We used a spin-1 Bose Hubbard model (SBHM) that includes an additional spin-dependent interaction potential due to the presence of the hyperfine degrees of freedom. 84 3.8 The phase diagrams in the AF case with different perturbation strengths 86 3.9 The phase diagram in the AF case with spin-dependent perturbation.
Bose Einstein condensation (BEC)
Above the critical temperature on the left and far below the critical temperature on the right. However, for interacting Bose gas, the ground state of many bodies in the system results in a nonlinear term in the order parameter and such interacting BEC in optical lattices is described by a mean field Gross Pitaevskii equation (GPE) [55,56] .
Optical lattice
Optical lattice potential
Further using two pairs of laser beams in the y and z direction, a three-dimensional optical lattice potential is formed as shown in [Fig.1.2] [2],. The potential height of the optical lattice, V0i is often expressed in terms of the return energy via ER = ~2k2/2m.
Laser cooling
The tunability of different parameters that are agents of the phase transition therefore gives an advantage over experiments carried out in the context of ordinary condensed matter physics.
Atom-atom interaction: Feshbach resonance
In a magnetically tuned Feshbach resonance, the length of the atomic scattering waves, as is now a function of the magnetic field, B as [62],. Here abg is the background scattering length as a result of the background collision in the open channel represents an off-resonance value.
Ultracold atoms in optical lattices
Bose Hubbard model (BHM)
Thus, in the MI phase, the many-body ground state is the product of a local Fock state with a fixed number of atoms (n) per lattice site, which is given by [54]. Visual representations of the occupation densities in the MI lobe (ρ= 1) in (b) and in the SF phase in (c) of bosons on the optical lattice.
Superfluid to Mott insulator transition: QPT
A point in the vicinity of two MI lobes with ρ = n and ρ = n + 1 is degenerate along the vertical axis (µ) and corresponds to the SF phase. With a further increase of the potential height above the critical value V0c '13Er [2], the sharp peaks in the absorption spectra disappear, indicating a destructive interference pattern and thereby signaling the transition to the MI phase.
Optical dipole trap (ODT)
To see how a dipole force arises, consider a spatially varying electric field of a laser beam that oscillates with the shape frequency ωl. Here the unit vector is in the direction of the electric field and|gi= |F, mFi is the atom's ground state and the summation takes over all the excited states.
Spin-1 Bose gas: An era of quantum magnetism
The application of an external magnetic field, which breaks the rotational symmetry of the system, shows a transition between easy-axis and easy-plane ferromagnetism. Furthermore, a synthetic dimension can be used to generate a high synthetic magnetic field of the order of unit flux per plaquette to overcome the rotational limitation of the neutral atoms in optical lattices.
A comparison between spin-0 and spin-1 Bose gases
Recently, it has been shown that the different hyperfine spin states can be sequentially coupled to form an extra dimension, known as a synthetic dimension, similar to the spatial dimension of an optical lattice [47,74] [Fig.1.9]. For 87Rb (F = 1) atoms, a Raman transition that splits the three spin states mF = ±1.0, thus providing the synthetic dimension together with an optical lattice dimension can be used to create a synthetic magnetic flux as a result of Peierl's coupling [ 47].
Various aspects of ultracold atoms in optical lattices
- Disorder optical potential
- Synthetic magnetic field
- Dipole-dipole interaction
- Multi-body interaction potential
Study of the BG phase in the context of ultracold Bose gas in optical lattices has attracted much attention due to the ability to introduce disorder in a controlled manner. At higher values of the three body interactions, we found different order of phase transition from the MI to the SF phase corresponding to the first and third MI lobes.
Spin-1 Bose Hubbard model (SBHM)
On-site spin-independent and spin-dependent interactions can be positive or negative depending on the value of the atomic scattering lengths, ao,2. The SBHM ground state [equation (2.15)] exhibits different types of symmetry depending on the value of U2/U0.
