Transition from band insulator to Mott insulator and formation of local moment in half-filled ionic SU($N$) Hubbard model

Quantum spin models are a class of physical models describing “spins” or “local moments” which originate from strong correlations between fermions (e.g. electrons or cold atoms) such that the “charge” (fermion number) degrees of freedom are frozen [1]. For instance, the Heisenberg model is a low energy description of the Hubbard model only when the Hubbard $U$ is large enough to drive the system into the Mott insulating phase [2, 3]. In the literature, the SU(2) Heisenberg model has been generalized to the SU($N$) case [4] with the spin operators satisfying the SU($N$) algebra. The SU($N$) Heisenberg models provide a vast area to explore many new phenomena [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. Among different SU($N$) representations for the local spins, a conjugate representation with $m$ fermions on A-sublattice and $(N-m)$ fermions on B-sublattice [4] is mostly studied. It may support a generalized Neel order: for instance, the first $m$ and remaining $(N-m)$ flavors are occupied on the two sublattices, respectively. Hence, the SU($N$) symmetry is broken into $\mathrm{SU}(m)\times\mathrm{SU}(N-m)$. The gapless fluctuations above this Neel order fall into the Grassmannian manifold $\mathrm{Gr}(N,m)=\mathrm{U}(N)/[\mathrm{U}(m)\times\mathrm{U}(N-m)]$ [31, 32, 33, 34], which is reduced to the $N$-component Ginzburg-Landau theory [35] or equivalently the famous CP^N-1 model in the special case of $m=1$.

One early motivation for doing the SU($N$) generalization of Heisenberg model is to perform $1/N$ expansion around the saddle point at $N=\infty$, providing a controllable way to reach the SU(2) model [36]. However, physically speaking, the SU($N$) spin models should derive from the SU($N$) Hubbard model in the limit that charge fluctuations are completely frozen. The SU($N$) Hubbard model is widely studied,[37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50], and is now within experimental reach, thanks to the fast technical development, mostly in the field of cold atoms [51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64]. This brings the large-$N$ model to life, but not just a gedanken model, and opens up a new field in the study of the finite but large $N$ versions of such models, in the search for novel quantum spin states.

However, it is important to ask under what condition is the system aptly described by the quantum spin model for which the local moments have to be well established. In our previous work, we have proposed to add a staggered ionic potential to the SU($N$) Hubbard model [65]. In this work, we will examine the specific conditions for the Hubbard $U$ and ionic potential $V$ under which the local moments can exist, with immediate relevance to experimental realization. We develop and apply an SU($N$)-symmetric renormalized mean field theory (RMFT) based on the variational approach of the Gutzwiller projection approximation [66, 67, 2, 68, 69, 70, 71]. The RMFT developed here may be applied or extended straightforwardly for general models with a large number of fermion flavors subject to any internal symmetry. For the ionic SU($N$) Hubbard model, we find the local moments, with quantized integer $m$ fermions on the A-sublattice and $(N-m)$ fermions on the B-sublattice, are well established when $U$ is above a critical value $U_{c}$, which depends on $N$, $m$, and the ionic potential $V$. For large $N$, $U_{c}$ is found to depend on $N$ linearly with fixed $m/N$, but sublinearly with fixed $m$. In addition, the local moment formation is accompanied by a peculiar transition from the band insulator to the Mott insulator [72, 73], at which the ionic potential and quasiparticle weight are renormalized to zero simultaneously. Finally, we show that the low energy physics of local moments is described by the widely studied SU($N$) quantum spin model inside the Mott insulating phase. Our results shed light on the realization of such models in, e.g., cold atoms.

