Ground-state and finite-temperature phase diagrams of the Bose-Hubbard model under artificial magnetic fields

Abstract

We investigate the effects of artificial magnetic fields on the ground-state of the two-dimensional Bose-Hubbard model. Using an asymmetric Bose-Hubbard model, we demonstrate that anisotropic hopping amplitude localizes bosons and enlarges the insulating and supersolid regions. We show these increases in the presence of artificial gauge fields up to the symmetric point of the field. Moreover, our calculations exhibit real-space modulations of the superfluid and supersolid phases. The bosonic current exhibits vortices in these phases, whose configurations depend on the commensuration between the magnetic field and the lattice. Although the magnetic field explicitly breaks the translational symmetry of the square lattice, this symmetry is restored in the Mott insulator phase. The local densities shows a checkerboard pattern in the density-wave and supersolid phases, regardless of the magnetic field strength. We investigate thermal fluctuations and demonstrate the robustness of insulating and supersolid phases up to temperatures comparable to the interaction energy, which supports the feasibility of observing such phases in experiments.

Introduction

Optical lattices are clean and highly controllable systems that provide the experimental realization of quantum phases^1,2,3. The versatility of their adjustable parameters makes them a suitable playground for simulating quantum phases that are difficult to access in solid-state systems. The Bose-Hubbard model (BHM) is a widely used model for simulating quantum phase transitions in optical lattices^1,4. Recent experiments suggest the possible emergence of topological phases in this model. The Coriolis force generated by a rotating reference frame mimics the Lorentz force in systems of charged particles^5,6,7,8. As this magnetic field is generated in charge-neutral systems, it is called an artificial magnetic field (AMF).

Laser-induced tunneling is another method to realize artificial magnetic field in bosonic systems^{9,10,11,12,13}. This synthetic field induces Peierls phase factors in the hopping terms of the BHM and leads to modifications in the single-particle spectrum and the emergence of nontrivial spatial structures¹⁴. The effective magnetic field generated in these systems is strong enough to reach the quantum Hall regime^{15,16,17,18,19,20}. The ability to control dimensionality and interaction parameters, in addition to the tunability of the magnetic field, makes these systems more interesting for generating integer^21,22,23 and half-integer^16,17,19 quantum Hall effects. Moreover, due to the confinement of atoms in a small volume, the interaction energies and the excitation gap are larger in trapped atoms, making the realization of the fractional quantum Hall effect easier¹⁷.

Theoretically, the artificial magnetic fields enlarge the Mott-insulator (MI) phase in the Bose-Hubbard model^{24,25,26,27,28}. This enlargement is associated with the localization of the single-particle dynamics, even in non-interacting systems²⁶. Moreover, AMFs lead to a superfluid (SF) phase with simultaneous spatial order, whose configuration depends on the commensuration between the magnetic field and the lattice¹⁴. The transition between MI and SF phases is known to be mediated by a chiral Mott insulator (CMI) phase in BHM+AMF two-leg ladders^29,30; however, no such phase has been reported for square lattices so far. The CMI is a gapped phase characterized by circulating currents within each plaquette and is related to exciton condensation. This bound state is significantly weaker in two spatial dimensions compared to one dimension. Furthermore, on the square lattice, it is very shallow and at least an order of magnitude weaker than on the triangular lattice³¹. These observations suggest that such a phase is unlikely to exist in the square lattices.

Nearest-neighbor (NN) interaction, which result from dipole-dipole interactions³², can break translational symmetry in the extended Bose-Hubbard model (eBHM)^33,34,35. Therefore, density-wave (DW) and supersolid (SS) phases appear in the phase diagram of this model^33,34,35. The SS phase results from the coexistence of broken translational and U(1) symmetries^36,37. The realization of this phase was debated³⁸ for many years. Including AMFs in the eBHM increases the SS regions, which makes observing this phase easier in experiments. Moreover, excess particles in the eBHM with NN repulsion represent a bosonic analog of the fractional quantum Hall effect at particular filling factors³⁹.