SBHM in the atomic limit (t = 0)
Finally, the resulting phase diagram of the MI lobes in the atomic boundary is shown in Figure 2.1. For smaller values of U2/U0, both odd and even MI lobes exist, and the chemical potential width of the even MI lobes increases with increasing values of the spin-dependent interaction and completely occupies the entire phase diagram above U2/U0 >0.5. In the ferromagnetic case, the MI lobes are the same as those in the spin 0 case with the same chemical potential width at U2 = 0 and gradually shrink at U2 = −U0.
Tools to solve SBHM
- Single site mean field theory (SSMFT)
- Ground state energy variation with occupation densities 61
- Order parameter and variational energy behaviour
- Spin eigenvalue and spin nematic order parameter
- Perturbative mean field approximation (PMFA)
- Quantum Monte Carlo (QMC)
- Density matrix renormalization group (DMRG)
- Multi-site mean field theory (MMFT)
Such asymmetry is due to the formation of the spin-singlet (nematic) phase corresponding to the even (odd) MI lobes and therefore leads to the stability of the even MI phase. For the odd MI lobes (μ/U0 = 0.5), the finite values of the Qzz imply a spin anisotropy in situ for the spin nematic state and then a second-order transition to the SF phase is observed [ Fig. 2.6 (d) ]. So far we have presented the numerical phase diagrams of the mean field for both values of the spin-dependent interaction.
Results
- The behavior of the SF order parameter and com-
- Spin eigenvalue and spin nematic order parameter
- Indicator for the MI, BG and SF phases
- Observation of the percolation phenomena
- Percolation probability and finite size scaling
- Phase diagrams
- Phase diagram with disorder in spin dependent inter-
For the lobe at MI, the MI region begins to shrink due to the appearance of the BG phase at ∆/U0 = 0.3 and shows a continuous phase transition to the SF phase [Fig.3.2 (left)]. For equal lobe MI, the MI phase starts to decrease due to the appearance of the BG phase and the MI-SF phase transition continues to be first order for ∆/U0= 0.3. These values are set for numerical convergence of the parameters corresponding to the MI phase.
- The behaviour of the order parameters and the indicator 87
At ∆/U0=0.3, χ remains zero until zt/U0= 0.085, above which χ starts to increase and finally takes a value of unity in the SF phase. At ∆/U0 = 0.5, χ only has a value between 0 and 1, which means that the system consists only of BG and SF phases. For ∆/U0 = 0.5, the MI lobes disappear because the perturbation strength is higher than the critical perturbation strength and only the BG and SF phases remain in the system.
Conclusions
In the next section, we provide a detailed theoretical formalism of the MFA on a square lattice for a spinor Bose gas in the presence of the synthetic and the external magnetic fields. The spin properties of the polar SF phase and the symmetry of the ground state were studied in detail with and without an external magnetic field in Ref. In the non-interacting limit, the single-particle energy spectrum of the Hamiltonian demonstrates a Hofstadter butterfly [162].
Results
Antiferromagnetic case
- Phase diagrams
- SF order parameter variation
- Energy spectrum and magnetization
At higher values of magnetic flux, the even MI lobes become unstable due to the suppression of spin-singlet pair formation. In Fig.4.4 we study the one-dimensional behavior of the SF order parameter corresponding to the even and the odd MI lobes in the AF case for different values of φ corresponding to λ = 1. While for the even MI lobes ( μ/U0 = 1.4) and for low flux strengths, i.e. φ ≤ U2/U0, the MI-SF phase transition has a first order character due to the jump of the order parameter.
Ferromagnetic case
AtBs= Be = 0.05 (blue lines), M¯tr is zero for both MI lobes, but M¯l shows a first-order transition as a function of jumping force for the even MI lobes and a transition of the second order for the odd MI lobes [Fig.4.6]. For the ferromagnetic interaction, the phase diagrams are similar to those of the spin-0 (scalar) system, except that only the chemical potential width now rescales with the Zeeman interaction strength, ηasµ+η→µ0 [148,159]. Unlike the AF case, in the ferromagnetic case all MI phases become more stable with increasing magnetic flux strengths due to the phase decoherence of the SF order parameter at the transition point, as discussed above.
