Criterion for Stability of Goldstone Modes and Fermi Liquid Behavior in a Metal with Broken Symmetry

Criterion for stability of Goldstone modes and Fermi liquid behavior in a metal with broken symmetry

Haruki Watanabea,1,2 and Ashvin Vishwanatha,b,1,2

aDepartment of Physics, University of California, Berkeley, CA 94720; and bMaterials Science Division, Lawrence Berkeley National Laboratories, Berkeley, CA 94720

Edited by Steven Allan Kivelson, Stanford University, Stanford, CA, and approved September 24, 2014 (received for review August 13, 2014) There are few general physical principles that protect the low- In contrast, coupling electrons in a metal to NGBs typically energy excitations of a quantum phase. Of these, Goldstone’s theo- leads to a much more benign outcome. We know, from examples rem and Landau–Fermi liquid theory are the most relevant to solids. of magnons in ferromagnets and phonons in crystals, that NGBs We investigate the stability of the resulting gapless excitations— are typically underdamped even in a metallic environment and Nambu–Goldstone bosons (NGBs) and Landau quasiparticles—when the FL theory remains valid. In other words, in these cases the coupled to one another, which is of direct relevance to metals with coupling between NGBs and FLs is irrelevant, leading to effec- a broken continuous symmetry. Typically, the coupling between tively independent fermionic excitations and NGBs at low ener- NGBs and Landau quasiparticles vanishes at low energies, leaving gies. This is because interactions involving NGBs are very strongly the gapless modes unaffected. If, however, the low-energy coupling restricted by both broken and unbroken symmetries. In particular, is nonvanishing, non-Fermi liquid behavior and overdamped bosons for these cases the scattering amplitude of electrons off NGBs are expected. Here we prove a general criterion that specifies when in the limit of small energy-momentum transfer must vanish. In the coupling is nonvanishing. It is satisfied by the case of a nematic contrast, quantum critical modes and gauge bosons couple di- Fermi fluid, consistent with earlier microscopic calculations. In addi- rectly to fermions, without derivatives acting on the bosonic field. tion, the criterion identifies a new kind of symmetry breaking—of However, there is one known exception to this rule. When the magnetic translations—where nonvanishing couplings should arise, continuous spatial rotation in d = 2 + 1 dimensions is spontane- opening a previously unidentified route to realizing non-Fermi ously broken by a Fermi surface distortion (20–22), the resulting PHYSICS liquid phases. orientational NGB strongly couples to electrons; i.e., their coupling does not vanish in the limit of small energy-momentum non-Fermi liquids | spontaneous symmetry breaking | Goldstone modes | transfer. We refer to this type of coupling as nonvanishing cou- strong magnetic fields pling. In this context, Oganesyan et al. (20) discussed non-Fermi liquid (NFL) behavior and Landau damping of NGBs, in close ccording to the Goldstone theorem, spontaneous breaking analogy with the case of critical bosons or gauge bosons coupled Aof a continuous symmetry leads to gapless Nambu–Goldstone to a FL. However, the deeper reason why this example violates bosons (NGBs). In a Lorentz invariant theory, these bosons are the standard rule of vanishing NGB–electron couplings in the expected to be well-defined excitations, even in the presence of infrared has been left unclear. Also, whether this is the only pat- other gapless fields, such as massless Dirac fermions, providing a tern of symmetry breaking with nonvanishing coupling remains an powerful general mechanism for low-energy excitation (1). A key open question. ingredient leading to their stability is the fact that interactions with In this article we formulate a simple criterion that allows one NGBs are strongly constrained by symmetry, leading to suppressed to diagnose the nature of the NGB–electron coupling. If the couplings at small momentum transfer. broken symmetry generator fails to commute with translations, In nonrelativistic systems, though, such general results are not the coupling is anomalous and is nonvanishing in the infrared. applicable. A particularly important scenario is spontaneous symmetry breaking in a metallic environment, of which there are Significance numerous examples such as magnetic order in a metal. Do the NGBs then survive as well-defined low-energy modes? Or does A remarkable feature of many particle systems is that although coupling to the high density of gapless fermionic excitations of they are described by equations respecting various symmetries, the metal lead to overdamped excitations? In this work we es- they may spontaneously organize into a state that explicitly tablish a general criterion to answer this question based on the breaks symmetries. An example is a crystal that breaks the pattern of symmetry breaking. translation symmetry of space. In such cases, a celebrated A closely related question has to do with the stability of the theorem predicts an excitation, the Goldstone mode. In this Fermi liquid (FL) when coupled to gapless bosonic modes. Be- paper we examine whether this continues to hold inside sides NGBs, gauge bosons can be gapless over an entire phase, a metal, where electrons can collide with the Goldstone exci- i.e., photons of the electromagnetic field or emergent gauge tations. Our result is a one-equation criterion that specifies bosons of spin liquids or quantum Hall states. Alternately, one whether the interactions between electrons and Goldstone modes can be ignored or whether it completely changes their can tune to a quantum critical point where gapless critical modes character. In the latter case, unusual phases of matter such as centered at wave vector q = 0 interact with the FL. The latter two non-Fermi liquids or superconductors may arise. cases, of gauge bosons or q = 0 quantum critical bosons coupled – to a Fermi sea, have been studied in many works (2 18). These Author contributions: H.W. and A.V. wrote the paper. d = + studies conclude that, for example in 2 1 dimensions, the The authors declare no conflict of interest. lifetime of excitations near the Fermi surface is significantly re- This article is a PNAS Direct Submission. duced, leading to an absence of well-defined quasiparticles and 1H.W. and A.V. contributed equally to this work. a breakdown of FL theory. Similarly, the bosonic modes get 2To whom correspondence may be addressed. Email: [email protected] or overdamped and can no longer be observed as well-defined [email protected]. particle-like excitations. In some cases, however, superconduc- This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. tivity intervenes at low energies (19). 1073/pnas.1415592111/-/DCSupplemental.