In this manuscript, we provide new insights into the effects of AMFs on the ground-state phase diagrams of the two-dimensional BHM and eBHM. In order to predict the effects of AMFs, we use asymmetric BHM and eBHM. These results show that anisotropic hopping enlarges the insulator and SS regions. The single-particle spectrum of the BHM+AMF is considered, which shows the Hofstadter-butterfly structure and exhibits degeneracies related to the field strength. The effects of different ranges of AMFs on the BHM and eBHM are considered, and the results show that the insulating and SS regions increase with AMF up to the symmetric point of the field. Although AMFs explicitly break the translational symmetry of the square lattice, this symmetry is restored in the Mott insulator phase. We show the checkerboard pattern of the local densities in the DW and SS phases. However, there are real-space modulations of the bosonic current in the SF and SS phases that depend on the magnetic field strength. Finally, we demonstrate the stability of different phases up to temperatures that are comparable to the interaction energies. To the best of our knowledge, the effects of AMFs on the properties of SS phase have not been considered elsewhere.

This paper is organized as follows. In the first part, we present the phase diagrams of the BHM and eBHM with asymmetric hopping energies. In the Model section, we introduce the BHM and incorporate AMFs as phase factors in the hopping terms. In the next section, we calculate the single-particle spectrum for different ranges of magnetic fields and demonstrate that the phase factors break the translational symmetry of the square lattice, thereby modulating the superfluid order parameter. In the Methods section, we describe the single-site Gutzwiller mean-field (SGMF) approach, and in the next section we present the ground-state phase diagrams for different interaction strengths. The effects of thermal fluctuations are discussed in the Finite Temperature Phase Diagrams section. Finally, concluding remarks are provided in the Conclusions.

Asymmetric Bose-Hubbard model

The inclusion of an artificial magnetic field in the Bose-Hubbard model results in anisotropic hopping amplitudes along different lattice directions, thereby frustrating the coherent motion of bosons. In order to obtain an initial insight into the influence of the magnetic field on the ground-state phase diagram of the BHM, we study the asymmetric Bose-Hubbard model. Previous investigations of the asymmetric BHM suggest that weak anisotropy tends to enlarge the Mott-insulating regions^40,41,42,43. This model is described by the following Hamiltonian:

$$\begin{aligned} H ~=~ \sum _{i} ~ [-t ~ (b^\dag _{i+\hat{x}} ~ b_{i} + \frac{1}{2} ~ b^\dag _{i+\hat{y}} ~ b_{i} + h.c.) + \frac{U}{2} ~ n_{i} (n_{i} -1) ~ - \mu ~ n_{i}], \end{aligned}$$

(1)

where $b^\dag _{i} (b_{i})$ is the boson creation (annihilation) operator at lattice site i, and $n_{i} = b^\dag _{i} b_{i}$ is the boson number operator. U is the on-site interaction energy, which is set as the energy scale ($U=1$). t and $\mu$ are the hopping amplitude between nearest neighbors and the chemical potential, respectively. In this equation, the hopping amplitude along the $\hat{y}$ direction is smaller than that along the $\hat{x}$ direction.

The ground-state phase diagram of this model, together with the standard BHM, is plotted in the top panel of the Fig. 1. In the absence of hopping amplitude t, the incompressible Mott insulator appears, in which translational and rotational symmetries are preserved. The average boson number exhibits a plateau in this phase, while the filling factor increases with $\mu$ in different MI lobes. As the hopping amplitude t increases, the system undergoes a phase transition in which the U(1) symmetry is spontaneously broken and a superfluid phase emerges. Asymmetric hopping energy reduces the coherence of bosonic motion, which enlarges the insulating regions compared to the standard symmetric BHM. Therefore, the MI phases are broadened in the asymmetric BHM, see top panel of the Fig. 1.

Fig. 1

SGMF $t-\mu$ ground-state phase diagrams in the units of on-site interaction ($U=1$) for top: BHM and bottom: eBHM with NN interaction $V_1/U=0.5$, and symmetric (circle) and asymmetric (dot) hopping energies. SGMF is explained in the Method section. MI-SF and DW-SS borders are plotted with red color and SS-SF borders with blue.

In the bottom panel of Fig. 1, the effects of nearest-neighbor interaction $V_1$ are added to the asymmetric BHM as follows:

$$\begin{aligned} V_1 \sum _{i} ~( n_{i} ~ n_{i+\hat{x}} + n_{i} ~ n_{i+\hat{y}}), \end{aligned}$$

In the absence of hopping, the NN interaction breaks the translational symmetry of the system, and the incompressible density-wave phase with a checkerboard pattern appears. The fillings factor of the DW phases increases with the chemical potential $\mu$ in each lobe. As t/U increases, bosons can move coherently through the lattice, and the supersolid phase emerges, where both translational and U(1) symmetries are broken simultaneously. At larger hopping amplitudes, the translational symmetry is restored, and the system enters the superfluid phase. Asymmetric hopping frustrates bosonic motion and stabilizes the solid order up to larger t/U, which enlarges the solid and SS phases.

