J Low Temp Phys (2016) 182:1–12 DOI 10.1007/s10909-015-1346-2

Phonon-Mediated Electron–Phonon Interaction in Hubbard–Holstein Model S. Nath1,2 · N. K. Ghosh1

Received: 21 March 2015 / Accepted: 3 October 2015 / Published online: 14 October 2015 © Springer Science+Business Media New York 2015

Abstract We explore the influence of inter-site electron–phonon (EP) interaction λ on some ground state and finite temperature properties of Hubbard–Holstein model using exact diagonalization technique. Both the adiabatic and non-adiabatic limits have been considered. Results show that the EP correlation function χ increases gradually for the adiabatic case with λ while a sharp transition is observed at λ = λc (U/t) for the non-adiabatic case. On-site (S0) and inter-site (S1) bipolarons are formed which tend to bind at λ ≥ λc (U/t). Schematic phase diagram has been shown. Entropy calculations show that the system goes to a more ordered state when λ ≥ λc (U/t) due to the formation of EP pairs. Broad peaks are observed in the specific heat curves which move to higher temperature region with λ suggesting an effective increase in exchange energy. Keywords Phonons

High Tc superconductivity · Electronic structure · Lattice dynamics ·

1 Introduction It is widely accepted that the interactions of the electrons with the local lattice deformation play a crucial role in determining the exotic properties of high-Tc cuprates [1,2], and it can even lead to the formation of small polarons for undoped cuprates [3]. While the electron–phonon (EP) interaction is well understood in metals, the role of phonons in strongly correlated systems is quite unclear due to the interplay of strong electron–

B

S. Nath [email protected]

1

Department of Physics, University of Kalyani, Kalyani, West Bengal 741235, India

2

Department of Physics, Krishnagar Government College, Krishnagar, West Bengal 741101, India

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electron (e–e) and EP interactions leading to a competing ordered phase. Despite its difficulty, this is an open problem to solve as various experiments significantly reveal lattice effects in the cuprates. To mention a few, angle-resolved photoemission spectroscopy experiments on high-Tc cuprate superconductors revealed ‘kinks’ in the band dispersion, which has been attributed to EP interactions [4], as well as small polaron formation in undoped Ca2−x Na x CuOCl2 [3]. Recently inelastic scattering [5] and optical conductivity [6] experiments strongly support the relevance of EP interaction. Also, long-range Fröhlich type EP interaction playing significant role in high temperature superconductivity (HTSC) has been suggested [7]. Besides cuprates, sizeable EP interaction is observed in other materials like colossal magneto-resistance manganites [8], fullerenes [9], V2 O3 [10] along with strong e–e interaction. The experimental as well as the theoretical studies on the EP interaction in HTSC have attracted constant interest of the scientific community, but still several aspects are not fully understood and are quite challenging. The theoretical difficulty results from the fact that in the HTSC materials the interaction of the charge carriers with the antiferromagnetically correlated Cu2+ spin background leads to a substantial reduced characteristic energy scale for coherent motion of the dressed quasiparticles. As a consequence, the EP effects become much more important in the unusual electronic environment and the standard Migdal approximation might breakdown [11]. The simple Hubbard–Holstein model (HHM) has played an important role in explaining the variety of interesting phases and phenomena like superconductivity, phase separation, charge- and spin-density waves, polaron and bipolaron formations. This HHM has been extensively studied (in one, two, and infinite dimensions and at various fillings) by employing various approaches such as exact diagonalization [12,13], density matrix renormalization group method (DMRG) [14], quantum Monte Carlo (QMC) simulation [10,15], semi-analytical slave boson approximation [16,17], dynamical mean field theory (DMFT) [18–20] and large-N expansion [21], variational methods based on Lang-Firsov transformation [22], Gutzwiller approximation [23] and cluster approximation [24]. Each of these approaches has its own advantages, and typical range of validity. Recently, Giustino et al., using density functional theory (DFT) showed that in the high-temperature superconductor La1.85 Sr 0.15 CuO4 , crystal lattice vibrations (phonons) should have a negligible effect on the observed photoemission spectra and concluded that phonons might not play a significant role [25]. But, it has been argued that the presence of on-site Coulomb repulsion U makes DFT inappropriate to explain accurately the electronic contribution to phonon breathing [26]. Many-body calculations established that the presence of strong correlation effects enhances the coupling to bond stretching phonons compared to DFT [27]. In the past, superconductivity due to on-site EP interaction has been a focus in a number of studies [15,28,29]. Calculations have been carried out using HHM and Holstein t − J [30] model at half or very close to half filling. However, we are interested to a quite different situation. Our focus is on the phonon-mediated interaction between electrons at the nearest neighbor sites i.e., inter-site EP interaction [30,31] (rather than on-site EP interaction studied earlier) on a low density system. Employing an exact diagonalization method, we have studied the role of inter-site EP interaction on the formation of different types of quasiparticles as well as electron pairing on a low-

