Excitonic condensation in semimetal – Semiconductor transition systems

∑ Vνµ,ν′µ′nνµ′nµν′ , (2.23) where nνν′ = 〈c†νcν′〉 and nνµ′ = 〈c†νcµ′〉 with c†ν , cν are particle creation and annihilation op- erators with the quantum number ν, respectively. The (+) mark applies to the boson particle system, and the (−) mark applies to the fermion system. 2.3. Broken symmetry 2.3.1. The concept of phase transition and broken symmetry At the critical temperature, the thermodynamical state of the system develops non-zero expectation value of some macroscopic quantities which have a symetry lower than the orig- inal Hamiltonian, it is called spontaneous breaking of symmetry. Those quantities are called order parameters that indicate the phase transition. For the mean field theory, we select the finite mean field through order parameters, then we derive a set of self-consistent equations determining the order parameters. 2.3.2. The Heisenberg model of ionic ferromagnets Applying the MF theory to the Hamiltonian of Heisenberg ferromagnetic model, we obtain MF Hamiltonian which is diagonalized in the site index HMF = −2 ∑ i mSi +mN〈Sz〉. (2.28) We can easily derive the equation α = tanh(bα), (2.31) where α = m/nJ0 và b = nJ0β. This equation can be numerically solved and the result given the temperature dependence of the magnetization m. 5 2.3.3. The Stoner model of metallic ferromagnets Applying HFA to the metallic ferromagnet model, based on the Hubbard model, the MF Hamiltonian becomes HMF = ∑ kσ εMFkσ c † kσckσ − UV 2 ∑ σ nσn−σ + UV 2 ∑ σ n2σ, (2.39) where εMFkσ = εk + U(n↑ + n↓ − nσ) = εk + Unσ¯, (2.40) with nσ = 1V ∑ k〈c†kσckσ〉 is the spin density. From this Hamiltonian, we can find self-consistent equations for the spin density. Then we find the solutions of the model. 2.3.4. BCS theory One of the most striking examples of symmetry breaking is the superconducting phase transition. c†kσ and ckσ are creation and annihilation operators with momentum k and spin σ =↓, ↑, respectively, BCS Hamiltonian in HFA is HMFBCS = ∑ kσ εkc † kσckσ − ∑ k (∆kc † k↑c † −k↓ + H.c.), (2.51) where ∆k = − ∑ k′ Vkk′〈c−k′↓ck′↑〉, (2.52) is called the gap equation. This Hamiltonian is solved by the Bogoliubov transformation de- termining new fermionic operators αk↑ and α † −k↓ which are called creation and annihilation quasiparticle operators. αk↑ = u∗kck↑ + vkc † −k↓, α†−k↓ = −v∗kck↑ + ukc†−k↓. (2.56) where u2k + v 2 k = 1. Finally, BCS Hamiltonian can be diagonalized in a form HMFBCS = ∑ k Ek(α † k↑αk↑ + α † k↓αk↓), (2.59) where Ek = √ ε2k + |∆k|2. Using this Hamiltonian, we can find solutions of the gap equation. Then we get the BCS prediction that the ratio of gap to critical temperature which agrees qualitatively with extracted data from experiments. 2.3.5. The excitonic insulator – EI ApplyingMF approximation to the electronic system in the two-band model with Coulomb interaction between them 1. Similar to the superconducting state survey in BCS theory, exci- tonic condensation state is characterized by quantity 〈c†kfk〉 6= 0. In HFA, neglecteing constants 1Note that, the electronic representation is completely equivalent to the hole representation by electronic transformation – hole. Then the annihilation operator of electron is replaced with the creation operator of hole and vice versa. 6 we can rewrite Hamiltonian HMF = ∑ k ε˜ckc † kck + ∑ k ε˜fkf † kfk + ∑ k (∆kf † kck + H.c.), (2.71) where ∆k = ∑ k′ Vk−k′〈c†k′fk′〉. (2.72) acts as an energy gap, or EI state order parameter. ε˜ck and ε˜ f k are the dispersion energies of c electrons and f electrons having contribution of Hartree-Fock energy shift. In order to diagonalize the Hamiltonian, we use the Bogoliubov transformation to define the new fermion operators αk and βk. The Hamiltonian of the system in the MF approximation will be completely diagonalized HMFEI = ∑ k Eαkα † kαk + ∑ k Eβkβ † kβk, (2.79) where E α/β k = ε˜ck + ε˜ f k 2 ∓ √ ξ2k + |∆k|2. (2.80) with ξk = 12 [ε˜ c k − ε˜fk] and E2k = ξ2k + |∆k|2. This Hamiltonian allows us