Lattice Bose Polaron

• Lattice Bose polaron is a dressed quasiparticle formed by a mobile impurity interacting with a bosonic lattice, incorporating both band structure and collective modes.
• Models range from Bogoliubov phonon dressings in weakly interacting BECs to strongly correlated Bose-Hubbard systems featuring Higgs modes and particle-hole fluctuations.
• Theoretical frameworks like QGW, QMC, and variational methods reveal key observables such as effective mass, dispersion, and spectral functions that diagnose SF–MI transitions.

A lattice Bose polaron is the dressed quasiparticle formed when a mobile impurity interacts with a bosonic environment in which lattice structure is an essential part of the problem. In one class of models, the impurity is confined to an optical lattice while the bath is a homogeneous weakly interacting Bose-Einstein condensate treated within Bogoliubov theory, so the lattice enters through the impurity band structure and quasimomentum-resolved dressing by phonons (Grusdt et al., 2014, Christ et al., 2024, Yin et al., 2015). In a second, more strongly correlated class, the impurity moves in the same lattice as a Bose-Hubbard or hard-core boson bath, and the dressing cloud is built not only from Bogoliubov-like Goldstone modes but also from amplitude/Higgs modes, particle-hole excitations, doublon-holon fluctuations, and beyond-Fröhlich two-excitation processes (Colussi et al., 2022, Santiago-García et al., 2024, Alhyder et al., 2024). Across these settings, the topic is defined by the interplay of impurity mobility, bosonic collective modes, lattice band structure, and, in the Bose-Hubbard case, proximity to the superfluid (SF)–Mott-insulator (MI) transition.

1. Microscopic formulations

A representative strongly correlated lattice formulation is the Bose-Hubbard impurity problem

$\hat H=\hat H_B+\hat H_I+\hat H_{IB},$

with

$$\hat{H}_B = -J \sum_{\langle {\bf r, r'} \rangle} \hat{a}^\dag_{\bf r}\hat{a}_{\bf r'} + \frac{U}{2} \sum_{\bf r} \hat{n}_{\bf r}\left(\hat{n}_{\bf r}-1\right) - \mu \sum_{\bf r} \hat{n}_{\bf r},$$

$$\hat{H}_I = -J \sum_{\langle {\bf r, r'} \rangle} \hat{a}^\dagger_{I,{\bf r}}\hat{a}_{I,{\bf r'}}, \qquad \hat{H}_{IB} = U_{12} \sum_{\bf r} \hat{n}_{\bf r}\hat{n}_{I,{\bf r}},$$

where the impurity and bath occupy the same lattice and interact locally. In this setting the bare impurity dispersion is

$$\varepsilon_{\bf k}=4J\sum_{i=1}^d \sin^2(k_i a/2),$$

with bare effective mass $M^{-1}=2Ja^2$ (Colussi et al., 2022).

A numerically exact two-dimensional square-lattice version, formulated in the grand canonical ensemble and extended to a harmonic trap, uses

$H = H_B + H_I + H_{IB} + H_\Omega ,$

with

$$H_B = -t \sum_{\langle i, j \rangle} a_i^\dagger a_j + \frac{U}{2} \sum_i n_{B,i}(n_{B,i}-1),$$

$$H_I = -\tilde t \sum_{\langle i,j\rangle} b_i^\dagger b_j, \qquad H_{IB} = U_{IB}\sum_i b_i^\dagger b_i\, a_i^\dagger a_i,$$

and

$$H_\Omega = -\mu \sum_i n_{B,i} + \Omega \sum_i r_i^2\,(n_{B,i}+n_{I,i}),$$

with most calculations at $t=\tilde t$. This formulation makes explicit the distinction between homogeneous and trapped realizations and between attractive $$\hat{H}_B = -J \sum_{\langle {\bf r, r'} \rangle} \hat{a}^\dag_{\bf r}\hat{a}_{\bf r'} + \frac{U}{2} \sum_{\bf r} \hat{n}_{\bf r}\left(\hat{n}_{\bf r}-1\right) - \mu \sum_{\bf r} \hat{n}_{\bf r},$$0 and repulsive $$\hat{H}_B = -J \sum_{\langle {\bf r, r'} \rangle} \hat{a}^\dag_{\bf r}\hat{a}_{\bf r'} + \frac{U}{2} \sum_{\bf r} \hat{n}_{\bf r}\left(\hat{n}_{\bf r}-1\right) - \mu \sum_{\bf r} \hat{n}_{\bf r},$$1 impurity-bath coupling (Hartweg et al., 1 Aug 2025).

