Green’s Function of a Dressed Particle

Abstract

We present a new, highly efficient yet accurate approximation for the Green's functions of dressed particles, using the Holstein polaron as an example. Instead of summing a subclass of diagrams (e.g. the non-crossed ones, in the self-consistent Born approximation (SCBA)), we sum all the diagrams, but with each diagram averaged over its free propagators' momenta. The resulting Green's function satisfies exactly the first six spectral weight sum rules. All higher sum rules are satisfied with great accuracy, becoming asymptotically exact for coupling both much larger and much smaller than the free particle bandwidth. Possible generalizations to other models are also discussed.PACS numbers: 72.10.Di, 63.20.Kr One of the most fundamental problems in both highenergy and condensed matter physics is to understand what happens when a particle couples to an environment, in particular what are the properties of the resulting object, consisting of the bare particle dressed by a cloud of excitations. This type of problem arises again and again as couplings to new kinds of environments are studied.The most desirable quantity to know is the Green's function G( k, ω) of the dressed particle -its poles mark the eigenspectrum, while the associated residues contain information on the eigenfunctions. Moreover, the spectral weight A( k, ω) = − 1 π ImG( k, ω) can be directly measured experimentally using Angle-Resolved Photoemission Spectroscopy [1]. Recently, such work has reignited a debate on whether the carriers in high-T c cuprates are polarons, that is, electrons dressed by phonons [2].G( k, ω) is the sum of an infinite number of diagrams corresponding to an expansion to all orders in the coupling strength [3]. Diagrammatic Monte Carlo (DMC) can perform the numerical summation of all diagrams [4]. Other ways to find G( k, ω) are from exact diagonalizations (ED) of small systems, variational methods, Density Matrix Renormalization Group in one-dimension, etc [5,6,7,8]. However, these methods require considerable computational resources, are time consuming, and often limit themselves to finding only the low-energy properties, such as the ground-state energy.To our knowledge, there are only two easy-to-estimate approximations for G( k, ω). One is the the SCBA which consists in summing only the non-crossed diagrams. Because the percentage of diagrams kept decreases fast with increasing order, SCBA fails badly at strong couplings. The other approximation, obtained from a modified Lang-Firsov (MLF) approach [9], is exact both for zero coupling and for zero bandwidth, however results for finite bandwidth/coupling are rather poor (see below).In this Letter we find a new approximation for G( k, ω), which is as easy to estimate as SCBA and MLF, but is highly accurate over most of the parameter space. We validate this both by comparison against numerical results, and by investigating its sum rules. Most of the discussion here is limited to the Holstein model, for which many numerical results are available. Possible generalizations for...

The Transient Localization Scenario for Charge Transport in Crystalline Organic Materials

“…(18). Studies have been performed to assess the validity of such approximation scheme, both at zero temperature via numerical Monte Carlo calculations combined with an analytical expansion in powers of 1/λ, [86] and at finite temperature using a generalization of the so-called momentum-average approximation [87] to non-translationally invariant systems. [48] Such studies have shown that, as was originally devised by Holstein, the band-narrowing approximation can only be applied if the vibrational frequencies are much larger that the unrenormalized electron bandwidth (the so called anti-adiabatic limit, J ω 0 ), because only in this case the phonon cloud can instantaneously re-arrange to follow the motion of the carriers as encoded in Eq.…”

Properties and temperature evolution of the spectrum of localized quasi-particles interacting with polarization phonons in two models

“…For example, within the new diagrammatic Monte Carlo method [8][9][10], the important problems of high-temperature superconductivity [11] have been solved, and the excited states of structure and structureless Fröhlich polaron [1,2] have been studied exactly and consistently. Recently in [12][13][14][15], a new powerful approach for the investigation of different quasi-particles interacting with phonons in homogeneous and inhomogeneous systems has been developed. It is based on the non-perturbative method of momentum average approximation.…”

Single-polaron properties of the one-dimensional breathing-mode Hamiltonian

“…30−35 Analytic approximations have also progressed over the years. 36,37,38,39 For some materials, a more appropriate model is provided by the breathing-mode Hamiltonian. For example, consider a half-filled 2D copper-oxygen plane of a parent cuprate compound.…”