We provide a rigorous derivation of Gutzwiller mean-field dynamics for lattice bosons, showing that it is exact on fully connected lattices. We apply this formalism to quenches in the interaction parameter from the Mott insulator to the superfluid state. Although within mean-field the Mott insulator is a steady state, we show that a dynamical critical interaction U d exists, such that for final interaction parameter U f > U d the Mott insulator is exponentially unstable towards emerging long-range superfluid order, whereas for U f < U d the Mott insulating state is stable. We discuss the implications of this prediction for finite-dimensional systems.
We provide a rigorous derivation of Gutzwiller mean-field dynamics for lattice bosons, showing that it is exact on fully connected lattices. We apply this formalism to quenches in the interaction parameter from the Mott insulator to the superfluid state. Although within mean-field the Mott insulator is a steady state, we show that a dynamical critical interaction U d exists, such that for final interaction parameter U f > U d the Mott insulator is exponentially unstable towards emerging long-range superfluid order, whereas for U f < U d the Mott insulating state is stable. We discuss the implications of this prediction for finite-dimensional systems.
In the context of out of equilibrium dynamics of isolated interacting quantum many-body systems, we study the mean-field discretized and nonlinear Gross-Pitaevskii equation (GPE) of motion for a fully connected (or all-to-all coupled) Bose-Hubbard model describing a large population of bosons on a lattice of generic dimension V, with potential experimental applications ranging from ultra-cold atomic gases in optical traps to systems of Josephson junctions. Considering a quench on the system Hamiltonian, we describe the time evolution of a system initialized to the Mott insulating phase and taken in the strong superfluid regime, finding a sharp change in the dynamical behaviour when varying the quench intensity across a critical value, that is a dynamical phase transition. Our results generalize the study presented in some previous papers focusing on the particular cases of V = 2, V = 3 and 1D lattice, and place particular emphasis on the relaxation of some dynamical order parameters in the large V limit. On the side of the dynamical phase transition corresponding to the strong superfluid regime, we find such relaxation to consist of a π-synchronization of the mean field bosonic phases, that we find to compete against the eventual site-dependent disorder of the Hamiltonian in what, in analogy with the notorious Kuramoto model for nonlinear coupled oscillators, can be referred to as a synchronization transition. Additionally, in support of the validity of the mean field approximation far from the transition, we report a study beyond mean field based on the Bogolioubov-de Gennes method.
We study the physics of interacting spin-1 bosons in an optical lattice using a variational Gutzwiller technique. We compute the mean-field ground state wave function and discuss the evolution of the condensate, spin, nematic, and singlet order parameters across the superfluid-Mott transition. We then extend the Gutzwiller method to derive the equations governing the dynamics of low energy excitations in the lattice. Linearizing these equations, we compute the excitation spectra in the superfluid and Mott phases for both ferromagnetic and antiferromagnetic spin-spin interactions. In the superfluid phase, we recover the known excitation spectrum obtained from Bogoliubov theory. In the nematic Mott phase, we obtain gapped, quadratically dispersing particle and hole-like collective modes, whereas in the singlet Mott phase, we obtain a nondispersive gapped mode, corresponding to the breaking of a singlet pair. For the ferromagnetic Mott insulator, the Gutzwiller mean-field theory only yields particle-hole-like modes but no Goldstone mode associated with long-range spin order. To overcome this limitation, we supplement the Gutzwiller theory with a Schwinger boson mean-field theory which captures superexchange-driven fluctuations. In addition to the gapped particle-hole-like modes, we obtain a gapless quadratically dispersing ferromagnetic spin-wave Goldstone mode. We discuss the evolution of the singlet gap, particle-hole gap, and the effective mass of the ferromagnetic Goldstone mode as the superfluid-Mott phase boundary is approached from the insulating side. We discuss the relevance and validity of Gutzwiller mean-field theories to spinful systems, and potential extensions of this framework to include more exotic physics which appears in the presence of spin-orbit coupling or artificial gauge fields.
A large-scale dynamical simulation of the superfluid--Mott-insulator transition in a gas of ultracold atoms placed in an optical lattice is performed using the time-dependent Gutzwiller mean-field approach. This approximate treatment allows us to take into account most of the details of the recent experiment [Greiner et al., Nature (London) 415, 39 (2002)] where by changing the depth of the lattice potential an adiabatic transition from a superfluid to a Mott insulator state has been reported. Our simulations reveal a significant excitation of the system with a transition to insulator in restricted regions of the trap only.
