Charge-density wave induced by combined electron-electron and electron-phonon interactions in 1 T -TiSe 2 : A variational Monte Carlo study
aa r X i v : . [ c ond - m a t . s t r- e l ] M a y Charge-density wave induced by combined electron-electron and electron-phononinteractions in 1 T -TiSe : A variational Monte Carlo study Hiroshi Watanabe , ∗ Kazuhiro Seki , and Seiji Yunoki , , Computational Quantum Matter Research Team,RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan Computational Condensed Matter Physics Laboratory, RIKEN, Wako, Saitama 351-0198, Japan Computational Materials Science Research Team,RIKEN Advanced Institute for Computational Science (AICS), Kobe, Hyogo 650-0047, Japan (Dated: June 1, 2015)To clarify the origin of a charge-density wave (CDW) phase in 1 T -TiSe , we study the groundstate property of a half-filled two-band Hubbard model in a triangular lattice including electron-phonon interaction. By using the variational Monte Carlo method, the electronic and lattice degreesof freedom are both treated quantum mechanically on an equal footing beyond the mean-fieldapproximation. We find that the cooperation between Coulomb interaction and electron-phononinteraction is essential to induce the CDW phase. We show that the “pure” exciton condensationwithout lattice distortion is difficult to realize under the poor nesting condition of the underlyingFermi surface. Furthermore, by systematically calculating the momentum resolved hybridizationbetween the two bands, we examine the character of electron-hole pairing from the viewpoint ofBCS-BEC crossover within the CDW phase and find that the strong-coupling BEC-like pairingdominates. We therefore propose that the CDW phase observed in 1 T -TiSe originates from aBEC-like electron-hole pairing. PACS numbers: 71.10.-w, 71.45.Lr, 71.35.Lk, 71.27.+a
I. INTRODUCTION
Charge-density wave (CDW) is widely observed in low-dimensional solids and has been extensively studied bothexperimentally and theoretically [1, 2]. Transition metaldichalcogenides MX (M=transition metal, X=S, Se, Te)are one of the typical CDW materials with a layeredtriangular lattice structure. They show various CDWpatterns, depending on the combination of M and X [3–5], and quite often superconductivity (SC) is observednext to the CDW phase by applying pressure [6, 7], dop-ing [8, 9], or intercalation [10]. However, the origin ofCDW and SC has not been fully understood and it hasbeen still under debate in spite of the long and extensivestudies so far.Recently, 1 T -TiSe , one of the old transition metaldichalcogenides, has again attracted much interests inthe context of exciton condensation. This material is asemimetal or a semiconductor in room temperature [11–13] and shows a commensurate CDW transition with a2 × × T c ∼ d and Se 4 p bands and the flat energy spectrum just belowthe Fermi energy have been observed [16], strongly sug- ∗ [email protected] gesting the possibility of exciton condensation. On theother hand, another possible mechanism for the CDWtransition is a band Jahn-Teller effect which results fromelectron-phonon interaction [18–21]. A large lattice dis-tortion of several percent observed below T c [4] indicatesthe strong coupling between electronic and lattice de-grees of freedom [22]. Very recently, it is proposed thatboth mechanisms work cooperatively for the CDW tran-sition [23–27].Exciton condensation is a quantum state expected inlow carrier density systems such as a semimetal or asemiconductor and has been extensively studied since1960s [28–31]. The exciton is a bound pair of an electronand a hole in different bands mediated by the interbandCoulomb interaction. When the binding energy of anexciton exceeds the band gap, the system has an insta-bility toward condensation of excitons, namely, a spon-taneous hybridization between different bands. Since therepulsive Coulomb interaction is attractive between anelectron and a hole, the exciton condensation is expectedto occur in principle without considering any additional“glue” of the electron-hole pair. Although extensive ef-forts have been devoted for half a century and severalcandidates have been proposed [16, 32–35], the