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arXiv:2508.00486

The Bose-Hubbard polaron from weak to strong coupling

Tom Hartweg, Tanul Gupta, Guido Pupillo

Published Aug 1, 2025• 0 citations
arXivHTMLPDF

Abstract

We investigate the zero-temperature properties of a mobile impurity immersed in a bath of bosonic particles confined to a square lattice. We analyze the regimes of attractive and repulsive coupling between the impurity and the bath particles for different strengths of boson-boson interactions in the bath, using exact large-scale quantum Monte-Carlo simulations in the grand canonical ensemble. For weak coupling, the polaron mass ratio is found to decrease around the Mott insulator (MI) to superfluid (SF) transition of the bath, as predicted by recent theory, confirming the possible use of the impurity as a probe for the transition. For strong coupling in the MI regime, instead, the impurity is found to modify the bath density by binding to an extra bath particle or a hole, depending on the sign of the polaron-bath interactions. While the binding prevent the aforementioned use of the polaron mass ratio as an MI-SF transition probe, we show that it can be used instead as a probe of the binding itself. Our exact numerical results provide a benchmark for comparing lattice Bose polaron theories and are relevant for experiments with cold atoms trapped in optical lattices, where the presence of a confining harmonic potential can be modeled by a slowly varying local chemical potential.