From the above analysis, we anticipate that hopping anisotropy suppresses coherent bosonic motion, which in turn enhances the extent of the insulating regions. However, including the magnetic field in Eq. 2 is not just a simple change of the hopping amplitudes in different directions. The gauge field introduces a phase factor in the hopping along one direction, such that hopping around a closed loop accumulates a nontrivial phase.

Model

We consider the Bose-Hubbard model on a two-dimensional square lattice in the presence of an artificial magnetic field. Previous experiments have simulated effective Landau gauge potentials, in which the vector potential is aligned with one of the square lattice axes^10,44. We take the vector potential along $\hat{y}$ direction for convenience:

$$\begin{aligned} H ~=~ \sum _{i} ~[ -t ~ (b^\dag _{i+\hat{x}} b_{i} + e^{- \textrm{i} 2 \pi \alpha x_i} ~ b^\dag _{i+\hat{y}} b_{i} + h.c.) + \frac{U}{2} ~ n_{i} (n_{i} -1) ~ - ~ \mu ~ n_{i}] , \end{aligned}$$

(2)

where $2 \pi \alpha$ is the dimensionless flux per plaquette, which defines the strength of the magnetic field, with AAAA, where $\textbf{A}$ is the vector potential related to the synthetic magnetic field as $\textbf{B} = \nabla \times \textbf{A}$. Here $\alpha =\frac{1}{q}$, where q is an integer and specifies the period of the AMF on the lattice. This shows that the magnetic field explicitly breaks translational symmetry along the $\hat{x}$ direction, and the unit cell becomes $q \times 1$. Therefore, in the condensed phases, this specific modulation appears on the square lattice. On the other hand, as the number operator $n_i$ is gauge invariant, translational symmetry is preserved in the insulating phases with large on-site interaction U. Also, the system displays the checkerboard pattern in the DW phase with large NN interactions.

Single-particle spectrum

Including AMFs in the Bose-Hubbard model (Eq. 2) explicitly breaks translational symmetry in the $\hat{x}$ direction and enlarges the unit cell to $q \times 1$ for $\alpha = 1/q$. Therefore, the magnetic field splits the dispersion relation into q subbands, and the first Brillouin zone reduces to $k_x \in [\frac{-\pi }{q} : \frac{\pi }{q} ]$ and $k_y \in [-\pi : \pi ]$.

In order to find the effect of AMFs on the excitation spectrum of the BHM, in this section we calculate the single-particle spectrum ($U=0$) of this model. We use the Fourier transformation of the creation and annihilation operators at site i:

$$\begin{aligned} b_{i}= & \frac{1}{N} \sum _{\varvec{k}} e^{\textrm{i} \varvec{k} r_i} a_{\varvec{k}} \nonumber \\ b^\dag _{i}= & \frac{1}{N} \sum _{\varvec{k}} e^{-\textrm{i} \varvec{k} r_i} a^\dagger _{\varvec{k}}. \end{aligned}$$

(3)

We transform the Hamiltonian in Eq. 2 into momentum space:

$$\begin{aligned} H(\varvec{k}) = -t ~\begin{pmatrix} \varepsilon _0 & e^{\textrm{i} k_x} & 0 & \cdots & e^{-\textrm{i} k_x} \\ \\ e^{-\textrm{i} k_x} & \varepsilon _1 & e^{\textrm{i} k_x} & \ddots & 0 & \\ \\ 0 & e^{-\textrm{i} k_x} & \varepsilon _2 & \cdots & 0 & \\ \\ \vdots & \vdots & \ddots & \ddots & \vdots \\ \\ e^{\textrm{i} k_x} & 0 & \cdots & e^{-\textrm{i} k_x} & \varepsilon _{q-1}, \end{pmatrix}, \end{aligned}$$

(4)

where N is the total number of bosons and $\varepsilon _q = 2 cos(k_y + 2 \pi \alpha (q-1)) + \mu /t$. Paradiagonalization⁴⁵ of this relation results excitation spectrum of the BHM+AMF.

