones generators satisfy be associated ( seen with in boosts, nature. dubbed the “framons”, are nowhere and yet the Goldstonesthen still due do to not theboost appear. fact Goldstone. that If [ But particle ifthe number there boost is is Goldstone spontaneously no in IHM broken, breaking patterns involving the called the analysis. type-I boost and generator, type-II In one “framids”. [ must The include former is a system in which the on the fields which isHiggs consistent Constraints with (IHC) the which symmetries. is This associated constraint with is the1.1 called IHM. the Inverse The missing Goldstones As was pointed out in [ so that bothtranslation rotations [ and boosts can be compensated for by a Goldstone dependent thus, assuming that the mass JHEP05(2018)014 5 ]. 5 (2.1) is the symmetry to write down the H ] and later general- U 23 , ]. We refer the reader to 22 28 , 27 ] in the unbroken phase. 6 ~ X · i~π e = – 5 – the corresponding broken generators. The un- . This parameterization will be generalized when U ~ X ~ T ] states that there can be no Goldstone associated with boost 5 ] in two ways. Ref. [ 5 is the symmetry group of the microscopic action and G are the Goldstone fields and ~π where We generalize [ 5 broken generators will bewe denoted break by space-time symmetries. As discussed below,invariance due we the may non-vanishing commutator use betweenfor boosts the and boost the Hamiltonian. Goldstoneto Here and consider we show relativistic show that generalizations. the need it couples non-derivatively. The coset methodology also allows us sub-group left unbroken by the vacuum. The vacuum manifold is parameterized by where A powerful method for generating actionssymmetries with was the appropriate developed non-linearly realized forized broken internal to symmetries space-time by symmetriesoriginal CCWZ by literature Volkov [ and for Ogievetsky details [ construction. and here The only method rapidly usesG/H review the the fact salient that points the of Goldstones this coordinatize coset the coset space scenarios in the context offor degenerate fermions. boosts We and will the show other two examples for of dilatations. DIHMs, one 2 Review coset construction We see that thereconstraints are are three applied paths andSome to the or space-time all system symmetry of retains the realization: oneto constraints are the Goldstone no applied existence for and inverse of we each Higgs IHCs,the have broken a or application reduced generator. a of number Goldstone of other can Goldstones inverse be due Higgs eliminated constraints. via In the this DIHM paper with or we without will consider all three see that again, theyfield do theory. not, This but analysis their wasin absence recently the used greatly unitary to constrains limit prove the that can form degenerate not electrons of behave interacting the like1.2 a effective Fermi liquid The [ paths to symmetry realization the Landau conditions in canonicalsymmetry Fermi realization. liquids, which when We imposed, dubbecause lead this the to the Goldstones the “Dynamical are proper further Inverse absent and Higgs discuss but a Mechanism” not more (DIHM), for generaland symmetry algebraic Schrodinger breaking reasons. pattern invariance where are both WeIn spontaneously boost will this broken, invariance case then which one go might we a expect call both step a a “type-III framon and framid”. a non-relativistic dilaton to arise. We will Furthermore, we will usewith the the coset symmetry methodology breaking to patternliquid construct of without/with the type nematic I/II theory order. framidsfollows of as from Fermi realized We liquids the by will dynamics aas see of canonical a that the Fermi Lagrange the effective multiplier field resolution we theory. of will generate By the a treating framon set the issue of Goldstone constraints, boson that are generalizations of allow us to generalize the criteria for non-derivatively coupled Goldstones given in [ JHEP05(2018)014 (2.3) (2.4) (2.5) (3.1) (2.2) ) frame. and consis- G/H 6 . ) . invariants. For a complete b X T H b A ] = A ψ ψ. ) + A T,X q A a ∇ A X . ~x q · a action. X . · π iT ]. A by forming iπ ]. In such cases the direct product of the im~v e + − 6= 0 32 ∇ t x 29 – · G ] µ 2 ¯ ]. P + ∂ ~ 30 P i µ mv – 6 – , A e A i 32 i 2 ¯ E P e = X ( ( [ A µ ≡ U → E non-Goldstone ψ ψ = A A ∇ ∇ U ) and use these objects to construct our action which will µ ∂ A 1 − U such that 7 ), relates the global frame to the transformed (acted upon by µ ) we can extract the vierbein, the covariant derivative of Goldstone fields ¯ E P 2.3 ] the criteria necessary to generate theories with non-derivatively coupled Goldstones From ( The Maurer-Cartan (MC) form decomposes into a set of well defined geometrical Once space-time symmetries are broken, the symmetry group is no longer compact. ) and the Gauge fields ( 5 A consequence of thisWhen fact translations is are that broken it by is localized semi-classical not objects longer (i.e. true defects) that the [ coordinate is lifted to π 6 7 ∇ is given by the status of a dynamical variable see for instance [ multiple contexts, we refer the reader to [ 3 Non-derivatively coupled (NDC) GoldstoneIn bosons [ ( be invariant under thediscussion full of symmetry the group coset construction and its application to broken spacetime symmetries in In this way, the covariant derivatives on the matter fields in the local frame are written as such that under a boost ground states with delocalized particles. objects, The vierbein The number of unbroken translationssymmetries may be as enhanced in if the thereinternal case exist and of internal space-time translational solids translations orthis are fluids broken work to [ we the will diagonal not subgroup be by considering the such solid. cases In as we are interested in zero temperature is a useful tool in determining this As such the structuretency constants requires can that not onetranslations necessarily generalize ( be the vacuum fully parameterization antisymmetric to include the unbroken the Goldstone couples tothe other coset gapless construction (non-Goldstone) seems fieldsHowever, this to in need imply the not that theory. beconstruct there the Notice an case, must that as invariant be action mentioned at above.Higgs without least constraint. It the one could We need very Goldstone will for well show boson. be a that that Goldstone, if we even can this without is an indeed inverse possible then the coset construction most-general action consistent with the symmetry breaking pattern, including terms where JHEP05(2018)014 , 9 T (3.4) (3.2) (3.3) ∼ ] ) allows formally then the 3.3 ¯ ¯ X P P,X ∼ ] ¯ P,X is the Poincare or Galilean ) is the fact that ( ) only involves the unbroken G . Alternatively if [ 3.3 3.3 , not only because of the zero µ ∂ P ) . π ( µ a ) and ( ... E . 