A broader usage of the term arises when only the impurity is on a lattice while the bath is a homogeneous BEC. There the lattice Bose polaron is described by a Fröhlich-type Hamiltonian in which the impurity has a tight-binding band and couples to Bogoliubov phonons,

$$\hat{H}_B = -J \sum_{\langle {\bf r, r'} \rangle} \hat{a}^\dag_{\bf r}\hat{a}_{\bf r'} + \frac{U}{2} \sum_{\bf r} \hat{n}_{\bf r}\left(\hat{n}_{\bf r}-1\right) - \mu \sum_{\bf r} \hat{n}_{\bf r},$$2

$$\hat{H}_B = -J \sum_{\langle {\bf r, r'} \rangle} \hat{a}^\dag_{\bf r}\hat{a}_{\bf r'} + \frac{U}{2} \sum_{\bf r} \hat{n}_{\bf r}\left(\hat{n}_{\bf r}-1\right) - \mu \sum_{\bf r} \hat{n}_{\bf r},$$3

with $$\hat{H}_B = -J \sum_{\langle {\bf r, r'} \rangle} \hat{a}^\dag_{\bf r}\hat{a}_{\bf r'} + \frac{U}{2} \sum_{\bf r} \hat{n}_{\bf r}\left(\hat{n}_{\bf r}-1\right) - \mu \sum_{\bf r} \hat{n}_{\bf r},$$4, and impurity-phonon coupling through a form factor $$\hat{H}_B = -J \sum_{\langle {\bf r, r'} \rangle} \hat{a}^\dag_{\bf r}\hat{a}_{\bf r'} + \frac{U}{2} \sum_{\bf r} \hat{n}_{\bf r}\left(\hat{n}_{\bf r}-1\right) - \mu \sum_{\bf r} \hat{n}_{\bf r},$$5 (Grusdt et al., 2014). A one-dimensional Bogoliubov-level extension retains both single- and two-phonon terms after the Lee-Low-Pines transformation and emphasizes the dressed dispersion $$\hat{H}_B = -J \sum_{\langle {\bf r, r'} \rangle} \hat{a}^\dag_{\bf r}\hat{a}_{\bf r'} + \frac{U}{2} \sum_{\bf r} \hat{n}_{\bf r}\left(\hat{n}_{\bf r}-1\right) - \mu \sum_{\bf r} \hat{n}_{\bf r},$$6 across the Brillouin zone rather than only an effective mass near $$\hat{H}_B = -J \sum_{\langle {\bf r, r'} \rangle} \hat{a}^\dag_{\bf r}\hat{a}_{\bf r'} + \frac{U}{2} \sum_{\bf r} \hat{n}_{\bf r}\left(\hat{n}_{\bf r}-1\right) - \mu \sum_{\bf r} \hat{n}_{\bf r},$$7 (Christ et al., 2024).

Within the literature considered here, a narrower and more strongly correlated usage reserves “genuine lattice Bose polaron” for the case where the bath itself is a tunable lattice boson system rather than a weak continuum gas. This usage is motivated by the fact that the impurity is then dressed by the full spectrum of lattice collective modes rather than only by Bogoliubov phonons (Colussi et al., 2022).

2. Collective modes and dressing channels

The structure of the bosonic bath determines what dresses the impurity. In a Bose-Hubbard bath treated by the Quantum Gutzwiller (QGW) formalism, fluctuations around the Gutzwiller product state

$$\hat{H}_B = -J \sum_{\langle {\bf r, r'} \rangle} \hat{a}^\dag_{\bf r}\hat{a}_{\bf r'} + \frac{U}{2} \sum_{\bf r} \hat{n}_{\bf r}\left(\hat{n}_{\bf r}-1\right) - \mu \sum_{\bf r} \hat{n}_{\bf r},$$8

are quantized in terms of collective modes $$\hat{H}_B = -J \sum_{\langle {\bf r, r'} \rangle} \hat{a}^\dag_{\bf r}\hat{a}_{\bf r'} + \frac{U}{2} \sum_{\bf r} \hat{n}_{\bf r}\left(\hat{n}_{\bf r}-1\right) - \mu \sum_{\bf r} \hat{n}_{\bf r},$$9, yielding a quadratic bath Hamiltonian

$$\hat{H}_I = -J \sum_{\langle {\bf r, r'} \rangle} \hat{a}^\dagger_{I,{\bf r}}\hat{a}_{I,{\bf r'}}, \qquad \hat{H}_{IB} = U_{12} \sum_{\bf r} \hat{n}_{\bf r}\hat{n}_{I,{\bf r}},$$0

