The ground-state electron density of a polaron bound to a Coulomb potential in a homogeneous magnetic field—the transverse coordinates integrated out—converges pointwise and weakly in the strong magnetic field limit to the square of a hyperbolic secant function.
A non-relativistic hydrogen atom in a strong magnetic field interacting with the quantized longitudinal optical modes of an ionic crystal is considered within the framework of Fröhlich’s 1950 polaron model.26 Starting with Platzman’s variational treatment in 1962, the polaron hydrogenic atom has been of interest for describing an electron bound to a donor impurity in a semiconductor.49 Its first rigorous examination, however, came much later in 1988 from Löwen who disproved several longstanding claims about a self-trapping transition.45
A study of the polaron hydrogenic atom in strong magnetic fields was initiated by Larsen in 1968 for interpreting cyclotron resonance measurements in InSb.35 The model has since been considered in formal analogy to the hydrogen atom in a magnetic field though the latter was understood rigorously again much later in 1981 by Avron et al. who proved several properties including the non-degeneracy of the ground state.3 Whether or not these hydrogenic properties indeed persist when a coupling to a quantized field is turned on remains to be seen.
Polarons are the simplest quantum field theory models, yet their most basic features such as the effective mass, ground-state energy, and wave function cannot be evaluated explicitly. And, quite unfortunately, while several successful theories have been proposed over the years to approximate the energy and effective mass of various polarons, usually they are built entirely on unjustified, even questionable, Ansätze for the wave function (see Refs. 15, 26, and 48); the only exceptions are the rigorous treatments1,2,6–8,13,17,19–23,27,30,42,43,45–47,53,57 including the recent work on the effective mass.14,38,39 The paper provides now for the first time an explicit description of the wave function.
For the polaron hydrogenic atom in a homogeneous magnetic field, its ground-state electron density in the magnetic-field direction is shown to converge in the strong field limit to the square of a hyperbolic secant function: a sharp contrast to the paradigmatic Gaussian variational wave functions (see Ref. 58 and also Ref. 54 and references therein). The explicit limiting function is realized as a density of the minimizer of a one-dimensional problem with a delta-function potential describing the second leading-order term of the ground-state energy (cf. Refs. 5, 17, and 41).
II. MODEL AND MAIN RESULT
The Fröhlich model is defined by the Hamiltonian
acting on the Hilbert space , where is a symmetric phonon Fock space over . The creation and annihilation operators for a phonon mode and ak act on and satisfy . The energy of the phonon field is described by the operator . The kinetic energy of the electron is described by the operator acting on , where is the two-dimensional Landau Hamiltonian with the magnetic vector potential corresponding to a homogeneous magnetic field of strength B ≥ 0 in the x3-direction; the transverse coordinates are denoted by x⊥ = (x1, x2). Furthermore, inf spec HB = B. The parameters α ≥ 0 and β > 0 denote the strengths of the Coulombic electron–phonon interaction and the localizing Coulomb potential; the coupling function is proportional to the square root of the Fourier transform of the Coulomb interaction. The ground-state energy is
where is a magnetic Sobolev space of order one. A ground state exists since −i∇ − β|x|−1 has a negative energy bound state in .29
Unlike previous treatments, here, the arguments remain valid for all values of the parameters α ≥ 0 and β > 0. First, the large B asymptotics of the ground-state energy is derived; the main result is given as Theorem 2.2. Since the pioneering work of Larsen, the model has been considered only in the perturbative regime α ≪ β, and the ground-state energy E0(B) has been approximated as the hydrogenic energy,
with a supposedly small correction from the electron–phonon interaction. The large B asymptotics of the hydrogenic energy was derived rigorously in 1981 by Avron et al.3 using ideas from Refs. 9 and 55,
with γE being the Euler–Mascheroni constant, and the expansion can be carried out to arbitrary order. The first three terms are understood heuristically: For large B, the electron is tightly bound in the transverse plane to the lowest Landau orbit while localized in the magnetic-field direction by a one-dimensional effective Coulomb potential that behaves to leading order like a delta well of strength β ln(B/(ln B)2)12,51,52 [see (4.1) and Appendix B below]. The electron motion is effectively one-dimensional (cf. Ref. 11). The second and third leading-order terms describe the dominant asymptotic behavior of the ground-state energy of this one-dimensional electron confined along the magnetic field. The pronounced anisotropy in the system is reflected by the characteristic length scales of the electron density in the transverse and the magnetic-field direction and 1/ln B, respectively.