2 ). First ( 3.1 ) we can see that if [ i → ∞ E 0 3.1 6= 0 k ~ √ 2.3 ] | x 4 . However, the Goldstone will often be absent d X – 7 – ,X E | are the unbroken space-time translations. The µ is not a well defined operator at infinite volume, Z ¯ k P ~ ) is a necessary but not a sufficient criteria for the ~ [ h P X = k ~ 3.1 → S 0 lim k ~ which can differ from ¯ P will arise in X ), i.e. 1 . 3 ) is a sufficient criteria for NDC, assuming the Goldstone boson asso- ). Also we will see that whether or not 8 3.3 is not removed by the inverse Higgs mechanism, we note that the veirbein X contains term linear in the Goldstone follows from the fact that the Goldstone acts as the is a broken generator and i E X To see that ( Within the coset formalism the search for non-derivative couplings starts with un- The non-relativistic case beingThat of particular importance below. 8 9 then the Goldstone will showthe up covariant derivative in acting the on connection, the in matter which fields. case the NDC will arise from transformation parameter. so that as long asthere the determinant will of be the a vierbeinGoldstone NDC contains associated a to with term matter linear fields. infrom the the From Goldstone, volume ( factor as income the from case of the broken boosts covariantization of or rotations. the Thus derivatives the first NDC will ciated with will contribute to the measure via commute. But an important distinctionfor between ( the non-commutation withAs the a Hamiltonian matter of asas fact, being this non-relativistic a explains criteria the forgroup NDC is nature NDC of of Goldstones. no the consequence dilaton as (both far relativistic the as the well criteria for non-derivative coupling is concerned. Note the distinction between thiscanonical criteria and spatial ( translations component, but morehowever, generally a if distinction there without are a internal difference translational because symmetries. internal and This space-time symmetries is derstanding how thedetermine Goldstones under couple what conditions to aany generic Goldstone derivatives arises matter acting in upon fields. thethe it. vierbein generalization or As Thus of connection a ( such, without necessary we condition need for non-derivative to coupling is and that the limitingcoset procedure construction is supports not the wellcriteria authors defined. to claims relativistic and However, systems. allows weexistence us Eq. will of to, ( see trivially, a below generalize non-derivatively thatnot their coupled the be Goldstones removed since via we an IHM. must also ensure that it can authors argue that the forward scatteringdiverge matrix elements of broken generators which compensates for the explicit factormay of be the Goldstone concerned momentum with in the the coupling. fact One that where JHEP05(2018)014 (4.1) ]. The ) if the 35 1.11 ) leads to a redundancy or 1.5 for consistency. Moreover, according . µ n ]). However, we know that condensed matter = i 36 – 8 – necessary µ A h ] is a system where boost invariance is spontaneously 10 ) we should expect the coupling to the framid to be at least is the Galilean group then due to relation the eq. ( ] the choice of order parameters can affect how the symmetry is the Poincar´egroup then the boost will be in the vierbein. But 3.3 G 20 G ], where a four vector gets a time-like expectation value. 34 It is tempting to disregard boost Goldstones since the associated generator does not Cases where the framid are not collective modes correspond to speculative theories Finally, note that if commute with the Hamiltonian and henceof there the is relativistic dilaton no immediately flat dispels direction.framid this into However, notion. the the existence Furthermore, the cosetto inclusion parameterization of the the is criteria formarginal. NDC ( lack of the evidencebounds for on a the Goldstone couplings arising (seesystems for break in instance boost, Einstein-Aether [ and theoryaround if allows to the us symmetry eliminate to breaking the place their pattern framids fate. is from such the that spectrum there it are is no incumbent IHC upon us to determine theory [ The resulting theory contains 3 Goldstone modes corresponding to the framons [ the vacuum is not annihilatedexcitations by of some the conserved material charge. responsiblesets for Put apart the another say breaking way, the the of pion modes boost in are invariance. QCD This from definition thebeyond plasmon the in a standard metal. model of particle physics and Relativity, such as Einstein-Aether the order parameter will beorder parameter, irrelevant. i.e. Nonetheless, those we whosevacuum are commutator expectation interested with in value boost a (e.g. generators certainyield have the class a Goldstones momentum of non-vanishing which density). aremean This collective a class excitations. quasi-particle of pole order Whereby (or parameters a resonance) “collective which exists excitation” as we a consequence of the fact that As was emphasized inis [ realized if therecompared exist to gapped the Goldstones cut-off). (assumingdetermine the whether In gap or particular size not the isa the representation gap. hierarchically the of small inverse However, the Higgs here order conditions we ( parameter(s) are only will interested in the truly gapless modes, so in this respect 4.1 Non-relativistic framids The type I framidbroken, as but defined all in other [ spaceabout time the symmetries symmetry are breaking intact. pattern The and coset not construction the only definite cares choice of the order parameter. U(1) particle number isconnection. unbroken, Whereas then if thein boost either Goldstone case will framid be will be associated non-derivatively with coupled. the 4 Framids JHEP05(2018)014 (4.8) (4.4) (4.5) (4.6) (4.7) (4.2) (4.3)  2 ) η · . ∂ ( = 0 2 L u 0 i . 1 2 j 2 ,E η i i − ~η ) η ∂ x 2 1 2 ( ) F ~η j . = = − · 3 η E i ~ 0 j i K / i = 1 η ∂ i − ( −  0 e 2 T n ∇ ? x . For a Fermi liquid the critical temperature · u ,E ,A c Λ j i i – 9 – ∼ 1 2 T iP i δ Fermi liquids superconduct, even if the coupling η e η ∼ C − − = c T = 2 i all = ˙ j T i ˙ = η i i U E 1 2 η 0 A  ∇ xdt = 1 symmetry d 0 0 d H E Z = S ] effect ensures that 37 is the strong coupling scale which is typically exponentially suppressed. Thus ), and the critical temperature ? F E I framid The vacuum manifold is parameterized by Thus we have narrowed our search for framons to degenerate Fermi gases whose phe- To manifest framids in the laboratory we need systems which break boosts yet whose The free action forunder the the Goldstone linearly realized follows by writing down all terms which are invariant The gauge field is given by and the covariant derivatives of the framids are (up to lowest order in fields and derivatives) Calculating the MC-form, we can extract the vierbein 4.2 Coset construction of Fermi liquid EFTWe begin with our investigation by rotational building the symmetry:with coset construction type broken for type boosts I framidsinvariance, but (i.e. as systems unbroken the relativistic rotations). case will We follow consider in a the similar case manner. of broken Galilean and the boost