Excitonic Instability at the Spin-State Transition in the Two-Band Hubbard Model
aa r X i v : . [ c ond - m a t . s t r- e l ] M a r Excitonic Instability at the Spin-State Transition in the Two-Band Hubbard Model
Jan Kuneˇs and Pavel Augustinsk´y Institute of Physics, Academy of Sciences of the Czech republic,Cukrovarnick´a 10, Praha 6, 162 53, Czech Republic (Dated: February 6, 2018)Using linear response theory with the dynamical mean-field approximation we investigate theparticle-hole instabilities of the two-band Hubbard model in the vicinity of the spin-state transition.Besides the previously reported high-spin–low-spin order we find an instability towards triplet ex-citonic condensate. We discuss the strong and weak coupling limits of the model, in particular, aconnection to the spinful hard-core bosons with a nearest-neighbor interaction. Possible realizationin LaCoO at intermediate temperatures is briefly discussed. PACS numbers: 71.35.Lk,71.27.+a,05.30.Jp,75.45+j
I. INTRODUCTION
Search for new states of matter is one of the centraltopics of condensed matter physics. While the develop-ment of cold atom techniques allowed the construction ofmany exotic phases in particular in systems of interact-ing bosons, electronic order parameters other than spin,charge and orbital densities or s-wave pairing supercon-ductivity are rather rare in real materials. We report ob-servation of an off-diagonal order close to the spin-statetransition in the two-band Hubbard model with Hund’scoupling and show that such electronic system providesrealization of some of the phases observed with interact-ing bosons.The role of Hund’s coupling in correlated electron sys-tems has been recently theoretically studied in the con-text of Hund’s metals and the spin-state transitionsdriven by pressure as well as temperature or dop-ing . Competition of different spin states was also linkedto the peculiar magnetic properties of iron pnictides .The two-band Hubbard model at half filling provides aminimal lattice realization of the spin-state transitionin correlated electron systems . Recently, a reentranttransition of Ising type to a two-sublattice order of high-spin (HS) and low-spin (LS) states was reported on abipartite lattice in the vicinity of the spin-state tran-sition . It was proposed that such ordered state canexplain properties of the notorious spin-state transitioncompound LaCoO at intermediate temperatures.In this article, we report a systematic investigationof the particle-hole instabilities in the normal phase ofthe two-band Hubbard model. Besides the previouslyreported Ising instability we find that an excitonic in-stability which breaks a continuous symmetry dominatesover a broad range of parameters. The idea of an insta-bility due to the long-range part of the Coulomb interac-tion in small gap semiconductors leading to so the calledexcitonic insulator phase appeared fifty years ago andmore recently was applied to the physics of LaB . Fol-lowing the work of Batista on electronic ferroelectric-ity, the excitonic instability was studied in the extendedFalicov-Kimball model as well as the two-band Hub- ∆/ (t a +t b )24 U / ( t a + t b ) -2 0 2 Energy (eV) S p ec t r a l d e n s it y HS Mott metal LS bandinsulator insulator
FIG. 1: (color online) Left: the conceptual phase diagram ofthe two-band Hubbard model for U = 4 J . The shaded areamarks the parameter range visited while varying the bandasymmetry ζ and crystal field ∆. Right: 1P spectral den-sities obtained at the points marked by stars (upper panelcorresponds to the upper star) at temperatures just abovethe leading T c . bard model without Hund’s coupling .The connection to the bosonic physics arises in thestrong-coupling limit. As was shown by Batista , theextended Falicov-Kimball model at half filling maps ontospinless hard-core bosons with nn repulsion, a problemmuch studied in the context of solid, superfluid andpossibly a supersolid phase . We show that in thestrong-coupling limit of the two-band Hubbard modelwith Hund’s coupling the mapping generalizes to thespinful hard-core bosons with some additional nn terms,a much less studied problem with a rich phase dia-gram.The paper is structured as follows. In Section IIwe state the problem and describe the computationalmethod. In Section III we summarize our numerical re-sults. In Section IV we derive the strong- and weak-coupling limits of the studied model in order to elucidatethe nature of the instabilities reported in Section III. We χ λ [0,0] [0, π ] [ π , π ] [0,0] [0, π ] [ π , π ] [0,0] [0, π ] [ π , π ] FIG. 2: (color online) The typical q-dependence of the lead-ing eigenvalues of the susceptibility matrix: spin longitudinal(red), OD (green) and OO (blue) in a system with a largeband asymmetry ζ = 0 .
22, ∆ = 3 .
40 at temperatures 773 K,644 K and 580 K (left to right). briefly discuss the classical limit, which provides the sim-ple understanding of the HS-LS phase, and then focus onvarious aspects of the excitonic phase. In Section V wesummarize our main findings.