www.pnas.org/cgi/doi/10.1073/pnas.1415592111 PNAS Early Edition | 1of5 Downloaded by guest on September 30, 2021 Spontaneous formation of a crystal breaks this symmetry, resulting in phonons. Now, the magnetic translation operator P~ generates NGBs (phonons) and satisfies the nonabelian algebra, ½Px; Py = −ieBN. Thus, electron–phonon interactions under a uniform magnetic field are predicted to have nonvanishing coupling as we verify by explicit calculation. This surprising conclusion may be rationalized by imagining the fermions to hop between sites of the corresponding tight-binding model. The external magneticR field affects the phase of the hopping matrix ~xj ~ tij → tij i A ~x′; t · ~x′ ~u as expð ~xi ð Þ d Þ. However, a phonon fluctuation that changes the local flux per unit cell produces a fluctuation of tij, as illustrated in Fig. 1. One can imagine this as resulting from a fluctuating gauge field δB = B∇ ·~u = ∇ × δA. Therefore, for electrons, some part of phonon fluctuation under a magnetic field is equivalent to that of a vector potential δA = B^z ×~u and the problem resembles that of NFL behavior arising from min- imal coupling to a fluctuating gauge field.

Fig. 1. A previously unidentified route to strongly coupling NGBs and quasi- Proof of the General Criterion. The total Hamiltonian of the particles—electron–phonon interaction in a uniform magnetic field. A lattice system can be split into three pieces, = + + , ~ Htot Hel HNGB Hint distortion (phonon fluctuation) with ∇ ·~u ≠ 0 changes the local flux threading and each of these terms commutes with symmetry generators. each unit cell (darker blue indicates a larger flux), inducing fluctuations of the We are mainly concerned with Hint, which we expand as a se- phase of the hopping matrix tij . The phonon therefore couples like a gauge field a ð0Þ ð1Þ ries in the NGB fields π , Hint = H + H + ⋯. Note that to the fermions, without spatial derivatives. ≡ ψ †; ψ + ð0Þ ψ †; ψ int int H0 Helð Þ Hint ð Þ is the mean-field Hamiltonian that defines a one-electron problem by picking a symmetry-broken ground state. The interaction with NGBs is then determined by Furthermore, armed with this criterion we are able to identify a a new physical setting, distinct from the spontaneous breaking symmetry; e.g., the linear coupling for a constant π is simply of rotation symmetry, that also leads to nonvanishing cou- obtained by rotating the mean-field Hamiltonian by the corre- Q SI Text plings and thus, if we follow standard arguments, to a NFL and sponding symmetry generator a ( ): overdamped NGBs. ð1Þ = − iπaQ ; : [2] Hint a H0 Results General Criterion for Nonvanishing Couplings. Let us assume that To set up the perturbation theory, we first solve the single- we are at zero temperature and we make no assumption about particle electron problem described by H0 and obtain simulta- ~ ~ spatial dimensionality except that it allows for spontaneous neous eigenstates nk of H0 and the momentum P, symmetry breaking. NGBs can be associated with symmetry E E E E generators that are spontaneously broken, which we label Qa. ~ ~ ~ ~ ~ ~ H nk = e ~nk ; Pnk = knk ; [3] Furthermore, to sharply define a Fermi surface we assume the 0 nk existence of a conserved momentum ~P. This could be either the n conserved momentum of continuous translation symmetry or where is the band index. When the translation symmetry is crystal momentum (of discrete translation symmetry). Let discrete, we replace the second relation with E E ~ ~ ~ ik·~ai ~ iP ·~ai Qa; Pi = iΛai: [1] Tink = e nk ; Ti = e ; [4]

