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arXiv:2210.09208 · Physical Review Research

Chebyshev pseudosite matrix product state approach for the spectral functions of electron-phonon coupling systems

Pei-Yuan Zhao, Ke Ding, Shuo Yang

Published Apr 14, 2023• 7 citations
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Abstract

The electron-phonon $(e\text{\ensuremath{-}}\mathrm{ph})$ coupling system often has a large number of phonon degrees of freedom, whose spectral functions are numerically difficult to compute using matrix product state (MPS) formalisms. To solve this problem, we propose a practical method that combines the Chebyshev MPS and the pseudosite density matrix renormalization group (DMRG) algorithm. The Chebyshev vector is represented by a pseudosite MPS with global $U(1)$ fermion symmetry, which maps ${2}^{{N}_{p}}$ bosonic degrees of freedom onto ${N}_{p}$ pseudosites, each with two states. This approach can handle arbitrary $e$-ph coupling Hamiltonians where pseudosite DMRG performs efficiently. We use this method to study the spectral functions of the doped extended Hubbard-Holstein model in a regime of strong Coulomb repulsion. Key features of the excitation spectra are captured at a modest computational cost. Our results show that weak extended $e$-ph couplings can increase the spectral weight of the holon-folding branch at low phonon frequencies, in agreement with angle-resolved photoemission observations on one-dimensional cuprates.

