MFA phase diagrams

5.2 Results

5.2.1 Antiferromagnetic case

5.2.1.1 MFA phase diagrams

reasonable to break the entire lattice into two sublattices, such as the A and B sublattices. So the unit cell has two types of atoms, namely A and B, where an A atom has Batom as all its neighbours and vice versa. In the CDW phase, one type of sublattice has higher occupancy than the other, so without much loss of generality, we assumen_A > n_B.

To begin with, let us consider VT = 0. In the absence of trapping, we can diagonalize Eq.(5.3) over the unit cell consisting of two sites namely A and B.

Thus for Atype of sublattice (µA =^µB= µ), Eq.(5.3) can be written as, H_A^MF = ^ztX

σ

(−ψBσaAσ−h.c+^ψBσψAσ)+ ^U0

2 nA(nA−₁)

− µnA+ ^U2

2 (S²_A−_2nA)+^zVρB(nA−ρA) 5.4 Similarly one can easily obtain H_B^MF by changing the indices from A to B (and B to A) and the total mean field Hamiltonian is just sum of two mean field Hamil- tonians, that is, H_i^MF = ^H_A^MF +^H_B^MF. The self consistent ground state energy and the eigenfunction are obtained by diagonalizing H_i^MF in the occupation basis |n_iσi withn_i =7 starting with some guess values forψ_(A/B)σ andρ_(A/B)σ as discussed in section2.4.1 of chapter 2.

We present our phase diagrams corresponding to both the weak and strong interaction limits ofzV/U₀in Fig.5.1(a)-(d). The reason for choosing four different valueszV/U₀ is justified in the upcoming discussion.

0 0.1 0.2 0.3 zt/U

0

0 1 2 3 4

µ/U 0

DW(10) MI(1) DW(21)

MI(2) DW(32)

MFA

SS

Perturbed

SS SS

SF

zV=0.7U

0

(a)

0 0.1 0.2 0.3 0.4 zt/U

0

0 1 2 3 4

µ/U

DW(10) DW(20) DW(21) MI(2)

DW(32)

MFA

SS PMFA SS

SF zV=0.95U

0

(b)

0

0 0.2 zt/U 0.4 0.6

0

0 1 2 3 4

µ/U

DW(10) DW(20) DW(30) DW(40)

MFA

SS

PMFA SS

SF

zV=1.15U

0

(c)

0

0 0.5 1 zt/U 1.5

0 0

1 2 3 4

µ/U

DW(10) DW(20) DW(30) DW(40)

MFA PMFA

SS

0

SF

zV=1.7U

0

(d)

Figure 5.1: Phase diagrams in AF case with U2/U0 = 0.03 for different values of zV/U0 are shown from (a)-(d). The dashed lines denote the results of the perturba- tion calculation (PMFA) and the circles represent the mean field (MFA) phase dia- grams. At zV/U0=0.7[Fig.(a)], the phase diagram consists of various CDW phases, the even MI(2) and odd MI(1) phases. In Fig.(b), for zV/U0 =^0.95>1−2U2/U0, the odd MI(1) lobes now gripped by the CDW(20) phase. In Fig.(c), the MI(2) phase is now occupied by the CDW(40) phase since zV/U0= 1.15>1+2U2/U0. In Fig.(d), at strong interaction limit, zV/U0= 1.7, all insulating phase are now the CDW phases.

In Fig.5.1(a), the phase diagram presented forzV/U₀ =0.7 shows that the CDW phase appears in between the MI lobes, and thus a direct transition from the CDW to the SF is interrupted due to the appearance of the SS phase (region enclosed by the * symbol). The symbol DW(ρAρ_B) implies the CDW phase having occupation densities, ρ_A (corresponding to sublattice A) and ρ_B (corresponding to sublattice B) with average occupation density, ρ¯ = (1/2)[ρ_A + ^ρB]. The SS phase, which not only appears concomitantly with the DW(21) and DW(32) phases, but also

exists along with the DW(10) phase forρ <¯ 1/2, as was predicted earlier through quantum Monte Carlo (QMC) studies in Ref. [101]. The MI(1) and MI(2) are the Mott insulating phases with occupation densities ρ_A = ^ρB = ^{1 and ρ}A = ^ρB = ² respectively. The odd and even MI phases which form the spin nematic and singlet phases are the distinguishing features of a spinor Bose gas compared to a spin-0 Bose gas and the stabilization of the singlet phase over the nematic phase has been studied extensively without extended interaction in Refs. [24,25].

In Fig.5.1(b), we found that corresponding tozV/U0= 0.95, although the phase diagram consists of all the compressible and incompressible phases, however, interestingly, the MI(1) phase is now completely occupied by the DW(20) phase and the chemical potential widths of the DW phases increase with zV/U0, which now has a value 0.95 [Fig.5.1(b)] compared to 0.7 in Fig.5.1(a).

