(2021) Exciton Condensation in Bilayer Spin-Orbit Insulator

PHYSICAL REVIEW RESEARCH 3, 013224 (2021)

Exciton condensation in bilayer spin-orbit insulator

Hidemaro Suwa ,1,2 Shang-Shun Zhang,2 and Cristian D. Batista 2,3 1Department of Physics, University of Tokyo, Tokyo 113-0033, Japan 2Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA 3Quantum Condensed Matter Division and Shull-Wollan Center, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA

(Received 5 October 2020; revised 26 January 2021; accepted 11 February 2021; published 9 March 2021)

We investigate the nature of the magnetic excitations of a bilayer single-orbital Hubbard model in the intermediate-coupling regime. This model exhibits a quantum phase transition (QPT) between a paramagnetic (PM) and an insulating antiferromagnetic (AFM) phase at a critical value of the coupling strength. By using the random phase approximation, we show that the QPT is continuous when the PM state is a band insulator and that the corresponding quantum critical point (QCP) arises from the condensation of preformed excitons. These low-energy excitons re-emerge on the other side of the QCP as the transverse and longitudinal modes of

the AFM state. In particular, the longitudinal mode remains sharp for the model parameters relevant to Sr3Ir2O7 because of the strong easy-axis anisotropy of this material.

DOI: 10.1103/PhysRevResearch.3.013224

I. INTRODUCTION on the spin orbit coupling. Similarly to the large U/|t| limit, where the transition from the QPM to the AFM state can be Low-dimensional Mott insulators play a central role in cor- described as a triplon condensation, the QPT from the band related electron physics because of the novel states of matter insulator to the AFM state corresponds to condensation of and phase transitions that they can host. In the strong-coupling preformed magnetic excitons. These bound states reemerge on limit, U/|t|1, the on-site repulsion U “freezes out” the the other side of the QCP as longitudinal and transverse modes charge degrees of freedom, turning the Mott insulator into a (magnons) of the AFM state. In contrast, the longitudinal quantum magnet described by an effective spin Hamiltonian. mode (LM) is absent in the AFM phase induced by a metal- The lattice connectivity can then be exploited to generate insulator transition with perfect Fermi surface (FS) nesting. competing exchange interactions that induce quantum phase In this article, we study the possible QPTs induced by transitions (QPTs) between different states of matter. One of reducing the U/|t| ratio in bilayer materials. Although the the simplest examples is provided by bilayer materials [1–6], transition into the band insulator has some similarities with where the competition between intralayer and interlayer hop- the transition into the QPM of pure spin systems (U/|t|1) ping amplitudes, t and t , can induce a transition between an z [7–16], there are also some important differences associated AFM state and a quantum paramagnet (QPM) comprising lo- with the enhanced charge fluctuations. By applying our anal- cal singlet states on the interlayer dimers. This QPT, which is ysis to the easy-axis bilayer antiferromagnet Sr Ir O , with typically induced by applying pressure, has been extensively 3 2 7 = 130 meV [23] and W = 70 meV [24], we reveal the studied in multiple quantum magnets to understand different existence of a LM in some regions of the Brillouin zone, properties of the QCP, such as the emergence and decay of the which arises from the band-insulating character of the non- longitudinal mode (LM) that is present in the antiferromag- interacting limit of the model: The Néel phase is induced netic (AFM) state [7–16]. by condensation of preformed Sz = 0 excitons at a critical The discovery of low-dimensional intermediate-coupling coupling strength U = U . This exciton reemerges in the Néel 4d- and 5d-electron correlated insulators introduces a knob, c phase (U > U ) as a LM, whose energy scale and dispersion U/t, that can be used to unleash the charge degrees of c are consistent with previous resonant inelastic x-ray scattering freedom via pressure or strain. For instance, the iridate (RIXS) measurements [24–27]. It is important to note that the materials [17–22] have a charge gap that is comparable to LM that emerges near the same QCP in the strong coupling the magnon bandwidth W . The reduction of U/t leads to the limit of the Hubbard model [25]existsover the whole Bril- suppression of the AFM ordering in favor of a paramagnetic louin zone and its energy is much lower than the charge gap state. In bilayer materials, such as Sr Ir O , the paramagnetic 3 2 7 because it is obtained from a pure spin model. In contrast, state can either be metallic or a band insulator depending the LM that we are proposing for Sr3Ir2O7 only exists in finite regions of momentum space, around the wave vectors q = 0 and q = (π,π), because it is induced by strong charge fluctuations (U ∼ Uc). Indeed, the mode disappears inside the Published by the American Physical Society under the terms of the particle-hole continuum for wave vectors that are far enough Creative Commons Attribution 4.0 International license. Further from q = 0 and q = (π,π). distribution of this work must maintain attribution to the author(s) Our results then suggest that Sr3Ir2O7 is a realization of and the published article’s title, journal citation, and DOI. the long-sought excitonic insulator that was predicted almost

2643-1564/2021/3(1)/013224(11) 013224-1 Published by the American Physical Society SUWA, ZHANG, AND BATISTA PHYSICAL REVIEW RESEARCH 3, 013224 (2021)