II. COMPUTATIONAL PROCEDURE
We consider the two-band Hubbard mode with nearest-neighbor (nn) hopping on a bipartite (square) lattice withthe kinetic H t and the interaction H int = H ddint + H ′ int terms given by H t = ∆2 X i,σ (cid:0) n aiσ − n biσ (cid:1) + X i,j,σ (cid:0) t a a † iσ a jσ + t b b † iσ b jσ (cid:1) + X h ij i ,σ (cid:0) V a † iσ b jσ + V b † iσ a jσ + c.c. (cid:1) H ddint = U X i (cid:0) n ai ↑ n ai ↓ + n bi ↑ n bi ↓ (cid:1) + ( U − J ) X i,σ n aiσ n bi − σ + ( U − J ) X iσ n aiσ n biσ H ′ int = J X iσ a † iσ b † i − σ a i − σ b iσ + J ′ X i (cid:0) a † i ↑ a † i ↓ b i ↓ b i ↑ + c.c. (cid:1) . (1)Here a † iσ , b † iσ are the creation operators of fermions withspin σ = ↑ , ↓ and n ciσ = c † iσ c iσ . Symbol P i,j impliessummation over ordered nn pairs, while P h ij i impliessummation over nn bonds (pairs without order). Themodel is studied at half filling, two electrons per site onaverage. The crystal field ∆ and the Hund’s exchange J are chosen so that the system is in the vicinity of theLS-HS transition.The numerical calculations were performed in the dy-namical mean-field approximation with the density-density interaction H ddint only. The effect of adding H ′ int χ λ [0,0] [0, π ] [ π , π ] χ OO [0,0] [0, π ] [ π , π ] FIG. 3: (color online) Left: Leading eigenvalues for equalbandwidths ( ζ = 1) and ∆ = 3 .
40 eV at 1160 K. The blueOO mode diverges faster than the green OD mode. Right:Splitting of the OO mode from (b) due to added cross-hopping V , = 0 . eV . The leading mode (two-fold degenerate) hasthe form a † σ b − σ + b † σ a − σ with σ = ↑ , ↓ . is considered in Section IV. We use the hybridizationexpansion continuous time quantum Monte Carlo (CT-HYB) to solve the auxiliary impurity problem andobtain the local one-particle (1P) and two-particle (2P)propagators. For selected parameters we have bench-marked the CT-HYB results against those obtained withthe Hirsch-Fye implementation of the present proce-dure .In order to study phase transitions, we search numer-ically for divergent static particle-hole susceptibilities inthe disordered high temperature phase. The lattice sus-ceptibility χ αβ,γδ ( T, q ) is a q -dependent matrix functionindexed by pairs of spin-orbital indices. It is calculatedfrom the Bethe-Salpeter equation as a function of the full1P propagator and the 2P-irreducible vertex. The cru-cial DMFT simplification consists in the fact that the 2Pirreducible vertex is k -independent and equals the impu-rity 2P irreducible vertex . Therefore the momentumdependence of χ ( T, q ) comes entirely from the 1P prop-agator.We calculate χ ( T, q ) on dense q -mesh in the Brillouinzone, diagonalize for every q , and identify the largesteigenvalues with the corresponding eigenvectors. Thetransition temperature is obtained from the zero cross-ing χ − λ ( T c ) = 0 of the inverse of the largest eigenvalue χ − λ ( T, q ) = 0. The advantage of this approach is thatno prior assumptions about the symmetry of the orderedphase is needed. III. NUMERICAL RESULTS
In this section we present the DMFT results obtainedfor the Hamiltonian H t + H ddint . Following Ref. 11, weset U =4, J =1 and use eV as energy units to allow fora straightforward comparison. The basic phase diagramof model (1) at half filling was computed by Werner and ∆ (eV) T e m p e r a t u r e ( K ) OOOD (a) ∆ c
400 600 800
Temperature (K) -0.100.10.2 χ - ( a r b . un it s ) (b) FIG. 4: (color online) (a) Representative dependencies of theinstability temperatures on the crystal field ∆: T OD (squares)for ζ = 0 .
28 and T OO (circles) for ζ = 0 .
55. The opensquare marks the position of the reentrant transition takenfrom Ref. 11. The blue line marks the estimated position of∆ c . (b) The T -dependence of the inverse eigenvalues χ − (circles) and χ − (squares) of the susceptibility at selectedvalues of ∆. The parameters ζ = 0 .
28, ∆ = 3 .
44 eV (blue)correspond to ∆ > ∼ ∆ c where the OD instability already disap-peared. For ζ = 0 .
55, ∆ = 3 .
38 eV (black) the OD instabilityexists only in a finite interval of temperatures. In both casesthe OO is the leading instability, which is physically realized.
Millis and its cartoon version is presented in Fig. 1. Weare interested in a small region close to the boundarybetween HS Mott insulator and LS band insulator, whichfixes the ∆ of interest to 3 J approximately. Our mainvariable parameter will be the asymmetry between a and b derived band characterized by ζ = t a t b t a + t b . For reasonthat becomes apparent in the discussion of the strongcoupling limit, we choose to vary ζ while keeping the sum t a + t b fixed. Consequently, the point representing oursystem moves slightly, covering the red region of Fig. 1when going between symmetric bands, ζ = 1, and theflat-band limit, ζ = 0.First, we discuss the eigenmodes of χ ( q ) for t a =0 .
45 eV, t b = 0 .
05 eV ( ζ = 0 . V , = 0, and∆ = 3 .
