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arXiv:2104.09207 · SciPost Physics

Downfolding the Su-Schrieffer-Heeger model

Arne Schobert, Jan Berges, Tim Oliver Wehling, Erik G. C. P. van Loon

Published Oct 20, 2021• 0 citations
arXivHTMLPDF

Abstract

Charge-density waves are responsible for symmetry-breaking displacements of atoms and concomitant changes in the electronic structure. Linear response theories, in particular density-functional perturbation theory, provide a way to study the effect of displacements on both the total energy and the electronic structure based on a single ab initio calculation. In downfolding approaches, the electronic system is reduced to a smaller number of bands, allowing for the incorporation of additional correlation and environmental effects on these bands. However, the physical contents of this downfolded model and its potential limitations are not always obvious. Here, we study the potential-energy landscape and electronic structure of the Su-Schrieffer-Heeger (SSH) model, where all relevant quantities can be evaluated analytically. We compare the exact results at arbitrary displacement with diagrammatic perturbation theory both in the full model and in a downfolded effective single-band model, which gives an instructive insight into the properties of downfolding. An exact reconstruction of the potential-energy landscape is possible in a downfolded model, which requires a dynamical electron-biphonon interaction. The dispersion of the bands upon atomic displacement is also found correctly, where the downfolded model by construction only captures spectral weight in the target space. In the SSH model, the electron-phonon coupling mechanism involves exclusively hybridization between the low- and high-energy bands and this limits the computational efficiency gain of downfolded models.