to determine all expectation values. At T = 0, ∆k is deter- mined by the gap equation ∆k = ∑ k′ Vk−k′ ∆k′ 2Ek′ . (2.81) This equation is similar to the gap equation of superconducting in BCS theory. ∆k 6= 0 in- dicates the hybridization between electrons in the valence band and the conduction band. Therefore, the system turn into the excitonic insulator state. CHAPTER 3. EXCITONS CONDENSATE IN THE TWO-BAND MODEL INVOLVING ELECTRON – PHONON INTERACTION 3.1. The two-band electronic model involving electron – phonon interaction The Hamiltonian for the two-band electronic model involving electron – phonon inter- action can be written H = ∑ k εckc † kck + ∑ k εfkf † kfk + ω0 ∑ q b†qbq + g√ N ∑ kq [c†k+qfk ( b†−q + bq ) + H.c.], (3.1) where c†k (ck); f † k (fk) and b † q(bq) are creation (annihilation) operators of c, f electrons carry- ing momentum k and phonons carrying momentum q, respectively; g is a electron – phonon coupling constant; N is the number of the lattice sites. εc,fk = ε c,f − tc,fγk − µ, (3.2) 7 where εc,f are the on-site energies; tc,f are the nearest-neighbor particle transfer amplitudes. In a 2D square lattice, γk = 2 (cos kx + cos ky) and µ is the chemical potential. At sufficiently low temperature, the bound pairs with finite momentumQ = (pi, pi) might condense, indicated by a non-zero value of dk = 〈c†k+Qfk〉 and d = 1 N ∑ k (〈c†k+Qfk〉+ 〈f †kck+Q〉), (3.4) These quantities express the hybridization between c and f electrons so they are called the order parameters of the excitonic condensate. The order parameter is nonzero representing the system stabilize in excitonic condensation state. 3.2. Applying mean field theory Applying MF theory with mean fields ∆ = g√ N 〈b†−Q + b−Q〉, (3.9) h = g N ∑ k 〈c†k+Qfk + f †kck+Q〉, (3.10) act as the order parameters which specify to spontaneous broken symmetry, Hamiltonian in (3.1) is reduced to Hamiltonian Hartree-Fock involving two parts, the electronic part (He) and the phononic part (Hph) are as follows HHF = He +Hph, (3.11) where He = ∑ k εckc † kck + ∑ k εfkf † kfk + ∆ ∑ k (c†k+Qfk + f † kck+Q), (3.12) Hph = ω0 ∑ q b†qbq + √ Nh(b†−Q + b−Q), (3.13) The phononic part is diagonalized by defining a new phonon operator B†q = b † q + √ N h ω0 δq,Q. (3.14) Meanwhile, the electronic part can be diagonalized itself by using a Bogoliubov transforma- tion with the new quasi-particle fermionic operators C1k and C2k. Then finally, we are led to a completely diagonalized Hamiltonian Hdia = ∑ k E1kC † 1kC1k +E 2 kC † 2kC2k +ω0 ∑ q B†qBq, (3.17) where the electronic quasiparticle energies read as E1,2k = εfk + ε c k+Q 2 ∓ sgn(εfk − εck+Q) 2 Wk, (3.18) 8 withWk = √ (εck+Q − εfk)2 + 4|∆|2. The quadratic form of Eq. (3.17) allows us to compute all expectation values, resulting in nck+Q = 〈c†k+Qck+Q〉 = ξ2knF (E1k) + η2knF (E2k), nfk = 〈f †kfk〉 = η2knF (E1k) + ξ2knF (E2k), (3.22) dk = 〈c†k+Qfk〉 = −[nF (E1k)− nF (E2k)] sgn(εfk − εck+Q) ∆ Wk , here nF (E 1,2 k ) are the Fermi-Dirac distribution functions; ξk and ηk are the prefactors of the Bogoliubov transformation which satisfy ξ2k + η 2 k = 1. The lattice displacement in the EI state at momentum Q xQ = 1√ N 1√ 2ω0 〈b†−Q+bQ〉 = − h ω0 √ 2 ω0 , (3.24) 3.3. Numerical results and discussion For the two-dimensional system consisting of N = 150× 150 lattice sites, the numerical results are obtained by solving self-consistently Eqs. (3.9), (3.10), (3.22) and (3.24) starting from some guessed values for 〈b†Q〉 and dk with a relative error 10−6. In what follows, all energies are given in units of tc and we fix tf = 0.3 to consider the half-filled band case, i.e. nc + nf = 1. The chemical potential µ has to be adjusted such that this equation is satisfied. 3.3.1. The ground state 0.5 1.0 1.5 2.0 2.5 3.0 -0.3 -0.2 -0.1 0.0 d  0 g=0.2 g=0.4 g=0.5 g=0.6 Fig. 3.2: The order parameter d as functions of phonon frequency ω0 for different values of g at εc − εf = 1 in the ground state. