superfluid-Mott Insulator Research Articles

Specialising neural-network quantum states for the Bose Hubbard model

Abstract Projected variational wavefunctions such as the Gutzwiller, many-body correlator and Jastrow ansatzes have provided crucial insight into the nature of superfluid-Mott insulator transition in the Bose Hubbard model (BHM) in two or more spatial dimensions. However, these ansatzes have no obvious tractable and systematic way of being improved. A promising alternative is to use Neural-network quantum states (NQS) based on Restricted Boltzmann Machines (RBMs). With binary visible and hidden units NQS have proven to be a highly effective at describing quantum states of interacting spin- 1 2 lattice systems. The application of NQS to bosonic systems has so far been based on one-hot encoding from machine learning where the multi-valued site occupation is distributed across several binary-valued visible units of an RBM. Compared to spin- 1 2 systems one-hot encoding greatly increases the number of variational parameters whilst also making their physical interpretation opaque. Here we revisit the construction of NQS for bosonic systems by reformulating a one-hot encoded RBM into a correlation operator applied to a reference state, analogous to the structure of the projected variational ansatzes. In this form we then propose a number of specialisations of the RBM motivated by the physics of the BHM and the ability to capture exactly the projected variational ansatzes. We analyse in detail the variational performance of these new RBM variants for a 10 × 10 BHM, using both a standard Bose condensate state and a pre-optimised Jastrow + many-body correlator state as the reference state of the calculation. Several of our new ansatzes give robust results as nearly good as one-hot encoding across the regimes of the BHM, but at a substantially reduced cost. Such specialised NQS are thus primed tackle bosonic lattice problems beyond the accuracy of classic variational wavefunctions.

Phases and coherence of strongly interacting finite bosonic systems in shallow optical lattice

We explore the ground states of strongly interacting bosons in the vanishingly small and weak lattices using the multiconfiguration time-dependent Hartree method for bosons (MCTDHB) which calculate numerically exact many-body wave function. Two new many-body phases: fragmented or quasi superfluid (QSF) and incomplete fragmented Mott or quasi Mott insulator (QMI) are emerged due to the strong interplay between short-range contact interaction and lattice depth. Fragmentation is utilized as a figure of merit to distinguish these two new phases. We utilize the eigenvalues of the reduced one-body density matrix and define an order parameter that characterizes the pathway from a very weak lattice to a deep lattice. We provide a detailed investigation through the measures of one- and two-body correlations and information entropy. We find that the structures in one- and two-body coherence are good markers to understand the gradual built-up of intra-well correlation and decay of inter-well correlation with increase in lattice depth. For the dipolar interaction, the many-body features become more distinct and true Mott state can appear even in a shallow lattice. Whereas, for incommensurate fraction of particles, incomplete localization happens that exhibits distinct features in the measure of two-body coherence.

Non-Hermitian superfluid–Mott-insulator transition in the one-dimensional zigzag bosonic chains

Non-Hermitian superfluid–Mott-insulator transition in the one-dimensional zigzag bosonic chains

Superfluid Mott-insulator quantum phase transition in an open cavity optomagnonic system

We investigated the superfluid Mott-insulator phase transition within an open coupled cavity optomagnonic array system. Based on the quasibosonic approach and employing perturbation approximation and mean-field theory, we obtain the time-dependent analytical superfluid order parameter. Our analysis revealed that dissipation induces decoherence within the system, while a strong magnon-cavity coupling enhances coherence significantly. Moreover, we explored the mechanisms by which increasing the magnon-cavity coupling strength drives the system towards the superfluid phase. The work holds significant guidance for the study of superfluidMott insulator phase transition under dissipative conditions, and providing a reliable reference for experimental implementation in open quantum systems.

Exact Solution of the Bose-Hubbard Model with Unidirectional Hopping.

