Quantum phase transitions and the degree of nonidentity in the system with two different species of vector bosons

We address the system with two species of vector bosons in an optical lattice. In addition to the the standard parameters characterizing such a system, we are dealing here with the"degree of atomic nonidentity", manifesting itself in the difference of tunneling amplitudes and on-site Coulomb interactions. We obtain a cascade of quantum phase transitions occurring with the increase in the degree of atomic nonidentity. In particular, we show that the phase diagram for strongly distinct atoms is qualitatively different from that for (nearly) identical atoms considered earlier. The resulting phase diagrams evolve from the images similar to the"J. Mir\'o-like paintings"to"K. Malewicz-like"ones.

The typical example is the system of vector bosons. This case corresponds to Bose-Hubbard model that is absent in the standard solid state theory and is the topic of intensive research last time [22][23][24][25][26][27][28][29][30][31][32][33][34], and [35][36][37] for review. Numerous striking effects, in particular, induced by the multispecie nature of boson system have been found recently including quantum phase transitions, many-body localization, and topological order, as well as the superfluidity and supersolidity of ultracold atomic systems [35][36][37]. Here, we focus on still unexplored physical phenomena in the systems with different species of vector bosons originating from the tunable interplay of spin degrees of freedom and of those identifying different sorts of atoms. Then the situation becomes quite intriguing: in addition to the standard parameters, there appear nontrivial ones related to the 'degree of atomic nonidentity': the difference of tunneling amplitudes and on-site interactions. We address here new physical effects including quantum phase transitions driven by atom nonidentity.
Vector cold atoms in optical lattices are characterized in general by the following parameters: intersite hopping amplitudes t α , where α=1, 2 labels different atoms, on-site U , a a¢ Coulomb interactions, and spin channel interaction parameters U s [31,38].
In our recent paper [31], we have considered the simplest limiting case of nearly identical bosons in the Mott insulating state: U 12 ;U 11 ;U 22 =U 0 and t 1 ;t 2 =U 0 . This model differs from the case of perfectly identical bosons just by the absence of cross-tunneling term: tunneling with the change of boson identity was forbidden. It has been shown in [31] that such model can be reduced to the Kugel-Khomskii [39] type spin-pseudospin model, but in contrast to the usual version of the latter model, here spin is S=1 and pseudospin-1/2 (where pseudospin labels different bosons). New effects in [31] have been found, in particular, that the ground state of the system always has the pseudospin domain structure while sign change of U s switches the spin arrangement of the ground state within domains from ferro to aniferromagnetic (AFM).
Here we also consider two species of vector bosons in an optical lattice, but we get rid of the identity conditions U 12 ;U 11 ;U 22 =U 0 , t 1 ;t 2 and trace the evolution of the system quantum state with the increase in the degree of atomic nonidentity (β=t 2 /t 1 , ξ 12 =U 12 /U 11 and ξ 22 =U 22 /U 11 ) starting from nearly identical atoms of [31].
Surprisingly, instead of some naive and predictable modification of our previous results [31], we find that the boson system evolves between six quantum states instead of two in [31] and the manifold of quantum phase transitions is so nontrivial that one can even compare the resulting phase diagrams with modern art paintings (we find some similarity with 'J Miró-like paintings' or 'K Malewicz-like' paintings depending on atom nonidentity), see figures 1-9.
The rest of our paper is organized as follows: in section 2, we introduce the model Hamiltonian, then in section 3, we reduce the initial general Hamiltonian for vector bosons to the effective Hamiltonian appearing to be anisotropic spin-pseudospin model of the Kugel-Khomskii type [39]; in section 4, we investigate different possible configurations in spin and pseudospin spaces and find their energy; in section 5, we address the ground state energy of the system and the quantum phase transitions. In particular, in section 5, we illustrate the evolution of phase diagrams with the degree of atomic nonidentity, which looks like the transformation from the Joan Miró style artistic image to that of Kazimir Malewicz. Finally in the appendix, we present the analysis of several special limiting cases of the model Hamiltonian that relate our model system to some well-known results.

Hamiltonian for two species of vector bosons
We consider two types of boson atoms with S=1 in the optical lattice with sites labeled by index i. The corresponding creation operators c ias † where σ={−1, 0, 1} is the spin index and α=1, 2 labels the type of boson.  The Hamiltonian includes three terms: The interaction between bosons is given by two terms: . Now we assume that the interaction constants can strongly differ from each other, in contrast to the case considered in [31].   The spin-dependent interaction term is taken in the standard form [40,41]: where n i =n i,1 +n i,2 is the total number of bosons at site i.     where i j , á ñ means the summation only over the nearest-neighbor sites and h t is the hopping Hamiltonian for one link. As usual, the repeated indices imply summation.

Effective Hamiltonian
Below we focus on the Mott insulating state, where cold atoms are localized at the sites of optical lattice with the number of bosons at each site equal to unity. We remind that in such case, the hopping terms where a=x, y, z.
Below we outline the algorithm of transforming the initial Hamiltonian to the effective one.

