A variational Monte Carlo study of exciton condensation
aa r X i v : . [ c ond - m a t . s t r- e l ] M a y A variational Monte Carlo study of excitoncondensation
Hiroshi Watanabe , Kazuhiro Seki , and Seiji Yunoki , , Computational Quantum Matter Research Team, RIKEN Center for Emergent MatterScience (CEMS), Wako, Saitama 351-0198, Japan Computational Condensed Matter Physics Laboratory, RIKEN, Saitama 351-0198, Japan Computational Materials Science Research Team, RIKEN Advanced Institute forComputational Science (AICS), Kobe, Hyogo 650-0047, JapanE-mail: [email protected]
Abstract.
Exciton condensation in a two-band Hubbard model on a square lattice is studiedwith variational Monte Carlo method. We show that the phase transition from an excitonicinsulator to a band insulator is induced by increasing the interband Coulomb interaction. Toexamine the character of the exciton condensation, the exciton pair amplitudes both in k -spaceand in real space are calculated. Using these quantities, we discuss the BCS-BEC crossoverwithin the excitonic insulator phase.
1. Introduction
Exciton condensation is one of the most interesting phenomena in strongly correlated electronsystems and it has been extensively studied since 1960s [1, 2, 3]. The exciton is a bound electron-hole pair mediated by the (attractive) Coulomb interaction between them. When the bindingenergy of an exciton exceeds the energy gap between the electron and hole bands, the system hasan instability towards condensation of excitons. Despite the extensive studies for half a century,the example of exciton condensation which is generally accepted is quite limited. One of thereasons is that the exciton is a neutral quasiparticle and difficult to be detected in experiments.Therefore, any conclusive evidence for exciton condensation in real materials is essential forfurther progress.Among the limited candidates, the transition metal dichalcogenide of TiSe is one of themost notable materials for exciton condensation. It shows a commensurate CDW transition at T c ∼ exhibits superconductivity with dome-shaped transition temperature induced by intercalatingCu atoms between layers [10] or by applying pressure [11]. The origin of the superconductivityhas not been clarified yet and the relation between the CDW and the superconductivity hasattracted much interests in the context of the quantum critical point.ost of the theoretical studies for exciton condensation are limited to an ideal case, e.g.,considering models in one dimensional, ignoring an intraband Coulomb interaction, a long-range Coulomb interaction, and a spin degrees of freedom. Although these models are useful todescribe exciton condensation alone, it is generally difficult to discuss the competition betweenother competing phases such as magnetism and superconductivity due to the lack of realisticcondition. Our aim is to establish a calculation scheme which can describe such a competitionand to clarify the emerging mechanism of exciton condensation and superconductivity in lowcarrier density systems such as TiSe . The variational Monte Carlo (VMC) method is one ofthe powerful methods for a strongly correlated electron systems and is suitable to this kind ofproblem. In this paper, we study a two-band Hubbard model with intra- and interband Coulombinteractions to investigate the detailed character of exciton condensation. The VMC methodis used for the calculation of physical quantities in the ground state. We show that the phasetransition from an excitonic insulator to a band insulator is induced by increasing the interbandinteraction. The importance of the intraband interaction is also discussed. Furthermore, theBCS-BEC crossover within the excitonic insulator is studied by calculating the exciton pairamplitudes both in k -space and in real space.