Conclusion
Later, possible ground state structures of spin nematic and spin singlet MI phases and the transition between them were investigated in Refs. We then present the phase diagrams, which include CDW and other phases corresponding to different values of the extended interaction strengths. Here we have considered a square lattice with dimension L×L and obtained the ground state energy and the eigenfunction of the system.
Results
Antiferromagnetic case
- MFA phase diagrams
- Spin eigenvalue behaviour
- PMFA phase diagrams
- Effects of harmonic trapping potential
The variation of the SF order parameter, ψA/Bin (b) and the ground state energy, EgA/Bin (d) for zV/U0=1.7. While for the DW(10) or DW(30) phases, hS2Ai=2,hSB2i=0 in the CDW phase and shows continuous transition to the SS and SF phases. From Fig.5.5(a) it is clear that they are symmetric about the center ofxi and moving from the center in both directions we find, ψi = 0 while ρi = 1 which signals the MI(1) phase .
Ferromagnetic case
We also considered the two-dimensional scenario with the inclusion of ky and found that the peaks in ρk correspond to (kx, ky) = (2πj,2πm) for the SF or SS phases in addition to (kx, ky)= (πj, πm) for the SS or CDW phases, where j, m are integers and kx,y are defined in the lattice constant unit. The pulse profile of ψk and S2k shows similar peak positions to ρk in the SF and SS phases, but no peak is observed at (kx, ky) = (πj, πm) for the CDW phase. Furthermore, the resulting boundary equations for the MI-SF and CDW-SS phase are identical to those obtained for different values of zV/U0 in Ref.
Conclusion
The explicit form of the three-body interaction was derived by Tiesinga et al. [192] which consists of both the spin-independent and the dependent terms, similar to that of the two-body interaction. In the next section we sketch the SBHM in the presence of the three-body interaction potential and a quick recapitulation of the mean field formalism.
Results
Antiferromagnetic case
- MI lobes width in the atomic limit with (W > 0 )
- MFA phase diagrams (W > 0 )
- Spin population fraction and spin eigenvalue ( W > 0 ) 130
- Phase diagrams for attractive three body interaction
- Phase diagrams for purely three body interaction ( U 0 =
The variation of the spin eigenvalue, hS2i with different values of W/U0 corresponding to the even and the odd MI lobes in the AF case is shown in Fig. 6.4(b). The second and fourth MI lobes show a first-order transition from the MI to the SF phase [Fig.6.4 (b)]. The most intriguing fact is that the odd–even asymmetry in the MI lobes remains intact in the presence of the attractive three-body interaction potential.
Ferromagnetic case
Conclusion
In the ferromagnetic case the phase diagrams are similar to those of the spin-0 Bose gas for the repulsive [190,198] three-body interaction. The next section introduces the mean field Hamiltonian in the presence of an attractive three-body interaction potential. Initially, we presented the mean field phase diagrams for different values of the attractive three-body force.
Results
Antiferromagnetic case
- MI lobes in the atomic limit
- MFA phase diagrams
- Spin eigenvalue and spin nematic order parameter
- PMFA phase diagrams
- Phase diagram for U 3 = − 1.4U 0
It shows that at a representative value for U3, namely U3/U0 = −0.5, although there is no change in the first and fifth lobes of MI, surprisingly, the third lobe of MI expands significantly to encompass the second (ρ = 2) and fourth ( ρ = 4) lobe MI. Furthermore, for a larger value of U3, namely U3/U0= −1.1, the third lobe of MI grows relative to the other lobes of MI so that it completely encroaches on the second and fourth lobes of MI. However, we verified that for smaller values of U2/U0 ≤ 0.03, the MFA and the phase diagrams obtained by PMFA differ slightly from each other at the tip of the MI lobes, especially this is visible at the third MI lobe.
Ferromagnetic case
Conclusion
However, an asymmetry is observed for the spin singlet phase at higher values of the flux strength. The spin eigenvalue and the spin nematic order parameter show the formation of the spin singlet phase for the straight MI lobes and indicate a first-order transition to the SF phase. This was confirmed by carefully examining the behavior of the varying ground state energy and the spin eigenvalue and spin nematic order parameter.