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 d , x Q c- f   =0.5   =1.0   =1.5   =2.0 Fig. 3.5: The order parameter d (filled sym- bols) and the lattice displacement xQ (open symbols) as functions of εc− εf for some val- ues of ω0 at g = 0.5, T = 0. Fig. 3.2 shows the dependence of the excitonic condensate order parameter d at T = 0 on the phonon frequency ω0 for different values of g at εc − εf = 1. For a given value of the coupling constant, the order parameter decreases when increasing phonon frequency. This is 9 also shown in Fig. 3.5 the dependence of the order parameter d and the lattice displacement xQ on εc − εf for some values of ω0 at g = 0.5, T = 0. The diagram shows that d and xQ are intimately related. When increasing ω0, both d and xQ decrease significantly, indicating a weakened condensation state. d and xQ are non-zero, the systems thus stabilize in the excitonic condensation state with the charge density wave state (EI/CDW). g=0.4 0.0 0.5 1.0 1.5 2.0 2.5 3.0   g=0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 EI/CDW   c-f g=0.6 g=0.7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 c-f Fig. 3.6: Ground-state phase diagram of the model in the (εc − εf , ω0) plane for different values of g. The excitonic condensation phase is indicated in orange. Fig. 3.6 shows the phase diagram of the model in the (εc−εf , ω0) plane in the ground state for different g. If g is large enough, we always find the excitonic condensate regime EI/CDW (orange) when the phonon frequency is less than the critical value ωc0. This critical value increases when increasing g. The excitonic condensation regime is narrowed if decreasing the two energy bands overlap and the electron – phonon interaction constant. 3.3.2. The effect of thermal fluctuations Fig. 3.7 describes the dependence of the order parameter d on the phonon frequency ω0 when varying the temperature at εc− εf = 1 and g = 0.5. For a given value of temperature, the value of the order parameter decreases rapidly when increasing the phonon frequency. The dependence of the order parameter d the lattice displacement xQ on the electron – phonon interaction when the temperature changes for εc − εf = 1 and ω0 = 0.5 are shown in Fig. 3.8. d and xQ are always closely related, they are non-zero i.e. the system exists in EI/CDW state 10 0.5 1.0 1.5 2.0 2.5 3.0 -0.2 -0.1 0.0 d  0 T=0 T=0.1 T=0.2 T=0.3 Fig. 3.7: The order parameter d as functions of the phonon frequency ω0 for different values of temperature at εc − εf = 1 and g = 0.5. 0.0 0.1 0.2 0.3 0.4 0.5 -0.5 0.0 0.5 1.0 1.5 0.6 d , x Q g T=0 T=0.1 T=0.2 T=0.3 Fig. 3.8: The order parameter d (filled sym- bols) and the lattice displacement xQ (open symbols) as functions of g for some values of T at εc − εf = 1 and ω0 = 0.5. T=0 0.0 0.2 0.4 0.6 0.8 1.0 g T=0.1 T=0.3 0.5 1.0 1.5 2.0 2.5 3.0 0.1  0 T=0.2 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.2 0.4 0.6 0.8 1.0 EI/CDW 0.1 g  0 Fig. 3.9: The phase diagram of the model in the (ω0, g) plane at εc − εf = 1 for some values of temperature. The excitonic condensation phase is indicated in orange. when the electron – phonon coupling is larger than a critical value gc. Fig. 3.9 shows the phase diagram in the (ω0, g) plane when εc−εf = 1 for some values of temperature. The larger phonon frequency, the greater critical value gc for phase transition of the excitonic condansation state. The higher temperature, the narrower condensation region. 11 T=0.1 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0   EI/CDW c-f 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 EI/CDW c-f   T=0 Fig. 3.13: The phase diagram of the excitonic condansation state of the model in the (εc − εf , ω0) plane at g = 0.5 for T varies. The excitonic condensation phase is indicated in orange. Fig. 3.13 shows the relationship of the phonon frequency and the c and f bands overlap (the external pressure) when T changes at g = 0.5. The diagram shows that if the temperature increases, the critical value ωc0 decreases and the exciton condensation regime shrinks. Fig. 3.15: The dependence of the order parameter |dk| on the