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AIP Advances

Renormalization effects and phonon density of states in high temperature superconductors

Vinod Ashokan, B. D. Indu

Published Feb 1, 2013• 24 citations
Publisher

Abstract

Using the versatile double time thermodynamic Green's function approach based on many body theory the renormalized frequencies, phonon energy line widths, shifts and phonon density of states (PDOS) are investigated via a newly formulated Hamiltonian (does not include BCS type Hamiltonian) that includes the effects of electron-phonon, anharmonicities and that of isotopic impurities. The automatic appearance of pairons, temperature, impurity and electron-phonon coupling of renormalized frequencies, widths, shifts and PDOS emerges as a characteristic feature of present theory. The numerical investigations on PDOS for the YBa2Cu3O7 − δ crystal predicts several new feature of high temperature superconductors (HTS) and agreements with experimental observations.

DOI
10.1063/1.4791762

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Using the versatile double time thermodynamic Green's function approach based on many body theory the renormalized frequencies, phonon energy line widths, shifts and phonon density of states (PDOS) are investigated via a newly formulated Hamiltonian (does not include BCS type Hamiltonian) that includes the effects of electron-phonon, anharmonicities and that of isotopic impurities. The automatic appearance of pairons, temperature, impurity and electron-phonon coupling of renormalized frequencies, widths, shifts and PDOS emerges as a characteristic feature of present theory. The numerical investigations on PDOS for the YBa2Cu3O7 − δ crystal predicts several new feature of high temperature superconductors (HTS) and agreements with experimental observations.
The renormalized mode frequencies are obtained with the help of quantum dynamical approach of many body phonon Green’s function technique via a general Hamiltonian (excluding BCS Hamiltonian) including the effects of phonons and electrons, anharmonicities and electron-phonon interactions. The numerical estimates have been carried out to study the renormalized mode frequency of high temperature cuprate superconductor (HTS) YBa2Cu3O7-δ using modified Born-Mayer-Huggins interaction potential (MBMHP) best applicable to study the dynamical properties of all HTS.
The evaluation of one electron thermodynamic Green’s functions using the equation of motion technique of quantum dynamics via newly formulated Hamiltonian and using Dyson’s equation approach. This involves approximation free approach and different cooper pairs are emerged automatically in the system and stands as an ab-initio approach. The detailed description of enhancement in the electron density of states (EDOS) for high temperature superconductors has been investigated. The investigated expressions of EDOS in the new framework are found responsible to describe a large number of dynamical properties of high temperature superconductors. The temperature dependence of EDOS has been found as a unique feature of the theory, which certainly becomes the outcome of the anharmonic interactions. The presence of electron-phonon interaction parameter in each term is an additional and new feature of the theory.
We have recently developed a three-temperature model to simulate the time evolution of the electron and phonon temperatures in high-temperature superconductors. This is the first model that is valid in the superconducting state. Based on this model, we calculated the time-resolved spectral function via the double-time Green's functions. Our calculations show that the theory not only reveals the familiar dip-hump structure in the normal state, but also a new signature of the electron-phonon coupling in the superconducting state. We further show that this signature can only be observed when the phonons are pumped directly. In this proceedeing, we provide detailed derivation of our theory.
The pairon spectral function for the high-[Formula: see text] cuprate superconductor [Formula: see text] has been evaluated through renormalized pairon frequency, analogous to the frequency (energy) of excited pairons. The exercise has been carried out with the help of many-body quantum dynamical Green’s function theory via a model Hamiltonian that includes the effects of the electrons, phonons, impurities and anharmonic fields. The analysis of the obtained results reveals that renormalized pairon frequency progressively increases up to the critical temperature [Formula: see text] and afterwards, instead of dropping to zero, it decreases smoothly with increasing temperature, capturing the essence of the pseudogap in high-[Formula: see text] cuprates. A further analysis of temperature-dependent spectral function also establishes the presence of the pseudogap above [Formula: see text]. The spectral function displays a conscious behavior with the variation of electron–phonon coupling constant ([Formula: see text]).
We develop a three-temperature model to simulate the time dependence of the electron and phonon temperatures in high-temperature superconductors displaying strong anisotropic electron-phonon coupling. This model not only takes the tight-binding band structure into account but also is valid in the superconducting state. Based on this model, we calculate the time-resolved spectral function via the double-time Green's functions. We find that the dip-hump structure evolves with the time delay. More interestingly, an unusual structure is obtained when the phonons are pumped directly. This finding may serve as a direct evidence for electron-vibration mode coupling.
Abstract Green-function techniques are among the most versatile tools for investigating quantum many-body systems at finite temperature. Yet, standard textbook treatments often fragment the formalism—separating bosons and fermions, systems with massless and massive particles, or restricting proofs to special Hamiltonians—thereby obscuring the structural unity of the theory. This work develops a general, Hamiltonian-independent framework for both real- and imaginary-time Green-functions, valid for any combination of bosonic and fermionic degrees of freedom, including mixed systems such as interacting electron–phonon models. The key conceptual advance is the nature index η , which complements the conventional statistical sign ζ . While ζ encodes particle statistics, η distinguishes canonical quantization of systems with massless particles ( η = 0) from second quantization of systems with massive particles ( η = 1). This classification removes ambiguities in chemical-potential terms and unifies the treatment of photons, phonons, electrons, and their mixtures under a single notation. From this basis, we formulate generalized ensemble constructions, generalized Lehmann-representations, and a fluctuation-dissipation theorem valid for arbitrary Hamiltonians. Several results—such as the monotonicity of particle number in the chemical potential, an explicit formula for the quadratic contribution to the internal energy from the retarded Green-function, and the Hartree-contribution to the self-energy in electron–phonon systems—are either absent from, or only sparsely mentioned in, existing literature. We further introduce Green’s Chart, a visual and algebraic map that systematically connects all Green-function types through analytic continuation, Fourier-transforms, and Wick-rotations, enabling all to be generated from the Matsubara Green-function. Applications to both non-interacting and interacting models demonstrate the framework’s ability to compute densities of states, thermodynamic potentials, and spectral functions with a consistent formalism. The results provide a complete and reproducible toolkit for equilibrium many-body theory, consolidating scattered results into a unified structure suitable for both pedagogical and computational purposes.