symmetry breaking scale is of thenomenology certainly same shows order. no signstempted of to non-derivatively coupled interpret Goldstone. zero sound Oneelectrons as might due be the to boost zero Goldstone, sound however, exchange the vanishes interaction in between the forward scattering limit. where Λ there is a rangeThis of is temperatures as where opposed the todensity framid the should bosonic contribute case to where the the heat critical temperature capacity. is set by the number function is repulsive ina all framon is channels for inscale there the ( to UV. be However, ascales all temperature as we window really between the need boost to symmetry manifest breaking ground state doesstraint. not break Thus any we symmetryper(fluids/conductors) may eliminate from which electrons the would moving list leadto in of to degenerate a possibilities. electrons an crystal in It inverseand background the would Higgs Luttinger as seem [ unbroken con- well that phase. as we su- One are might relegated be concerned that the Kohn JHEP05(2018)014 is g (4.9) (4.10) (4.11) (4.13) (4.12) , breaks v  i k ) scale as “resid- x i ψ ( X v  ) h i  d δ ] the quasi-particle self mη ) t 41 ( + – 4 is non-derivatively coupled, i k 39 ψ i∂ η ) ( . t ) ε ( 3 † k + , x, t ) ψ 2 ( x ) ( t ψ ( v ~x m~η · 2 h k 1 2 x · ψ ~v. im~v are technically irrelevant (see below). ) t − + t ( ) + imv ψ ]. Derivatives acting on i 2 1 e ~η † k ∂ i 42 m~v ψ v η – 10 – i ) 2 → X e + ~η . The vacuum of the system, labeled by m~η 0 → ) = ∂ m ( + ) x i i (  k  ~ φ † ( x, t ( k is a scalar. Notice that the ψ dt g g i undergoes a shift xdt ψ k d , e.g., is well described by Fermi liquid theory, the framid must d η d 3 d Z Z = i,a Y 0 ), the coupling for the Goldstone to matter fields via the covariant deriva- ) which obey S = k 2.4 int S defines a superselection sector [ transforms as ) coupled to gauge fields. To power count this theory it is useful to introduce the v φ is the unknown dispersion relation that is fixed by the dynamics. Due to the ~v term will be sub-leading and not play role in the remainder of our discussion. ε 2 η As in the standard EFT description of Fermi liquids [ such that ual momenta” ( boost invariance and so we expect that framid should exist as an independent degree of realizations of the symmetry.particle An ( example of thisnotion is of a the field case labelone of is as a interested was massive in introduced the complex incarrying dynamics scalar Heavy of momenta Quark a much massive Effective less source Theoryre-phased which than (HQET) interacts field the where with quark light gauge mass. fields The label is introduced by defining a Before moving onto further discussionlight about the a framids subtle in point Fermisame liquids, about symmetry we non-linear want breaking realizations to pattern high- of canent lead broken particle to symmetries, content. contrasting which physical This is theories usuallyHowever, with that below happens very the differ- we when show there that are two even different with order same parameters. order parameter we can have two different as expected from ourbehavior. considerations Given of that the He somehow decouple, algebra, yet which it must can do lead so in to such a4.3 non way that Fermi the liquid theory Multiple remains realizations boost invariant. of broken symmetry Higher order polynomials inthe the coupling matter field function whichspherical now symmetry formally implies depends upon the framon. The assumption of The interaction is most conveniently written in momentum space velocity while the Goldstone field tive is given by where central extension of the Galilean algebra, the fermion under a boost transformation with Following eq. ( JHEP05(2018)014 can 1 . The (4.15) (4.14)  c ) F k t, ~x = ( | φ ) , then θ )] η ( k ~ to be completely t, ~x | ( 1 † c φ ) 2 m~η i 2 ). The matter fields (which , ) − x 4.9 ~ ∂ ( ) · θ ( ~η k ~ ψ + ~x t · ) ∂ . θ ) ( [( and we get the standard non-relativistic k ~ i φ − , and the theory should be invariant. This − t, ~x λ ) ( e t φ ) ) F t, ~x k ( – 11 – ( φ ] implies that the action can only be a function of iε ) im~η e 2 44 ) are completely different and we have no reason to − η θ ~ ∂ m~η X i 2 ))( decouples from + ) = can equally well be fixed by Reparametrization Invariance x ~ η ∂ t, ~x 1 ( ( · c † ψ ~η φ . which is invariant under translations and rotations, ) i + φ iP t ∂ im~η in the spectrum and the theory will still respect all the symmetries. ] where the labels change due to Coulomb exchange. Notice that )( = 1 then . The fact that theory should not depend upon the bin size imposes η + ] = 43 i 1 F ~ ∂ c t, ~x ( (( † ) is the large momenta around which we expand. As opposed to the HQET φ 1 H,K E/E m c θ 2  ( ∼ i k 2 ~ + ], which is related to the freedom in splitting the heavy quark momentum into λ = 44 . In general RPI generates relations between leading order and sub-leading Wilson φ ~ ∂ If we choose Using the covariant derivatives derived in the previous section, we can write down the L + F scale as constraints on the action.fixed That value is, we wish, we by shouldre-parameterization an invariance be (RPI) amount able [ scaling deform as k the momentum around any the assumption of rotationalfield invariance implying label that thecase, magnitude of here the binsakin are to dynamical and NRQCD therethere [ is is no a super-selectionwe sum rule. have over This effectively the case tessellated is labels the more as Fermi surfaces opposed into to “bins”. an integral, The size this of illustrates each the bin fact will that we will call electrons fromremoving the here large on) energy are and effectively momentum expanded components around via the the Fermi redefinition surface, by a Fermi liquid lies in the change in4.4 the number density from Power counting one toTo determine Avogadro’s the number. possible symmetrysystematics realization in of a the Fermi relevant liquid, EFT we must whose first action discuss is the given by ( The two theories (with andbelieve without they will leadbreaking pattern to and same the same physicsmultiple order in parameter ways (local of the momentum realizing IR density). thetrivial and Thus example, boost there yet we are symmetry. they note While that have it only the would difference same seem between the symmetry that HQET this ground is state a and rather that of be fixed by requiringcommutator the [ theory to obey(RPI) Galilean [ algebra, ina particular large by and satisfying small the arbitrary piece and (more keep on this below). However we can leave kinetic term for a free particle. Had we started with a theory without the system which are clearly absentboost as the invariance must choice of be vacuum non-linearly is realized. not dynamical. Nonethelessmost the general action for freedom. Typically, the Goldstone modes are associated with collective excitations of a ~ JHEP05(2018)014 ]). 41 . In – ~ ∂ (4.21) (4.19) (4.20) (4.16) (4.17) (4.18) i . With . 