Abstract
Samarium hexaboride is a classic threedimensional mixed valence system with a hightemperature metallic phase that evolves into a paramagnetic charge insulator below 40 K. A number of recent experiments have suggested the possibility that the lowtemperature insulating bulk hosts electrically neutral gapless fermionic excitations. Here we show that a possible ground state of strongly correlated mixed valence insulators—a composite exciton Fermi liquid—hosts a three dimensional Fermi surface of a neutral fermion, that we 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 in SmB_{6}.
Similar content being viewed by others
Introduction
Electronic solids where the valence of one of the constituent elements is nonintegral show a number of fascinating properties^{1,2} arising from the Coulomb interaction between electrons. Of interest to us in this paper is a class of mixedvalence (MV) systems, a classic example being SmB_{6}^{3,4}, where a high temperature metallic state evolves into an insulator at low temperatures. Attention has been refocused on this material in recent years following the proposal^{5} that it may be an interactiondriven topological insulator (TI)^{6,7}. There is compelling evidence now for metallic surface states in this material (of possibly topological origin) despite an electrically insulating bulk at low temperatures from predominantly transport^{8,9,10,11,12,13} and other measurements^{14}. A different fascinating aspect of a number of MV insulators, including SmB_{6}^{3,4}, are various thermodynamic and transport anomalies at low temperatures, apparently at odds with an insulating behavior in the bulk. Traditionally, these anomalies have often been attributed to the presence of ingap states. An interesting development was the observation of quantum oscillations (QO) in magnetization, first reported in SmB_{6} by Li et al.^{15} and interpreted as additional evidence for the twodimensional metallic surface states.
However, subsequent measurements of QO in magnetization in SmB_{6} by Tan et al.^{16} observed frequencies corresponding to almost half of the bulk Brillouin zone. Tan et al.^{16} found that the frequencies, the cyclotron mass and the amplitude of the oscillations are quite similar to the measured quantumoscillations in the other metallic hexaborides RB_{6} (R ≡ La, Pr, or Ce)^{17,18,19,20}. Moreover the measured density of states from the lowtemperature specific heat is in good agreement with the value obtained from quantum oscillations^{21}. Based on these observations ref. ^{16} raised the surprising possibility that the quantum oscillations are a property of the electrically insulating bulk. They also suggested that the oscillations originate from the same ingap states responsible for the low temperature anomalies which have since been reexamined closely. However it has also been argued more recently^{22} that some of the same QO results can be explained using a purely twodimensional model of the metallic surface states. The low temperature anomalies include a finite linear specific heat coefficient^{16,23,24} and bulk optical conductivity below the chargegap^{25}. Furthermore, a fieldinduced thermal conductivity proportional to 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 the presence of a Fermi surface of electrically neutral fermions in the bulk that nevertheless couple to the external magnetic, but not to weak DC electricfields.
Inspired by the current baffling experimental situation, we are led to a number of theoretical questions. Can MV insulators host Fermi surfaces of neutral fermionic quasiparticles? If so, what is the origin of the neutral (fermionic) excitation and what constrains the volume of the Fermi surface? What are the thermodynamic and transport signatures of phases with such neutral fermionic excitations? Can Fermi surfaces of neutral fermions, that do not couple directly to the external magneticfield, give rise to quantum oscillations? In a separate development, it has been pointed out^{28,29} that under certain conditions, even bandinsulators with gaps smaller than the cyclotron energy can exhibit quantum oscillations.
In recent years, a number of triangular lattice organic materials close to the Mott transition have been shown to act as chargeinsulators but thermal metals^{30,31,32,33}, where the electron appears to have splintered apart into fractionalized excitations (“partons”)^{34,35}; while the charge degree of freedom can remain gapped, the spinful, neutral spinon can form a Fermi surface. The possibility of observing quantum oscillations for such neutral spinon Fermi surfaces has been addressed previously by Motrunich^{36}. However, strongly correlated mixedvalent insulators are far from being a Mott insulator, thereby requiring a different microscopic mechanism to stabilize such a neutral Fermi surface.
Here we show that in the limit of strong Coulomb interactions in a mixedvalence insulator, there is a well defined mechanism for the formation of an electrically neutral fermionic quasiparticle—dubbed the fermionic composite exciton (ce)—that can form a Fermi surface; the resulting phase—the composite exciton Fermi liquid (CEFL)—is electrically insulating but will have a neutral fermi surface. We show that the CEFL shares a number of features with the observed phenomenology in SmB_{6}. We also note that while the present work is motivated by the recent experiments in SmB_{6}, it is potentially relevant to other mixedvalence insulators^{2,14}, such as SmS under pressure, YbB_{12} etc.
Results
Electronic structure
The electronic configuration of Sm is [Xe]4f^{6}5d^{0}6s^{2}. In SmB_{6}, the valence of Sm is known to fluctuate between Sm^{2+} and Sm^{3+} with an average valence of ~ 2.6^{37,38}. There is strong spin–orbit coupling in this material and the sixfold degeneracy of the \(J = \frac{5}{2}\) orbital is lifted due to crystal field splitting, giving rise to a quartet (Γ_{8}) and a doublet (Γ_{7}). The fivefold degenerate d–orbitals split up into a doublet (e_{ g }) and a triplet (t_{2g}). The ground state of Sm in SmB_{6} is in a coherent superposition of 5d^{1} (e_{ g }) + 4f^{5} (Γ_{8}) ⇋ 4f^{6}. In contrast, the ground state of La in metallic LaB_{6} has an electronic configuration of [Xe]5d^{1}6s^{2} and there are no f–electrons.
