Theory of Competing Excitonic Orders in Insulating Wte $ 2 $ Monolayers

Theory of competing excitonic orders in insulating WTe2 monolayers

Yves H. Kwan,1 T. Devakul,2 S. L. Sondhi,2 and S. A. Parameswaran1 1Rudolf Peierls Centre for Theoretical Physics, Clarendon Laboratory, Oxford OX1 3PU, UK 2Department of Physics, Princeton University, Princeton, New Jersey 08540, USA (Dated: September 24, 2021) We develop a theory of the excitonic phase recently proposed as the zero-field insulating state observed near charge neutrality in monolayer WTe2. Using a Hartree-Fock approximation, we numerically identify two distinct gapped excitonic phases: a spin density wave state for weak non- zero interaction strength and spin spiral order at stronger interactions, separated by a narrow window of non-excitonic quantum spin Hall insulator. We introduce a simplified model capturing key features of the WTe2 band structure, in which these phases appear as distinct valley ferromagnetic orders. We link the competition between the excitonic phases to the orbital structure of electronic wavefunctions at the Fermi surface and hence its proximity to the underlying gapped Dirac point in WTe2. We briefly discuss collective modes of the two excitonic states, and comment on implications for experiments.

I. INTRODUCTION

When the ground state of a semimetal or narrow- gap semiconductor becomes unstable to electron-hole Coulomb attraction, it is replaced by an equilibrium condensate of electron-hole pairs (excitons) [1–7]. This new excitonic state of matter is typically insulating, but separated by a phase transition from a conventional band insulator. Although theoretically proposed over half a cen- tury ago, the excitonic state has proven to be remarkably elusive experimentally, with significant progress towards this goal only coming in the past decade or so [8–15]. Recent transport and tunneling measurements on ultr- aclean monolayers of the transition-metal dichalcogenide WTe2 have been argued to be consistent with an excitonic insulating ground state near the charge neutrality point [16, 17]. Signatures of this state develop only at FIG. 1. Bottom: Hartree-Fock phase diagram and evolu- tion of excitonic pairing scale ∆exc with interaction strength low temperatures, indicating an electron ordering transi- −1 tion. Strikingly, despite being insulating at zero field [16– (star indicates estimated experimental value). Excitonic order is present (absent) in the SDW and spin spiral (SS) 20], WTe2 shows robust Shubnikov-de Haas oscillations (semimetal (SM) and insulator (I)) phases. Top: charge/spin at high magnetic fields [17]. This suggests that the insu- order and schematic pairing structure for SDW/SS. Fermi lator is either highly unconventional, or else transitions to pockets of the non-interacting k · p model are schematically a conductor with increasing field. Improving the under- shown with dotted grey lines. standing of the zero-field insulating state is a necessary first step to exploring this intriguing system. An obstacle to this goal is posed by the complex band ing its phase structure. structure of WTe2 which, in the absence of interactions, Here, we explore the phase diagram of WTe2, focus- consists of a pair of over-tilted Dirac cones [21], weakly ing on spin and valley pseudospin order in the excitonic gapped by spin-orbit coupling (SOC). This leads to a states and its interplay with the orbital structure of the

arXiv:2012.05255v2 [cond-mat.str-el] 23 Sep 2021 pair of conduction band minima (electron pockets) at energy bands. We first map out the phase diagram an incommensurate wavevector ±qc, flanking a single numerically (Fig.1) within a Hartree-Fock (HF) treat- valence band maximum (hole pocket) at the Brillouin ment of interactions. We find two distinct excitonic in- zone Γ point. The anisotropic pockets, strong SOC, the sulators, corresponding to spin-density wave (SDW) and twofold ‘valley’ index labelling electron pockets, and pos- spin spiral (SS) orders, for different interaction strengths. sibly nontrivial orbital structure on the Fermi surface We introduce a simplified analytically tractable model (FS) due to the near-Dirac band structure are in stark that captures the low-energy structure of the interaction- contrast to the simplified starting point that, with few ex- renormalized bands in WTe2. This gives an intuitive ceptions [22, 23], underlies studies of the excitonic state. picture where individual SDW/SS excitons are degen- A theory of excitonic insulators in WTe2 must incorpo- erate, but compete due to exciton interactions. We link rate these complexities and clarify their role in influenc- this competition to the orbital content of the conduction 2

0 0 0 nn ;σ nn ;σ nn ;¯σ ∗ band, which can be tuned experimentally. We sketch F−k,−k0 and Fkk0 = [F−k,−k0 ] ), the latter can generi- qualitative features of the SDW/SS collective modes and cally vary as the bands traverse the BZ. discuss their experimental signatures. We close by spec- ulating on possible implications for high-ﬁeld transport.