40 eV, the parameters of Ref. 11. The full 16 × χ ( q ) can be, in a standard way using the spin-conservation law, block-diagonalized to ↑↑ − ↓↓ , ↑↑ + ↓↓ , ↑↓ and ↓↑ blocks (channels), each having 4 × χ λ ( q ) withsizable magnitude. These correspond to i) the spin lon-gitudinal mode P σ σ ( n aσ + n bσ ) in the ↑↑ − ↓↓ channel,ii) the orbital diagonal (OD) mode P σ ( n aσ − n bσ ) in the ↑↑ + ↓↓ channel, and iii) four degenerate orbital off-diagonal (OO) modes a †↑ b ↓ , b †↑ a ↓ , a †↓ b ↑ , b †↓ a ↑ in the ↑↓ and ↓↑ channels. In Fig. 2, the q dependence of the cor-responding eigenvalues in the 2D Brillouin zone is plot-ted for several temperatures. Similar plot for symmetricbands, ζ = 1, is shown in Fig. 3.The leading instability for ζ = 0 .
22 is identified in theOD mode at ( π, π ). The corresponding transition tem-perature agrees well with the onset of the HS-LS checker-board order found in Ref. 11. Increasing the crystal field∆ rapidly suppresses the transition temperature T OD ,see Fig. 4a, and the OD instability eventually disappears ζ T e m p e r a t u r e ( K ) gap closing FIG. 5: (color online) Instability of the normal phase as afunction of band asymmetry ζ for various CF parameters ∆.Open circles denote the divergence T OO of the OO mode, filledsquares mark the divergence T OD of the OD mode. The linesare guides to the eye. The dashed vertical lines mark the ζ ’sfor which the ∆ dependences of T OO and T OD are shown inFig. 4a. above some ∆ c . For ∆ < ∼ ∆ c the OD instability disap-pears at low temperatures as shown in Fig. 4b, leadingto a reentrant transition. For ∆ > ∼ ∆ c , the proximity ofthe ordered phase at an intermediate temperature givesrise to a peak in the susceptibility, Fig. 4b. These resultsprovide the same picture as the calculations of Ref. 11performed in the ordered HS-LS phase. However, in ad-dition to that, one can see that the OO susceptibility alsoexhibits a substantial increase at ( π, π ) with decreasingtemperature.Next, we vary the band asymmetry ζ while keepingthe cross-hopping V , = 0. For more symmetric bands adifferent result is obtained, as shown in Fig. 3, where thedominant χ λ ( q ) are plotted for ζ = 1. In this case, theOO mode at ( π, π ) is the leading instability. This impliesformation of an ordered state with spontaneous local off-diagonal hybridization characterized by non-zero value of h a † i,σ b i, − σ i and anti-ferro periodicity.In Fig. 5, we show the calculated instability lines in the ζ - T plane for several values of ∆. The actual calculationswere performed for t b ≤ t a , but the results hold also for t a ≤ t b , since on a bipartite lattice at half-filling the lat-ter can be mapped on the former by exchange of a and b followed by the particle-hole transformation and the signreversal of a and b operators on one sublattice. Severalobservations can be made. For the studied parametersthere are two possible instabilities corresponding to theOO and OD modes. The OO mode, favored by moresymmetric bands, is the leading instability over a broadrange of band asymmetries. The OO instability is sup-pressed when one of the bands becomes narrow, in whichcase the instability line T OO ( ζ ) extrapolates linearly tozero. The OD mode is the leading instability only forstrongly asymmetric bands. For constant t a + t b , the T OD ( ζ ) is insensitive to ζ within the accuracy of our cal-culation. For all ζ , the T OO is less sensitive to the crystalfield ∆ than T OD .The OO instability shows little sensitivity to the pres-ence of a charge gap in the disordered state as there isno apparent change in the behavior of T OO ( ζ ) when thegap disappears. In Fig. 5, we mark closing of the chargegap above the LS state. The actual 1P spectral functionsat temperatures just above T c close to both ends of the ζ -range are shown in Fig. 1.The results obtained for positive t a and t b can be read-ily extended to an arbitrary combination of ± t a , ± t b bythe transformation c i → ( − i c i ( c = a and/or b ). This isbecause for V , = 0 the orbital diagonal and orbital off-diagonal modes do not mix even within the same channel.The OD susceptibility χ OD ( q ) is then insensitive to thesigns of t a and t b , i.e. the OD divergence always takesplace at ( π, π ). The OO susceptibility χ OO ( q ) is shiftedby ( π, π ) if t a t b <
0, i.e. the OO divergence is at the zonecenter in this case.For small non-zero cross hopping V , the location ofthe divergent modes are still determined by the signs of t a and t b . The main effect of such a finite V , is a partiallifting of the degeneracy of χ OO ( q ), as shown in Fig. 3for V , = 0 . a † σ b − σ and b † σ a − σ modes formsymmetric and anti-symmetric combinations which fol-low distinct q dependences. The degeneracy of ↑↓ and ↓↑ channels is not affected by the spin preserving hop-ping. IV. DISCUSSION
Before discussing various limits of the studied model,we point out formal equivalence between the excitoniccondensation and superconductivity. This can be seenby exchanging the notion of particle and hole for one ofthe fermionic species, e.g. b i → b † i , which turns a - b repul-sion into attraction. This equivalence obviously breaksdown when electromagnetic response is concerned sincethe excitons carry no charge. Nevertheless, it is usefulto consider the analogy to superconductivity, which ismore familiar to most physicists. The excitonic order pa-rameter in our study is local, i.e. has no k -dependence,which is analogous to s -wave superconductivity. An h ab i order parameter, composed of different orbitals, is un-usual for a superconductor, due to the weakness of theelectron-electron attraction, but can be easily realized inan excitonic condensate, as the electron-hole attraction isstrong. Consisting of two distinct orbitals, the spin part h ab i order parameter is not restricted by Pauli principleand can be both singlet or triplet. It is the J >
A. Strong-coupling limit
The strong-coupling limit is characterized by the LSand HS states being separated from the remainingatomic states by energy E i − E HS / LS ≫ | t a | , | t b | , | V , | .In this case an effective model without charge fluc-tuations can be formulated using the Schrieffer-Wolfftransformation , which provides a simplified picture ofthe low-energy physics. The resulting effective Hamilto-nian with hopping treated to the second order is derivedin Appendix A. In the following, we discuss some of itsaspects.