2. Model and method
We consider a two-band Hubbard model on a two-dimensional square lattice defined by thefollowing Hamiltonian, H = X k σ ε a ( k ) a † k σ a k σ + X k σ ε b ( k ) b † k σ b k σ + U aa X i n ai ↑ n ai ↓ + U bb X i n bi ↑ n bi ↓ + U ab X i n ai n bi (1)where α † k σ ( α = a, b ) denotes the creation operator of an electron with wave vector k andspin σ (= ↑ , ↓ ) on band α . U αα is an intraband Coulomb interaction and U ab is an interbandCoulomb interaction between bands a and b . n αiσ = α † iσ α iσ denotes the number operator and n αi = P σ n αiσ . We set the non-interacting band energy as ε a ( k ) = 2 t a (cos k x + cos k y ) and ε b ( k ) = 2 t b (cos k x + cos k y ) + E G . The Fermi surface and the energy dispersion are shown inFigs. 1(a) and (b), respectively. The electron and hole Fermi surfaces are perfectly nested withwave vector Q = ( π, π ) and have instability towards exciton condensation.Here, we introduce the following Gutzwiller-Jastrow type trial wave function: | Ψ i = P J c P (2)G | Φ i . (2)The one-body part | Φ i is obtained by diagonalizing the mean-field Hamiltonian, H MF = X k σ ( a † k σ , b † k + Q σ ) (cid:18) ε a ( k ) ˜∆ k ˜∆ k ˜ ε b ( k + Q ) (cid:19) (cid:18) a k σ b k + Q σ (cid:19) . (3)Since we fix t a = t as an energy unit, ε a ( k ) is unchanged through the VMC calculation. On theother hand, ˜ t b and ˜ E G in ˜ ε b ( k ) are optimized so as to minimize the variational energy. ˜∆ k is avariational parameter for the exciton condensation and generates the spontaneous hybridizationbetween bands a and b with wave vector Q = ( π, π ). Although there are several types of excitoncondensation depending on charge and spin degrees of freedom, they are energetically degeneratefor the Hamiltonian without spin-dependent interactions such as Hund’s coupling. Therefore,we assume a spin-independent variational parameter as follows,˜∆ k = ˜∆ exp (cid:20) − ˜ A ( ε a ( k ) − ˜ ε b ( k + Q )) (cid:21) . (4) igure 1. (color online) (a) Fermi surfaces and (b) energy dispersions of the noninteractingtight-binding energy band with electron density n = 2. (c) Ground state phase diagram in the U aa /t − U ab /t plane. PM, EI, and BI denote paramagnetic metal, excitonic insulator, and bandinsulator, respectively. PM is stabilized only for U ab /t = 0 due to the perfect nesting condition.˜∆ is an amplitude of the variational parameter and ˜ A controls the extent of exciton in k -space.We have found that the exciton condensation can be described without ˜ A (i.e., ˜∆ k = ˜∆) but theintroduction of ˜ A greatly improves the variational energy and gives a better trial wave function.The Gutzwiller operator P (2)G = Y i,γ [1 − (1 − g γ ) | γ i h γ | i ] (5)in | Ψ i is the one extended for the two-band system. i is a site index and γ represents possibleelectron configurations at each site, namely, | i = | i , | i = | ↑i , · · · , | i = |↑↓ ↑↓i .The variational parameters g γ ’s vary from 0 to 1, which control the weight of each electronconfiguration. Here, we classify the possible 16 local electron configurations into 10 groups bythe local energy and set the same value of g γ ’s for electron configurations with the same energy.The explicit grouping is shown elsewhere.The remaining operator P J c = exp − X i = jαβ v αβij n αi n βj (6)in | Ψ i is the charge Jastrow factor, which controls the long-range charge correlations. Sincethe on-site Coulomb interactions U aa , U bb , and U ab affect not only the on-site but also theoff-site (long-range) charge correlation, the charge Jastrow factor is essential for describing theappropriate trial wave function [12]. The band dependence is fully considered, namely, v aaij , v bbij ,and v abij are independently optimized. For the spatial dependence, we assume that v αβij dependsonly on the distances (not on the direction), v αβij = v αβ ( | r i − r j | ), and consider the range of R = | r i − r j | < L/ N = L × L .These variational parameters, ˜ ε b , ˜∆ k , g γ , v αβij are simultaneously optimized so as to minimizethe variational energy. For the optimization, we employ the stochastic reconfigurationmethod [13] which works quite effectively for the VMC method.