momentum and the temperature along the (k, k) direction in the first Brillouin zone for some values of ω0 at εc − εf = 1 and g = 0.5. The Fermi momenta are indicated white dashed lines. Fig. 3.15 shows the nature of the excitonic condensation state in the system, indecating the dependence of the order parameter |dk| on T for some values of ω0 at g = 0.5 and εc−εf = 1 in the first Brillouin zone. At below the critical temperature Tc, |dk| is strongly peaked at momenta close to the Fermi momentum kF (described by the white dashed lines) which shows that excitons condense in the BCS-type. Increasing ω0, |dk| decreases and Tc also decreases. The influence of the temperature and the phonon frequency on the excitonic condensation state in the model is shown on the phase diagram (ω0, T ) for two values of the electron – phonon coupling g = 0.5 (Fig. 3.16a) and g = 1.0 (Fig. 3.16b) at εc − εf = 1. The excitonic 12 condensation regime is expanded when increasing electron – phonon coupling constant. a) b) g=0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.2 0.4 0.6 0.8 1.0 E I/C D W T  0 g=1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.2 0.4 0.6 0.8 1.0 EI/CDW T  0 Fig. 3.16: The phase diagram of the excitonic condansation state of the model in the (ω0, T ) plane at εc − εf = 1 for g = 0.5 (Fig. a) and g = 1.0 (Fig. b). The excitonic condensation phase is indicated in orange. 0.0 0.1 0.2 0.3 0.4 0.5 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3  0 =2.5 (b) d , x Q T g=1.0 g=1.1 g=1.2 0.0 0.1 0.2 0.3 0.4 0.5 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 T (a)  0 =0.5 d , x Q g=0.2 g=0.4 g=0.5 Fig. 3.17: The order parameter d (filled symbols) and the lattice displacement xQ (open symbols) as functions of T at ω0 = 0.5 (Fig. a) and ω0 = 2.5 (Fig. b) for some values of g with εc− εf = 1. Fig. 3.17 shows that d and xQ are still intimately related. For a given ω0 and g, d and xQ is only non-zero when the temperature is smaller than the critical temperature value Tc. The temperature dependence of the lattice displacement agrees qualitatively with extracted data from neutron diffraction experiments at low temperatures as follows Tc. The temperature dependence of the order parameter is similar to the superconducting parameter. This once again reminds us of a similar relevance to the BCS theory of the superconductivity where Cooper pairs are formed. Then, the phase diagram of the model in the (g, T ) plane when fixing εc − εf = 1 for the phonon frequency ω0 = 0.5 (the adiabatic regime) and ω0 = 2.5 (the anti-adiabatic regime) is shown in Fig. 3.19. When the temperature increases, a large thermal fluctuation destroys the bound state of c − f electrons, the excitonic condensation state thus is weakened. The diagram also shows that, when increasing the phonon frequency from the 13 adiabatic limit (Fig.a) to the anti-adiabatic limit (Fig.b), the critical value of the electron – phonon coupling constant also increases. The excitonic condensation regime thus narrows. a) b)  0 =2.5 0.0 0.4 0.8 1.2 1.6 0.0 0.2 0.4 0.6 0.8 1.0 EI/CDW T g  0 =0.5 0.0 0.4 0.8 1.2 1.6 0.0 0.2 0.4 0.6 0.8 1.0 EI/CDW T g Fig. 3.19: The phase diagram of the excitonic condansation state of the model in the (g, T ) plane at εc − εf = 1 for ω0 = 0.5 (Fig. a) và ω0 = 2.5 (Fig. b). The excitonic condensation phase is indicated in orange. a) b) g=0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.2 0.4 0.6 0.8 1.0 EI/CDW T e-h g=0.7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.2 0.4 0.6 0.8 1.0 EI/CDW T e-h Fig. 3.21: The phase diagram of the excitonic condansation state of the model in the (εc − εf , T ) plane at ω0 = 0.5 and g = 0.5 (Fig. a) or g = 0.7 (Fig. b). The excitonic condensation phase is indicated in orange. Finally, the phase diagram of the model in the (εc−εf , T ) plane for the electron – phonon coupling constant g = 0.5 (Fig. a) and g = 0.7 (Fig. b) at ω0 = 0.5 is shown in Fig. 3.21. The phase diagram shows that for each given value of g, we always find the EI/CDW state (indi- cated by the orange regime) below the critical temperature Tc. This critical value Tc decreases as εc − εf increases, thus the excitonic condensation regime shrinks. Our results the temperature dependence of the excitonic condensation state in the system fit quite well with the recent experimental observation of C.Monney et. al.. The results also confirm the important influence of temperature and phonon on excitonic condensation state. 