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arXiv:2608.04979

Delocalized Coupled-Cluster Theory for Polaron Structure and Dynamics

Hamlin Wu, Moritz K. A. Baumgarten, Tong Jiang, Joonho Lee

Published Aug 5, 2026• 0 citations
arXivHTMLPDF

Abstract

Polaron ground states and finite-temperature dynamics remain challenging to simulate because existing methods struggle to combine nonperturbative accuracy, systematic improvability, and scalability from models to materials-specific Hamiltonians. We introduce a translationally invariant variational coupled-cluster (CC) theory for polarons, termed delocalized CC (dCC), with closed-form energies at cost as low as $\mathcal{O}(N^3)$ and no phonon-number cutoff. dCC accurately describes the ground states of the one- and two-dimensional Holstein and Su--Schrieffer--Heeger (optical and bond) models and the Fr{ö}hlich model, in close agreement with density matrix renormalization group (DMRG) and diagrammatic Monte Carlo benchmarks. A projected tangent-space response formalism built on the same ansatz yields electron-addition spectral functions and optical conductivities at zero and finite temperature. The resulting spectra agree well with DMRG, Lanczos, and neural-network quantum-state benchmarks while retaining a physically interpretable excitation hierarchy, and extend to two-dimensional lattices at finite temperature beyond the practical reach of these methods. The same framework applies directly to \textit{ab initio} electron--phonon matrix elements, yielding LiF electron- and hole-polaron binding energies that match state-of-the-art many-body calculations. These results establish dCC as a unified variational framework for polaron ground states and dynamics, from model systems to real materials.