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doped quarter filled system. The effect of inter-site EP interaction on thermodynamic properties like entropy and specific heat has also been examined. We have used the fact that important phonons couple only to charge fluctuations in the copper-oxide plane [32]. The present study is in the weak coupling regime of EP interaction with both the adiabatic and non-adiabatic phonon energies.

2 Formulation The Hamiltonian for the strongly correlated electron systems in a 2D plane in presence of inter-site EP coupling can be written in the following form: H = HHubb + Hph + Hel−ph ,

(1)

where HHubb is the pure Hubbard Hamiltonian, Hph and Hel−ph describe the purely phononic part and the phonon-mediated interaction between nearest neighbors (NN) respectively. HHubb = −t



† (ciσ c jσ + H.c.) + U

σ



n i↑ n i↓

(2)

i

Hph = ω0



bi† bi

(3)

i



Hel−ph = −g

(bi† + bi )n j

(4)

† i, j are the nearest neighbor sites on a simple two-dimensional square lattice; ciσ (ciσ ) are the electron creation (annihilation) operators with spin σ at the site i; t, U are the NN hopping amplitude and the on-site Coulomb repulsion, respectively. ω0 is the dispersionless phonon energy; g is the NN phonon-mediated (inter-site EP) interaction energy. bi(†) are the phonon annihilation (creation) operators, and n i is the number of electrons associated with site i. g2 as the ratio between We define the dimensionless EP coupling constant λ = 2tdω 0 2

the polaronic binding energy E P = − ωg 0 and the bare average kinetic energy of the electrons which is of the order of half the bandwidth D(∼ −2td), d is the dimensionality. λ is used to measure the strength of the inter-site electron–phonon  interactions. The eigenfunction of the Hamiltonian (1) can be taken as |  = e,p aep | eel ⊗ | pph , where e and p label electronic and bosonic basic states, respectively, and aep are the coefficients of the wave function. Since, in the present work, we focus only on the weak coupling regime of the electron–phonon coupling constant, the infinitedimensional bosonic part of the Hilbert space has been truncated by considering one phonon (Nph = 1) per site at most. We operate phonon operators like as bi† | 1i  =| 1i , bi† | 0i  =| 1i , bi | 1i  =| 0i , bi | 0i  = 0; where i denotes position of sites and 0(1) denotes the absence (presence) of a phonon [30].

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In contrast to our single-band single-phonon model incorporating NN hopping interaction, Piekarz et al. [33] considered two phonon modes (breathing and buckling). They also argued that the diagonal EP interactions are small and do not consider the hopping part of the Hamiltonian. In Ref. [34], the authors studied the converging behavior of the ground state energy with phonon numbers and suggested that one-phonon calculation is a good approximation in the weak coupling regime of EP interaction (λ ≤ 0.3) for ω0 = 0.2t. It has also been argued that, qualitatively the behaviors of various physical quantities of interest remain insensitive to the number of phonons though Nph = 1 generally under-estimate the role of phonons [35]. Moreover, it has been shown that with the increase of phonon frequency the weight of m-phonon state decreases to smaller m max = 0 . For weak couplings (λ ≤ 1) for fixed EP coupling λ and for ω0 → ∞, Nph and ω0 = 0.4t, the contribution of one-phonon state in the 2-electron ground state is dominant and it vanishes for Nph > 3 [36]. Analytically, the number of phonons in the zeroth order in the initial basis for a Holstein polaron is approximately found to be [37] g2 2dλt (5) Nph =< bi† bi >= 2 = ω0 ω0 To begin with, we first compute the correlation function between the electron and the phonon displacement (lattice deformation) per electron [38] in the ground state, χ ( p − q) =< 0 |(c†p c p (bq† + bq )|0 >