gener-ally accepted materials for the exciton condensation arestill absent. Therefore, any conclusive evidence for theexciton condensation in real materials is highly desiredfor further progress and 1 T -TiSe would be one of thepromising examples.In the pioneering studies for exciton condensation [29,30], an isotropic band dispersion with perfect FS nest-ing is assumed, for which the excitonic instability is al-ways present in a semimetallic case. The extension toanisotropic band dispersions shows that the degree of FSnesting greatly affects the instability of exciton condensa-tion [36]. Moreover, the mean-field approximation gener-ally overestimates the instability toward ordered states,including the exciton condensation. Therefore, for dis-cussion of the exciton condensation in real materials, therealistic band dispersion and the appropriate method be-yond the mean-field approximation are both required.In this paper, a half-filled two-band Hubbard modelwith electron-phonon interaction in a triangular latticeis studied to understand the origin of the CDW phasein 1 T -TiSe . The ground state properties are calculatedusing the variational Monte Carlo (VMC) method formultiorbital systems [37]. We find that the cooperationbetween Coulomb interaction and electron-phonon inter-action is essential to induce the CDW phase. The CDWphase is observed between the normal metal and bandinsulator phases with intermediate interband Coulombinteraction. We show that the “pure” exciton condensa-tion without lattice distortion is difficult to realize un-der the poor FS nesting condition in a triangular lattice.Furthermore, we systematically calculate the momentumresolved hybridization between the two bands to showthat the strong-coupling BEC-like pairing dominates inthe CDW phase. Our results therefore suggest that theCDW phase observed in 1 T -TiSe originates from thestrong-coupling BEC-like electron-hole pairing.The rest of this paper is organized as follows. In Sec. II,a two-band Hubbard model in a two-dimensional trian-gular lattice is introduced as a low energy effective modelfor 1 T -TiSe . The detailed explanation of the VMCmethod and the variational wave functions are also givenin Sec. II. The numerical results are then provided inSec. III. Finally, the implication of our results for 1 T -TiSe is discussed in Sec. IV, followed by the summaryin Sec. V. II. MODEL AND METHOD
We consider a two-band Hubbard model in a two-dimensional triangular lattice defined as H = X k ,σ ε c k c † k σ c k σ + X k ,σ ε f k f † k σ f k σ + U c X i n ci ↑ n ci ↓ + U f X i n fi ↑ n fi ↓ + U ′ X i n ci n fi + 1 √ N X k , q ,σ h g ( k , q )( b q + b †− q ) c † k + q σ f k σ + H.c. i + X q ω ( q ) (cid:18) b † q b q + 12 (cid:19) , (1)where c † k σ ( f † k σ ) creates an electron in c ( f ) band withmomentum k and spin σ (= ↑ , ↓ ). The band dispersions FIG. 1. (color online) (a) Fermi surfaces (blue circle andred ellipses) and (b) energy dispersions in the noninteract-ing limit for the two-band Hubbard model with electrondensity n = 2. A set of tight-binding parameters used is( t f , t c , t ′ c , µ c ) = (1 . , . , . , . t with t = t f as an energyunit. (c) Schematic real-space figure of a 2 × q , q ,and q are indicated in (a). Dashed lines in (a) represent thefolded Brillouin zone due to the formation of the 2 × c - f plane along one particular di-rection, e.g., AB direction indicated in (c). u i represents thedisplacement of c atom at site i from its original position. ε c k and ε f k are given as ε c k = 2 t c cos k x + 2 cos 12 k x cos √ k y ! + 2 t ′ c cos √ k y + 2 cos 32 k x cos √ k y ! + µ c (2)and ε f k = 2 t f cos k x + 2 cos 12 k x cos √ k y ! , (3)respectively. We introduce the next-nearest-neighborhopping t ′ c to locate the bottom of c band at M points. U c ( U f ) is an on-site intraband Coulomb interaction within c ( f ) band and U ′ is an on-site interband Coulomb interac-tion between c and f bands. n αiσ is a number operator of α (= c, f ) electron at site i with spin σ and n αi = n αi ↑ + n αi ↓ . g ( k , q ) is an electron-phonon coupling constant and b † q isa creation operator of phonon with momentum q and fre-quency ω ( q ). In this