Bandstructures for the surface as well as the insulating bulk^{39} have been modeled using multiorbital tightbinding models^{40,41}, but we focus here on the simplest twoband model for a mixedvalence compound in three dimensions^{2} to illustrate the key ideas. In particular, we will restrict ourselves to the situation where both the d and f orbitals 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 at site 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 crystalfield multiplet index α (both σ, α = ↑, ↓ and we drop the distinction between the two from now on). As discussed above, it is appropriate to consider a model where the f–valence fluctuates between n^{f} = 1 and n^{f} = 2. With respect to the n^{f} = 2 state, the above configurations can be interpreted as an empty state and a state with one hole respectively. We therefore carry out the following particlehole (PH) transformation \(f_\alpha \to \varepsilon _{\alpha \beta }f_\beta ^\dagger = \tilde f_\alpha\) where ε_{ αβ } is the fully antisymmetric tensor and we have introduced \(\tilde f\) as the f–hole. The standard periodic Anderson Hamiltonian^{42}, but now written in terms of the \(\tilde f\)–hole is given by,
where \(n_{\bf{r}}^{\tilde f}\) = \(\mathop {\sum}\nolimits_\alpha {\kern 1pt} \tilde f_{{\bf{r}}\alpha }^\dagger \tilde f_{{\bf{r}}\alpha }\) = \(2  n_{\bf{r}}^f\), with \(n_{\bf{r}}^f\) = \(\mathop {\sum}\nolimits_\alpha {\kern 1pt} f_{{\bf{r}}\alpha }^\dagger f_{{\bf{r}}\alpha }\) and \(n_{\bf{r}}^d\) = \(\mathop {\sum}\nolimits_\alpha d_{{\bf{r}}\alpha }^\dagger d_{{\bf{r}}\alpha }\). U_{ df } is a repulsive density–density interaction between the f and d–electrons (or equivalently, it represents an attractive interaction between the \(\tilde f\)–hole and the d–electron) and U_{ ff } represents a large onsite Coulomb repulsion between the \(\tilde f\) holes. The hoppings for the d–electron (\(\tilde f\)–hole) are given by \(t_{{\bf{rr}}\prime }^d\) (\(t_{{\bf{rr}}\prime }^f\)), with \(\left {t^d} \right \gg \left {t^f} \right\) and μ_{ d } represents the chemicalpotential for d–electrons. The hybridization, V_{ αβ }, between the d and f electrons, has odd parity V_{ αβ }(−k) = −V_{ αβ }(k).
We are interested in the limit of U_{ ff } → ∞, and U_{ df } large but finite. We use here a slightly different variant of the standard slaveboson representation^{43},
where we have fractionalized the \(\tilde f\)–hole into a spinless boson (“holon”), b, that carries the physical, negative (−1) electromagnetic charge (i.e., opposite to electron charge) under the external gaugefield, A_{ μ } and a neutral fermion (“spinon”), χ_{ α }, that carries the spin (α); see Fig. 1a. There is a redundancy associated with the above parametrization χ → χ_{ α }e^{−iθ}, b → be^{iθ} which leaves f_{ α } invariant. We therefore assume that the holon (spinon) carries a unit positive (negative) charge under an emergent U(1) gaugefield a_{ μ } = (a_{0}, a). We are interested in describing phases with a chargegap (i.e., insulators) where the holon remains uncondensed, \(\left\langle b \right\rangle = 0\) and where the Fermi surface of the d–electrons is absent. The above definition in terms of the partons is to be supplemented with a gaugeconstraint, that ensures restriction to gaugeinvariant states in the Hilbert space, of the form \(b_{\bf{r}}^\dagger b_{\bf{r}}\) = \(\chi _{{\bf{r}}\alpha }^\dagger \chi _{{\bf{r}}\alpha }\). We impose an additional hardcore constraint on the bosons, i.e., \(b_{\bf{r}}^\dagger b_{\bf{r}} \le 1\), which ensures no double occupancy of the \(\tilde f\)–hole; the total density of doped holes is then \(\mathop {\sum}\nolimits_{\bf{r}} {\kern 1pt} {\kern 1pt} b_{\bf{r}}^\dagger b_{\bf{r}}\) = \(\mathop {\sum}\nolimits_{\bf{r}} {\kern 1pt} \tilde f_{{\bf{r}}\alpha }^\dagger \tilde f_{{\bf{r}}\alpha }\). (See the Methods section for a comparison to the standard slaveboson representation.)
