Mixed-valence insulators with neutral Fermi surfaces
MMixed-valence insulators with neutral Fermi surfaces
Debanjan Chowdhury, ∗ Inti Sodemann,
1, 2 and T. Senthil Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA. Max-Planck Institute for the Physics of Complex Systems, Dresden, 01187, Germany.
Abstract-
Samarium hexaboride is a classic three-dimensional mixed valence system with a high-temperaturemetallic phase that evolves into a paramagnetic charge insulator below 40 kelvin. A number of recent exper-iments have suggested the possibility that the low-temperature insulating bulk hosts electrically neutral gap-less fermionic excitations. Here we show that a possible ground state of strongly correlated mixed valenceinsulators—composite exciton Fermi liquid— hosts a three dimensional Fermi surface of a neutral fermion, thatwe name the “composite exciton”. We describe the mechanism responsible for the formation of such excitons,discuss the phenomenology of the composite exciton Fermi liquids and make comparison to experiments inSmB . Introduction
Electronic solids where the valence of one of the constituentelements is non-integral show a number of fascinating prop-erties [1, 2] arising from the Coulomb interaction betweenelectrons. Of interest to us in this paper is a class of mixed-valence (MV) systems, a classic example being SmB [3, 4],where a high temperature metallic state evolves into an in-sulator at low temperatures. Attention has been refocusedon this material in recent years following the proposal [5]that it may be an interaction-driven topological insulator (TI)[6, 7]. There is compelling evidence now for metallic surfacestates in this material (of possibly topological origin) despitean electrically insulating bulk at low temperatures from pre-dominantly transport [8–13] and other measurements [14].A di ff erent fascinating aspect of a number of MV insula-tors, including SmB [3, 4], are various thermodynamic andtransport anomalies at low temperatures, apparently at oddswith an insulating behavior in the bulk. Traditionally, theseanomalies have often been attributed to the presence of in-gap states. An interesting development was the observationof quantum oscillations (QO) in magnetization, first reportedin SmB by Li et al. [15] and interpreted as additional evi-dence for the two-dimensional metallic surface states.However, subsequent measurements of QO in magnetiza-tion in SmB by Tan et al. [16] observed frequencies corre-sponding to almost half of the bulk Brillouin zone. Tan etal. [16] found that the frequencies, the cyclotron mass andthe amplitude of the oscillations are quite similar to the mea-sured quantum-oscillations in the other metallic hexaborides R B ( R ≡ La, Pr, Ce) [17–20]. Moreover the measured den-sity of states from the low-temperature specific heat is ingood agreement with the value obtained from quantum os-cillations [21]. Based on these observations Ref. 16 raisedthe surprising possibility that the quantum oscillations are aproperty of the electrically insulating bulk. They also sug-gested that the oscillations originate from the same in-gapstates responsible for the low temperature anomalies whichhave since been re-examined closely. However it has also ∗ Corresponding author: [email protected] been argued more recently [22] that some of the same QO re-sults can be explained using a purely two-dimensional modelof the metallic surface states. The low temperature anomaliesinclude a finite linear specific heat coe ffi cient [16, 23, 24] andbulk optical conductivity below the charge-gap [25]. Fur-thermore, a field-induced thermal conductivity proportionalto the temperature has been reported in some samples [21](though this feature does not seem to be present universally[26, 27]). Taken together these measurements suggest thepresence of a Fermi surface of electrically neutral fermionsin the bulk that nevertheless couple to the external magnetic,but not to weak DC electric-fields.Inspired by the current ba ffl ing experimental situation, weare led to a number of theoretical questions. Can MV insula-tors host Fermi surfaces of neutral fermionic quasiparticles?If so, what is the origin of the neutral (fermionic) excitationand what constrains the volume of the Fermi surface? Whatare the thermodynamic and transport signatures of phaseswith such neutral fermionic excitations? Can Fermi surfacesof neutral fermions, that do not couple directly to the externalmagnetic-field, give rise to quantum oscillations? In a sepa-rate development, it has been pointed out [28, 29] that undercertain conditions, even band-insulators with gaps smallerthan the cyclotron energy can exhibit quantum oscillations.In recent years, a number of triangular lattice organic ma-terials close to the Mott transition have been shown to actas charge-insulators but thermal metals [30–33], where theelectron appears to have splintered apart into fractionalizedexcitations (“partons”) [34, 35]; while the charge degree offreedom can remain gapped, the spinful, neutral spinon canform a Fermi surface. The possibility of observing quantumoscillations for such neutral spinon Fermi surfaces has beenaddressed previously by Motrunich [36]. However, stronglycorrelated mixed-valent insulators are far from being a Mottinsulator, thereby requiring a di ff erent microscopic mecha-nism to stabilize such a neutral Fermi surface.Here we show that in the limit of strong Coulomb in-teractions in a mixed-valence insulator, there is a well de-fined mechanism for the formation of an electrically neu-tral fermionic quasiparticle—dubbed the fermionic compos-ite exciton (ce)—that can form a Fermi surface; the resultingphase - the composite exciton Fermi liquid (CEFL) - is elec-trically insulating but will have a neutral fermi surface. We a r X i v : . [ c ond - m a t . s t r- e l ] M a y show that the CEFL shares a number of features with the ob-served phenomenology in SmB . We also note that whilethe present work is motivated by the recent experiments inSmB , it is potentially relevant to other mixed-valence insu-lators [2, 14], such as SmS under pressure, YbB etc. ResultsElectronic structure-