(6)

where p and q label the sites. This correlation function can be considered as a measure of the size of the polaron [39] and is quite different from the polar on radius in the extreme adiabatic limit, which refers to the spatial extent of a symmetry-breaking localized state. The two electron binding energy in the presence of phonons is a useful parameter to show the formation of bipolaron. It is defined as the difference of the ground state energy for two electrons with opposite spins i.e., bipolaronic ground state and twice the ground state energy for a single electron [40] bp = E bi − 2p We consider the entropy per lattice site   H  1 kB lnZ + S= N T

(7)

(8)

where N is the number of lattice sites. Finally, we have calculated the low-temperature specific heat defined as C = kB β 2 where Z =

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α

∂2 lnZ ∂β 2

(9)

e−β E α , the sum is taken over all eigenstates, E α s are the eigenvalues .

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√ √ In the present study, we follow the exact diagonalization (ED) method on a 8× 8 site [41] 2D square cluster with periodic boundary conditions. All our calculations are at quarter filling of the cluster (i.e., n = 0.5) and basis states used are with total z = 0. Since the ground state of the HHM is spinprojection of electronic spin Stot z singlet, our calculations on Stot = 0 subspace are good enough to study the above quantities in the ground state.

3 Results and Discussions In this communication, we have compared the results of the adiabatic (ω0 < t) and non-adiabatic (ω0 > t) regions of phonon energies. For this, we have chosen the values of ω0 = 0.4t and 3.0t, respectively. Moreover, calculations are for U/t = 0–2 for ω0 = 0.4t, and U/t = 0, 4, 8, 12 for ω0 = 3.0t. Using Eq. (5), the maximum value of the EP interaction obtained, that is valid for one-phonon approximation (Nph = 1) are λ ∼ 0.8 for ω0 = 3.0t, and λ ∼ 0.1 for ω0 = 0.4t. Keeping view of the facts discussed in the last section, we argue that our one-phonon approximation holds quite well for λ ≤ 0.3 for ω0 = 0.4t and λ ≤ 0.8 for ω0 = 3.0t and the qualitative behavior of the observable does not change apprehensively in this limit with further increase of phonon numbers . 3.1 Ground State Properties The variation of the EP correlation function χ (| p − q|) with inter-site EP interaction λ is shown in Fig. 1a for ω0 = 0.4t and U/t = 1.0. Figure 1b shows the results for U/t = 0.1. It is evident from the figures that on-site correlation is much dominant over other sites for negligible or small values of EP interactions. The NN site (intersite) correlation gradually increases and for λ ∼ 0.25 (U/t = 0.1) the inter-site correlations also dominate along with the on-site correlations. However, only the onsite correlation dominates for U/t = 1.0 in our range of EP interaction. Correlations between other sites are negligible. Thus, EP correlation is maximum just up to two lattice spacings for U/t = 0.1. Both the on-site and inter-site correlations gradually increase with the phonon-mediated interaction λ for ω0 = 0.4t. In this region, there is a competition between phonon-mediated interaction λ and the Hubbard repulsion U/t, as a result the correlation increases gradually. No sharp transition in the correlation function is observed. The on-site and inter-site lattice deformations may trap a nearby electron forming a small or on-site (S0) and large or inter-site (S1) bipolaron (singlet or triplet), respectively. The formation of the bipolarons is slower for U/t = 1.0 than U/t = 0.1, and no inter-site bipolarons are formed in the considered range of EP interaction, as stronger Hubbard repulsion suppresses the formation of bipolaron. Compared to inter-site EP correlation, smaller λ is necessary for the growth of on-site EP correlation. In Fig. 2a, we plot χ (| p − q|) with inter-site EP interaction λ for U/t = 8.0 and ω0 = 3.0t. Figure 2b shows the same for U/t = 4.0. Here also, on-site and NN site correlation functions are dominant suggesting the formation of S0 and S1 bipolaron, respectively. However, a sharp transition in the correlation has been observed at a