Basis states
In what follows, when we consider the link i j , á ñ between the nearest-neighbor sites, we focus on the basis of possible states for two bosons with spins S 1 =1 and S 2 =1 at neighboring sites i=1 and j=2. We are interested in the case with single occupation, i.e. when one boson of either type is located at each lattice site, n n 1 We can pass now to the basis of the eigenstates of the total spin squared S S S We designate these states as SM S , 0, 1, 2 ñ = | and M=−S, K, S. This basis can be written as follows where f=1, K, 4 enumerates the ways to distribute two types of bosons over two sites. The coordinate part S f f ñ | ( ) is given explicitly in [31].
Applying h t , see equation (4), to the basis states (6), we obtain two kinds of intermediate (virtual) states. The first type will be realized for two identical bosons at one site (i or j), the second type is for two nonidentical bosons at one site. Intermediate energies depend on the spin and types of bosons. They are There are no intermediate states corresponding to the total spin S=1 for two identical a-or b-bosons due to the symmetry of the total wave function [40].
The energy of the virtual states with the double site occupancy is much larger than the energy of the states with the single site occupancy we are focusing at. To find corrections to the energy of the single occupancy states related to the hoppings, we need the second-order terms of the perturbation theory. So, further it will be convenient to work with the operator In the basis of states (6), the matrix of h can be presented in the following block form Here I n , n=1, 3, 5, are the identity n×n matrices. I 1 accounts for one state with S=0, I 3 accounts for three states with S=1, and I 5 accounts for five states with S=2.
In what follows, we identify the single occupancy of site i with a-or b-boson by the pseudospin-1/2 states respectively. It will be convenient to introduce explicitly two types of creation operators: a is † for bosons of type α=1 and b is † for bosons of type α=2. For this purpose, we rewrite the pseudospin operator i  g at sites i, see equation (5), in the form We can rewrite, as usual, the set of  g operators in the other equivalent form: To describe the occupancy of sites i and j, we introduce the basis of pseudospin states i j ab a b ñ = ñ ñ | | | . Then, we find the correspondence between two-boson orbital states (6) and pseudospin states abñ In what follows, we map the matrix h, equation (9), onto an effective spin-pseudospin operator in the space SM . This operator will be given in terms of spin S=1 operators S i , S j and pseudospin T=1/2 operators T i , T j and it has the same structure as matrix h, equation (9).
Next, we introduce the projection operator Q S and P T onto the combination of states SMñ | and TM T ñ | corresponding to the total spin S=0, 1, 2 and pseudospin T=0, 1 at the link i j Similarly, in the pseudospin space the projectors onto the singlet T=0 and triplet T=1 states are It is also convenient to introduce the following projectors in the pseudospin space Here, we introduce the dimensionless parameter β=t 2 /t 1 that characterizes the difference of the tunnel amplitudes for different boson species.
The energy E of the ground state can be formally written as the average of the effective Hamiltonian over the ground state wave function (we will find it later using the variational approach) is the number of nearest neighbors for the D-dimensional cubic lattice. Within the mean-field approximation, we can neglect any correlations between spin and p-spin degrees of freedom. Then, for example, R Q R Q We take the trial wave function in the p-spin space as the 'mixed orbital state' on two-sublattices A and B i A cos sin , , 30 The FM and AFM p-spin states will occur at 4 q = p for η=+1 and η=−1, respectively. These states are polarized in x direction: For other angles θ, the mixed orbital state is a some intermediate state between the FM and AFM ones. That is why, it is more correctly to refer to this mixed orbital state as the mixed orbital state with η=+1, or η=−1 rather than the FM and AFM states. Now, we find averages over the trial mixed orbital state of the p-spin operators entering R i , i=0, 1, 2 Later on, we will minimize the energy E with respect to the angle θ.
In contrast to the p-spin space, the effective Hamiltonian in the spin space is isotropic. So, we do not expect any exotic states there. Hence, we take for trial wave functions in the spin space the usual FM, AFM, and nematic (NEM) states. It should be noted that NEM naturally appears for spin-1 isotropic Heisenberg Hamiltonian, it has zero average site spin, and it is discussed in detail in [40]. Finally, we have found all the averages and correlation functions entering equation (28) for the energy. We minimize numerically the energy E(θ, η), find the ground stats at different values of the parameters, and draw the corresponding phase diagrams.