We now state the general criterion. If Λai = 0, this is the usual ~a Λ ≠ where f igi=1;...;d are primitive lattice vectors. The interaction situation where the coupling does vanish. However, if ai 0, of electrons and NGBs to lowest order can then be written as then the coupling between the NGB and electrons does not (Fig. 2) (1) vanish in the limit of small energy-momentum transfer. Note that this criterion is very general and involves only the pattern of symmetry breaking. For any internal symmetry (e.g., spin rotation or number conservation), the commutator is zero. Thus, for nonvanishing couplings one must consider a space-dependent symmetry. The simple case of broken space translation symmetry has Qa = Pa and the commutator is again zero, which implies that the corresponding Goldstone modes, the phonons, have vanishing coupling to electrons at small momentum transfer, as is well known. However, for the case of rotational symmetry breaking, Qa = Lz,whichsatisfies½Lz; Pi = ieijPj ≠ 0, where i; j ∈ fx; yg. Thus, nonvanishing couplings are expected in this case, consistent with the results of Oganesyan and coworkers in the context of nematic order in a 2D Fermi fluid (20–22). This general criterion allows us to identify an entirely new example of nonvanishing coupling. The criterion for nonvanishing coupling is also fulfilled by spontaneous breaking of magnetic translations. That is, begin with charged particles in Fig. 2. (1) The bare vertex with one NG line. (2a and 2b) One-loop diagrams a uniform magnetic field, with magnetic translation symmetry. for the self-energy of the boson. (3) The same for electrons.