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(a)

0.4

|p-q|=0 |p-q|=1 |p-q|=sqrt2 |p-q|=2 |p-q|=sqrt5 |p-q|=3

χ(|p-q|)

0.3

0.2

U=1.0t ω0=0.4t

0.1

0.0 0.00

0.05

0.10

0.15

0.20

0.25

0.30

λ

(b) 0.4

χ(|p-q|)

0.3

U=0.1t ω0=0.4t 0.2

|p-q|=0 |p-q|=1 |p-q|=sqrt2 |p-q|=2 |p-q|=sqrt5 |p-q|=3

0.1

0.0 0.00

0.05

0.10

0.15

0.20

0.25

0.30

λ

Fig. 1 Variation of electron–phonon correlation χ (| p − q|) with phonon-mediated interaction λ for ω0 = 0.4t and a U/t = 1.0 and b U/t = 0.1

certain (critical) value of λ = λc (U/t). With the increase of U/t, the value of the critical EP interaction λc increases. For U/t = 4.0, the transition occurs at λc ∼ 0.2, whereas for U/t = 8.0, transition takes place at λc ∼ 0.5. However, on-site correlation leading to the formation of S0 bipolaron grows at slightly lower values of λ than that for inter-site correlation leading to S1 bipolaron formation. In the range 0 ≤ λ ≤ 0.4 for U/t = 8.0, the strong Coulomb repulsion and the phonon-mediated interaction compensate each other and thus χ (| p − q|) is negligible. With λ > 0.4, the EP interaction dominates over the Hubbard repulsion, increasing on-site and intersite χ . Above λc , two polarons tend to bind each other forming bipolarons which is also confirmed by the bipolaron binding energy. However, complete saturation in the correlation is not observed for U/t = 8.0, but is observed for U/t = 4.0 suggesting well-formed stable bipolarons. Moreover, with the increase of Coulomb repulsion,

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(a) 0.8 0.7 0.6

χ(|p-q|)

0.5

U=8.0t ω0=3.0t

0.4

|p-q|=0 |p-q|=1 |p-q|=sqrt2 |p-q|=2 |p-q|=sqrt5 |p-q|=3

0.3 0.2 0.1 0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

λ

(b) 0.8 0.7 0.6

U=4.0t ω0=3.0t

χ(|p-q|)

0.5 0.4

|p-q|=0 |p-q|=1 |p-q|=sqrt2 |p-q|=2 |p-q|=sqrt5 |p-q|=3

0.3 0.2 0.1 0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

λ Fig. 2 Variation of EP correlation χ (| p.q|) with phonon-mediated interaction λ for ω0 = 3.0t and a U/t = 12.0 and and b U/t = 4.0

higher magnitude of λ is required for the formation electron pair. Thus, it is evident that phonon-mediated interaction generates an effective attraction between the electrons against the Coulomb repulsion resulting in the formation of bipolarons. Recently, the existence of such critical EP coupling ∼ U/W (W = bandwidth = 8t) for crossover from antiferromagnetic to charge-density-wave state is also suggested by Johnston et al. [42] using Determinantal Quantum Monte Carlo method. From Figs. 1 and 2, it is also evident that correlation for ω0 = 3.0t is much more pronounced than that for ω0 = 0.4t. In the non-adiabatic regime, the lattice fluctuations are fast and phonons are able to follow immediately, and thus substantial EP correlation appears at a small EP interaction. For U/t = 8.0 (ω0 = 3.0t), χ (| p − q|) →0. In the infinite system, the sharp crossover observed in Fig. 2 can become a phase transition from the normal state to bipolaronic state. The bipolarons are considered as

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(a) 0.8 0.6 0.4 0.2

Δbp

0.0 -0.2 -0.4

U/t=0.0 U/t=0.5 U/t=1.0 U/t=1.5 U/t=2.0

-0.6 -0.8 -1.0 0.00

0.05

ωo=0.4t 0.10

0.15

0.20

0.25

0.30

λ

(b)