model, the lattice distortion changesthe c - f bond length as shown in Fig. 1(d), which couplesto the c - f hybridization modulated with wave vector q through g ( k , q ). The total number of sites is indicatedby N .The noninteracting tight-binding parameters,( t f , t c , t ′ c , µ c ) = (1 . , . , . , . t, (4)are set to mimic the electronic structure of 1 T -TiSe witha hole pocket at the Γ point and electron pockets at the Mpoints as shown in Figs. 1(a) and 1(b). The ordering wavevectors connecting Γ and M points are denoted as q , q ,and q . Although the ordering wave vectors observed in1 T -TiSe connect Γ and L points with a finite k z compo-nent [38], here we consider a pure two-dimensional modelfor simplicity and the limitation of the model is discussedlater.The effect of Coulomb interaction and electron-phononinteraction is treated on an equal footing using a VMCmethod. We consider the trial wave function as follows: | Ψ i = P e-ph | Ψ ph i | Ψ e i . (5) | Ψ e i = P (2)G P J c | Φ i is an electron wave function con-sisting of three parts. | Φ i is a Slater determinant con-structed by diagonalizing the one-body part of Hamilto-nian H including the variational tight-binding parame-ters (˜ t f = 1, ˜ t c , ˜ t ′ c , ˜ µ c ) and the off-diagonal element V which induces the c - f hybridization. Here, we assume V = V exp[ − A (˜ ε c k + q i − ˜ ε f k ) ], and V and A are bothvariational parameters. V is an amplitude of the c - f hybridization and A controls the internal extent of ex-citon in k space. We have found that the variationalenergy is improved by introducing A and the behavior of A is related to the BCS-BEC crossover of exciton con-densation [39]. P (2)G is a Gutzwiller factor extended fortwo-band systems [39, 40]. In P (2)G , possible 16 patternsof charge and spin configuration at each site | Γ i , i.e., | i = | i , | i = | ↑i , · · · , | i = |↑↓ ↑↓i , are differentlyweighted and their weight { g Γ } are optimized as varia-tional parameters. P J c = exp[ − P i = j P αβ v αβij n αi n βj ] isa charge Jastrow factor which controls long-range chargecorrelations. Here, v αβij = v αβ ( | r i − r j | ) is assumed and r i is the position of site i .The trial wave function for phonon is assumed to bea Gaussian in the normal coordinate { Q q } representa-tion [41, 42],Ψ ph ≡ (cid:10) { Q q } (cid:12)(cid:12) Ψ ph (cid:11) = exp " − X q
12 ( Q q − β q ) α q , (6)where Q q = P i u i e − i q · r i / √ N is Fourier transform of realspace lattice distortion { u i } at site i . Since the orderingwave vectors q , q , and q are exactly half of the recip-rocal lattice vectors of the normal phase [see Fig. 1(a)],the corresponding normal coordinate Q q i ( i = 1 , , α q and β q all real.Notice that α q controls the extent of the Gaussian wavefunction, i.e., the amplitude of lattice vibration, and that FIG. 2. (color online) Ground state phase diagram of thetwo-band Hubbard model in (a) U - U ′ plane ( g/t = 0 .
19) and(b) U - g plane ( U ′ = U/ U c = U f = U and ω/t =0 .
1. NM, CDWI, and BI denote normal metal, charge-density-wave insulator, and band insulator, respectively. The electrondensity is n = 2, i.e., at half filling. β q corresponds to the average value of Q q and thus thereexists a static lattice distortion with h u i i 6 = 0 for a finite β q . The Monte Carlo update scheme for { Q q } and theestimation of phonon energy are the same as in Ref. 42.The remaining part is an electron-phonon projectionoperator: P e-ph = exp h γ P i u i n ci (2 − n fi ) i . This oper-ator controls the attraction between c electrons and f holes which results from the electron-phonon interactionand γ is a variational parameter.The variational parameters in | Ψ i are therefore ˜ t c , ˜ t ′ c ,˜ µ c , V , A , { g Γ } , { v αβij } , { α q } , { β q } , and γ , and theyare simultaneously optimized using stochastic reconfig-uration method [43]. The system sizes are varied from L × L = 12 ×
12 to 24 ×
24 with antiperiodic boundaryconditions in both directions of primitive lattice vectorsfor the triangular lattice.
III. RESULT
Figure 2(a) shows the ground state phase diagramwhere U c = U f = U and U ′ are varied for fixed g ( k , q ) /t = g/t = 0 .