II. WTe2 MODEL

III. HARTREE-FOCK PHASE DIAGRAM We begin with a k · p theory [16, 24] valid near the P † WTe2 Γ-point, H0 = k hαβ(k)ckαckβ, where α, β are composite spin-orbital indices, and The Hamiltonian H = H0 +Hint defined by (1) and (2) captures the key features of WTe relevant to studying its ˆ z 2 h(k)=ε+(k)+[ε−(k)+δ]τ + vxkxτxsy + vykyτys0. (1) low-energy behaviour near charge neutrality. We numerically study the phase diagram via self-consistent HF cal- The Pauli matrices τ µ, sµ act in orbital and spin space culations with momentum cutoffs |k | < 3qc , |k | < Gy , with τ z = ±1 (sz = ±1) referring to d, p orbitals x 2 y 4 −1 1 where Gy = 1.01 A˚ is the reciprocal lattice vector in (↑, ↓ spins) respectively, ε±(k) = 2 (εd(k) ± εp(k)) where 2 2 the y direction. Anticipating a possible excitonic insta- ε (k) = ak2 +bk4 and ε (k) = − k , with a = −3 eVA˚ , d p 2m bility, we allow for translational symmetry breaking at ˚4 −1˚−2 ˚ b = 18 eVA , m = 0.03 eV A , vx = 0.5 eVA, wavevector qc. More details of the self-consistent Hartree vy = 3 eVA˚ are chosen to match the ab initio band struc- Fock calculations can be found in AppendixA. ture of Ref. [16]. δ controls the band overlap, with δ < 0 (δ > 0) corresponding to a semimetal (semiconductor). Fig.1 shows the phase diagram as a function of the in- −1 We take δ = −0.45 eV which sets the Fermi energy (EF ) teraction strength . In the non-interacting limit, the of the noninteracting bands at charge neutrality to EF ' system starts off with three Fermi pockets, as sketched −0.493 eV, yielding a hole pocket at Γ, and two electron by the grey dotted lines in the middle row of Fig.1. −1 −1 pockets with minima at qc = ±0.3144 A˚ xˆ, incommen- As is increased, the system remains semimetallic un- −1 −1 surate with the reciprocal lattice vector Gx = 1.81 A˚ . til c0 ≈ 0.05 where it transitions into a gapped SDW H0 respects parity Pˆ = τz and time-reversal Tˆ = isyKˆ phase. This phase possesses non-trivial excitonic or- ˆ dering, diagnosed by the integrated q = qc coherence symmetries (K is complex conjugation), and hence its q bands are twofold degenerate. Absent SOC (v = 0), ∆ ≡ P |hc† c i|2 and exhibits both SDW x exc α,β kα k+qcβ ˚−1 H0 has overtilted Dirac cones at qD = ±0.2469 A xˆ. order in the xz spin plane (orthogonal to the SOC axis) SOC gaps the Dirac point, and yields an indirect neg- at wavevector qc, and charge density (CDW) wave or- ative band gap; however it retains U (1) spin rotation s der at 2qc (see Fig.3). The SDW preserves combined symmetry about the y axis, that we assume henceforth. PˆTˆ symmetry, so its bands remain doubly degenerate. We now rewrite interactions, which are density-density Excitonic order is suppressed in a small window around in spin and orbital space, in the band eigenbasis defined −1 ≈ 0.11 in favor of a non-excitonic quantum spin Hall P n † c1 by diagonalizing (1), H0 = knσ εkdknσdknσ. Here, insulator, which yields to a second excitonic PˆTˆ-broken † P α † dknσ = α uknσckα where the sum is over spins/orbitals phase that we dub the spin spiral (SS). Unlike the SDW, α and uknσ are the relevant Bloch functions, which we make SS has no CDW order (at least at purely electronic level), diagonal in spin space by choosing the spin quantization and the local spin polarization is of constant magnitude axis along y, and n = a, b correspond to valence and and rotates in the xz spin-plane with wavevector qc. Fi- conduction bands. In this basis, we find −1 −1 nally, for ≥ c2 ≈ 0.35, ∆exc vanishes and the system is a non-excitonic trivial insulator. For the experimental 1 X H = U(q):ρ† ρ :, (2) interaction strength −1 ≈ 0.28, the ground state is a int 2A q q q SS exciton insulator with an indirect gap of ∼ 230 meV −2 (see Fig.4) and local spin polarization of ∼ 0.002 µBA˚ . where A is the system area, : ... : denotes normal or- ˆ ˆ dering with respect to the Fock vacuum, and U(q) = [Enforcing P and T gives a single excitonic phase with 2 slightly higher energy [16, 25].] e tanh qξ is a dual-gate screened interaction. [Ex- 20q 2 −1 periments correspond to a screening length ξ = 25 nm, At large , ∆exc receives contributions from k-states and relative permittivity ' 3.5.] We consider −1 across much of the folded BZ, suggesting that it distorts as a parameter that tunes the overall strength of in- the bands even far from EF . However throughout the −1 teractions. Eq. (2) introduces the band-projected den- SDW and in the SS near c1 , excitonic coherence is local- P nn0;σ † ized in k-space around the centers of the Fermi pockets. sities ρq ≡ nn0kσ Fk−q,kdnσ,k−qdn0σk and form factors nn0;σσ0 This suggests that we can understand excitonic coher- Fk,k0 = huknσ|uk0n0σ0 i. Apart from constraints im- ence and competition between the SDW/SS phases in ˆ ˆ nn0;σ posed by P and T (which respectively require Fkk0 = this regime by focusing only on states near EF . 3

