New results for the free energy of a polaron in a strong magnetic field are presented. The method used here is the extended version of Feynman's pathintegral formalism in which the most general quadratic form containing an infinite number of variational parameters is employed to simulate the variational trial action. A set of nonlinear integral equations is established to determine the upper bound for the free energy in the entire range of electron-phonon coupling constants, temperatures and magnetic fields. Analytical solutions are provided for some limiting cases of interest, and are compared critically with existing calculations of the ground-state energy at zero temperature. The free energy thus obtained is exact to the lowest order in the coupling constant and provides the good interpolation up to the strong coupling regime.
A detailed numerical analysis is made of the analytic results presented in paper I. Numerical results are presented for the mass of the Feynman polaron, parallel (${M}_{\ensuremath{\parallel}}$) and perpendicular (${M}_{\ensuremath{\perp}}$) to the magnetic field, and for the following thermodynamic quantities: the magnetization, the susceptibility, the internal energy, the entropy, and the specific heat. Those quantities are studied for different values of the electron-phonon coupling ($\ensuremath{\alpha}$), temperature (
T), and magnetic field strength (
H). We found that an ideal gas of polarons undergoes a phase transition. In the physical parameter space ($\frac{1}{\ensuremath{\alpha}},\frac{1}{\mathcal{H}},T$) the points of first-order phase transition define a two-dimensional surface which is circumscribed by a line of second-order phase transitions. At the transition point the polaron transforms in the direction perpendicular to the magnetic field, and with increasing magnetic field strength, from a polaron state (${M}_{\ensuremath{\parallel}}\ensuremath{\approx}{M}_{\ensuremath{\perp}}$) to an almost free Landau state (${M}_{\ensuremath{\parallel}}>>{M}_{\ensuremath{\perp}}\ensuremath{\sim}1$). This transition can be viewed as a magnetic-field-induced two-dimensional stripping of the polaron. The experimental consequences of this phase transition on the thermodynamic quantities and the magneto-optical absorption spectrum are discussed.
The ground-state energy of the free polaron is calculated, using the Fock approximation. A new model Hamiltonian involving an electron in a harmonic potential of cylindrical symmetry in the presence of a variational magnetic field, internal in origin, is used. This magnetic field, which has no physical existence, introduces the possibility of symmetry breaking. This approach is shown to be equivalent to that of Peters and Devreese (1982), who used path integrals to study the properties of the polaron in a magnetic field, if the variational parameters of their model action are chosen to match those of the author's model and if the magnetic field in the model action is used as a variational parameter whose origin is internal. Using that approach, a strong-coupling ground-state energy much lower than the usual one is found (E 0 =-0.1658 alpha 2 ). It is shown that such a low value cannot be obtained from the adiabatic approximation because a model spectrum having a degenerate ground state has been used. His results call into question the assumption that the energies obtained from the Fock approximation or from the Feynman integrals are upper bounds to the polaron ground-state energy in the case of model actions or Hamiltonians involving magnetic fields.
We present a novel path integral Monte Carlo scheme to solve the Fröhlich polaron model. At intermediate and strong electron-phonon couplings, the polaron self-trapping is properly taken into account at the level of an effective action obtained by a preaveraging procedure with a retarded trial action. We compute the free energy at several couplings and temperatures in three and two dimensions. Our results show that the accuracy of the Feynman variational upper bound for the free energy is always within a few percent. The method can be generalized to the N-polarons case.
The Feynman inequality
EG\ensuremath{\le}
Etrial+${\mathrm{lim}}_{\mathrm{\ensuremath{\beta}}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$〈S-
Strial〉/\ensuremath{\beta} for path integrals provides a powerful upper bound on the ground-state energy
EG of a large variety of systems.
Etrial is the ground-state energy of some trial system with action
Strial for imaginary values of the time variable, and S is the action (also expressed in imaginary time variables) of the system under study. \ensuremath{\beta}=1/
kBT, where
kB is the Boltzmann constant and T the temperature. However, the Feynman inequality is not a priori justified for a system in a magnetic field, because imaginary terms subsist in the action also after transforming to imaginary time variables. Replacing or extending this inequality when magnetic fields are present has therefore been a long-standing problem. In the present paper we solve this problem. We first derive an inequality, providing an upper bound for the ground-state energy, that is valid even in the case of a nonzero magnetic field,
EG\ensuremath{\le}
Etrial+〈\ensuremath{\infty}\ensuremath{\Vert}scrT{
markUtrial (\ensuremath{\infty},-\ensuremath{\infty})[V(0)-
Vtrial(0)]}-\ensuremath{\infty}〉,for a system with Hamiltonian
H0+V. T time-ordering operator, and
Utrial is the time evolution operator of a trial system with Hamiltonian
H0+
Vtrial in the interaction representation, with the interactions V(t) and
Vtrial(t) switched on adiabatically. Because of the time ordering, retardation effects are also properly taken into account. The contribution of the magnetic field is included in the unperturbed Hamiltonian
H0. If the time-dependent integrands occurring in the matrix element in the right-hand side of our generalized inequality satisfy certain analyticity conditions in the complex-time plane, this inequality reduces to the Feynman inequality for path integrals. If these analyticity conditions are not satisfied, our generalized inequality may introduce supplementary terms
EDB in the right-hand side of the Feynman upper bound,
EG\ensuremath{\le}
Etial+${\mathrm{lim}}_{\mathrm{\ensuremath{\beta}}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$〈S-
Strial〉/\ensuremath{\beta} +
EDB,because different branch lines or singularities have to be taken into account in the transformation to imaginary time variables. As an important illustration, our generalized inequality is applied to the problem of the Fr\"ohlich polaron in a magnetic field. From the generalization of the Feynman an inequality derived in the present paper, we determine the conditions to be imposed on the variational parameters in the trial action, such that the Feynman upper bound in its original form remains valid for a polaron in a magnetic field. Some limiting cases are studied analytically to illustrate the relevance of our additional constraints on the variational parameters of the trial system. In the free-particle limit and for a particular value of one of the variational parameters, we explicitly derive the contributions from the branch lines in the complex-time plane which arise if these additional constraints are not satisfied.
In the expansion of the ground-state energy of a polaron in a weak magnetic field, the zeroth-order term is the polaron self-energy, while the first-order term is inversely proportional to the polaron effective mass. The effective mass so obtained is exactly equivalent to the freepolaron effective mass as defined by Fr\"ohlich. This equivalence principle is used to approximate the polaron effective mass by employing an approximate expression for the ground-state energy of a polaron in a weak magnetic field obtained by applying Feynman's path-integral variational method. The resultant polaron effective mass is found to be higher than Feynman's result by less than 1%.