Composite excitons
The global requirement for obtaining a mixedvalence insulator, that is also consistent with the known electronic count in SmB_{6} is \(\mathop {\sum}\nolimits_{\bf{r}} {\kern 1pt} \tilde f_{{\bf{r}}\alpha }^\dagger \tilde f_{{\bf{r}}\alpha }\) = \(\mathop {\sum}\nolimits_{\bf{r}} {\kern 1pt} d_{{\bf{r}}\sigma }^\dagger d_{{\bf{r}}\sigma }\) (equivalently, \(\mathop {\sum}\nolimits_{\bf{r}} \left[ {d_{{\bf{r}}\sigma }^\dagger d_{{\bf{r}}\sigma } + f_{{\bf{r}}\alpha }^\dagger f_{{\bf{r}}\alpha }} \right]\) = 2), which when combined with the above constraints automatically implies n^{b} = n^{d}. As a result of the attractive interaction (Eq. (1)) between the \(\tilde f\)–holes and d–electrons (U_{ df } > 0), there is now an attractive interaction between the holons and the conduction electrons. For sufficiently strong attraction, it is therefore possible to form bound states of the conduction electrons and the holons to form a neutral fermionic composite exciton (fCE),
The above quasiparticle is electrically neutral but is charged under the internal U(1) gauge field associated with the slave boson construction (see Fig. 1b); at a finite density it can form a Fermi surface that is minimally coupled to the emergent gaugefield a_{ μ }. Note that in our specific example, as a result of the hardcore constraint, the number of bosons are guaranteed to be equal to the number of conduction electrons and therefore, the number of fCE is identical to the number of conduction electrons, i.e., n^{ψ} = n^{d}. The volume of the Fermi surface of the ψ fermions will then be identical to the volume of the original conduction (d–)electron Fermi surface.
The effective Hamiltonian that describes the low energy physics, after the conduction electrons have formed bound states with the holons can be expressed as,
where ε_{CE} is the fCE dispersion (see Methods section for an estimate of the nearest neighbor fCE hopping) and ε_{χ,k} is the spinon dispersion. Note that, by construction, the holon is gapped. On the other hand as a result of the complete binding of all the d–electrons to form fCE, the charged d–excitations are also gapped. The ellipses denote various allowed terms; one such term (among others) is the exchange interaction between the f moments,
which also modifies the dispersion for the spinon bands, with the hopping t_{ χ } set by t^{f}, J_{ H } and the holon hopping (see Supplementary Note 1).
For a finite V, the fCE band hybridizes with the spinon band to yield renormalized bands as shown in Fig. 1c (see Methods section). It is convenient to carry out a PH transformation on \(\chi _{\mathbf{r}\alpha } \to \tilde \chi _{\mathbf{r}\alpha } \equiv \varepsilon _{\alpha \beta }\chi _{\mathbf{r}\beta }^\dagger\). Then,
A finite t_{ χ } is necessary to get crossings at the fermilevel; one then obtains an electrically neutral semi“metal” with “particle” and “hole” pockets with equal volume. The Fermi surfaces thus obtained have both fCE and spinon character; from now on we do not distinguish between the two. Note that the hopping amplitudes for the fCE and spinon are not individually gaugeinvariant, unlike the gaugeinvariant ratio ζ = t_{CE}/t_{ χ }. Which sign of ζ is preferred depends on various microscopic details; if ζ < 0 (ζ > 0) the ground state will in fact be an insulator (semimetal) of fCE and spinons.
Let us now briefly describe a possible mechanism that allows the insulating bulk hosting a CEFL to coexist with a metallic surface. Previously, it has been argued^{44} that the Kondoscreening can be reduced significantly near the surface leading to “Kondobreakdown,” in which the f–moments decouple from the conduction electrons, giving rise to quasiparticles that are lighter. As a result of surfacereconstruction and screening effects^{45}, it is also possible that the ratio U_{ df }/t_{ d } is smaller close to the boundaries than in the bulk. The weaker attraction between the holon and the conduction electrons can then lead to an unbinding of the fCE close to the surface, thus liberating the holon and the conduction electron within a length scale, ξ, from the surface (Fig. 2). The unbound holons can then Bose condense near the surface, confining the gaugefield, thereby rendering the originally neutral fermions with physical charge. In this way, one may recover metallic quasiparticles at the surface as a result of unbinding of the fCE. Moreover, depending on the details of the fCE dispersion (which may be itself topological) and the oddparity hybridization, V_{ αβ }(k), it is possible for the metallic quasiparticles at the surface to realize topologically protected surfacestates. We leave a discussion of the detailed quantitative theory for future work.
Phenomenology of CEFL
Returning now to a description of the bulk, the lowtemperature specific heat is dominated by the fluctuation of the fermiongauge field system. As a result of the gaugefield fluctuations (see Methods section and the Supplementary Note 2 for a discussion of the lowenergy field theory) the lowT specific heat^{46,47} is given by,
Measurements of specific heat in SmB_{6} do report a linear in T specific heat^{16,23,24}. Moreover the gapless fCE excitations along the neutral Fermi surface contribute to the NMR spinlattice relaxation rate, 1/T_{1}, in the usual way,
Measurements on SmB_{6} support such metallic 1/T_{1}T behavior (V. Mitrovic, personal communication)^{48}. Note that however as a result of strong spinorbit effects, the above quantity need not be related to the Knightshift by Korringa’s relation.
The mere presence of a chargegap in the system does not imply a lack of subgap optical conductivity^{49}; the only physical requirement is that the conductivity vanish as ω → 0. We are interested here in the form of Re[σ(ω)] at low, but finite, frequencies. We apply the IoffeLarkin rule to the (fCE + holon) system^{50} (see Supplementary Note 3 for details) and relate the holonresponse to a dielectric constant, \(\epsilon _b\). We expect the response of the fCE to be similar to that of a metal at low but nonzero frequencies with \({\mathrm{Re}}\left[ {\sigma _{{\mathrm{CE}}}(\omega )} \right] \gg \omega\) and \({\mathrm{Im}}\left[ {\sigma _{{\mathrm{CE}}}(\omega )} \right] \ll {\mathrm{Re}}\left[ {\sigma _{{\mathrm{CE}}}(\omega )} \right]\). Then,
where the fCE conductivity can be expressed in the generalized Drude form σ_{CE}(ω) = ρ/(Γ(ω) − iω), with Γ(ω) a frequency dependent scattering rate and ρ is defined to be the total optical weight. At low ω, where \(\left {{ {\Gamma }}(\omega )} \right \ll \omega\), the real part of the conductivity can be evaluated as^{51,52}
Depending on the mechanism responsible for relaxation of currents, one can then obtain different results for Γ(ω); we discuss the different regimes in the Methods section. Recent measurements of opticalconductivity in the THz regime in SmB_{6}^{25} find appreciable spectral weight below the insulating gap, much larger than any imaginable impurity band contribution.
