One-dimensional Bose-Hubbard Model Research Articles

Kibble-Zurek Scaling with Matrix Product States

We study the dynamics of second-order phase transitions that may have taken place in the early universe and analyze the Kibble-Zurek (KZ) scaling as well as the formation of topological defects emerging from a quench in the one-dimensional Bose-Hubbard model. We describe the universal dynamics of the KZ mechanism (KZM) and the topological defect formation by simulating the Bose–Hubbard model at zero temperature. We study the effects of thermalization on the ground state by analyzing the transition between Mott and superfluid phases. The out-of-equilibrium dynamics induced by a quench of the hopping parameter is presented. We find a KZ scaling behaviour at zero temperature and analyze the KZ dynamics.

Many-body localization from a one-particle perspective in the disordered one-dimensional Bose-Hubbard model

We numerically investigate 1D Bose-Hubbard chains with onsite disorder by means of exact diagonalization. A primary focus of our work is on characterizing Fock-space localization in this model from the single-particle perspective. For this purpose, we compute the one-particle density matrix (OPDM) in many-body eigenstates. We show that the natural orbitals (the eigenstates of the OPDM) are extended in the ergodic phase and real-space localized when one enters into the MBL phase. Furthermore, the distributions of occupations of the natural orbitals can be used as measures of Fock-space localization in the respective basis. Consistent with previous studies, we observe signatures of a transition from the ergodic to the many-body localized (MBL) regime when increasing the disorder strength. We further demonstrate that Fock-space localization, albeit weaker, is also evidently present in the distribution of the physical densities in the MBL regime, both for soft- and hardcore bosons. Moreover, the full distribution of the densities of the physical particles provides a one-particle measure for the detection of the ergodic-MBL transition which could be directly accessed in experiments with ultra-cold gases.

Haldane insulator in the 1D nearest-neighbor extended Bose-Hubbard model with cavity-mediated long-range interactions

In the one-dimensional Bose-Hubbard model with on-site and nearest-neighbor interactions, a gapped phase characterized by an exotic non-local order parameter emerges, the Haldane insulator. Bose-Hubbard models with cavity-mediated global range interactions display phase diagrams, which are very similar to those with nearest-neighbor repulsive interactions, but the Haldane phase remains elusive there. Here we study the one-dimensional Bose-Hubbard model with nearest-neighbor and cavity-mediated global-range interactions and scrutinize the existence of a Haldane Insulator phase. With the help of extensive quantum Monte-Carlo simulations we find that in the Bose-Hubbard model with only cavity-mediated global-range interactions no Haldane phase exists. For a combination of both interactions, the Haldane Insulator phase shrinks rapidly with increasing strength of the cavity-mediated global-range interactions. Thus, in spite of the otherwise very similar behavior the mean-field like cavity-mediated interactions strongly suppress the non-local order favored by nearest-neighbor repulsion in some regions of the phase diagram.Graphical abstract

Neural-network quantum states at finite temperature

We propose a method to obtain the thermal-equilibrium density matrix of a many-body quantum system using artificial neural networks. The variational function of the many-body density matrix is represented by a convolutional neural network with two input channels. We first prepare an infinite-temperature state, and the temperature is lowered by imaginary-time evolution. We apply this method to the one-dimensional Bose-Hubbard model and compare the results with those obtained by exact diagonalization.

Superfluid-to-Mott transition in a Bose-Hubbard ring: Persistent currents and defect formation

We revisit here the Kibble-Zurek mechanism for superfluid bosons slowly driven across the transition towards the Mott-insulating phase. By means of a combination of the Time-Dependent Variational Principle and a Tree-Tensor Network, we characterize the current flowing during annealing in a ring-shaped one-dimensional Bose-Hubbard model with artificial classical gauge field on up to 32 lattice sites. We find that the superfluid current shows, after an initial decrease, persistent oscillations which survive even when the system is well inside the Mott insulating phase. We demonstrate that the amplitude of such oscillations is connected to the residual energy, characterizing the creation of defects while crossing the quantum critical point, while their frequency matches the spectral gap in the Mott insulating phase. Our predictions can be verified in future atomtronics experiments with neutral atoms in ring shaped traps. We believe that the proposed setup provides an interesting but simple platform to study the non-equilibrium quantum dynamics of persistent currents experimentally.

