Delocalized Coupled-Cluster Theory for Polaron Structure and Dynamics

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

REVIEW 3 major objections 5 minor 148 references

Delocalized Coupled-Cluster Theory for Polaron Structure and Dynamics

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper introduces dCC, a translationally invariant variational coupled-cluster theory for single polarons that gives closed-form energies at $O(N^3)$ cost and a tangent-space response formalism for spectra and optical response.

desk verdict A genuinely new variational coupled-cluster hierarchy for polarons with strong ground-state benchmarks; the finite-temperature response is the main soft spot, and the authors mostly concede that themselves. read the letter →

arxiv 2608.04979 v1 pith:YRJUT6YZ submitted 2026-08-05 cond-mat.mtrl-sci physics.chem-ph

Hamlin Wu , Moritz K.A. Baumgarten , Tong Jiang , Joonho Lee This is my paper · ORCID

classification cond-mat.mtrl-sciphysics.chem-ph PACS 71.38.-k

keywords polaroncoupled-clusterelectron-phononinteractioncoherentstatemomentumprojectionspectralfunctionopticalconductivityLiF

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Delocalized coupled-cluster theory (dCC) is introduced as a variational, translationally invariant wavefunction method for single polarons: a coherent-state reference is dressed by electron–phonon cluster excitations and projected onto definite crystal momentum, giving closed-form variational energies at $O(N^3)$ cost with no phonon-number cutoff. The same excitation hierarchy is reused in a projected tangent-space response space whose eigenvalues and matrix elements yield electron-addition spectral functions and optical conductivities at zero and finite temperature, with thermal phonon traces evaluated analytically. The paper claims that the first-order theory dCC-1-S1 reproduces DMRG, DiagMC, and NNQS benchmarks across the Holstein, Su–Schrieffer–Heeger, and Fröhlich models, and yields LiF electron- and hole-polaron binding energies of 0.397 and 1.948 eV that are close to published many-body results. If correct, this gives materials scientists a single systematically improvable framework for polaron ground states and dynamics, from model Hamiltonians to ab initio electron–phonon matrix elements.

What carries the argument

The central object is the momentum-projected dCC wavefunction $|\Psi^K_{\mathrm{dCC}}\rangle = \hat{\Xi}_K e^{\hat{T}} |\Psi_{\mathrm{D2}}\rangle$, where $|\Psi_{\mathrm{D2}}\rangle$ is a Davydov D2 coherent-state electron–phonon product state, $\hat{T}$ contains electronic, phononic, and coupled electron–phonon excitations, and $\hat{\Xi}_K$ projects onto crystal momentum $K$. Because the system has a single carrier, the electronic exponentials truncate, and the variational energy reduces to $E_K = \sum_R e^{iK\cdot R} H(R) / \sum_R e^{iK\cdot R} S(R)$, where $H(R)$ and $S(R)$ are translated Hamiltonian and overlap matrix elements evaluated in closed form by coherent-state algebra. The same one- and two-phonon excitation classes are then used to build a nonorthogonal tangent-space basis for a generalized eigenvalue problem whose solutions provide the excited states entering the Lehmann and Kubo expressions for spectral functions and optical conductivities.

What would settle it

Compare the integrated spectral weight and line shape of the finite-temperature electron-addition spectral function for the six-site Holstein model at $T = 0.4$, $0.6$, and $1.0$ using finite-temperature Lanczos on the same Hamiltonian; if the dCC-1-S1-(1,1) weight falls below the Lanczos reference by more than the depletion already visible at $T = 0.2$, the tangent-space completeness assumption fails. Alternatively, run TD-DMRG or exact diagonalization on a small two-dimensional Holstein lattice (for example $4\times4$ sites at $g = 1.8$) at $T = 0.1$ and $0.2$ and compare the spectra at the zone-corner momentum.

Watch

Extended reading notes

Core claim

The central discovery is that a single-carrier coupled-cluster ansatz built on a momentum-projected coherent state admits a closed-form variational energy, so the exponential cluster operator can be optimized variationally rather than projectively, and the same cluster hierarchy can be reused as a tangent-space response basis. Because the fermionic sector contains only one electron, the electronic part of the exponential truncates exactly; because the bosonic sector is built on coherent states, all phonon sums are resummed analytically through coherent-state overlaps and Gaussian integrals. Momentum projection restores translational symmetry while preserving the localized electron–phonon correlations that define the polaron. With only one-phonon coupled excitations (dCC-1-S1), the paper argues, the resulting ground states track benchmark results for the Holstein, optical and bond Su–Schrieffer–Heeger, and Fröhlich models, and the same ansatz applied to LiF gives binding energies of 0.397 eV (electron) and 1.948 eV (hole).

Load-bearing premise

The load-bearing premise is that the truncated tangent-space manifold built from first- and second-order excitation operators around the dCC ground state is complete enough to represent the thermally populated spectrum, a premise the paper itself sees strained at high temperature in one dimension and uncheckable in two dimensions.

Editorial extensions

If this is right

dCC-1-S1 reproduces DMRG, DiagMC, and NNQS ground-state benchmarks for one- and two-dimensional Holstein, OSSH, and BSSH models at $O(N^3)$ cost per momentum sector, with no explicit phonon-number cutoff.
Polaron band structures follow by variationally optimizing in each crystal-momentum sector, correcting the zone-edge error of the momentum-projected coherent-state reference in the dispersive-phonon Holstein model.
The tangent-space response hierarchy reproduces Lanczos and TD-DMRG spectral functions and optical conductivities in one dimension at zero and finite temperature, and its (2,2) manifold redistributes spectral weight into multiphonon satellites.
Applied directly to ab initio electron–phonon matrix elements, the same ansatz yields LiF electron- and hole-polaron binding energies of 0.397 and 1.948 eV, close to published DiagMC, NNQS, and Green's-function results.
Finite-temperature momentum-resolved spectra and optical conductivities extend to two-dimensional lattices, beyond the practical reach of the DMRG and Lanczos benchmarks used in one dimension.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The closed-form variational energy makes dCC a natural starting point for bipolaron extensions, though the two-carrier fermionic sector would no longer truncate linearly.
The analytic coherent-state thermal trace used for the electron-addition Green's function could be adapted to finite-temperature mobility and transport coefficients in materials, where the same phonon-bath trace appears.
In the moderate-temperature regime where the tangent-space manifold is accurate, dCC spectra could serve as reference data for cheaper approximate methods such as cumulant or mixed quantum–classical approaches, giving an independent benchmark beyond TD-DMRG's one-dimensional reach.
The same excitation hierarchy could be combined with time-dependent variational principles for nonequilibrium polaron dynamics after a pump or quench, a direction the paper names but does not implement.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a real advance, not a repackaging. The dCC hierarchy—momentum-projected coherent-state reference plus a truncated coupled-cluster excitation manifold with closed-form variational energies—is genuinely new, and the ground-state benchmarks are strong. The weakest link is the finite-temperature response formalism, and the authors are upfront about it.