Th ionic SU($N$) Hubbard model we consider is described by the Hamiltonian $H=H_{t}+H_{U}$, with

and

where $i$ labels the lattice site, $\langle ij\rangle$ denotes a nearest-neighbor bond, $a=1,2,\cdots,N$ labels the flavor of the fermions, $\hat{n}(i)=\sum_{a}\hat{n}_{a}(i)=\sum_{a}c_{a}^{\dagger}(i)c_{a}(i)$ is the local density operator, $U$ denotes the Hubbard interaction, and $V$ is the staggered ionic potential. The model is clearly SU($N$)-symmetric in the internal flavor space. In real space, it breaks the translational symmetry, since A- and B-sublattices are distinct. However, a particle-hole transformation $c_{ia}\to(-1)^{i}c_{ia}^{\dagger}$ interchanges these sublattices, so the system is invariant under A-B sublattice transformation combined with the particle-hole transformation. As a result, the charge density is exactly staggered and the system is at half filling on average. Such a symmetry can be used to reduce the variational parameters, as can be seen in later discussions.

The $H_{U}$-term can be rewritten as

where $n_{0}(i)=\frac{N}{2}-(-1)^{i}\frac{V}{U}$ may be understood as the local ground charge tunable continuously by the staggered gate voltage $V$. We ask whether $\hat{n}(i)$ can be quantized to integers $m$ ($0<m<N$) on the A-sublattice and $(N-m)$ on the B-sublattice, i.e., forming local moments, called $m$-tuple moments, for large enough $U$. The concept of local moment is a natural generalization of the SU(2) case for which only one kind of local moment with $m=1$ is possible. Here, however, the states with different $m$-tuple moments should belong to different Mott insulating states. On the other hand, in the limit of $U=0$, a nonzero $V$ always yields a band insulator. For large enough $U$, the system is expected to enter different Mott insulating states labeled by $m$. Whether these $m$-tuple moments exist and how to describe these band-to-Mott insulator transitions are the main concerns of the present work. To answer these questions clearly, and for simplicity, we shall focus on the paramagnetic case in the following.

Local moment formation is beyond any Hartree-Fock mean field description. We employ the standard Gutzwiller projection approximation to treat the correlation effect. The SU(2) version of such a theory has been applied widely [67, 2, 68, 70, 71], and will be extended here to the SU($N$)-symmetric case in which all the $N$-flavors are equivalent. For sufficient generality, we present the theory for an arbitrary case of the applied potential in this section, and will specify the ionic potential in the next section.

Specifically, we consider a variational theory with the following trial wave function,

where $|\psi_{0}\rangle$ is the ground state of a free variational Hamiltonian to be specified, and $\hat{{\cal P}}$ is the Gutzwiller projection operator in the grand canonical ensemble

where $\hat{Q}_{k}(i)$ is the projection operator for the $k$-tuple state (with $k$-fermions),

Clearly, in the absence of projection, we have $\hat{{\cal P}}_{i}=\sum_{k}\hat{Q}_{k}(i)=1$. The idea of the Gutzwiller projection is to reassign weights to the basis states, and this is how correlations (at least the local ones) can be captured. Here, the weight for the $k$-tuple is assumed to be

where $g_{i}$ is the site-dependent Gutzwiller projection parameter to punish multi-fermion occupations (but can be chosen to be uniform in our case). In the spirit of density functional theory [74], the ground state energy is a unique functional of the density distribution. Therefore we have introduced a fugacity $y_{i}$ to maintain the fermion density before and after projection in the grand canonical ensemble [70]. The fermion density on each site before projection is

where $f_{i}$ is the average occupation number per flavor, and $\langle\cdot\rangle_{0}$ indicates the average performed with respect to $|\psi_{0}\rangle$. We have defined $q_{k0}(i)$ as the average of $\hat{Q}_{k}(i)$ in the unprojected state,

where $C_{N}^{k}$ is the combinatorial factor. After projection, we still require

with

where $\langle\cdot\rangle$ denotes average with respect to $|\psi\rangle$. Exact evaluation on the right hand side is difficult. To make analytical progress, we resort to the usual Gutzwiller approximation [67]: the projection operator unrelated to the target operator under average can be Wick-contracted separately. This approximation can be shown to be exact in infinite dimensions [75] for general on-site interactions [69] , and turns out to work satisfactorily in finite dimensions [68, 69, 71]. Under the Gutzwiller approximation, we have

Note the projection operator ${\cal P}$ is simplified to ${\cal P}_{i}$. After substituting $\hat{{\cal P}}_{i}$ in Eq. 5, we obtain

where we have used $\hat{Q}_{k}^{2}(i)=\hat{Q}_{k}(i)$ as a property of the projection operator $\hat{Q}_{k}(i)$. The fugacity $y_{i}$ (or $\ln y_{i}$ in practice) is then tuned to satisfy the density restriction Eq. 10.