The above hydrogenic heuristics still apply when a coupling to the phonon field is introduced, i.e., α > 0. For large B, the phonons cannot follow the electron’s rapid motion in the transverse plane and so resign themselves to dressing its entire Landau orbit: not only is the electron again localized in the magnetic-field direction by the one-dimensional effective Coulomb potential but also the electron–phonon coupling function is now proportional to the square root of the Fourier transform of the same effective Coulomb interaction [cf. Refs. 34 and 56 and property (k) in Ref. 51]; the system behaves as a one-dimensional strongly coupled polaron to the leading order with interaction strength α ln(B/(ln B)2) confined along the magnetic field by a delta well of strength β ln(B/(ln B)2), i.e., in the effective one-dimensional model, the electron–phonon coupling is mediated by the magnetic field. The analogous large B asymptotics of the polaron hydrogenic energy is derived to second order.
Here, the second leading-order term describes the dominant asymptotic behavior of the ground-state energy of the effective one-dimensional strongly coupled polaron confined along the magnetic field. It is evaluated explicitly by minimizing a nonlinear functional. Furthermore, the cross term in (2.6) indicates that for large B, the effect of the electron–phonon interaction is not perturbative.
The large B asymptotics for the polaron hydrogenic energy is argued differently from the proof of (2.3) given by Avron et al. and generalizes the result of Frank and Geisinger who proved (2.4)–(2.6) when β = 0 using upper and lower bounds to the ground-state energy.17 Their upper bound is established with a trial wave function. Their lower bound is established by showing that the Hamiltonian when restricted to the lowest Landau level is bounded from below in the sense of quadratic forms by an essentially one-dimensional strong-coupling Hamiltonian; the strategy from Ref. 42 is then used to arrive at the nonlinear minimization problem for the second leading-order term along with lower order error terms (cf. Ref. 28).
For proving Theorem 2.1, Frank and Geisinger's strategy in Ref. 17 applies mutatis mutandis. Here, the argument for the upper bound is simplified using the effective Coulomb potential, given in (4.1), which plays an essential role in the proof of (2.3) by Avron et al. but is conspicuously absent from Ref. 17. Here, the argument for the lower bound uses the bathtub principle37 to extract a delta function from the Coulomb potential in the lowest Landau level; just like in Ref. 17, the error terms are bounded by , but—and this has been demonstrated recently in Ref. 24 for a strongly coupled polaron—the error terms in the lower bound should be much smaller. Moreover, the upper bound suggests that the third leading-order term again analogously to its hydrogenic counterpart in (2.3) is .
The expansion in (2.4) should be carried out to higher order. I conjecture the first six leading-order terms behave analogously to their hydrogenic counterparts in (2.3): these should describe the leading asymptotics for the minimization of a nonlinear functional arising naturally in the proof of the upper bound and given below in (4.3), which is a classical approximation to the Fröhlich model of the strongly coupled one-dimensional polaron confined along the magnetic field: in this classical approximation, the electron–phonon interaction is replaced with the one-dimensional effective Coulomb self-interaction of the electron. The second leading-order term above arises from this classical approximation by estimating the one-dimensional effective Coulomb interaction as a delta interaction of strength ln B. Furthermore, in the seventh leading-order term, there should be an order-one quantum correction to the classical approximation (cf. Refs. 24 and 32). These higher-order asymptotics should be provable with better control of the error terms when arguing the lower bound (cf. Ref. 24) and by making full use of the one-dimensional effective Coulomb potential: the electron–phonon interaction of the one-dimensional Hamiltonian that is derived both here and in Frank and Geisinger's proof of the lower bound is described using an artificial coupling function, given in (5.9) below, when instead it should really be argued that the coupling is proportional to the square-root of the Fourier transform of the effective Coulomb potential in (4.1); then, the strategy from Ref. 42 can be used to arrive at the classical approximation in (4.3) now as a lower bound.
The two-term asymptotics of the ground-state energy achieved in Theorem 2.1 suffices for arguing the main result.
For proving Theorem 2.2, the strategy is to add to the Hamiltonian ϵ times the one-dimensional potential W scaled appropriately in the magnetic-field direction (cf. Refs. 4, 18, 33, 40, 41, and 44). For
and Eϵ(B) being the corresponding ground-state energy, it is argued vis-à-vis Theorem 2.1 that
By the variational principle, , and the expectation value on the right-hand side evaluates to
Then, for ϵ > 0,
and taking the limit B → ∞, by Theorem 2.1 and (2.9),
For ϵ < 0, the above inequality is reversed with “lim sup” replaced by “lim inf.” Hence, the theorem follows if the quotient on the left-hand side has a limit as ϵ → 0. The map is differentiable at ϵ = 0 for all W if and only if the one-dimensional problem for the energy in (2.5) admits up to the complex phase a unique minimizer: Uniqueness is established by explicitly solving the corresponding Euler–Lagrange equation (cf. Ref. 41) and by the variational principle and a compactness argument,
In Sec. III, the differentiation of the one-dimensional energy (2.11) is proved as Theorem 3.3. In Sec. IV, an upper bound to the ground-state energy is proved as Theorem 4.1. In Sec. V, a lower bound to the ground-state energy is proved as Theorem 5.1, and Theorem 2.1 follows from Theorems 4.1 and 5.1. In Sec. VI, the main result Theorem 2.2 is proved.