Fig. 2

Lowest band of the non-interacting BHM+AMF with different gauge fields: $\alpha = \frac{1}{2}$ (top left), $\frac{1}{3}$ (top right), $\frac{1}{4}$ (bottom left), and $\frac{1}{5}$ (bottom right).

In Fig. 2, we have plotted the lowest single-particle spectrum of the BHM+AMF in the SF phase. This figure shows that the lowest band structure is degenerate for $k_y$s values separated by $2 \pi /q$. Therefore, the Brillouin zone is reduced to $[\frac{-\pi }{q} : \frac{\pi }{q} ]$ for both directions. This means that, in the SF phase, the order parameters also exhibit this specific modulation along the $\hat{y}$ direction too. Using the order parameters, we show this fact in the next sections.

Fig. 3

The Hofstadter-butterfly⁴⁶: The single-particle energies E of BHM+AMF as a function of flux $\alpha$.

The single-particle energy of this model exhibits a fractal structure as a function of $\alpha$, known as the Hofstadter-butterfly⁴⁶. It is symmetric around $\alpha = 1/2$ for $0 \le \alpha \le 1$, see Fig. 3. Therefore, we expect that the results on both sides of the symmetric point are the same. For example, we verified (not shown) that the phase diagrams for $\alpha =1/5$ and 4/5 are the same, except for an evident reversal in the direction of the bosonic current. Therefore, we have limited our discussion to magnetic fields up to 1/2 for simplicity.

Method

We use the single-site Gutzwiller mean-field (SGMF) method^47,48,49,50 to obtain the ground-state phase diagrams. In this method, the creation and annihilation operators are decomposed as:

$$\begin{aligned} b_{i}= & \phi _{i}+\delta b_{i} \nonumber \\ b^\dag _{i}= & \phi _{i}^*+\delta b^\dag _{i}, \end{aligned}$$

(5)

at a particular lattice site i, respectively. The mean-field part $\phi _{i}=\langle b_{i} \rangle$ is the SF order parameter, and $\phi _{i}^*$ is its complex conjugate, which are defined with respect to the ground-state of the system. $\delta b_{i}$ ($\delta b^\dag _{i}$) is quantum fluctuations around the mean-field results. Using these relations, the Hamiltonian in Eq. (2) can be written as the mean-field Hamiltonian:

$$\begin{aligned} H^{MF} ~=~ \sum _{i} H^{MF}_{i}, \end{aligned}$$

(6)

up to linear quantum fluctuations. $H^{MF}_{i}$ is the single-site mean-field Hamiltonian:

$$\begin{aligned} H^{MF}_{i} ~=~ -t ~ [ b_{i} ~( \phi ^*_{i+\hat{x}} + \phi ^*_{i-\hat{x}} + e^{-\textrm{i} 2 \pi \alpha x_i} ~ \phi ^\star _{i+\hat{y}} + e^{\textrm{i} 2 \pi \alpha x_i} ~ \phi ^\star _{i-\hat{y}})+h.c.] + ~ \frac{U}{2} ~ n_{i} (n_{i} -1) -\mu ~ n_{i}. \end{aligned}$$

(7)

The ground-state of the system, $\left| {\Psi _{G}}\right\rangle$, is expressed as a direct product of the normalized single-site ground-states $\left| {\psi }\right\rangle _{i}$ and is written in the Fock space:

$$\begin{aligned} \left| {\Psi _{G}}\right\rangle = \prod _{i} \left| {\psi }\right\rangle _{i} = \prod _{i} ~ \sum _{n=0}^{n_b} c_{n}^{i} \left| {n}\right\rangle _{i}. \end{aligned}$$

(8)

$n_b$ is the maximum allowed occupation number, and $c_{n}^{i}$ are the complex coefficients corresponding to the occupation state $\left| {n}\right\rangle _{i}$ at lattice site i. The ground-state is obtained from the self-consistent solution of the single-site energy.