2 0 -2 -4 -6

Δbp

-8

ω0=3.0t U/t=0.0 U/t=4.0 U/t=8.0 U/t=12.0

-10 -12 -14 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

λ Fig. 3 Variation of bipolaron binding energy bp versus λ for various U/t and a ω0 = 0.4t and b ω0 = 3.0t

localized Cooper pairs in the limit of strong EP coupling constant [43]. However, this crossover may not lead to a superconducting state. A superconducting state emerges in such a system due to the presence of very light mobile bipolarons [44]. The mass of the bipolarons is inversely proportional to the size of the bipolarons. Since the size of the bipolarons thus formed is small and hence they are heavy, and extremely localized to the sites and thus no superconducting phase emerges. The bipolaron binding energy (bp ) is a good parameter to examine the formation of electron pairs mediated by phonons. A negative value of bp indicates the stability of a two electron bound state [45]. In Fig. 3a, b, we plot the variation of the bipolaron binding energy bp versus λ for ω0 = 0.4t and 3.0t, respectively. It is clear from Fig. 3a that no bipolaron is formed for U/t = 2.0 and ω0 = 0.4t in our range of EP interaction. However, for ω0 = 3.0t, bipolarons exist even in the presence of strong

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Hubbard repulsion U/t = 8.0. But, for U/t = 12.0 no stable bipolarons are formed in the range of considered EP interaction. In both the cases, the stable bipolarons are formed after a certain value of λ = λc . This critical λc of Fig. 3b matches well with Fig. 2. The critical value of λ for adiabatic case (ω0 = 0.4t) is determined from the calculation of binding energy only. For both the adiabatic and anti-adiabatic limits, and U/t = 0.0 stable bipolarons form for zero or negligible value of EP interaction λ, although |bp | is greater for higher phonon energy. For U/t = 0.0, phonons are the only source of interaction (attractive) and this effective attraction mediated by phonons pairs two electrons to form bipolarons even at a very low λ. For finite U/t and λ < λc , the Coulomb repulsion U and the inter-site EP interaction compete with each other and the electrons can be considered as free. But, for λ > λc phonon-mediated interaction overcomes the Coulomb repulsion leading to an effective attraction between the electrons forming electron–electron pair i.e., bipolarons. The S0 bipolarons formed are stable under the condition U < 2E p (E p is the single polaron energy). In this limit, all the polarons are paired in the ground state with an effective bipolaron–bipolaron and bipolaron–polaron repulsive situation originating from Pauli’s exclusion principle (two bipolarons, and three or more polarons cannot occupy the same site). Though the phonon-mediated interaction leads to the formation of both S0 and S1 bipolaron, the binding energy calculated is the resultant of the two. In Fig. 4a, b, we plot the schematic phase diagram of the HHM with phononmediated interaction and Coulomb repulsion for ω0 = 0.4t and 3.0t, respectively, separating the regions of formation of different quasiparticles as a function of g/t (g 2 = 2tdω0 λ) and U/t. The phase diagram is obtained by bp = 0.0. In both the figures, we consider the region where the one-phonon approximation is valid. The region of formation of both types of bipolarons is shown in the phase diagram. Results clearly indicate that no bipolarons are formed for U/t > 1.58 for ω0 = 0.4t. For ω0 = 3.0t the maximum value of U/t for which bipolarons can form in our range of EP interaction is U/t ∼ 11.3.

3.2 Thermodynamic Properties Figure 5 shows the temperature dependence of entropy for various values of λ and U/t = 8.0. The inset shows the result for U/t = 4.0. Here, we consider only the anti-adiabatic limit (ω0 = 3.0t) as bipolaron formation is more prominent in this limit. Initially, the increase of λ slightly increases disorder (entropy) of the system at low temperature. The slight increase in entropy of the system at low T for small λ (< λc ) is due to the competition between Coulomb repulsion and attractive EP interaction. But, for λ ≥ λc , the polarons tend to bind to form bipolarons and the system goes to a more ordered phase. A close observation reveals that the transition to the ordered state is in conformity with Fig. 2 for both values of U/t. Hence, we can expect that at larger values of the inter-site EP interaction λ (> λc ), on-site Coulomb repulsion is suppressed, as a result, the two electrons coalesce on a single site (S0 bipolaron) and/or on neighboring sites (S1 bipolaron).