14 The excitonic condensation state is only formed when the system is at low temperatures and the electron – phonon interaction is large enough. CHAPTER 4. EXCITONS CONDENSATE IN THE EXTENDED FALICOV-KIMBALL MODEL INVOLVING ELECTRON – PHONON INTERACTION 4.1. The extended Falicov-Kimball model involving electron – phonon interaction The Hamiltonian for the extended Falicov-Kimball model involving electron – phonon interaction can be written H = H0 +Hint, (4.1) where H0 discribes the non-interacting part of electron – phonon system H0 = ∑ k εckc † kck + ∑ k εfkf † kfk + ω0 ∑ q b†qbq. (4.2) here c†k (ck); f † k (fk) and b † q (bq) are creation (annihilation) operators of c, f electrons carrying momentum k and phonons carrying momentum q, respectively. The c(f) electronic excitation energies are still given by equation (3.2). The interacting part Hamiltonian reads Hint = U N ∑ k,k′,q c†k+qck′f † k′−qfk+ g√ N ∑ kq [c†k+qfk(b † −q + bq) + H.c.], (4.4) where U is the Coulomb interaction and g is the electron – phonon coupling constant. 4.2. Applying mean field theory Using Hartree-Fock approximation is similar to chapter 3, and diagonalizing Hamilto- nian, we have a completely diagonalized Hamiltonian Hdia = ∑ k E+k α † 1kα1k + ∑ k E−k α † 2kα2k + ω0 ∑ q B†qBq, (4.10) where α†1k (α1k) and α † 2k (α2k) are the Bogoliubov quasi-particle fermionic creation (annihi- lation) operators, respectively, with the electronic quasiparticle energies E±k = εfk + ε c k+Q 2 ∓ sgn(ε f k − εck+Q) 2 Γk, (4.11) here Γk = √ (εck+Q − εfk)2 + 4|Λ|2, (4.12) and the electronic excitation energies now have acquired Hartree shifts ε f/c k = ε f/c k + Un c/f , (4.7) with nc(f) is the c(f) electron density; Λ also acts as the order parameters of the excitonic condensation state which is given by Λ = g√ N 〈b†−Q + b−Q〉 − U N ∑ k 〈c†k+Qfk〉. (4.9) 15 We also obtain the system of self-consistently equations from the average values nck+Q = 〈c†k+Qck+Q〉 = u2knF (E+k ) + v2knF (E−k ), (4.13) nfk = 〈f †kfk〉 = v2knF (E+k ) + u2knF (E−k ), (4.14) nk = 〈c†k+Qfk〉 = − [ nF (E + k )− nF (E−k ) ] sgn(εfk − εck+Q) Λ Γk , (4.15) 〈b†q〉 = − √ Nh ω0 δq,Q,, (4.16) where nF (E±k ) is the Fermi-Dirac distribution function; uk and vk are the prefactors of the Bogoliubov transformation which satisfy u2k+v 2 k = 1. The lattice displacement and the single- particle spectral functions of c and f electrons are therefore also determined by xQ = 1√ N 1√ 2ω0 〈b†−Q + bQ〉 = − h ω0 √ 2 ω0 , (4.19) Ack (ω) = u 2 k−Qδ ( ω − E+k−Q ) + v2k−Qδ ( ω − E−k−Q ) , (4.23) Afk (ω) = v 2 kδ ( ω − E+k ) + u2kδ ( ω − E−k ) . (4.24) 4.3. Numerical results and discussion For the two-dimensional system consisting of N = 150× 150 lattice sites, the numerical results are obtained by solving self-consistently Eqs. (4.7) – (4.9) and (4.13) – (4.16) starting from some guessed values for 〈b†Q〉 and nk with a relative error 10−6. In what follows, all energies are given in units of tc and we fix tf = 0.3; εc = 0; ω0 = 2.5. The chemical potential µ has to be adjusted such that the system is in the half-filled band state, i.e., nf + nc = 1. 