DOI
10.21468/scipostphys.11.4.079

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Charge-density waves are responsible for symmetry-breaking displacements of atoms and concomitant changes in the electronic structure. Linear response theories, in particular density-functional perturbation theory, provide a way to study the effect of displacements on both the total energy and the electronic structure based on a single ab initio calculation. In downfolding approaches, the electronic system is reduced to a smaller number of bands, allowing for the incorporation of additional correlation and environmental effects on these bands. However, the physical contents of this downfolded model and its potential limitations are not always obvious. Here, we study the potential-energy landscape and electronic structure of the Su-Schrieffer-Heeger (SSH) model, where all relevant quantities can be evaluated analytically. We compare the exact results at arbitrary displacement with diagrammatic perturbation theory both in the full model and in a downfolded effective single-band model, which gives an instructive insight into the properties of downfolding. An exact reconstruction of the potential-energy landscape is possible in a downfolded model, which requires a dynamical electron-biphonon interaction. The dispersion of the bands upon atomic displacement is also found correctly, where the downfolded model by construction only captures spectral weight in the target space. In the SSH model, the electron-phonon coupling mechanism involves exclusively hybridization between the low- and high-energy bands and this limits the computational efficiency gain of downfolded models.
Ab initio downfolding describes the electronic structure of materials within a low-energy subspace, often around the Fermi level. Typically starting from mean-field calculations, this framework allows for the calculation of one- and two-electron interactions, and the parametrization of a many-body Hamiltonian representing the active space of interest. The subsequent solution of such Hamiltonians can provide insights into the physics of strongly-correlated materials. While phonons can substantially screen electron-electron interactions, electron-phonon coupling has been commonly ignored within ab initio downfolding, and when considered this is done only for short-range interactions. Here we propose a theory of ab initio downfolding that accounts for all mechanisms of electron-phonon coupling on equal footing, regardless of the range of the interactions. Our practical computational implementation is readily compatible with current downfolding approaches. We apply our approach to polar materials MgO and GeTe, and we reveal the importance of both short-range and long-range electron-phonon coupling in determining the magnitude of electron-electron interactions. Our results show that in the static limit, phonons reduce the on-site repulsion between electrons by 40% for MgO, and by 79% for GeTe. Our framework also predicts that overall attractive nearest-neighbor interactions arise between electrons in GeTe, consistent with superconductivity in this material.
Most nonperturbative numerical studies of electron-phonon interactions focus on model Hamiltonians where the electrons interact with a phonon branch via a single type of microscopic mechanism. Two commonly explored couplings in this context are the Holstein and Su-Schrieffer-Heeger (SSH) interactions, which describe phonons modulating the on-site energy and intersite electron hopping, respectively. Many materials, however, have multiple phonon branches that can each interact with electronic degrees of freedom in different ways. We present here a determinant quantum Monte Carlo study of the half-filled two-dimensional (bond) SSH-Holstein Hamiltonian, where electrons couple to different phonon branches via either the Holstein or SSH mechanism. We map the model's phase diagram and determine the nature of the transitions between charge-density wave, bond-order wave, and antiferromagnetic order.
We present a density matrix renormalization group study of the doped one-dimensional (1D) Hubbard-Su-Schrieffer-Heeger (Hubbard-SSH) model, where the atomic displacements linearly modulate the nearest-neighbor hopping integrals. Focusing on an optical variant of the model in the strongly correlated limit relevant for cuprate spin chains, we examine how the SSH interaction modifies the model's ground- and excited-state properties. The SSH coupling weakly renormalizes the model's single- and two-particle response functions for electron-phonon $(e\text{\ensuremath{-}}\mathrm{ph})$ coupling strengths below a parameter-dependent critical value gc{g}_{\mathrm{c}}gc​. For larger $e\text{\ensuremath{-}}\mathrm{ph}$ coupling, the sign of the effective hopping integrals changes for a subset of orbitals, which drives a lattice dimerization distinct from the standard nesting-driven picture in 1D. The spectral weight of the one- and two-particle dynamical response functions are dramatically rearranged across this transition, with significant changes in the ground-state correlations. We argue that this dimerization results from the breakdown of the linear approximation for the $e\text{\ensuremath{-}}\mathrm{ph}$ coupling and thus signals a fundamental limitation of the linear SSH interaction. Our results have consequences for our understanding of how SSH-like interactions can enter the physics of strongly correlated quantum materials, including the recently synthesized doped cuprate spin chains.
This study explores the ground-state phase diagram and topological properties of the spinless 1D Su-Schrieffer-Heeger (SSH) model with nearest-neighbor (NN) interactions at quarter filling. We analyze key physical quantities such as the local electron density distribution, correlation functions for bond-order-wave (BOW) and charge-density-wave (CDW) -- by integrating twisted boundary conditions with the Lanczos technique and employing high-precision numerical diagonalization methods, complemented by a mean-field approximation (MFA) based on bond-order and charge-density modulation analysis. This approach enables precise identification of phase transition critical points. Our results indicate that the system exhibits a topologically trivial band insulating (BI) phase for strong attractive interactions, with its upper boundary forming a downward-opening curve peaking at V/t≃−2.3V/t\simeq-2.3V/t≃−2.3 and extending to V/t≃−2.6V/t\simeq-2.6V/t≃−2.6. Within −2.6≤V/t≤−0.5-2.6 \leq V/t \leq -0.5−2.6≤V/t≤−0.5, a BOW phase emerges for ∣δt/t∣>0.45\left|δt/t\right| > 0.45∣δt/t∣>0.45, with its boundaries converging as ∣δt/t∣\left|δt/t\right|∣δt/t∣ decreases, terminating at a single point at ∣δt/t∣≃0.45\left|δt/t\right|\simeq0.45∣δt/t∣≃0.45. In other parameter regions, a CDW phase is realized. Through this analysis, we elucidate the topological properties of the interacting spinless SSH model at quarter filling, highlighting the competition among CDW, BOW, and BI phases. By tuning VVV and δtδtδt, the system exhibits diverse correlated phenomena, offering new insights into one-dimensional quantum phase transitions and the interplay between topology and order.
The Su-Schrieffer-Heeger (SSH) model, with bond phonons modulating electron tunneling, is a paradigmatic electron-phonon model that hosts an antiferromagnetic order to bond order transition at half-filling. In the presence of a repulsive Hubbard interaction, the antiferromagnetic phase is enhanced, but the phase transition remains first order. Here, we explore the physics of the SSH model with an attractive Hubbard interaction, which hosts an interesting interplay among charge order, sss-wave pairing, and bond order. Using the numerically exact determinant quantum Monte Carlo method, we show that both charge order, present at weak electron-phonon coupling, and bond order, at large coupling, give way to sss-wave pairing when the system is doped. Furthermore, we demonstrate that the SSH electron-phonon interaction competes with the attractive Hubbard interaction and reduces the sss-wave pairing correlation.
Over the past several years, a new generation of quantum simulations has greatly expanded our understanding of charge density wave phase transitions in Hamiltonians with coupling between local phonon modes and the on-site charge density. A quite different, and interesting, case is one in which the phonons live on the bonds, and hence modulate the electron hopping. This situation, described by the Su-Schrieffer-Heeger (SSH) Hamiltonian, has so far only been studied with quantum Monte Carlo in one dimension. Here we present results for the 2D SSH model, show that a bond ordered wave (BOW) insulator is present in the ground state at half filling, and argue that a critical value of the electron-phonon coupling is required for its onset, in contradistinction with the 1D case where BOW exists for any nonzero coupling. We determine the precise nature of the bond ordering pattern, which has hitherto been controversial, and the critical transition temperature, which is associated with a spontaneous breaking of Z_{4} symmetry.