The present work reports on the dynamical measures of order, disorder and complexity for the interacting bosons in optical lattice. We report results both for the relaxed state as well as quench dynamics. Our key observations are: (1) Lattice depth can be taken as order-disorder parameter. (2) The superfluid to Mott insulator transition can be treated as `order-disorder' transition. Our main motivation is to find how the system organize by itself during quench and how it optimizes the complexity. We find dynamical measures of order and disorder are more sensitive tool than entropy measures. We specifically calculate the time scale of entry and exit of different phases during time evolution. Initially the system exhibits collapse revival trend, however gradually looses its ability to turn back to superfluid phase and finally Settle to Mott insulator phase.
The phase diagrams and phase transitions of bosons with short-ranged repulsive interactions moving in periodic and/or random external potentials at zero temperature are investigated with emphasis on the superfluid-insulator transition induced by varying a parameter such as the density. Bosons in periodic potentials (e.g., on a lattice) at T=0 exhibit two types of phases: a superfluid phase and Mott insulating phases characterized by integer (or commensurate) boson densities, by the existence of a gap for particle-hole excitations, and by zero compressibility. Generically, the superfluid onset transition in d dimensions from a Mott insulator to superfluidity is ``ideal,'' or mean field in character, but at special multicritical points with particle-hole symmetry it is in the universality class of the (d+1)-dimensional XY model. In the presence of disorder, a third, ``Bose glass'' phase exists. This phase is insulating because of the localization effects of the randomness and analogous to the Fermi glass phase of interacting fermions in a strongly disordered potential.The Bose glass phase is characterized by a finite compressibility, no gap, but an infinite superfluid susceptibility. In the presence of disorder the transition to superfluidity is argued to occur only from the Bose glass phase, and never directly from the Mott insulator. This zero-temperature superfluid-insulator transition is studied via generalizations of the Josephson scaling relation for the superfluid density at the ordinary \ensuremath{\lambda} transition, highlighting the crucial role of quantum fluctuations. The transition is found to have a dynamic critical exponent z exactly equal to d and correlation length and order-parameter correlation exponents \ensuremath{\nu} and \ensuremath{\eta} which satisfy the bounds \ensuremath{\nu}\ensuremath{\ge}2/d and \ensuremath{\eta}\ensuremath{\le}2-d, respectively. It is argued that the superfluid-insulator transition in the presence of disorder may have an upper critical dimension dc which is infinite, but a perturbative renormalization-group calculation wherein the critical exponents have mean-field values for weak disorder above d=4 is also discussed. Many of these conclusions are verified by explicit calculations on a model of one-dimensional bosons in the presence of both random and periodic potentials. The general results are applied to experiments on 4He absorbed in porous media such as Vycor. Some measurable properties of the superfluid onset are predicted exactly [e.g., the exponent x relating the \ensuremath{\lambda} transition temperature to the zero-temperature superfluid density is found to be d/2(d-1)], while stringent bounds are placed on others. Analysis of preliminary data is consistent with these predictions.
Abstract This paper is a first step toward rigorously establishing the validity of mean-field approximations for the dynamics of quantum systems in a large dimensional limit. Our result is a derivation of the large d mean-field limit of the Bose–Hubbard model on a square lattice. The Bose–Hubbard model effectively describes bosons on a lattice with on-site interactions and nearest-neighbour hopping, serving as a foundational framework for understanding strong particle interactions and the superfluid to Mott insulator phase transition. We prove a trace norm estimate between the one-lattice-site reduced density of the Schrödinger dynamics and the mean-field dynamics in the limit of large dimension. Here, the mean-field approximation is in the hopping amplitude and not in the interaction, leading to a very rich and non-trivial mean-field equation. This mean-field equation does not only describe the condensate, as is the case when the mean-field description comes from a large particle number limit averaging out the interaction, but it allows for a phase transition since it contains the full non-trivial interaction. Our work is a rigorous justification of a simple case of the highly successful dynamical mean-field theory (DMFT) for bosons, which somewhat surprisingly yields many qualitatively correct results in three dimensions.