After integrating out all the matterfields, the groundstate energy of the system in the limit of weak fields follows from gaugeinvariance,
where μ_{ b }, μ_{CE}, μ_{ v } represent the permeability of the gapped holons, compositeexcitons and the background “vacuum” respectively; all of these quantities depend on the UV details of the underlying theory. u_{osc}(b) is the oscillatory component, relevant for our discussion on quantumoscillations and the ellipses denote additional higher order terms. In the limit of a small B, the internal b can optimize itself in order to minimize the energy; ignoring the oscillatory component in Eq. (11), the optimum value is
an O(1) number that is a priori unknown. In the regime where \(\mu _{{\mathrm{CE}}} \gg \mu _b\), b locks almost perfectly to the external B (i.e., α → 1).
The period of the oscillations is then (see Methods section)^{53},
where \(S_ \bot ^i\) is the crosssectional area of the fCE Fermi surface sheet i perpendicular to B. In the limit where α → 1 (i.e., where b → B), the period is directly related to the volume of the composite exciton fermi surface, but in general it can be significantly different depending on the value of α. Including the effect of impurities broadens the Landaulevels and the oscillation amplitude has an additional Dingle suppression ~exp(−1/ω_{ c,i }τ_{ i })^{54}, where τ_{ i } is the elastic lifetime and ω_{ c,i } is the effective cyclotron energy in sheet i.
The low temperature thermal conductivity, κ, is dominated by the fermionic contribution (i.e., the holon, the gaugefield and the phonon contributions are expected to be small compared to the fCE contribution) and there is no difference between the physical thermal conductivity and the conductivity due to the fCE. Let us first estimate the longitudinal thermal conductivity, \(\kappa _{xx} \approx \kappa _{xx}^{{\mathrm{CE}}}\), due to the compositeexcitons. We assume that the fermionic composite excitons form a state akin to an ordinary metal for thermal transport^{55}, such that the longitudinal conductivity is given by
where ε_{ F } is the Fermienergy, m_{ i } represent the masses for the two pockets and we have allowed for different lifetimes, τ_{ i }, for the two pockets. At zero magneticfields, all of the experiments on SmB_{6} find a value of κ_{ xx }/T that extrapolates to zero as T → 0^{10,26}. There is no consensus yet on whether κ_{ xx }/T extrapolates to a finite value in the limit of T → 0 at a finite magnetic field^{21,26}. The presence of a zerofield Tlinear specific heat combined with an absence of a finite Tlinear thermal conductivity suggests the presence of either a small zerofield gap that closes at higher fields, or, the presence of localized states. Within the former scenario, it is plausible that at zerofield and at low temperatures, the CE Fermi surfaces undergo pairing to yield a gapped state with a small insulating gap, that can be significantly smaller than the chargegap.
We note that experimentally, the thermal conductivity measurements have been carried out at very low temperatures (<1 K) while the coefficient of the linear in T specific heat is typically extrapolated from higher temperatures. The opening of a small insulating pairing gap at a temperature T_{ p } corresponds to an actual phase transition in (3 + 1)–dimensions (in the Ising universality class) with an associated divergence in the specific heat. Interestingly, a number of experiments report a strong upturn in the specific heat around ~1 K, which is believed to be inconsistent with the usual Schottky contribution. Within the above scenario, it is plausible that the upturn in the specific heat is associated with the onset of the divergence around T_{ p } ≈ 1 K.
It is also useful to estimate the thermal Hall conductivity, κ_{ xy }. In the weakfield regime, as noted previously, the composite excitons move essentially under the effect of an effective magnetic field \(\overline {\bf{b}}\) and are subject to the Lorentz force associated with this field. However note that the two pockets contribute to the thermal Hall response with opposite signs. We know semiclassically that for each pocket
where \(\omega _{\mathrm{c},i} = e\left {\bf{b}} \right{\mathrm{/}}m_i\) = \(\alpha e\left {\bf{B}} \right{\mathrm{/}}{\bf{m}}_i\) and \(\kappa _{xx}^i\) can be read off from Eq. (14) above. The total thermal Hall response is the difference of the response for the “particle” and “hole”like pockets. Observation of a nonzero thermal Hall effect is a good indicator that the parameter α—which determines the magnitude of orbital effects of the external magnetic field—is not too small. In SmB_{6}, if the quantum oscillations truly arise from the bulk neutral fermi surface of composite excitons as a result of the mechanism proposed above, then that necessarily implies a finite bulk thermal Hall response. However, we note that since the system is analogous to a compensated semimetal, the thermal Hall effect is expected to be vanishingly small at higher fields when ω_{c,i}τ_{ i } ≳ 1.