38 ) F + x . . E ( ~ P ) ⊥  θ i ( k ∼ l k ~ ψ i X  /λ ) 1  F d k ∼ δ ( ) ε ) 4 k p − t, l ( − )) 4 x k k ( k ψ ( ) 1 3 . m~η − ) d t, l x + ) δ ( ( l ) ) 3 . ~ ∂ θ ⊥ † − i k , ( ~ P p 2 ψ k ~ t, / · + ) ( 1 ψ − 2 ~v ) k ~ − ~ P θ − ⊥ ( λ t, l ( + k ε ψ ( k ~ ( 0 ) 2 ∼ δ + k F ) i∂ – 12 – 2 ~v ψ ) ) = ) · 1 → x x k, t } ∼ ~ ⊥ ( ( ( 0 l ) ~ t, l ψ θ ( ( i∂ + m~η does not change the quasi-particle label. The reasons 1 k ~ = 0) † 0 k 1 2 ψ η ψ ~ i∂ P ) ~p,t i )( ( ) + Λ, where the theories’ breakdown scale is Λ † k ~ 2 ~ ∂ ( i t, l g ) E/ ( , ψ x † k ~ + ( dt ∼ ) ~ i P θ l ( ( = 0) λ d † k ~ · ldt ψ d ) d k, t x ~ d Z ( ( ~η ψ i xdt ψ Z k ~ { d X − d k ~ 0 X + ) scale as Z i∂ 10 ) ⊥ =  θ k ( does not scale. Thus ignoring the framon for the moment, the leading order action X k × ~ k FL k ] such that S = The scaling of the electron field We are ignoring spin as it will not play a role in our discussion. 45 0 [ 10 S since is given by follows from the equal time commutator We first review the EFTIn of the Fermi EFT, liquids and thesurface its power ( power counting counting is (forthis details such scaling, see that the [ most thecoupling relevant momenta function terms around perpendicular in the to Fermi the surface the action and Fermi come keeping from the leading expanding term the in energy and 4.4.1 Review of EFT of Fermi liquids scalings Notice that the interactionfor with this will be discusseda below boost once the we labels have area fixed left boost the invariant power the but counting time the systematics. residual derivative transforms momentum Under as shifts. Furthermore under P Full theory derivatives thenthis decompose way into we theconservation may RPI at drop each invariant vertex. combination the The exponential action factors becomes as long as we assume label momentum coefficients. It is convenient for power counting purposes to introduce a label operator JHEP05(2018)014 k 0 l λ . A one loop l g are either marginally rele- of the Fermi surface. l λ g : the BCS configuration (back to /λ . However, there are two configurations 1. The latter scaling might seem odd given scales as the bin size, which does not play λ ∼ k is constant on a spherically symmetric Fermi l , in the EFT language this would be called a k l F k – 13 – k | i ⊥  and ∂ε ∂k λ E = ∼ F ⊥ l ~v ), the operator will scale like 3 λ . Figure (a) shows the interaction of a quasi-particle with a framid that is far ∼ . Allowed kinematic configuration for quasi-particle scattering. Diagram (a) is the BCS 2 , where it can be seen that these are the only two possible configurations that allow 1 It is convenient to decompose the BCS coupling into partial waves Naively the interaction terms looks irrelevant because the delta function scale as off its mass shell“potential in framid” the and sense canwith that be those of integrated the out.pling. phonon and Thus Note screened these that electromagnetic interactions theoutgoing interactions, are quasiparticles potential into can a is swept, not non-local effectively along be local cou- the because same and the hence labels it on is analytic the in incoming (the and small) residual 4.4.2 Power counting inLet the us coset now construction derivekinematics of the the power framid interactions. countingin The from figure two the allowed scattering coset configurations are construction. shown We begin with the vant/irrelevant for attractive/repulsive UV initialdoes data. not run, The but forwardwill plays scattering show an coupling that important Galilean roleBCS in invariance kinematics is the are IR sufficient thewithout nonetheless. to only the prove possible need Interestingly, that below to marginal/relevant consider we the interactions.the the four forward effects This Fermi scattering of operator. result the and follows special kinematics on the power counting of back incoming momenta) andfigure forward scattering. These twofor configurations momentum conservation are that shown keep in all momentum within calculation (which is exact) of the beta function shows that integral can be absorbed into the label sum. for generic kinematic configurationsaccount so that, ( once thefor scaling of which the one measure of is taken the into delta function will scale as 1 The Fermi velocity defined as surface. In the lastThe term residual there momenta is scale a as that Kronecker delta it for is the a labela residual momenta role that momentum. for is implied. However, Fermi surfaces which are featureless Another way of saying this is that the Figure 1 back to back configurationwhich which the final leads state to momenta Cooper lie on pair top condensation. of the (b) initial Forward state scattering, momenta. in JHEP05(2018)014 , F , so λ ... (4.23) (4.25) (4.22) (4.24) E/E ∼ ... ) + x λ ... ( ) ) + θ ( x ], which leaves ) + k ~ ( ) x ψ 46 θ ( ( i ) k ~ θ ~ ∂ ( ψ · k ~ and the framons can i ) ψ ~ 2 ∂ θ  ( · λ F ) for the theory to be boost ∂ε ∂k θ ∼ i~v ( λ · In two dimensions we must dn k F λ . )) i~v ) + ? F x ? ∼ k ( ~ m m ij ) + δ ? = m~η ) − x m + F ( m d ~v ~ − ∂ δ )( i θ = m ( ∼ F )( k ~ ) + ( ~v θ ∂ε ∂ by considering the canonical commutator +2 ( · θ n is yet to be determined. However, symmetries ( n F – 14 – in the interaction as a Wilson coefficient. λ k ~ ~v n 2 (0) · · ∼ ~η ) ) /k x x − ( ( ~η ~η 0 (0)] j ˙ i∂ − − η If we define our power counting parameter as , where h , ) 0 we can fix 0 n . ) ) λ x F n i∂ x i∂ ( (  h λ i E ) 4.17 ∼ ) ) θ η ( . [ ∼ x x † k k ~  λ ( ( ) ) ∂ θ θ E ∼ ( ( † † k k ~ ~ must scale in the same way as the residual momentum of order k ∼ xdt ψ η 2 k d xdt ψ xdt ψ d d Z . Thus we see that in two spatial dimensions ) d d Figure (b) shows the interaction with an on-shell framon whose momentum θ 1 ( 2 − X k ~ Z Z is the effective mass defined by . Given that d 11 ) ) λ θ θ . Allowed kinematic configuration for framid-quasi-particle scattering. Diagram (a) in- ? = ( ( = X X k k ~ ~ m ∼ as the covariant derivative must scale homogeneously in n =2 η = = From here on to simplify the notation we will be dropping the label sum and the bold Expanding the action ( d 0 0 In this sense it is better to think of the 1 n S S 11 font for labels as all momenta unless stated otherwise will be labels. where multipole expand the framononly field the to coupling preserve to manifest the power framon counting zero [ mode. The leading order action is given by not change the (residual)relevant. momentum This of however the isresidual quasi-particles momentum not and the only case their in zero three mode dimensions is where the framon carries off invariant. That is, that thus leading to near forward scattering. momenta. is necessarily soft then we only knowfix that Figure 2 volves an off-shell framid, which can be integrated out. (b) shows the interaction with a soft framid JHEP05(2018)014 and (4.27) (4.26) λ . Recall . 