Using these relations, the superfluid order parameter is defined as follows:

$$\begin{aligned} \phi _{i}= \left\langle {\Psi _{G}}\right| b_{i} \left| {\Psi _{G}}\right\rangle = \sum _{n=0}^{n_b} ~ c_{n-1}^ {i *} ~ c_{n}^ {i} ~ \sqrt{n}. \end{aligned}$$

(9)

According to this expression, it is clear that $\phi _{i}$ is zero in the DW phase, where only one occupation number state is present at each lattice sites and a single non-zero coefficient $c^{i}_n$ contributes to the single-site wave function. Moreover, $\phi _{i}$ also vanishes in the MI phase, where these coefficients $c^{i}_n$ are identical across all lattice sites. However $\phi _{i}$ is finite in the SF and SS phases due to number fluctuations. The occupancy and number fluctuations at each lattice site are defined as follows:

$$\begin{aligned} \langle n_{i} \rangle = \sum _{n=0}^{n_b} ~ |{c}^{i}_{n}|^2 ~ n, \end{aligned}$$

(10)

and

$$\begin{aligned} \delta n _{i} = \sqrt{\langle n^2_{i} \rangle -\langle n_{i} \rangle ^2 }, \end{aligned}$$

(11)

respectively. The plateaus of the order parameter $\langle n_{i} \rangle$ can distinguish insulating phases from SF and SS phases. Local compressibility $\delta n_{i}$ is zero in the insulating phases, which makes them incoherent. It is large in the SF phase, which makes it a strongly phase-coherent state. Due to the SF component, this order parameter is also finite in the SS phase.

Ground state phase diagrams

Using the SGMF method, we solve the BHM in the presence of an artificial magnetic field for different values of the gauge field $\alpha$. Adding an AMF to the BHM explicitly breaks the translational symmetry of the square lattice, and the unit cell enlarges to $q \times 1$ for $\alpha = 1/q$. The SGMF ground-state phase diagrams of the BHM+AMF model are plotted in the top panel of Fig. 4 for different values of the magnetic field $\alpha$.

Fig. 4

SGMF $t-\mu$ ground-state phase diagrams for different gauge fields $\alpha = 0$ (triangle), $\frac{1}{5}$ (circle) and $\frac{1}{2}$ (dot) in the units of on-site interaction ($U=1$) for top: BHM and bottom: eBHM with NN interaction $V_1/U=0.5$. MI-SF and DW-SS borders are plotted with red color and SS-SF borders with blue.

Fig. 5

Schematic SGMF plot of different phases for $\alpha = 1/5$ at various parameters ($t/U, \mu /U$): top left (0.01,1.5) in MI phase; top right (0.7,0.1) in SF phase; bottom left (0.1,1.5) in DW phase and bottom right (0.2,2) in SS phase. The radius of circles is proportional to the average number of bosons on every lattice sites and is unique in every frame. SF phase exhibits $q \times q$ pattern while SS phase has checkerboard one.

In the absence of hopping amplitude, the strong on-site interaction U suppresses number fluctuations and eliminates the coherence between neighboring lattice sites. Therefore, the incompressible MI phase appears, where the original translational symmetry of the square lattice returns to the system. In Fig. 5, we have plotted schematic pictures of the different phases. The filling factor of the MI phases increases with the chemical potential $\mu /U$, and their borders shrink with increasing hopping energy t/U.

At larger hopping amplitudes, the SF phase emerges, where U(1) symmetry is simultaneously broken. Moreover, as the Fig. 5 shows, the system exhibits broken translational symmetry in both directions. The unit cell becomes $q \times q$ for $\alpha = 1/q$, something that was seen in the Single-particle spectrum section and is absent in the standard BHM.

Considering the phase diagrams of the BHM+AMF with different values of $\alpha$, we find that the insulating boundaries increase with the magnetic fields by localizing bosons through reducing the coherence of their motion. Thus, larger magnetic fields result in larger MI boundaries, see top panel of Fig. 4 for $\alpha = 1/5$ and 1/2. In order to investigate the effects of the gauge field on the bosonic motion in the condensed phases, we use the following directional operator, known as the bosonic current operator, which calculates the current from site i to site j:

$$\begin{aligned} J_{i \rightarrow j} = \textrm{i} ~ t ~ e^{\textrm{i} \phi _{i,j}} ~ b^\dag _j b_i + h.c., \end{aligned}$$

(12)

where $\phi _{i,j}$ measures the phase factor that bosons acquire by hopping from i to j. According to Eq. 2, $\phi _{i,i+\hat{x}}=0$ and $\phi _{i,i+\hat{y}}=2 \pi \alpha x_i$. Bosonic current patterns for different values of the AMF are plotted in Fig. 6.