IV. EFFECTIVE MODEL an enhanced U(4) symmetry and hence does not distin- guish between exciton phases with distinct spin and val- We now construct a simplified effective model that cap- ley orders. This degeneracy is resolved at beyond-DTA tures low-energy features of the renormalized symmetry- (bDTA) level, where the neglected q ∼ qc, 2qc interac- preserving band structure most relevant to the excitonic tions split the various states, with the orbital structure of order. The use of self-consistent HF bands (without exci- the bands playing a crucial role. The influence of orbital tonic distortion) as a starting point is crucial: since (2) is structure on energetic competition is evident already in normal-ordered with respect to the Fock vacuum, bands the few-exciton problem about the insulating state where will naturally deform due to self-exchange when the k · p the valence (conduction) bands of Heff are filled (empty). model is half-filled (consistent with charge neutrality). As demonstrated in AppendixE, single excitons with the Therefore it is only sensible to consider the interplay of symmetries of SS/SDW are degenerate, but are split at the band structure with excitonic condensation after in- two-exciton level by inter-valley 2qc interactions of their corporating such renormalization effects. Consequently, constituent electrons, which are sensitive to orbital struc- the parent state for the excitonic phases is an insulator ture via form factors. with a small indirect band gap (even though we began with a semimetal at −1 = 0). Since we are in the regime where excitonic pairing is V. VARIATIONAL STATES only peaked around Γ and ±qc, we restrict attention to Bloch states within three small ‘pockets’ centered about To fully explore excitonic order, we consider the ex- these special momenta [2]. We treat electrons from the tended HF states (generalizing [2]) two conduction band minima as separate species distin- Y † X σ guished by a ‘valley’ pseudospin λ = +, −, and describe |Φi = αkσ |0i , αkσ = ukakσ + vk wsλbksλ, (5) † k,σ sλ them using independent creation operators bkσλ, where † k is measured from λqc. We also introduce a , an elec- kσ where the parameters are chosen to minimise hHeffiΦ tron creation operator for the valence band maximum. 2 2 with uk, vk real and even in k, uk + vk = 1, The momentum takes all values within some pocket cut- P σ σ0 w w¯ = δ 0 , and overbar denotes complex conju- off |k| < k large enough to encompass the region of sλ sλ sλ σσ cut gation. Eq. (5) describes a state obtained by first folding excitonic pairing. We approximate the dispersions by a,b the BZ by qc, so that all three pockets are centred at Γ, best-fit effective-mass parabolas k at extrema of the −1 and then introducing excitonic coherence between the va- self-consistent bands at (ignoring small ‘teardrop’ lence band and a specific spin-valley combination in the corrections to the band structure). We model the form conduction band parametrized by w. At DTA level, the factors of the valleys as arising from a gapped Dirac energy is w-independent, and uk, vk (which set the mo- cone, with the Dirac point displaced by some wavevector mentum structure of excitonic coherence) are determined −λk0xˆ from the minimum of the dispersion to capture by self-consistently solving the coupled integral equations the tilted structure relevant to WTe2. Accordingly, in √ 1/2 p 2 2 P 0 λ 2v = 1 − ξ / ξ + ∆ , ∆ = U(k−k )g 0 , our model we take the valley-λ Bloch function |u˜kbσi at k k k k k k0 k k (see Fig.2) to be the positive eigenvector of 1 b a a a where gk = ukvk, ξk = 2 (¯k − ¯k), with ¯k = k + P 0 2 b b P 0 2 U(k − k )v 0 and ¯ = − U(k − k )v 0 . For h (k) = λv˜ (k + λk )τ +v ˜ k τ + λσmτ˜ . (3) k0 k k k k0 k σ,λ x x 0 z y y y x small −1 in the exciton phases of Fig.1, we find that † Though valence band Bloch functions can be modeled gk ∼ hab i, which is a direct measure of excitonic pair- similarly in principle, in practice they do not affect the ing, qualitatively matches the momentum-resolved con- SDW-SS competition. The effective Hamiltonian, tributions to ∆exc in the HF, justifying the use of the effective model. The w-dependence is restored upon per- turbatively evaluating the bDTA splitting terms: X a † X b † U(q) † Heff = qaqσaqσ+ qbqσλbqσλ + :ρqρq :, (4) 2A +− 2 X ++ −− ˜ q λ δE[w] = D| Tr W | − Jss0 Wss0 Ws0s + E[w], (6) ss0 is normal ordered with respect to the filled valence band. X 2 ↑ 2 X 2 ↓ 2 The natural hierarchy of inter- and intra-pocket in- D = U(2qc)| vkFkk| = U(2qc)| vkFkk| ,(7) k k teractions U(qc)/U(0) ' 0.025 admits a physically intuitive separation of scales [2]. In the dominant term ap- 1 X 2 2 0 s∗ s0 J 0 = v v 0 U(k − k + 2q )[F 0 F 0 + c.c](8), ss 2 k k c k k k k proximation (DTA), we retain only the band dispersion kk0 and interaction terms with small intra-pocket momen- λλ0 P σ σ∗ s − + tum transfers q qc (form factor effects are negligible where Wss0 ≡ σ wsλws0λ and Fkk0 ≡ hu˜kbs|u˜k0bsi, the within each pocket since we can choose a smooth gauge sole potentially nontrivial form factor in our model, de- nn;σ pends on the valley Bloch parametrization (3). Terms in where Fk,k+q → 1 as q → 0). At this order, we determine the existence of an excitonic instability and the E˜[w] give identical energies for SDW/SS; as these are not momentum structure of excitonic pairing. The DTA has central to our discussion we relegate them to AppendixD. 4