In the SF phase, the two lowest branches are the gapless Goldstone mode and the gapped amplitude (Higgs) mode; in the MI phase, the elementary excitations are particle-hole excitations. At the tip of the $$\hat{H}_I = -J \sum_{\langle {\bf r, r'} \rangle} \hat{a}^\dagger_{I,{\bf r}}\hat{a}_{I,{\bf r'}}, \qquad \hat{H}_{IB} = U_{12} \sum_{\bf r} \hat{n}_{\bf r}\hat{n}_{I,{\bf r}},$$1 Mott lobe, corresponding to the commensurate $$\hat{H}_I = -J \sum_{\langle {\bf r, r'} \rangle} \hat{a}^\dagger_{I,{\bf r}}\hat{a}_{I,{\bf r'}}, \qquad \hat{H}_{IB} = U_{12} \sum_{\bf r} \hat{n}_{\bf r}\hat{n}_{I,{\bf r}},$$2 critical point, both Goldstone and Higgs modes become gapless, whereas along the lobe edges the transition is of commensurate-incommensurate (CI) type, with a quadratic Goldstone mode and a gapped Higgs mode (Colussi et al., 2022).

The impurity couples to bath density fluctuations. Retaining only the linear fluctuation term $$\hat{H}_I = -J \sum_{\langle {\bf r, r'} \rangle} \hat{a}^\dagger_{I,{\bf r}}\hat{a}_{I,{\bf r'}}, \qquad \hat{H}_{IB} = U_{12} \sum_{\bf r} \hat{n}_{\bf r}\hat{n}_{I,{\bf r}},$$3 yields a Fröhlich-like lattice polaron model. The essential beyond-Fröhlich extension is the quadratic density fluctuation term $$\hat{H}_I = -J \sum_{\langle {\bf r, r'} \rangle} \hat{a}^\dagger_{I,{\bf r}}\hat{a}_{I,{\bf r'}}, \qquad \hat{H}_{IB} = U_{12} \sum_{\bf r} \hat{n}_{\bf r}\hat{n}_{I,{\bf r}},$$4, whose pair-emission and pair-absorption channel is weighted by $$\hat{H}_I = -J \sum_{\langle {\bf r, r'} \rangle} \hat{a}^\dagger_{I,{\bf r}}\hat{a}_{I,{\bf r'}}, \qquad \hat{H}_{IB} = U_{12} \sum_{\bf r} \hat{n}_{\bf r}\hat{n}_{I,{\bf r}},$$5. At zero temperature this two-excitation channel becomes crucial near the $$\hat{H}_I = -J \sum_{\langle {\bf r, r'} \rangle} \hat{a}^\dagger_{I,{\bf r}}\hat{a}_{I,{\bf r'}}, \qquad \hat{H}_{IB} = U_{12} \sum_{\bf r} \hat{n}_{\bf r}\hat{n}_{I,{\bf r}},$$6 critical point and in the MI, where the one-particle density vertices $$\hat{H}_I = -J \sum_{\langle {\bf r, r'} \rangle} \hat{a}^\dagger_{I,{\bf r}}\hat{a}_{I,{\bf r'}}, \qquad \hat{H}_{IB} = U_{12} \sum_{\bf r} \hat{n}_{\bf r}\hat{n}_{I,{\bf r}},$$7 vanish. In the MI the impurity therefore dresses only through two-particle processes such as particle-hole pair creation, which is a defining lattice effect (Colussi et al., 2022).