A one-dimensional Bose-Hubbard model with unidirectional hopping is shown to be exactly solvable. Applying the algebraic Bethe ansatz method, we prove the integrability of the model and derive the Bethe ansatz equations. The exact eigenvalue spectrum can be obtained by solving these equations. The distribution of Bethe roots reveals the presence of a superfluid-Mott insulator transition at the ground state, and the critical point is determined. By adjusting the boundary parameter, we demonstrate the existence of a non-Hermitian skin effect even in the presence of interaction, but it is completely suppressed for the Mott insulator state in the thermodynamical limit. Our result represents a new class of exactly solvable non-Hermitian many-body systems, which has no Hermitian correspondence and can be used as a benchmark for various numerical techniques developed for non-Hermitian many-body systems.

Operator growth and Krylov complexity in Bose-Hubbard model

We study Krylov complexity of a one-dimensional Bosonic system, the celebrated Bose-Hubbard Model. The Bose-Hubbard Hamiltonian consists of interacting bosons on a lattice, describing ultra-cold atoms. Apart from showing superfluid-Mott insulator phase transition, the model also exhibits both chaotic and integrable (mixed) dynamics depending on the value of the interaction parameter. We focus on the three-site Bose Hubbard Model (with different particle numbers), which is known to be highly mixed. We use the Lanczos algorithm to find the Lanczos coefficients and the Krylov basis. The orthonormal Krylov basis captures the operator growth for a system with a given Hamiltonian. However, the Lanczos algorithm needs to be modified for our case due to the instabilities instilled by the piling up of computational errors. Next, we compute the Krylov complexity and its early and late-time behaviour. Our results capture the chaotic and integrable nature of the system. Our paper takes the first step to use the Lanczos algorithm non-perturbatively for a discrete quartic bosonic Hamiltonian without depending on the auto-correlation method.

Neural-network quantum states for a two-leg Bose-Hubbard ladder under magnetic flux

Quantum gas systems are ideal analog quantum simulation platforms for tackling some of the most challenging problems in strongly correlated quantum matter. However, they also expose the urgent need for new theoretical frameworks. Simple models in one dimension, well studied with conventional methods, have received considerable recent attention as test cases for new approaches. Ladder models provide the logical next step, where established numerical methods are still reliable, but complications of higher dimensional effects like gauge fields can be introduced. In this paper, we investigate the application of the recently developed neural-network quantum states in the two-leg Bose-Hubbard ladder under strong synthetic magnetic fields. Based on the restricted Boltzmann machine and feedforward neural network, we show that variational neural networks can reliably predict the superfluid-Mott insulator phase diagram in the strong coupling limit comparable with the accuracy of the density-matrix renormalization group. In the weak coupling limit, neural networks also diagnose other many-body phenomena such as the vortex, chiral, and biased-ladder phases. Our work demonstrates that the two-leg Bose-Hubbard model with magnetic flux is an ideal test ground for future developments of neural-network quantum states.

Classical-quantum correspondence of special and extraordinary-log criticality: Villain's bridge

There has been much recent progress on exotic surface critical behavior, yet the classical-quantum correspondence of special and extraordinary-log criticality remains largely unclear. Employing worm Monte Carlo simulations, we explore the surface criticality at an emergent superfluid-Mott insulator critical point in the Villain representation, which is believed to connect classical and quantum O(2) critical systems. We observe a special transition with the thermal and magnetic renormalization exponents $y_t=0.58(1)$ and $y_h=1.690(1)$ respectively, which are close to recent estimates from models with discrete spin variables. The existence of extraordinary-log universality is evidenced by the critical exponent $\hat{q}=0.58(2)$ from two-point correlation and the renormalization-group parameter $\alpha=0.28(1)$ from superfluid stiffness, which obey the scaling relation of extraordinary-log critical theory and recover the logarithmic finite-size scaling of critical superfluid stiffness in open-edge quantum Bose-Hubbard model. Our results bridge recent observations of surface critical behavior in the classical statistical mechanical models [Parisen Toldin, Phys. Rev. Lett. 126, 135701 (2021); Hu $et$ $al.$, $ibid.$ 127, 120603 (2021); Parisen Toldin $et$ $al.$, $ibid.$ 128, 215701 (2022)] and the open-edge quantum Bose-Hubbard model [Sun $et$ $al.$, Phys. Rev. B 106, 224502 (2022)].