Let us finally address the fate of the fCE semimetal phase as it is doped away from the mixedvalence limit by excess d–electrons or holes (e.g., by chemical substitution or by gating thin films). There are two natural outcomes: if the holon remains uncondensed, the d–electrons (or holes) can form a “small” Fermi surface while the neutral fCE Fermi surface continues to be present. This phase is the familiar (mixedvalence) fractionalized Fermiliquid (FL*)^{56}. On the other hand, if the holons condense as a result of doping away from the mixedvalence limit, the CE fermi surfaces become Fermi surfaces of physical electrons (and holes) as a result of confinement. The exact outcome is sensitive to microscopic details and is beyond the scope of our discussion here.
The mechanism responsible for the formation of the fermionic exciton is physically distinct from the one responsible for the conventional bosonic exciton^{1}. A few recent theoretical studies have tried to address the origin of the lowenergy bulk excitations in SmB_{6} using a variety of different approaches^{57,58,59}. The CEFL is strikingly distinct from these previous proposals—unlike refs. ^{58,59} the composite exciton is not a Majorana fermion, and unlike ref. ^{57}, the composite exciton has Fermi statistics and forms a Fermi surface (see Supplementary Note 4 for a more detailed comparison).
Discussion
We have described a phase of matter with a neutral Fermi surface of composite excitons in a mixedvalent insulator with a chargegap. A number of properties associated with such a phase resembles the experimental results in bulk SmB_{6}. Future numerical studies of the periodic Anderson model in the insulating regime and in the presence of strong interactions may be able to shed light on questions related to energetics and stability of various phases. We also note that more recent measurements on a mixed valence insulator compound different from SmB_{6}, that displays clear bulk quantum oscillations and has metallic longitudinal thermal conductivity down to the lowest measurable temperatures at zero field, in a clear indication of the formation of a Fermi surface of neutral fermions (L. Li, Y. Matsuda, and T. Shibauchi, personal communication).
Methods
Slaveboson representation
In order to motivate the rationale behind choosing the prescription in Eq. (2), recall that the standard slaveboson representation proceeds as,
where h_{ r } is a spinless bosonic holon with the constraint \(h_{\bf{r}}^\dagger h_{\bf{r}} + \mathop {\sum}\nolimits_\alpha {\kern 1pt} \chi _{{\bf{r}}\alpha }^\dagger \chi _{{\bf{r}}\alpha } = 1\). Consider now the scenario where the h–holons are perturbed away from a Mottinsulating state with \(\left\langle h \right\rangle = 0\) and \(\left\langle {h_{\bf{r}}^\dagger h_{\bf{r}}} \right\rangle = 1  x\) (where x represents the density of doped holes away from the 4f^{6} configuration). The two representations are then physically equivalent if we make the transformation \(h_{\bf{r}}^\dagger \to b_{\bf{r}}\) and \(\left\langle {b_{\bf{r}}^\dagger b_{\bf{r}}} \right\rangle = x\); for a concrete scenario, consider, e.g., the quantum rotor model where \(h_{\bf{r}}^\dagger = e^{i\theta _{\bf{r}}}\) and \(n_{\bf{r}}^h\) is the boson density conjugate to θ_{ r }.
Fermionic composite exciton hopping
Consider the limit where there is a clear hierarchy of scales: \(U_{ff} \gg U_{df} \gg t_d \gg V\) and where t_{ d } is the nearest neighbor hopping for the d–electrons. In this regime, the nearest neighbor hopping amplitude for a single fCE is approximately given by (see Supplementary Note 1)
where t_{ f } is the effective nearest neighbor \(\tilde f\)–hole hopping. There is, in principle, a very strong onsite repulsion set by U_{ ff } for the fCE, as a result of the constraint of no double occupancy for the hardcore holons. However, if the binding is not purely onsite and has some finite extent, the repulsion between the fCE can be renormalized down from the bare U_{ ff } and the resulting state can be described within a weakly interacting CEFL.
The fermionic exciton is clearly significantly different from the more conventional bosonic exciton^{1,57,60} that has been discussed in the context of semimetals and narrow gap semiconductors. The latter arises in the limit where U_{ df } dominates over U_{ ff }. In contrast, as shown above, the fermionic exciton is expected to arise naturally in the limit where \(U_{ff} \gg U_{df}\), which is a more realistic regime for mixedvalent systems. Fermionic composite excitons have also been discussed recently in the context of multicomponent quantum hall states^{61}.
Low energy field theory for CEFL
Let us describe the lowenergy effective field theory for the CEFL phase described in the main text^{56}. The composite exciton, ψ_{kα,i}, with i = 1, 2 representing the two pockets, is coupled minimally to a_{ μ } and the nonrelativistic b holons at a finite chemical potential, μ_{ b } > 0, are coupled minimally to Δa_{ μ } = a_{ μ } − A_{ μ } (see Supplementary Note 3 for a more complete discussion). Let us first discuss the form of the gaugefield propagator, D_{ ij }(iω_{ n }, q)≡\(\left\langle {a_i\left( {i\omega _n,{\bf{q}}} \right) a_j\left( {  i\omega _n,  {\bf{q}}} \right)} \right\rangle\) where we choose to work in the Coulomb gauge ∇ · a = 0, with a being purely transverse. As a result of the minimal coupling, integrating out the fCE excitations leads to a Landaudamped form of the propagator,
where Ξ, β are constants determined by details of the fCE dispersion.