Hard-core boson baths generate a related but distinct structure. In a two-dimensional square-lattice condensate of hard-core bosons, the impurity-bath interaction contains a mean-field term, a one-excitation Fröhlich-like term, and explicit beyond-Fröhlich two-excitation terms. A central observation is that as $$\hat{H}_I = -J \sum_{\langle {\bf r, r'} \rangle} \hat{a}^\dagger_{I,{\bf r}}\hat{a}_{I,{\bf r'}}, \qquad \hat{H}_{IB} = U_{12} \sum_{\bf r} \hat{n}_{\bf r}\hat{n}_{I,{\bf r}},$$8 and $$\hat{H}_I = -J \sum_{\langle {\bf r, r'} \rangle} \hat{a}^\dagger_{I,{\bf r}}\hat{a}_{I,{\bf r'}}, \qquad \hat{H}_{IB} = U_{12} \sum_{\bf r} \hat{n}_{\bf r}\hat{n}_{I,{\bf r}},$$9, the mean-field and Fröhlich terms vanish, but the beyond-Fröhlich term remains finite; the low-density strong-coupling behavior is therefore controlled by two-mode processes rather than by ordinary single-mode coupling (Santiago-García et al., 2024).

In one-dimensional Bose-Hubbard systems with a spin impurity, the dressing cloud is naturally visualized as a hole bound to the impurity. In the Mott regime this impurity-hole composite is tightly localized and follows from the effective Heisenberg description with superexchange $$\varepsilon_{\bf k}=4J\sum_{i=1}^d \sin^2(k_i a/2),$$0. In the superfluid regime the same picture survives in a more extended form: the impurity still suppresses the density of bath bosons near it, but the depletion cloud broadens and can eventually merge into an uncorrelated particle-hole continuum (Dutta et al., 2013).

For continuum-BEC-based lattice polarons, the bath degrees of freedom remain Bogoliubov phonons, but beyond-Fröhlich terms can still be important. In the extended lattice Bogoliubov-Fröhlich Hamiltonian, the impurity couples through $$\varepsilon_{\bf k}=4J\sum_{i=1}^d \sin^2(k_i a/2),$$1 to single-phonon processes and through $$\varepsilon_{\bf k}=4J\sum_{i=1}^d \sin^2(k_i a/2),$$2 and $$\varepsilon_{\bf k}=4J\sum_{i=1}^d \sin^2(k_i a/2),$$3 to number-conserving two-phonon scattering and pair creation/annihilation. Because the lattice kinetic term becomes $$\varepsilon_{\bf k}=4J\sum_{i=1}^d \sin^2(k_i a/2),$$4 after the Lee-Low-Pines transformation, the dressing cloud can reshape the entire Bloch band (Christ et al., 2024).

3. Theoretical frameworks

The QGW approach provides a semi-analytical route through the entire Bose-Hubbard phase diagram. One first extracts the bath modes and impurity-bath vertices from QGW, then computes the impurity Green’s function perturbatively to second order in weak $$\varepsilon_{\bf k}=4J\sum_{i=1}^d \sin^2(k_i a/2),$$5. In this scheme the self-energy separates into a Hartree shift, a one-mode Fröhlich-like term, and a genuine beyond-Fröhlich two-mode term, and observables are then obtained from the on-shell self-energy (Colussi et al., 2022).

Exact large-scale quantum Monte Carlo (QMC), based on the worm algorithm and specifically a multi-species worm algorithm in the grand canonical ensemble, has been used to benchmark the lattice Bose polaron from weak to strong coupling in two dimensions. The effective mass is extracted from an imaginary-time diffusion estimator in the $$\varepsilon_{\bf k}=4J\sum_{i=1}^d \sin^2(k_i a/2),$$6 limit, and the same-site density-density correlator $$\varepsilon_{\bf k}=4J\sum_{i=1}^d \sin^2(k_i a/2),$$7 diagnoses particle or hole binding. Within this framework an important conclusion is that, in the thermodynamic limit, a single impurity does not shift the MI–SF transition point of the bath and does not alter bulk Mott properties such as the excitation gap or compressibility (Hartweg et al., 1 Aug 2025).

