On the Best Quadratic Approximation in Feynman’s Path Integral Treatment of the Polaron

The best quadratic approximation to the retarded polaron action due to Adamowski et al. and Saitoh is investigated numerically for a wide range of coupling constants. The non-linear variational equations are solved iteratively with an efficient method in order to obtain the ground state energy and the effective mass of the polaron. The virial theorem and expansions for small and large couplings are used to check the high accuracy of the numerical results. Only small improvements over Feynman’s (non-optimal) results are observed. For a moving polaron it is shown that the most general quadratic trial action may contain anisotropic terms which, however, do not lead to improvements for the ground state energy and effective mass. 1. The polaron problem is a non-relativistic field theory for an electron moving in a crystal and has received a lot of attention in the past decades (for reviews see [1], [2]). Among the many theoretical treatments Feynman’s approach [3] is still outstanding: he first integrated out the phonons to obtain an effective action for the electron which for large Euclidean times reads S eff = ∫ β 0 dt 1 2 ẋ − α 2 √ 2 ∫ β 0 dt dt′ e−(t−t ) | x(t)− x(t) | . (1) He then performed a variational approximation with a quadratic retarded trial action St = 1 2 ∫ β 0 dt ẋ + 1 2 ∫ β 0 dt dt′ f(t− t′) [ x(t)− x(t′) 2 , (2) choosing fF (σ) = C exp(−wσ) where C and w are two variational parameters (the strength parameter C is usually written as w(v2 −w2)/4 ). This yields one of the best analytical approximations for the ground state energy of the polaron for all values of the dimensionless coupling constant α. The best possible, rotationally invariant, quadratic trial action is obtained by replacing the exponential retardation by an arbitrary function f(σ) and was proposed by Adamowski et al. [4] and Saitoh [5]. Surprisingly for both small and strong coupling this best isotropic quadratic approximation only yields very small improvements for the ground state energy E0 E0 α→0 −→ −α− aα +O(α) (3) α→∞ −→ −ā α +O(1) . (4) One finds at small coupling aF = 0.0123457, aiso = 0.0125978 whereas the exact value is a = 0.0159196 and for strong coupling āF = āiso = 0.106103 compared to ā = 0.108513 . Probably discouraged by these analytical results the best isotropic quadratic approximation has never been investigated systematically for a whole range of couplings, in particular for intermediate coupling. It is the purpose of the present note to do this for the ground state energy and the effective mass of the polaron and to point out that an easy, efficient method exists to solve the non-linear variational equations. Although numerical results have been reported in the literature [4, 6] it will turn out that they are unreliable and considerably overestimate the improvement on Feynman’s approach. We also investigate the question whether the inclusion of anisotropic terms in the quadratic trial action leads to further improvements. This work is an outgrow of recent attempts to generalize Feynman’s polaron approach to four-dimensional field theories [7-11] in the context of the worldline formalism [12]. The nomenclature is the one used in Ref. [7]. 2. We start with the expression for the ground state energy using the quadratic trial action (2)