What's new and good: the variational energy evaluation at O(N^3) with no phonon cutoff is a clean trick that exploits the single-carrier structure, and the Appendix B-C derivations are careful and internally consistent. The 1D and 2D ground-state comparisons against DMRG, DiagMC, and NNQS are appropriate and honestly presented; the hierarchy systematically improves, and even dCC-1-S1 does well except where the paper notes residual error, such as the 2D OSSH model at strong coupling. The LiF binding energies are a useful demonstration with ab initio matrix elements, and the comparison to prior methods is fair.

Soft spots, in proportion. First, the Fröhlich continuum results are not strict variational upper bounds because of the auxiliary finite-size/cutoff corrections and the shared-exponent extrapolation. The paper says this explicitly, and the error analysis with exponent ranges is reasonable, but the word "variational" does not carry over to the continuum numbers. Second, the finite-temperature Green's function rests on Eq. (28), which assumes the truncated tangent-space states are eigenstates. The paper's own 1D benchmark shows spectral-weight depletion at high T, and the 2D finite-T spectra have no independent reference. The authors qualify the claim to low and moderate temperature, so the advertised capability is narrower than the abstract suggests; a referee should probe how quantitatively reliable the (1,1) manifold is at the temperatures shown (T = 0.1, 0.2 in 2D) and whether the (2,2) manifold would change the line shapes. That is a real question, not a fatal flaw. Third, the data repository is referenced but there is no code or commit hash; for a methods paper that is a minor reproducibility gap.

The citation pattern is fine: comparisons are to external benchmarks, and the self-citations are to the authors' own prior dD2/NNQS/CSPT2 work, which is appropriate context rather than circular support.

Bottom line: a serious referee should engage with this. The ground-state framework will be useful regardless of how the finite-T question settles, and the response formalism is clearly enough presented to be tested. I would send it out, with the referee focused on Section IV C and the 2D finite-T claims.

Referee Report

3 major / 5 minor

Summary. The manuscript introduces delocalized coupled-cluster (dCC) theory for single polarons, combining a momentum-projected coherent-state (Davydov D2) reference with a coupled-cluster excitation hierarchy. The authors derive closed-form variational ground-state energies, with the lowest truncation dCC-1-S1 scaling as O(N^3) and requiring no phonon-number cutoff for lattice models. Using the same excitation hierarchy, they build a projected tangent-space response formalism and compute electron-addition spectral functions and optical conductivities at zero and finite temperature, benchmarking against DMRG, Lanczos, TD-DMRG, NNQS, and DiagMC results. The framework is then applied to ab initio electron-phonon matrix elements, yielding LiF electron- and hole-polaron binding energies close to published many-body results.

Significance. If the results hold, dCC is a significant methodological advance: it provides a systematically improvable variational ansatz spanning model and material electron-phonon Hamiltonians, with polynomial scaling, restored translational symmetry, and access to momentum-resolved dynamics. The paper is unusually strong in its closed-form first-order energy derivation (Appendices B-C), its extensive benchmarking across Holstein, OSSH, BSSH, and Fröhlich models, and its public data repository. The main risk is not the ground-state theory but the finite-temperature response formalism, whose truncated tangent-space eigenstate assumption is acknowledged by the authors to lose spectral weight at high temperature (Section V B 3). The Fröhlich continuum energies are also corrected by a non-variational auxiliary procedure whose regime of validity deserves closer scrutiny. These are load-bearing issues for the paper's broadest claims, but they are addressable within the manuscript's scope.

major comments (3)

[Section IV C 1, Eq. (28)] The finite-temperature Green's function rests on treating the truncated dCC tangent-space states as eigenstates and inserting an approximate resolution of identity. The paper's own 1D benchmark in Section V B 3 (Fig. 6) shows spectral-weight depletion relative to TD-DMRG at high temperature, which the text attributes to 'the incompleteness of the underlying linear-response manifold.' Because Eq. (28), together with Eqs. (29)-(30), is the basis for all finite-temperature results, including the 2D spectra in Fig. 7 and conductivities in Fig. 9 that have no independent reference, the central finite-temperature dynamics claim is not established at the temperatures and couplings shown. I request either a convergence demonstration across balanced manifolds (e.g., dCC-1-S1-(2,2)) for at least one 1D finite-temperature benchmark at T where depletion is observed, or a clear restriction of the finite-temperature claims with explicit error estimates.
[Section V A 3 and Appendix D] The continuum Fröhlich energies in Fig. 3 and Table A1 are not strict variational upper bounds because they use E_corrected = E_dCC + (E∞_aux - E_grid_aux), and the auxiliary selection in Appendix D 2 is partly a posteriori (the 'flattest sequence' criterion). The statement that the dCC discretization error coincides with that of second-order perturbation theory to leading order is an assumption that is not demonstrated in the strong-coupling regime, where Table A1 reports deviations from DiagMC up to 2.54%. To make the Fröhlich benchmark load-bearing, the authors should report per-α exponent fits, quantify the sensitivity of E∞ to the auxiliary choice at each α, and state explicitly which conclusions depend on the non-variational correction rather than on the raw grid energies.
[Section IV A and Eq. (21)] The response hierarchy is described as systematically improvable, but all finite-temperature results are computed at the lowest-order dCC-1-S1-(1,1) manifold. The 1D optical conductivity comparison in Fig. 8 shows that unbalanced manifolds (1,2) and (2,1) separate strongly as temperature increases, and the balanced (2,2) results are not shown for the finite-temperature regime where the depletion appears. Without a finite-temperature convergence study, the 'systematically improvable' claim for dynamics is demonstrated only at zero and low temperature, not for the finite-temperature predictions that are advertised as the method's unique reach.

minor comments (5)

[Figure 6 caption] The caption uses 'dCC-1-(1,1)' while the text and other figures use 'dCC-1-S1-(1,1)'; please unify the notation.
[Figure 4 inset] The cumulative integrated spectral weight I is not defined in the caption; specify the integration window and normalization used to compute it.
[Eq. (28)] The approximate resolution of identity inserted in Eq. (28) is not written explicitly, and because the tangent-space basis is nonorthogonal, clarify how the overlap matrix S_K of Eq. (22) enters the Lehmann weights.
[Appendix D, Table A1] The entry for α=1 is described as reported without extrapolation using kmax=4, while the protocol states infrared floor kmax≥4 and cutoff ratios kmax/α={1.4,1.75,2,2.5,3}; reconcile these statements or explain why α=1 is treated separately.
[Section IV C] The sentence limiting expected accuracy to low-to-moderate temperature is an important caveat and should be reflected in the abstract and conclusions, where the finite-temperature reach is currently advertised without that qualification.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: dCC energies come from variational minimization with external benchmarks, and the finite-temperature response assumption is an acknowledged approximation rather than a circular reduction.