T 60 years ago [28–30]. However, higher resolution RIXS ex- ck ≡ (cA↑,k, cA↓,k, cB↑,k, cB↓,k ) and periments are needed to confirm this prediction. k + k k − k (1) =−2t cos 1 2 + cos 1 2 . (5) k 2 2 II. MODEL HMF is diagonalized by a unitary 4 × 4matrixU (k), We consider a bilayer single-orbital Hubbard model H = k −H + H , with H = U n ↑n ↓ and K U U r r r = ψ , cγσ,k U(γσ),n(k) n,k (6) α † † i σ n H = + + 2 r z , + . ., K t cr cr aν tz c(r⊥,1)e c(r⊥ 2) H c (1) r,ν r⊥ where n ≡ (s,σ) with s =±, σ =↑, ↓ and each column of U (k) is an eigenvector of HMF: , ,α, ∈ † ≡ † , † k where t tz U R, cr [c↑,r c↓,r], is the Nambu spinor of ⎛ ⎞ ⎛ ⎞ ≡ , = , the electron field [r (r⊥ l), l 1 2 is the layer index, and 1+sz↑k 0 ↑ = + ν = , ⎜x k 2 ⎟ ⎜ ⎟ r⊥ r1a1 r2a2], and aν ( 1 2) are the primitive vectors 1−sz↓k ⎜ ⎟ ⎜ ↓ ⎟ ⎜ 0 ⎟ ⎜x k 2 ⎟ of the square lattice of each layer. The sign r takes the values ↑ = , ↓ = , Xs (k) ⎜ − ⎟ Xs (k) ⎜ ⎟ (7) 1(−1) for r ∈ A (B) sublattice of the bipartite bilayer system. ⎝ s 1 sz↑k ⎠ ⎝ 0 ⎠ 2 + This Hamiltonian is an effective model for a bilayer system 1 sz↓k 0 s 2 with finite SOC, which is realized in ruthenates and iridates. For example, the large SOC of the bilayer iridates splits the with 5dt orbitals of the Ir4+ ion into J = 1/2 and 3/2 multiplets 2g bσ k δ [31]. Consequently, H becomes a low-energy model for the xσ k = , zσ k = , |bσ | δ2 +| |2 5d hole (5d5 electronic configuration) of the strontium iridates k bσ k (1) −iσ α after projecting the relevant multiorbital Hubbard model onto bσ k = − tz cos(kz )e 2 , (8) the lowest energy J = 1/2 doublet. The phase α arises from k δ = ε = hopping matrix elements between dxz and dyz orbitals allowed and UM. The corresponding eigenenergies, sσ (k) δ2 +| |2, | |2 = 2 by staggered octahedral rotations. The phase of the intralayer s bσ k are independent of the spin flavor, bσ k bk hopping is gauged away by applying a sublattice-dependent and zσ k ≡ zk, because of the U(1) invariance of H under gauge transformation. global spin rotations about the z axis. The noninteracting system is then metallic when the band gap at each k point, = δ2 + 2 = | | = III. MEAN FIELD APPROXIMATION k 2 bk 2 bk , closes on a nodal line bk 0of For α = 0, the model (1) has an easy z-axis spin anisotropy band crossing points that coincides with the FS [see Fig. 1(a)]. and the ground state of the AFM phase can have Néel Because we are considering a bilayer system, the half-filled μ μ γr condition corresponds to an integer number of electrons per ordering, Sr =(−1) Mδμz, where γr = (1 + r)/2, Sr = 1/2c†σ μc (μ = x, y, z), and M is the magnetization. A mean unit cell, implying that the ground state of the noninteracting r r = field decoupling of H leads to (U 0) system can either be a metal or a band insulator, U as illustrated in Fig. 1(a).Forα = 0, the metal to insulator γ 1 transition occurs via a semimetallic state with small and iden- HMF =−UM (−1) r c†σ c + Un c†c + C, (2) U r z r 2 r r tical electron and hole pockets that shrink into a quadratic r r Fermi point. For instance, for tz = 0, the FS is the square + =±π − =±π where C = UM2N − 1Un2N , N is the number of lattice defined by the equations k1 k2 and k1 k2 . s 4 s s | | α = sites, n = N −1 c†c is the electron density, and the sec- Upon increasing tz , while keeping 0, the square shrinks s r r r | | > | | ond term can be absorbed into the chemical potential. into a circular pocket that finally disappears for tz 4 t .In contrast, the noninteracting system is always a band insulator By Fourier transforming annihilation and creation opera- α = tors, for 0. The on-site repulsion U induces AFM ordering via a QPT 1 i(k⊥·r⊥+k l) that depends on the nature of the noninteracting state. In both cσ, = √ e z cγσ, , r ∈ sublattice γ, (3) r N k cases, noninteracting metal and insulator, the longitudinal u ∈ k BZ AFM susceptibility (q = 0)atω = 0 is given by 1 N = N k ≡ k⊥, k where u 2 s is the number of unit cells, and ( z ) γ , γ , γ +γ 1 1 χ ( 1 z);( 2 z) = (−1) 1 2 , (9) runs over the first Brillouin zone (BZ): k⊥ = k1b + k2b , 0 1 2 2Nu |bk| = / , − / = / , / , ∈ , π k∈BZ with b1 (1 2 1 2), b2 (1 2 1 2), k1 k2 [0 2 ), and k = 0,π, we obtain the momentum space representation of z where γ1,2 = A, B are sublattice indices. Note that the antiferromagnetic susceptibility can diverge at q = 0 because HMF = †HMF , ck k ck (4) the nonmagnetic unit cell contains one site of each sublat- k∈BZ tice. From the mean field calculation, the critical interaction with strength is given by (1) − α σ σ − i 2 z −1 1 1 HMF = UM z k tz cos(kz )e , U = . (10) k (1) α σ c N | | − i 2 z − σ 2 u bk k tz cos(kz )e UM z k∈BZ

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U = Uc, because the integral (10) is now convergent. This phase diagram clearly shows the significance of the spin-orbit coupling: The proximity to the critical point is caused by a finite α even for small tz/t. The physical interpretation of the difference between the metallic and the band-insulating cases will become clearer upon analyzing the behavior of the magnetic excitations. The order parameter is determined by solving the self- consistent mean field equation 1 M = zk. (11) 2Nu k

−1/(ρ0U ) Near the critical point U = Uc, we obtain M ∝ (t/U )e for the metal, where ρ0 the density of states at the Fermi = − / 4 level, and M 2 (U Uc) (DUc ) for the band insulator 1 −3 with D = ∈ |bk| . Nu k BZ

IV. EXCITON CONDENSATION In view of the U (1) invariance of H and the AFM ground state, the transverse and longitudinal spin fluctuations are decoupled from each other. Within the random phase approximation (RPA), the magnetic susceptibilities of the transverse and the longitudinal modes are given by 1 FIG. 1. (a) Schematic picture for the band structure of noninter- χ +−(q, iω ) = χ +−(q, iω ), (12) n τ 0 − χ +− , ω 0 n acting metal and band insulator. (b) Phase boundaries in the mean U 0 (q i n ) field approximation for α = 0, 10−4, 10−3, 10−2, 10−1, 0.5, and 1.45 1 χ zz(q, iω ) = χ zz(q, iω ), (13) (from top to bottom). For α = 0andU = 0, the system is a metal n τ 0 − U χ zz , ω 0 n 2 0 (q i n ) for |tz/t| < 4 and a band insulator for |tz/t| > 4. The metal be- α = comes an AFM insulator for an infinitesimally small U.For 0, respectively, where τ0 is the 2 × 2 identity matrix. Here the present model (1) exhibits a continuous phase transition from the χ +− and χ zz refer to 2 × 2 matrices in the sublattice space, band insulator to the AFM insulator at a finite U/t. The solid circle χ +− χ zz while 0 and 0 are the bare magnetic susceptibilities. See symbol indicates the estimated parameter set for the bilayer iridate Appendix A for details of the RPA calculation. Sr3Ir2O7 with α = 1.45. (c) Energy diagram as a function of U for The eigenfrequencies ωq of the collective transverse modes the parameter set relevant to Sr3Ir2O7. The red dashed line indicates (magnons) can be extracted from the poles of the transverse Sz =± +− the energy of the 1 exciton that becomes the transverse mode RPA susceptibility: det[τ 0 − Uχ (q,ω )] = 0. The spec- for U > U . The blue dotted line indicates the energy of the Sz = 0 0 q c trum is fully gapped because the U(1) symmetry of H is not exciton that becomes a longitudinal mode for U > U . The gray area c spontaneously broken by the AFM ordering along the z axis. indicates the particle-hole continuum. From a comparison with RIXS = . As shown in Fig. 1(c), the transverse modes remain gapped data [24–27], we estimate that U 0 33 eV for Sr3Ir2O7. z at U = Uc and they become the S =±1 exciton modes of nonmagnetic band insulator for U < Uc. Note that this gap The resultant phase boundaries for several values of α are closes in the absence of spin-orbit coupling (α = 0) because shown in Fig. 1(b). the Hamiltonian becomes SU(2) invariant. α = = | / | < For 0 and U 0, the system is metallic for tz t 4. The most interesting feature is the emergence of a LM In this case, the perfect nesting due to the coincidence of below the particle-hole continuum around the point of the the particle and hole Fermi surfaces leads to a logarithmic second Brillouin zone [q = (π,π,π)]. The origin of this γ , γ , χ ( 1 z);( 2 z) ,ω = = divergence of 0 (q 0) at q 0. Note that the mode can be understood by analyzing the excitation spec- 2 divergence becomes ln q for a square FS (tz = 0) because of trum of the band insulator for U < Uc. The bare magnetic the Van Hove singularity in the density of states at the Fermi susceptibility at q = 0, which is equivalent to (π,π,π), is χ zz ,ω = ω τ 0 − τ x τ x level (the corners of the square correspond to saddle points of 0 (0 ) ( )( ), where is the Pauli matrix and the dispersion relation). In other words, the metal becomes 1 1 1 an AFM insulator for an infinitesimally small value of U, (ω) = . (14) N | | ω 2 implying that the critical interaction strength Uc is equal to 2 u bk − k∈BZ 1 2b zero. In contrast, the system becomes a band insulator (finite k charge gap) for α = 0 because the sublattice symmetry is no The function (ω) is real, and it increases monotonically longer present. As shown in Fig. 1(b), the metal-insulator with ω up to the lower edge of the particle-hole continuum transition at U = 0 is then replaced by a continuous QPT ωph = 2min(bk ), where it diverges: (ω = 0) = 1/Uc and ω =∞ between the band and the AFM insulators at a finite U value, limω→ωph ( ) . The putative pole of the longitudinal