2of5 | www.pnas.org/cgi/doi/10.1073/pnas.1415592111 Watanabe and Vishwanath Downloaded by guest on September 30, 2021 Z X d d ν ab ð1Þ d kd k′ a † a Π ν;~q = −iπ γ q^ ; = v c c ~π ; [5] Hint 2d n′~k′;n~k n′~k′ nk ~q ab ~q n′;n;a ð2πÞ Z [10] dk ~q =~k′ −~k va γab q^ = d va vb δ e δ q^ · ∇~ e : where and ~ ~ is the (bare) vertex function, which is d ~k;~k ~k;~k ~k ~k ~k n′k′;nk ð2πÞ the matrix element of Hð1Þ. This can be written via Eq. 2 as int ~ The first delta function puts the electron momentum k on the D E ~ Fermi surface and the second one further restricts k into a sub- πava = −iπa n′~k′ Q ; n~k ~q n′~k′;n~k ~q a H0 space where ~q is tangential to the Fermi surface. Note that the D E [6] 10 va = correction in Eq. vanishes if ~~ 0 (we have suppressed the = iπa n′~k′ Q n~k e − e ; k;k ~q a n′~k′ n~k band index n). The one-loop corrected boson propagator D−1 = D−1 − Π ν ∝−iq3 0 has overdamped poles due to the singu- 10 which, for low energy scattering n = n′ and ~q → 0, is larity in Eq. . Thus, the NGBs are destroyed (overdamped) by interaction with the Fermi surface at this order. D E Now, one can study the lifetime of fermionic quasiparticles by va ≈ i n~kQ n~k ~q · ∇~ e : [7] n~k′;n~k a ~k n~k evaluating diagram 3 of Fig. 2 with the corrected propagator D; the result is (23) ~ ~ Clearly, as long as nk Qa nk is finite, the vertex vanishes as − d= ~q → 0. This is why scatterings of electrons off NGBs usually τ 1 ≡−2 Im Σ ∝ ω 3: [11] vanish at ~q = 0, leaving behind well-defined NGBs and Fermi liquid quasiparticles. Therefore, Landau’s criterion ωτ → ∞ as ω → 0 does not hold d ≤ However, we can evade this conclusion if, and only if, the when 3, implying the breakdown of the FL theory. ~ ~ ~ ~ matrix element nk′ Qa nk diverges as k′ → k. To ensure an ap- Thus, this one-loop treatment at least shows the instability of FLs and NGBs against infinitesimal couplings with ~v~~ ≠ 0. The propriate divergence, we will need ½Qa; Pi ≠ 0 as we now explain. k;k ultimate fate of these interacting systems continues to be an ~ ~ PHYSICS If ½Qa; P i =iΛai ≠ 0, then the matrix element is nk′ Qa nk = ~ ~ active area of research (10, 12–14) and we do not expand further −i nk′ Λai nk = k′i − ki (SI Text). Substituting this in Eq. 7 on that aspect here. We merely establish the condition when ~q → and setting 0, we have the coupling interactions with NGBs are relevant and render the decoupled D E fixed point unstable, similarly to other well-studied cases. X va = n~kΛ n~k ∂ e ; [8] Below, we demonstrate our general criterion through examples. n~k;n~k ai ki n~k i Examples which is generically nonvanishing. This proves our claim that Internal Symmetries—Conventional Coupling. Let us first discuss when the symmetry generator corresponding to the NGB fails interactions between electrons and magnons in ferromagnets (in to commute with translations, nonvanishing couplings result. If the absence of spin-orbit interactions). The coupling between the translation symmetry is discrete rather than continuous, we the ferromagnetic order parameter ~n and the electron spin i † simply replace P bythediscretetranslationoperatorTi and ~s = ψ ð~σ=2Þψ (~σ is the Pauli matrix) may not contain any deriv- require ½Qa; Ti ≠ 0. Then the matrix element atives; e.g., D E ~ ~ D E ~ ~ Hel−magnon = Jn · s: [12] nk′ Qa; Ti nk n~k′ Q n~k = − [9] a ~ ~ eik′ ·~ai − eik ·~ai Hence it is not obvious that the electron–magnon vertex vanishes in the limit of small momentum transfer. However, we know it ~ ~ ~ is inversely proportional to ðk′ − kÞ ·~ai, leading again to a non- must from our general criterion, as the spin S and the momen- ~ ~ ~ vanishing coupling in Eq. 7 even at k′ = k. Note that ½Qa; Ti ≠ 0 tum P commute. follows from ½Qa; Pi ≠ 0. To see this more explicitly, we perform a local SUð2Þ rotation † Before turning to specific examples we discuss consequences Uð~x; tÞ defined by U ð~x; tÞ~nð~x; tÞ ·~σUð~x; tÞ = σz. Now, the spin–spin of the nonvanishing couplings. interaction becomes a Zeeman field along sz, while electron– magnon interactions are included in derivatives of the rotated Non-Fermi Liquid and Overdamped Goldstone Bosons. A non- † electron field ∂μψ = Uð∂μ + iAμÞψ′ through Aμ ≡−iU ∂μU.Ifwe vanishing coupling connects our problem to well-studied expand Aμ in series of NGB fields, each term contains one de- problems of a Fermi surface interacting with gauge or critical rivative acting on them. Therefore, electron–magnon interac- ~v = −e ~k′ +~k = m bosons. The vertex ~k′;~k ð Þ 2 of the gauge coupling tions vanish in the limit of small energy-momentum transfer −e~A ·~j ~k′ =~k does not vanish at . Similarly, the Yukawa in- (more details in SI Text). q = teraction between 0 critical bosons and electrons is not In general, generators Qa of internal symmetries commute with severely restricted by symmetries and nonvanishing couplings ~P and therefore we always obtain vanishing couplings. are expected (e.g., Yukawa couplings). We can readily argue, via the one-loop calculation below, that nonvanishing cou- Continuous Space Rotation—Nonvanishing Coupling. Our first non- plings destabilize the fixed point of free NGBs and a decou- trivial example is the spontaneous breaking of continuous spatial pled Fermi liquid. The actual fate of this strongly coupled rotation symmetry. For concreteness, consider nematic order in problem—a non-Fermi liquid, a superconductor, or some other 2 + 1 dimensions, in which a circular Fermi surface is distorted state—requires a case-by-case analysis and is currently under active into an ellipse in the ordered phase. The generator of SOð2Þ investigation. spatial rotations is Lz, which does not commute with the mo- The boson self-energy correction Πabðν;~qÞ from diagrams 2a mentum operator, ½Lz; Pi = ieijPj ≠ 0 (where i; j ∈ fx; yg). Hence, and 2b of Fig. 2 is dominated by (SI Text) from Eq. 8, we expect a nonvanishing coupling