full rationale

Walking the paper's derivation chain, the dCC ground-state energy is obtained by direct variational minimization of the closed-form projected energy E_K_dCC = (Σ_R e^{iK·R} H(R)) / (Σ_R e^{iK·R} S(R)) with respect to the coupled-cluster amplitudes; no benchmark energy or experimental value enters the optimization. The response spectrum is obtained by diagonalizing the projected tangent-space Hamiltonian (Eqs. 22-23), and the finite-temperature expressions (Eqs. 28-30) are explicitly approximate: the text states 'assuming our dCC states are eigenstates' after inserting a truncated tangent-space resolution of identity. This is a documented accuracy limitation, and the paper itself notes that 'incompleteness of the underlying linear-response manifold' causes spectral-weight depletion at high temperature (Fig. 6), but this is not a circular identification of the prediction with its input. The central benchmarks are external: DMRG, DiagMC, Lanczos, NNQS, and TD-DMRG. Several compared values (dD2, CSPT2, NNQS) come from papers with overlapping authorship, but they are used as references rather than to define or fit the dCC amplitudes, and the dCC wavefunction remains an independent variational ansatz. No fitted parameter is renamed as a prediction, and no equation is defined in terms of the quantity it claims to predict. The minor self-citations are not load-bearing for the derivation, so the circularity score is low.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The central derivation rests mainly on standard single-carrier electron-phonon Hamiltonians, coherent states, and CC truncations. The nonstandard assumptions are the tangent-space eigenstate approximation at finite temperature and the Fröhlich auxiliary-correction protocol. No new physical entities are introduced; all parameters are variational or extrapolation-fit parameters.

free parameters (3)

Fröhlich momentum-cutoff extrapolation exponent p = p ≈ 2.5, uncertainty band 1.6-3.6
Used in E(kmax)=E∞+c k^{-p} (Appendix D 2, Eq. A34) to obtain continuum Fröhlich energies; the exponent is fitted to the computed grid energies across couplings, so the reported E∞ depends on this fit.
Fröhlich reference orbital scale parameter w or Gaussian width σ = Optimized by minimization with kinetic-energy matching constraint
Selects the coherent-state electronic orbital for the Fröhlich reference family (Appendix D 1); affects the strong-coupling extrapolation and is not derived from the Hamiltonian alone.
Fröhlich kmax/α cutoff ratios and infrared floor = kmax/α ∈ {1.4, 1.75, 2, 2.5, 3}, kmax ≥ 4
Selected set of cutoffs for the power-law extrapolation; the floor kmax ≥ 4 excludes small cutoffs at α=1 and shapes the dataset used for the continuum limit.

assumptions (7)

domain assumption The electron-phonon Hamiltonian is restricted to harmonic phonons and linear coupling (Eq. 1).
Higher-order couplings and anharmonicity are neglected; this excludes some strong-distortion regimes, as acknowledged for OSSH in Section V A 2.
domain assumption There is exactly one carrier, so the fermionic sector of the CC expansion truncates exactly.
The entire dCC derivation depends on a single electron or hole; bipolarons and multi-carrier generalizations are left to future work (Section VI).
domain assumption The coherent-state Davydov D2 reference plus CC excitations spans the relevant polaron physics at each truncation.
The ansatz is a variational manifold; its completeness at first or second order is assumed and tested against benchmarks rather than proven.
standard math Momentum projection over supercell translations restores translational symmetry with an idempotent projector Ξ_K.
Used in Eqs. (11)-(14); assumes the Hamiltonian is invariant under discrete lattice translations.
ad hoc to paper In the finite-T Green's function, dCC states are treated as eigenstates and a tangent-space resolution of identity is inserted (Eq. 28).
This is an uncontrolled approximation; accuracy depends on manifold completeness and is known to degrade at high temperature (Section V B 3).
ad hoc to paper For the Fröhlich model, the discretization error of the dCC energy coincides with that of an auxiliary quantity to leading order, justifying E_corrected.
Auxiliary selection rules and the shared power-law fit are specific to this paper's extrapolation protocol (Appendix D 2).
domain assumption The linear-response hierarchy is balanced only for equal phonon and electron-phonon response orders (n,n).
Unbalanced manifolds (1,2) and (2,1) are used for diagnosis but not as a convergence sequence at finite temperature (Section IV A).

how reviews work

0 comments

Cite this review

Pith. "Pith review of Delocalized Coupled-Cluster Theory for Polaron Structure and Dynamics." pith.science (2026). https://pith.science/paper/YRJUT6YZ

@misc{pith2026260804979,
  author       = {Pith},
  title        = {Pith review of: Delocalized Coupled-Cluster Theory for Polaron Structure and Dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YRJUT6YZ}},
  note         = {Machine review of arXiv:2608.04979}
}

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{\"o}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.

Figures

Figures reproduced from arXiv: 2608.04979 by the authors.

**Figure 1.** shows the ground-state energy obtained at various levels of dCC theory for the 32-site Holstein, BSSH, and OSSH models across a range of electron– phonon coupling strengths compared with DMRG calculations. We use the standard dimensionless coupling parameter λ = g 2/(2ωt) for the Holstein and BSSH models, and λ = g 2/ωt for the OSSH model. These definitions compare the phonon-mediated interaction scale with the el… view at source ↗

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: shows zero-temperature spectral functions for the Holstein, OSSH, and BSSH models on 32-site onedimensional lattices at intermediate coupling (λ = 1). Across all three coupling types, the dCC tangent-space spectra are consistent with the dominant momentumresolved fea…

**Figure 5.** Figure 5: compares zero-temperature electron-addition spectra of the two-dimensional Holstein and BSSH models computed in the weak-coupling regime, corresponding to the dimensionless coupling λ = 0.5 previously defined. This coupling is smaller than the λ = 1 value used to ex…

**Figure 6.** Figure 6: shows the finite-temperature spectral function for a six-site Holstein model (with g = 1), benchmarked against finite-temperature TD-DMRG calculations. We chose this six-site system to enable comparison with previous finite-temperature Lanczos results of Bonˇca et al.…

**Figure 7.** Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]

**Figure 9.** Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]

**Figure 10.** Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

148 extracted references · 67 canonical work pages

[1]

Here we work directly in the frequency do- main, which allows us to handle the regular component and zero-frequency singular part separately

One dimension Figure 8 shows the real part of the optical conductiv- ity for the 1D Holstein model with 32 sites at intermedi- ate coupling (λ= 1) at several temperatures, compared with Lanczos (only atT= 0) and TD-DMRG calcula- tions [54]. Here we work directly in the frequency do- main, which allows us to handle the regular component and zero-frequency ...
[2]

Two dimensions Finite-temperature optical conductivity remains par- ticularly challenging in two dimensions. Recent mixed quantum–classical Green–Kubo calculations have been applied to large two-dimensionalab initiosystems [51], but a systematically improvable fully quantum treatment at comparable scales remains unavailable and mixed quantum–classical app...
[3]