The disappearance of the MI(1) phase can be understood by considering the atomic limit, that is, t = 0 where the system only consists of the MI and CDW phases. In the atomic limit, the ground state energyEg(nAnB)from Eq.(5.1) in the CDW phase is given by,

E_g(n_An_B)= ^U0 4

X

i=^A,B

[n_i(n_i−₁)]− ^µ 2

X

i=^A,B

n_i+^U2 4

X

i=^A,B

[S_i(S_i+1)−_2ni]+^zV

2 n_An_B 5.5 Following the calculations carried out in Ref. [107], we found that the chemical potential width for the odd MI lobes isU0−_2U2, while for the even MI lobes, it is U0+2U2. Also we have checked that for the CDW phase, the chemical potential width iszV using the same assumption that for the odd occupation densities, we shall consider the spin eigenvalues to beS = 1 and for even occupation densities, S = 0 will be considered. So for zV/U0 = 1± _2U2/U0, there are possibilities of coexistence of different CDW and MI phases because of the degeneracy in their ground state energies. As a result, atU2/U0= 0.03, whenzV/U0>0.94, the MI(1) phase now gets absorbed by the DW(20) phase and this applies for the other odd MI lobes as well for U2/U0<0.5.

At larger value of the extended interaction, that is, for zV/U0 = 1+2U2/U0 = 1.06, the MI(2) phase becomes degenerate with the DW(40) phase and beyond this critical value, all insulating phases become CDW phases and the SS phase has now expanded significantly with increasing zt/U0 [Fig.5.1(c)]. We have also ob- tained the phase diagrams at stronger values of the extended interaction strength, that is, zV/U0 = 1.7 in Fig.5.1(d), which indicates that the system is more likely to be in the SS phase compared to the CDW or the SF phases.

Since the formation of spin singlet pairs corresponding to even occupation densities and their stabilization over the odd MI lobes has been studied in Refs.

[24,25,110] without the extended interaction, one can ask, is it possible to have also the spin singlet phase corresponding to integer ρ¯ for the CDW phase. If

we look at Fig.5.1(a)-(b), it can be concluded that although the critical tunneling strength,zt_c/U₀for transition from the incompressible to the compressible phases still occurs at higher values of zt/U₀ for the MI(2) phase compared to that of the MI(1) phase, but zt_c/U₀ for the DW(20) phase is now enhanced with increasing zV/U₀ compared to the other insulating phases except for the DW(10) phase.

Interestingly, from Fig.5.1(c)-(d), we found that thezt_c/U₀for a transition from the CDW to the SS phase corresponding to the DW(20) or DW(40) are higher than that of the DW(10) or DW(30), thereby indicating a possibility of spin singlet formation in these CDW phases which we shall confirm by calculating the local spin eigenvalue in the subsequent discussion.

5.2.1.2 SF order parameter and occupation density variation

It is also helpful to study the nature of phase transition for different phases in the AF case. In section2.2 in chapter 2, the order parameters variation show

0 0.1 0.2 0.3

zt/U

0

0 0.5 1

ρ

ρ_A

ρ_B

A/B

µ/U

0=0.3

(a)

zV/U0=0.7

0 0.5 1 zt/U

0 0

0.5 1

Ψ

Ψ_A

Ψ_B zV/U0=1.7

µ/U₀=1.5

(b)

A/B

0 0.25 0.5 0.75 1

zt/U

0

0 1 2

ρ

0.57062 0.5708 2.01

2.02

ρ

_A

0.5706⁰ 0.5708 0.03

ρ

_B

ρ_A

ρ_B

µ/U₀=1.5

(c)

zV/U0=1.7

A/B

0 0.5 zt/U₀ 1 1.5

-3 -2 -1 0

E

0.56 0.57 0.58 -2.058

-2.052

E

E_g E_g^B

A

zV/U₀=1.7 µ/U₀=1.5

(d)

A g

gA/B

Figure 5.2: The one dimensional behaviour of the occupation densities ρ_A/B in the AF case with zV/U0= 0.7in (a) and zV/U0 =1.7in (c). The variation of the SF order parameter, ψA/Bin (b) and the ground state energy, E_g^A/Bin (d) for zV/U0=^{1.7. The} discontinuity in the order parameters and E_g^A/Bindicate a first order transition for the DW(20)-SS transition while second order for the DW(10)-SS transition respectively.

that the MI-SF phase transition is first order for the even MI phase and second order for the odd MI lobes without an extended interaction in Refs. [24,25].

In Fig.5.2(a), we have shown the one dimensional behaviour ofρ_A/BforzV/U₀= 0.7 at µ/U₀ = 0.3 and it shows a continuous transition from DW(10) to the SS and then to the SF phases, indicating a second order transition. The same holds true for other DW phases. However, we have checked that the transition from DW to SF phase is found to be first order in nature, and the SF-MI transition for the even and odd MI phases still show first and second order transition respectively forzV/U₀ =^{0.7 andzV/U}0 =^0.95.

ForzV/U₀>1.15, the CDW phases indicate a first or a second order transition depending upon the even or the odd occupation densities respectively. The SF order parameters and occupation densities show a continuous transition from DW(10) or DW(30) to the SS and then SF phase for both values ofzV/U₀, namely, zV/U₀= ^{1.15 and zV/U}0= 1.7. While for the DW(20) or DW(40),ψ_A/B [Fig.5.2(b)]

and ρ_A/B [Fig.5.2(c)] show a first order transition to the SS phase. The ground state energy, E_g^A/B of the DW(20) or DW(40) also shows discontinuous transition to the SS phase, a behaviour similar to the spin singlet MI phase [Fig.5.2(d)].