For the specific nonrelativistic form of the theory for the holons, there are no holons in the ground state and the only holon selfenergy, Σ_{ b }, contribution arises as a result of the coupling to the gaugefield and \({ {\Sigma }}_b\left( {i\omega _n,{\bf{q}}} \right)\) ~ \(q^2\left( {1 + c\left {\omega _n} \right{\mathrm{ln}}\left( {1{\mathrm{/}}\left {\omega _n} \right} \right) + \ldots } \right)\) at T = 0, where c is a constant. The above correction is less important than the bare terms in the holon action and can therefore be ignored.
Finally, as a result of the coupling to the gaugefield fluctuations, the fermions have a selfenergy,
upto additional logarithmic corrections.
Alternative route to CEFL
We demonstrate here an alternate route toward arriving at a description of the bulk CEFL phase from a different starting point. Consider a compensated semimetal with (physical) d–electron and f–hole pockets. We are interested in driving the semimetal into an insulating phase in the presence of strong interactions. The Hamiltonian is given by,
where the hoppings t^{d}, t^{f} are positive and V denotes the hybridization. We will specify the form of H_{int} momentarily.
Consider setting V = 0 for now and using the slaverotor formalism to represent the electronic operators as,
where the rotor field, \(e^{i\theta _{\bf{r}}}\), carries physical charge and the spinful fermions ψ_{ α }, χ_{ α } are electrically neutral. Let n_{ r } be the boson density conjugate to the rotor field. Then the gaugeinvariant states satisfy the constraint: \(n_{\bf{r}} + n_{\bf{r}}^\psi  n_{\bf{r}}^\chi = 0\), where \(n_{\bf{r}}^d = n_{\bf{r}}^\psi , n_{\bf{r}}^f = n_{\bf{r}}^\chi\). Let us then consider the interaction term to be of the form,
It is then clear that at small U (compared to the hoppings), the rotor fields condense \(\left\langle {e^{i\theta _{\bf{r}}}} \right\rangle \ne 0\) and we recover the compensated semimetal phase. At strong U, one can drive a “Mott”transition to a phase where the rotorfields are gapped \(\left\langle {e^{i\theta _{\bf{r}}}} \right\rangle = 0\) (i.e., to an insulator) where the ψ, χ fermions can form Fermi surfaces, inherited from the original d, f Fermi surfaces. This is the CEFL phase. Both phases are stable to having a small V.
Optical conductivity of CEFL
As introduced in Eq. (10), in the regime where Γ(ω) arises primarily due to scattering of the fermions off the gaugefield fluctuations and where the effects of static disorder can be ignored (i.e., the meanfree path, \(\ell _{{\mathrm{mf}}}\), is long), Γ(ω) ~ ω^{5/3}. In three dimensions, this arises from the fCE selfenergy, ImΣ_{CE}(ω) ~ ω (upto additional logarithms) and includes two extra powers of \(\left {\bf{q}} \right\sim \omega ^{1/3}\). Hence, under these set of assumptions, Re[σ(ω)] ~ ω^{2.33}.
On the other hand, in the regime where Γ(ω) still arises due to scattering of the fermions off the gaugefield fluctuations, but the finite \(\ell _{mf}\) modifies the \(\left \omega \right{\mathrm{/}}q\) form in the gaugefield propagator (Eq. (18)) around \(q\sim \ell _{{\mathrm{mf}}}^{  1}\), Γ(ω) ~ ω^{2}, and Re[σ(ω)] ~ ω^{2}.
Finally note that the fCE density can couple to the local disorderpotential and Γ may be dominated entirely by a frequency independent elastic scatteringrate (Γ_{0}); then Re[σ_{CE}(ω)] ≈ ρ/Γ_{0} which leads to Re[σ(ω)] ~ ω^{2}. Similarly, as a result of localization effects, 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 disordered metals will apply.
Quantum oscillations in CEFL phase
For small fields the energy in Eq. (11) can be rewritten as,
and \(\mu _{{\mathrm{eff}}}^{  1} = \mu _b^{  1} + \mu _{{\mathrm{CE}}}^{  1}\). At zero temperature, the oscillatory component is given by^{53},
where \(\chi _{{\mathrm{osc}}}^i\) sets the scale for the overall amplitude of the oscillations from the Fermi surface sheet i.
Data availability
All relevant data are available from the authors upon reasonable request.
References
Mott, N. F. Rareearth compounds with mixed valencies. Philos. Mag. 30, 403–416 (1974).
Varma, C. M. Mixedvalence compounds. Rev. Mod. Phys. 48, 219 (1976).
Menth, A., Buehler, E. & Geballe, T. H. Magnetic and semiconducting properties of SmB_{6}. Phys. Rev. Lett. 22, 295–297 (1969).
Nickerson, J. C. et al. Physical properties of SmB_{6}. Phys. Rev. B 3, 2030–2042 (1971).
Dzero, M., Sun, K., Galitski, V. & Coleman, P. Topological Kondo insulators. Phys. Rev. Lett. 104, 106408 (2010).
Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).
Qi, X.L. & Zhang, S.C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011).
Allen, J. W., Batlogg, B. & Wachter, P. Large low temperature hall effect and resistivity in mixedvalent SmB_{6}. Phys. Rev. B 20, 4807–4813 (1979).
Zhang, X. et al. Hybridization, interion correlation, and surface states in the Kondo insulator SmB_{6}. Phys. Rev. X 3, 011011 (2013).
Kim, D. J. et al. Surface hall effect and nonlocal transport in SmB_{6}: evidence for surface conduction. Sci. Rep. 3, 3150 (2013).