Fr¨ ohlich, Electrons in lattice fields, Advances in Physics3, 325 (1954)

H. Fr¨ ohlich, Electrons in lattice fields, Advances in Physics3, 325 (1954)

1954
[4]

L. D. Landau, Electron motion in crystal lattices, Physikalische Zeitschrift der Sowjetunion3, 664 (1933)

1933
[5]

S. I. Pekar, Local quantum states of electrons in an ideal ion crystal, Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki16, 341 (1946)

1946
[6]

Alexandrov, ed.,Polarons in Advanced Materials, Vol

A. Alexandrov, ed.,Polarons in Advanced Materials, Vol. 103 (Springer Netherlands, 2007)

2007
[7]

Holstein, Studies of polaron motion: Part i

T. Holstein, Studies of polaron motion: Part i. the molecular-crystal model, Annals of Physics8, 325 (1959)

1959
[8]

Holstein, Studies of polaron motion: Part ii

T. Holstein, Studies of polaron motion: Part ii. the “small” polaron, Annals of Physics8, 343 (1959)

1959

Show all 148 references

[9]

Mandal, M

A. Mandal, M. A. Taylor, B. M. Weight, E. R. Koessler, X. Li, and P. Huo, Theoretical Advances in Polariton Chemistry and Molecular Cavity Quantum Electrody- namics, Chemical Reviews123, 9786 (2023)

2023
[10]

Emin,Polarons(Cambridge University Press, 2012)

D. Emin,Polarons(Cambridge University Press, 2012)

2012
[11]

T. S. Haugland, E. Ronca, E. F. Kjønstad, A. Rubio, and H. Koch, Coupled cluster theory for molecular po- laritons: Changing ground and excited states, Phys. Rev. X10, 041043 (2020)

2020
[12]

Ruggenthaler, N

M. Ruggenthaler, N. Tancogne-Dejean, J. Flick, H. Ap- pel, and A. Rubio, From a quantum-electrodynamical light–matter description to novel spectroscopies, Nature Reviews Chemistry2, 0118 (2018)

2018
[13]

Mordovina, C

U. Mordovina, C. Bungey, H. Appel, P. J. Knowles, A. Rubio, and F. R. Manby, Polaritonic coupled-cluster theory, Phys. Rev. Res.2, 023262 (2020)

2020
[14]

Flick, M

J. Flick, M. Ruggenthaler, H. Appel, and A. Rubio, Atoms and molecules in cavities, from weak to strong coupling in quantum-electrodynamics (qed) chemistry, Proceedings of the National Academy of Sciences114, 3026 (2017)

2017
[15]

Schmitt-Rink, C

S. Schmitt-Rink, C. M. Varma, and A. E. Ruckenstein, Spectral function of holes in a quantum antiferromag- net, Phys. Rev. Lett.60, 2793 (1988)

1988
[16]

C. L. Kane, P. A. Lee, and N. Read, Motion of a single hole in a quantum antiferromagnet, Phys. Rev. B39, 6880 (1989). 20

1989
[17]

Martinez and P

G. Martinez and P. Horsch, Spin polarons in the t-j model, Phys. Rev. B44, 317 (1991)

1991
[18]

Koepsell, J

J. Koepsell, J. Vijayan, P. Sompet, F. Grusdt, T. A. Hilker, E. Demler, G. Salomon, I. Bloch, and C. Gross, Imaging magnetic polarons in the doped Fermi–Hubbard model, Nature572, 358 (2019)

2019
[19]

Dagotto, Correlated electrons in high-temperature superconductors, Rev

E. Dagotto, Correlated electrons in high-temperature superconductors, Rev. Mod. Phys.66, 763 (1994)

1994
[20]

Grusdt, A

F. Grusdt, A. Bohrdt, and E. Demler, Microscopic spinon-chargon theory of magnetic polarons in thet−j model, Phys. Rev. B99, 224422 (2019)

2019
[21]

Grusdt and E

F. Grusdt and E. Demler, New theoretical approaches to bose polarons, Quantum matter at ultralow temper- atures191, 325 (2015)

2015
[22]

Prembabu, R

S. Prembabu, R. Sahay, S. Divic, and A. Vishwanath, Spin polarons in flat band ferromagnets (2025)

2025
[23]

Schmidt and M

R. Schmidt and M. Lemeshko, Rotation of quantum im- purities in the presence of a many-body environment, Phys. Rev. Lett.114, 203001 (2015)

2015
[24]

N. B. Jørgensen, L. Wacker, K. T. Skalmstang, M. M. Parish, J. Levinsen, R. S. Christensen, G. M. Bruun, and J. J. Arlt, Observation of attractive and repulsive polarons in a bose-einstein condensate, Phys. Rev. Lett. 117, 055302 (2016)

2016
[25]

Massignan, M

P. Massignan, M. Zaccanti, and G. M. Bruun, Polarons, dressed molecules and itinerant ferromagnetism in ul- tracold Fermi gases, Reports on Progress in Physics77, 034401 (2014)

2014
[26]

Y. E. Shchadilova, R. Schmidt, F. Grusdt, and E. Dem- ler, Quantum dynamics of ultracold bose polarons, Phys. Rev. Lett.117, 113002 (2016)

2016
[27]

R. A. Marcus, Chemical and electrochemical electron- transfer theory, Annual Review of Physical Chemistry 15, 155 (1964)

1964
[28]

M.-G. Hu, M. J. Van de Graaff, D. Kedar, J. P. Cor- son, E. A. Cornell, and D. S. Jin, Bose polarons in the strongly interacting regime, Phys. Rev. Lett.117, 055301 (2016)

2016
[29]

R. A. Marcus, Theory of electron-transfer reaction rates of solvated electrons, The Journal of Chemical Physics 43, 3477 (1965)

1965
[30]

Zhang, N

C. Zhang, N. V. Prokof’ev, and B. V. Svistunov, Peierls/su-schrieffer-heeger polarons in two dimensions, Phys. Rev. B104, 035143 (2021)

2021
[31]

Chandler, Y

D. Chandler, Y. Singh, and D. M. Richardson, Excess electrons in simple fluids. i. general equilibrium theory for classical hard sphere solvents, The Journal of Chem- ical Physics81, 1975 (1984)

1984
[32]

J. P. Donley, D. R. Heine, C. A. Tormey, and D. T. Wu, Liquid-state polaron theory of the hydrated electron re- visited, The Journal of Chemical Physics141, 024504 (2014)

2014
[33]

Baroni, S

S. Baroni, S. de Gironcoli, A. Dal Corso, and P. Giannozzi, Phonons and related crystal properties from density-functional perturbation theory, Rev. Mod. Phys.73, 515 (2001)

2001
[34]

Kagan and M

Y. Kagan and M. Klinger, Role of fluctuational barrier” preparation” in the quantum diffusion of atomic parti- cles in a crystal, Zh. Eksp. Teor. Fiz70, 255 (1976)

1976
[35]

M. R. Carbone, A. J. Millis, D. R. Reichman, and J. Sous, Bond-peierls polaron: Moderate mass enhance- ment and current-carrying ground state, Physical Re- view B104, 10.1103/physrevb.104.l140307 (2021)

2021 doi
[36]

Giannozzi, O

P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau, M. Buongiorno Nardelli, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, M. Cococcioni, N. Colonna, I. Carnimeo, A. Dal Corso, S. de Gironcoli, P. Delu- gas, R. A. DiStasio, A. Ferretti, A. Floris, G. Fratesi, G. Fugallo, R. Geba...