by D) is lowered sufficiently, the SDW can beat out the SS. Revisiting the HF numerics, we indeed find that the effective positions of the gapped Dirac points shift away from the minima towards Γ as −1 is increased, consistent with this scenario. This clarifies the relevance of Dirac and spin-orbit physics to the excitonic order in WTe2.

FIG. 2. Sketch of λ = + valley dispersion and orbital struc- VI. COLLECTIVE MODES ture for the two limits of (3) appropriate to (a) SDW and (b) SS phases; red star marks location of gapped Dirac point. The distinct broken symmetries in the two excitonic phases lead to distinctive collective mode spectra [2, 27]. We now identify the SDW and SS phases in terms of The three candidate continuous global symmetries of the w. As shown in Fig.1, the in-plane SDW is described by system (besides charge conservation, assumed throughout) are: (i) the ‘excitonic’ U(1)eh symmetry of treating ieiα 0 e−iφ electrons and holes (alternatively, conduction and valence SDW: wσ = √ , wσ =w ¯s , (9) s+ iφ s− σ+ electrons) as separately conserved [28–30], generated by 2 e 0 σs iθ −iθ aσ → aσe eh , bσλ → bσλe eh ; (ii) the U(1)v symme- s which has spin density ρ (r) ∼ sin(qcx − try corresponding to independent conservation of elec- c iλθv α)[sin φ, 0, cos φ], charge density ρ (r) ∼ cos[2(qcx − α)] trons in the two valleys (bσλ → bσλe ); and (iii) the J↑↑ U(1) spin rotation symmetry about the y-axis, manifest and bDTA splitting δE = D − 2 . (Though Ref. [2] s iσθs iσθs identifies SDW is the ground state for F → 1, we find in H0 + Hint (aσ → aσe , bσλ → bσλe ). The first that in reality SS is favored in this limit.) In the SS, two of these are explicitly broken at a microscopic level, since interactions mix bands, but are present in the DTA. ↓ −iα ↑ iα SS: w↑+ = e , w↓− = e (10) However as we show in AppendixD, at bDTA level, in- † † teraction terms ∼ a a b+b− (in E˜[w]) destroy the U(1)eh s with spin density ρ (r) ∼ [sin(qcx + α), 0, cos(qcx + α)] symmetry, while U(1)v is preserved. Accordingly in the and δE = 0. Inversion exchanges valleys, yielding a DTA/bDTA regime (throughout the SDW and in the SS −1 −1 spiral with opposite handedness. [Up to global rota- for ∼ c1 ) we expect that interactions only weakly tions/translations, the most general xz-spin order is an gap any U(1)v pseudo-Goldstone modes but strongly gap s elliptic spiral, ρ (r) ∼ [cos χ sin(qcx), 0, sin χ cos(qcx)]. the ‘excitonic’ U(1)eh mode. At this level, we take the y As discussed in AppendixF, assuming H has S -rotation U(1)s symmetry of H0 + Hint to be exact, though as it is symmetry, this generates a CDW with Fourier compo- not symmetry-required it may be weakly broken beyond c s 2 nent ρ (2qc) ∼ |ρ (qc)| ∝ cos(2χ), but here energetics the k · p limit. Finally, since exciton condensation oc- force χ → ±π/4, 0, corresponding to a circular SS with curs at a finite wavevector the breaking of U(1)s and/or no CDW, or a pure SDW without spiral order [26] – the U(1)v is intertwined with translational symmetry break- cases we consider.] ing at qc. Hence we expect a minimum (or gapless point) Using (9) and (10) in (6), we find that the competi- in the collective mode dispersion at q ' qc in the original tion between SDW and SS is tuned by D and J↑↑: as WTe2 BZ. in our two-exciton warmup problem (AppendixE), these The SDW breaks U(1)s, with the corresponding free- describe |q| ∼ 2qc interactions between the valleys. D is dom parameterized by φ in (9), leading to a standard a Hartree term that directly penalizes 2qc CDW order, magnon mode (gapless in the interacting k · p model). while J reflects intervalley exchange. SS is unaffected by In addition, since the conduction band minima are not both contributions due to its perfect spin-valley locking, at high-symmetry points and hence generically incom- but the SDW is energetically favored if 2D < J↑↑. The mensurate with the undistorted lattice, we expect a pha- central role played by the orbital structure/form factors son mode (captured by α) generated by U(1)v rotations, is apparent in two limiting cases of (3). For k0 → ∞, which will be weakly gapped as the latter is only an where the Dirac physics is invisible to low-lying conduc- approximate symmetry. In contrast the SS only has a tion electrons near the band minimum (recall EF is in phason mode, since spin and valley are locked. We an- the renormalized band gap) that participate in excitonic ticipate this will again be weakly gapped, with the gap pairing, the Bloch functions are uniform and identical in increasing for higher interactions. Owing to the broken both valleys, because hσ,λ(k) ∼ τz. Since F' 1, we inversion symmetry in the SS, it also hosts domain walls have D ' J↑↑, hence stabilizing the SS state. On the separating regions with opposite handedness of spin ro- other hand for k0 = 0, there is a cancellation of phases in tation. An out-of-plane magnetic field explicitly breaks the k-sum in Eq.7, suppressing D: the orbital content U(1)s; on symmetry grounds this allows SS to induce a of the valleys winds in a manner that suppresses the 2qc small CDW amplitude, which might be one route to its † CDW even when hb+b−i= 6 0. J generically remains non- detection. Since spectroscopy of collective modes can be zero, so if the Hartree cost for charge order (parametrized a diagnostic of exciton condensation [9, 28–30], investi- 5 gation of the collective excitation spectrum in WTe2 is namic quantities but not magnetoresistance. A more ex- likely to be a fruitful avenue of study. otic explanation invokes a fractionalized Fermi surface of neutral ‘composite exciton’ quasiparticles, whose quantum oscillations can give a weak metallic contribution VII. DISCUSSION to charge transport superimposed on an activated background [37, 38]. Given the typical fragility of fraction- We have proposed that two distinct gapped excitonic alized phases, it seems unlikely to be energetically com- phases can be generated by interactions in monolayer petitive at zero field with the large-gap broken-symmetry WTe2, with one (SS) likely relevant to recent experi- states found here. We therefore conjecture that if such ments [16]. Using an effective model, we have linked a fractionalized phase exists, some yet-unknown mecha- energetic competition between SS and a proximate SDW nism must stabilize it at high fields. Potential alternative phase to the orbital structure of the renormalized bands explanations invoking the field-induced CDW order in SS near the Fermi energy. Since these depend on both the may also be interesting to pursue. Investigating the high- interaction strength and the initial semi-metallic Fermi field phase structure is a subtle and urgent question for surfaces — which can be tuned by adjusting the interac- future work. tion screening length and the electrostatic displacement field respectively — it is possible that the SDW phase can be stabilized experimentally. The SDW and SS may ACKNOWLEDGMENTS be distinguished by their broken symmetries (especially the presence or absence of a 2qc charge modulation, and We thank B.A. Bernevig, Nick Bultinck, L. Fu, their distinct qc-spin orders) and the resulting differences N.P. Ong, Z. Song, and Sanfeng Wu for helpful dis- in their collective excitations. cussions and correspondence. We acknowledge support The unusual oscillations in high field magnetoresis- from the European Research Council under the Euro- tance [17] occur on a scale (∼ 100 MΩ) typical of in- pean Union Horizon 2020 Research and Innovation Pro- sulators, yet their temperature dependence is not ac- gramme, Grant Agreement No. 804213-TMCS (YHK, tivated. The latter fact appears to rule out explana- SAP). Additional support was provided by the Gordon tions centred on modulation of the excitonic gap [31–33]. and Betty Moore Foundation through Grant GBMF8685 Other proposed mechanisms for quantum oscillations towards the Princeton theory program. YHK acknowl- in insulators [34–36] would manifest only in thermody- edges the hospitality of Princeton University.

Appendix A: k · p Model and Hartree-Fock (HF)

The setup here closely follows the supplement of Ref. [16]. In the basis {|d ↑i , |d ↓i , |p ↑i , |p ↓i}, the k·p Hamiltonian of monolayer WTe2 is 1 0 0 0 0 0 0 0 δ 0 1 0 0 k2 δ 0 0 0 0 H = ak2 + bk4 +   + − −   + v k τ s + v k τ s (A1) 0 2 0 0 0 0 2m 2 0 0 1 0 x x x y y y y 0 0 0 0 0 0 0 0 1

a = −3, b = 18, m = 0.03, δ = −0.9, vx = 0.5, vy = 3 (A2) where energies are measured in eV and lengths in A.˚ The symmetries are inversion Pˆ = τz and time-reversal Tˆ = isyKˆ , leading to a two-fold degeneracy of the bands under PˆTˆ. With the above parameters, the bandstructure at charge neutrality consists of a hole pocket at the zone centre, and two electron pockets with minima at qc = ±0.3144ˆx. The undistorted lattice has reciprocal lattice vector lengths Gx = 1.81 and Gy = 1.01. The Fermi energy is EF ' −0.493. Without the SOC term, the bandstructure contains two overtilted Dirac cones at qD = ±0.2469ˆx. The U(1)s symmetric SOC term gaps the Dirac point, leading to an indirect negative band gap. The interaction Hamiltonian is taken to be density-density in spin and orbital space 1 X X H = U(q)c† c† c c (A3) int 2NΩ k+q,α p−q,β p,β k,α k,p,q α,β e2 qξ U(q) = tanh (A4) 20q 2 where N is the total number of unit cells in the system, Ω is the real-space unit cell area, and α, β are combined orbital/spin indices. The interaction potential is of dual-gate screened form, with ξ the gate distance and the relative permittivity of the encapsulating hBN. Experimentally relevant parameters are ' 3.5 and ξ ' 250. Since we are 6

FIG. 3. Self-consistent HF results for the excitonic, spin and charge order parameters as a function of interaction strength −1. Note that ρSDW and ρSS in the middle panel measure the average local spin population imbalance (number per square nanometer) due to the corresponding order.