013224-3 SUWA, ZHANG, AND BATISTA PHYSICAL REVIEW RESEARCH 3, 013224 (2021) susceptibility is determined by the condition of the RIXS data [25], based on a pure spin model (i.e., ω = / , strong-coupling limit), that reports the existence of a LM ( ) 1 U (15) over the whole Brillouin zone. The effective Hamiltonian (1) which implies that a pole must exist in the energy window for Sr3Ir2O7 can be obtained by constructing a tight-binding = / 0 ω ωph for 0 < U < Uc. As long as Uc is finite, i.e., the model of the t2g orbitals and projecting it onto Jeff 1 2 noninteracting system is a band insulator, the pole appears just lowest energy doublet [32]. The resulting parameters of the below the gap = ωph that signals the onset of the particle- effective single-orbital Hubbard model can be optimized to hole continuum. This pole corresponds to the formation of an reproduce the experimentally observed magnon dispersion z α = . = . , = . , = . S = 0 exciton. Upon examining the transverse susceptibility, [24–27]: 1 45 and t 0 11 tz 0 09 U 0 33 in units we also find a doubly degenerate pole associated with the of eV. This value of α ≈ π/2 produces a strong easy-axis formation of Sz =±1 excitons [see Fig. 1(c)], whose energy anisotropy, realized in a tetragonal elongation of octahedra z | | > | | is higher than the energy of the S = 0 exciton because of the consistent with t tz , as estimated for Sr3Ir2O7 [32]. easy-axis anisotropy. We note that the three excitonic states The dynamical spin structure factor become degenerate in the absence of spin-orbit interaction ∞ because H is SU(2) invariant in that limit. The binding en- μν ,ω = itω μ ν , S (q ) dte Sq (t )S−q(0) (17) ergy exhibits the singular behavior Eb ∝ ωph exp(−κωph/U ), ∞ κ characteristic of 2D systems in the weak-coupling limit ( is a μ 1 μ iq·r z where S = √ S e , is obtained from the dynamical nonuniversal number). As shown in Fig. 1(c) ,theS = 0 exci- q N r r ton becomes soft at U = U and ω 2 2(U − U )/(DU 2 ) spin susceptibility given in Eqs. (12) and (13). Figure 2 c L c c = π for U Uc. The condensation of this mode signals the onset shows the out-of-phase (qz ) transverse (OT) response, xx yy of the Néel phase with magnetic moments pointing along the S (q,ω) = S (q,ω), and the out-of-phase longitudinal (OL) zz ,ω = . ≈ . z axis due to the effective easy-axis anisotropy. response, S (q ), for U 0 33, Uc 0 28, and 0.23 eV. = In the AFM phase (U > Uc), the bare magnetic suscepti- The in-phase, or qz 0, transverse (IT) response is not shown bility at q = 0 is given by because it is practically identical to the OT response [24]. This is a direct consequence of the strong easy-axis effec- 1 1 1 − z2 (ω) = z k . (16) tive interlayer exchange that suppresses the single magnon k ω 2 2UM Nu − 2 tunneling between the two layers. In the large-U limit, k∈BZ 1 z 2UM k the effective interlayer exchange includes only Ising and ω / ω ω ω = Once again, ( ) is real, and d d 0for ph Dzyaloshinskii-Moriya exchange interactions that do not split 2 2 2 (UM) + min(bk ) . If the noninteracting state is a band the two degenerate single-layer modes. This anisotropy is insulator (finite Uc), it holds that bk > 0 for any k,imply- also responsible for the large spin gap revealed by RIXS ing that a pole must then exist in the window 0 ω ωph measurements [24]. No excitation peak is found below the because (ω = 0) < 1/U and (ωph ) > 1/U.ForU Uc, particle-hole continuum in the in-phase longitudinal response. π π the exciton energy scales as ω = 4 (U − U )/(DU 2 ). Upon To match the RIXS peak at ( , ), we added the intralayer L c c 2 2 ≈ / further increasing U, ωL increases quickly and approaches next nearest neighbor hopping 0.012 eV t 10, which only the lower edge of the particle-hole continuum asymptotically. changes the eigenenergy in the above argument. The overall This behavior can be understood by considering the large-U dispersion curve for U = 0.33 eV, shown in Figs. 2(a) and limit. In this limit, the effective particle-hole interaction that 2(b), is consistent with the RIXS measurements [24–27]. The provides the “glue” for the formation of the LM is the ex- resultant charge gap (≈130 meV) is also consistent with the 2/ experimental observation [23]. The structure of the continuum change interaction, on the order of tz U, along the vertical bonds, implying that the binding energy of the LM must reflects the underlying electron bands. Unfortunately, the res- vanish for U →∞. The particle and the hole break only olution of the reported RIXS data [24–27] is not enough to one AFM link when they occupy the two sites of the same extract structures in the continuum. It would be of interest to vertical bond, while they break two vertical links when they compare the structure and the onset of the continuum to our occupy two different vertical bonds. In contrast, the binding calculation. energy of the low-energy transverse modes is on the order of The most salient feature of the results shown in Fig. 2 is U because the particle and the hole occupy the same site. See the sharp OL mode near q⊥ = (0, 0) and (π,π). Interestingly, Appendix B for the analysis of the exciton wave function. the sharp OL mode appears only at restricted wave vectors The situation is qualitatively different for a metallic non- because the particle-hole continuum exists in the same energy = interacting system with perfect FS nesting.Ifmin(bk ) = 0, scale. The OL mode becomes gapless at U Uc,whilethe the condition (15) cannot be fulfilled because (ω<ω ) OT mode remains gapped, as shown in Figs. 2(c) and 2(d); ph < (ωph ) = 1/U. Therefore, the LM of the AFM phase is ab- the exciton peaks in the band insulator for U Uc remain sent in this case. sharp, as shown in Figs. 2(e) and 2(f).TheU dependence of the energy of this OL mode has important consequences for its stability. While the kinematic constraints do not allow V. LONGITUDINAL MODE OF BILAYER IRIDATE for its decay into two transverse modes for the parameters of 2 From the above analysis, we predict that the bilayer iri- Sr3Ir2O7, the fact that ωL (ωT ) is of order U (t /U)inthe date Sr3Ir2O7 should exhibit a LM in a relatively small large-U/t limit implies that the decay becomes kinematically region around the center of the Brillouin zone. This predic- allowed above a certain value of U. In other words, the sharp tion is qualitatively different from a previous interpretation OL mode that we predict for Sr3Ir2O7 is protected by the