Fig. 6

SGMF bosonic currents (Eq. 12) for the SF phase at $t/U = 0.8$, $\mu /U = 0.2$ and different gauge fields: $\alpha = \frac{1}{2}$ (top left), $\frac{1}{3}$ (top right), $\frac{1}{4}$ (bottom left), and $\frac{1}{5}$ (bottom right). The circles are positions of the lattice sites which their radius are proportional to the density $n_i$. The directions of arrows show the current $J_{i \rightarrow j}$ between lattice site i and j. The length of the arrows indicate the magnitude of current in every subplots. The length scale is unique in every frame. Apart from the $\alpha = \frac{1}{2}$, SS phase has similar patterns.

As discussed in the previous sections, including the AMF leads to SF phases with simultaneous spatial orders in both directions, whose configurations depend on the commensuration between the magnetic field and the lattice size. In Fig. 6, the bosonic currents exhibit a $q \times q$ periodicity and construct special vortex configurations for each $\alpha$. The number of vortices decreases and they become more separated as q increases, resembling Abrikosov lattices, especially in the larger q¹⁴.

Using the SGMF method, we consider the effects of NN interactions in the BHM+AMF (see the bottom panel of Fig. 4). In the absence of hopping amplitude, a checkerboard solid orders appears for different values of $\alpha$. These DW phases persist up to larger t/U, where the U(1) symmetry is spontaneously broken and the supersolid order, with broken translational and rotational symmetries, appears. The local densities and the absolute value of the superfluid order parameter exhibit a checkerboard structure in the SS phase. Apart from the $\alpha = \frac{1}{2}$, the current operator represents vortex patterns with a $q \times q$ periodicity (for $\alpha = \frac{1}{q}$) in this phase (see Fig. 6). The bosonic current vanishes in the SS phase with the gauge field $\alpha = \frac{1}{2}$. Finally, at large hopping amplitudes, the SF phase appears. The local densities and bosonic current has $q \times q$ (for $\alpha = \frac{1}{q}$) pattern in the SF phase (see Fig. 6). As the bottom panel of Fig. 4 shows, the solid and SS phase boundaries increase with the magnetic filed $\alpha$.

Finite temperature phase diagrams

Fig. 7

Top: SGMF $t-\mu$ phase diagram for BHM+AMF at $\alpha = 1/2$ and temperature $T/U = 0.04$. Bottom: SGMF order parameter $\langle n \rangle$ for BHM+AMF at $\alpha = 1/2$, $t/U = 0.02$ and different temperatures.

Fig. 8

Top: SGMF $t-\mu$ phase diagram for eBHM+AMF at $\alpha = 1/2$ and temperature $T/U = 0.04$. Bottom: SGMF order parameter $\langle n \rangle$ for eBHM+AMF at $\alpha = 1/2$, $V_1/U=0.5$, $t/U = 0.15$ and different temperatures.

In the ground-state, there is long-range phase coherence in the SF phase, such that each boson is spread out over the entire lattice. On the other hand, the number fluctuation vanishes in the insulating phases.

In this section, we examine thermal fluctuations on the ground-state phase diagrams, which are important for realizing the BHM+AMF in experiments. At finite temperature T, phase diagrams are obtained from the thermal averages of the order parameters, which are computed by considering all the single-site eigenvectors $\left| {\psi }\right\rangle _i^l$ and their corresponding eigenvalues $E_l$. The partition function is written as:

$$\begin{aligned} Z = \sum _{l}^{n_b} e^{- \beta E_l}; ~~ \beta = 1/k_B T, \end{aligned}$$

(13)

where the Boltzmann constant $k_B =1$ for simplicity. Therefore, at finite temperature T, the superfluid order parameter and the occupancy are defined as follows:

$$\begin{aligned} \phi _{i}= \frac{1}{Z} \sum _{l} ~ _{i}^l \left\langle {\psi }\right| ~ b_i ~ e^{- \beta E_l} \left| {\psi }\right\rangle _{i}^l, \end{aligned}$$

(14)

and

$$\begin{aligned} \langle n_{i} \rangle = \frac{1}{Z} \sum _{l} ~ _{i}^l \left\langle {\psi }\right| ~ n_i ~ e^{- \beta E_l} \left| {\psi }\right\rangle _{i}^l. \end{aligned}$$

(15)