The electronic configuration of Smis [Xe]4 f d s . In SmB , the valence of Sm is known tofluctuate between Sm + and Sm + with an average valenceof approximately ∼ . J = orbital is lifted due to crystal field splitting, giving riseto a quartet ( Γ ) and a doublet ( Γ ). The five-fold degener-ate d − orbitals split up into a doublet ( e g ) and a triplet ( t g ).The ground state of Sm in SmB is in a coherent superposi-tion of 5 d ( e g ) + f ( Γ ) (cid:11) f . In contrast, the groundstate of La in metallic LaB has an electronic configurationof [Xe]5 d s and there are no f − electrons.Band-structures for the surface as well as the insulat-ing bulk [39] have been modeled using multi-orbital tight-binding models [40, 41], but we focus here on the simplesttwo-band model for a mixed-valence compound in three di-mensions [2] to illustrate the key ideas. In particular, we willrestrict ourselves to the situation where both the d and f or-bitals are treated as doublets instead of quartets. Model-
We start with a band of d − conduction electrons,where d r σ is the annihilation operator for a d − electron atsite r with spin σ , and a heavy band of f electrons, where f r α is the annihilation operator for an f − electron at site r and crystal-field multiplet index α (both σ, α = ↑ , ↓ and wedrop the distinction between the two from now on). As dis-cussed above, it is appropriate to consider a model where the f − valence fluctuates between n f = n f =
2. Withrespect to the n f = f α → ε αβ f † β = ˜ f α where ε αβ is thefully antisymmetric tensor and we have introduced ˜ f as the f − hole. The standard periodic Anderson Hamiltonian [42],but now written in terms of the ˜ f − hole is given by, H = (cid:88) rr (cid:48) ,α ( − t d rr (cid:48) − µ d δ rr (cid:48) ) d † r α d r (cid:48) α − (cid:88) rr (cid:48) ,α t f rr (cid:48) ˜ f † r α ˜ f r (cid:48) α + (cid:88) r , r (cid:48) (cid:20) ε βγ V αβ ( r − r (cid:48) ) d † r α ˜ f † r (cid:48) γ + H.c. (cid:21) − U d f (cid:88) r n ˜ f r n d r + U f f (cid:88) r n ˜ f r ( n ˜ f r − , (1)where n ˜ f r = (cid:80) α ˜ f † r α ˜ f r α = − n f r , with n f r = (cid:80) α f † r α f r α and n d r = (cid:80) α d † r α d r α . U d f is a repulsive density-density inter-action between the f and d − electrons (or equivalently, itrepresents an attractive interaction between the ˜ f − hole andthe d − electron) and U f f represents a large on-site Coulombrepulsion between the f − electrons. The hoppings for the d − electron ( ˜ f − hole) are given by t d rr (cid:48) ( t f rr (cid:48) ), with | t d | (cid:29) | t f | ≡ composite excitonb-holon 𝝌 -spinonf-hole (a)(b) Out[8]= k ε ε CE ε χ (c) FIG. 1.
Route to composite exciton Fermi liquids. (a)
Slave-boson representation for the ˜ f − hole in terms of a holon (blue cir-cle) and spinon (black arrow), coupled mutually to a (zigzag line). (b) Strong binding of the conduction d − electron (red circle witharrow) to the holon leads to formation of a fermionic compos-ite exciton coupled to the same a . (c) Two-band model when the f − valence fluctuates between n f = ε CE (blue dashed line) and a narrow spinon dispersion ε χ (orange dashed line) shown for the gauge-invariant combination ζ = t CE / t χ >
0. The hybridization between the two gives rise to twobands (orange and green solid lines) and as a result of the fillingleads to a semi-metallic state (yellow shaded regions), where thevolumes of the two pockets are equal. For ζ <
0, the resulting statewould be an insulator. and µ d represents the chemical-potential for d − electrons.Thehybridization, V αβ , between the d and f electrons, has oddparity V αβ ( − k ) = − V αβ ( k ).We are interested in the limit of U f f → ∞ , and U d f largebut finite. We use here a slightly di ff erent variant of the stan-dard slave-boson representation [43],˜ f r α = b r χ r α , (2)where we have fractionalized the ˜ f − hole into (i) a spinlessboson (“holon”), b , that carries the physical, negative ( − A µ , and, (ii) a neutral fermion(“spinon”), χ α , that carries the spin ( α ); see Fig.1a. Thereis a redundancy associated with the above parametrization χ → χ α e − i θ , b → be i θ which leaves f α invariant. Wetherefore assume that the holon (spinon) carries a unit pos-itive (negative) charge under an emergent U (1) gauge-field a µ = ( a , a ). We are interested in describing phases witha charge-gap (i.e. insulators) where the holon remains un-condensed, (cid:104) b (cid:105) = d − electrons is absent. The above definition in terms of thepartons is to be supplemented with a gauge-constraint, thatensures restriction to gauge-invariant states in the Hilbertspace, of the form b † r b r = χ † r α χ r α . We impose an additionalhard-core constraint on the bosons, i.e. b † r b r ≤