Measurement-induced phase transition: A case study in the nonintegrable model by density-matrix renormalization group calculations

We study the effect of local projective measurements on the quantum quench dynamics. As a concrete example, a one-dimensional Bose-Hubbard model is simulated by the matrix product state and time-evolving block decimation. We map out a global phase diagram in terms of the measurement rate in spatial space and time domain, which demonstrates a volume-to-area law entanglement phase transition. When the measurement rate reaches the critical value, we observe a logarithmic growth of entanglement entropy as the subsystem size or evolved time increases. Moreover, we find that the probability distribution of the single-site entanglement entropy distinguishes the volume and area law phases, similar to the case of disorder-induced many-body localization. We also investigate the scaling behavior of entanglement entropy and mutual information between two separated sites, which is indicative of a single universality class and thus suggests a possible unified description of this transition.

Information measures for local quantum phase transitions: Lattice bosons in a one-dimensional harmonic trap

We study ground-state quantum entanglement in the one-dimensional Bose-Hubbard model in the presence of a harmonic trap. We focus on two transitions that occur upon increasing the characteristic particle density: the formation of a Mott-insulating domain with site occupation one at the center of the trap (lower transition) and the emergence of a superfluid domain at the center of the Mott-insulating one (upper transition). These transitions generate discontinuities in derivatives of the total energy and have been characterized by local (nonextensive) order parameters, so we refer to them as local quantum phase transitions. We show that a second derivative of the total energy is continuous with a kink at the lower transition, and that it is discontinuous at the upper transition. We also show that bipartite entanglement entropies are order parameters for those local quantum phase transitions. We use the density-matrix renormalization group and show that the transition points extracted from entanglement measures agree with the predictions of the local density approximation in the thermodynamic limit. We discuss how to determine the transition points from results in small systems, such as the ones realized in recent optical lattice experiments that measured the second-order Renyi entanglement entropy.

Ground state phase diagram of the one-dimensional Bose-Hubbard model from restricted Boltzmann machines

Motivated by recent advances in the representation of ground state wavefunctions of quantum many-body systems using restricted Boltzmann machines as variational ansatz, we utilize an open-source platform for constructing such ansatz called NetKet to explore the extent of applicability of restricted Boltzmann machines to bosonic lattice models. Within NetKet, we design and train these machines for the one-dimensional Bose-Hubbard model through a Monte Carlo sampling of the Fock space. We vary parameters such as the strength of the onsite repulsion, the chemical potential, the system size and the maximum site occupancy and use converged equations of state to identify phase boundaries between the Mott insulating and superfluid phases. We compare the average density and the energy to results from exact diagonalization and map out the ground state phase diagram, which agrees qualitatively with previous finding obtained through conventional means.

Universal shift of the fidelity susceptibility peak away from the critical point of the Berezinskii-Kosterlitz-Thouless quantum phase transition

We show that the peak which can be observed in fidelity susceptibility around the Berezinskii-Kosterlitz-Thouless transition is shifted from the quantum critical point (QCP) at $J_c$ to $J^*$ in the gapped phase by a value $|J^* - J_c| = B^2/36$, where $B^2$ is a transition width controlling the asymptotic form of the correlation length $\xi \sim \exp(-B/\sqrt{|J-J_c|})$ in that phase. This is in contrast to the conventional continuous QCP where the maximum is an indicator of the position of the critical point. The shape of the peak is universal, emphasizing the close connection between fidelity susceptibility and the correlation length. We support those arguments with numerical matrix product state simulations of the one-dimensional Bose-Hubbard model in the thermodynamic limit, where the broad peak is located at $J^*=0.212$ that is significantly different from $J_c=0.3048(3)$. In the spin-$3/2$ XXZ model the shift from $J_c=1$ to $J^*=1.0021$ is small but the narrow universal peak is much more pronounced over the non-universal background.

Dynamical quantum phase transitions in collapse and revival oscillations of a quenched superfluid

In this work we revisit collapse and revival oscillations in superfluids suddenly quenched by strong local interactions for the case of a one-dimensional Bose-Hubbard model. As the main result we identify the inherent nonequilibrium quantum many-body character of these oscillations by revealing that they are controlled by a sequence of underlying dynamical quantum phase transitions in the real-time evolution after the quench, which manifest as temporal nonanalyticities in return probabilities or Loschmidt echos. Specifically, we find that the time scale of the collapse and revival oscillations is, firstly, set by the frequency at which dynamical quantum phase transitions appear, and is, secondly, of emergent nonequilibrium nature, since it is not only determined by the final Hamiltonian but also depends on the initial condition.