3qc Gy working with a k·p model, our calculations require a momentum cutoff, which is taken to be |kx| < 2 , |ky| < 4 . The prefactor in Eq A3 is set by the density of momentum points, which would be ABZ /N. However in our calculations ABZ our momentum cutoff has area Akp with Nkp points, so we should replace NΩ → NkpΩ . Akp Anticipating excitonic pairing at ±qc, we perform self-consistent HF calculations allowing for coherence by multiples of q , i.e. hc† c i can take non-zero values for integer n. Representative (folded) band structures are shown in c kα k+nqcβ Figure4. We allow the HF solution to break various symmetries such as time-reversal and inversion. To ensure that we find the lowest energy mean-field solution, we minimize over multiple random seeds for the initial density matrix. The q 1 P † 2 presence of excitonic condensation is diagnosed by the integrated order parameter ∆exc ≡ |hc c i| . Nkp α,β kα k+qcβ The spin-valley nature of the ordering is diagnosed by computing (finite-momentum) charge/spin densities

1 X ρc = hc† c i (A5) Q ˜ k−Qσa kσa A σka

s 1 X † ρ = σ 0 hc c i (A6) Q ˜ σσ k−Qσa kσ0a A σσ0ka

ABZ where a runs over the orbital degree of freedom, and A˜ = NkpΩ . Since there are no ferromagnetic states, the Akp relevant densities for characterising the excitonic order are ρc , ρc , ρs , ρs . In real space, the number densities for qc 2qc qc 2qc spin and charge are

c X iq·r c ρ (r) = e ρQ (A7) Q s X iq·r s ρ (r) = e ρQ. (A8) Q

Self-consistent HF reveals that there are two energetically competitive types of excitonic insulator, the spin spiral (SS) and spin density wave (SDW) which both occur at qc. Consider the magnitude of the local spin density in the presence of spin order which occurs at a single wavevector qc

ρs(r) · ρs(r) = 2ρs · ρs∗ + (ρs · ρs eiqcx + c.c.). (A9) qc qc qc qc Therefore the following quantities are useful to diagnose the presence of SS and SDW order q ρSDW = 2|ρs · ρs | (A10) qc qc q q ρSS = 2ρs · ρs∗ − 2|ρs · ρs |. (A11) qc qc qc qc

A pure SS state has a local spin polarization that rotates with constant magnitude (comparing points with the same intra unit cell coordinates), while a pure SDW state has an oscillating local spin magnitude whose oscillation wavelength is longer than the monolayer unit cell. 7

FIG. 4. Folded HF band structures for diﬀerent interaction strengths—in order of increasing −1, the phases are semimetal, SDW, non-excitonic quantum spin Hall insulator (diagnosed by using the Fu-Kane formula [39]), spin spiral, spin spiral (at −1 0 0 experimentally relevant ), trivial insulator. Labeled momentum points are M = (−qc/2,Gy/4),Y = (0,Gy/4),X = (−qc/2, 0). Given the momentum cutoﬀ of the k · p theory, there are 12 bands per momentum in the folded BZ. All phases except the spin spiral have doubly-degenerate bands due to PˆTˆ symmetry. Calculations were done on a 75 × 25 momentum grid.

Self-consistent Hartree-Fock results for the various order parameters are shown in Figure3. Note that SDW order is accompanied by charge order at 2qc. HF calculations were also performed without allowing for excitonic coherence, in order to obtain the ‘parent’ self- consistent states appropriate for an analytic weak-coupling treatment of excitonic pairing. Representative results are shown in Figure5.

Appendix B: Trial Excitonic Insulator States

For generality, consider the situation with one valence pocket at Γ, and Nλ equivalent conduction valleys at Q(λ). The insulating excitonic trial states considered in the main text are of the following form

Y † X σ |Φi = αkσ |0i , αkσ = ukakσ + vk wsλbksλ, (B1) k,σ sλ

2 2 P σ σ0 where uk, vk are real and even, uk + vk = 1, sλ wsλw¯sλ = δσσ0 , and overbar denotes complex conjugation. uk, vk σ parameterizes the (small)-momentum structure of exciton coherence, while wsλ parameterizes the spin-valley structure of pairing. Note that, for a ﬁxed choice of gauge for the Bloch operators, the choice of orthonormal complex vectors w↑, w↓ in C2Nλ uniquely speciﬁes the trial state without any redundancy—there is no gauge redundancy corresponding to unitary rotation within occupied orbitals. Now specialize to the case of two valleys λ = ±, so that Q(λ) = λqc. The trial states considered by Ref. [2] are a strict subset of Eqn. B1, and can be parameterized by

σ (+) (−) wsλ = lMσs δλ+ + mMσs δλ− (B2) where l2 + m2 = 1, and the M matrices are unitary. This can describe the SDW, but can only describe the spin spiral if l and m are allowed to be σ-dependent. 8

FIG. 5. HF band structures along the kx axis, when the HF is restricted to forbid translation symmetry breaking. Calculations were done on a 75 × 25 momentum grid.

It can be shown that global spin-rotation Uˆ s and valley-rotation Uˆ v act as

ˆ s σ X † σ0 U : wsλ → Uσσ0 ws0λUs0s (B3) σ0s0 ˆ v σ X σ U : wsλ → wsλ0 Uλ0λ (B4) λ0

iθ where U = exp 2 nˆ · σ is a SU(2) unitary. U(1)eh rotations corresponding to separate conservation of conduction σ σ iθ and valence populations act as wsλ → wsλe .

Appendix C: Dominant Term Approximation (DTA) Equations

The DTA equations [2,5] determine the internal momentum structure of excitonic coherence (i.e. the coeﬃcients uk, vk). The starting point is an eﬀective model that describes the band extrema of a self-consistent non-excitonic band structure, which can be semimetallic or insulating. For simplicity consider the ‘two-pocket’ case (one conduction † † minimum bkσ and valence maximum akσ)—the multi-valley case can be treated analogously. In the DTA, only intra- nn;σ pocket interactions are included, and the gauge is chosen smooth so that Fk,k+q ' 1 for small momentum transfer q, leading to the Hamiltonian

X 1 X Hˆ = nd† d + U(q)d† d† d d . (C1) DTA k nkσ nkσ 2 n,k+q,σ n0,k0−q,σ0 n0,k0,σ0 n,k,σ knσ kk0qnn0σσ0

Therefore there is an emergent U(1) symmetry corresponding to separate conservation of conduction and valence band electrons. There is also now global SU(2)s spin rotation symmetry, as well as SU(2)v valley rotation symmetry. We consider an insulating excitonic ansatz |Φ(w)i described by the operator for the ﬁlled bands

X σ X σ σ0∗ αkσ = ukakσ + vk ws bks, ws ws = δσσ0 . (C2) s s

2 2 where uk, vk are real and even, and uk + vk = 1. We now recall that the parameters of the model are extracted from a self-consistent band structure. Therefore when counting the interactions of any distorted state, we need to measure 9 the density relative to the reference self-consistent state Φ0:

X 1 X E [Φ(w)] = const + 2 v2(b − a) − U(k − k0)hd† d i0hd† d i0 (C3) DTA k k k 2 nk0σ n0k0σ0 n0kσ0 nkσ k kk0nn0σσ0 † 0 2 0 hakσakσ0 i = (uk − Nak)δσσ0 (C4) † 0 σ0 hbkσakσ0 i = gkwσ (C5) † 0 2 0 hbkσbkσ0 i = (vk − Nbk)δσσ0 . (C6)