1, which en-sures no double occupancy of the ˜ f − hole; the total density ofdoped holes is then (cid:80) r b † r b r = (cid:80) r ˜ f † r α ˜ f r α . (See the Methodssection for a comparison to the standard slave-boson repre-sentation.) Composite excitons-
The global requirement for obtain-ing a mixed-valence insulator, that is also consistent with theknown electronic count in SmB is (cid:80) r ˜ f † r α ˜ f r α = (cid:80) r d † r σ d r σ (equivalently, (cid:80) r [ d † r σ d r σ + f † r α f r α ] = n b = n d . As a result of the attractive interaction (Eq.1) between the˜ f − holes and d − electrons ( U d f > ffi ciently strong attraction, it is therefore possi-ble to form bound states of the conduction electrons and theholons to form a neutral fermionic composite exciton (fCE), ψ k α ≡ b d k α , ψ † k α ≡ b ∗ d † k α . (3)The above quasiparticle is electrically neutral but is chargedunder the internal U (1) gauge field associated with the slaveboson construction (see fig.1a); at a finite density it can forma Fermi surface that is minimally coupled to the emergentgauge-field a µ . Note that in our specific example, as a resultof the hard-core constraint, the number of bosons are guar-anteed to be equal to the number of conduction electrons andtherefore, the number of fCE is identical to the number ofconduction electrons, i.e. n ψ = n d . The volume of the Fermisurface of the ψ fermions will then be identical to the volumeof the original conduction ( d − )electron Fermi surface.The e ff ective Hamiltonian that describes the low energyphysics, after the conduction electrons have formed boundstates with the holons, can be expressed as, H (cid:48) = (cid:88) k ,α ε CE ψ † k α ψ k α + (cid:88) k ,α ε χ, k χ † k α χ k α + (cid:88) rr (cid:48) ,αβ (cid:20) ε βγ V αβ ( r − r (cid:48) ) ψ † r α χ † r (cid:48) γ + H.c. (cid:21) + ..., (4)where ε CE is the fCE dispersion (see Methods section for anestimate of the nearest neighbor fCE hopping) and ε χ, k is thespinon dispersion. Note that, by construction, the holon isgapped. On the other hand as a result of the complete bindingof all the d − electrons to form fCE, the charged d − excitationsare also gapped. The ellipses denote various allowed terms;one such term (among others) is the exchange interaction be-tween the f moments, H ex = J H (cid:88) (cid:104) r , r (cid:48) (cid:105) S r · S r (cid:48) , (5)which also modifies the dispersion for the spinon bands, withthe hopping t χ set by t f , J H and the holon hopping (see sup-plementary note 1).For a finite V , the fCE band hybridizes with the spinonband to yield renormalized bands as shown in fig. 1(b) (seeMethods section). It is convenient to carry out a PH transfor-mation on χ r α → ˜ χ r α ≡ ε αβ χ † r β . Then, (cid:88) r n ψ r = (cid:88) r n χ r = (cid:88) r (2 − n ˜ χ r ) . (6)A finite t χ is necessary to get crossings at the fermi-level;one then obtains an electrically neutral semi-“metal” with‘particle’ and ‘hole’ pockets with equal volume. The Fermisurfaces thus obtained have both fCE and spinon character; ξ FIG. 2.
Metallic surface with insulating CEFL bulk.
The bulkrealizes a CEFL, a compensated semi-metal with particle and hole-like pockets (as shown in fig. 1c), that have both fCE and spinon-like character (see fig. 1a, b for a carricature of the excitations).Upon approaching the surface, it possible for the fCE to unbind asa result of reduced U d f / t d in a region of typical size ∼ ξ , thereby lib-erating the d − electrons and holons, which may Bose condense. Thelatter leads to confinement and the resulting state is then a decou-pled metallic surface. Only the top and bottom surfaces are shownfor clarity. from now on we do not distinguish between the two. Notethat the hopping amplitudes for the fCE and spinon are notindividually gauge-invariant, unlike the gauge-invariant ratio ζ = t CE / t χ . Which sign of ζ is preferred depends on variousmicroscopic details; if ζ < ζ >
0) the ground state will infact be an insulator (semi-metal) of fCE and spinons.Let us now briefly describe a possible mechanism thatallows the insulating bulk hosting a CEFL to coexist witha metallic surface. Previously, it has been argued [44]that the Kondo-screening can be reduced significantly nearthe surface leading to “Kondo-breakdown”, in which the f − moments decouple from the conduction electrons, givingrise to quasiparticles that are lighter. As a result of surface-reconstruction and screening e ff ects [45], it is also possiblethat the ratio U d f / t d is smaller close to the boundaries thanin the bulk. The weaker attraction between the holon andthe conduction electrons can then lead to an unbinding ofthe fCE close to the surface, thus liberating the holon andthe conduction elecrton within a length scale, ξ , from thesurface (fig.2). The unbound holons can then Bose condensenear the surface, confining the gauge-field, thereby renderingthe originally neutral fermions with physical charge. In thisway, one may recover metallic quasiparticles at the surface asa result of unbinding of the fCE. Moreover, depending on thedetails of the fCE dispersion (which may be itself topolog-ical) and the odd-parity hybridization, V αβ ( k ), it is possiblefor the metallic quasiparticles at the surface to realize topo-logically protected surface-states. We leave a discussion ofthe detailed quantitative theory for future work. Phenomenology of CEFL-