A strong-coupling, quantum-critical framework combines QGW with diagrammatic field theory and an infinite-order resummation of a generalized ladder or $$\varepsilon_{\bf k}=4J\sum_{i=1}^d \sin^2(k_i a/2),$$8-matrix class of diagrams. The impurity Green’s function

$$\varepsilon_{\bf k}=4J\sum_{i=1}^d \sin^2(k_i a/2),$$9

and spectral function

$M^{-1}=2Ja^2$0

are then computed from coupled Bethe-Salpeter equations for multimode scattering matrices $M^{-1}=2Ja^2$1. This resummation captures strong impurity-boson correlations, in-medium bound states, multiple polaron branches, and the branch rearrangements that appear near the $M^{-1}=2Ja^2$2 critical point (Alhyder et al., 2024).

Variational methods remain central in several regimes. A Chevy-like ansatz with zero- and one-excitation sectors has been used for an impurity in a hard-core boson condensate,

$M^{-1}=2Ja^2$3

and a momentum-resolved impurity-hole ansatz has been used in one-dimensional Bose-Hubbard systems,

$M^{-1}=2Ja^2$4

These constructions interpolate between tightly bound Mott polarons and more extended superfluid polarons (Santiago-García et al., 2024, Dutta et al., 2013).

Lang-Firsov-based approaches generalize the lattice Bose polaron beyond the single-impurity limit. A variational polaron transformation for a two-component Bose-Hubbard mixture yields an effective heavy-species Hamiltonian

$M^{-1}=2Ja^2$5

with phonon-dressed hopping $M^{-1}=2Ja^2$6 and induced interaction $M^{-1}=2Ja^2$7, thereby extending impurity dressing ideas to a finite-density many-polaron problem near a Mott transition (Benjamin et al., 2014). A two-band extension with a variational Lang-Firsov transformation produces band-dependent hopping renormalization, band shifts, induced interactions, and a Lindblad description of residual inter-band relaxation (Yin et al., 2015).

For impurity-on-lattice plus homogeneous-BEC problems, a lattice-adapted Lee-Low-Pines transformation together with coherent-state mean-field theory yields both static dressed bands and non-equilibrium dynamics. An operator-valued flow equation approach has further been developed to diagonalize the extended lattice Bogoliubov-Fröhlich Hamiltonian with operator-valued coefficients, allowing explicit comparison between Fröhlich and beyond-Fröhlich dispersions (Grusdt et al., 2014, Christ et al., 2024).

4. Observables and quasiparticle characterization

The central observables are the impurity dispersion, the bath-induced energy shift, the effective mass, the quasiparticle residue, the decay rate, local impurity-bath correlations, and the spectral function. In the QGW perturbative framework the on-shell polaron dispersion is

$M^{-1}=2Ja^2$8

with low-momentum expansion

$M^{-1}=2Ja^2$9

and

$H = H_B + H_I + H_{IB} + H_\Omega ,$0

The quasiparticle is well defined when $H = H_B + H_I + H_{IB} + H_\Omega ,$1 and $H = H_B + H_I + H_{IB} + H_\Omega ,$2 remains near unity (Colussi et al., 2022).

QMC work characterizes the two-dimensional Bose-Hubbard polaron through the small-$H = H_B + H_I + H_{IB} + H_\Omega ,$3 expansion

$H = H_B + H_I + H_{IB} + H_\Omega ,$4

with bare lattice mass $H = H_B + H_I + H_{IB} + H_\Omega ,$5 for $H = H_B + H_I + H_{IB} + H_\Omega ,$6, and studies the mass ratio $H = H_B + H_I + H_{IB} + H_\Omega ,$7 as the principal mobility diagnostic. The same-site correlator

$H = H_B + H_I + H_{IB} + H_\Omega ,$8

distinguishes bound regimes through the limits $H = H_B + H_I + H_{IB} + H_\Omega ,$9, $$H_B = -t \sum_{\langle i, j \rangle} a_i^\dagger a_j + \frac{U}{2} \sum_i n_{B,i}(n_{B,i}-1),$$0, and $$H_B = -t \sum_{\langle i, j \rangle} a_i^\dagger a_j + \frac{U}{2} \sum_i n_{B,i}(n_{B,i}-1),$$1 for an extra boson, no extra defect, and a hole, respectively (Hartweg et al., 1 Aug 2025).