3. Results
First, we study the stability of the exciton condensation and its competing phase by changingthe intra- and interband interactions. The system size is 28 ×
28 through the paper. Here, we igure 2. (color online) Electron-hole pair amplitude in k -space φ ( k ) for (a) U ab /t = 4 . U ab /t = 5 . k denotes the original (noninteracting) Fermi wave vector shown in Fig. 1(a)with a red circle. (c) U ab /t dependence of the effective energy gap ˜ E G /t . Dotted line correspondsto the original value E G /t = 6 shown in Fig. 1(b). In all figures, U aa /t = U bb /t = 8 . U aa = U bb for simplicity. Figure 1(c) shows the obtained phasediagram in the U aa /t − U ab /t plane. The order parameter of exciton condensation is definedas ∆ = P k ,σ (cid:10) b † k + Q σ a k σ + H.c. (cid:11) and the finite ∆ leads to a broken translational symmetrywith ordering vector Q = ( π, π ). (Note that ∆ is different from the variational parameter ˜∆ k defined in eq. (4).) Since the Fermi surfaces of bands a and b are perfectly nested as shown inFig. 1(a), an infinitesimally small U ab induces the exciton condensation and the system becomesfully gapped. This phase is called excitonic insulator which has been extensively studied sofar. As for the trial wave function, we have confirmed that the charge Jastrow factor P J c isquite important for the stability of exiton condensation through lowering the variational energyof excitonic insulator phase. This is because the exciton pair has nonnegligible long-rangecomponent (at least several lattice constant) even with on-site Coulomb interaction alone. Thedetails are discussed later in Fig. 3. When U ab increases further, the bands a and b are completelydecoupled and the band insulator is realized. In the band insulator phase, the band a is fullyfilled ( n a = 2) and the band b is empty ( n b = 0) to avoid the energy loss of U ab despite the energyloss of U aa . Therefore, the region of the excitonic insulator becomes larger as U aa increases [14].This phase diagram is consistent with the previous variational cluster approximation (VCA)study both qualitatively and quantitatively [15]. It is interesting to note that the excitonicinsulator phase is restricted to a quite narrow region when the intraband interaction is absent( U aa = 0). This indicates the importance of the intraband interaction to stabilize the excitoncondensation. Since most of the previous studies for exciton condensation neglect the intrabandCoulomb interactions, its stability might be underestimated.Next, we calculate the electron-hole pair amplitude in k -space defined as, φ ( k ) = X σ (cid:10) b † k + Q σ a k σ + H . c . (cid:11) (7)Figure 2 shows the behavior of φ ( k ) in the first Brillouin zone. For U ab /t = 4 . φ ( k ) has a sharppeak at k = ˜ k F , which is an effective (or renormalized) Fermi wave vector in the wave function | Ψ i . A sharp peak structure in k -space corresponds to a widely extended electron-hole pair inreal space. The dotted line in Fig. 2(a) corresponds to the original (noninteracting) Fermi wavevector k and we can see that | ˜ k F | > | k | . This is because the effective energy gap ˜ E G in | Ψ i is reduced from the original value ( ˜ E G /t < E G /t = 6) and the Fermi surface is enlarged. Onthe other hand, as shown in Fig. 2(b), the peak structure is rather broad for U ab /t = 5 . igure 3. (color online) Electron-hole pair amplitude in real space φ ( R ) and R | φ ( R ) | for (a) U ab /t = 4 . U ab /t = 5 . U aa /t = U bb /t = 8 . | ˜ k F | < | k | , and the effective energy gap is larger than theoriginal one, ˜ E G /t > E G /t . The U ab dependence of ˜ E G /t is shown in Fig. 2(c) and this behavioris explained as follows. The character of the exciton condensation is determined by the balancebetween intra-( U aa ) and interband ( U ab ) interactions. When the effect of U aa is dominant( U ab /t . E G /t < E G /t , todecrease n a and avoid the loss of U aa . As U ab increases for fixed U aa , ˜ E G /t increases andexceeds the original value around U ab /t ∼ .
8. For U ab /t & .
8, the effect of U ab is dominantand the effective energy gap is renormalized to a larger value, ˜ E G /t > E G /t , to decrease n b and avoid the loss of U ab . Finally, the bands a and b are completely decoupled and the bandinsulator is realized for U ab /t & .