4 ) t λ ( 1 p . ψ  ) i t k ( 2 p i = 0 where ψ X ) t  B i ( d 3 O δ † p ) t ψ ( ) 1 t k ( 4 ψ † p ) t ψ ( 2  as the measure scales as k ) 2 a ψ ) i,a p λ t ( ( ∂p 3 † ∂g k ψ ) ) i t t X ( ( p 4 † k ψ ψ – 15 –  i  ) p j i p j k k ~ ( i ∂p ∂ε ∂ gives the operator constraint X ∂g m η  · ) − d ~η ( i δ p 2 m d a  ) p ) π j d t X ( d (2 † p dt ψ i 4 =1 k Y a d d ) p d d π Z d while the last term naively scales as (2 Z 2 m 0 i λ Z − Y = = B i Since the kinetic piece of the Framon action vanishes for the constant zero mode int O S . However, symmetries forbid such contributions and it must be that if we do not 2 12 The power counting of the terms in this constraint deserve attention. The first two Expanding the action for the four-Fermi interaction term leads to the coupling Before we move on to determine the consequences of the multipole expansion let us = 0 as shown in the appendix where we also derive the relativistic generalization of λ Even though our arguments in this section are strictly valid only for d=2, we keep d arbitrary to plays the role of a Lagrange multiplier. 12 B i (being conserved charges) are not.the order Thus it we are would seem working).the that time However, the dependence if last we will term insert cancel. must the vanish quartic (to term ingeneralize a it two latter point to d=3. function that at this point wefunction have does not not made any scale. assumptionhave We these about are special special trying kinematics kinematics. to soterm Thus derive the in we the delta the might constraint. fact naively that InWe think general the begin that this only by we is relevant noticing can true, couplings that but drop there the the is quartic quartic an term exception as is we time now dependent explain. while the quadratic terms constraint is non-local inis the a sense function that of it theO is Noether integrated. charges. This Indeed, current isthis algebra crucial, constraint. imposes as this the same constraint constraint terms scale as This is a strong operator constraint both technically and colloquially. Notice that the Using the equations of motion for Let us consider thesions. ramifications of the multipoleη expansion of the framon in two dimen- incompatible. Thus werealized see via a that Goldstone inthe as two theory, the spatial as framon dimensions, will equationsnot of be the follow. motion discussed symmetries allow below. can us not to In eliminate be three4.4.3 it spatial from dimensions The this framid conclusion as does Lagrange multiplier and the Landau relation pause to clarify thisfrom unusual scaling. the Typically scaling in ofwould an EFT the be the momenta useful scaling notnot to of the the understand fields other what follows waymultipole happens around expand to as the loops in framon with this interaction, momenta case. that scaling power as Indeed, counting and it boost invariance are JHEP05(2018)014 k ~ (4.31) (4.29) (4.30) . i p ∂p ∂ε ) ε ) (4.32) ) (4.28) p − x, t F − ( ) ε 2 ( F θ δ p ( ) ( k θ ψ .  ) ) p, k ) θ ( g x, t ( d ) ] for a Fermi Liquid i ) (cos p 2 l θ d π ( 47 P d ∂k † k l (2 k, p, p, k g ψ ( we get ) 1 l Z ∂g g X 14 2 x, t m ? ) θ ( ), + ) π θ m 1 ) θ ) in a 1-particle state with momentum 2 ( (2 cos ) + 2 k 1 3 (cos ψ i l 4.27 ) dθ P p, k l ( ∂p – 16 – g Z k, p, p, k g x, t l ( ) ( 2 ) = 1 + p ) 1 F P ∂g θ ? π p ( − 2 † m k  m (2 F p ψ ) = p d ) θ + ( 2 d ( θ g F , θ Z v d 1 ) p θ d d ( π = ) g d m π (2 ) F 2 i m (2 k θ ( Z X k + i is actuality scaling as order one. This is a consequence of the power ) ∂ λ i ∂k k xdt ( d m ∂k d ∂ε Z + 2 m i 13 k = = ) i F ∂k ∂ε F k S k m We can glean more information from the Landau criteria by utilizing the fact that the It is interesting to ask whether or not more information can be extracted from the Using this result we get the famous Landau relation [ Next using the assumption of rotational invariance, and expanding the coupling func- We re-write this result in the form Consider taking matrix of element of ( = = In canonical EFT’s one uses dimensional regularization exactlyWe take to d=2 avoid for this sake mixing of issue simplicity which but com- the results are valid for arbitrary d. | i 13 14 k ~ k | plicates the power counting. equation is RG invariant.beta Differentiating function it vanishes. with respect Forlogarithmically. generic to To momenta the avoid the RG this conclusion four scaleone we point implies impose loop one that a result. loop the kinematic If interaction constraint we diverges to consider suppress the the forward scattering interaction, constraint by considering a twoof body state. the However, constraint as operator isLandau seen condition on by is external inspection imposed, the linesfour and insertion will point furthermore amplitude will the will insertion be automatically suppressed ofits be since the scaling. there satisfied quartic is no function once power into divergence the a that can enhance Notice that at this point ittheory. is not clear that this result will hold to all orders in perturbation The second term on the r.h.s. vanishes by sphericaltion symmetry. in Legendre polynomials, naively scaling as divergence of the integral. Suchwhen mixing a cut-off of regulator orders is is used.is commonplace physical. However, in here effective the field cut-off theories (the radius of the Fermi surface) ( We can now see why that the interaction term is enhanced because the radial integral, JHEP05(2018)014 (5.1) (5.2) (5.3) . = 0 we have not been able to 0 i 15 ) x ,E . j 2 Θ( η ~η iL 2 1 − (Θ) − e ], although to our knowledge its non- ) ij x = ( R 48 ~η 0 · = ~ K i i 0 − e ,A j x · η ,E – 17 – iP e (Θ) (Θ) ij j i ) = R R − , x = since the constraints imply that the loop involves no sum Θ = j i i ~η, E ( A U = 1 vanishes ] as well as in neutron stars [ 0 0 1 E ] algebraically. 49 (Θ) is the two dimensional rotation matrix. The gauge fields are R Calculating the MC-form we may extract the vierbein Finally recall that this result assumed that the framon acts as a Lagrange multiplier. Recall at this point the result in the section only hold at one loop. However, now Thus we have reached the conclusion that the only allowed interactions are BCS and It would seem that we have ruled out the possibility of a BCS interaction which has a In cases where the Fermi surface is singular there are other relevant interactions whose self contractions 15 where would vanish [ The rotational Goldstone boson (Θ) isof called electronic the systems “angulon” [ has beenlinear studied self in interactions the have context not been previously derived. (the typeII framid). Weto work avoid in an two algebraic spatial inverseis dimensions Higgs unbroken. for constraints, The the we vacuum sake assume will of that be simplicity. the parameterized Again, by U(1) particle number play. This will be discussedin below three when dimensions. we list the possible paths to5 symmetry realization Fermi liquid withLet broken us rotational now consider invariance the case where the rotational symmetry is broken by the Fermi surface that tadpole corrections toand the vanish. one loop insertion of the constraint areHowever, pure this counter-term was onlythe forced logical upon possibility us