1. Density-density interaction ( H ′ int = 0 ) First, we consider model (1) with the density-densityinteraction only for which the DMFT calculations, re-ported in preceding section, were performed. The effec-tive Hamiltonian then has the form H ddeff = X i µn i + K ⊥ X ij,s d † i,s d j,s + X h ij i (cid:0) K k n i n j + K S zi S zj (cid:1) + K X h ij i ,s (cid:16) d † i,s d † j, − s + d i,s d j, − s (cid:17) . (2)describing two flavors s = ± n i = P s d † i,s d i,s ≤
1, corresponding to HSstates created by d † = a †↑ b ↓ and d †− = a †↓ b ↑ out of theLS vacuum. Neglecting the cross-hopping contributionthe coupling constants have a simple form µ = ∆ − J − Z t a + t b U − J , K ⊥ = t a t b U − J , K k = ( t a + t b ) U +4 J ( U − J )( U + J ) , and K = t a + t b U + J , where Z = 4 is the number of nearest neigh-bors. The last term appears only for finite cross hoppingand has the form K = − V V U − J ( U + J − ∆)( U − J +∆) .
2. Classical limit ( ζ = 0 ) The behavior of model (1) as revealed by the DMFTcalculations strongly depends on the band asymmetry ζ .The OD instability was found only for rather asymmetricbands t a t b ≪ t a + t b , which leads to K ⊥ ≪ K k in (2).In the limit t a t b = 0 the hopping K ⊥ disappears, andthe effective model (2) reduces to the classical Blume-Emmery-Griffiths (BEG) model . Assigning s i = 0 to | LS i and s i = ± d †± | LS i one arrives at its usual form H BEG = µ X i s i + X h ij i (cid:0) K k s i s j + K s i s j (cid:1) . (3)With our choice of the parameters U , J , t a + t b and ∆,we have µ ≈ µ = 0 corresponds to ∆ = 3 .
41) and K k /K = 4. According to Ref. 30, for K k /K = 4 and µ between µ min < < µ max the BEG model exhibits a solid(S) order, characterized by a checker-board arrangementof HS and LS sites. This is equivalent to a staggereddensity h n i i in the language of the bosonic model (2).For µ < µ > T . The solidorder as well as the reentrant transition was found alsoin previous DMFT simulations of 2BHM with asym-metric bands. Proximity to the BEG limit thus providesa simple explanation of the OD instability in the strongcoupling and asymmetric bands region of model (1). Theanalysis of the BEG model suggests that for µ ≈ µ min competition between the anti-ferromagnetic and the solidphase gives rise to a rather complicated phase diagram.This parameter range is, however, beyond the scope ofthis work.
3. Superfluid phase
For general ζ , the hopping K ⊥ cannot be neglected.Much studied in the context of cold atoms, the spinlessversion of (2) is known to host a superfluid (SF) phasein addition to the solid (S) phase discussed above. Ex-istence of a supersolid order at the boundary betweenS and SF phases is a subject of intense research on themodel generalizations . The spinless model (2) can alsobe derived as the strong-coupling limit of the extendedFalicov-Kimball model .The SF phase is characterized by a finite value of h d i,s i ,which corresponds to spontaneous appearance of an off-diagonal expectation value h a † i,σ b i, − σ i in 2BHM, and thuscan be identified with the observed OO instability. With-out cross-hopping, V , = 0, the SF phase of (2) is similarto the spinless case in the sense that it consists of twocopies of the latter coupled only by amplitude fluctua-tions. Inclusion of the cross-hopping has a very differenteffect on the spinless and spinful models. In the spinlesscase , the cross-hopping must have the same form as the d operator and thus non-zero V introduces a source term V ∗ d + V d † to the Hamiltonian, removing the distinctionbetween the normal and SF phases. In the spinful case(2), however, the spin-preserving cross hopping has a dif-ferent spin symmetry than the d s operators and thereforenon-zero V introduces the K term instead. Finite K locks together the phases of h d i, i and h d i, − i . This isreflected in the partial lifting of the degeneracy of theOO mode. The distinction between the normal and SFphases is thus preserved irrespective of the cross hopping.