DOI
10.48550/arxiv.2608.04979

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The Journal of Chemical Physics•May 7, 2013•23 citations
Polaron ground states and finite-temperature dynamics remain challenging to simulate because existing methods struggle to combine nonperturbative accuracy, systematic improvability, and scalability from models to materials-specific Hamiltonians. We introduce a translationally invariant variational coupled-cluster (CC) theory for polarons, termed delocalized CC (dCC), with closed-form energies at cost as low as O(N3)\mathcal{O}(N^3)O(N3) and no phonon-number cutoff. dCC accurately describes the ground states of the one- and two-dimensional Holstein and Su--Schrieffer--Heeger (optical and bond) models and the Fr{ö}hlich model, in close agreement with density matrix renormalization group (DMRG) and diagrammatic Monte Carlo benchmarks. A projected tangent-space response formalism built on the same ansatz yields electron-addition spectral functions and optical conductivities at zero and finite temperature. The resulting spectra agree well with DMRG, Lanczos, and neural-network quantum-state benchmarks while retaining a physically interpretable excitation hierarchy, and extend to two-dimensional lattices at finite temperature beyond the practical reach of these methods. The same framework applies directly to \textit{ab initio} electron--phonon matrix elements, yielding LiF electron- and hole-polaron binding energies that match state-of-the-art many-body calculations. These results establish dCC as a unified variational framework for polaron ground states and dynamics, from model systems to real materials.
We develop and compare several analytical approximations for the ground state and zero-temperature polaron dispersion in finite-width, nonparabolic conduction bands. The main focus of the work is an extension of the Feynman variational method to a tight-binding lattice, where the effective-mass approximation is no longer applicable. The resulting variational formulation is not restricted to a specific phonon dispersion or electron-phonon interaction and provides a uniform description across weak-, intermediate-, and strong-coupling regimes. In addition, we revisit and generalize other analytical approaches traditionally formulated for continuum polarons, including canonical transformations and self-consistent Wigner-Brillouin-type approximations. For lattice polarons, these methods exhibit qualitative features absent in the continuum case, such as a nontrivial connection between weak- and strong-coupling limits. We show that an improved Wigner-Brillouin scheme yields a momentum-dependent polaron self-energy free of resonances and in good agreement with numerically exact results. All methods are applied to the Holstein model on a tight-binding lattice and are benchmarked against numerically exact calculations, including diagrammatic Monte Carlo (both our calculations and preceding works) and exact diagonalization results. Furthermore, the analytical approaches are extended to polarons with Rashba-type spin-orbit coupling, providing a stringent test of their applicability in systems with nontrivial band structure. Our results demonstrate that the modified Feynman variational method yields ground-state energies and dispersions with accuracy comparable to, and in many cases exceeding, that of other established analytical approaches. The developed framework offers a versatile and reliable analytical description of lattice polarons beyond the continuum approximation.
Polarons—quantum particles dressed by a cloud of bath excitations—arise whenever a particle interacts strongly with its host medium. Electron and hole polarons, formed when carriers bind to a phonon cloud through electron-phonon (e-ph) interactions, govern charge transport in transition metal oxides, organic semiconductors, and halide perovskites. The concept generalizes to exciton-phonon interactions, giving rise to exciton polarons that produce the large Stokes shifts and broad photoluminescence (PL) lineshapes central to light-emitting applications. A predictive treatment of polarons has remained elusive. Numerically exact methods such as diagrammatic Monte Carlo (DMC) have been restricted to simplified model Hamiltonians, while first-principles approaches based on density functional theory provide material-specific e-ph interactions but treat the e-ph correlation approximately. This thesis bridges that gap by developing first-principles diagrammatic Monte Carlo (FPDMC), which stochastically sums all e-ph Feynman diagrams using accurate ab-initio interactions, yielding numerically exact polaron properties in real materials. Two methodological advances make FPDMC possible. A matrix-product formalism sums over electronic band indices exactly, alleviating the multi-band sign problem in realistic band structures. A data-driven compression of e-ph interactions based on singular-value decomposition (SVD) in the Wannier basis then accelerates the method by more than 10³×. The SVD compression also reveals a hidden low-dimensional structure of quantum interactions, maintaining accuracy across charge transport, spin relaxation, band renormalization, and superconductivity calculations. Extending this philosophy to phonon-phonon interactions via a permanent CP (PCP) decomposition yields 10³–10⁴× compression of anharmonic force constants and orders-of-magnitude speedups in thermal conductivity calculations. Applying FPDMC, we study polarons in LiF, SrTiO₃, and TiO₂, finding excellent agreement with experimental mobilities and resolving the contrasting small- and large-polaron transport in the two TiO₂ polymorphs. To make the microscopic picture concrete, we introduce an e-ph correlation function that visualizes lattice distortion and polaron orbital character. Extending FPDMC to exciton polarons, we obtain first-principles exciton-phonon interactions from finite-momentum Bethe-Salpeter equation calculations and derive a Fermi-golden-rule PL theory. Applied to CrBr₃, FPDMC predicts a broad PL spectrum agreeing with experiment in both peak position and lineshape.
We introduce a method, numerically exact in principle, for computing the momentum- and frequency-resolved single-particle Green's function of a polaron at finite temperature. The method, which we refer to as the finite-temperature Green's function cluster expansion, combines two ingredients: the generalized Green's function cluster expansion, a numerically exact extension of the momentum average family of methods that solves the polaron problem at zero temperature through a hierarchy of equations of motion for restricted phonon cloud configurations; and the thermofield double formalism, which maps the thermal trace onto a pure-state expectation value over a doubled Hilbert space. The resulting equations of motion have the same algebraic structure as those of the multi-boson zero-temperature theory, with the temperature entering through a Bogoliubov-type mixing angle that controls the coupling to a set of fictitious bath bosons. We implement the method in our open-source software package and benchmark it on the one-dimensional Holstein polaron, computing spectral functions, dispersions, lifetimes, and effective masses across coupling regimes and temperatures up to T/Ω∼1T/Ω\sim 1T/Ω∼1. Where finite-temperature density matrix renormalization group results are available, we find quantitative agreement at affordable computational cost. The method recovers momentum-resolved spectra directly in frequency space, with no time evolution or analytic continuation. We also discuss the practical costs of the approach. In particular, since the doubled phonon Hilbert space has a non-trivial configuration structure in which real and fictitious clouds compete, convergence in the corresponding cloud parameters requires care.
Ab initio calculations of the phonon-induced band structure renormalization are currently based on the perturbative Allen-Heine theory and its many-body generalizations. These approaches are unsuitable to describe materials where electrons form localized polarons. Here, we develop a self-consistent, many-body Green's function theory of band structure renormalization that incorporates localization and self-trapping. We show that the present approach reduces to the Allen-Heine theory in the weak-coupling limit, and to total energy calculations of self-trapped polarons in the strong-coupling limit. To demonstrate this methodology, we reproduce the path-integral results of Feynman and diagrammatic Monte Carlo calculations for the Fröhlich model at all couplings, and we calculate the zero point renormalization of the band gap of an ionic insulator including polaronic effects.
How a polaron loses its quasiparticle coherence with increasing temperature is a long-standing open problem. Holstein addressed it in 1959 only in the extreme antiadiabatic, strong-coupling limit, while more recent numerically exact approaches are largely restricted to one dimension. Here we solve this problem on square and simple cubic lattices across the weak-, intermediate-, and strong-coupling regimes. We show that the polaron effective mass m∗m^*m∗ and inverse lifetime 1/τ1/τ1/τ increase monotonically with temperature until, at T∼0.5 ΩT \sim 0.5\,ΩT∼0.5Ω, the quasiparticle peak dissolves into a broad incoherent thermal continuum. The phonon frequency ΩΩΩ therefore defines a universal coherence scale, independent of dimensionality and coupling strength, validating Holstein's prediction far beyond the regime in which it was derived. These results follow from a finite-temperature generalization of the Momentum Average (MA) approximation, yielding a closed-form, diagrammatically derived self-energy that is asymptotically exact in the strong-coupling limit at all temperatures. Benchmark comparisons demonstrate excellent quantitative agreement of the resulting 1D spectral functions with the numerically exact Variational Exact Diagonalization-Finite-Temperature Lanczos Method (VED-FTLM) and finite-TTT Density Matrix Renormalization Group (DMRG).
An efficient, yet very accurate trial wave function, constructed from projecting the well-known Davydov D1 Ansatz onto momentum eigenstates, is employed to study the ground state properties of the generalized Holstein Hamiltonian with simultaneous diagonal and off-diagonal coupling. Ground-state energies have been obtained with a precision matching that of the computationally much more demanding density-matrix renormalization group method. The delocalized D1 Ansatz lowers the ground-state energies at the Brillouin zone boundary significantly compared with the Toyozawa and Global-Local Ansätze in the weak coupling regime, while considerable improvement is demonstrated to have been achieved over the entire Brillouin zone in the strong coupling regime. Unique solutions are obtained with the delocalized D1 for different initial conditions when the transfer integral is 20 times the phonon frequency at the zone center, implying the absence of any self-trapping discontinuity. The scaled correlation variance is found to fit satisfactorily well with the predictions of the perturbation theories.