0 0 where gk = ukvk, and Nnk is the ﬁlling of Φ0 (for the insulating parent state in the main text, we have Nak = 1). The direct contributions with q = 0 are canceled by the neutralizing background. Evaluating the interaction term, we obtain the DTA energy

X 2 b a X 0 2 0 2 0 2 0 2 0 EDTA = const + 2 vk(k − k) − U(k − k ) (uk − Nak)(uk0 − Nak0 ) + (vk − Nbk)(vk0 − Nbk0 ) + 2gkgk0 (C7) k kk0 which is independent of w, leading to a huge degeneracy at DTA level. We minimize this energy with respect to vk   2 b a X 2 0 0 1 − 2vp 0 = ∂vp EDTA = 4(p − p)vp − 4 U(k − p) (2vk − 1 + Nak − Nbk)vp + gk  (C8) q 2 k 1 − vp

" # 2 X 2 0 X 2 0 1 − 2vp X → pb − U(k − p) vk − Nbk − pa + U(k − p) uk − Nak vp = U(k − p)gk (C9) q 2 k k 1 − vp k 2 1 − 2vp → 2ξpvp = ∆p (C10) q 2 1 − vp where we have deﬁned

a X 0 2 0 ¯k = ka − U(k − k )(uk0 − Nak0 ) (C11) k0 b X 0 2 0 ¯k = kb − U(k − k )(vk0 − Nbk0 ) (C12) k0 1 ξ = (¯b − ¯a) (C13) k 2 k k X 0 ∆k = U(k − k )gk0 . (C14) k0

The minimization condition can be recast as the coupled integral equations v u ! u1 ξk v = t 1 − (C15) k 2 p 2 2 ξk + ∆k which are solved by iteration. The energy bands of the excitonic state are given by

¯a + ¯b q E = k k − ξ2 + ∆2 (C16) kα 2 k k ¯a + ¯b q E = k k + ξ2 + ∆2 . (C17) kβ 2 k k

b If we have two valleys, we will have an additional energy band Ekγ = k which remains unaltered. In this case it is possible that the excitonic state remains semimetallic if the parent state is semimetallic. 10

Appendix D: Beyond Dominant Term Approximation (bDTA) Splitting Terms

While the DTA equations determine uk, vk, the choice of w can only be resolved by considering the neglected inter-pocket interactions [2]. Assuming a good DTA/bDTA separation of scales, we can use ﬁrst-order perturbation theory to evaluate the neglected terms of hΦ(w)|Hˆ |Φ(w)i. In the two-valley case, we obtain

X σ σ¯ σ0 σ¯0 σ0 σ δE[w] = Bσσ0 (wσ+ +w ¯σ¯−)(w ¯σ0+ + wσ¯0−) − 2Re Cσσ0 wσ+wσ0− (D1) σσ0 X σ σ σ0 σ0 σ σ σ0 σ0 + Dws+w¯s−w¯s0+ws0− − Jss0 ws+w¯s0+ws0−w¯s− (D2) σσ0ss0

X ab;σ∗ X ab;σ0 B 0 = U(q ) g F g 0 F 0 0 (D3) σσ c k k,k+qc k k ,k +qc k k0 X 0 ab;σ∗ ba;σ0 C 0 = g g 0 U(k − k + q )F 0 F 0 (D4) σσ k k c k ,k+qc k −qc,k kk0 X D = U(2q )| v2F bb;σ |2 (D5) c k k−qc,k+qc k

1 X 2 2 0 bb;s∗ bb;s0 Jss0 = v v 0 U(k − k + 2qc)[F 0 F 0 + c.c.]. (D6) 2 k k k −qc,k+qc k −qc,k+qc kk0 where B is Hermitian, J, C are symmetric, and the spin-quantization axis is chosen along the preserved direction (SOC is U(1)s preserving). The F refer to the form factors of the eﬀective model, and the momentum labels are bb;σ absolute momenta measured from the zone center. For example, F 0 is an intervalley form factor because k−qc,k +qc the k, k0 always represent ‘small’ momenta. The B-term and D-term are Hartree terms that penalize charge density wave modulations at wavevector qc and 2qc respectively. The C-term and J-term are exchange terms at momentum transfer q ∼ qc and 2qc respectively. In the limit of vanishing SOC, and specializing to a restricted class of states (that includes the SDW but not the spin spiral), we recover the bDTA expression of Ref. [2]. Using the transformations in Eqns B3,B4, it can be shown that δE[w] is invariant under U(1)s and U(1)v symmetries. U(1)s is present because the starting model was already assumed to have this symmetry. U(1)v can be seen by investigating the possible inter-pocket interaction terms which conserve momentum. This symmetry ceases to be sensible once excitonic coherence remains strong out to momenta k ∼ qc/2 in the folded BZ, since then the division of the relevant low-energy Bloch states into small ‘pockets’ fails, and the weak-coupling perspective is no longer useful. There is no U(1)eh symmetry corresponding to separate conservation of valence/conduction electrons, because bDTA † † contains interaction terms ∼ a a b+b−. These U(1)eh-violating terms are reﬂected in the B- and C-terms of the bDTA energy functional.

Appendix E: Spiral/SDW Competition for Two Excitons

In this section we argue that the spin spiral vs SDW competition outlined in the main text is invisible to a single Q † exciton, and is a selection mechanism at the many-exciton level. Let |Φ0i = kσ akσ |vaci be the parent insulating state of the eﬀective model. An exciton creation operator (with net momentum q = 0 in the folded BZ) can be parameterized as a linear combination of single particle-hole operators:

† X ∗ † Bσ(w) = fkwsλbksλakσ (E1) ksλ

P 2 where fk, satisfying k fk = 1, is real and even, and parameterizes the exciton momentum structure (predominantly determined by q ∼ 0 interactions), while w indicates the valley/spin structure of the electron. We focus on the q ∼ 2qc components of the interaction Hamiltonian, since these were found to mediate the SDW/spiral competition. Consider a single exciton state

† |σwi = Bσ(w) |Φ0i . (E2)

† † This vanishes under the action of q ∼ 2qc interaction terms, since b b bb will always annihilate the above state. Hence a single exciton is not sensitive to the competition described in the main text. 11

Now we consider the interaction energy of two-exciton states. For simplicity we assume the quasi-boson approximation and consider the following two-exciton states

