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Shamit Kachru Gapless ultra-quantum matter Shamit Kachru (Stanford) Simons UQM collaboration kickoff meeting Some of the most intriguing phases of quantum matter — motivating much of the research carried out by members of the collaboration — involve gapless theories. Two prototype sources of related questions in Nature: — 2DEG at even denominator filling fraction: — (normal state of) unconventional superconductors: I can briefly describe idealized theorists’ versions of both systems. The first problem starts with electrons in a plane subjected to a magnetic field. Flux attachment suggests that we try to reinterpret a state at filling fraction 1 ⌫ = 2 by attaching two units of flux to each electron. This new quasiparticles still have fermionic statistics, but move in a vanishing effective magnetic field. Naively these “composite fermions” fill out a Fermi surface. Interactions can have various interesting effects: qy qy Fermi interactions. We describe the correlation functions empty states of both the boson and fermion degrees of freedom at the (a) (b) non-Fermi liquid fixed point; they di⇥er from the results obtained in alternative treatments. In §4, we re-introduce q q the four-Fermi interactions and describe subtleties associ- x filled states x ated with log2 divergences that arise in their presence. In §5, we discuss controlled large N theories where the sub- tleties of §4 do not arise, and we find fixed points which generalize those of 3 to include four-Fermi interactions. § q The lesson I willWe show take that these from fixed pointsthis haveis the no superconducting following: it will be(c) empty states instabilities. We close with a discussion of open issues in filled states useful to find§ 6.controlled Explicit calculations approaches which we refer toto in non-Fermi the main liquid k + q fixed points usingbody are toy presented models, in several even appendices. unrealistic toy models. k II. EFFECTIVE ACTION AND SCALING ANALYSIS II. Our toy model & RG philosophy FIG. 1. Summary of tree-level scaling. High energy modes Let ⌥σ denote a fermion field with spin ⌅ = , , and (blue) are integrated out at tree level and remaining low en- dispersion (k), defined relative to the Fermi level.⇤ ⌅ Let ⌃ ergy modes (red) are rescaled so as to preserve the boson and be theEffective scalar boson field theory: corresponding Fermion-boson to the order pa- problemfermion kinetic terms. The boson modes (a) have the low rameter for the quantum phase transition. The e⇥ective energy locus at a point whereas the fermion modes (b) have We willlow energystudy Euclidean the actiontheory⌫ consists=1/ 2with of a purelynon UV fermionicFermi action: liquid?their low energy locus on the Fermi surface. The most rele- ! − vant Yukawa coupling (c) connects particle-hole states nearly term, aStarting purely bosonicUV action: term and a Yukawa coupling be- tween bosons and fermions: ⌫ =5/2 ?? perpendicular to the Fermi surface; all other couplings are ! irrelevant under the scaling. d = d⇧ d x = S⌅⌫ +=(2S⇤ +k S+⌅ 1)⇤/2 stripes? S L − ! ⇤ ⇤ ⌅ = ⌥¯σ [∂⇥ + µ (i )] ⌥σ + ⇥⌅⌥¯σ⌥¯σ ⌥σ ⌥σ strongFermions repulsive forces. These interactions are decoupled L − ⇧ Obtaining a precise2 understanding⇥ byof an these auxiliary states boson field ⌃ representing a fermion bilin- = m2 ⌃2 +(∂ ⌃)2 + c2 ⌃ + ⇤ ⌃4 L⇤ ⇤ remains⇥ a problem⇧ 4! of significantear,bosons interest. and the partition function is obtained by averaging d+1 d+1 ⇥ over all possible values of both the fermion and boson d kd q ¯ S⌅,⇤ = 2(d+1) g(k, q)⌥(k)⌥(k + q)⌃(q), (1) fields.“Yukawa” Initially, coupling the auxiliary field has no dynamics and ⇤ (2⇤) is massive. However, as high energy modes of the ma- where repeatedThe second spin indices problem are summed. suggests The first term, a differentterial of theoretical interest are integrated out, radiative corrections This⌅, represents is the a Landau theory Fermi of liquid, Landau with weak Fermi residual liquidsinduce and dynamics Wilson- for the bosons. In a Wilsonian theory, self-interactionsL g=0:question. fermi incorporated liquid One decoupled in starts forward from with and fluctuating BCSa metal, scat- order butthe parameter. dynamicsin candidate are encapsulated only in local, analytic cor- Wednesday, June 26, 13 Fisher scalar order parameters, interacting in the simplest tering amplitudes. The second term represents an in- rections to the bare action. This mode elimination is con- theories,non-zero g: thenon-trivial Fermi feedback surface between interacts bosons with and fermions.an (emergent) teracting scalar boson field with speed c and mass m tinued until eventually, the UV cuto⇥ E represents order parameter mediatingpossible way: a second⇤ order transition. ⇥ F (which corresponds to the inverse correlation length that the scale up to which the quasiparticle kinetic energy can vanishes as the system is tuned to the quantum criti- be linearized about the Fermi level. At these low ener- cal point). The third term is the Yukawa coupling be- gies, and in the vicinity of the quantum critical point tween the fermion and boson