Quantum phase transitions in the dimerized extended Bose-Hubbard model

We present an unbiased numerical density-matrix renormalization group study of the one-dimensional Bose-Hubbard model supplemented by nearest-neighbor Coulomb interaction and bond dimerization. It places the emphasis on the determination of the ground-state phase diagram and shows that, besides dimerized Mott and density-wave insulating phases, an intermediate symmetry-protected topological Haldane insulator emerges at weak Coulomb interactions for filling factor one, which disappears, however, when the dimerization becomes too large. Analyzing the critical behavior of the model, we prove that the phase boundaries of the Haldane phase to Mott insulator and density-wave states belong to the Gaussian and Ising universality classes with central charges $c=1$ and $c=1/2$, respectively, and merge in a tricritical point. Interestingly we can demonstrate a direct Ising quantum phase transition between the dimerized Mott and density-wave phases above the tricritical point. The corresponding transition line terminates at a critical end point that belongs to the universality class of the dilute Ising model with $c=7/10$. At even stronger Coulomb interactions the transition becomes first order.

Kibble-Zurek scaling of the one-dimensional Bose-Hubbard model at finite temperatures

We use tensor network methods - Matrix Product States, Tree Tensor Networks, and Locally Purified Tensor Networks - to simulate the one dimensional Bose-Hubbard model for zero and finite temperatures in experimentally accessible regimes. We first explore the effect of thermal fluctuations on the system ground state by characterizing its Mott and superfluid features. Then, we study the behavior of the out-of-equilibrium dynamics induced by quenches of the hopping parameter. We confirm a Kibble-Zurek scaling for zero temperature and characterize the finite temperature behavior, which we explain by means of a simple argument.

Rhombi-chain Bose-Hubbard model: Geometric frustration and interactions

We explore the effects of geometric frustration within a one-dimensional Bose-Hubbard model using a chain of rhombi subject to a magnetic flux. The competition of tunnelling, self-interaction and magnetic flux gives rise to the emergence of a pair-superfluid (pair-Luttinger liquid) phase besides the more conventional Mott-insulator and superfluid (Luttinger liquid) phases. We compute the complete phase diagram of the model by identifying characteristic properties of the pair-Luttinger liquid phase such as pair correlation functions and structure factors and find that the pair-Luttinger liquid phase is very sensitive to changes away from perfect frustration (half-flux). We provide some proposals to make the model more resilient to variants away from perfect frustration. We also study the bipartite entanglement properties of the chain. We discover that, while the scaling of the block entropy pair-superfluid and of the single-particle superfluid leads to the same central charge, the properties of the low-lying entanglement spectrum levels reveal their fundamental difference.

Symmetries and Many-Body Excitations with Neural-Network Quantum States

Artificial neural networks have been recently introduced as a general ansatz to represent many-body wave functions. In conjunction with variational MonteCarlo calculations, this ansatz has been applied to find Hamiltonian ground states and their energies. Here, we provide extensions of this method to study excited states, a central task in several many-body quantum calculations. First, we give a prescription that allows us to target eigenstates of a (nonlocal) symmetry of the Hamiltonian. Second, we give an algorithm to compute low-lying excited states without symmetries. We demonstrate our approach with both restricted Boltzmann machines and feed-forward neural networks. Results are shown for the one-dimensional spin-1/2 Heisenberg model, and for the one-dimensional Bose-Hubbard model. When comparing to exact results, we obtain good agreement for a large range of excited-states energies. Interestingly, we find that deep networks typically outperform shallow architectures for high-energy states.

Accessing finite-momentum excitations of the one-dimensional Bose-Hubbard model using superlattice-modulation spectroscopy

We investigate the response to superlattice modulation of a bosonic quantum gas confined to arrays of tubes emulating the one-dimensional Bose-Hubbard model. We demonstrate, using both time-dependent density matrix renormalization group and linear response theory, that such a superlattice modulation gives access to the excitation spectrum of the Bose-Hubbard model at finite momenta. Deep in the Mott-insulator, the response is characterized by a narrow energy absorption peak at a frequency approximately corresponding to the onsite interaction strength between bosons. This spectroscopic technique thus allows for an accurate measurement of the effective value of the interaction strength. On the superfluid side, we show that the response depends on the lattice filling. The system can either respond at infinitely small values of the modulation frequency or only above a frequency threshold. We discuss our numerical findings in light of analytical results obtained for the Lieb-Liniger model. In particular, for this continuum model, bosonization predicts power-law onsets for both responses.