Returning now to a descrip-tion of the bulk, the low-temperature specific heat is domi-nated by the fluctuation of the fermion-gauge field system.As a result of the gauge-field fluctuations (see Methods sec-tion and the supplementary note 2 for a discussion of thelow-energy field theory) the low- T specific heat [46, 47] isgiven by, C = γ T , where γ ∼ ln(1 / T ) . (7)Measurements of specific heat in SmB do report a linearin T specific heat [16, 23, 24]. Moreover the gapless fCEexcitations along the neutral Fermi surface contribute to theNMR spin-lattice relaxation rate, 1 / T , in the usual way,1 T T = const . (8)Measurements on SmB support such metallic 1 / T T behav-ior (V. Mitrovic, personal communication) [48]. Note how-ever that as a result of strong spin-orbit e ff ects, the abovequantity need not be related to the Knight-shift by Korringa’srelation.The mere presence of a charge-gap in the system does notimply a lack of sub-gap optical conductivity [49]; the onlyphysical requirement is that the conductivity vanish as ω →
0. We are interested here in the form of Re[ σ ( ω )] at low, butfinite, frequencies. We apply the Io ff e-Larkin rule to the (fCE + holon) system [50] (see supplementary note 3 for details)and relate the holon-response to a dielectric constant, (cid:15) b . Weexpect the response of the fCE to be similar to that of a metalat low but non-zero frequencies with Re[ σ ce ( ω )] (cid:29) ω andIm[ σ ce ( ω )] (cid:28) Re[ σ ce ( ω )]. Then,Re[ σ ( ω )] = ω (cid:18) (cid:15) b − π (cid:19) σ ce ( ω )] , (9)where the fCE conductivity can be expressed in the general-ized Drude form σ ce ( ω ) = ρ/ ( Γ ( ω ) − i ω ), with Γ ( ω ) a fre-quency dependent scattering rate and ρ is defined to be thetotal optical weight. At low ω , where | Γ ( ω ) | (cid:28) ω , the realpart of the conductivity can be evaluated as [51, 52], ω Re[ σ ce ( ω )] ≈ ρ Re[ Γ ( ω )] . (10)Depending on the mechanism responsible for relaxation ofcurrents, one can then obtain di ff erent results for Γ ( ω ); wediscuss the di ff erent regimes in the methods section. Recentmeasurements of optical-conductivity in the THz regime inSmB [25] find appreciable spectral weight below the insu-lating gap, much larger than any imaginable impurity bandcontribution.After integrating out all the matter-fields, the ground-stateenergy of the system in the limit of weak fields follows fromgauge-invariance, u ( b , B ) = u + ( b − B ) µ b + b µ ce + B µ v + u osc ( b ) + .., (11)where µ b , µ ce , µ v represent the permeability of the gappedholons, composite-excitons and the background ‘vacuum’respectively; all of these quantities depend on the UV de-tails of the underlying theory. u osc ( b ) is the oscillatory com-ponent, relevant for our discussion on quantum-oscillations and the ellipses denote additional higher order terms. In thelimit of a small B , the internal b can optimize itself in orderto minimize the energy; ignoring the oscillatory componentin Eq.11, the optimum value is b = b = α B , with α = µ CE ( µ CE + µ b ) (12)an O (1) number that is a priori unknown. In the regime where µ CE (cid:29) µ b , b locks almost perfectly to the external B (i.e. α → ∆ (cid:18) B (cid:19) = π S i ⊥ (cid:32) + µ b µ CE (cid:33) − = πα S i ⊥ , (13)where S i ⊥ is the cross-sectional area of the fCE Fermi surfacesheet i perpendicular to B . In the limit where α → b → B ), the period is directly related to the volume ofthe composite exciton fermi surface, but in general it can besignificantly di ff erent depending on the value of α . Includingthe e ff ect of impurities broadens the Landau-levels and theoscillation amplitude has an additional Dingle suppression ∼ exp( − /ω ci τ i ) [54], where τ i is the elastic lifetime and ω ci is the e ff ective cyclotron energy in sheet i .The low temperature thermal conductivity, κ , is dominatedby the fermionic contribution (i.e. the holon, the gauge-fieldand the phonon contributions are expected to be small com-pared to the fCE contribution) and there is no di ff erence be-tween the physical thermal conductivity and the conductivitydue to the fCE. Let us first estimate the longitudinal thermalconductivity, κ xx ≈ κ ce xx , due to the composite-excitons. Weassume that the fermionic composite excitons form a stateakin to an ordinary metal for thermal transport [55], suchthat the longitudinal conductivity is given by κ xx = (cid:88) i = , k B τ i m i (cid:18) m i ε F (cid:126) (cid:19) / T , (14)where ε F is the Fermi-energy, m i represent the masses for thetwo pockets and we have allowed for di ff erent lifetimes, τ i ,for the two pockets. At zero magnetic-fields, all of the ex-periments on SmB find a value of κ xx / T that extrapolates tozero as T → κ xx / T extrapolates to a finite value in the limit of T → T − linear specific heat combined with an absence of a finite T − linear thermal conductivity suggests the presence of ei-ther a small zero-field gap that closes at higher fields, or, thepresence of localized states. Within the former scenario, it isplausible that at zero-field and at low temperatures, the CEFermi surfaces undergo pairing to yield a gapped state witha small insulating gap, that can be significantly smaller thanthe charge-gap.We note that experimentally, the thermal conductivitymeasurements have been carried out at very low temper-atures ( < ffi cient of the linear in T specific heat is typically extrapolated from higher temper-atures. The opening of a small insulating pairing gap at atemperature T p corresponds to an actual phase transition in(3 + − dimensions (in the Ising universality class) with anassociated divergence in the specific heat. Interestingly, anumber of experiments report a strong upturn in the specificheat around ∼ T p ≈ κ xy . In the weak-field regime, as noted previously, the com-posite excitons move essentially under the e ff ect of an ef-fective magnetic field b and are subject to the Lorentz forceassociated with this field. However note that the two pocketscontribute to the thermal Hall response with opposite signs.We know semi-classically that for each pocket κ ixy = ( ω c , i τ i ) κ ixx , (15)where ω c , i = e | b | / m i = α e | B | / m i and κ ixx can be read o ff fromEq.14 above. The total thermal Hall response is the di ff er-ence of the response for the ‘particle’ and ‘hole’-like pock-ets. Observation of a non-zero thermal Hall e ff ect is a goodindicator that the parameter α - which determines the mag-nitude of orbital e ff ects of the external magnetic field - is nottoo small. In SmB , if the quantum oscillations truly arisefrom the bulk neutral fermi surface of composite excitons asa result of the mechanism proposed above, then that neces-sarily implies a finite bulk thermal Hall response. However,we note that since the system is analogous to a compensatedsemi-metal, the thermal Hall e ff ect is expected to be vanish-ingly small at higher fields when ω c , i τ i (cid:38) d − electrons or holes (e.g. by chemical substitution or bygating thin films). There are two natural outcomes: if theholon remains uncondensed, the d − electrons (or holes) canform a ‘small’ fermi surface while the neutral fCE fermisurface continues to be present. This phase is the familiar(mixed-valence) fractionalized Fermi-liquid (FL*) [56]. Onthe other hand, if the holons condense as a result of dopingaway from the mixed-valence limit, the CE fermi surfacesbecome Fermi surfaces of physical electrons (and holes) as aresult of confinement. The exact outcome is sensitive to mi-croscopic details and is beyond the scope of our discussionhere.The mechanism responsible for the formation of thefermionic exciton is physically distinct from the one respon-sible for the conventional bosonic exciton [1]. A few recenttheoretical studies have tried to address the origin of the low-energy bulk excitations in SmB using a variety of di ff erentapproaches [57–59]. The CEFL is strikingly distinct fromthese previous proposals - unlike Refs. 58 and 59 the com-posite exciton is not a Majorana fermion, and unlike Ref. 57,the composite exciton has fermi statistics and forms a fermisurface (see supplementary note 4 for a more detailed com-parison). Discussion-
We have described a phase of matter witha neutral Fermi surface of composite excitons in a mixed-valent insulator with a charge-gap. A number of properties associated with such a phase resembles the experimentalresults in bulk SmB . Future numerical studies of theperiodic Anderson model in the insulating regime and in thepresence of strong interactions may be able to shed lighton questions related to energetics and stability of variousphases. We also note that more recent measurements on amixed valence insulator compound di ff erent from SmB ,that displays clear bulk quantum oscillations and has metal-lic longitudinal thermal conductivity down to the lowestmeasurable temperatures at zero field, in a clear indicationof the formation of a Fermi surface of neutral fermions (L.Li, Y. Matsuda, and T. Shibauchi, personal communication). MethodsSlave-boson representation-
In order to motivate the ratio-nale behind choosing the prescription in Eq.2, recall that thestandard slave-boson representation proceeds as,˜ f r α = h † r χ r α , (16)where h r is a spinless bosonic holon with the constraint h † r h r + (cid:80) α χ † r α χ r α =
1. Consider now the scenario where the h − holons are perturbed away from a Mott-insulating statewith (cid:104) h (cid:105) = (cid:104) h † r h r (cid:105) = − x (where x represents thedensity of doped holes away from the 4 f configuration).The two representations are then physically equivalent if wemake the transformation h † r → b r and (cid:104) b † r b r (cid:105) = x ; for a con-crete scenario, consider e.g. the quantum rotor model where h † r = e i θ r and n h r is the boson density conjugate to θ r . Fermionic composite exciton hopping-
Consider thelimit where there is a clear hierarchy of scales: U f f (cid:29) U d f (cid:29) t d (cid:29) V and where t d is the nearest neighbor hop-ping for the d − electrons. In this regime, the nearest neighborhopping amplitude for a single fCE is approximately givenby (see supplementary note 1) t CE ∼ t d (cid:18) t f U d f (cid:19) , (17)where t f is the e ff ective nearest neighbor ˜ f − hole hopping.There is, in principle, a very strong on-site repulsion set by U f f for the fCE, as a result of the constraint of no doubleoccupancy for the hard-core holons. However, if the bindingis not purely on-site and has some finite extent, the repulsionbetween the fCE can be renormalized down from the bare U f f and the resulting state can be described within a weaklyinteracting CEFL.The fermionic exciton is clearly significantly di ff erentfrom the more conventional bosonic exciton [1, 57, 60] thathas been discussed in the context of semimetals and narrowgap semiconductors. The latter arises in the limit where U d f dominates over U f f . In contrast, as shown above,the fermionic exciton is expected to arise naturally in thelimit where U f f (cid:29) U d f , which is a more realistic regimefor mixed-valent systems. Fermionic composite excitonshave also been discussed recently in the context of multi-component quantum hall states [61]. Low energy field theory for CEFL-