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FIG. 2. Dynamical spin structure factors of the out-of-phase mode: (a) in-plane or transverse Sxx(q,ω) = Syy(q,ω) component and (b) out- zz xx yy zz of-plane or longitudinal component S (q,ω)forU = 0.33 eV; (c) S (q,ω) = S (q,ω)and(d)S (q,ω)forU = Uc ≈ 0.28 eV; and (e) Sxx(q,ω) = Syy(q,ω) and (f) Szz(q,ω)forU = 0.23 eV. The hopping parameters were chosen to reproduce the RIXS data of the bilayer iridate Sr3Ir2O7 [24–27]: t = 0.11 eV, tz = 0.09 eV, α = 1.45, and the intralayer next nearest neighbor hopping 0.012 eV. The broadening factor is η = 10−4 eV. Intensity larger than the maximum value in the scale (color) bar is plotted in the same color. large spin gap generated by the easy-axis anisotropy and by transition occurs at a finite value of U [32], implying that the relatively small value of U/t. Thus, the sharpness of the there are two alternative scenarios for suppressing the AFM OL peak and the fact that ωL and ωT are comparable energy order via reduction of the coupling strength (U/t) in bilayer scales are clear indicators of the proximity of Sr3Ir2O7 to iridates. For the case of Sr3Ir2O7, we predict that, if the the QCP at U = Uc. Because the real material has a small material transitions into a band insulator, a QCP must exist interbilayer hopping, the dimension of the effective theory at Uc ≈ 0.28 eV, which is only 15% smaller than the esti- that describes this QCP, D = 3 + 1, coincides with the upper mated value (U = 0.33 eV) at ambient pressure. The coupling critical dimension (Gaussian fixed point). This means that the strength U/t can be reduced by applying high pressure [33]. mean field theory adopted here is correct up to logarithmic In this scenario, the LM becomes soft at U = Uc. In con- corrections. trast, if the transition is of first order, the material becomes Finally, we note that higher order processes through the metallic without the softening of the LM. While absent in a z Jeff = 3/2 orbitals induce further neighbor hopping terms metallic phase, the characteristic S = 0 exciton peak appears that can make the noninteracting system semimetallic without in the bilayer spin-orbit band insulator for U < Uc,asshown the FS nesting. In this case, a first-order metal-to-insulator in Fig. 2(f).

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VI. SUMMARY AND DISCUSSION We have reported the existence of a sharp LM in antiferromagnetic bilayer Mott insulators with large spin-orbit coupling. Whenever the noninteracting system is a band insulator, Sz = 0, ±1 excitons emerge from the particle-hole continuum for an infinitesimally small value of U.Forma- terials with easy-axis anisotropy, such as the bilayer iridate Sr3Ir2O7, the Ising-like AFM ordering results from the con- z densation of S = 0 excitons at a critical value U = Uc.If the noninteracting system is a metal with FS nesting, the formation and condensation of the particle-hole pairs that pro- + duce the magnetic moments occur simultaneously at Uc = 0 . Similarly to the case of single-layer Mott insulators, such as FIG. 3. (a) Free fermion propagator in the background of the Sr2IrO4, the LM is absent in the resulting AFM state. magnetic order: −G0(q), where q ≡ (q, iωn ). (b) Bare interac- 1 We emphasize that the LM that we discussed in this work tion vertex − Vσ σ σ σ . (c) Bare and (d) RPA spin-charge Nuβ 1 4; 2 3 exists as a sharp mode only close enough to the QCP between susceptibility. the AFM phase and the paramagnetic band insulator. A similar QCP still exists in the large-U limit, where it divides the AFM ACKNOWLEDGMENTS state from a quantum paramagnetic spin state, whose mean field description is a product state of rung singlets [25], which We thank Jian Liu and Mark Dean for helpful discussions. is adiabatically connected with the band insulator. Note that H.S. acknowledges support from JSPS KAKENHI Grant No. such a mean field state can only be captured by expanding the JP19K14650. S.-S.Z. and C.D.B. are supported by fund- variational space of product single-site sates that is assumed ing from the Lincoln Chair of Excellence in Physics. This by the conventional AFM mean field approach used in this research used resources of the Oak Ridge Leadership Com- work. However, our estimate of the hopping amplitudes for puting Facility at the Oak Ridge National Laboratory, which Sr3Ir2O7, consistent with the effective model obtained from is supported by the Office of Science of the US Department of 2 the three-orbital model [32], leads to Jz/J = (tz/t ) ≈ 0.67 in Energy under Contract No. DE-AC05-00OR22725. the strong coupling limit. This value of the exchange coupling / ratio is significantly smaller than the critical value (Jz J)c. APPENDIX A: RANDOM PHASE APPROXIMATION (RPA) The bond-operator mean field theory gives (Jz/J)c = 4forthe isotropic and the easy-axis cases [16,25], while the exact value In view of the U (1) invariance of H and of the ground state, for the SU(2) invariant case is very close to (Jz/J)c = 2.5221 the transverse and longitudinal spin fluctuations are decoupled according to quantum Monte Carlo simulations [34]. There- from each other. The same is true for the charge fluctuations. fore, as we discussed in the previous section, the sharp LM Consequently, we rewrite the (bare) interaction vertex in three mode does not survive in the large-U limit of the Hubbard different forms which account for the longitudinal and trans- model of Sr3Ir2O7 (the LM loses its sharpness far enough verse spin fluctuations, respectively, from the QCP because the kinematic constraints allow it to 1 † † H = Vσ σ σ σ c c cσ , cσ , , (A1) decay into pairs of transverse modes). Moreover, as shown in int 1 4; 2 3 σ1,r σ2,r 3 r 4 r 2 {σ } Fig. 2, the sharp LM appears only at restricted wave vectors r i because the particle-hole continuum exists in the same energy where the interaction vertex takes three equivalent forms, scale. This characteristic feature of Sr Ir O indicates that 3 2 7 this material is an excitonic insulator [28–30] near the critical 1 0 0 z z Vσ σ σ σ = U σ σ − σ σ (A2) 1 4; 2 3 σ1σ4 σ2σ3 σ1σ4 σ2σ3 point U = Uc. In other words, the proximity of Sr3Ir2O7 to 2 quantum criticality is caused by strong charge fluctuations, 1 0 0 + − = Uσσ σ σσ σ − Uσσ σ σσ σ (A3) which are absent in the pure spin model. 2 1 4 2 3 1 4 2 3 Finally, it is worth mentioning that the longitudinal mode 1 can also exist in bilayer Mott insulators with easy-plane = Uσ 0 σ 0 − Uσ − σ + , (A4) σ1σ4 σ2σ3 σ1σ4 σ2σ3 anisotropy. The corresponding XY-AFM state results from 2 the condensation of the Sz =±1 excitons, while the Sz = 0 σ 0 × σ ± = 1 σ x ± σ y is the 2 2 identity matrix, and 2 ( i ). exciton remains gapped at U = Uc and becomes the LM for We are ready to compute the dynamic spin-charge suscepti- U > Uc. The main difference is that this LM, also known bility within the RPA. Figures 3(a) and 3(b) show the fermion as “Higgs mode,” is critically damped in (3+1)D because it propagator and the bare vertex, respectively. The free fermion is allowed to decay into two transverse magnons (Goldstone Green’s function modes) of the AFM state [11–13,15,35,36]. It is of great Pnσ (q) interest to seek the sharp LM and the Higgs mode in other G0(q, iωn ) = (A5) iω − ε σ (q) physical systems. Recently, fermionic systems described by n=±,σ =↑,↓ n n bilayer Hubbard models have been realized in cold atoms [37], is represented by a solid line, where P σ (q) = implying that the exciton condensation that we have discussed n |X σ (q) X σ (q)| is the projector on the eigenstate |X σ (q) of here can also be realized in these systems. n n n the mean field Hamiltonian.