2017
[37]

Ponc´ e, E

S. Ponc´ e, E. Margine, C. Verdi, and F. Giustino, Epw: Electron–phonon coupling, transport and super- conducting properties using maximally localized wan- nier functions, Computer Physics Communications209, 116 (2016)

2016
[38]

Giustino, Electron-phonon interactions from first principles, Rev

F. Giustino, Electron-phonon interactions from first principles, Rev. Mod. Phys.89, 015003 (2017)

2017
[39]

P. J. Robinson, J. Lee, A. Mahajan, and D. R. Reich- man, Ab initio polaron wave functions, Phys. Rev. B 111, 054304 (2025)

2025
[40]

Damascelli, Z

A. Damascelli, Z. Hussain, and Z.-X. Shen, Angle- resolved photoemission studies of the cuprate supercon- ductors, Rev. Mod. Phys.75, 473 (2003)

2003
[41]

J. A. Sobota, Y. He, and Z.-X. Shen, Angle-resolved photoemission studies of quantum materials, Rev. Mod. Phys.93, 025006 (2021)

2021
[42]

W. H. Sio, C. Verdi, S. Ponc´ e, and F. Giustino, Po- larons from first principles, without supercells, Phys. Rev. Lett.122, 246403 (2019)

2019
[43]

Lee, H.-Y

N.-E. Lee, H.-Y. Chen, J.-J. Zhou, and M. Bernardi, Facile ab initio approach for self-localized polarons from canonical transformations, Phys. Rev. Mater.5, 063805 (2021)

2021
[44]

B. K. Chang, I. Timrov, J. Park, J.-J. Zhou, N. Marzari, and M. Bernardi, First-principles electron-phonon in- teractions and polarons in the parent cuprate la 2cuo4, Phys. Rev. Res.7, L012073 (2025)

2025
[45]

Lafuente-Bartolome, C

J. Lafuente-Bartolome, C. Lian, W. H. Sio, I. G. Gur- tubay, A. Eiguren, and F. Giustino, Ab initio self- consistent many-body theory of polarons at all cou- plings, Phys. Rev. B106, 075119 (2022)

2022
[46]

W. H. Sio, C. Verdi, S. Ponc´ e, and F. Giustino, Ab initio theory of polarons: Formalism and applications, Phys. Rev. B99, 235139 (2019)

2019
[47]

Y. Luo, B. K. Chang, and M. Bernardi, Comparison of the canonical transformation and energy functional formalisms for ab initio calculations of self-localized po- larons, Phys. Rev. B105, 155132 (2022)

2022
[48]

should enable material properties to be converged to the TDL. The variational structure also invites exten- sions to bipolarons, where two carriers interact with a shared phonon environment, and to time-dependent vari- ational principle algorithms for excited-state and non- eq...
[49]

M. K. A. Baumgarten, H. Wu, T. Jiang, and J. Lee, A scalable translationally invariant variational theory of ab initio polarons (2026)

2026
[50]

Y. Luo, J. Park, and M. Bernardi, First-principles dia- grammatic monte carlo for electron–phonon interactions and polaron, Nature physics21, 1275 (2025)

2025
[51]

Y. Luo, D. Desai, B. K. Chang, J. Park, and M. Bernardi, Data-driven compression of electron- phonon interactions, Phys. Rev. X14, 021023 (2024). 21

2024
[52]

Z. Dai, J. Lafuente-Bartolome, and F. Giustino, Po- larons from first principles (2025)

2025
[53]

Franchini, M

C. Franchini, M. Reticcioli, M. Setvin, and U. Diebold, Polarons in materials, Nature reviews. Materials6, 560 (2021)

2021
[54]

Jiang and J

T. Jiang and J. Lee, First-principles origins of charge transport in molecular semiconductors, arXiv preprint arXiv:2607.25089 (2026)

2026 arXiv
[55]

Jeckelmann and S

E. Jeckelmann and S. R. White, Density-matrix renormalization-group study of the polaron problem in the holstein model, Phys. Rev. B57, 6376 (1998)

1998
[56]

Bonˇ ca, S

J. Bonˇ ca, S. Trugman, and I. Batisti´ c, Holstein polaron, Physical Review B60, 1633 (1999)

1999
[57]

Jansen, J

D. Jansen, J. Bonˇ ca, and F. Heidrich-Meisner, Finite- temperature optical conductivity with density-matrix renormalization group methods for the holstein polaron and bipolaron with dispersive phonons, Phys. Rev. B 106, 155129 (2022)

2022
[58]

Jansen, J

D. Jansen, J. Bonˇ ca, and F. Heidrich-Meisner, Finite- temperature density-matrix renormalization group method for electron-phonon systems: Thermodynam- ics and holstein-polaron spectral functions, Physical Re- view B102, 165155 (2020)

2020
[59]

A. S. Mishchenko, N. Nagaosa, N. V. Prokof’ev, A. Sakamoto, and B. V. Svistunov, Optical conductiv- ity of the fr¨ ohlich polaron, Phys. Rev. Lett.91, 236401 (2003)

2003
[60]

Toyozawa, Self-trapping of an electron by the acous- tical mode of lattice vibration

Y. Toyozawa, Self-trapping of an electron by the acous- tical mode of lattice vibration. i:, Progress of Theoretical Physics26, 29 (1961)

1961
[61]

Zhao, The hierarchy of davydov’s ans¨ atze: From guesswork to numerically “exact” many-body wave functions, The Journal of Chemical Physics158, 080901 (2023)

Y. Zhao, The hierarchy of davydov’s ans¨ atze: From guesswork to numerically “exact” many-body wave functions, The Journal of Chemical Physics158, 080901 (2023)

2023
[62]

J. Sun, L. Duan, and Y. Zhao, Delocalized davydov d1 ansatz for the holstein polaron, The Journal of Chemical Physics138, 174116 (2013)

2013
[63]

Mahajan, P

A. Mahajan, P. J. Robinson, J. Lee, and D. R. Reich- man, Structure and dynamics of electron-phonon cou- pled systems using neural quantum states, Phys. Rev. B112, 184310 (2025)

2025
[64]

Coester, Bound states of a many-particle system, Nu- clear Physics7, 421 (1958)

F. Coester, Bound states of a many-particle system, Nu- clear Physics7, 421 (1958)

1958
[65]