A first order superfluid-Mott insulator transition for a Bose–Hubbard model in an emergent lattice

The combination of strong correlation and emergent lattice can be achieved when quantum gases are confined in a superradiant Fabry–Perot cavity. In addition to the discoveries of exotic phases, such as density wave ordered Mott insulator and superfluid, a surprising kink structure is found in the slope of the cavity strength as a function of the pumping strength. In this article, we show that the appearance of such a kink is a manifestation of a liquid–vapour-like transition between two superfluids with different densities. The slopes in the immediate neighborhood of the kink become divergent at the liquid–vapour critical points and display a critical scaling law with a critical exponent 1 in the quantum critical region.

Superfluid–Mott-insulator quantum phase transition in a cavity optomagnonic system

The emerging hybrid cavity optomagnonic system is a very promising quantum information processing platform for its strong or ultrastrong photon-magnon interaction on the scale of micrometers in the experiment. In this paper, the superfluid-Mott insulator quantum phase transition in a two-dimensional cavity optomagnonic array system has been studied based on this characteristic. The analytical solution of the critical hopping rate is obtained by the mean field approach, second perturbation theory and Landau second order phase transition theory. The numerical results show that the increasing coupling strength and the positive detunings of the photon and the magnon favor the coherence and then the stable areas of Mott lobes are compressed correspondingly. Moreover, the analytical results agree with the numerical ones when the total excitation number is lower. Finally, an effective repulsive potential is constructed to exhibit the corresponding mechanism. The results obtained here provide an experimentally feasible scheme for characterizing the quantum phase transitions in a cavity optomagnonic array system, which will offer valuable insight for quantum simulations.

Nonlocal Correlation and Entanglement of Ultracold Bosons in the 2D Bose–Hubbard Lattice at Finite Temperature

AbstractThe temperature‐dependent behavior emerging in the vicinity of the superfluid (SF) to Mott‐insulator (MI) transition of interacting bosons in a 2D optical lattice, described by the Bose–Hubbard model is investigated. The equilibrium phase diagram at finite‐temperature is computed using the cluster mean‐field (CMF) theory including a finite‐cluster‐size‐scaling. The SF, MI, and normal fluid (NF) phases are characterized as well as the transition or crossover temperatures between them are estimated by computing physical quantities such as the superfluid fraction, compressibility and sound velocity using the CMF method. It is found that the nonlocal correlations included in a finite cluster, when extrapolated to infinite size, leads to quantitative agreement of the phase boundaries with quantum Monte Carlo (QMC) results as well as with experiments. Moreover, it is shown that the von Neumann entanglement entropy within a cluster corresponds to the system's entropy density and that it is enhanced near the SF–MI quantum critical point (QCP) and at the SF–NF boundary. The behavior of the transition lines near this QCP, at and away from the particle‐hole (p–h) symmetric point located at the Mott‐tip, is also discussed. The results obtained by using the CMF theory can be tested experimentally using the quantum gas microscopy method.

Criticality of the excess energy cost due to the unit-flux-quantum external field for the (2 + 1)D superfluid–insulator transition

The two-dimensional (2D) spin-S = 1 XY model was investigated numerically as a realization of the (2 + 1)D superfluid–Mott-insulator (SF–MI) transition. The interaction parameters are extended so as to suppress corrections to finite-size scaling. Thereby, the external field of a unit flux quantum (Φ = 2π) is applied to the 2D cluster by incorporating the phase factor (ϕ ij : gauge angle between the i and j sites) into the hopping amplitudes. Taking the advantage in that the exact-diagonalization method allows us to treat such a complex-valued matrix element, we evaluated the excess energy cost ΔE(2π) due to the magnetic flux Φ = 2π explicitly in the SF (XY) phase. As a result, we found that the amplitude ratio ρ s/ΔE(2π) (ρ s: spin stiffness) makes sense in proximity to the critical point, exhibiting a notable plateau in the SF-phase side. The plateau height is estimated, and compared to the related studies.