fields and is more naturally where the field ⌃ condenses, it is legitimate to view ⌃ described in momentum space. The quantity g(k, q)is as an independent, emergent fluctuating field that cou- a generic coupling function that depends both on the ples to the low energy fermions via a Yukawa coupling as fermion momentum k, as well as the momentum trans- written above23. This will be the point of departure of fer q (we have suppressed spin indices for clarity). For a our analysis below. Tuesday,spherically January 27, 15 symmetric Fermi system, the angular depen- We first describe a consistent scaling procedure for the dence of g(k, q) for k = kF can be decomposed into dis- action in Eq. 1. The key challenge stems from the tinct angular momentum| | channels, each of which marks fact that the boson and fermion fields have vastly dif- adi⇥erent broken symmetry. Familiar examples include ferent kinematics. Our bosons have dispersion relation 2 2 2 2 ferromagnetism (angular momentum zero) and nematic k0 = c k +m⇤, so that low energies correspond as usual order (angular momentum 2). More generally, the cou- to low momentum, and their scaling is that of a standard pling can be labelled by the irreducible representation of relativistic field theory where all components of momen- the crystal point group and it respects symmetry trans- tum scale the same way as k0. By contrast, the fermion formations under which ⌃ and ⌥⌥¯ both change sign. dispersion relation is k0 = (k) µ,sotheirlowen- Before proceeding, we make a few comments on the ori- ergy states occur close to the Fermi− surface (Fig. 1). gins of the e⇥ective action above. One starts with a the- Moreover, the Yukawa coupling between the two sets of ory involving fermions interacting at short distances with fields must conserve energy and momentum in a coarse- 2 fields with spin σ = , interacting at short dis- σ " # EF tances with strong repulsive forces. These interactions are decoupled by an auxiliary boson field φ representing Wilson-Fisher a fermion bilinear, and the partition function is obtained + dressed non-Fermi liquid by averaging over all possible values of both the fermion andfields boson σ fields.with Initially, spin σ = the, auxiliaryinteracting field at has short no dy- dis- 1 Scale where Landau tances with strong repulsive" # forces. These interactions !LD gEF EF namics and is massive. However, as high energy modes ⇠ pN damping sets in of theare decoupled material of by interest an auxiliary are integrated boson field out,φ representing radiative Wilson-Fisher a fermion bilinear, and the partition function is obtained + dressed non-Fermi liquid corrections induce dynamics for the bosons. ??? Inby a averaging Wilsonian over theory, all possible the dynamics values of are both encapsulated the fermion onlyand in bosonlocal, analytic fields. Initially, corrections the auxiliary to the bare field action has.This no dy- 1 Scale where Landau !LD gEF namics and is massive. However, as high energy modes ⇠ pN damping sets in mode elimination is continued until eventually, the UV cutoof↵⇤ the materialE represents of interest the are integrated scale up to out, which radiative the corrections⌧ induceF dynamics for the bosons. FIG. 1. This figure depicts the regime of energy scales over quasiparticle kinetic energy ✏(k) can be linearized about which our description is controlled. The??? physics below the the FermiIn a Wilsonian level. At theory,these low the energies, dynamics and are in encapsulated the vicin- only in local, analytic corrections to the bare action.This parametrically low scale of Landau damping remains to be ity of the quantumIII. critical Our point problem where the field φ con- mode elimination is continued until eventually, the UV understood. denses, it is legitimate to view φ as an independent, cuto↵⇤ E represents the scale up to which the emergent fluctuating⌧ F field. The resulting e↵ective low FIG. 1. This figure depicts the regime of energy scales over quasiparticle kinetic energy ✏(k) can be linearized about energy Euclidean action consists of a purely fermionic rotationalwhich symmetry our description is broken is controlled. whereas The translation physics below sym- the the Fermi level. At these low energies, and in the vicin- Havingterm, introduced a purely bosonic the term ingredients, and a Yukawa it becomes coupling be- clear metryparametrically remains preserved. low scale of In Landau the case damping of continuous remains to beA theorists’ity of the idealization quantum critical of pointthe problem where the — field solveφ con- for understood. whattween our bosonstoy problem and fermions: of interest should be. We start Pomeranchuk transitions, the bosons condense at zero denses,the dynamics it is legitimate of this to quantum view φ as field an independent, theory: emergent fluctuating field. The resulting e↵ective low momentum and therefore couple to fermions at every with the decoupledd Landau theories: energy= Euclideand⌧ d actionx = consistsS + Sφ of+ aS purely φ fermionic pointrotational of the Fermi symmetry surface.
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