Coester and H

F. Coester and H. K¨ ummel, Short-range correlations in nuclear wave functions, Nuclear Physics17, 477 (1960)

1960
[66]

ˇC ´ ıˇ zek, On the correlation problem in atomic and molecular systems

J. ˇC ´ ıˇ zek, On the correlation problem in atomic and molecular systems. calculation of wavefunction compo- nents in ursell-type expansion using quantum-field the- oretical methods, The Journal of Chemical Physics45, 4256 (1966)

1966
[67]

R. J. Bartlett and M. Musia l, Coupled-cluster theory in quantum chemistry, Rev. Mod. Phys.79, 291 (2007)

2007
[68]

Bishop, An overview of coupled cluster theory and its applications in physics, Theoretica chimica acta80, 95 (1991)

R. Bishop, An overview of coupled cluster theory and its applications in physics, Theoretica chimica acta80, 95 (1991)

1991
[69]

I. Y. Zhang and A. Gr¨ uneis, Coupled cluster theory in materials science, Frontiers in MaterialsV olume 6 - 2019, 10.3389/fmats.2019.00123 (2019)

2019
[70]

Gruber, K

T. Gruber, K. Liao, T. Tsatsoulis, F. Hummel, and A. Gr¨ uneis, Applying the coupled-cluster ansatz to solids and surfaces in the thermodynamic limit, Phys. Rev. X8, 021043 (2018)

2018
[71]

McClain, Q

J. McClain, Q. Sun, G. K.-L. Chan, and T. C. Berkel- bach, Gaussian-based coupled-cluster theory for the ground-state and band structure of solids, Journal of chemical theory and computation13, 1209 (2017)

2017
[72]

H. J. Monkhorst, Calculation of properties with the coupled-cluster method (1977)

1977
[73]

Arponen, R

J. Arponen, R. F. Bishop, E. Pajanne, and N. I. Robinson, Extended coupled-cluster method. iii. zero- temperature hydrodynamics of a condensed bose fluid, Phys. Rev. A37, 1065 (1988)

1988
[74]

Leidemann and G

W. Leidemann and G. Orlandini, Modern ab initio ap- proaches and applications in few-nucleon physics with A≥4, Progress in Particle and Nuclear Physics68, 158 (2013)

2013
[75]

Hagen, T

G. Hagen, T. Papenbrock, M. Hjorth-Jensen, and D. J. Dean, Coupled-cluster computations of atomic nuclei, Reports on Progress in Physics77, 096302 (2014)

2014
[76]

Yang, Z.-H

J. Yang, Z.-H. Cui, A. Mahajan, H. Zhai, D. R. Reich- man, and G. K.-L. Chan, Benchmarking the exponential ansatz for the holstein model, The Journal of Chemical Physics161, 104105 (2024)

2024
[77]

Y. Qiu, T. M. Henderson, J. Zhao, and G. E. Scuseria, Projected coupled cluster theory, The Journal of Chem- ical Physics147, 064111 (2017)

2017
[78]

J. F. Stanton and R. J. Bartlett, The equation of motion coupled-cluster method. a systematic biorthogonal ap- proach to molecular excitation energies, transition prob- abilities, and excited state properties, The Journal of Chemical Physics98, 7029 (1993)

1993
[79]

A. I. Krylov, Equation-of-motion coupled-cluster meth- ods for open-shell and electronically excited species: The hitchhiker’s guide to fock space, Annual Review of Physical Chemistry59, 433 (2008)

2008
[80]

H. Koch, H. J. A. Jensen, P. Jo/rgensen, and T. Hel- gaker, Excitation energies from the coupled cluster sin- gles and doubles linear response function (ccsdlr). appli- cations to be, ch+, co, and h2o, The Journal of Chemi- cal Physics93, 3345 (1990)

1990
[81]

Hirata, M

S. Hirata, M. Nooijen, and R. J. Bartlett, High-order de- terminantal equation-of-motion coupled-cluster calcula- tions for electronic excited states, Chemical Physics Let- ters326, 255 (2000)

2000
[82]

G. D. Mahan,Many-particle physics(Kluwer Aca- demic/Plenum Publishers, 2000)

2000
[83]

Bruus and K

H. Bruus and K. Flensberg,Many-body quantum the- ory in condensed matter physics : an introduction(New York, 2004)

2004
[84]

Bernardi, First-principles dynamics of electrons and phonons, The European physical journal

M. Bernardi, First-principles dynamics of electrons and phonons, The European physical journal. B, Condensed matter physics89, 1 (2016)

2016
[85]

W. P. Su, J. R. Schrieffer, and A. J. Heeger, Solitons in polyacetylene, Phys. Rev. Lett.42, 1698 (1979)

1979
[86]

D. J. J. Marchand, G. De Filippis, V. Cataudella, M. Berciu, N. Nagaosa, N. V. Prokof’ev, A. S. Mishchenko, and P. C. E. Stamp, Sharp transition for single polarons in the one-dimensional su-schrieffer- heeger model, Phys. Rev. Lett.105, 266605 (2010)

2010
[87]

Devreese, Fr\” ohlich polarons

J. Devreese, Fr\” ohlich polarons. lecture course in- cluding detailed theoretical derivations–, arXiv preprint arXiv:1611.06122 (2016)

2016 arXiv
[88]

E. J. Heller and D. Kim, Schr¨ odinger correspondence applied to crystals, J. Phys. Chem. A.123, 4379 (2019)

2019
[89]

D. Kim, A. Aydin, A. Daza, K. N. Avanaki, J. Keski- Rahkonen, and E. J. Heller, Coherent charge carrier dynamics in the presence of thermal lattice vibrations, Phys. Rev. B106, 054311 (2022). 22

2022
[90]

Y. Zhao, D. W. Brown, and K. Lindenberg, Varia- tional energy band theory for polarons: Mapping po- laron structure with the toyozawa method, The Journal of Chemical Physics107, 3159 (1997)

1997
[91]

Capek and D

V. Capek and D. Krausova, To the relation between en- ergies of molecular soliton and extended states in chains, Czechoslovak journal of physics37, 1201 (1987)

1987
[92]

Bolsterli, Translation invariance and localized states, Phys

M. Bolsterli, Translation invariance and localized states, Phys. Rev. D13, 1727 (1976)

1976
[93]

A. F. White, Y. Gao, A. J. Minnich, and G. K.-L. Chan, A coupled cluster framework for electrons and phonons, The Journal of Chemical Physics153, 224112 (2020)

2020
[94]

T. D. Crawford and H. F. Schaefer III, An introduction to coupled cluster theory for computational chemists, inReviews in Computational Chemistry(John Wiley & Sons, Ltd, 2000) pp. 33–136

2000
[95]

H. J. Monkhorst, Chemical physics without the born- oppenheimer approximation: The molecular coupled- cluster method, Phys. Rev. A36, 1544 (1987)

1987
[96]