Enhanced nonlinear quantum metrology with weakly coupled solitons in the presence of particle losses

The estimation of physical parameters with Heisenberg sensitivity and beyond is one of the crucial problems for current quantum metrology. However, unavoidable lossy effect is commonly believed to be the main obstacle when applying fragile quantum states. To utilize the lossy quantum metrology, we offer an interferometric procedure for phase parameters estimation at the Heisenberg (up to 1/N) and super-Heisenberg (up to 1/N^3) scaling levels in the framework of the linear and nonlinear metrology approaches, respectively. The heart of our setup is the novel soliton Josephson Junction (SJJ) system providing the formation of the quantum probe, i.e, the entangled Fock (N00N-like) state, beyond the superfluid-Mott insulator quantum phase transition point. We illustrate that such states are close to the optimal ones even with moderate losses. The enhancement of phase estimation accuracy remains feasible both for the linear and nonlinear metrologies with the SJJs, and allows further improvement for the current experiments performed with atomic condensate solitons with a mesoscopic number of particles.

Evidence from on-site atom number fluctuations for a quantum Berezinskii-Kosterlitz-Thouless transition in the one-dimensional Bose-Hubbard model

We study the one-dimensional Bose-Hubbard model describing the superfluid-Mott insulator quantum phase transition of cold atoms in optical lattices. We show that derivatives of the variance of the on-site atom number occupation, computed with respect to the parameter driving the transition, have extrema that are located off the critical point even in the thermodynamic limit. We discuss whether such extrema provide solid evidence of the quantum Berezinskii-Kosterlitz-Thouless transition taking place in this system. The calculations are done for systems with the mean number of atoms per lattice site equal to either one or two. They also characterize the nearest-neighbor correlation function, which is typically discussed in the context of time-of-flight images of cold atoms.

Orbital order in a bosonic p -band triangular lattice

We present a detailed study of the Bose-Hubbard model in a $p$-band triangular lattice by focusing on the evolution of orbital order across the superfluid-Mott insulator transition. Two distinct phases are found in the superfluid regime. One of these phases adiabatically connects the weak interacting limit. This phase is characterized by the intertwining of axial $p_\pm=p_x \pm ip_y$ and in-plane $p_\theta=\cos\theta p_x+\sin\theta p_y$ orbital orders, which break the time-reversal symmetry and lattice symmetries simultaneously. In addition, the calculated Bogoliubov excitation spectrum gaps the original Dirac points in the single-particle spectrum but exhibits emergent Dirac points. The other superfluid phase in close proximity to the Mott insulator with unit boson filling shows a detwined in-plane ferro-orbital order. Finally, an orbital exchange model is constructed for the Mott insulator phase. Its classical ground state has an emergent SO$(2)$ rotational symmetry in the in-plane orbital space and therefore enjoys an infinite degeneracy, which is ultimately lifted by the orbital fluctuation via the order by disorder mechanism. Our systematic analysis suggests that the in-plane ferro-orbital order in the Mott insulator phase agrees with and likely evolves from the latter superfluid phase.

Stability of supercurrents in a superfluid phase of spin-1 bosons in an optical lattice

We study collective modes and superfluidity of spin-1 bosons with antiferromagnetic interactions in an optical lattice based on the time-dependent Ginzburg-Landau (TDGL) equation derived from the spin-1 Bose-Hubbard model. Specifically, we examine the stability of supercurrents in the polar phase in the vicinity of the Mott insulating phase with even filling factors. Solving the linearized TDGL equation, we obtain gapless spin-nematic modes and gapful spin-wave modes in the polar phase that arise due to the breaking of $S^2$ symmetry in spin space. Supercurrents exhibit dynamical instabilities induced by growing collective modes. In contrast to the second-order phase transition, the critical momentum of mass currents is finite at the phase boundary of the first-order superfluid-Mott insulator (SF-MI) phase transition. Furthermore, the critical momentum remains finite throughout the metastable SF phase and approaches zero towards the phase boundary, at which the metastable SF state disappears. We also study the stability of spin currents motivated by recent experiments for spinor gases. The critical momentum of spin currents is found to be zero, where a spin-nematic mode causes the dynamical instability. We investigate the origin of the zero critical momentum of spin currents and find it attributed to the fact that the polar state becomes energetically unstable even in the presence of an infinitesimal spin current. We discuss implications of the zero critical momentum of spin currents for the stability of the polar state.