Spectral formulations emphasize

$$H_B = -t \sum_{\langle i, j \rangle} a_i^\dagger a_j + \frac{U}{2} \sum_i n_{B,i}(n_{B,i}-1),$$2

or, at zero momentum in the hard-core bath problem,

$$H_B = -t \sum_{\langle i, j \rangle} a_i^\dagger a_j + \frac{U}{2} \sum_i n_{B,i}(n_{B,i}-1),$$3

together with the residue

$$H_B = -t \sum_{\langle i, j \rangle} a_i^\dagger a_j + \frac{U}{2} \sum_i n_{B,i}(n_{B,i}-1),$$4

Because both impurity and bath excitations live on a lattice, the one-excitation continuum is bounded; for $$H_B = -t \sum_{\langle i, j \rangle} a_i^\dagger a_j + \frac{U}{2} \sum_i n_{B,i}(n_{B,i}-1),$$5 its width is

$$H_B = -t \sum_{\langle i, j \rangle} a_i^\dagger a_j + \frac{U}{2} \sum_i n_{B,i}(n_{B,i}-1),$$6

This bounded continuum is a distinct lattice signature (Santiago-García et al., 2024).

In multiband lattice-polaron formulations, the coherent polaron bands are

$$H_B = -t \sum_{\langle i, j \rangle} a_i^\dagger a_j + \frac{U}{2} \sum_i n_{B,i}(n_{B,i}-1),$$7

with

$$H_B = -t \sum_{\langle i, j \rangle} a_i^\dagger a_j + \frac{U}{2} \sum_i n_{B,i}(n_{B,i}-1),$$8

These quantities encode the band-dependent binding energy and mass enhancement produced by phonon dressing (Yin et al., 2015).

Across the surveyed literature, the same basic interpretation recurs: lowering the energy, reducing mobility, suppressing $$H_B = -t \sum_{\langle i, j \rangle} a_i^\dagger a_j + \frac{U}{2} \sum_i n_{B,i}(n_{B,i}-1),$$9, transferring spectral weight into continua, or producing new poles in $$H_I = -\tilde t \sum_{\langle i,j\rangle} b_i^\dagger b_j, \qquad H_{IB} = U_{IB}\sum_i b_i^\dagger b_i\, a_i^\dagger a_i,$$0 all indicate stronger dressing, but the microscopic content of the dressing cloud depends sharply on whether the bath is a weak BEC, a hard-core condensate, or a Bose-Hubbard system near or inside a Mott regime (Colussi et al., 2022, Alhyder et al., 2024).

5. Phase structure, strong coupling, and quantum criticality

In weak impurity-bath coupling, the lattice Bose polaron can act as a probe of the MI–SF transition. In the QGW theory of Colussi and collaborators, reducing $$H_I = -\tilde t \sum_{\langle i,j\rangle} b_i^\dagger b_j, \qquad H_{IB} = U_{IB}\sum_i b_i^\dagger b_i\, a_i^\dagger a_i,$$1 from the deep SF side makes the impurity more dressed, lowers its energy below the Hartree shift, and increases its effective mass, with behavior that depends strongly on the universality class of the transition. At the $$H_I = -\tilde t \sum_{\langle i,j\rangle} b_i^\dagger b_j, \qquad H_{IB} = U_{IB}\sum_i b_i^\dagger b_i\, a_i^\dagger a_i,$$2 transition, $$H_I = -\tilde t \sum_{\langle i,j\rangle} b_i^\dagger b_j, \qquad H_{IB} = U_{IB}\sum_i b_i^\dagger b_i\, a_i^\dagger a_i,$$3 and $$H_I = -\tilde t \sum_{\langle i,j\rangle} b_i^\dagger b_j, \qquad H_{IB} = U_{IB}\sum_i b_i^\dagger b_i\, a_i^\dagger a_i,$$4 evolve smoothly and reach an absolute minimum on the SF side; at the CI transition, the impurity properties show a sharp, nonanalytic change when one-particle processes are discontinuously suppressed on entering the MI. Along noninteger-filling lines approaching the hard-core superfluid regime, both $$H_I = -\tilde t \sum_{\langle i,j\rangle} b_i^\dagger b_j, \qquad H_{IB} = U_{IB}\sum_i b_i^\dagger b_i\, a_i^\dagger a_i,$$5 and $$H_I = -\tilde t \sum_{\langle i,j\rangle} b_i^\dagger b_j, \qquad H_{IB} = U_{IB}\sum_i b_i^\dagger b_i\, a_i^\dagger a_i,$$6 diverge, $$H_I = -\tilde t \sum_{\langle i,j\rangle} b_i^\dagger b_j, \qquad H_{IB} = U_{IB}\sum_i b_i^\dagger b_i\, a_i^\dagger a_i,$$7, and the behavior is identified as a bosonic orthogonality catastrophe (Colussi et al., 2022).