Generation of atypical hopping and interactions by kinetic driving

We study the effect of time-periodically varying the hopping amplitude in a one-dimensional Bose–Hubbard model, such that its time-averaged value is zero. Employing Floquet theory, we derive a static effective Hamiltonian in which nearest-neighbor single-particle hopping processes are suppressed, but all even higher-order processes are allowed. Unusual many-body features arise from the combined effect of nonlocal interactions and correlated tunneling. At a critical value of the driving, the system passes from a Mott insulator to a superfluid formed by two quasi-condensates with opposite non-zero momenta. This work shows how driving of the hopping energy provides a novel form of Floquet engineering, which enables atypical Hamiltonians and exotic states of matter to be produced and controlled.

Unconventional Universality Class of One-Dimensional Isolated Coarsening Dynamics in a Spinor Bose Gas.

By studying the coarsening dynamics of a one-dimensional spin-1 Bose-Hubbard model in a superfluid regime, we analytically find an unconventional universal dynamical scaling for the growth of the spin correlation length, which is characterized by the exponential integral unlike the conventional power law or simple logarithmic behavior, and numerically confirmed with the truncated Wigner approximation.

Out-of-time-order correlation at a quantum phase transition

In this paper we numerically calculate the out-of-time-order correlation functions in the one-dimensional Bose-Hubbard model. Our study is motivated by the conjecture that a system with Lyapunov exponent saturating the upper bound $2\pi/\beta$ will have a holographic dual to a black hole at finite temperature. We further conjecture that for a many-body quantum system with a quantum phase transition, the Lyapunov exponent will have a peak in the quantum critical region where there exists an emergent conformal symmetry and is absent of well-defined quasi-particles. With the help of a relation between the R\'enyi entropy and the out-of-time-order correlation function, we argue that the out-of-time-order correlation function of the Bose-Hubbard model will also exhibit an exponential behavior at the scrambling time. By fitting the numerical results with an exponential function, we extract the Lyapunov exponents in the one-dimensional Bose-Hubbard model across the quantum critical regime at finite temperature. Our results on the Bose-Hubbard model support the conjecture. We also compute the butterfly velocity and propose how the echo type measurement of this correlator in the cold atom realizations of the Bose-Hubbard model without inverting the Hamiltonian.

Scrambling and thermalization in a diffusive quantum many-body system

Out-of-time ordered (OTO) correlation functions describe scrambling of information in correlated quantum matter. They are of particular interest in incoherent quantum systems lacking well defined quasi-particles. Thus far, it is largely elusive how OTO correlators spread in incoherent systems with diffusive transport governed by a few globally conserved quantities. Here, we study the dynamical response of such a system using high-performance matrix-product-operator techniques. Specifically, we consider the non-integrable, one-dimensional Bose–Hubbard model in the incoherent high-temperature regime. Our system exhibits diffusive dynamics in time-ordered correlators of globally conserved quantities, whereas OTO correlators display a ballistic, light-cone spreading of quantum information. The slowest process in the global thermalization of the system is thus diffusive, yet information spreading is not inhibited by such slow dynamics. We furthermore develop an experimentally feasible protocol to overcome some challenges faced by existing proposals and to probe time-ordered and OTO correlation functions. Our study opens new avenues for both the theoretical and experimental exploration of thermalization and information scrambling dynamics.

Driven Bose-Hubbard model with a parametrically modulated harmonic trap

We investigate a one-dimensional Bose-Hubbard model in a parametrically driven global harmonic trap. The delicate interplay of both the local atom interaction and the global driving allows to control the dynamical stability of the trapped quantum many-body state. The mechanism is illustrated for weak interaction by a discretized Gross-Pitaevskii equation within a Gaussian variational ansatz, yielding to a Mathieu equation for the condensate width. The parametric resonance condition can be tuned by the atom interaction strength. For stronger interaction, this mechanism is confirmed by results of the numerically exact time-evolving block decimation scheme. The global modulation also induces an effective time-independent inhomogeneous hopping strength for the atoms.