that in the two framon dimensions. remains in In the three spectrum dimensions, and there there is is no DIHM at configurations, however, assuming afind featureless any Fermi sensible surface examples. that we have restricted ourthe interactions to Landau BCS relation and forward holds scattering to we know all that orders. that This well known result follows from the fact forward scattering. Wesumed are that not the claiming onlyFurthermore, that sensible our coupling this argument with is regarding vanishing athe beta the rigorous fact function acceptability proof is that of forward since the our(with scattering. we BCS no arguments have coupling associated allow as- is counter-term). for based any It on coupling is which possible leads that to there a are other vanishing allowed tadpole kinematic over the large label bins leading to a powernon-vanishing beta suppressed function result. at one loop.would However, not this is contribute not the toBCS case the as interaction. such Landau an interaction relation since the tadpole diagram vanishes for the then the one loop result JHEP05(2018)014 ∼ ) x (5.4) (5.5) (5.6) (5.7) thus the ψ. λ  )) j ) must hold for ψ. ... i mη ij 4.27  + Θ + j j j ∂ ∂ i F i∂ ( Θ v ij ij Θ  i (Θ) , ) + R Θ + ( i Θ) ε ∂ j m~η ∇ + . Unlike the framid, the angulon + p 2 ) is no longer justified, as the angulon ~ ∂ Θ = i Θ)( j i ∼ ( m~η ∂ · ), a Goldstone boson mass is forbidden 1 2 ∇ 4.31 ( E F 5.6 (Θ) ij ~v ij ) + D i + R ∂ ~ – 18 – ∂ + j · η 2 in three. ] but they did not consider the framon. Given that ) and ( ˙ i~η λ Θ = Θ) 5 i (Θ) + 4.8 ij ∇ 0 = ( R i∂ , + h and following the same arguments as above the field Θ( KE ) and keeping on the leading order piece we have Θ † 0 λ ~ L ∂ ∂ ( · 5.5 i ~η  † xdt ψ + d i d ˙ = 3 it is classically marginal and the fate of Fermi liquid behavior is Θ d Z xdt ψ d = 2, the interaction with the angulon is relevant and thus destroys Fermi = d Θ = d 0 ψ Z S ∇ = ) imposed by the non-linearly realization of boost invariance. However, at least ψ S 4.31 in two spatial dimensions and as Notice that the breaking of rotational symmetry does not effect the operator rela- Expanding the action ( 2 / 1 no gapped Goldstones asanism a for consequence our of chosenexpect the symmetry that breaking fact pattern. this that masslessness thereit If is should should there no hold persist is non-perturbatively. inverse no toangulon Higgs anomaly Vishwanath all mass mech- then and correction orders we Watanabe at in showed should one perturbation the loop cancellation theory, [ of indeed ask whether or not one can impose a5.1 DIHC to eliminate The the stability angulon ofAs from Goldstone the can boson spectrum. be massdespite seen under the from renormalization fact the that actions the ( Goldstone boson need not be derivatively coupled. There are coupling becomes strong in thesions IR and it quasi-particle is picture possible breakstheory down. that remains In a weakly three coupled. perturbative dimen- the result In system for any to the be case boost thephysical Landau invariant. operator ramifications. relation However, constraint could in It ( follow strongprediction would coupling if in be it the systems is interesting with not to easy broken utilize to rotational this deduce symmetry. constraint the to In generate particular new it is interesting to liquid behavior. In determined by the sign of the beta function fortion this ( coupling. in two spatial dimensions, the Landau relation ( We see that for scaling is not fixed by symmetrymomentum and transfer its consistent momentum scaling with iswith the determined an effective by the theory, angulon maximum i.e. shouldangulon the momentum leave scales scattering as the of electron anλ electron near the Fermi surface to within The kinetic piece ofvariance the is angulon given by Lagrangian consistent with time reversal and parityso in- the angulon is a “type I” Goldstone, i.e. The quadratic piece of the quasiparticle action is given by the covariant derivatives of the angulons are JHEP05(2018)014 ) C (6.1) (6.2) (5.8) . Note this should 3 )] ~p ) )] θ ]. As such, we will study systems ~p ( ( ε 26 R ( − ε i ω − ), special conformal transformations ( [ iK ω D [ iD − pLog d ] = ] = pLog – 19 – d ,C dωd i H,C P R [ [ dωd R i ] = θ Γ[ as a consequence rotational invariance of the measure. This result θ ). The relevant commutators of the Schrodinger group (the non-relativistic ) implies that if dilatations are broken then so are the special conformal i K 6.1 . Diagram a) could contribute to wave function renormalization whereas both a) and b) , whereas the angulon only knows about the shape of the Fermi surface and not its F E as these relationsnote imply that a ( reduction in the naive number of Goldstones. Furthermore, invariance in non-relativistic systems isdilaton unique appears to as be the in non-relativisticfor tension which kinetic with the term broken boost symmetries for invariance are [ the and dilatations boosts ( ( conformal group) are when using a cut-off. Such counter-terms should not be6 considered fine tuning. Broken conformal symmetry:As eliminating mentioned the in non-relativistic dilaton the introduction, consequences of spontaneous breaking of conformal NOT be expected forscale the framon, sincedepth. boost Of symmetry course, breaking boost isinvariant invariance regulator, dictates sensitive i.e. the to not framon the mass awhose UV must cut-off. mass vanish corrections The if vanish situation we in use is dimensional a regularization analogous but boost to necessitates the counter-terms case of the dilaton which is independent of tells us that, at the level oftwo the diagrams integrals, which there contribute must be to an the algebraic mass cancellation at between the one loop shown in figure we have constructed theNonetheless full is action, it the instructive tofrom all study the orders the angulon. proof one The follows loopof Goldstone from case the mass the in effective can order Ward be action tobackground. identity. read generated distinguish off For by the the by integrating angulon framon considering out we the the find quadratic electrons piece in a constant Goldstone Figure 3 could contribute to aboost mass. invariance. At zero external momentum the two diagrams cancel as dictated by JHEP05(2018)014 ]. 