Let us describe thelow-energy e ff ective field theory for the CEFL phase de-scribed in the main text [56]. The composite exciton, ψ k α, i ,with i = , a µ and the non-relativistic b holons at a finite chemi-cal potential, µ b >
0, are coupled minimally to ∆ a µ = a µ − A µ (see supplementary note 3 for a more complete discussion).Let us first discuss the form of the gauge-field propagator, D i j ( i ω n , q ) ≡ (cid:104) a i ( i ω n , q ) a j ( − i ω n , − q ) (cid:105) where we choose towork in the Coulomb gauge ∇ · a =
0, with a being purelytransverse. As a result of the minimal coupling, integratingout the fCE excitations leads to a Landau-damped form ofthe propagator, D i j ( i ω n , q ) = δ i j − q i q j / q Ξ | ω n | / q + β q , (18)where Ξ , β are constants determined by details of the fCEdispersion.For the specific non-relativistic form of the theory forthe holons, there are no holons in the ground state and theonly holon self-energy, Σ b , contribution arises as a resultof the coupling to the gauge-field and Σ b ( i ω n , q ) ∼ q (cid:18) + c | ω n | ln(1 / | ω n | ) + ... (cid:19) at T =
0, where c is a constant. Theabove correction is less important than the bare terms in theholon action and can therefore be ignored.Finally, as a result of the coupling to the gauge-field fluc-tuations, the fermions have a self-energy,Im Σ ce ( ω ) ∼ ω, (19)upto additional logarithmic corrections. Alternative route to CEFL-
We demonstrate here an al-ternate route towards arriving at a description of the bulkCEFL phase from a di ff erent starting point. Consider a com-pensated semi-metal with (physical) d − electron and f − holepockets. We are interested in driving the semi-metal into aninsulating phase in the presence of strong interactions. TheHamiltonian is given by, H csm = − (cid:88) rr (cid:48) t d rr (cid:48) d † r α d r (cid:48) α + (cid:88) rr (cid:48) t f rr (cid:48) f † r α f r (cid:48) α + (cid:88) rr (cid:48) (cid:15) αβ V rr (cid:48) d † r α f † r (cid:48) β + H int , (20)where the hoppings t d , t f are positive and V denotes the hy-bridization. We will specify the form of H int momentarily.Consider setting V = d r α ≡ e i θ r ψ r α , f r α ≡ e − i θ r χ r α , (21)where the rotor field, e i θ r , carries physical charge and thespinful Fermions ψ α , χ α are electrically neutral. Let n r bethe boson density conjugate to the rotor field. Then thegauge-invariant states satisfy the constraint : n r + n ψ r − n χ r = n d r = n ψ r , n f r = n χ r . Let us then consider the interactionterm to be of the form, H int = U (cid:88) r ( n dr − n fr ) → U (cid:88) r n r . (22) It is then clear that at small U (compared to the hoppings),the rotor fields condense (cid:104) e i θ r (cid:105) (cid:44) U , one can drive a‘Mott’-transition to a phase where the rotor-fields are gapped (cid:104) e i θ r (cid:105) = ψ, χ fermions canform Fermi surfaces, inherited from the original d , f Fermisurfaces. This is the CEFL phase. Both phases are stable tohaving a small V . Optical conductivity of CEFL-
As introduced in Eq. 10,in the regime where Γ ( ω ) arises primarily due to scatteringof the fermions o ff the gauge-field fluctuations and wherethe e ff ects of static disorder can be ignored (i.e. the mean-free path, (cid:96) mf , is long), Γ ( ω ) ∼ ω / . In three dimensions,this arises from the fCE self-energy, Im Σ ce ( ω ) ∼ ω (uptoadditional logarithms) and includes two extra powers of | q | ∼ ω / . Hence, under these set of assumptions, Re[ σ ( ω )] ∼ ω . .On the other hand, in the regime where Γ ( ω ) still arisesdue to scattering of the fermions o ff the gauge-field fluctua-tions, but the finite (cid:96) mf modifies the | ω | / q form in the gauge-field propagator (Eq.18) around q ∼ (cid:96) − , Γ ( ω ) ∼ ω andRe[ σ ( ω )] ∼ ω .Finally note that the fCE density can couple to the lo-cal disorder-potential and Γ may be dominated entirely bya frequency independent elastic scattering-rate ( Γ ); thenRe[ σ ce ( ω )] ≈ ρ/ Γ which leads to Re[ σ ( ω )] ∼ ω . Sim-ilarly, as a result of localization e ff ects, it is possible for Σ ce ( ω ) to vanish much faster than ω such that Re[ σ ( ω )] ≈ Re[ σ ce ( ω )], in which case results for strongly disorderedmetals will apply. Quantum oscillations in CEFL phase-
For small fieldsthe energy in Eq.11 can be rewritten as, u ( b , B ) = u (cid:48) ( B ) + ( b − b ) µ e ff + u osc ( b ) , (23) u (cid:48) ( B ) = u + (cid:18) µ v + µ b + µ CE (cid:19) B , (24)and µ − ff = µ − b + µ − . At zero temperature, the oscillatorycomponent is given by [53], u osc ( b ) = χ i osc (cid:32) | b | π S i ⊥ (cid:33) / f (cid:32) π S i ⊥ | b | (cid:33) , (25) f ( x ) = ∞ (cid:88) n = ( − n n / cos(2 π nx − π/ , (26)where χ i osc sets the scale for the overall amplitude of the os-cillations from the Fermi surface sheet i . Data availability-
All relevant data are available from theauthors upon reasonable request.