After obtaining all of $q_{k0}(i)$ and $q_{k}(i)$, we are in a position to evaluate the variational energy in the projected state under the Gutzwiller approximation. The local charging energy is obtained most straightforwardly, given the fact that the total charge operator and the projectors $\hat{Q}_{k}(i)$ share the same local basis states as eigenstates,

The kinetic energy is slightly more difficult to evaluate. Since the fermion hopping involves two sites, we need to keep two projectors, say, $\hat{{\cal P}}_{i}$ and $\hat{{\cal P}}_{j}$ in the hopping on the $\langle ij\rangle$ bond,

Since the fermion operator is self-projective, we need to remove over projections before taking the quantum average. For a given site $i$, we observe that

where we have defined a partial projection operator $Q_{k}^{\hat{a}}(i)$ for $k$ fermions in the local Fock space excluding flavor $a$,

Its average in $|\psi_{0}\rangle$ is evaluated to be

for any flavor $a$ in our SU($N$)-symmetric case. Inserting the above relations in Eq. 15, we obtain

where $g_{t}(i,j)=z(i)z(j)$ is the renormalization of the hopping by the projection, and

In the second line we have used Eq.(13) to trade $\eta_{k}(i)y_{i}^{k}/\sqrt{{\cal D}_{i}}$ for $\sqrt{q_{k}(i)/q_{k0}(i)}$.

Combining the potential and kinetic energies, we obtain the total variational energy $E$ in the projected state,

where $\chi_{ij0}=\left\langle c_{a}^{\dagger}(i)c_{a}(j)+{\rm H.c.}\right\rangle_{0}$ is the average of hopping operator in the unprojected state. This energy is understood as a functional of (i) the fermion density $f_{i}$ which in turn depends on the trial wavefunction $|\psi_{0}\rangle$, and (ii) the Gutzwiller projection parameter $g_{i}$. The fugacity parameters $y_{i}$ are taken as Lagrange multipliers that are eliminated by forcing the invariance of the local fermion density under the projection. The variational Gutzwiller approximation is closely related to the RMFT. Minimizing $E$ with respect to $|\psi_{0}\rangle$, i.e., $\delta E/\delta\langle\psi_{0}|=0$, with fixed fermion density $Nf_{i}$, we obtain

where ${\cal E}$ is introduced as the Lagrange multiplier enforcing normalization of the wave function, and $H_{\rm RMFT}$ is a free Hamiltonian yet encoded with the renormalization effect from the Gutzwiller projection,

where the variational local chemical potential $\mu_{i}$ is introduced to enforce $Nf_{i}=\langle\hat{n}(i)\rangle_{0}$. It can be shown that the single-particle spectrum of $H_{\rm RMFT}$ is just the quasipaticle excitation spectrum beyond the correlated variational ground state, with the quasiparticle weight renormalized by $g_{t}$ [70].

In this section, we apply the variational Gutzwiller approximation and RMFT developed in the previous section to the ionic SU($N$) Hubbard model in our interest.

We have found the conditions for different Mott insulating states in which different types of local moments are formed and charge degrees of freedom are frozen. The low energy effective theory inside these Mott lobes should be described by these local moments, or equivalently the SU($N$) “spins”.