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The (bare) dynamic spin-charge susceptibility of the noninteracting mean ﬁeld Hamiltonian is given by γ ,μ γ ,ν χ ( 1 );( 2 ) , ω 0 (q i n ) 1 n (ε σ (k + q)) − n (ε σ (k)) γ ,μ γ ,ν = F l F k ( 1 );( 2 ), Fkσ ;lσ Nu iωn − (εlσ (k + q) − εkσ (k)) k k,l (A6) where γ1,γ2 = A, B are sublattice indices and μ, ν = 0, ±, z are charge-spin indices. Here we have introduced the polar- ization factor FIG. 4. The particle-hole Green’s function and the T matrix un- (γ1,μ);(γ2,ν) μ ν F (k; q) = Tr P σ (k)γ P σ (k + q)γ kσ ;lσ k 1 l 2 der the RPA.

μ ν = X σ (k)| |X σ (k + q) X σ (k + q)| |X σ (k) , k γ1 l l γ2 k order ∼U. The wave function of the pair is anticipated to (A7) be a tightly bound state because the mean field bandwidth of an electron/hole ∼t 2/U is comparable to the binding energy μ 1 μ ≡ τ + γ τ ⊗ σ γ 2 with γ 2 ( 0 z ) , where takes the values 1 ζt /U, where ζ 0.88 obtained by solving the pole equation − γ = A B τ τ × z ( 1) for ( ), and 0 and z are the 2 2 identity for the relevant set of the hopping parameters. and the Pauli matrix of the sublattice space, respectively. The To determine the size of the bound state, we consider the χ diagram of 0 is shown in Fig. 3(c). Because of the U (1) spin particle-hole Green’s function (γ1,μ);(γ2,ν) rotation symmetry about the z axis, Fkσ ;lσ (k; q) has the (2) iG γ σ ,γ σ γ σ ,γ σ (Q,ω; k, q) following structure: ( 1 1 1 1 );( 2 2 2 2 ) ∞ (γ ,μ);(γ ,ν) F 1 2 iωt † Q Q = dte G|Tt cγ σ k − , t cγ σ k + , t ⎛ ⎞ 1 1 1 1 γ , γ , −∞ 2 2 F ( 1 0);( 2 0) 000 ⎜ γ ,+ γ ,− ⎟ 00F ( 1 );( 2 ) 0 † Q Q ⎜ ⎟ × + , γ σ − , | . = γ ,− γ ,+ . cγ σ q 0 c 2 2 q 0 G (B1) ⎝ 0 F ( 1 );( 2 ) 00⎠ 2 2 2 2 γ , γ , 000F ( 1 z);( 2 z) The exciton gives rise to a pole at ω = ωL (Q) for the center- (A8) of-mass momentum Q: (γ ,μ);(γ ,ν) (2) χ 1 2 iG γ σ ,γ σ γ σ ,γ σ (Q,ω; k, q) The same structure holds for 0 , confirming that ( 1 1 1 1 );( 2 2 2 2 ) the transverse spin fluctuations, longitudinal spin fluctuations, ∗ ψ ψ γ1σ1;γ σ (k) γ σ ,γ σ (q) and charge fluctuations are decoupled from each other. In the ∼ 1 1 2 2 2 2 + , + regular term (B2) following, we focus on the spin channel, which is of main ω − ωL (Q) + i0 interest for this work. † Q Q where ψγ σ ;γ σ (k) = G|cγ σ (k − )cγ σ (k + )|bQ , |bQ Figure 3(d) represents the magnetic susceptibility at the 1 1 1 1 1 1 2 1 1 2 is the exciton eigenstate with center of mass momentum Q, RPA level. The results are and ωL (Q) is the exciton energy measured from the ground +− 1 +− state. The probability amplitude to find a hole with spin σ1 at χ (q, iωn ) = +− χ (q, iωn ), (A9) RPA τ 0 − χ , ω 0 r1 and an electron with spin σ2 at r2 is obtained by Fourier U 0 (q i n ) transforming the wave function ψγ σ γ σ (k): 1 1 1; 1 1 χ −+ (q, iω ) = χ −+(q, iω ) (A10) RPA n τ 0 − Uχ −+(q, iω ) 0 n ψ ∈ γ , ∈ γ 0 n σ1;σ2 (r1 1 r2 2 ) † for the transverse channel and = G|c (r )cσ (r )|b σ1 1 2 2 1 χ zz , ω = χ zz , ω iQ·(r +r )/2 1 ik·(r −r ) RPA(q i n ) 0 (q i n ) (A11) = 1 2 ψγ σ γ σ 2 1 . τ 0 − U χ zz , ω e 1 1; 2 2 (k)e (B3) 2 0 (q i n ) Nu k χ +− χ −+ χ zz for the longitudinal channel. Note that RPA, RPA, and RPA As shown in Fig. 4, the particle-hole Green’s function is × χ +− χ −+ are 2 2 matrices in the sublattice space, while 0 , 0 , determined by the T matrix: χ zz and 0 are the corresponding bare magnetic susceptibilities, (2) iG γ σ ,γ σ γ σ ,γ σ (Q,ω; k, q) respectively. ( 1 1 1 1 );( 2 2 2 2 ) = (2) ,ω , iG(γ σ ,γ σ );(γ σ ,γ σ )(Q ; k q) APPENDIX B: EXCITON WAVE FUNCTION 1 1 1 1 2 2 2 2 (2) + iG γ σ ,γ σ γ σ ,γ σ (Q,ω; k, k ) We here investigate the wave function of the exciton ( 1 1 1 1 );( 3 3 3 3 ) z formed by multiple particle-hole pairs with S = 0. In the × − ,ω , ( i)T(γ3σ3,γ σ );(γ4σ4,γ σ )(Q ; k q ) large-U limit, it is mainly composed of a single particle-hole 3 3 4 4 × (2) ,ω , . pair because the energy cost of creating a particle-hole is of iG γ σ ,γ σ γ σ ,γ σ (Q ; q q) (B4) ( 4 4 4 4 );( 2 2 2 2 )