III. DIFFERENTIATING THE ONE-DIMENSIONAL ENERGY
The one-dimensional problem for the energy in (2.5) is denoted as
In the following, Lemmas 3.1, 3.2, and Theorem 3.3 apply for all α ≥ 0 and β > 0.
When α = 0, the corresponding minimizer in (3.2) is .
If is a minimizing sequence for , then converges in to the minimizer ϕ0 given in (3.2).
IV. UPPER BOUND TO THE GROUND-STATE ENERGY
Theorem 4.1 will follow from Lemmas 4.2 and 4.3.
V. LOWER BOUND TO THE GROUND-STATE ENERGY
The Proof of Theorem 5.1 is given at the end of Subsection V B.
A. Extracting a delta-function potential
Below in Lemma 5.2, the Coulomb potential restricted to the lowest Landau level is approximated as a delta well of strength ln(B/(ln B)2). First, some remarks about the notation are in order. For , its electron density in the magnetic-field direction is denoted as , i.e.,
By Hölder’s inequality, and . Furthermore, the integral operator acting on with kernel
is the projection onto the lowest Landau level, i.e., the ground state of the Landau Hamiltonian HB, and . The operators and acting on and , respectively, are also denoted .
Lemma 5.2 is used to prove the following corollary.
B. Restricting to the lowest Landau level
Below in Lemma 5.8, a Hamiltonian restricted to the lowest Landau level is derived as a lower bound. First, an ultraviolet cutoff is needed. With and , the cutoff Hamiltonian is denoted as
If , then .
The lemma follows from Lemma 5.1 in Ref. 17.□
Lemma 5.4 is used to prove the following lemma.
Lemma 5.5 and Corollary 5.3 from Subsection V A are used to prove the following lemma.
The following observation is immediate from Theorem 4.1: For every and large B, there exist wave functions satisfying
The estimate in (5.8) and Lemma 5.6 are used to prove the following corollary.
The corollary follows from Lemma 5.6.□
Corollary 5.7 and Lemma 5.4 are used to prove Lemma 5.8.
The Hamiltonian realized as a lower bound in Lemma 5.8 is estimated from below.
Proposition 5.9 and Lemma 5.8 are used to prove Theorem 5.1.
Proposition 5.9 is proved in Subsection V C.
C. Proof of Proposition 5.9
1. Reduction to one dimension
Below in Lemma 5.10, a one-dimensional Hamiltonian is derived as a lower bound. In Ref. 17, the authors consider a one-dimensional Hamiltonian with and ,
acting on with and κ as in the statement of Proposition 5.9, and
satisfying and for .
2. Localization and decomposition
Below in Lemma 5.11, the one-dimensional Hamiltonian is bounded from below by a Hamiltonian describing an electron interacting with only finitely many modes. In Ref. 17, the authors decompose the mode space into intervals, indexed with b, each of length , and consider for and 0 < γ < 1 the block Hamiltonian
with kb being a mode in block b and the block creation and annihilation operators and acting on , where
Furthermore, localizing the electron in the x3-direction on an interval of length J, the following lemma is proved.
The lemma follows from Lemma 6.5 in Ref. 17.□
3. Error estimates
The block Hamiltonian is bounded from below. Similarly, as in Refs. 17, 28, and 42 representing the block creation and annihilation operators by coherent state integrals and completing the square, it follows for a suitably chosen kb that
will be estimated from below in Lemma 5.12. Combining (5.10) with Lemma 5.11,
Now, it follows from Lemma 5.10 that for some constant C > 0,
The argument for Lemma 5.2 from Subsection V A is used to prove Lemma 5.12.
From Lemma 5.12 and further assuming that
it can be seen that there is a constant C > 0 such that
It is shown in Ref. 17 choosing and that
Now choosing L = 1/ln B and ϵ = 1/ln B, since for B ≥ C,
thereby concluding the Proof of Proposition 5.9.
The above arguments apply mutatis mutandis (see also Ref. 31).□
VI. PROOF OF THEOREM 2.2
This paper is based on the work supported by the NSF (Grant No. DMS-1600560). I thank Michael Loss for several valuable suggestions, Rupert Frank for the general strategy used in the Proof of Theorem 2.2, and the referee for helpful comments about the presentation.
Data sharing is not applicable to this article as no new data were created or analyzed in this study.
APPENDIX A: COMPACTNESS OF MINIMIZING SEQUENCES
In the following, Theorem A.1 applies for all α ≥ 0 and β > 0.
If a sequence satisfies with the functional as given in (3.1), then there exists a subsequence and some such that , , and as k → ∞.
APPENDIX B: BOUND ON THE EFFECTIVE COULOMB POTENTIAL
Recall the effective Coulomb potential from (4.1).
The corollary follows from Lemma B.1.□
Now, recall the potential from (5.5).