Wolgast, S. et al. Lowtemperature surface conduction in the Kondo insulator SmB_{6}. Phys. Rev. B 88, 180405 (2013).
Phelan, W. A. et al. Correlation between bulk thermodynamic measurements and the lowtemperatureresistance plateau in SmB_{6}. Phys. Rev. X 4, 031012 (2014).
Hatnean, M. C., Lees, M. R., Paul, D. M. & Balakrishnan, G. Large, high quality singlecrystals of the new topological Kondo insulator, SmB_{6}. Sci. Rep. 3, 3071 (2013).
Dzero, M., Xia, J., Galitski, V. & Coleman, P. Topological Kondo insulators. Annu. Rev. Condens. Matter Phys. 7, 249–280 (2016).
Li, G. et al. Twodimensional fermi surfaces in Kondo insulator SmB_{6}. Science 346, 1208–1212 (2014).
Tan, B. S. et al. Unconventional fermi surface in an insulating state. Science 349, 287–290 (2015).
Ishizawa, Y., Tanaka, T., Bannai, E. & Kawai, S. de haasvan alphen effect and fermi surface of LaB_{6}. J. Phys. Soc. Jpn. 42, 112–118 (1977).
Harima, H., Sakai, O., Kasuya, T. & Yanase, A. New interpretation of the de haasvan alphen signals of LaB_{6}. Solid State Commun. 66, 603–607 (1988).
Ånuki, Y., Nishihara, M., Sato, M. & Komatsubara, T. Fermi surface and cyclotron mass of PrB_{6}. J. Magn. Magn. Mater. 52, 317–319 (1985).
Onuki, Y., Komatsubara, T., Reinders, P. H. P. & Springford, M. Fermi surface and cyclotron mass of CeB_{6}. J. Phys. Soc. Jpn. 58, 3698–3704 (1989).
Hartstein, M. et al. Fermi surface in the absence of a fermi liquid in the Kondo insulator SmB_{6}. Nat. Phys. 14, 166–172 (2017).
Denlinger, J. D. et al. Consistency of photoemission and quantum oscillations for surface states of SmB_{6}. Preprint at http://arxiv.org/abs/1601.07408 (2016).
Flachbart, K. et al. Specific heat of SmB_{6} at very low temperatures. Phys. B Condens. Matter 378, 610–611 (2006).
Wakeham, N. et al. Lowtemperature conducting state in two candidate topological Kondo insulators: SmB_{6} and Ce_{3}Bi_{4}Pt_{3}. Phys. Rev. B 94, 035127 (2016).
Laurita, N. J. et al. Anomalous threedimensional bulk ac conduction within the Kondo gap of SmB_{6} single crystals. Phys. Rev. B 94, 165154 (2016).
Xu, Y. et al. Bulk fermi surface of chargeneutral excitations in SmB_{6} or not: a heattransport study. Phys. Rev. Lett. 116, 246403 (2016).
Boulanger, M. et al. Fielddependent heat transport in the Kondo insulator SmB_{6}: phonons scattered by magnetic impurities. Preprint at http://arxiv.org/abs/1709.10456 (2017).
Knolle, J. & Cooper, N. R. Quantum oscillations without a fermi surface and the anomalous de HaasVan Alphen effect. Phys. Rev. Lett. 115, 146401 (2015).
Zhang, L., Song, X.Y. & Wang, F. Quantum oscillation in narrowgap topological insulators. Phys. Rev. Lett. 116, 046404 (2016).
Shimizu, Y., Miyagawa, K., Kanoda, K., Maesato, M. & Saito, G. Spin liquid state in an organic mott insulator with a triangular lattice. Phys. Rev. Lett. 91, 107001 (2003).
Yamashita, S. et al. Thermodynamic properties of a spin1/2 spinliquid state in a κtype organic salt. Nat. Phys. 4, 459–462 (2008).
Yamashita, M. et al. Highly mobile gapless excitations in a twodimensional candidate quantum spin liquid. Science 328, 1246–1248 (2010).
Yamashita, S., Yamamoto, T., Nakazawa, Y., Tamura, M. & Kato, R. Gapless spin liquid of an organic triangular compound evidenced by thermodynamic measurements. Nat. Commun. 2, 275 (2011).
Lee, S.S. & Lee, P. A. U(1) gauge theory of the Hubbard model: Spin liquid states and possible application to κ–(BEDT–TTF)_{2}cu_{2}(CN)_{3}. Phys. Rev. Lett. 95, 036403 (2005).
Motrunich, O. I. Variational study of triangular lattice spin1/2 model with ring exchanges and spin liquid state in κ–(ET)_{2}cu_{2}(CN)_{3}. Phys. Rev. B 72, 045105 (2005).
Motrunich, O. I. Orbital magnetic field effects in spin liquid with spinon fermi sea: possible application to κ–(ET)_{2}cu_{2}(CN)_{3}. Phys. Rev. B 73, 155115 (2006).
Chazalviel, J. N., Campagna, M., Wertheim, G. K. & Schmidt, P. H. Study of valence mixing in SmB_{6} by Xray photoelectron spectroscopy. Phys. Rev. B 14, 4586–4592 (1976).
Mizumaki, M., Tsutsui, S. & Iga, F. Temperature dependence of Sm valence in SmB_{6} studied by Xray absorption spectroscopy. J. Phys.: Conf. Ser. 176, 012034 (2009).