4.3.1. The momentum dependence of the quasiparticle energies and the order parameter Fig. 4.1 and Fig. 4.2 show the momentum dependence along the (k, k) direction in the -1.0 -0.5 0.0 0.5 1.0 -0.4 -0.2 0.0 n k k/ U=0 U=1.0 U=1.5 -2 0 2 4 6 E + k , E - k U=0 U=1.0 U=1.5 |n k | Fig. 4.1: The quasiparticle energies E+k (solid lines); E−k (dash lines) and |nk| for small val- ues of U at g = 0.6; T = 0. -2 0 2 4 6 8 E + k , E - k U=3.5 U=3.8 U=4.2 -1.0 -0.5 0.0 0.5 1.0 -0.4 -0.2 0.0 n k k/ U=3.5 U=3.8 U=4.2 |n k | Fig. 4.2: The quasiparticle energies E+k (solid lines); E−k (dash lines) and the order parame- ter |nk| for large values of U at g = 0.6; T = 0. first Brillouin zone of the quasiparticle energy bands E+k ; E − k and the order parameter |nk| for 16 some values of U in the weak and strong interaction limit at g = 0.6, εf = −2.0 in the ground state. In Fig. 4.1, the Fermi surface plays an important role to form the condensation state of excitons. We affirm that excitons in system condensate in the BCS-type, like the Cooper pairs in superconductivity BCS theory. Fig. 4.2 shows that large Coulomb interaction binds an electron in the conduction band and an electron in the valence band in a tightly bound state. Therefore, |nk| has a maximum value at zero momentum, this confirms that excitons conden- sate in BEC-type, like normal bosons. The investigation similarly the momentum dependence of the quasiparticle energies and the order parameter when g or T changes. The results con- firmed that the condensation state is only formed when the temperature is low enough and the electron – phonon coupling constant and Coulomb interaction are large enough. 4.3.2. The EI order parameter and the lattice displacement Fig. 4.5: Λ (solid lines) and xQ (dash lines) as functions of U for different g at εf = −2.0; T = 0. Fig. 4.6: Λ (solid lines) and xQ (dash lines) as functions of U for different εf at g = 0.6; T = 0. Fig. 4.8: Λ (solid lines) and xQ (dash lines) as functions of T for different g at U = 1.5; εf = −2.0. In Fig. 4.5, the EI order parameter Λ and the lattice displacement xQ are shown as func- tions of U for some values of g at T = 0 and εf = −2.0. And Fig. 4.6 shows Λ and xQ as a function of U at zero temperature when g = 0.6 for different values of εf . The results confirm that excitonic condensation state exists only in a limited range of Coulomb interactions. In presence of the electron – phonon interaction, we observe the EI/CDW state. Fig. 4.8 shows Λ and xQ depending on T when changing g. At g is greater than the critical value gc, Λ always exists simultaneously with xQ. At T ≤ Tc, both are nonzero and the system exists in excitonic condensation state with a finite lattice distortion. Increasing g, the EI transition temperature Tc increases. The temperature dependence of the lattice displacement fits quite well with experimental results obtained from neutron diffraction experiments at low temperatures or the recent experimental observation in the quasi-two-dimensional 1T -TiSe2. 4.3.3. The nature of excitonic condensation state in the model Fig. 4.10 shows the momentum dependence of the excitonic condensation order parame- ter |nk| in the ground state for some values of U at g = 0.6 and εf = −2.0 in the first Brillouin 17 Fig. 4.10: The order parameter |nk| depending on momentum k in the first Brillouin zone for some values of U at g = 0.6; εf = −2.0; T = 0. The Fermi momenta are determined by the white dashed lines. Fig. 4.11: The order parameter |nk| depending on momentum along the (k, k) direction and Coulomb interaction in the first Brillouin zone for g = 0.6 and εf = −2.0 at T = 0. zone. The excitons with low Coulomb interaction condense in the BCS-type in which the Fermi surface plays an important role in the formation and condensation of excitons. The ex- 18 Fig. 4.12: The order parameter |nk| depending on momentum k in the first Brillouin zone for different temperatures U = 1.5 (left panels) and U = 3.7 (right panels) at g = 0.6 and εf = −2.0. The Fermi momenta are determined by the white dashed lines. citons with strong Coulomb interaction will condense in the BEC-type. The value U = 3.39 can be called the critical value for the BCS-BEC crossover of the excitonic condensation for the set of parameters chosen in Fig. 4.10. The excitonic condensation state is only established when the Coulomb interaction is in between Uc1 and Uc2 as shown in Fig. 4.11. Fig. 4.12 shows in detail the nature of the excitonic