19 and ω ( q ) /t = ω/t = 0 . U ′ islarge enough, the c band is lifted above the Fermi energyand the BI phase with the empty c band and the fully-occupied f band is stabilized. No static lattice distortionis observed in both NM and BI phases.Between the NM and BI phases, the CDWI phaseemerges where the c - f hybridization parameter∆ q = X k ,σ (cid:10) c † k + q σ f k σ + H.c. (cid:11) (7)is finite [45] for q corresponding to the three orderingwave vectors q , q , and q , simultaneously, implying a FIG. 3. (color online) (a) Average c -electron density h n c X i (redbars) and f -hole density 2 − h n f X i (blue bars) in 2 × c - f bond length u A and u B at A and B sites, respectively, asa function of Monte Carlo (MC) step. The model parametersused are ( U/t, U ′ /t, g/t, ω/t )=(4.0, 2.0, 0.19, 0.1) for L = 24in the CDWI phase. triple- q CDW state. Here, hOi = h Ψ |O| Ψ i / h Ψ | Ψ i withthe optimized | Ψ i . Thus, the first Brillouin zone is foldedas indicated in Fig. 1(a). It leads to the charge dispro-portionation in 2 × U/t in spite of a finite g/t . This is in sharp contrastwith the case of a square lattice where the NM phase ap-pears only at U ′ = 0 and the exciton condensation phaseis widely observed even without the electron-phonon in-teraction [39, 47, 48]. This difference is caused by thedifferent FS nesting condition: the FS nesting is better(perfect if only with the nearest-neighbor hopping) in thesquare lattice but poor in the triangular lattice. There-fore, the electron-phonon interaction is indispensable tomanifest the CDWI phase under the poor FS nesting [2].We also show the phase diagram in Fig. 2(b) where U and g are varied with U ′ = U/
2. The CDWI re-gion is enlarged with increasing U and g , implying thatboth Coulomb interaction and electron-phonon interac-tion stabilize the CDWI phase. This result is thus qual-itatively consistent with previous study [26]. We alsofind that the CDWI phase is not stabilized, but onlythe NM and BI phases appear, when g = 0, at least,in a realistic parameter region. This suggests that the“pure” exciton condensation induced by the Coulombinteraction alone, the original idea of exciton conden-sation [28–30], is difficult to realize in our model. Thepure exciton condensation certainly occurs in particularmodels such as one-dimensional models [23, 49] or two-dimensional models with perfectly nested electron andhole FSs [39, 47, 48, 50]. Therefore, the stability of the FIG. 4. (color online) Momentum resolved c - f hybridization φ ( k ) for (a) ( U/t, U ′ /t )=(8.0, 4.0) and (b) ( U/t, U ′ /t )=(3.0,1.5). The noninteracting Fermi momentum k c F (folded aroundΓ point at the center) and k f F are shown with white solid andblack dashed curves, respectively [see also Fig. 1(a)]. g/t =0 .
19 and ω/t = 0 . L = 24. pure exciton condensation depends strongly on the lat-tice structure and the underlying FS.Let us now examine the detailed properties of theCDWI phase. Figure 3(a) shows the distribution of aver-age c -electron density h n c X i and f -hole density 2 −h n f X i in2 × c electrons and f holes, i.e., mobile carriers, are con-centrated mostly at A site with h n c A i + h n f A i > c - f hybridization energy coupled with the lattice dis-tortion, while the number of these mobile carriers aresmall and h n c X i + h n f X i < q , q ,and q are equivalent in a hexagonal lattice structure.Next, let us discuss the lattice degrees of freedom in theCDWI phase. In the VMC calculation, the bond length u i always fluctuates around the average value during theMonte Carlo steps [41]. As shown in Fig. 3(b), we findthat the bond length at A site is shortened from theoriginal one ( u A < u B > c - f hybridization φ ( k ) defined as φ ( k ) = X q ,σ (cid:10) c † k + q σ f k σ + H . c . (cid:11) . (8)Figure 4 shows φ ( k ) for ( U/t, U ′ /t )=(8.0, 4.0) and (3.0,1.5), both being located in the CDWI phase in Fig. 2.For ( U/t, U ′ /t )=(8.0, 4.0), φ ( k ) is extended in the wholeBrillouin zone, indicating the strong-coupling BEC-likepairing due to the large Coulomb interaction. Although φ ( k ) becomes less extended with decreasing the Coulombinteraction, it still has a broad structure away fromthe Fermi momentum k F , as shown in Fig. 4(b) for( U/t, U ′ /t )=(3.0, 1.5). Indeed, the CDWI region rapidlydecreases