013224-7 SUWA, ZHANG, AND BATISTA PHYSICAL REVIEW RESEARCH 3, 013224 (2021)

Here iG(2) is the noninteracting particle-hole Green’s function ∞ (2) iωt † Q Q † Q Q iG (Q,ω; k, q) = dte G|T c k − , t cγ σ k + , t c q + , 0 cγ σ q − , 0 |G (γ σ ,γ σ );(γ σ ,γ σ ) t γ1σ1 1 1 γ σ 2 2 1 1 1 1 2 2 2 2 −∞ 2 2 2 2 2 2 (B5) ∞ ω Q Q = δ i t + , − , − k,q dte Gγ σ ;γ σ k t Gγ2σ2;γ1σ1 k t (B6) −∞ 1 1 2 2 2 2 ∞ dν Q ω Q ω = δ + ,ν+ − ,ν− k,q Gγ σ ;γ σ k Gγ2σ2;γ1σ1 k (B7) −∞ 2π 1 1 2 2 2 2 2 2 Q Q P ,σ k + P− ,σ k − 1 1 γ σ γ σ 1 1 γ σ γ σ 2 1 1; 2 2 2 2 2; 1 1 = iδ , δσ σ δσ σ k q 1 2 1 2 ω − ε , + Q − ε− , − Q 1 k 2 1 k 2 Q Q P− ,σ k + P ,σ k − 1 1 2 γ σ ;γ σ 1 1 2 γ σ ;γ σ − 1 1 2 2 2 2 1 1 , (B8) ω + ε , − Q − ε− , + Q 1 k 2 1 k 2 where Ps,σ (k) =|Xs,σ (k) Xs,σ (k)| is the projector to the (s,σ) eigenstate at k.IntheRPA(seeFig.4), the T matrix is given by

U z z U z (γ3z,γ4z) U z − γ σ ,γ σ γ σ ,γ σ ,ω , = σ σ δγ γ δγ γ δγ γ + σ δγ γ − χ ,ω σ δγ γ . iT( 3 3 );( 4 4 )(Q ; k q ) i σ σ σ σ 3 4 3 4 i σ σ 3 ( i) RPA (Q )i σ σ 4 (B9) 3 3 4 4 2 3 3 4 4 3 4 2 3 3 3 2 4 4 4

Note that the bare Hubbard interaction is written in terms γ4 (2) × (−1) G γ σ ,γ σ γ σ ,γ σ (0,ωL; q, q) H = ( 4 2 4 2 );( 2 2 2 2 ) of the longitudinal component of the spin operator, int γ / 2 − z 2 = † z = 4 r[(14)(nr) (Sr ) ] with nr σ cσ (r)cσ (r) and Sr † + regular terms. (B13) (1/2) σ σ cσ (r) cσ (r). Under the RPA, the T matrix that describes the renormalized interaction between electrons in- The spectral weight of the exciton pole is equal to the projec- cludes contributions from longitudinal spin ﬂuctuations. tion of the exciton wave function to the two-magnon sector Let us consider the exciton at Q = 0, which has the lowest energy ωL (Q = 0) ≡ ωL. Near the exciton pole, the longitudi- ψγσ;γ σ (k) nal RPA susceptibility is given by σ γ (2) = √ δ − 3 ,ω , . σσ ( 1) G(γσ;γ σ );(γ σ,γ σ )(0 L; k k) γ2+γ ω 3 3 (γ z,γ z) (−1) 2 1 ( L ) γ χ 2 2 (0,ω) ∼− , (B10) 3 RPA 2 + U (ωL ) ω − ωL + i0 (B14) where For each combination of sublattices, it reads ω 3 − 2 σ L 1 zk 1 zk 2 (ω ) = (B11) ψAσ ;Aσ (k) = √ δσσ 1 − z L 3 2 ω k 4(UM) Nu ω 2 2 ( L ) k∈BZ − L 2 1 2UM zk ε , − ε− , × 1 k 1 k , (B15) > ω2 − ε − ε 2 for U Uc. In the vicinity of the critical point Uc,it L ( 1,k −1,k ) ω 1 | |−3 ∝ takes the asymptotic form ( L ) UMN bk √ u k ψBσ Bσ (k) =−ψAσ Aσ (k), (B16) − ; ; U Uc. The longitudinal RPA susceptibility,√ therefore, ac- ∝ / − quires a diverging spectral weight 1 U Uc, implying 1 − z2 σ ∗ k strong longitudinal spin ﬂuctuations. In the large-U limit, ψ = √ δ Aσ ;Bσ (k) σσ xσ k (ωL ) 2 1 2b2 ω → k ∼ O , − σ + σ ( L ) 2 (1) (B12) 1 zk 1 zk Nu ζ 2 + 2 × + , k t 2bk ωL − (ε1,k − ε−1,k ) ωL + ε1,k − ε−1,k and the spectral weight of the longitudinal RPA susceptibility (B17) approaches zero as U −2. ∗ ψBσ ;Aσ (k) = ψAσ Bσ (k). (B18) According to Eq. (B9), the exciton mode also appears as a ¯ ; ¯ −1 pole of the particle-hole Green’s function. For instance, In the large-U limit, ψAσ ;Aσ (k) ∼ O(U ), ψA↑;B↑(k) = ψ∗ ∼ O −2 ψ = ψ∗ ∼ O (2) B↓;A↓(k) (U ), and A↓;B↓(k) B↑;A↑(k) (1). G γ σ ,γ σ γ σ ,γ σ (0,ω; k, q) ( 1 1 1 1 );( 2 2 2 2 ) The asymptotic form of the nonzero amplitude reads 1 1 ∗ ψ = ψ ∼ σ1σ2δσ σ δσ σ A↓;B↓(k) B↑ A↑(k) ω − ω + i0+ (ω ) 1 1 2 2 ; L L ∗ 1 2bk γ3 (2) = √ , × (−1) G (0,ωL; k, k) (B19) (γ1σ1,γ σ1 );(γ3σ1,γ3σ1 ) ω ζ 2 + 2 − 2 1 ( L ) t 2 bk minq bq γ3

013224-8 EXCITON CONDENSATION IN BILAYER SPIN-ORBIT … PHYSICAL REVIEW RESEARCH 3, 013224 (2021)