General properties of the phase diagrams
We address here the ground state of the system and possible quantum phase transitions. Performing the energy minimization, we find the phase diagrams for different ranges of the parameters. The key parameters are β=t 2 /t 1 , λ=U s /U 11 , ξ 12 =U 12 /U 11 , and ξ 22 =U 22 /U 11 . All these parameters (except λ responsible for spin channel interaction) are related to the difference between the types of bosons.
There are six different phases: three phases have the FM pseudospin arrangement, while three others correspond to the AFM pseudospin state. Evolution of these phases is illustrated in figures 1-9; figure 1 is supplemented by the colorbar, where the correspondence between colors and phases is shown. Points at the colorbar correspond to the following orders in spin and pseudospin systems: Here, for example, color '1' corresponds to FM spin and pseudospin orders.
In figures 1-9, we demonstrate the evolution of the phase diagrams within the wide range of parameters β=t 2 /t 1 , λ=U s /U 11 , ξ 12 =U 12 /U 11 and ξ 22 =U 22 /U 11 . The most interesting phase transition is that accompanied by the change of atom distribution over the optical lattice: p-spin FM ↔ p-spin AFM. Note that the cold colors correspond to FM p-spin, while the warm ones-to p-spin AFM. So the transitions with p-spin change can be found in the phase diagram at the lines, where cold colors change to warm ones.
Evolution of the phase diagrams for ξ 12 =U 12 /U 11 =0 with ξ 22 =U 22 /U 11 on (λ, β)-plane is shown in figure 1. In this case, there is no Coulomb interaction between different boson species. We see that there is always quantum phase transition at the line λ=0. This is true not only for figure 1, but also for all phase diagrams in figures 1-9. This phase transition is driven by the sign change of spin channel interaction U s . This transition has been recently revealed in [31] for nearly identical vector bosons. Here, we show that this transition is quite robust with respect to the evolution of the degree of nonidentity.
One can also see in figure 1, that the 'left color' is always blue. It corresponds to FM spin ordering (with FM p-spin ordering). This situation is intuitively obvious: since the 'left' phase is determined by large negative λthe interaction in the spin channel. In all the next figures, the 'left color' also corresponds to the FM spin ordering, sometimes with the AFM p-spin ordering (yellow).
Looking through the complete set of phase diagrams, one can notice the absence of exact symmetry with the respect to reflections λ →−λ and β →−β, though some traces are detectable. The β-symmetry is restored, when ξ 12 ?ξ 22 .
Let us mention that the intuitive speculations useful, for example, in the case of simple Heisenberg model, can be misleading here, because the effective Hamiltonian is rather nontrivial.
We also underline the evolution of the artistic image of the phase diagrams. Namely, at small ξ 12 their style resembles the JMiró paintings, while at large ξ 12 -those of KMalewicz. Figures 1 and 2 are the most multicolored-there are quantum phase transitions nearly between all the possible phases. These pictures correspond to the low or moderate interspecies Coulomb interaction U 12 as compared to the single-species one. This feature can be attributed to small or moderate difference in the parameters characterizing their nonidentity. One can also notice that there are many reentrant phase transitions in figures 1-9, especially for moderate ξ 12 .

Phase diagrams: specific features
When ξ 22 becomes sufficiently large, then the NEM spin phase prevails. The ξ 22 -threshold for this behavior is the smallest at large ξ 12 , as can be seen in figures 1-9.

Conclusions
To conclude, we have investigated the evolution of the quantum state of vector two species bosons in optical lattices with the 'degree of atomic nonidentity' that drives the cascade of quantum phase transitions. Surprisingly, this quite simple model exhibits rather rich and complicated manifold of quantum phase transitions between six different phases when interaction parameters of the model change in rather limited subspace. In fact, one can say that the system is 'unstable' with the respect to small variation of parameters.
Technically, we have reduced the initial general Hamiltonian for vector bosons, using the Mott insulating state natural small parameters, to the anisotropic spin-pseudospin model of the Kugel-Khomskii type that served as the effective Hamiltonian. This procedure gave a chance for analytical progress in investigations of the system ground state. The variational approach have been used to uncover the phase diagram of the system in hand. Finally, we have investigated also limiting cases of the effective Hamiltonian and demonstrated the relation of our rather complicated Hamiltonian to the well-known results. There are further perspectives to find new results in the two species fermion systems in optical lattices and hybrid boson-fermion ones looking at the 'degree of atomic nonidentity' as the driving parameter.

Acknowledgments
AM and AB are grateful to Wu-Ming Liu for interest to this work and hospitality in Beijing National Laboratory for Condensed Matter Physics. NC is grateful to APekovic for stimulating discussions at the initial stage of this work, to Laboratoire de Physique Théorique, Toulouse, where this work has been initiated, for the hospitality, and to CNRS. This work was supported by the Russian Foundation for Basic Research (projects Nos. 16-02-00295, 16-02-00304, 17-52-53014, and 17-02-00323). We also express our gratitude to Government of the Russian Federation (project No. 05.Y09.21.0018) for support of the preparation of this paper revised version and the Computational Centers of Russian Academy of Sciences and National Research Center 'Kurchatov Institute' for providing the access to URAL, JSCC, and HPC facilities. (22)

Appendix. Special cases of Hamiltonian
is the projector onto the pseudospin state T M 1, 0 , when the matrix h can be reduced to the form Note that due to the presence of the projector P t0 , the states with spin S=1 will be automatically symmetric in the orbital space, while the antisymmetric combination of orbital states is automatically excluded from the effective Hamiltonian.
Next, we rewrite the Hamiltonian h in terms of the spin operators. We use the relation The systems with identical bosons with odd and even number bosons per site were discussed in [40,41].