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Polaron models with regular interactions at strong coupling
Mathematical Physics · Submitted 17 Jun 2021
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We show that for all such models the effective mass diverges in the strong coupling limit, in all spatial dimensions.
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The abstract, sentence by sentence
We study a class of polaron-type Hamiltonians with sufficiently regular form factor in the interaction term. We investigate the strong-coupling limit of the model, and prove suitable bounds on the ground state energy as a function of the total momentum of the system. These bounds agree with the semiclassical approximation to leading order. The latter corresponds here to the situation when the particle undergoes harmonic motion in a potential well whose frequency is determined by the corresponding Pekar functional. We show that for all such models the effective mass diverges in the strong coupling limit, in all spatial dimensions. Moreover, for the case when the phonon dispersion relation grows at least linearly with momentum, the bounds result in an asymptotic formula for the effective mass quotient, a quantity generalizing the usual notion of the effective mass. This asymptotic form agrees with the semiclassical Landau--Pekar formula and can be regarded as the first rigorous confirmation, in a slightly weaker sense than usually considered, of the validity of the semiclassical formula for the effective mass.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We study a class of polaron-type Hamiltonians with sufficiently regular form factor in the interaction term. | contribution 0.80 | 0.83 | 0.03 | asserted 0.89 |
| s1 | We investigate the strong-coupling limit of the model, and prove suitable bounds on the ground state energy as a function of the total momentum of the system. | contribution 0.91 | 0.96 | 0.13 | asserted 1.00 |
| s2 | These bounds agree with the semiclassical approximation to leading order. | result 0.84 | 0.82 | 0.12 | asserted 1.00 |
| s3 | The latter corresponds here to the situation when the particle undergoes harmonic motion in a potential well whose frequency is determined by the corresponding Pekar functional. | interpretation 0.97 | 0.24 | 0.06 | asserted 0.99 |
| s4 | We show that for all such models the effective mass diverges in the strong coupling limit, in all spatial dimensions. | result 1.00 | 0.96 | 0.38 | asserted 1.00 |
| s5 | Moreover, for the case when the phonon dispersion relation grows at least linearly with momentum, the bounds result in an asymptotic formula for the effective mass quotient, a quantity generalizing the usual notion of the effective mass. | result 0.97 | 0.93 | 0.14 | asserted 1.00 |
| s6 | This asymptotic form agrees with the semiclassical Landau--Pekar formula and can be regarded as the first rigorous confirmation, in a slightly weaker sense than usually considered, of the validity of the semiclassical formula for the effective mass. | interpretation 0.94 | 0.86 | 0.31 | asserted 0.94 |
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Original abstract
We study a class of polaron-type Hamiltonians with sufficiently regular form factor in the interaction term. We investigate the strong-coupling limit of the model, and prove suitable bounds on the ground state energy as a function of the total momentum of the system. These bounds agree with the semiclassical approximation to leading order. The latter corresponds here to the situation when the particle undergoes harmonic motion in a potential well whose frequency is determined by the corresponding Pekar functional. We show that for all such models the effective mass diverges in the strong coupling limit, in all spatial dimensions. Moreover, for the case when the phonon dispersion relation grows at least linearly with momentum, the bounds result in an asymptotic formula for the effective mass quotient, a quantity generalizing the usual notion of the effective mass. This asymptotic form agrees with the semiclassical Landau--Pekar formula and can be regarded as the first rigorous confirmation, in a slightly weaker sense than usually considered, of the validity of the semiclassical formula for the effective mass.
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The model has a long history tracing back to the thirties [ref] but due to its basic character it remains a model of reference in many problems, and is still under active investigation in condensed matter physics; we refer to [ref] for an overview and further references.
New theoretical approaches to Bose polarons · The model · 1.00
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