DOI
10.1103/physrevresearch.5.023026

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The electron-phonon (e\ensuremath-ph)(e\text{\ensuremath{-}}\mathrm{ph})(e\ensuremath-ph) coupling system often has a large number of phonon degrees of freedom, whose spectral functions are numerically difficult to compute using matrix product state (MPS) formalisms. To solve this problem, we propose a practical method that combines the Chebyshev MPS and the pseudosite density matrix renormalization group (DMRG) algorithm. The Chebyshev vector is represented by a pseudosite MPS with global U(1)U(1)U(1) fermion symmetry, which maps 2Np{2}^{{N}_{p}}2Np​ bosonic degrees of freedom onto Np{N}_{p}Np​ pseudosites, each with two states. This approach can handle arbitrary eee-ph coupling Hamiltonians where pseudosite DMRG performs efficiently. We use this method to study the spectral functions of the doped extended Hubbard-Holstein model in a regime of strong Coulomb repulsion. Key features of the excitation spectra are captured at a modest computational cost. Our results show that weak extended eee-ph couplings can increase the spectral weight of the holon-folding branch at low phonon frequencies, in agreement with angle-resolved photoemission observations on one-dimensional cuprates.
We introduce the Chebyshev pseudosite matrix product state approach (ChePSMPS) as a solver for cluster perturbation theory (CPT), crucial for simulating spectral functions in two-dimensional electron-phonon (eee-ph) coupling systems. ChePSMPS distinguishes itself from conventional exact diagonalization solvers by supporting larger clusters, thereby significantly mitigating finite-size effects. Free from the fermion sign problem, ChePSMPS enhances its ability to explore eee-ph effects and generate high-resolution spectral functions in doped Mott insulators. We use this method to simulate the spectra for both one- and two-dimensional Hubbard-Holstein models, highlighting its superiority over other methods. Our findings validate ChePSMPS as a powerful and reliable Green's function solver. In conjunction with embedding methods, ChePSMPS emerges as an essential tool for simulating strongly correlated eee-ph coupling systems.
The appearance of certain spectral features in one-dimensional (1D) cuprate materials has been attributed to a strong, extended attractive coupling between electrons. Here, using time-dependent density matrix renormalization group methods on a Hubbard-extended Holstein model, we show that extended electron-phonon ({\it e-ph}) coupling presents an obvious choice to produce such an attractive interaction that reproduces the observed spectral features and doping dependence seen in angle-resolved photoemission experiments: diminished 3kF3k_F3kF​ spectral weight, prominent spectral intensity of a holon-folding branch, and the correct holon band width. While extended {\it e-ph} coupling does not qualitatively alter the ground state of the 1D system compared to the Hubbard model, it quantitatively enhances the long-range superconducting correlations and suppresses spin correlations. Such an extended {\it e-ph} interaction may be an important missing ingredient in describing the physics of the structurally similar two-dimensional high-temperature superconducting layered cuprates, which may tip the balance between intertwined orders in favor of uniform ddd-wave superconductivity.
Abstract The appearance of certain spectral features in one-dimensional (1D) cuprate materials has been attributed to a strong, extended attractive coupling between electrons. Here, using time-dependent density matrix renormalization group methods on a Hubbard-extended Holstein model, we show that extended electron-phonon (e-ph) coupling presents an obvious choice to produce such an attractive interaction that reproduces the observed spectral features and doping dependence seen in angle-resolved photoemission experiments: diminished 3kF3k_F3kF​ spectral weight, prominent spectral intensity of a holon-folding branch, and the correct holon band width. While extended e-ph coupling does not qualitatively alter the ground state of the 1D system compared to the Hubbard model, it quantitatively enhances the long-range superconducting correlations and suppresses spin correlations. Such an extended e-ph interaction may be an important missing ingredient in describing the physics of the structurally similar two-dimensional high-temperature superconducting layered cuprates, which may tip the balance between intertwined orders in favor of uniform ddd-wave superconductivity.
Abstract The appearance of certain spectral features in one-dimensional (1D) cuprate materials has been attributed to a strong, extended attractive coupling between electrons. Here, using time-dependent density matrix renormalization group methods on a Hubbard-extended Holstein model, we show that extended electron-phonon (e–ph) coupling presents an obvious choice to produce such an attractive interaction that reproduces the observed spectral features and doping dependence seen in angle-resolved photoemission experiments: diminished 3kF spectral weight, prominent spectral intensity of a holon-folding branch, and the correct holon band width. While extended e–ph coupling does not qualitatively alter the ground state of the 1D system compared to the Hubbard model, it quantitatively enhances the long-range superconducting correlations and suppresses spin correlations. Such an extended e–ph interaction may be an important missing ingredient in describing the physics of the structurally similar two-dimensional high-temperature superconducting layered cuprates, which may tip the balance between intertwined orders in favor of uniform d-wave superconductivity.
We show that recursively generated Chebyshev expansions offer numerically efficient representations for calculating zero-temperature spectral functions of one-dimensional lattice models using matrix product state (MPS) methods. The main features of this Chebyshev matrix product state (CheMPS) approach are as follows: (i) it achieves uniform resolution over the spectral function's entire spectral width; (ii) it can exploit the fact that the latter can be much smaller than the model's many-body bandwidth; (iii) it offers a well-controlled broadening scheme that allows finite-size effects to be either resolved or smeared out, as desired; (iv) it is based on using MPS tools to recursively calculate a succession of Chebyshev vectors $|{t}_{n}\ensuremath{\rangle}$, (v) the entanglement entropies of which were found to remain bounded with increasing recursion order nnn for all cases analyzed here; and (vi) it distributes the total entanglement entropy that accumulates with increasing nnn over the set of Chebyshev vectors $|{t}_{n}\ensuremath{\rangle}$, which need not be combined into a single vector. In this way, the growth in entanglement entropy that usually limits density matrix renormalization group (DMRG) approaches is packaged into conveniently manageable units. We present zero-temperature CheMPS results for the structure factor of spin-12\frac{1}{2}21​ antiferromagnetic Heisenberg chains and perform a detailed finite-size analysis. Making comparisons to three benchmark methods, we find that CheMPS (a) yields results comparable in quality to those of correction-vector DMRG, at dramatically reduced numerical cost; (b) agrees well with Bethe ansatz results for an infinite system, within the limitations expected for numerics on finite systems; and (c) can also be applied in the time domain, where it has potential to serve as a viable alternative to time-dependent DMRG (in particular, at finite temperatures). Finally, we present a detailed error analysis of CheMPS for the case of the noninteracting resonant level model.
We propose a density-matrix renormalization-group (DMRG) approach to study lattices including bosons. The key to the approach is an exact mapping of a boson site containing 2N{2}^{N}2N states to NNN pseudosites, each with 2 states. The pseudosites can be viewed as the binary digits of a boson level. We apply the pseudosite DMRG method to the polaron problem in the one- and two-dimensional Holstein models. Ground-state results are presented for a wide range of electron-phonon coupling strengths and phonon frequencies on lattices large enough (up to 80 sites in one dimension and up to $20\ifmmode\times\else\texttimes\fi{}20$ sites in two dimensions) to eliminate finite-size effects, with up to 128 phonon states per phonon mode. We find a smooth but quite abrupt crossover from a quasi-free-electron ground state with a slightly renormalized mass at weak electron-phonon coupling to a polaronic ground state with a large effective mass at strong coupling, in agreement with previous studies.