2 FIG. 5. Probability distribution |ψ↓↓(0, r2 )| for several strengths of the on-site coupling U. A hole with spin ↓ is assumed located at r1 = 0 on layer 1. The critical coupling is Uc ≈ 0.277 eV for the hopping parameters relevant to Sr3Ir2O7, while the on-site Hubbard interaction is estimated to be U = 0.33 eV (= 1.19Uc ) for the material. M is the magnitude of the local ordered magnetic moment. The distribution function 2 |ψ↓↓(0, r2 )| is normalized for each value of U, so that the sum over r2 is equal to one.

where ζ 0.88 determines the exciton binding energy Eb ≡ at r1 = 0 (sublattice A and layer 1), the distribution of the ω − ω ζ 2/ ψ ∈ A, ∈ B |ψ , |2 ph L tz U. The amplitude ↓↓(r1 r2 ) de- spectral weight, i.e., ↓↓(0 r2 ) , is shown in Fig. 5.Inthe termines the spectral weight of the configuration with one large-U limit, a hole and an electron occupy different sublat- hole with spin ↓ at r1 ∈ A and one electron with spin ↓ at tices [Fig. 5(h)]. The bound state with the particle-hole pair r2 ∈ B. Since the ordered moment is −M on sublattice A and occupying a vertical bond has the largest spectral weight and M on sublattice B, the exciton is created by moving either the probability of finding the particle and the hole on different ↓ spin from sublattice A to B or a ↑ spin from sublattice layers is higher than the probability of finding them on the B to A. On the condition that a hole with spin ↓ is pinned same layer. As a result of the strong binding energy relative to

013224-9 SUWA, ZHANG, AND BATISTA PHYSICAL REVIEW RESEARCH 3, 013224 (2021) the bandwidth of the single electron or hole (∼Eb), the size of either sublattice). Note that the magnetic moments on the two the bound state is comparable to one lattice constant. sublattices must be opposite because of the relative “−”sign Figure 5 shows the evolution of the real space distribution between ψσ ;σ (r1 ∈ A, r2 ∈ A) and ψσ ;σ (r1 ∈ B, r2 ∈ B); i.e., of the particle-hole pair as a function of U. The ordered the system develops Néel magnetic ordering for U > Uc. =| | z| | magnetic moment, M G Sr G decreases upon reduc- The time-reversal symmetry is restored as U de- ing U/t within the ordered phase [Figs. 5(e)–5(g)] and the creases further below Uc (in the band insulator), and probability of ﬁnding the particle and the hole on the same the wave function of the exciton becomes extended [see z| site increases. Thus, Sr G provides a reasonably good ap- Figs. 5(a)–5(c)], as expected from the reduction of the bind- proximation of the exciton eigenstate. At U = Uc, we ﬁnd ing energy. For zero binding energy (ζ = 0), the integral ∗ ψ = ψ = 1 ik·(r2−r1 ) A↓;B↓(k) B↑ A↑(k) 0 [Fig. 5(d)]. The magnetically ψA↓;B↓(k)e that determines the real space ; Nu k ordered state emerges as a superposition of the nonmag- wave function ψ↓↓(r1 ∈ A, r2 ∈ B) is dominated by the sin- netic ground state of the band insulator and states containing gular points k0 that minimize the particle-hole excitation multiple coherent excitons. The choice of the phase factor car- δ2 + 2 energy 2 bk. In the simplest case where there is a unique · − ried by each condensing exciton, equal to 0 or π, corresponds ψ ∈ A, ∈ B ∝ 1 ik0 (r2 r1 ) singular point k0, ↓↓(r1 r2 ) N e takes > < u to the Z2 time-reversal symmetry breaking (M 0or 0on the form of a plane wave.

[1] M. Jaime, V. F. Correa, N. Harrison, C. D. Batista, N. [12] D. Podolsky and S. Sachdev, Spectral functions of the Higgs Kawashima, Y. Kazuma, G. A. Jorge, R. Stern, I. Heinmaa, mode near two-dimensional quantum critical points, Phys. Rev. S. A. Zvyagin, Y. Sasago, and K. Uchinokura, Magnetic- B 86, 054508 (2012). Field-Induced Condensation of Triplons in Han Purple Pigment [13] M. Lohöfer, T. Coletta, D. G. Joshi, F. F. Assaad, M. Vojta, S.

BaCuSi2O6, Phys. Rev. Lett. 93, 087203 (2004). Wessel, and F. Mila, Dynamical structure factors and excita- [2] S. E. Sebastian, N. Harrison, C. D. Batista, L. Balicas, M. Jaime, tion modes of the bilayer Heisenberg model, Phys. Rev. B 92, P. A. Sharma, N. Kawashima, and I. R. Fisher, Dimensional 245137 (2015). reduction at a quantum critical point, Nature (London) 441, 617 [14] T. Hong, M. Matsumoto, Y. Qiu, W. Chen, T. R. Gentile, S. (2006). Watson, F. F. Awwadi, M. M. Turnbull, S. E. Dissanayake, [3] C. D. Batista, J. Schmalian, N. Kawashima, P. Sengupta, S. E. H. Agrawal, R. Toft-Petersen, B. Klemke, K. Coester, K. P. Sebastian, N. Harrison, M. Jaime, and I. R. Fisher, Geometric Schmidt, and D. A. Tennant, Higgs amplitude mode in a two- Frustration and Dimensional Reduction at a Quantum Critical dimensional quantum antiferromagnet near the quantum critical Point, Phys.Rev.Lett.98, 257201 (2007). point, Nat. Phys. 13, 638 (2017). [4] V. Zapf, M. Jaime, and C. D. Batista, Bose-Einstein condensa- [15] Y. Q. Qin, B. Normand, A. W. Sandvik, and Z. Y. Meng, Ampli- tion in quantum magnets, Rev. Mod. Phys. 86, 563 (2014). tude Mode in Three-Dimensional Dimerized Antiferromagnets, [5] M. B. Stone, M. D. Lumsden, S. Chang, E. C. Samulon, C. D. Phys. Rev. Lett. 118, 147207 (2017). Batista, and I. R. Fisher, Singlet-Triplet Dispersion Reveals Ad- [16] Y. Su, A. Masaki-Kato, W. Zhu, J.-X. Zhu, Y. Kamiya, and S.-Z. ditional Frustration in the Triangular-Lattice Dimer Compound Lin, Stable Higgs mode in anisotropic quantum magnets, Phys.