Grand-canonical QMC confirms the weak-coupling dip of $$H_I = -\tilde t \sum_{\langle i,j\rangle} b_i^\dagger b_j, \qquad H_{IB} = U_{IB}\sum_i b_i^\dagger b_i\, a_i^\dagger a_i,$$8 around the MI–SF transition in a two-dimensional Bose-Hubbard bath, validating the use of the impurity as a probe of the critical region. The same calculations show that the strong-coupling physics is qualitatively different. Deep in the Mott phase there are three regimes: a free-polaron window

$$H_I = -\tilde t \sum_{\langle i,j\rangle} b_i^\dagger b_j, \qquad H_{IB} = U_{IB}\sum_i b_i^\dagger b_i\, a_i^\dagger a_i,$$9

an attractive bound regime

$$H_\Omega = -\mu \sum_i n_{B,i} + \Omega \sum_i r_i^2\,(n_{B,i}+n_{I,i}),$$0

in which the impurity binds an extra bath boson, and a repulsive bound regime

$$H_\Omega = -\mu \sum_i n_{B,i} + \Omega \sum_i r_i^2\,(n_{B,i}+n_{I,i}),$$1

in which the impurity binds a hole. In these bound regimes $$H_\Omega = -\mu \sum_i n_{B,i} + \Omega \sum_i r_i^2\,(n_{B,i}+n_{I,i}),$$2 becomes very small, so the mass ratio ceases to be a clean MI–SF probe and instead becomes a probe of binding itself (Hartweg et al., 1 Aug 2025).

A strong-coupling diagrammatic treatment near the quantum critical regime enriches this picture further. At integer filling near the $$H_\Omega = -\mu \sum_i n_{B,i} + \Omega \sum_i r_i^2\,(n_{B,i}+n_{I,i}),$$3 critical point, the bath modes reorganize from MI particle/hole excitations into SF Goldstone and Higgs modes, and the impurity spectrum develops multiple quasiparticle branches. One branch exhibits a sharp cusp in its energy, while a new ground-state polaron appears exactly at the critical point and then evolves continuously into the usual SF Bose polaron deeper in the superfluid. Away from integer filling these nonanalytic features become smooth crossovers inherited from varying “Mottness.” The same work emphasizes a physically important distinction between canonical and grand-canonical settings: at fixed total particle number, local impurity-boson or impurity-hole binding changes the effective filling of the remaining bath and can smear or remove sharp unit-filling critical features (Alhyder et al., 2024).

One-dimensional lattice Bose polarons display an additional instability structure. In the variational study motivated by the experiment of Fukuhara et al., stable polarons exist for all momenta when $$H_\Omega = -\mu \sum_i n_{B,i} + \Omega \sum_i r_i^2\,(n_{B,i}+n_{I,i}),$$4, whereas for weaker interactions the bound impurity-hole branch first merges with the particle-hole continuum near $$H_\Omega = -\mu \sum_i n_{B,i} + \Omega \sum_i r_i^2\,(n_{B,i}+n_{I,i}),$$5. The diagnostic signature is a sharp drop in the on-site hole density $$H_\Omega = -\mu \sum_i n_{B,i} + \Omega \sum_i r_i^2\,(n_{B,i}+n_{I,i}),$$6, and the same framework predicts stable bipolarons for $$H_\Omega = -\mu \sum_i n_{B,i} + \Omega \sum_i r_i^2\,(n_{B,i}+n_{I,i}),$$7 (Dutta et al., 2013).

Hard-core boson baths reveal a different strong-coupling asymmetry. At strong attraction and low filling, the attractive polaron retains a finite energy because the beyond-Fröhlich term survives even when mean-field and Fröhlich terms vanish, yet its residue collapses and the spectral weight is transferred into the bounded continuum. By contrast, in the same low-density limit the repulsive polaron remains coherent with $$H_\Omega = -\mu \sum_i n_{B,i} + \Omega \sum_i r_i^2\,(n_{B,i}+n_{I,i}),$$8. Near unit filling the coherence pattern is qualitatively reversed: the attractive branch can remain sharp while the repulsive branch loses spectral weight (Santiago-García et al., 2024).