Watanabe and Vishwanath PNAS Early Edition | 3of5 Downloaded by guest on September 30, 2021 ~ D E T ≡ eiP ·~ai ; lattice translations i , where ðPx PyÞ is given by v = e ~kp ~k ∂ e ; [13] −i∂ ; −i∂ + eBx ~A = −By^x ~k;~k ij j ki ~k ð x y Þ in the Landau gauge . Due to an effective periodic potential, the electron band structure becomes where we assume a single band and, for simplicity, ignore spin. dispersive (Fig. 3). We are interested in coupling the NGBs To see this from an explicit calculation, suppose that the (phonons) to quasiparticle excitations near the Fermi surface of spatial SOð2Þ rotation is spontaneously broken by the order pa- a partially filled band. T ~ rameter h~ni = ð1; 0Þ . The Goldstone fluctuation θ of the order In this case, the conserved (magnetic) momenta P also play T ~ Qa = a a = x; y parameter n = ðcos θ; sin θÞ can couple to the spinless electron the role of broken generators P ð Þ that produce ψ = χ= m ~n · ∇~ψ2 ∇ψ ~n phonons. Hence, we should look at the commutation relation field via, e.g., Hint ð 2 Þ as both and are ~ T ½Pa; Pb = −ieabeB ≠ 0. For discrete translations, we have ½P; Ti = vectors. Expanding the interaction to first order in θ ½~n ∼ ð1; θÞ , −eB^z ×~aiTi (no sum over i), and we expect nonvanishing coupling we have from Eqs. 7 and 9: † ð0Þ ∇xψ ∇xψ H = χ ; [14] ~v = eB^z ×~a ~b · ∇~ e : [18] int 2m n~k;n~k i i ~k n~k

† † ~ ~ ∇xψ ∇yψ + ∇yψ ∇xψ Here fbigi= ; are reciprocal lattice vectors ~ai · bj = δij. Therefore, Hð1Þ = χθ : [15] 1 2 int 2m electrons may show NFL behaviors as a result of the nonvanishing interaction with phonons. As we know, in the absence of the Hence, the single-particle electron Hamiltonian is given by magnetic field B = 0, the electron–phonon coupling is conventional as one can see from Eq. 18. 18 ~p2 p2 Let us confirm the nonvanishing coupling in Eq. from a = + χ x ; [16] H0 m m direct calculation. For simplicity, we assume one flux quantum 2 2 per square lattice unit cell and assume the mean-field lattice e = + χ k2 + k2 = m potential experienced by electrons is leading to a single-particle dispersion ~k ½ð1 Þ x y 2 for plane waves with wave vector k. 15 πx πy By directly evaluating the matrix element of Eq. for plane V ~x = −V 2 + 2 : [19] waves, we get 0 cos a cos a χ v = k′k + k k′ : [17] The electrons and phonons interact with each other through ~k′;~k m x y x y ~ ~ † ~ 2 the potential Hint = Vðx − uÞψ ψ. Expanding Hint in series of u, we have This is consistent with our criterion in Eq. 13. Indeed, using ~ ~ the above dispersion e~ and nk pj nk = kj in Eq. 13, one gets k ~ ~ 2 v~~ = χkxky=m, which agrees with Eq. 17 in the limit k′ → k. ~p − e~A k;k ~ ~ Note that the vertex does not vanish at k′ = k for generic points = + V ~x ; [20] H0 m on the Fermi surface, except for few high-symmetry points with 2 kx = 0orky = 0. Therefore, at most of a part of the Fermi surface, ð1Þ = −~u · ∇~V: [21] the quasiparticle lifetime is heavily suppressed by the interaction Hint with the NGB θ that originated from spontaneously broken continuous rotation. A nematic order of an elliptically distorted We diagonalize H0 in the strong magnetic field limit, perturba- Fermi surface (20, 21) and a ferromagnetic order in the presence tively taking into account the lattice potential to the lowest order of a Rashba interaction (22, 24) are known examples of this in mV0=eB. In the Landau gauge, the lowest Landau level wave mechanism. functions that simultaneously diagonalize Ti are given by ref. 29 Finally, let us remark on a subtlety regarding space–time (SI Text): symmetries. In certain cases, even if the spatial rotation is X 2 spontaneously broken, NGBs associated with the broken rotation −ð1=2Þðy=ℓ + kxℓ + ð2πℓ=aÞmÞ +iðkx+ð2π=aÞmÞx−ikyam Ψ~ ∝ e : [22] may not appear. Suppose translations px;y are spontaneously broken k m∈Z in 2 + 1 dimensions. Although rotation symmetry is also broken, it does not lead to independent NGBs. Phonons originating from px;y ℓ θ Therefore, the lowest electron band to first order in perturbation playtheroleoftheNGBof z as well, and the fluctuation asso- theory is ciated with ℓz is related to displacement fields by θ = ∂xuy −∂yux. Although the field θ can couple strongly to electrons, these addi- tional derivatives annihilate the scattering in the limit of small energy-momentum transfer. Even when only px or py is broken, ℓz AB cannot produce an independent NGB. For example, helimagnets in 3 + 1 dimensions with the spiral vector along the z axis break pz − ℓz and ℓx;y but the phonon associated with pz plays the role of NGBs of ℓx;y and orientational NGBs are absent (25–28).