with decreasing U [see Fig. 2(a)] and our sys-tematic calculations do not find a clear BCS-like region inthe CDWI phase shown in Fig. 2. Because of i) the poornesting between c -electron and f -hole FSs and ii) thesmall density of states around the Fermi energy for lowcarrier densities, the energy gain due to the gap open-ing induced by the c - f hybridization in the vicinity of k F is small and hence the weak-coupling BCS-like pair-ing is not favored. Even in such a case, the BEC-liketightly-bounded electron-hole pairing in real space canbe induced by the electron-phonon interaction with thehelp of Coulomb interaction and dominates the CDWIphase. In contrast, we have found a clear and wide BCS-like region for the same model but in a square latticewith perfectly nested FSs [39]. Therefore, the FS nestingis essential for the BCS-like pairing. IV. DISCUSSION
Finally, let us discuss the implication of our resultsfor 1 T -TiSe . In our model, Ti 3 d and Se 4 p bands aresimplified as c and f bands, respectively, and the or-bital characters are ignored. Moreover, our model onlyincludes the change of c - f bond length which coupleswith exciton condensation. Even with these simplifica-tions, our model captures the important energy scalesof 1 T -TiSe . The electron-phonon coupling used here is4 g /ω = 0 − . t ≈ − . T -TiSe [5] if we take t ≈ . . − . ∼ . T -TiSe is due to the strong-coupling BEC-like electron-hole pair-ing. Indeed, the BEC-like character is indicated by sev-eral theoretical works [13, 27] and experimental obser-vations such as a short coherence length estimated byKohn anomaly [5], lack of incommensurate CDW phase,relatively high electrical resistivity above T c , and a largevalue of 2∆ /k B T c (∆: the CDW gap) [38].On the other hand, the chiral CDW phase observedin 1 T -TiSe [55, 56] is beyond our model. In the chi-ral CDW phase, the charge density is modulated withclockwise or anticlockwise pattern. The proper descrip- tion of this phase requires three dimensionality [57] orhigher-order electron-phonon and phonon-phonon inter-actions [26] which induce the phase difference betweenthe three ordering wave vectors. The origin of the SC in-duced by applying pressure or intercalation of Cu atomsis also an interesting unresolved issue. The relation be-tween the CDW and the SC is still controversial [58, 59]and both conventional [20, 23, 60] and unconventionalSC [61] have been proposed. Our results suggest thatboth electronic and lattice degrees of freedom are crucialto understand the origin of the SC. Our study will bea first step toward the unified understanding of variousquantum phases observed in 1 T -TiSe . V. SUMMARY
In summary, we have studied the two-band Hubbardmodel in a triangular lattice for 1 T -TiSe with theelectron-phonon interaction. The VMC method is em-ployed to treat the electronic and lattice degrees of free-dom on an equal footing beyond the mean-field approxi-mation. We have shown that both Coulomb and electron-phonon interactions stabilize the CDW phase. We havefound that the “pure” exciton condensation without thelattice distortion is difficult to realize and the electron-phonon interaction is essential for the CDW phase. Thecharacter of electron-hole pairing within the CDW phasehas also been examined by calculating the momentum re-solved c - f hybridization. We have shown that the strong-coupling BEC-like pairing dominates the CDW phase.Under the poor FS nesting condition and with small den-sity of states around Fermi energy, the energy gain dueto the gap opening in the vicinity of k F is small andhence the weak-coupling BCS-like pairing is not favored.Our results thus conclude that the CDW phase observedin 1 T -TiSe originates from the strong-coupling BEC-likeelectron-hole pairing due to the cooperative Coulomb andelectron-phonon interactions. ACKNOWLEDGMENT
The authors thank Y. Fuseya, T. Shirakawa, T.Kaneko, and K. Imura for useful discussions. The compu-tation has been done using the RIKEN Cluster of Clus-ters (RICC) facility and the facilities of the Supercom-puter Center, Institute for Solid State Physics, Univer-sity of Tokyo. This work has been supported by JSPSKAKENHI Grant No. 26800198 and in part by RIKENiTHES Project. [1] G. Gr¨uner,
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