4. SU(2) symmetric interaction
Next, we discuss the effect of the spin-flip and pair-hopping terms in H ′ int , which were not included in theDMFT simulation. The spin-flip term renders model (1) SU (2) symmetric and a third boson d † = a †↑ b ↑ − a †↓ b ↓ √ ap- pears in the effective model H eff = X i µn i + K ⊥ X ij d † i d j + X h ij i (cid:0) K k n i n j + K S i · S j (cid:1) − K X h ij i (cid:16) d † i · d † j + d i · d j (cid:17) + K X i,j ( d i + d † i ) · S j . (4)Here, ( S i ) α = P ss ′ d † i,s S αss ′ d i,s ′ and n i = P s d † i,s d i,s ,where s = 0 , ± S αss ′ are spin S=1 operators. The d operators are arranged in a vector d = (cid:0) √ ( d − − d ) , i √ ( d − + d ) , d (cid:1) . As before, the hard-core con-straint n i ≤ is assumed. We are not aware of any specificstudies of the S = 1 model (4). On a mean field level onecan repeat the arguments used for the density-density in-teraction which lead to the expectation of solid order for K ⊥ ≪ K k . The SF order parameter generalizes to a 3-component vector the phase of which is again determinedby the K term. The K term is new and does not havean analogy in the density-density case.
5. Coupling constants
The full expressions for the coupling constants aregiven in Appendix A. Here, we consider their signs asfunctions the hopping parameters t a,b and V , and im-plications for the broken symmetry phases.Varying the chemical potential µ ≈ ∆ − J , we can tunebetween two ‘trivial’ limits: the vacuum state h n i i ≈ h n i i ≈ h n i i .The fact that K k is always positive, being proportionalto t a + t b , V + V , implies that, irrespective of the signs ofthe hoppings, the OD instability leads always to an anti-ferro (AF) order. Similarly, K ∼ t a + t b , V + V impliesthat there is always AF magnetic interaction between thenearest neighbors. The sign of K ⊥ ∼ t a t b depends on therelative sign of t a and t b . The cross-hopping contributionto K ⊥ is proportional to V V J ′ and thus may interfereboth constructively or destructively with the t a t b term. K ⊥ > K ⊥ < π, π ) to (0 ,
0) simply by changingthe sign of t a or t b .Non-zero K fixes the phase of h d i in the SF phase.Depending on the sign of K K ⊥ it selects h d i to be real orimaginary. This corresponds to divergence of either thesymmetric a † b + b † a or the anti-symmetric a † b − b † a OOmode. The K term appears when the pair-hopping J ′ =0 or the cross-hopping V , = 0 is present. Inspection ofthe formulas in Appendix A shows that for V , = 0 the K ∼ − J ′ t a t b contribution always favors real h d i , whilefor V , = 0 one can get either sign of K K ⊥ .Finally, K ∼ ( V t a + V t b ) appears only in the SU (2)symmetric case with the cross-hopping present. In case of h d i having a real component this term acts as an effectiveZeeman field and induces spin polarization along h d i . B. Weak-coupling limit
In the weak coupling limit, we consider almost empty(full) a ( b ) bands with a small mutual overlap and searchfor the divergencies of the static susceptibility using therandom phase approximation. The bare susceptibility, inthis case, is dominated by the diagonal elements χ ab,ab ,corresponding to formation of electron-hole pairs withdifferent orbital characters. The χ aa,aa and χ bb,bb ele-ments, as well as χ aa,ab which may appear due to thecross-hopping, are small and we can restrict our consid-erations to the 2 × t a t b the diagonal element χ ab,ab is peaked either at (0 ,
0) or ( π, π ) due to Fermi surfacenesting. If V , = 0 an off-diagonal element χ ab,ba ap-pears.We find divergent susceptibilities in the magnetic(triplet) channel which have the form χ S,A OO = χ ab,ab ± χ ab,ba − ( U − J ± J ′ )( χ ab,ab ± χ ab,ba ) (5)and belong to a symmetric a † b + b † a and an anti-symmetric a † b − b † a mode, respectively. Positive J al-ways favors χ S OO to be the leading divergence. The cross-hopping V , , which controls the sign of χ ab,ba , may se-lect χ S OO as well as χ A OO to be the leading instability.For J ′ = V , = 0 the two modes are degenerate. With-out Hund’s coupling ( J = 0) the singlet andtriplet channels become degenerate. In that case, non-zero cross-hopping V , may preclude the phase transitionin that the singlet excitonic pairing only enhances theexisting off-diagonal expectation values. With Hund’scoupling the triplet order parameter always represents atrue symmetry breaking as it has distinct symmetry foran arbitrary spin-preserving hopping.In the mean-field picture, assuming an F order for sim-plicity, we get H MF = (cid:18) ε a ( k ) σ V ( k ) σ + σ · φ V ∗ ( k ) σ + ( σ · φ ) ∗ ε b ( k ) σ (cid:19) , (6)with σ α being the Pauli matrices in the spin space. Di-vergence of χ S OO implies φ ∗ = φ while divergence of χ A OO implies φ ∗ = − φ (for details see Appendix B). Omit-ting the overall charge conservation, which is not brokenat the transition, the order parameter reduces the SU (2)symmetry of (1) into U(1) and thus behaves as a point on + − + − + − + − + − + − + − + − + − + − + − + − t b t a V V t a t a t b t b V V V V a)b)c) FIG. 6: An example of various combinations of the hoppingswith orbitals of s and p z symmetry. a ) t a,b > V , = 0, b ) t a,b > V = − V , and c ) t a > t b < V = − V . S sphere. If J ′ = V = 0 Hamiltonian (1) has additional U (1) symmetry associated with the relative phase of a and b states. Breaking this symmetry leads to a complexorder parameter that lives in S × S .Expressions (5, 6) hold also in the case of density-density interaction with the provision that divergent χ S,A OO are found only in the ↑↓ and ↓↑ channels (not in ↑↑ - ↓↓ )and φ z ≡ SU (2) symmetry of Hamiltonian(1) reduces to U (1) in case of the density-density inter-action. The order parameter for non-zero V , is a realor imaginary vector ( φ x , φ y ) living in S . If V , = 0 therelative phases of all spin-orbital flavors are independentleading to [ U (1)] symmetry, which is reduced to U (1) atthe transition. The order parameter is then a complexvector ( φ x , φ y ) living in S × S . C. Physical meaning of the excitonic orderparameter