Magnetic Translation—Nonvanishing Coupling. As a previously unidentified example of nonvanishing couplings, we discuss continuous translation under a uniform magnetic field in 2 + 1 Fig. 3. (A) Electron band structure under a uniform magnetic field (Landau dimensions. Suppose that a crystalline order with lattice vectors B V ~x ~a levels). ( ) A spontaneously generated periodic lattice potential ð Þ pro- f igi=1;2 is spontaneously formed, breaking the magnetic trans- duces dispersing bands. The quasiparticle excitation of the partially filled lation and giving birth to phonons (NGBs). We assume an in- band (filled states shaded in blue) has a reduced lifetime due to the non- teger flux quantum per unit cell for the commutativity of the vanishing electron–phonon interaction.

4of5 | www.pnas.org/cgi/doi/10.1073/pnas.1415592111 Watanabe and Vishwanath Downloaded by guest on September 30, 2021 D E eB results. A different but equally valid viewpoint on our result is to e = E + ~kV~k = − V~ k a + k a [23] ~k 0 m cos y cosð x Þ consider the magnetic field being applied after breaking the 2 translation symmetry (as in a crystal), which should modify the 2 – with V~ ≡ V e−ðπℓ=aÞ . Using our formula [18] and the disper- electron phonon coupling. Therefore, in a clean metal, a mag- 0 netic field should induce a nonvanishing coupling between sion above, we predict the nonvanishing coupling ~v~~ = ~ k;k eBVað−sin kya; sin kxaÞ. phonons and electrons. Although in principle this would have Now we directly compute the electron–phonon vertex by important consequences, in a typical solid, even at the highest available magnetic fields there is a wide separation ℓ a be- evaluating matrix elements of Eq. 21 with the zeroth-order − = i~q ·~x ℓ = eB 1 2 a wave function [22]andu = u~qe ,whichgives tween the magnetic length ð Þ and lattice spacing . The typical dispersion of the Landau levels induced by the lattice, and ! −Cðℓ=aÞ2 ′ hence the coupling constant, is e 1(SI Text). Thus, al- ðky + ky + iqxÞa ðqℓÞ2 kx′ +kx − − −iqy ℓ2 sin though a nonvanishing coupling is expected, its absolute mag- ~v = eBVae~ 4 2 2 : [24] ~k′;~k ′ ðkx + kx − iqyÞa nitude is extremely small. A more promising physical scenario is sin 2 the quantum Hall regime, where ℓ ∼ a, and where translation symmetry breaking into stripe and bubble phases is predicted ~ ~ This indeed agrees with our formula when k′ → k. and may have been observed in higher Landau levels (31–34). Let us now note some important physical consequences. For We leave an analysis of this interesting possibility to future work. our results, it is important that spontaneous breaking of magnetic translation symmetry occurs in a system with a uniform ACKNOWLEDGMENTS. We are grateful to Andrew Potter, Siddharth magnetic field. On the other hand, if the underlying symmetry is Parameswaran, Yasaman Bahri, Philipp Dumitrescu, and Tomáš Brauner for discussions and especially thank Max Metlitski for stimulating discussions at regular translation, and magnetic flux is spontaneously generated the early stages of this work and for useful comments on the draft. H.W. was in the symmetry-breaking process (as in a skyrmion lattice), this supported by the Honjo International Scholarship Foundation and A.V. is does not lead to nonvanishing coupling (30), and a Fermi liquid supported by National Science Foundation Grant DMR 1206728.

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