0 0 † † 0 |σw; σ w i = Bσ(w)Bσ0 (w ) |0i (E3) where we neglect the normalization. We will be interested in cases where the w, w0 describe excitons with spin/valley structures corresponding to spiral or SDW phases. Focusing on the q ∼ 2qc contributions again, we obtain after some algebra 0 0 2 X 2 ↑ 2 X 0 0 hσw; σw |Hˆ |σw; σw i = U(2q )| f F | (w w¯ w¯ 0 w 0 + c.c.) (E4) q∼2qc N c k k,k s+ s− s + s − k ss0 1 X 2 2 0 s∗ s0 0 0 − f f 0 U(k − k + 2q )(F 0 F 0 w w¯ 0 w 0 w¯ + c.c.) . (E5) 2 k k c k k k k s+ s + s − s− kk0ss0 For w, w0 corresponding to the spin spiral (Eqn 10 in main text), the above contributions vanish as expected since 0 there is no 2qc coherence. For w, w corresponding to the SDW (Eqn 9 in main text), we recover the competition between the direct and exchange terms, which take analogous forms to the D- and J-terms in the bDTA. We note that these calculations (involving q ∼ 0 and q ∼ qc terms as well) can be generalized to derive the interaction terms of an eﬀective quasi-boson Hamiltonian.

Appendix F: Elliptical Spin Spirals and Charge Order

In this section we consider the more general class of elliptical spin spirals, which encompasses the limiting cases of SDW and (circular) spin spiral discussed in the main text. The spin/valley structure of these states can be parameterized using the language of Eq B1 π w↑ = e−i(α+φ) sin(χ − ) (F1) ↓+ 4 π w↑ = ei(α−φ) cos(χ − ) (F2) ↓− 4 π w↓ = ei(−α+φ) cos(χ − ) (F3) ↑+ 4 π w↓ = ei(α+φ) sin(χ − ). (F4) ↑− 4 The spin spiral is recovered for χ = ±π/4 (corresponding to the two senses of rotation), while χ = 0 corresponds to the SDW. The spin and charge densities for these states are   sin χ cos φ cos qcx + α + cos χ sin φ sin qcx + α s ρ (r) ∼  0  (F5) sin χ sin φ cos qcx + α + cos χ cos φ sin qcx + α c ρ (r) ∼ cos 2χ cos 2(qcx + α) . (F6) Hence the principal axes of the elliptical spiral are controlled by φ and lie along [cos φ, 0, sin φ] and [− sin φ, 0, cos φ], while α controls the position along x. χ is related to the ellipticity of the spin order, and also controls the strength 2 ∼ cos 2χ of the associated 2qc charge density wave. To further understand the relation between the ellipticity of the spiral and the charge order, we can analyze their y coupling within Landau theory [26]. With the constraints given by TRS, U(1)s about s , and translation (there are no Umklapp processes since qc is not at a high-symmetry point), the lowest order coupling between charge density ρc and x − z spin density ρs,⊥ is Z c s,⊥ 2 X c s,⊥ s,⊥ F ∼ dxρ (x) ρ (x) ∼ ρ−p−p0 ρp · ρp0 (F7) p,p0 where we have used the fact that there is no order along the y-direction. Since the spin order has non-trivial 0 contributions for momenta p = ±qc, we focus on the case p = p which couples to the 2qc charge order. With appropriate choice of coordinate and spin axes, the spin order parameter of the elliptical spiral can be chosen as ρs,⊥(x) ∼ [cos χ sin(q x), sin χ cos(q x)], with Fourier components ρs,⊥ ∼ [±i cos χ, sin χ], leading to ρs,⊥ · ρs,⊥ ∼ c c ±qc ±qc ±qc 12 cos 2χ. Hence the coupling between 2qc charge order and qc spin order contains a multiplicative factor of cos 2χ. 2 This vanishes for the circular spiral, which can be intuited from the fact that ρs,⊥(x) is spatially uniform. This s,⊥ 2 argument holds for higher order terms in Landau theory, since in-plane spin must enter as ρ (x) due to U(1)s. 2 J In the bDTA, the energy of the elliptical spiral is EbDTA = −2Re C↑↓ +cos 2χ D − 2 . Hence the energetics mean J that we have χ → ±π/4, 0 depending on whether D − 2 is positive or negative. U(1)s symmetry breaking and CDW modulation in a circular spiral phase: Note that when U(1)s symmetry in the xz spin plane is broken (e.g., by a magnetic ﬁeld perpendicular to the monolayer), the reduction in symmetry admits additional terms such as (ρs,xρs,x − ρs,zρs,z)ρc, allowing even a circular spiral to generate a CDW modulation. 13