Finally, we discuss the physical meaning of the real,imaginary or complex excitonic order parameter. InFig. 6 we present simple realizations of these phases using s and p z orbitals: a) V , = 0 with complex order param-eter φ , b) t a t b V V > φ and c) t a t b V V < φ .Let us start by considering real φ = (0 , , φ z ). Thecorresponding operator a †↑ b ↑ − a †↓ b ↓ + b †↑ a ↑ − b †↓ a ↓ de-scribes the z-component of magnetization (spin) densitywith the distribution given by the product of a and b orbitals ϕ a ( x ) ϕ b ( x ). In present case, the product is a p function, i.e. the leading multipole of the distribution isa dipole and the above operator may be viewed as de-scribing an on-site magnetic quadrupole. The rotationof φ corresponds to changing the magnetization direc-tion while keeping its distribution fixed, i.e. cannot beviewed as a 3D rotation of the quadrupole as rigid object.The operator a †↑ b ↑ − a †↓ b ↓ − b †↑ a ↑ + b †↓ a ↓ correspondingto imaginary φ = (0 , , φ z ) describes an on-site patternof a magnetization current. Rotation of imaginary φ cor-responds to changing the magnetization direction whilekeeping the current pattern fixed. Complex φ is difficultto visualize. In this case it is possible to continuously ro-tate magnetic multipole into a local spin current withoutchanging the energy of the system.A model built on d z and d x − y orbitals may be morerealistic with respect to real materials. Similar consider-ations would apply leading to a finite value of magneticoctupole, in case of real, and more a complicated patternof the on-site spin current, in case of imaginary orderparameter. While the direct experimental detection ofthe magnetic multipoles may be experimentally difficult,presumably, the most experimentally accessible would bethe effect of excitonic order on the transport propertiesat weak to moderate coupling. D. Further work
Despite a narrow parameter range in the vicinity ofthe spin-state transition, the present results reveal arich phase diagram, nevertheless, other phases may existnearby. In the ζ = 0 limit and ∆ below the studied range,the BEG phase diagram contains anti-ferromagnetic HSphase separated from the solid HS-LS phase by a nar-row strip of a phase containing both magnetic and HS-LS order. For finite ζ the boundary between the S andSF provides an interesting possibility for a stable super-solid phase. Although it was excluded for 2D spinlessbosons with a simple nn repulsion, the effect of theadditional terms in (4) or the departure from the strong-coupling limit is unexplored. Another interesting ques-tion is the possibility of coexistence of the SF and AFmagnetic orders, observed in the bosonic t-J model withanisotropic exchange .Our investigation of the Hubbard model in the vicinityof spin-state transition was motivated by the physics ofLaCoO . While a two-band model ignoring the electron-lattice coupling is probably too simplistic to describe thiscomplicated multi-orbital material, some useful insightsare obtained. In particular, the present study shows thatthe excitonic condensation is in a broad range of param-eters preferred to the HS-LS order, an order which hasbeen discussed in LaCoO context and treated with first-principles LDA+U method . The proposal of excitoniccondensation in this material may be tested on the samelevel of approximation by introducing the ’excitonic’ in-stead of the standard mean-field decoupling of the on-site interaction in LDA+U. V. CONCLUSIONS
Using dynamical mean-field theory we have per-formed an unbiased numerical search probing all possibleparticle-hole instabilities of the two-band Hubbard modelin the parameter range close to the spin-state transition.Our main result is the observation of an instability to-wards condensation of spinful excitons. Together withthe previously reported solid HS-LS order, these are theonly instabilities of the model in the studied parameterrange. We have shown that keeping other parametersfixed the bandwidths ratio is the control parameter se-lecting the leading instability, an observation which hasa particularly simple explanation in the strong couplinglimit as tuning the ration of nn hopping and nn repulsionin a hard-core bosons model. The strong-coupling map-ping onto spinful hard-core bosons with nn interactionprovides a possibility of electronic realization of some ex-otic phases observed with cold atoms. Comparing thesolid HS-LS order and the superfluid excitonic order wefind that the former does not exist in the weak couplingregime and due to its Ising character can be easily sup-pressed by geometrical frustration, while the latter ex-ists both in strong and weak coupling limits and due tothe continuous character can better adapt to geometricalfrustration, e.g. by forming a 120 ◦ order on triangularlattice. The main implication for real materials is the factthat the excitonic condensation should be considered acompetitor to the HS-LS order in systems close to thespin-state transition. Acknowledgments
We thank D. Vollhardt, A. Kampf, A. Kauch, P.Nov´ak, R. T. Scalettar and J. Otsuki for discussions andvaluable suggestions. We acknowledge the support ofDeutsche Forschungsgemeinschaft through FOR1346 andthe Grant Agency of the Czech Republic through project13-25251S.