Christiansen, A second quantization formulation of multimode dynamics, The Journal of Chemical Physics 120, 2140 (2004)

O. Christiansen, A second quantization formulation of multimode dynamics, The Journal of Chemical Physics 120, 2140 (2004)

2004
[97]

M. D. Prasad, Time-dependent coupled cluster method: A new approach to the calculation of molecular absorp- tion spectra, The Journal of Chemical Physics88, 7005 (1988)

1988
[98]

C. C. Gerry and P. L. Knight,Introductory Quantum Optics(Cambridge University Press, 2023)

2023
[99]

J. J. Sakurai and J. Napolitano,Modern Quantum Me- chanics(Cambridge University Press, 2020)

2020
[100]

T. D. Lee, F. E. Low, and D. Pines, The motion of slow electrons in a polar crystal, Phys. Rev.90, 297 (1953)

1953
[101]

Marie, F

A. Marie, F. Kossoski, and P.-F. Loos, Variational cou- pled cluster for ground and excited states, The Journal of Chemical Physics155, 104105 (2021)

2021
[102]

T. Shi, E. Demler, and J. Ignacio Cirac, Variational study of fermionic and bosonic systems with non- gaussian states: Theory and applications, Annals of Physics390, 245 (2018)

2018
[103]

Hackl, T

L. Hackl, T. Guaita, T. Shi, J. Haegeman, E. Demler, and J. I. Cirac, Geometry of variational methods: dy- namics of closed quantum systems, SciPost Phys.9, 048 (2020)

2020
[104]

Marijanovi´ c, Y.-F

F. Marijanovi´ c, Y.-F. Qu, and E. Demler, Su-schrieffer- heeger bipolaron in one dimension: Spectrum and dynamical response, Physical Review B113, 125140 (2026)

2026
[105]

Prelovˇ sek and X

P. Prelovˇ sek and X. Zotos, Transport of strongly corre- lated electrons, AIP Conference Proceedings629, 161 (2002)

2002
[106]

Zotos, F

X. Zotos, F. Naef, and P. Prelovsek, Transport and con- servation laws, Phys. Rev. B55, 11029 (1997)

1997
[107]

Kohn, Theory of the insulating state, Phys

W. Kohn, Theory of the insulating state, Phys. Rev. 133, A171 (1964)

1964
[108]

Bonˇ ca and S

J. Bonˇ ca and S. A. Trugman, Dynamic properties of a polaron coupled to dispersive optical phonons, Phys. Rev. B103, 054304 (2021)

2021
[109]

Nocera, J

A. Nocera, J. Sous, A. E. Feiguin, and M. Berciu, Bipolaron liquids at strong peierls electron-phonon cou- plings, Phys. Rev. B104, L201109 (2021)

2021
[110]

N. V. Prokof’ev and B. V. Svistunov, Phonon mod- ulated hopping polarons: x-representation technique, Physical Review B106, L041117 (2022)

2022
[111]

Zhang, N

C. Zhang, N. V. Prokof’ev, and B. V. Svistunov, Bond bipolarons: Sign-free monte carlo approach, Physical Review B105, L020501 (2022)

2022
[112]

T. Hahn, S. Klimin, J. Tempere, J. T. Devreese, and C. Franchini, Diagrammatic monte carlo study of fr¨ ohlich polaron dispersion in two and three dimensions, Phys. Rev. B97, 134305 (2018)

2018
[113]

Lu and R

Y. Lu and R. Rosenfelder, Second-order correction to feynman’s path-integral calculation of the polaron ef- fective mass, Phys. Rev. B46, 5211 (1992)

1992
[114]

Lafuente-Bartolome, C

J. Lafuente-Bartolome, C. Lian, W. H. Sio, I. G. Gur- tubay, A. Eiguren, and F. Giustino, Unified approach to polarons and phonon-induced band structure renor- malization, Phys. Rev. Lett.129, 076402 (2022)

2022
[115]

R. P. Feynman, Slow electrons in a polar crystal, Phys. Rev.97, 660 (1955)

1955
[116]

Fehske, J

H. Fehske, J. Loos, and G. Wellein, Spectral properties of the 2d holstein polaron, Zeitschrift f¨ ur Physik. B, Condensed matter104, 619 (1997)

1997
[117]

Hohenadler, M

M. Hohenadler, M. Aichhorn, and W. von der Linden, Spectral function of electron-phonon models by cluster perturbation theory, Phys. Rev. B68, 184304 (2003)

2003
[118]

Gao and Y

Q. Gao and Y. Wan, Summing real time feynman paths of lattice polarons with matrix product states, Phys. Rev. B112, 165121 (2025)

2025
[119]

Dzsotjan, R

D. Dzsotjan, R. Schmidt, and M. Fleischhauer, Dynam- ical variational approach to bose polarons at finite tem- peratures, Phys. Rev. Lett.124, 223401 (2020)

2020
[120]

Bonˇ ca, S

J. Bonˇ ca, S. A. Trugman, and M. Berciu, Spectral func- tion of the holstein polaron at finite temperature, Phys. Rev. B100, 094307 (2019)

2019
[121]

Nguyen, A

H. Nguyen, A. Mandal, A. Mahajan, and D. R. Re- ichman, Mixed quantum-classical methods for polaron spectral functions, The Journal of Chemical Physics 163, 114105 (2025)

2025
[122]

P. J. Robinson, I. S. Dunn, and D. R. Reichman, Cu- mulant methods for electron-phonon problems. i. per- turbative expansions, Phys. Rev. B105, 224304 (2022)

2022
[123]

Fratini and S

S. Fratini and S. Ciuchi, Optical properties of small po- larons from dynamical mean-field theory, Phys. Rev. B 74, 075101 (2006)

2006
[124]

G. L. Goodvin, A. S. Mishchenko, and M. Berciu, Op- tical conductivity of the holstein polaron, Phys. Rev. Lett.107, 076403 (2011)

2011
[125]

Makov and M

G. Makov and M. C. Payne, Periodic boundary condi- tions in ab initio calculations, Phys. Rev. B51, 4014 (1995)

1995
[126]

J. Lee, S. Zhang, and D. R. Reichman, Constrained-path auxiliary-field quantum monte carlo for coupled elec- trons and phonons, Phys. Rev. B103, 115123 (2021)

2021
[127]

J. Ren, W. Li, T. Jiang, Y. Wang, and Z. Shuai, Time-dependent density matrix renormalization group method for quantum dynamics in complex systems, WIREs Computational Molecular Science12, e1614 (2022)

2022
[128]

Van Voorhis and M

T. Van Voorhis and M. Head-Gordon, A geometric ap- proach to direct minimization, Mol. Phys.100, 1713 (2002)

2002
[129]

Epifanovsky, A

E. Epifanovsky, A. T. B. Gilbert, X. Feng, J. Lee, Y. Mao, N. Mardirossian, P. Pokhilko, A. F. White, M. P. Coons, A. L. Dempwolff, Z. Gan, D. Hait, P. R. Horn, L. D. Jacobson, I. Kaliman, J. Kussmann, A. W. Lange, K. U. Lao, D. S. Levine, J. Liu, S. C. McKen- zie, A. F. Morri...