Structural superfluid–Mott-insulator transition for a Bose gas in multirods

We report on a structural superfluid--Mott-insulator (SF-MI) quantum phase transition for an interacting one-dimensional Bose gas within permeable multirod lattices, where the rod lengths are varied from zero to the lattice period length. We use the ab initio diffusion Monte Carlo method to calculate the static structure factor, the insulation gap, and the Luttinger parameter, which we use to determine if the gas is a superfluid or a Mott insulator. For the Bose gas within a square Kronig-Penney (KP) potential, where barrier and well widths are equal, the SF-MI coexistence curve shows the same qualitative and quantitative behavior as that of a typical optical lattice with equal periodicity but slightly larger height. When we vary the width of the barriers from zero to the length of the potential period, keeping the height of the KP barriers, we observe a way to induce the SF-MI phase transition. Our results are of significant interest, given the recent progress on the realization of optical lattices with a subwavelength structure that would facilitate their experimental observation.

Superfluid–Mott insulator quantum phase transition in hybrid cavity optomechanical arrays

A two-dimensional hybrid cavity optomechanical array formed by an optomechanical cavity embedded with a two-level atom in each site is introduced to discuss the superfluid–Mott insulator quantum phase transition of light. On account of the optomechanical coupling effects stemmed from the coupled movable oscillator, the energy spectrum of the system changes, and an effective on-site repulsive interaction is reformed by modulating the optomechanically controlled parameters. By adjusting the detuning, an intersection between different critical chemical potentials can be found together with a disappearance of certain Mott lobes physically. The pronounced optomechanical coupling strength is in favor of the Mott insulator phase due to the enhanced effective on-site interactions. Additionally, the Kerr-nonlinearity reduces the average excitation of the high Mott insulator state and diminishes the superfluid regime. The results obtained here provide a novel image for characterizing the quantum phase transitions in optomechanical array systems, which will offer valuable insight for quantum simulations.

Quantum Monte Carlo study of strongly interacting bosonic one-dimensional systems in periodic potentials

We present diffusion Monte Carlo (DMC) and path-integral Monte Carlo (PIMC) calculations of a one-dimensional Bose system with realistic interparticle interactions in a periodic external potential. Our main aim is to test the predictions of the Luttinger liquid (LL) theory, in particular with respect to the superfluid-Mott insulator transition at both zero and finite temperatures, in the predicted robust and fragile superfluid regimes. For that purpose, we present our results of the superfluid fraction $\rho_s/\rho_0$, the one-body density matrix, the two-body correlation functions, and the static structure factor. The DMC and PIMC results in the limit of very low temperature for $\rho_s/\rho_0$ agree, but the LL model for scaling $\rho_s/\rho_0$ does not fit the data well. The critical depth of the periodic potential is close to the values obtained for ultracold gases with different models of interaction, but with the same value of the bare LL parameter, demonstrating the universality of LL description. Algebraic decay of correlation functions is observed in the superfluid regime and exponential decay in the Mott-insulator one, as well as in all regimes at finite temperature for distances larger than a characteristic length.

Quantum-Granularity Effect in the Formation of Supermixed Solitons in Ring Lattices

We investigate a notable class of states peculiar to a bosonic binary mixture featuring repulsive intraspecies and attractive interspecies couplings. We evidence that, for small values of the hopping amplitudes, one can access particular regimes marked by the fact that the interwell boson transfer occurs in a jerky fashion. This property is shown to be responsible for the emergence of a staircase-like structure in the phase diagram of a mixture confined in a ring trimer and to resemble the mechanism of the superfluid-Mott insulator transition strongly. Under certain conditions, in fact, we show that it is possible to interpret the interspecies attraction as an effective chemical potential and the supermixed soliton as an effective particle reservoir. Our investigation is developed both within a fully quantum approach based on the analysis of several quantum indicators and by means of a simple analytical approximation scheme capable of capturing the essential features of this ultraquantum effect.