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arXiv:2311.12094 · Physical review. B./Physical review. B

Monte Carlo solver and renormalization of Migdal-Eliashberg spin chain

Yang-Zhi Chou, Zhentao Wang, S. Das Sarma

Published Feb 20, 2024• 0 citations
arXivHTMLPDF

Abstract

Motivated by the recently developed classical spin model for the Migdal-Eliashberg theory, we develop numerical and analytical methods based on this spin-chain representation and apply these methods to the Bogoliubov-Tomachov-Morel-Anderson pairing potential, which incorporates the phonon-mediated attraction and Coulomb repulsion. We show that the Monte Carlo method with heat bath updates can efficiently obtain the gap functions even for the situations challenging for the iterative solvers, suggesting an unprecedented robust approach for solving the full nonlinear Migdal-Eliashberg theory. Moreover, we derive the renormalization of all the couplings by tracing out the high-frequency spins in the partition function. The derived analytical renormalization equations produce the well-known ${\ensuremath{\mu}}^{*}$ effect for the Bogoliubov-Tomachov-Morel-Anderson pairing potential and can be generalized to other superconductivity problems. We further point out that several interesting features (e.g., sign changing in the frequency-dependent gap function) can be intuitively understood using the classical spin-chain representation for Migdal-Eliasherg theory. Our results show the advantage of using the spin-chain representation for solving Migdal-Eliashberg theory and provide new ways for tackling general superconductivity problems.

DOI
10.1103/physrevb.109.054514

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Motivated by the recently developed classical spin model for the Migdal-Eliashberg theory, we develop numerical and analytical methods based on this spin-chain representation and apply these methods to the Bogoliubov-Tomachov-Morel-Anderson pairing potential, which incorporates the phonon-mediated attraction and Coulomb repulsion. We show that the Monte Carlo method with heat bath updates can efficiently obtain the gap functions even for the situations challenging for the iterative solvers, suggesting an unprecedented robust approach for solving the full nonlinear Migdal-Eliashberg theory. Moreover, we derive the renormalization of all the couplings by tracing out the high-frequency spins in the partition function. The derived analytical renormalization equations produce the well-known \ensuremathμ∗{\ensuremath{\mu}}^{*}\ensuremathμ∗ effect for the Bogoliubov-Tomachov-Morel-Anderson pairing potential and can be generalized to other superconductivity problems. We further point out that several interesting features (e.g., sign changing in the frequency-dependent gap function) can be intuitively understood using the classical spin-chain representation for Migdal-Eliasherg theory. Our results show the advantage of using the spin-chain representation for solving Migdal-Eliashberg theory and provide new ways for tackling general superconductivity problems.
We present quantum simulation studies for a system of strongly correlated fermions coupled to local anharmonic phonons. The Monte Carlo calculations are based on a generalized version of the Projector Quantum Monte Carlo Method allowing a simultaneous treatment of fermions and dynamical phonons. The numerical simulations yield exact results in the electron-phonon parameter regime relevant for high-Tc superconductivity, which is not accessible by perturbative methods like Eliashberg-Theory. The class of electron-phonon-models covered in the simulations describes superconductivity exhibiting several features of the new high-Tc materials.
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Understanding the properties of strongly correlated materials is a tough work since most of the times analytical tools cannot be applied due to the complicated nature of the problem. Here we apply quantum Monte Carlo techniques in order to tackle with one of the most important and unknown phenomena of many materials: the mechanism of superconductivity. We study the Hubbard-Holstein model which is one of the most simple theoretical models including strong correlation and electron-phonon coupling. By using twist-averaging boundary conditions we eliminate finite-size errors in the Monte Carlo simulations. With this useful tool we first solve the Hubbard-Holstein model by variational Monte Carlo by reporting the phase diagram of the model for different phonon frequencies at half filling. Then we investigate the phase separation away from half filling. Finally we try to attack the problem with an essentially unbiased method based on the auxiliary field quantum Monte Carlo technique and accelerated Langevin dynamics. By curing the sign problem via Cauchy integration in the complex plane where the auxiliary fields have been defined, we report the effect of the electron-phonon coupling on some observables such as magnetization, spin and charge structure factors. We show preliminary results that are already meaningful to understand the nature of the transition between magnetism and charge order in the model at half filling.
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We present a method for solving impurity models with electron-phonon coupling, which treats the phonons efficiently and without approximations. The algorithm is applied to the Holstein-Hubbard model in the dynamical mean field approximation, where it allows access to strong interactions, very low temperatures, and arbitrary fillings. We show that a renormalized Migdal-Eliashberg theory provides a reasonable description of the phonon contribution to the electronic self-energy in strongly doped systems, but fails if the quasiparticle energy becomes of order of the phonon frequency.