Acknowledgements-
We thank Suchitra Sebastian forsharing many of their unpublished results and thank her andOlexei Motrunich for many stimulating discussions. We alsothank Peter Armitage, Nicholas Laurita, Lu Li, Yuji Mat-suda and Vesna Mitrovic for discussions and for sharing theirdata. D.C. is supported by a postdoctoral fellowship from theGordon and Betty Moore Foundation, under the EPiQS ini-tiative, Grant GBMF-4303, at MIT. While at MIT, I.S. wassupported by the Pappalardo Fellowship. T.S. is supportedby a US Department of Energy grant DE-SC0008739, and inpart by a Simons Investigator award from the Simons Foun-dation.
Author Contributions-
D.C., I.S. and T.S. contributed to the theoretical research described in the paper and the writingof the manuscript.
Competing financial interests-
The authors declare nocompeting financial or non-financial interests.
SUPPLEMENTARY NOTES 1
Fermionic composite exciton hopping:
Consider the situation where there is a single site with configuration 4 f d ,surrounded by sites with configuration 4 f . As U ff → ∞ , and within slave-boson theory, the single site is identified by n ˜ f = n b =
1. At large U d f ( (cid:28) U f f ), the lowest energy configuration will be the state where the binding has taken place withthe formation of the CE. In order for the fCE to hop, both the holon as well as the d − electron have to hop to the neighboringsites via virtual processes. Let us now estimate this hopping amplitude within a controlled approximation. For this purpose,consider the following related Hamiltonian, ˜ H = H d + H d f + H t f -J H + H hyb , (27)where H d and H d f represent the Hamiltonian for the d − electrons and the repulsive density-density interactions with strength U d f between the d and f − electrons respectively. We have introduced H t f -J H as an e ff ective model for the ˜ f − hole with, H t f -J H = P G (cid:20) − t f (cid:88) (cid:104) rr (cid:48) (cid:105) ˜ f † r σ ˜ f r (cid:48) σ + J H (cid:88) (cid:104) rr (cid:48) (cid:105) S r · S r (cid:48) (cid:21) P G , (28)where t f represents the nearest neighbor ˜ f − hole hopping amplitude and P G denotes the Gutzwiller projection operator thatforbids double occupancy at any given site. Finally, H hyb is the term ( ∼ V ) responsible for hybridization between the d and f − electrons. We begin by setting V =
0, which results in an enhanced U d (1) × U f (1) symmetry associated with the twoconserved fermion numbers. Furthermore, as mentioned earlier, let us restrict ourselves to the regime where U ff (cid:29) U df (cid:29) t d (cid:29) V . In this limit and within the slave-boson treatment introduced earlier, the fermionic composite exciton hopping isgiven by, t CE ∼ t d t b U df . (29)The holon hopping, t b , can now be estimated using a self-consistent mean-field treatment of the Hamiltonian defined in Eq.28above, H t f -J H →− t f (cid:88) (cid:104) rr (cid:48) (cid:105) (cid:104) χ † r α χ r (cid:48) α (cid:105) b † r b r (cid:48) − (cid:88) (cid:104) rr (cid:48) (cid:105) (cid:18) t f (cid:104) b † r b r (cid:48) (cid:105) + J H (cid:104) χ † r α χ r (cid:48) α (cid:105) (cid:19) χ † r σ χ r (cid:48) σ , (30)where we have ignored pairing terms for the spinon fields. The e ff ective holon hopping will then simply be given by t b ∼ t f with an O (1) coe ffi cient, as long as the spinons are in a state (e.g. with a Fermi-surface) where (cid:104) χ † k χ k (cid:105) = n χ k is not flat as afunction of k . This leads to the estimate of the fCE hopping in Eq. 17. In the presence of a small but finite V , the resultingfCE and spinon bands hybridize and yield a semi “metal”, as discussed in the main text. From the above equation, it is alsopossible to read o ff the self-consistently generated hopping for the spinons, t χ . SUPPLEMENTARY NOTES 2
Low-energy field theory:
In the CEFL phase, the low-energy degrees of freedom are the fCE, the spinons and the gappedholons that are all minimally coupled to a dynamical gauge-field. The fCE and the spinons hybridize to yield a compensatedsemi-metal with ‘particle-like’ and ‘hole-like’ pockets. Let us then write down the resulting low-energy e ff ective field theory.We continue to denote the resulting neutral fermions (which are superpositions of the fCE and the spinon) as ψ k α, i , where wehave introduced an additional label i = , i = i = ψ fermions couple minimally to a µ and the non-relativistic b holons couple minimally to ∆ a µ = a µ − A µ . TheLagrangian is given by L = L ψ + L b + L ψ b + L a , (31) L ψ = ψ † α, i ( ∂ τ − ia − µ i ) ψ α, i − m i ψ † α, i ( − i ∇ − a ) ψ α, i , (32) L b = b ∗ ( ∂ τ − i ∆ a − µ b ) b − m b b ∗ ( − i ∇ − ∆ a ) b + u | b | + ... (33) L ψ b = g | b | ψ † α, i ψ α, i + ..., (34) L a = e ( (cid:15) µνλ ∂ ν a λ ) . (35)The form of the above Lagrangian is similar to theories considered earlier in a di ff erent context [56]. Note that we haveallowed density-density interaction terms between the holon density and composite exciton density in L ψ b above; these termsare formally irrelevant close to the transition µ b = L a contains a Maxwell term for the emergent gauge-field, but asdiscussed in the methods section, integrating out the fermions leads to a landau-damped form of the propagator for thetransverse-component of the gauge-field, D i j , as in Eq.18. The time component of the gauge-field couples to the density anddoes not lead to any singular non-Fermi liquid behavior.In addition to the properties of the holon self-energy, Σ b , at T = T > Σ b (0 , ) ∼ uT / and therefore the properties associated with theholon are determined by an interplay of µ b and the above “thermal” mass. In three-dimensions, there is no