6 (6.3) (6.4) (6.9) (6.5) (6.13) (6.11) (6.12) . φ − ~ ∂ e (6.10) · ) j (6.8) i . ) 2 δ x φ tρ~η λ ρ~x , ) ρtλ (6.7) ) ~ = ∂φ ρ ρt ) 2 ~ηe iDφ + + i ( + · tρ∂ j i − (6.6) − − t − C ) e ~η − E g, π ~ i ∂η ρ~x ~ ∂λ ( ρ = · η + · h − tρ∂ j .... iCλ ) ~η ˙ ~ φ 2 ∂ A φ ~η − − + e ... , g + + e 0 + + ~η ~η i i · ) φ t i π j ˙ + · η x η λ λ 2 ~ ( K α α λ ( ˙ ( ( ˙ i e ( i φ ~ αλ φ ∂ λδ α∂ φ φ φ U α~η 2 – 20 – i = − − ( ∂ α∂ 2 3 2 4 are redundant degrees of freedom. The ensuing 2 3 1 − D ~η e ∂ φ φ i 0 e e e x 2 φ 3 1 2 · φ E e − − e − e = = ) = − ) iP ~ ∂ π 8) to zero. Linearizing yields the two possible inverse gives the resulting transformation properties of the ~ λ λ e . β ~ ( ======β · φ ( 0 0 0 = i i 2 1 and − ~ β 8) = λ λ φ φ ~ 0 η η . - K i i gU − 0 0 j 0 λ e A U ∇ ∇ ∇ ∇ ∇ ∇ t = x ~η λ φ ∂ ∂ 0 0 6) and (6 . Table . E H 6) and (6 . ∈ . Infinitesimal variation of Goldstones under broken charges. h and Table 1 G ∈ g Let us now address the question of the possible symmetry realizations. We will see that We see that there are two possible inverse Higgs constraints coming from setting the The invariance of these objects under boosts, dilatations and special conformal trans- no matter what path is chosen, the systems will not behave like a canonical Fermi liquid [ where Goldstones. covariant derivatives in (6 Higgs relations from (6 formations follows byGoldstones first via determining the relation the non-linear transformation properties of the The algebra implies thatvierbein both is given by The gauge fields are transformations. The vacuum is parameterized via and the covariant derivatives are given by JHEP05(2018)014 will gets using λ (6.16) (6.17) (6.14) (6.15) λ φ , ) ) which we t and ( p ~ in the action. η 4.26 ψ ). The invariant . F i ) µ t F ( 6.5 µ 4 p ~ ), ( ψ ) and ( ) φ )) + t and redefine the energy ) ( 6.4 d 3 4.17 − F † p ~ m~η µ (2 ψ e ) + ) t ( ~p 4 ( 2 p Thus, although we have two pos- φ p ~ − − ψ ) e iK, 16 3 t ( p ( ε 1 † p ~ ] = − + ˜ ψ 2 ) p ~ η p · · µ C,P φ + [ ~η ∂ − φ 1 2 p = , e ( – 21 – − i ) ˙ e φ d iD ( ]. If we use one IHC then again we will have the − m~η 5 φ 0 dtδ ] = − ∂ d e a φ ) 2 p π d + − ) is the energy of the quasi-particle measured from the d i H,C (2 p [ ie ( p ~ ( ). As far as the interactions are concerned we have ε φ h 4 p =1 ( ) − Y a t ) must be amended. If two of the relation involve the same ε e ( ( † Z p ~ g + ˜ 1.5 ψ 1 2 × F φ µ 2 = e d ) int ) = pdt p π S ( d ε (2 d Z = 0 S This non-locality in EFT arises due to a poor choice of variables and is not in any sense fundamental 16 since the underlying theory is local. For notational convience wefunctional will as drop the explicit factor of Here the energyFermi functional surface ˜ since we have explicitly included the chemical potential this is not anot choice play as a a role consequence as ofwritten it power down shows counting the up and most neither symmetry.now in general amend Notice the using boost that vierbein the invariant new nor interaction versionaction the of in for connection. the ( the vierbein We quasi-particle have and already is gauge field given ( by DIHMs as discussed in the next section. 6.1 Consequence ofTo broken derive conformal the symmetry relevant DIHCs viathe we the will framon DIHM again as build Lagrange the coset multipliers. and As treat both in the the dilaton previous and cases, in two spatial dimensions so that the two constraintselimination are stated linked establishing in thegenerator ( fact on that the the l.h.s. criteriacases then for where there Goldstone is this one happens. fewer allowable The constraint. final We possibility know is of that no we other eliminate both which would lead to asible theory constraints which we appears can non-local. onlyof impose the one fact while we maintaining have locality. This is a consequence gapped as (6.10) is timesquaring reversal it. invariant This and realization thus includes aninvalidate two a allowed non-derivatively coupled term Fermi Goldstones in liquid which the would same description action spectrum [ without and theconstraints same such conclusion that we is equate reached. Finally we may consider using both We may choose not to eliminate any Goldstones, however note that in this case, the JHEP05(2018)014 . ) φ 4.31 ) (6.20) (6.21) (6.18) (6.19) t ( 4 ) = 0 p ~ t ( ψ 4 ) p ~ t ( ψ 3 ) † p ~ t ( ψ 3 ) † p ~ t ) gives us the beta ( ) with respect to ψ 2 ) dt p ~ t a ( 6.21 ψ 2 p 6.17 ) dt . d p ~ t a ) ( d ψ p g 1 ) ( d † p ~ t Z ( d β ψ 1 ) we have the constraints in the coupling. The Landau ) and ( † p ~ 4  =1 − Z Y a ) ψ µ )  1 2 6.19 4 =1 , µ ) 6.16 Y a i , µ i i ) (i.e. its is not an effective theory) p ~ p ~ 1 2 ∂µ , µ p ~ ( ) + i ∂ ( t p ~ ( ∂µ ∂g 4.21 ( , p ~ ) + ∂g φ t µ = 0. ? ψ · due to the Landau condition in eq. ( ( ∂g 2 i φ p ~  − m p ? µ i p ~ 2 ψ O ) m −  ∂ε − ∂p = i = , µ i – 22 – i ) ) i = p ε p ~ φ ∂ε p ~ ∂p , µ i ∂ , µ ( δφ i i m − i δS p ~ p p ~ ) ) vanishes, and, as such, if we take the one particle p ~ ∂ ∂g ( ( p − )), · ( g ) ) ε i ∂g µ ( 6.19 p 2 d · fermions at unitarity are not properly described by Fermi p ~ ( g  i ε − ) − p ~ 2 t )= ) (  ) to leading order in − † p ~ ) t , µ ) i ( p, µ ( † p ~ p ~ 0 = (2 , µ 6.17 g to zero and varying the action ( ( i ). For higher angular momentum channels, ( g p ~ η ) g ( ( d g pdt φ ψ β ) pdt ψ d − d d d ) and ( d (2 − ) = Z  µ ) which ensures boost invariance remains unchanged but we generate a new Z ( (2 6.16 . φ k ~ g  k ~ ) X × × X 4.27 d = − = φ φ S We can also consider how these symmetry constraints can be utilized if we assume that Let us now see if a Fermi liquid description is consistent with these constraints. Given O For S-wave scattering ( and (2 function to all orders. states. Then taking the one particle matrix element of ( and liquid theory the microscopic theory isas defined done via in the simulations. action Inis this no ( case mechanism since by which there the is quadratic no term restriction can to cancel forward with scattering the there quartic for all choices of second term in thematrix last element we line see ofthe that ( the quadratic quadratic term and willand quartic depend terms the upon must quartic the vanish amplitude willrunning separately of which since not. the is incoming inconsistent In withis external three Fermi free. momentum liquid dimensions Thus theory, we we and conclude in see that: two that dimensions the the theory coupling has power law our assumption of rotationalthe marginal invariance coupling and is the only notion a of function of a the well angles defined which are Fermi scale surface, invariant. Thus the The constraint follows from imposing relation ( constraint by setting Expanding ( Here we have also introduced the renormalization scale JHEP05(2018)014 ]. 