DOI
10.48550/arxiv.2508.00486

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We investigate the zero-temperature properties of a mobile impurity immersed in a bath of bosonic particles confined to a square lattice. We analyze the regimes of attractive and repulsive coupling between the impurity and the bath particles for different strengths of boson-boson interactions in the bath, using exact large-scale quantum Monte-Carlo simulations in the grand canonical ensemble. For weak coupling, the polaron mass ratio is found to decrease around the Mott insulator (MI) to superfluid (SF) transition of the bath, as predicted by recent theory, confirming the possible use of the impurity as a probe for the transition. For strong coupling in the MI regime, instead, the impurity is found to modify the bath density by binding to an extra bath particle or a hole, depending on the sign of the polaron-bath interactions. While the binding prevent the aforementioned use of the polaron mass ratio as an MI-SF transition probe, we show that it can be used instead as a probe of the binding itself. Our exact numerical results provide a benchmark for comparing lattice Bose polaron theories and are relevant for experiments with cold atoms trapped in optical lattices, where the presence of a confining harmonic potential can be modeled by a slowly varying local chemical potential.
The authors perform here large-scale quantum Monte Carlo simulations in the grand-canonical ensemble to characterize a mobile impurity in a 2D Bose--Hubbard bath. They find that the polaron effective mass is a reliable probe of the Mott--superfluid transition only at weak coupling, while strong impurity--bath interactions induce MI\ifmmode\pm\else\textpm\fi{}1 bound states with suppressed mobility. The work delivers controlled benchmarks across interaction strengths and clarifies how impurity physics manifests in experimentally relevant trapped atom systems.
The problem of mobile impurities in quantum baths is of fundamental importance in many-body physics. There has recently been significant progress regarding our understanding of this due to cold atom experiments, but so far it has mainly been concerned with cases where the bath has no or only weak interactions, or the impurity interacts weakly with the bath. Here, we address this gap by developing a new theoretical framework for exploring a mobile impurity interacting strongly with a highly correlated bath of bosons in the quantum critical regime of a Mott insulator (MI) to superfluid (SF) quantum phase transition. Our framework is based on a powerful quantum Gutzwiller (QGW) description of the bosonic bath combined with diagrammatic field theory for the impurity-bath interactions. By resumming a selected class of diagrams to infinite order, a rich picture emerges where the impurity is dressed by the fundamental modes of the bath, which change character from gapped particle-hole excitations in the MI to Higgs and gapless Goldstone modes in the SF. This gives rise to the existence of several quasiparticle (polaron) branches with properties reflecting the strongly correlated environment. In particular, one polaron branch exhibits a sharp cusp in its energy, while a new ground-state polaron emerges at the O(2) O ( 2 ) quantum phase transition point for integer filling, which reflects the nonanalytic behavior at the transition and the appearance of the Goldstone mode in the SF phase. Smooth versions of these features are inherited in the polaron spectrum away from integer filling due to the influence of Mott physics on the bosonic bath. We furthermore compare our diagrammatic results with quantum Monte Carlo calculations, obtaining excellent agreement. This accuracy is quite remarkable for such a highly non-trivial case of strong interactions between the impurity and bosons in a maximally correlated quantum critical regime, and it establishes the utility of our framework. Finally, our results show how impurities can be used as quantum sensors and highlight fundamental differences between experiments performed at a fixed particle number or a fixed chemical potential.
We investigate how the coherence and spatial dressing of a single impurity evolve in the two-dimensional Bose-Hubbard model when the impurity couples attractively to the bath. Using large-scale, sign-problem-free worm-algorithm quantum Monte Carlo, we measure the impurity winding, bath superfluid response and compressibility, and impurity-bath density correlations. In a compressible superfluid bath (Ub/t=13.3U_{\mathrm{b}}/t=13.3Ub​/t=13.3), strengthening attraction drives an interaction-controlled \emph{winding-collapse self-trapping crossover}: a mobile light polaron evolves continuously into a heavy polaron and ultimately into a bound cluster with vanishing winding, while the bath remains globally superfluid. In incompressible Mott-insulating baths, by contrast, extended density rearrangements are suppressed and the dressing cloud collapses; we compare the resulting short-range deformation patterns for both attractive and repulsive couplings. Across the SF-MI transition at fixed moderate attraction (Uib/t=−8.0U_{\mathrm{ib}}/t=-8.0Uib​/t=−8.0), the impurity crosses over from a mobile polaron with an extended deformation cloud to a nearly free defect with minimal dressing in the Mott background; for this coupling it does not lock into a fully self-trapped defect even deep in the insulator. Together with our companion Letter [Impurity Self-Trapping in Lattice Bose systems] on repulsive couplings, these results provide a unified microscopic picture of impurity self-trapping in correlated lattice bosons, connecting interaction-driven winding collapse in the superfluid to compressibility-controlled undressing across the SF-MI transition.
We map out the global phase diagram of a single mobile impurity in the two-dimensional Bose-Hubbard model, spanning the bath evolution from a compressible superfluid (SF) to an incompressible Mott insulator (MI) and the full range of impurity-bath coupling. Using sign-problem-free worm-algorithm quantum Monte Carlo, we identify two distinct self-trapping mechanisms that organize the entire diagram. In the compressible SF, increasing impurity-bath coupling ∣Uib∣|U_{\mathrm{ib}}|∣Uib​∣ drives an interaction-driven self-trapping crossover signaled by a collapse of the \emph{impurity} winding number: a light, extended polaron evolves continuously into a heavy polaron and ultimately into a self-trapped state -- a repulsive \emph{saturated bubble} or an attractive \emph{bound cluster} -- even while the bath remains globally superfluid, demonstrating self-trapping without any bath phase transition. By contrast, when the bath is tuned across the SF-MI transition at fixed UibU_{\mathrm{ib}}Uib​, localization is compressibility controlled. The vanishing bath compressibility quenches long-wavelength density redistribution and suppresses polaronic dressing, converting the SF polaron into a weakly dressed, nearly free defect upon entering the MI when ∣Uib∣≤8.0|U_{\mathrm{ib}}| \le 8.0∣Uib​∣≤8.0. Then increasing ∣Uib∣|U_{\mathrm{ib}}|∣Uib​∣ triggers a distinct Mott-specific route: the impurity binds a quantized vacancy or particle excitation, manifested by discrete changes ΔNb=±1ΔN_b=\pm1ΔNb​=±1 in the total bath occupation. Together, our results provide a unified microscopic picture of impurity self-trapping in correlated lattice bosons, governed by winding collapse in the SF and by compressibility loss and defect quantization across the SF-MI boundary.
Bose polarons, quasiparticles composed of mobile impurities surrounded by cold Bose gas, can experience strong interactions mediated by the many-body environment and form bipolaron bound states. Here we present a detailed study of heavy polarons in a one-dimensional Bose gas by formulating a nonperturbative theory and complementing it with exact numerical simulations. We develop an analytic approach for weak boson-boson interactions and arbitrarily strong impurity-boson couplings. Our approach is based on a mean-field theory that accounts for deformations of the superfluid by the impurities and in this way minimizes quantum fluctuations. The mean-field equations are solved exactly in the Born-Oppenheimer approximation, leading to an analytic expression for the interaction potential of heavy polarons, which is found to be in excellent agreement with quantum Monte Carlo (QMC) results. In the strong coupling limit, the potential substantially deviates from the exponential form valid for weak coupling and has a linear shape at short distances. Taking into account the leading-order Born-Huang corrections, we calculate bipolaron binding energies for impurity-boson mass ratios as low as 3 and find excellent agreement with QMC results.
A mobile impurity particle immersed in a quantum degenerate gas leads to the formation of a quasiparticle, which can serve as a sensor for its environment. Here, we develop a unified wave function description of an impurity in a two-dimensional Bose-Hubbard model across the Mott-insulator to superfluid transition. Using a multi-mode variational ansatz based on the quantum Gutzwiller approach, we show that the impurity can form several kinds of polaronic and molecular states. In addition to their spectral properties, the wave function provides direct access to the microscopic structure of these states, and we demonstrate that strong correlations in the critical regime of the quantum phase transition give rise to several non-analytic features in the polaron properties. Increasing the impurity-boson interaction leads to transitions between polaron and molecular ground states, and the number of bosons in the impurity dressing cloud grows rapidly at the quantum critical point as the bath modes soften. We furthermore reveal qualitatively distinct non-equilibrium many-body dynamics after the injection of the impurity in the different phases of the Bose-Hubbard model. Our spectral, real space, and dynamical predictions describe complementary properties of polarons in a strongly correlated bosonic bath, which can be observed using current techniques with atoms in optical lattices.