6. Dynamics, extensions, and limits of validity

The lattice Bose polaron is not only a static quasiparticle but also a dynamical object. For an impurity on a one-dimensional lattice immersed in a homogeneous BEC and driven by a constant force, the dressed quasiparticle undergoes coherent Bloch oscillations together with interaction-induced drift caused by phonon radiation. In the adiabatic regime the center-of-mass trajectory directly measures the renormalized dispersion,

$$H_\Omega = -\mu \sum_i n_{B,i} + \Omega \sum_i r_i^2\,(n_{B,i}+n_{I,i}),$$9

while the drift velocity obeys a non-Esaki-Tsu scaling. In the heavy-polaron limit the analytic result yields $t=\tilde t$0 at small force, implying Ohmic behavior only for $t=\tilde t$1 and sub-Ohmic behavior for $t=\tilde t$2 (Grusdt et al., 2014).

Multi-band and finite-density generalizations show that the concept extends beyond a single impurity branch. In a two-band Fröhlich model, each impurity band acquires its own phonon dressing, its own renormalized hopping $t=\tilde t$3, and its own polaron shift $t=\tilde t$4, while residual incoherent coupling yields a polaron-renormalized inter-band relaxation rate rather than the bare Fermi Golden Rule rate. At strong coupling the phonon clouds in the two bands differ enough that the impurity can no longer tunnel between them by phonon processes, producing inter-band self-trapping (Yin et al., 2015). In a two-component Bose-Hubbard mixture, a variational polaron transformation generates reduced hopping and mediated interactions for an entire heavy-species lattice gas rather than only for a single impurity, and at $t=\tilde t$5 the interaction with the light superfluid favors the superfluid phase of the heavy species (Benjamin et al., 2014).

Several papers also identify directly relevant experimental settings. Two-dimensional Bose-Hubbard polarons in a harmonic trap exhibit the usual “wedding-cake” structure of Mott plateaus and superfluid shells, and the compressible shells act as effective particle reservoirs, so impurity-particle or impurity-hole binding can occur even for relatively weak coupling in trapped systems (Hartweg et al., 1 Aug 2025). Spectroscopic probes emphasized in the strong-coupling and hard-core-bath literature include RF spectroscopy, Ramsey spectroscopy, momentum-resolved spectroscopy, and direct access to $t=\tilde t$6; quantum gas microscopy and site-resolved density profiles provide complementary spatial information (Alhyder et al., 2024, Santiago-García et al., 2024). The hard-core boson setting has also been connected to moiré materials, where optical excitations may be described in terms of hard-core bosons (Santiago-García et al., 2024).

The limits of validity are equally explicit. QGW-based impurity calculations commonly assume a single impurity, weak or moderate impurity-bath coupling, zero temperature, and negligible modification of the bath ground state by the impurity; extremely close to criticality or in lower dimensions, nonlocal correlations may be stronger than the approximation captures (Colussi et al., 2022). Bogoliubov-based lattice-polaron descriptions require a weakly interacting homogeneous BEC and neglect strong bath back-action and higher-order phonon correlations (Grusdt et al., 2014, Yin et al., 2015). Chevy-like single-excitation truncations omit higher excitation sectors, although the hard-core constraint makes that truncation more reliable than in a soft bosonic bath (Santiago-García et al., 2024). The operator-valued flow equation method is designed for the subsonic regime and becomes unreliable near the attractive-repulsive transition or in supersonic regions where higher operator structures proliferate (Christ et al., 2024).

Taken together, these results show that “lattice Bose polaron” does not denote a single universal quasiparticle. In weak-BEC realizations it is a dressed Bloch particle with a renormalized band and radiative transport; in hard-core and Bose-Hubbard environments it is a probe of strong correlations, Mott physics, universality class, and local defect binding. This suggests that the decisive diagnostic is not merely whether the impurity is dressed, but which bath modes, which lattice constraints, and which ensemble conditions control the dressing cloud (Colussi et al., 2022, Alhyder et al., 2024).