6 – 23 – We then presented an example of a DIHC which can not be satisfied by studying the In general we do not know a priori if a given DIHC can be satisfied. In the case A Landau relation from GalileanThe Landau algebra relation canmanding also that be the derived Fermi (similar LiquidGalilean to action, boost without Landau’s invariant. including originalthe the derivation) This boost by Noether is de- Goldstone, charges equivalent should constructed to be from satisfying the the Galilean Fermi algebra Liquid by action. using The only commutator Goldstone of special conformalis transformation gets not gapped. consistentGoldstone However, with the would a leads ensuing to theory Fermi (at liquid least) description marginal Fermi as liquid the behavior [ non-derivative coupling of the symmetries can be realizedDIHC by (Landau the relation) inclusionwhich of to allows a eliminate for dilaton the the elimination andsatisfied of associated a the and dilaton Goldstone framon. is as the leads well. Using onlyleads to In the path to two another boost to dimensions a symmetry the DIHC DIHC realizationsymmetry constraint can while can be that in be three contradicts dimension consistently the the realized Lagrangian DIHC by dynamics. including a The dilaton full and Schrodinger a framon, while the and forward scattering. At theare same the time, canonical only Fermi possible liquidcoincidence. theory marginal tells interactions us that based these on power counting.realization Clearly of this the is Schrodinger group no broken by a Fermi sea. We showed how the broken the IR, strong dynamicsthe will framon set could in be and hiding change under the the relevant shroud degrees ofof of strong the freedom. coupling. breaking As of such, interactions to boost be invariance relegated to we particular showed kinematic that configurations, i.e. the back-to-back constraints (BCS) force all low energy the coupling to thesystems which effective manifest mass. aGoldstone) boost This path Goldstone, in to as symmetry in itselfno realization three does a seems dimensions priori not to the reason explain framon bewe why have (the why perfectly we shown, boost we consistent. would the framon expect know Thus is no of there non-derivatively such coupled, no is systems thus it to is arise natural in to nature. expect that However, in as can be generalized.(DIHM) We whereby a introduced strong operator the constraintrealization. notion (DIHC) is A of imposed simple a which example enforces of dynamicaltheory. symmetry a inverse DIHC The Higgs was missing presented mechanism DIHC here boost in is Goldstone the imposed. context is of seen In Fermi liquid this to case be the absent constraint as leads it to is the unnecessary Landau once relation the which relates The predicitve power of symmetriessymmetries is not are lost simply when realized the inboson ground a appears state non-linear is which fashion. not saturate invariant. Forare the internal The symmetries broken, relevant Ward Goldstone new identities. pathwaysthat When to the space-time inverse symmetry symmetries Higgs realization mechanism arise. which leads to We a have reduction shown in in the number particular, of Goldstones 7 Conclusions JHEP05(2018)014 , i i i k i K iP k (B.3) (B.1) (B.2) (A.1) is the ] = ) )) = where 00 4 ), (taking where i k T 2 ,H i g i p, k W − ( ( iP G 3 g + O ) k i p ) k ( 4 − ] = ) ε i k 4 2 ( is the generator of = i k k i i − ∂ − 3 + ∂p G K H,K k i 3 1 0 i k k | − ( p ) iP − 2 d ψ ( k 2 † p where δ k ψ 4 + ) and satisfying [ ] = | i k 0 i 0 + 1 p ψ h iP k 1 3 p p ( k k − ) d ψ ( d V,W ψ d ) ) is due to interactions. The Poincar`e p ( 2 d ( ] = p δ † k ( i i ( d 4 Z δ ] is a little more involved then compared ε ψ δ k 4 ] + [ W † p 1 i k ψ + † k 51 ψ 3 = ψ i ψ H,G k  3 p ) is the momentum operator. In terms of the } ) k V, k ψ i 0 p∂ ψ 2 i † p ψ k d i 2 † k ( 2 P d ∂ p, k † k ψ – 24 – g ). , ψ † p ] + [ ( 1 is the energy momentum tensor, and p i ψ † i g k Z 1 i ψ k i † k ψ p ψ 0 { d µν i 4.27 | ,W ψ 3 as the free Hamiltonian and V as the interaction d ∂ − p 0 T ) d ) p i i p 0 ψ H ψ k Z † p 2 k p ψ ) Z ( H ( i ψ p ψ g i | g p i ( † Y p 0 ] + [ i i ε p im h i where † † p p k k p + d d − d , k p ψ p d d 00 0 d p d p ψ p ψ ψ d T p d d ψ H ) Z Z i i [ Z d d p ψ p x i ( Z i i † p ) ε p t R Y Z Z Y † † p p k ( = = = = = − p ψ ε d i i i i 0 p ψ p ψ d V k d d P W H tP d d ) + Z k ( t = Z Z ε i is the correction to the boost operator due to presence of interactions. This is = = =  i i i K i H P ∂ G W ∂k ) Using anti-commutation relation k ( ε Assuming weak interactions between quasi-particles andone neglecting particle terms matrix of elements for a state with external momentum, k) because sum of free and theis interacting from Hamiltonian free density. part Soalgebra of we condition defined the now Hamiltonian becomes density and where to the Galilean case.is The the commutator generator we ofGalilean need the counterparts. to Lorentz Denoting satisfy boost’s is but still the [ Noether charges are different from their B Landau relation fromHere Poincar´ealgebra we derive the Landau relationsame for result a relativistic can Fermi be liquidLandau from reached relation current by for algebra. the using The relativistic the Fermi liquids relativistic [ coset construction. The derivation of we get back the operator relation in ( Galilean boost, H isquasi-particle the fields, Hamiltonian these and operators are given by of the Galilean algebra we need to satisfy is [ JHEP05(2018)014 = ), . i µ 0 ( | . (B.4) ]. , (1976) p D ψ † p ψ | SPIRE 0 B 105 IN h (1962) 965 ][ (2014) 16314 (1961) 154 (2006) 085101 and 127 (2010) 054403 19 k 111 ~ B 73 and (2013) 011602 Nonperturbative behavior B 81 Nucl. Phys. ]. ~p , ) in Legendre polynomials, θ 110 Phys. Rev. ]. ( , is the chemical potential and g Nuovo Cim. SPIRE ]. , µ Phys. Rev. arXiv:1501.03845 IN , [ Phys. Rev. SPIRE ][ , IN SPIRE  ][ 1 IN Proc. Nat. Acad. Sci. , where G ][ Zoology of condensed matter: Framids, , µ 1 3 is angle between Phys. Rev. Lett. (2015) 155 θ . , l ]. ) = Quantum Theory of a Nematic Fermi Fluid 1 + g 06 ) F Broken Symmetries  µ k – 25 – ( ( µ ε D arXiv:1204.1570 SPIRE = ) where IN θ JHEP ∗ = [ and the density of states at the Fermi surface, Symmetry Obstruction to Fermi Liquid Behavior in the Criterion for stability of Goldstone Modes and Fermi , l m cond-mat/0102093 ∗ hep-ph/0008116 G [ (cos [ m g Couplings of a light dilaton and violations of the equivalence ), which permits any use, distribution and reproduction in On How to Count Goldstone Bosons ]. Implications of Relativity on Nonrelativistic Goldstone Theorems: ) = SPIRE p, k (2013) 039901] [ ( IN (2000) 037 g ][ CC-BY 4.0 (2001) 195109 = 2 and expanded the coupling function 110 08 ) and defined θ arXiv:1712.07797 d This article is distributed under the terms of the Creative Commons Field Theories with Superconductor Solutions , ]. B 64 (cos JHEP l ]. ]. , P l Quantum critical points of helical Fermi liquids SPIRE g l ) to leading order in g. Using IN p [ P SPIRE SPIRE − Addendum ibid. IN IN arXiv:1404.3728 Gapped Excitations at Finite[ Charge Density ordinary stuff, extra-ordinary stuff 445 [ [ Liquid behavior in a[ metal with broken symmetry Unitary Limit of the quantum phase transition to a nematic Fermi fluid principle Phys. Rev. F J. Goldstone, J. Goldstone, A. Salam and S. Weinberg, A. Nicolis and F. Piazza, H. Watanabe and A. Vishwanath, I.Z. Rothstein and P. Shrivastava, M.J. Lawler, D.G. Barci, V. Fern´andez,E. Fradkin and L. Oxman, C. Xu, D.B. Kaplan and M.B. Wise, V. Oganesyan, S. Kivelson and E. Fradkin, A. Nicolis, R. Penco, F. Piazza and R. Rattazzi, H.B. Nielsen and S. Chadha, ) = p θ [7] [8] [9] [5] [6] [2] [3] [4] [1] ( [10] [11] References This work supported06ER41449. The by authors the thank Riccardo DOE Penco for contractsOpen comments on DOE Access. the manuscript. 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