Ba3Mn2O8, Phys.Rev.Lett.100, 237201 (2008). Rev. B 102, 125102 (2020). [6] M. Kofu, H. Ueda, H. Nojiri, Y. Oshima, T. Zenmoto, K. C. [17] R. Arita, J. Kuneš, P. Augustinský, A. V. Kozhevnikov, A. G. Rule, S. Gerischer, B. Lake, C. D. Batista, Y. Ueda, and S.-H. Eguiluz, and M. Imada, Mott versus Slater-type metal-insulator

Lee, Magnetic-Field Induced Phase Transitions in a Weakly transition in Sr2IrO4 and Ba2IrO4, JPS Conf. Proc. 3, 013023 Coupled s = 1/2 Quantum Spin Dimer System Ba3Cr2O8, (2014). Phys. Rev. Lett. 102, 177204 (2009). [18] J. G. Rau, E. K.-H. Lee, and H.-Y. Kee, Spin-orbit physics [7] A. V. Chubukov, S. Sachdev, and J. Ye, Theory of two- giving rise to novel phases in correlated systems: Iridates and dimensional quantum Heisenberg antiferromagnets with a related materials, Annu. Rev. Condens. Matter Phys. 7, 195 nearly critical ground state, Phys.Rev.B49, 11919 (1994). (2016). [8] S. Sachdev, Universal relaxational dynamics near two- [19] G. Cao and P. Schlottmann, The challenge of spin-orbit-tuned dimensional quantum critical points, Phys. Rev. B 59, 14054 ground states in iridates: A key issues review, Rep. Prog. Phys. (1999). 81, 042502 (2018). [9] M. Matsumoto, B. Normand, T. M. Rice, and M. Sigrist, Field- [20] L. Hao, D. Meyers, H. Suwa, J. Yang, C. Frederick, T. R. Dasa, and pressure-induced magnetic quantum phase transitions in G. Fabbris, L. Horak, D. Kriegner, Y. Choi, J.-W. Kim, D.

TlCuCl3, Phys.Rev.B69, 054423 (2004). Haskel, P. J. Ryan, H. Xu, C. D. Batista, M. P. M. Dean, and [10] C. Rüegg, B. Normand, M. Matsumoto, A. Furrer, D. F. J. Liu, Giant magnetic response of a two-dimensional antiferro- McMorrow, K. W. Krämer, H. U. Güdel, S. N. Gvasaliya, H. magnet, Nat. Phys. 14, 806 (2018). Mutka, and M. Boehm, Quantum Magnets Under Pressure: [21] L. Hao, Z. Wang, J. Yang, D. Meyers, J. Sanchez, G. Fabbris,

Controlling Elementary Excitations in TlCuCl3, Phys. Rev. Y. Choi, J.-W. Kim, D. Haskel, P. J. Ryan, K. Barros, J.-H. Lett. 100, 205701 (2008). Chu, M. P. M. Dean, C. D. Batista, and J. Liu, Anoma- [11] D. Podolsky, A. Auerbach, and D. P. Arovas, Visibility of the lous magnetoresistance due to longitudinal spin ﬂuctuations

amplitude (Higgs) mode in condensed matter, Phys.Rev.B84, in a jeff = 1/2 Mott semiconductor, Nat. Commun. 10, 5301 174522 (2011). (2019).

013224-10 EXCITON CONDENSATION IN BILAYER SPIN-ORBIT … PHYSICAL REVIEW RESEARCH 3, 013224 (2021)

[22]J.Bertinshaw,Y.K.Kim,G.Khaliullin,andB.J.Kim,Square [29] B. I. Halperin and T. M. Rice, Possible anomalies at a lattice iridates, Annu. Rev. Condens. Matter Phys. 10, 315 semimetal-semiconductor transistion, Rev. Mod. Phys. 40, 755 (2019). (1968). [23] Y. Okada, D. Walkup, H. Lin, C. Dhital, T.-R. Chang, S. [30] D. Jérome, T. M. Rice, and W. Kohn, Excitonic insulator, Phys. Khadka, W. Zhou, H.-T. Jeng, M. Paranjape, A. Bansil, Z. Rev. 158, 462 (1967). Wang, S. D. Wilson, and V. Madhavan, Imaging the evolution [31] B. J. Kim, H. Jin, S. J. Moon, J.-Y. Kim, B.-G. Park, C. S. Leem, of metallic states in a correlated iridate, Nat. Mater. 12, 707 J. Yu, T. W. Noh, C. Kim, S.-J. Oh, J.-H. Park, V. Durairaj, G.

(2013). Cao, and E. Rotenberg, Novel Jeff = 1/2 Mott State Induced [24] J. Kim, A. H. Said, D. Casa, M. H. Upton, T. Gog, M. by Relativistic Spin-Orbit Coupling in Sr2IrO4, Phys.Rev.Lett. Daghofer, G. Jackeli, J. van den Brink, G. Khaliullin, and 101, 076402 (2008). B. J. Kim, Large Spin-Wave Energy Gap in the Bilayer Iridate [32] J.-M. Carter and H.-Y. Kee, Microscopic theory of magnetism

Sr3Ir2O7: Evidence for Enhanced Dipolar Interactions Near the in Sr3Ir2O7, Phys.Rev.B87, 014433 (2013). Mott Metal-Insulator Transition, Phys. Rev. Lett. 109, 157402 [33] J. Zhang, D. Yan, S. Yesudhas, H. Deng, H. Xiao, B. Chen, (2012). R. Sereika, X. Yin, C. Yi, Y. Shi, Z. Liu, E. M. Pärschke, [25] M. Moretti Sala, V. Schnells, S. Boseggia, L. Simonelli, A. C.-C. Chen, J. Chang, Y. Ding, and H.-k. Mao, Lattice frustra-

Al-Zein, J. G. Vale, L. Paolasini, E. C. Hunter, R. S. Perry, D. tion in spin-orbit Mott insulator Sr3Ir2O7 at high pressure, npj Prabhakaran, A. T. Boothroyd, M. Krisch, G. Monaco, H. M. Quantum Mater. 4, 23 (2019). Rønnow, D. F. McMorrow, and F. Mila, Evidence of quantum [34] A. Sen, H. Suwa, and A. W. Sandvik, Velocity of excitations in

dimer excitations in Sr3Ir2O7, Phys.Rev.B92, 024405 (2015). ordered, disordered, and critical antiferromagnets, Phys.Rev.B [26] T. Hogan, R. Dally, M. Upton, J. P. Clancy, K. Finkelstein, Y.-J. 92, 195145 (2015). Kim, M. J. Graf, and S. D. Wilson, Disordered dimer state in [35] I. Afﬂeck and G. F. Wellman, Longitudinal modes in quasi-one-

electron-doped Sr3Ir2o7, Phys.Rev.B94, 100401(R) (2016). dimensional antiferromagnets, Phys.Rev.B46, 8934 (1992). [27] X. Lu, D. E. McNally, M. Moretti Sala, J. Terzic, M. H. Upton, [36] Y. Kulik and O. P. Sushkov, Width of the longitudinal magnon D. Casa, G. Ingold, G. Cao, and T. Schmitt, Doping Evolution of in the vicinity of the O(3) quantum critical point, Phys.Rev.B

Magnetic Order and Magnetic Excitations in (Sr1−xLax )3Ir2O7, 84, 134418 (2011). Phys. Rev. Lett. 118, 027202 (2017). [37] M. Gall, N. Wurz, J. Samland, C. F. Chan, and M. Köhl, [28] J. D. Cloizeaux, Exciton instability and crystallographic anoma- Competing magnetic orders in a bilayer Hubbard model with lies in semiconductors, J. Phys. Chem. Solids 26, 259 (1965). ultracold atoms, Nature (London) 589, 40 (2021).

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