2021
[130]

Bradbury, R

J. Bradbury, R. Frostig, P. Hawkins, M. J. John- son, Y. Katariya, C. Leary, D. Maclaurin, G. Nec- ula, A. Paszke, J. VanderPlas, S. Wanderman-Milne, and Q. Zhang, JAX: composable transformations of Python+NumPy programs (2018)

2018
[131]

Blondel, Q

M. Blondel, Q. Berthet, M. Cuturi, R. Frostig, S. Hoyer, F. Llinares-L´ opez, F. Pedregosa, and J.- P. Vert, Efficient and modular implicit differentiation, arXiv preprint arXiv:2105.15183 (2021)

2021 arXiv
[132]

J. W. Negele and H. Orland,Quantum Many-Particle Systems(1988)

1988
[133]

Chiesa, D

S. Chiesa, D. M. Ceperley, R. M. Martin, and M. Holz- mann, Finite-size error in many-body simulations with long-range interactions, Phys. Rev. Lett.97, 076404 (2006)

2006
[134]

L. M. Fraser, W. M. C. Foulkes, G. Rajagopal, R. J. Needs, S. D. Kenny, and A. J. Williamson, Finite- size effects and coulomb interactions in quantum monte carlo calculations for homogeneous systems with peri- odic boundary conditions, Phys. Rev. B53, 1814 (1996)

1996
[135]

Vasilchenko, A

V. Vasilchenko, A. Zhugayevych, and X. Gonze, Vari- ational polaron equations applied to the anisotropic fr¨ ohlich model, Phys. Rev. B105, 214301 (2022)

2022
[136]

Miglio, V

A. Miglio, V. Brousseau-Couture, E. Godbout, G. Anto- nius, Y.-H. Chan, S. G. Louie, M. Cˆ ot´ e, M. Giantomassi, and X. Gonze, Predominance of non-adiabatic effects in zero-point renormalization of the electronic band gap, npj computational materials6(2020)

2020
[137]

S. I. Pekar,Untersuchungen ¨ uber die Elektronentheorie der Kristalle(De Gruyter, 1954)

1954
[138]

S. J. Miyake, Strong-coupling limit of the polaron ground state, Journal of the Physical Society of Japan 38, 181 (1975)

1975
[139]

E. R. Davidson, The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices, Journal of Computa- tional Physics17, 87 (1975)

1975
[140]

R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users’ Guide(Society for In- dustrial and Applied Mathematics, 1998) https://epubs.siam.org/doi/pdf/10.1137/1.9780898719628. 24 APPENDICES Appendix A: Computational Details DMRG reference values are obtained with the Renormali...

1998 doi
[141]

implemented in a development version of Q–Chem [126], or a line-search limited memory BFGS (L-BFGS-LS) implementation in JAX [127, 128]. For theab initiocalculations, we work directly with theab initioband-dispersion ϵik , phonon frequenciesω νq, and electron–phonon couplingsg...
[142]

Reference choice The dCC calculations of this appendix employ the discrete machinery of the main text and require only a suitable choice of reference state. We use a coherent-state product reference in which the electron occupies an orbital drawn from a one-parameter family, w...
[143]

We apply three such auxiliary quantities

Finite-size correction and continuum convergence Each converged dCC energy is corrected by the exact discretization error of an auxiliary energy, whose continuum limit is known in closed form. We apply three such auxiliary quantities. With reference orbital coefficientsϕ k and...
[144]

A comparison of the ground-state energies obtained via different methods is shown in Fig

Ground-state energies and benchmark comparison Table A1 collects the dCC ground-state energies across the coupling range together with digitized diagrammatic Monte Carlo values [109]. A comparison of the ground-state energies obtained via different methods is shown in Fig. 3. ...
[145]

Dispersive-phonon band structure We augment the 1D Holstein model with a phonon hoppingt ph P i b† i+1bi + h.c., which yields the phonon dispersion ω(q) =ω 0 + 2tph cos(q) withω 0 the Einstein frequency. Figure A1 compares the resulting polaron band structure with numerically ...
[146]

Real-space observables For the Holstein model, the density-displacement correlation CH(r) :=⟨ X i ˆni ˆXi+r⟩,(A35) with ˆXi the displacement operatorb † i +b i, measures the lattice deformation around the carrier, while for the BSSH model the corresponding bond correlation fun...
[147]

Perturbative analysis In this appendix, we develop an intuition for the qualitative differences which arise in the observed 2D Holstein and BSSH optical conductivities. As the coupling we use (g= 1) is comparatively weak in the two-dimensional case, we utilize one-phonon pertu...
[148]

Sector decomposition The analysis above describes absorption out of the ground state. At finite temperature, initial states of nonzero total crystal momentum additionally contribute, and we characterize their absorption profiles with our tangent space response framework in Fig...

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Here we work directly in the frequency do- main, which allows us to handle the regular component and zero-frequency singular part separately

One dimension Figure 8 shows the real part of the optical conductiv- ity for the 1D Holstein model with 32 sites at intermedi- ate coupling (λ= 1) at several temperatures, compared with Lanczos (only atT= 0) and TD-DMRG calcula- tions [54]. Here we work directly in the frequency do- main, which allows us to handle the regular component and zero-frequency ...

[2] [2]

Two dimensions Finite-temperature optical conductivity remains par- ticularly challenging in two dimensions. Recent mixed quantum–classical Green–Kubo calculations have been applied to large two-dimensionalab initiosystems [51], but a systematically improvable fully quantum treatment at comparable scales remains unavailable and mixed quantum–classical app...

[3] [3]

Fr¨ ohlich, Electrons in lattice fields, Advances in Physics3, 325 (1954)

H. Fr¨ ohlich, Electrons in lattice fields, Advances in Physics3, 325 (1954)

1954

[4] [4]

L. D. Landau, Electron motion in crystal lattices, Physikalische Zeitschrift der Sowjetunion3, 664 (1933)

1933

[5] [5]

S. I. Pekar, Local quantum states of electrons in an ideal ion crystal, Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki16, 341 (1946)

1946

[6] [6]

Alexandrov, ed.,Polarons in Advanced Materials, Vol

A. Alexandrov, ed.,Polarons in Advanced Materials, Vol. 103 (Springer Netherlands, 2007)

2007

[7] [7]

Holstein, Studies of polaron motion: Part i

T. Holstein, Studies of polaron motion: Part i. the molecular-crystal model, Annals of Physics8, 325 (1959)

1959

[8] [8]

Holstein, Studies of polaron motion: Part ii

T. Holstein, Studies of polaron motion: Part ii. the “small” polaron, Annals of Physics8, 343 (1959)

1959