Given the ground state with $m$ fermions on the A-sublattice and $(N-m)$ fermions on the B-sublattice, we can perform a second order perturbation with respect to the kinetic Hamiltonian $H_{t}$, to obtain an effective Hamiltonian in the low energy sector,

subject to $n_{i}=\sum_{a}c_{a}(i)^{\dagger}c_{a}(i)=m$ on the A-sublattice and $n_{i}=N-m$ on the B-sublattice. This restriction suggests to define spin operators $S_{ab}(i)$ on site-$i$ as

such that the traceless condition ${\rm Tr}~{}S(i)=0$ is satisfied. Further, it can be checked that these $S_{ab}$ satisfy the SU($N$) algebra:

Using these spin operators, the above Hamiltonian can be rewritten as the SU($N$) Heisenberg model

where $J=\frac{4t^{2}}{(1+\delta^{2})U}$. Since $U\sim U_{c}$ here should be proportional to $N$, the above Hamiltonian has a natural large-$N$ limit. The SU($N$) Heisenberg model has been widely studied in the literature, as a mathematical generalization of the SU(2) Heisenberg model [4]. In this work, we have shown its relation to the ionic SU($N$) Hubbard model.

The above SU($N$) Hubbard or Heisenberg model supports an antiferromagnetic ground state with the Neel order. To represent the Neel order, we may select a specific spin axis, e.g., one of the diagonal Cartan base,

with $\ell_{m}^{a}=\frac{1}{m}$ for $a\leq m$ and $-\frac{1}{N-m}$ for $a>m$. The Neel order is then described by $\langle L_{m}(i)\rangle\sim(-1)^{i}{\cal N}$, with $m$ flavors of fermions on the A-sublattice and the remaining $(N-m)$ flavors on the B-sublattice.

Note that even if the local moments $L_{m}$ are ordered, the state still enjoys an internal symmetry group, $\mathrm{SU}(m)\times\mathrm{SU}(N-m)$, which becomes of merely gauge degrees of freedom if the charge is fully quantized. The Goldstone modes above the Neel ordered state fall into the Grassmannian manifold $\mathrm{Gr}(N,m)=\mathrm{U}(N)/[\mathrm{U}(m)\times\mathrm{U}(N-m)]$ [31]. Such fluctuations exchange the flavor content of the local moments without affecting the charge, in analogy to the spin rotation in the SU(2) system.

In summary, we have developed a Gutzwiller approximation and RMFT for the SU($N$)-symmetric fermionic systems. Applying to the ionic Hubbard model, we find the conjugate local moments, with $m$ fermions on the A-sublattice and $(N-m)$ fermions on the B-sublattice, are well established when the Hubbard $U$ is above a critical value $U_{c}$. We obtained an analytical solution to $U_{c}$ wich depends on the bare kinematics and a universal function of $m$, $N$ and the charge frustration $\delta$. For large $N$, $U_{c}$ is found to depend on $N$ linearly for fixed $m/N$ but sublinearly with fixed $m$ if $N$ is fixed. The local moment formation is accompanied by a peculiar band-insulator to Mott-insulator transition, where the ionic potential and quasiparticle weight are renormalized to zero simultaneously. Inside the Mott insulating phase, the system is effectively described by the SU($N$) Heisenberg model which is widely studied previously in the literature. Our results shed light on the realization of such models in cold atom systems.

Finally, several remarks on the Gutzwiller projection are in order. First, it can be improved by including additional Jastrow factors [71]. Second, in one dimension, the Gutzwiller projection is inaccurate or even fails, while a long-range Jastrow factor alone (without Gutzwiller projection) turns out to be able to capture the Mott insulating state correctly [77]. Third, even in infinite dimensions, the metal-Mott insulator transition is better described by the Gutzwiller projection followed by a partial Schrieffer-Wolff unitary transformation [78]. The latter two directions are intriguing and even challenge the notion of the Mott state defined by the absence of double occupancy, in the SU($2$) case. It would be interesting to improve our study of the slightly more complicated ionic SU($N$) Hubbard model along similar lines. However, we believe our results for two and higher dimensional ionic models should provide a qualitatively correct picture regarding the multiple transitions from the band insulator to Mott insulator, as well as the order of magnitude of the critical interactions.