24. Note that we cannot observe the limit of the Bose-Einstein condensation (BEC) of excitons where the effective Fermi wave vector ˜ k F vanishes,namely, ˜ E G /t > a and b ) and ∆ = 0. However, the behaviorof the variational parameters drastically changes across U ab /t ∼ . U ab /t . In the VCA study, the BCS-BEC crossover occursaround U ab /t ∼ . k F is observed at U ab /t = 5 . N = 28 ×
28, to describe the bound electron-hole pair of exciton at least in the BECregion. The electron correlation is exactly treated within a 2 × φ ij defined as φ ij = X σ (cid:10) b † iσ a jσ (cid:11) + H . c . (8)Although in general φ ij depends not only on the distance between r i and r j , R = | r i − r j | , butalso on the direction between r i and r j , we have found that the direction dependence of φ ij issmall at least for parameters studied here. Therefore, we assume φ ij = φ ( R ) in the following.Figure 3 shows the behavior of φ ( R ) for U aa /t = U bb /t = 8 .
0. For U ab /t = 4 . φ ( R ) gradually decreases with increasing R and exhibits oscillatory behavior. Asimilar oscillatory behavior can be observed in the electron-electron pair amplitude (cid:10) c † i ↑ c † j ↓ (cid:11) of theBCS superconductivity, which is proportional to sin( k F R ) /R . This is because the wave functionf exciton condensation is composed of a coherent electron-hole pair and essentially identicalto that of the BCS superconductivity with a coherent electron-electron pair [2]. On the otherhand, for U ab /t = 5 . φ ( R ) decreases more rapidly and the oscillatory behavior ishardly observed. As U ab increases, the attraction between electron and hole increases and theradius of exciton becomes smaller in real space. These behaviors are consistent with the k -spacepicture: a widely-extended (tightly-bounded) exciton in real space corresponds to a localized(an extended) exciton in k -space.Finaly, we calculate the coherence length of electron-hole pair defined as ξ = X R R | φ ( R ) | . X R | φ ( R ) | . (9)The coherence length is often used to distinguish the BCS region ( ξ >
1, weak coupling) fromthe BEC region ( ξ <
1, strong coupling). In Fig. 3, we plot the quantity R | φ ( R ) | appearing inEq.(9). For U ab /t = 5 .
2, this quantity is almost negligible for R &
4, indicating that the systemsize ( N = 28 ×
28) is large enough to describe the bound electron-hople pair of exciton. Theestimated coherence length ξ is 0.77. On the other hand, for U ab /t = 4 . R | φ ( R ) | graduallydecreases with increasing R , showing oscillatory behavior, and has significant contributionfor rather large R . We have also found that the size dependence of the calculation becomessevere around this U ab /t and below. This indicates that there is a BCS-BEC crossover aroundthis U ab /t , below which the coherence length rapidly increases in the BCS region where even N = 28 ×
28 is not large enough to estimate ξ . Although it is difficult to discuss the BCS-BECcrossover with φ ( R ) and ξ shown here, we can visualize the spatial distribution of exciton usingthese quantities, which helps for further understanding the exciton condensation. The accurateestimation of ξ and other physical quantities in real space is an important future problem.
4. Summary and discussion
In this paper, we have studied the two-band Hubbard model with perfectly nested electronand hole Fermi surfaces to discuss the stability of the exciton condensation with changing theCoulomb interactions. We have shown that the phase transition from an excitonic insulator toa band insulator occurs by increasing U ab and the region of the excitonic insulator is extendedfor larger U aa . We have also calculated the exciton pair amplitude both in k -space and in realspace and proposed that the BCS-BEC crossover occurs around U ab /t ∼ . U aa /t = 8 . P J c becomes more importantbecause the screening effect of long-range Coulomb interaction, which is an essential elementfor the exciton condensation, will be properly included through P J c . On the other hand, theexciton condensation is greatly suppressed when the perfect nesting condition is broken [16].Understanding the effect of the long-range Coulomb interaction and imperfect nesting on thestability of the exciton condensation and other competing phases such as a superconductivity isimportant to clarify the detailed properties of TiSe and other low carrier density systems. References [1] Mott N F 1961
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