[1] B. Halperin and T. Rice, Possible anomalies at a [17] P. Wang, G. Yu, Y. Jia, M. Onyszczak, F. A. Cevallos, semimetal-semiconductor transistion, Rev. Mod. Phys. S. Lei, S. Klemenz, K. Watanabe, T. Taniguchi, R. J. 40, 755 (1968). Cava, and et al., Landau quantization and highly mobile [2] B. Halperin and T. Rice, The excitonic state at fermions in an insulator, Nature 589, 225–229 (2021). the semiconductor-semimetal transition, in Solid State [18] S. Tang, C. Zhang, D. Wong, Z. Pedramrazi, H.-Z. Tsai, Physics, Vol. 21 (Elsevier, 1968) pp. 115–192. C. Jia, B. Moritz, M. Claassen, H. Ryu, S. Kahn, et al., 0 [3] D. Jérome,T. M. Rice, and W. Kohn, Excitonic insula- Quantum spin Hall state in monolayer 1T -WTe2, Nature tor, Phys. Rev. 158, 462 (1967). Physics 13, 683 (2017). [4] W. Kohn, Excitonic phases, Phys. Rev. Lett. 19, 439 [19] Z. Fei, T. Palomaki, S. Wu, W. Zhao, X. Cai, B. Sun, (1967). P. Nguyen, J. Finney, X. Xu, and D. H. Cobden, Edge [5] J. D. Cloizeaux, Exciton instability and crystallographic conduction in monolayer WTe2, Nature Physics 13, 677 anomalies in semiconductors, Journal of Physics and (2017). Chemistry of Solids 26, 259 (1965). [20] S. Wu, V. Fatemi, Q. D. Gibson, K. Watanabe, [6] J. Zittartz, Theory of the excitonic insulator in the pres- T. Taniguchi, R. J. Cava, and P. Jarillo-Herrero, Obser- ence of normal impurities, Phys. Rev. 164, 575 (1967). vation of the quantum spin Hall effect up to 100 kelvin [7] J. Kuneˇs,Excitonic condensation in systems of strongly in a monolayer crystal, Science 359, 76 (2018). correlated electrons, Journal of Physics: Condensed Mat- [21] L. Muechler, A. Alexandradinata, T. Neupert, and ter 27, 333201 (2015). R. Car, Topological nonsymmorphic metals from band [8] J. Eisenstein, Exciton condensation in bilayer quan- inversion, Phys. Rev. X 6, 041069 (2016). tum hall systems, Annual Review of Condensed Matter [22] F.-C. Wu, F. Xue, and A. H. MacDonald, Theory of two- Physics 5, 159 (2014). dimensional spatially indirect equilibrium exciton con- [9] A. Kogar, M. S. Rak, S. Vig, A. A. Husain, F. Flicker, densates, Phys. Rev. B 92, 165121 (2015). Y. I. Joe, L. Venema, G. J. MacDougall, T. C. Chiang, [23] S. S. Ataei, D. Varsano, E. Molinari, and M. Rontani, E. Fradkin, J. van Wezel, and P. Abbamonte, Signatures Evidence of ideal excitonic insulator in bulk MoS2 under of exciton condensation in a transition metal dichalco- pressure (2020), arXiv:2011.02380 [cond-mat.str-el]. genide, Science 358, 1314 (2017). [24] X. Qian, J. Liu, L. Fu, and J. Li, Quantum spin Hall ef- [10] L. Du, X. Li, W. Lou, G. Sullivan, K. Chang, J. Kono, fect in two-dimensional transition metal dichalcogenides, and R.-R. Du, Evidence for a topological excitonic insu- Science 346, 1344 (2014). lator in InAs/GaSb bilayers, Nature Communications 8, [25] Z. Song and B. Bernevig, private communication. 1971 (2017). [26] O. Zachar, S. A. Kivelson, and V. J. Emery, Landau the- [11] H. Cercellier, C. Monney, F. Clerc, C. Battaglia, L. De- ory of stripe phases in cuprates and nickelates, Phys. Rev. spont, M. G. Garnier, H. Beck, P. Aebi, L. Patthey, B 57, 1422 (1998). H. Berger, and L. Forró,Evidence for an excitonic in- [27] J. Nasu, T. Watanabe, M. Naka, and S. Ishihara, Phase sulator phase in 1T -TiSe2, Phys. Rev. Lett. 99, 146403 diagram and collective excitations in an excitonic insula- (2007). tor from an orbital physics viewpoint, Phys. Rev. B 93, [12] Y. Wakisaka, T. Sudayama, K. Takubo, T. Mizokawa, 205136 (2016). M. Arita, H. Namatame, M. Taniguchi, N. Katayama, [28] B. Remez and N. R. Cooper, Effects of disorder on the M. Nohara, and H. Takagi, Excitonic insulator state in transport of collective modes in an excitonic condensate, Ta2NiSe5 probed by photoemission spectroscopy, Phys. Phys. Rev. B 101, 235129 (2020). Rev. Lett. 103, 026402 (2009). [29] Y. Murakami, D. Goleˇz,T. Kaneko, A. Koga, A. J. Millis, [13] K. Seki, Y. Wakisaka, T. Kaneko, T. Toriyama, T. Kon- and P. Werner, Collective modes in excitonic insulators: ishi, T. Sudayama, N. L. Saini, M. Arita, H. Namatame, Effects of electron-phonon coupling and signatures in the M. Taniguchi, N. Katayama, M. Nohara, H. Takagi, optical response, Phys. Rev. B 101, 195118 (2020). T. Mizokawa, and Y. Ohta, Excitonic Bose-Einstein con- [30] D. Goleˇz,Z. Sun, Y. Murakami, A. Georges, and A. J. densation in Ta2NiSe5 above room temperature, Phys. Millis, Nonlinear spectroscopy of collective modes in an Rev. B 90, 155116 (2014). excitonic insulator, Phys. Rev. Lett. 125, 257601 (2020). [14] Y. Lu, H. Kono, T. Larkin, A. Rost, T. Takayama, [31] L. Zhang, X.-Y. Song, and F. Wang, Quantum oscillation A. Boris, B. Keimer, and H. Takagi, Zero-gap semicon- in narrow-gap topological insulators, Phys. Rev. Lett. ductor to excitonic insulator transition in Ta2NiSe5, Na- 116, 046404 (2016). ture Communications 8, 1 (2017). [32] H. K. Pal, F. Piéchon, J.-N. Fuchs, M. Goerbig, and [15] K. Fukutani, R. Stania, J. Jung, E. F. Schwier, K. Shi- G. Montambaux, Chemical potential asymmetry and mada, C. I. Kwon, J. S. Kim, and H. W. Yeom, Elec- quantum oscillations in insulators, Phys. Rev. B 94, trical tuning of the excitonic insulator ground state of 125140 (2016). Ta2NiSe5, Phys. Rev. Lett. 123, 206401 (2019). [33] P. A. Lee, Quantum oscillations in the activated con- [16] Y. Jia, P. Wang, C.-L. Chiu, Z. Song, G. Yu, B. Jäck, ductivity in excitonic insulators: Possible application to S. Lei, S. Klemenz, F. A. Cevallos, M. Onyszczak, monolayer wte2, Phys. Rev. B 103, L041101 (2021). N. Fishchenko, X. Liu, G. Farahi, F. Xie, Y. Xu, [34] J. Knolle and N. R. Cooper, Quantum Oscillations with- K. Watanabe, T. Taniguchi, B. A. Bernevig, R. J. Cava, out a Fermi Surface and the Anomalous de Haas–van L. M. Schoop, A. Yazdani, and S. Wu, Evidence for a Alphen Effect, Phys. Rev. Lett. 115, 146401 (2015). monolayer excitonic insulator (2020), arXiv:2010.05390 [35] J. Knolle and N. R. Cooper, Excitons in topological [cond-mat.mes-hall]. Kondo insulators: Theory of thermodynamic and trans- 14

port anomalies in SmB6, Phys. Rev. Lett. 118, 096604 valence insulators with neutral Fermi surfaces, Nature (2017). Communications 9, 1766 (2018). [36] S. Grubinskas and L. Fritz, Modiﬁcation of the Lifshitz- [38] I. Sodemann, D. Chowdhury, and T. Senthil, Quantum Kosevich formula for anomalous de Haas–van Alphen os- oscillations in insulators with neutral Fermi surfaces, cillations in inverted insulators, Phys. Rev. B 97, 115202 Phys. Rev. B 97, 045152 (2018). (2018). [39] L. Fu and C. L. Kane, Topological insulators with inver- [37] D. Chowdhury, I. Sodemann, and T. Senthil, Mixed- sion symmetry, Phys. Rev. B 76, 045302 (2007).