In the top panel of Fig. 7, we have plotted the phase diagram of the BHM+AMF at $\alpha =1/2$ and temperature $T/U=0.04$. As mentioned earlier, the AMF localizes bosons and enhances the insulating and SS phase boundaries. In contrast, thermal fluctuations reduce the coherence of the states and melt the insulating phases. Because quantum fluctuations are larger near the phase boundaries, thermal fluctuations gradually reduce the plateaus width from the edges, and the MI phase emerges around the ground-state insulating regions. In the MI phase, both the translational and U(1) symmetries are preserved, but the average number of bosons varies with $\mu$. Moreover, in the presence of thermal fluctuations, the superfluid density is suppressed, and the system undergoes a transition to a thermal insulating phase with a varying filling factor^33,34, where both the translational and the U(1) symmetries are preserved. In the bottom panel of Fig. 7, the average number of bosons is plotted for $\alpha =1/2$, $t/U=0.02$ and different temperatures T. The vertical lines determines the ground-state phase boundaries, where thermal fluctuations soften these transitions.

The thermal phase diagram of the eBHM+AMF at $\alpha =1/2$, $V_1/U=0.5$ and $T/U=0.04$ is plotted in the top panel of Fig. 8. The DW phase appears around the plateaus as a result of thermal fluctuations, which have a checkerboard structure, but the average number of bosons is not constant. The bottom panel of this figure shows that melting starts from the ground-state boundaries (vertical lines). Thermal fluctuations reduce the SS phase by washing out the superfluid component, but its boundaries remain large enough to be detected in experiments. At large T, which is comparable to the interaction terms, the SS phase disappears and the system enters the DW phase (see Fig. 9). Thermal fluctuations soften the transitions and eventually melt the plateaus at a transition temperature that depends on the strength of frustration and the interaction parameters. At sufficiently large T, the system enters the thermal insulator phase, where all the symmetries return to the system and the filling increases with $\mu$.

Fig. 9

SGMF $T-\mu$ phase diagram for eBHM+AMF at $\alpha = 1/2$, $V_1/U = 0.5$ and $t/U=0.1$.

Conclusions

We have considered the ground-state and finite-temperature phase diagrams of the BHM and eBHM in the presence of an AMF. First, using the asymmetric BHM and eBHM, we demonstrated the enlargement of the insulating and supersolid phases as a result of frustrated hopping energies. We related this increase to the disruption of the coherence of bosonic motions. This effect was observed in both the BHM and eBHM in the presence of the AMF, with $\alpha =1/2$ producing the largest regions.

It is noteworthy that both the asymmetric BHM and BHM+AMF break lattice symmetries, albeit in different ways. In the BHM+AMF, the Peierls phase factors associated with the magnetic field effectively enlarge the magnetic unit cell and break both translational and time-reversal symmetries. In contrast, in the asymmetric BHM, the lattice translational symmetry is preserved, but the C4 (fourfold rotational) symmetry is broken due to unequal hopping amplitudes along different directions.

Weak anisotropy tends to enlarge the MI regions, whereas strong anisotropy can lead to dimensional-crossover effects. Therefore, comparing the two models is most meaningful in the regime of weak anisotropy and relatively small hopping amplitudes, where both models primarily suppress coherent bosonic motion. In the strong-hopping or large-flux regimes, however, the BHM+AMF exhibits magnetic-field-specific phenomena-such as unit-cell enlargement, flux-induced currents, and time-reversal breaking that cannot be captured by a simple asymmetric-hopping model.

The synthetic magnetic fields modulated the particle numbers and bosonic currents at the lattice sites. The spatial structure of these configurations depends on the commensuration between the magnetic field and the lattice. In the MI phase, the translational symmetry of the square lattice is restored, while the DW and SS phases exhibit a checkerboard pattern. There is a $q \times q$ configuration in the SF phase. The bosonic current also exhibits vortex patterns with this symmetry in the SF and SS phases. Finally, we have shown that thermal fluctuations delocalize bosons, and melt the plateaus, and cause compressible phases to appear around the lobs. Moreover, superfluidity decreases with increasing temperature. At large T, the compressible normal fluid appears.

It is worth mentioning that although these results are obtained by mean-field theory, previous studies^51,52 have shown that quantum fluctuations are small enough to allow the ground-state of Eq. 2 to be obtained qualitatively using the SGMF. This method provides quick initial insights without the limitations associated with the large computational cost of exact diagonalization methods. For example, the cluster Gutzwiller method requires large clusters ($q \times q$) to correctly include broken translational symmetry. Moreover, due to the infamous sign problem, quantum Monte Carlo cannot be performed in the presence of magnetic fields²⁶. Nonetheless, including correlations between different sites with other methods results in quantitatively more accurate phase boundaries.