Appendix A: Strong coupling parameters
The parameters of the bosonic model were obtained bysecond order perturbation theory in the hopping using( H eff ) αβ = h α | H | β i +12 X i (cid:18) h α | H | i ih i | H | β i E α − E i + h α | H | i ih i | H | β i E β − E i (cid:19) , (A1)where | α i and | β i are the states built from the local LSand HS states and | i i is everything else. The formula wasevaluated in Mathematica using the SNEG package . µ = ∆ − J + Z (cid:0) t a + t b (cid:1) (cid:18) J ′ ∆ ′ ( U − J + 2∆ ′ ) − J ′ ′ (∆ ′ + ∆) ( U − J + ∆ ′ + ∆) − ∆ ′ + ∆2∆ ′ ( U − J + ∆ ′ − ∆) (cid:19) + Z V + V J ′ ∆ ′ (∆ ′ + ∆) ( U − J + 2∆ ′ + ∆) − U − J + ∆ ′ + (∆ ′ + ∆) ∆ ′ ( U − J + 2∆ ′ − ∆) ! K k = (cid:0) t a + t b (cid:1) (cid:18) − J ′ ∆ ′ ( U − J + 2∆ ′ ) − J ′ ( U + J )∆ ′ (∆ ′ + ∆) + J ′ ∆ ′ (∆ ′ + ∆) ( U − J + ∆ ′ + ∆)+ ∆ ′ ( U + J − ∆) + ∆(3 J + ∆)( U + J )∆ ′ ( U − J + ∆ ′ − ∆) (cid:19) + V + V − J ′ ∆ ′ (∆ ′ + ∆) ( U − J + 2∆ ′ + ∆) + 4 U − J + ∆ ′ − (∆ ′ + ∆) ∆ ′ ( U − J + 2∆ ′ − ∆) − U + J )( U + J ) − ∆ ! K ⊥ = t a t b (cid:18) J ′ ∆ ′ (∆ ′ + ∆) ( U − J + ∆ ′ + ∆) + ∆ ′ + ∆∆ ′ ( U − J + ∆ ′ − ∆) (cid:19) + V V J ′ ∆ ′ ( U − J + ∆ ′ ) K = (cid:0) t a + t b (cid:1) U + J + (cid:0) V + V (cid:1) U + J ( U + J ) − ∆ K = − t a t b J ′ ( U − J + ∆ ′ )( U + J )∆ ′ ( U − J + 2∆ ′ ) − V V (cid:18) J ′ ( U − J + ∆ ′ + ∆)∆ ′ (∆ ′ + ∆) ( U + J + ∆) ( U − J + 2∆ ′ + ∆)+ (∆ ′ + ∆) ( U − J + ∆ ′ − ∆)∆ ′ ( U + J − ∆) ( U − J + 2∆ ′ − ∆) (cid:19) K = − V t a + V t b √ p ∆ ′ (∆ ′ + ∆) (cid:18) J ′ (cid:18) U − J + ∆ ′ + 1 U − J + ∆ ′ + ∆ + 1 U + J + ∆ + 1 U + J (cid:19) + (∆ ′ + ∆) (cid:18) U − J + ∆ ′ + 1 U − J + ∆ ′ − ∆ + 1 U + J − ∆ + 1 U + J (cid:19)(cid:19) , where ∆ ′ = √ ∆ + J ′ . In Hamiltonian (1) we did dis-tinguish between J in H ddint and in H ′ int . Nevertheless,the above expressions apply to both the models withdensity-density interaction H ddint and the full interaction H ddint + H ′ int with the provision that in the density-densitycase K = 0 and the other expressions are evaluated for J ′ = 0. Appendix B: Mean-Field Decoupling
Here we show how a mean-field decoupling of the ( U − J ) P σ n a,σ n b, − σ term in the interaction gives rise tothe spontaneous hybridization in the SF phase. First, weconsider the J ′ = V , = 0 case with degenerate χ S OO and χ A OO modes. Writing the above term as − ( U − J ) ( a †↑ b ↓ )( b †↓ a ↑ ) − ( U − J ) ( a †↓ b ↑ )( b †↑ a ↓ ) (B1)we obtain decoupling φ a †↑ b ↓ + φ ∗ b †↓ a ↑ + φ − a †↓ b ↑ + φ ∗− b †↑ a ↓ , (B2)using complex fields φ and φ − , which acquire finitevalues φ = φ x + iφ y ∼ h b †↓ a ↑ i , φ − = φ x − iφ y ∼ h b †↑ a ↓ i (B3) in the SF phase.If the χ S OO and χ A OO are not degenerate the fields φ and φ − are not independent. In this case we use a decou-pling which based on the symmetric and anti-symmetricmodes starting from rewriting the interaction as − U − J (cid:16) a †↑ b ↓ + b †↑ a ↓ (cid:17) (cid:16) a †↓ b ↑ + b †↓ a ↑ (cid:17) − U − J (cid:16) a †↑ b ↓ − b †↑ a ↓ (cid:17) (cid:16) b †↓ a ↑ − a †↓ b ↑ (cid:17) (B4)leading to a decoupling φ S (cid:16) a †↑ b ↓ + b †↑ a ↓ (cid:17) + φ ∗ S (cid:16) a †↓ b ↑ + b †↓ a ↑ (cid:17) + φ A (cid:16) a †↑ b ↓ − b †↑ a ↓ (cid:17) + φ ∗ A (cid:16) b †↓ a ↑ − a †↓ b ↑ (cid:17) (B5)with φ S ∼ h a †↓ b ↑ + b †↓ a ↑ i , φ A ∼ h b †↓ a ↑ − a †↓ b ↑ i . (B6)Comparing the corresponding terms in H MF we see thatfinite φ S implies φ = φ ∗− and thus real φ x and φ y .Finite φ A on the other hand implies φ = − φ ∗− and thusimaginary φ x and φ y .Since the decoupled term appears in both the SU (2)and density-density interactions the above derivationsapplies to both cases. In the SU (2) interaction, whichincludes the spin-flip term, decoupling in terms of a †↑ b ↑ − a †↓ b ↓ is possible, which leads to the same mean-field equa- tions and gives rise to the φ z component of the orderparameter. A. Georges, L. de’Medici, and J. Mravlje, Annu. Rev. Con-dens. Matter Phys. , 137 (2013). Z. P. Yin, K. Haule, and G. Kotliar, Nat. Phys. , 294(2011). J. Kuneˇs, A. V. Lukoyanov, V. I. Anisimov, R. T. Scalet-tar, and W. E. Pickett, Nat. Mater. , 198 (2008). J. Kuneˇs, D. M. Korotin, M. A. Korotin, V. I. Anisimov,and P. Werner, Phys. Rev. Lett. , 146402 (2009). V. Kˇr´apek, P. Nov´ak, J. Kuneˇs, D. Novoselov, D. M. Ko-rotin, and V. I. Anisimov, Phys. Rev. B , 195104 (2012). R. Eder, Phys. Rev. B , 035101 (2010). P. Augustinsk´y and J. Kunesˇs, Computer Physics Commu-nications , 2119 (2013). J. Chaloupka and G. Khaliullin, Phys. Rev. Lett. ,207205 (2013). P. Werner and A. J. Millis, Phys. Rev. Lett. , 126405(2007). R. Suzuki, T. Watanabe, and S. Ishihara, Phys. Rev. B , 054410 (2009). J. Kuneˇs and V. Kˇr´apek, Phys. Rev. Lett. , 256401(2011). B. I. Halperin and T. M. Rice, Rev. Mod. Phys. , 755(1968). L. Balents and C. M. Varma, Phys. Rev. Lett. , 1264(2000). C. D. Batista, Phys. Rev. Lett. , 166403 (2002). B. Zenker, D. Ihle, F. X. Bronold, and H. Fehske, Phys.Rev. B , 121102 (2012). B. Zenker, D. Ihle, F. X. Bronold, and H. Fehske, Phys.Rev. B , 235123 (2011). K. Seki, R. Eder, and Y. Ohta, Phys. Rev. B , 245106(2011). B. Zocher, C. Timm, and P. M. R. Brydon, Phys. Rev. B , 144425 (2011). T. Kaneko, K. Seki, and Y. Ohta, Phys. Rev. B , 165135(2012). G. Schmid, S. Todo, M. Troyer, and A. Dorneich, Phys.Rev. Lett. , 167208 (2002). G. G. Batrouni and R. T. Scalettar, Phys. Rev. Lett. ,1599 (2000). A. Kuklov, N. Prokof’ev, and B. Svistunov, Phys. Rev.Lett. , 050402 (2004). M. Boninsegni and N. V. Prokof’ev, Phys. Rev. B ,092502 (2008). A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg,Rev. Mod. Phys. , 13 (1996). W. Metzner and D. Vollhardt, Phys. Rev. Lett. , 324(1989). P. Werner, A. Comanac, L. de’ Medici, M. Troyer, andA. J. Millis, Phys. Rev. Lett. , 076405 (2006). E. Gull, A. J. Millis, A. I. Lichtenstein, A. N. Rubtsov,M. Troyer, and P. Werner, Rev. Mod. Phys. , 349 (2011). J. R. Schrieffer and P. A. Wolff, Phys. Rev. , 491(1966). M. Blume, V. J. Emery, and R. B. Griffiths, Phys. Rev. A , 1071 (1971). W. Hoston and A. N. Berker, Phys. Rev. Lett. , 1027(1991). F. Mila, J. Dorier, and K. P. Schmidt, Prog. Theor. Phys.Supplement , 355 (2008). K. Kn´ıˇzek, Z. Jir´ak, J. Hejtm´anek, P. Nov´ak, and W. Ku,Phys. Rev. B , 014430 (2009). R. ˇZitko, Comput. Phys. Commun.182