Lu, F., Zhao, J., Weng, H., Fang, Z. & Dai, X. Correlated topological insulators with mixed valence. Phys. Rev. Lett. 110, 096401 (2013).
Alexandrov, V., Dzero, M. & Coleman, P. Cubic topological Kondo insulators. Phys. Rev. Lett. 111, 226403 (2013).
Baruselli, P. P. & Vojta, M. Distinct topological crystalline phases in models for the strongly correlated topological insulator SmB_{6}. Phys. Rev. Lett. 115, 156404 (2015).
Varma, C. M. & Yafet, Y. Magnetic susceptibility of mixedvalence rareearth compounds. Phys. Rev. B 13, 2950–2954 (1976).
Coleman, P. New approach to the mixedvalence problem. Phys. Rev. B 29, 3035–3044 (1984).
Alexandrov, V., Coleman, P. & Erten, O. Kondo breakdown in topological Kondo insulators. Phys. Rev. Lett. 114, 177202 (2015).
Baruselli, P. P. & Vojta, M. Surface reconstruction in a tightbinding model for the topological Kondo insulator SmB_{6}. 2D Mater. 2, 044011 (2015).
Holstein, T., Norton, R. E. & Pincus, P. de Haasvan Alphen effect and the specific heat of an electron gas. Phys. Rev. B 8, 2649–2656 (1973).
Reizer, M. Y. Relativistic effects in the electron density of states, specific heat, and the electron spectrum of normal metals. Phys. Rev. B 40, 11571–11575 (1989).
Caldwell, T. et al. Highfield suppression of ingap states in the Kondo insulator SmB6. Phys. Rev. B 75, 075106 (2007).
Ng, T.K. & Lee, P. A. Powerlaw conductivity inside the Mott gap: ato κ–(BEDT–TTF)_{2}cu_{2}(CN)_{3}. Phys. Rev. Lett. 99, 156402 (2007).
Ioffe, L. B. & Larkin, A. I. Gapless fermions and gauge fields in dielectrics. Phys. Rev. B 39, 8988–8999 (1989).
Rosch, A. & Howell, P. C. Zerotemperature optical conductivity of ultraclean fermi liquids and superconductors. Phys. Rev. B 72, 104510 (2005).
Rosch, A. Optical conductivity of clean metals. Ann. der Phys. 15, 526–534 (2006).
Sodemann, I., Chowdhury, D. & Senthil, T. Quantum oscillations in insulators with neutral fermi surfaces. Phys. Rev. B 97, 045152 (2018).
Shoenberg, D. Magnetic oscillations in metals (Cambridge University Press, Cambridge, 2009).
Katsura, H., Nagaosa, N. & Lee, P. A. Theory of the thermal hall effect in quantum magnets. Phys. Rev. Lett. 104, 066403 (2010).
Senthil, T., Vojta, M. & Sachdev, S. Weak magnetism and nonfermi liquids near heavyfermion critical points. Phys. Rev. B 69, 035111 (2004).
Knolle, J. & Cooper, N. R. Excitons in topological Kondo insulators: theory of thermodynamic and transport anomalies in SmB_{6}. Phys. Rev. Lett. 118, 096604 (2017).
Baskaran, G. Majorana Fermi sea in insulating SmB_{6}: a proposal and a theory of quantum oscillations in Kondo insulators. Preprint at http://arxiv.org/abs/1507.03477 (2015).
Erten, O., Chang, P.Y., Coleman, P. & Tsvelik, A. M. Skyrme insulators: insulators at the brink of superconductivity. Phys. Rev. Lett. 119, 057603 (2017).
Mott, N. F. The transition to the metallic state. Philos. Mag. 6, 287 (1961).
Barkeshli, M., Nayak, C., Papic, Z., Young, A. & Zaletel, M. Fractionalized exciton Fermi surfaces and condensates in twocomponent quantized Hall states. Preprint at http://arxiv.org/abs/1611.01171 (2016).
Acknowledgements
We thank Suchitra Sebastian for sharing many of their unpublished results and thank her and Olexei Motrunich for many stimulating discussions. We also thank Peter Armitage, Nicholas Laurita, Lu Li, Yuji Matsuda, and Vesna Mitrovic for discussions and for sharing their data. D.C. is supported by a postdoctoral fellowship from the Gordon and Betty Moore Foundation, under the EPiQS initiative, Grant GBMF4303, at MIT. While at MIT, I.S. was supported by the Pappalardo Fellowship. T.S. is supported by a US Department of Energy grant DESC0008739, and in part by a Simons Investigator award from the Simons Foundation.
Author information
Authors and Affiliations
Contributions
D.C., I.S. and T.S. contributed to the theoretical research described in the paper and the writing of the manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Electronic supplementary material
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Chowdhury, D., Sodemann, I. & Senthil, T. Mixedvalence insulators with neutral Fermi surfaces. Nat Commun 9, 1766 (2018). https://doi.org/10.1038/s41467018041632
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41467018041632
This article is cited by

The reverse quantum limit and its implications for unconventional quantum oscillations in YbB12
Nature Communications (2024)

Gapless fermionic excitation in the antiferromagnetic state of ytterbium zigzag chain
Communications Materials (2023)

Electromagnetic signatures of a chiral quantum spin liquid
npj Quantum Materials (2023)

Evidence for a monolayer excitonic insulator
Nature Physics (2022)

felectron hybridised Fermi surface in magnetic fieldinduced metallic YbB12
npj Quantum Materials (2022)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.