phase-transitionassociated with condensation of (cid:104) b (cid:105) at T > SUPPLEMENTARY NOTES 3 Io ff e-Larkin composition rules: We derive the Io ff e-Larkin sum rules [50] in this supplementary note. From the fieldtheory description introduced in the previous supplementary note, integrating out the vector potential a leads to the constraint j ψ + j b = , where the j represent the respective currents, i.e. the holons and fermions can only move subject to this constraint.As a result of this constraint, the net electrical response of the system will correspond to the sequential (i.e. ‘series’) circuitof the two individual ‘resistances’. Suppose we now integrate out the matter fields altogether and obtain an e ff ective actionpurely in terms of the external and internal gauge fields, S e ff [ A , a ] = ˆ d r d τ (cid:20) a Π ψ a + ( A − a ) Π b ( A − a ) + A Π A + ... (cid:21) , (36)where we have restricted ourselves to the simplest quadratic action and ... denote additional contributions which may arisee.g. from the singular re-arrangement of the Fermi surfaces in the presence of a finite b = ∇ × a . The coe ffi cients Π ψ and Π b denote the full response functions due to the neutral fermions and gapped holons respectively. We have also included Π ,that arises from the trivial background for the sake of completeness. The response functions to leading order in small ω, q aregiven by, Π ψ ( ω, q ) = i ω σ ψ ( ω, q ) − χ ψ q , (37) Π b ( ω, q ) = i ω σ b ( ω, q ) − χ b q , (38)where σ ψ, b and χ ψ, b denote the conductivities and diamagnetic susceptibilities respectively (Note that χ ψ, b ≡ µ − , b , as definedin the main text). Since a is a dynamical field and the path-integral sums over all allowed configurations of a , we can integrateit out and determine the e ff ective action purely in terms of the external vector potential, A . The resulting action is given by, S e ff [ A ] = ˆ d r d τ A (cid:20) Π + Π ψ Π b Π ψ + Π b (cid:21) A (39)Let us now first focus on the limit of q → ω (i.e. there is a uniform electric field). In this limit, the net conductivityof the system can be read o ff from the second term above as, σ ( ω ) = σ ψ ( ω ) σ b ( ω ) σ ψ ( ω ) + σ b ( ω ) , (40)which is the promised sequential response of the two resistors. Note that one can arrive at the same formula in terms of theshift of the gauge field a and its resulting ‘backflow’ e ff ect. In the limit of ω → T =
0, Re[ σ b ] = ω → q is finite (i.e. there is a uniform magnetic-field). Here, one can reado ff the net diamagnetic response as, χ = χ ψ χ b χ ψ + χ b , (41)in an electrical insulator. This is precisely the form of the diamagnetic response that one would obtain starting from thefree-energy in Eq. 11 for b = α B (Eq.12); the resulting locking mechanism is then responsible for Landau quantization of theneutral Fermi-surface and quantum oscillations [53]. SUPPLEMENTARY NOTES 4
Comparison to other theoretical proposals for SmB : In this supplementary note, we compare our theoretical proposalfor the CEFL with other proposals that have made an attempt to account for at least some of the anomalous features observedin experiments on SmB . The other proposals can be broadly classified as follows: (i) “Magnetic-breakdown” in a small-gapinsulator [28, 29] - If the typical cyclotron energy, at large enough magnetic fields, is larger than the insulating gap, thiscan lead to Landau quantization and an analogue of breakdown e ff ects (but as a function of energy). However within thispicture, in the low temperature and zero magnetic field limit, the system is still an insulator and does not have any of thethermodynamic or optical signatures associated with the experimental observations described earlier. (ii) “Majorana” Fermi-surfaces [58, 59] - Within this picture, there is a Fermi-surface in the bulk, where the excitations are Majorana-like, insteadof being complex fermions of the type we propose (i.e. fCE). In particular in the Majorana-based interpretation, the zeromagnetic field state in the limit of zero temperatures is a superconductor, and not an electrical insulator. It is important tonote that in an incompressible phase, all excitations carry a well defined charge; the Majorana fermions are objects where theanti-particle is identical to the particle itself and therefore carries no charge. Since there are no neutral fermionic excitationsin a system described by an electronic Hilbert space in the UV, these objects can only emerge in the IR and be necessarilynon-local. The only known route of doing this theoretically is to couple them to an emergent gauge-field; the non-localMajorana fermions can be coupled to a discrete (e.g. in the simplest case a Z ) gauge field. In three dimensions, there is thennecessarily a finite temperature phase transition associated with transition into the phase with deconfined, non-local Majoranaexcitations. This should be seen e.g. as a divergence in the specific heat at the transition. Within our framework, there is nofinite temperature phase transition into the CEFL phase with a deconfined U (1) gauge-field, even in three dimensions. (iii)Gapped (conventional) excitonic insulators [57] - As a result of a large joint density of states, it is possible that the system isclose to a conventional (bosonic) excitonic instability with a finite momentum, Q (governed by details of the band-structure),but with a small gap, ∆ . If the energy gap is small along a ring of momentum around Q (e.g. like a roton), in the limitof temperatures larger than the gap it can give rise to power-law features in some thermodynamic properties (as opposed toexponential in ∆ / T ). 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