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(a) 1.6 1.4

ω0=0.4t λc curve

1.2

Competition between intersite EP interaction and Hubbard repulsion, polaron

U/t

1.0 0.8 0.6

S0 bipolaron S1 bipolaron

0.4 0.2 0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

g/t

(b)

12

ω0=3.0t

10

U/t

8 6 4

λc curve

Competition between intersite EP interaction and Hubbard repulsion, polaron S0 bipolaron S1 bipolaron

2 0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

g/t Fig. 4 Schematic phase diagram for a ω0 = 0.4t and b ω0 = 3.0t. The phase diagram is obtained by bp = 0

In Fig. 6, the variation of specific heat with temperature for several values of λ is shown for U/t = 8.0 and ω0 = 3.0t. The inset shows the same for U/t = 4.0. It appears that C is strongly dependent on T with a maximum at T ∼ 1.45t for U/t = 8.0 and λ = 0.0. For U/t = 4.0, the maximum for λ = 0.0 is around T ∼ 1.9t The peak however moves to higher temperature region with the increase of λ. At low T , specific heat C ∼ T 2 is observed. With the increase of EP interaction, the peak becomes broader and the peak-height increases. In the spin system, the peak can be attributed to the thermal excitation of the spin degrees of freedom with the exchange energy scale persisting in the doped system. So, the contribution of phonon-meditated interaction is to increase the effective exchange energy.

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0.5 λ=0.0 λ=0.2 λ=0.4 λ=0.6 λ=0.8

0.4

0.3

S

U/t=8.0 ω0=3.0t

λ=0.0 λ=0.2 λ=0.4 λ=0.6 λ=0.8

0.5

0.2

0.4

S

0.1

0.3

U/t=4.0 ω0=3.0t

0.2

0.1

0.0 0.0

0.0 0.0

0.5

1.0

1.5

2.0

0.5

2.5

1.0

3.0

1.5

2.0

T/t

2.5

3.5

3.0

3.5

4.0

4.0

T/t Fig. 5 Temperature dependence of entropy S for ω0 = 3.0t and U/t = 8.0. Inset shows the same for U/t = 4.0 λ=0.0 λ=0.2 λ=0.4 λ=0.6 λ=0.8

2.8 2.4 2.0

1.6 U/t=8.0

ω0=3.0t

C

3.6

1.2

3.2 2.8 2.4 2.0

0.8

C 1.6

λ=0.0 λ=0.2 λ=0.4 λ=0.6 λ=0.8

U/t=4.0 ω0=3.0t

1.2 0.8

0.4

0.4 0.0 0.0

0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

T/t

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

T/t Fig. 6 Temperature dependence of specific heat C for ω0 = 3.0t and U/t = 12.0. Inset shows the same for U/t = 4.0

4 Conclusion We used ED technique to study HHM with inter-site EP interaction λ. It appears that there is a competition between Hubbard repulsion and inter-site EP interaction. EP correlation function χ increases gradually with λ for U/t = 1.0 and 0.1, ω0 = 0.4t (adiabatic limit). But, for U/t = 8.0 and 4.0, ω0 = 3.0t (non-adiabatic limit), EP correlation shows a sharp transition at a certain λ = λc (U/t). Only on-site and NN site χ ’ s are significant. On-site (S0) and inter-site (S1) bipolarons are formed which tend to bind at λ = λc (U/t). Schematic phase diagrams are drawn representing the parameter regions for polarons and bipolarons. The entropy curves shows that the system goes to

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a more ordered state with the formation of bipolarons for λ ≥ λc . Specific heat curves show broad peaks which can be attributed to the thermal excitation of the spin degrees of freedom. Peaks shift to higher temperature region with the increase of λ. Acknowledgments We are grateful to the Department of Science and Technology, Government of India for their financial support under DST-PURSE programme. S. N. thanks Department of Higher Education, Government of West Bengal for their moral support.

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