The thermopower properties of interacting systems

The gist: Additional interaction scales can indeed enhance the Seebeck coefficient and further induce multiple sign changes as a function of doping

Introduction and Motivation

The Seebeck coefficient quantifies the voltage generated across a material in response to a temperature gradient It is important to note that optimizing thermoelectric performance is challenging because it requires not only a large S, but also high electrical conductivity and low thermal conductivity Strong electronic correlations offer a promising new avenue for its enhancement because correlated systems exhibit non-rigid band structures which may be tuned to achieve more favorable combinations of S, σ, and κ Correlated narrow-gap semiconductors like FeSi and FeSb2 exhibit exceptionally large Seebeck coefficients that significantly exceed predictions of conventional electronic band structure theories Cuprates also exhibit an anomalous Seebeck coefficient, such as a pronounced decrease of S/T to negative values in underdoped YBa2Cu3O6+y (YBCO) at low temperatures

Methodology

The study investigates the Seebeck coefficient in the presence of attractive interactions, nearest-neighbor interactions, sublattice potentials, and electron-phonon coupling The Seebeck coefficient is obtained via the Kelvin formula using entropy data derived from density calculations within determinant quantum Monte Carlo (DQMC) The general Hamiltonian considered is H = HK + HU + HV + Hel−ph + Hph, which includes hopping energy, onsite and nearest-neighbor interactions, local electron-ion interaction, and optical phonon modes Thermodynamic and transport properties are obtained by performing Determinant Quantum Monte Carlo (DQMC) simulations The entropy is computed using the formula s(µ, T) = Z µ / µ0 dµ' ∂n / ∂T µ' The Seebeck coefficient is then calculated using the Kelvin formula SKelvin = -1/e dµ / ∂T n = 1/e dS / dn T

Results on Interaction Scales

The investigation covers four limits of the general Hamiltonian, including the attractive Hubbard model (AHM) where U < 0, the extended Hubbard model (EHM) which includes nearest-neighbor electron-electron interactions and electron-phonon effects, and the ionic Hubbard model which introduces a staggered on-site potential

1. In the attractive Hubbard model, for U/t = -6 at half-filling, the ground state supports a supersolid phase, while away from half-filling it exhibits s-wave superconducting order The Seebeck coefficient exhibits a sign change for n!= 1 driven by electronelectron interactions in the repulsive case, but in the attractive and noninteracting cases, the only sign change occurs at half-filling due to particle-hole symmetry

2. In the extended Hubbard model around half-filling, adding a nearest-neighbor repulsion V > 0 tends to localize pairs on one sublattice, producing a CDW pattern which opens a charge gap at half-filling The thermopower anomaly is larger in the case with NN interactions than in the on-site case

3. In the extended Hubbard model around quarter-filling, an additional sign change of the Seebeck coefficient is observed, which is robust for T/t ≲ 1 This sign change is associated with a Fermi surface reconstruction (FSR) driven by electronic interactions

4. In the Holstein-Hubbard model, electron-phonon coupling smooths out the Mott plateau already visible in the pure Hubbard model, and increasing λ at fixed U = 0 effectively narrows the bandwidth leading to behavior similar to that of the attractive Hubbard model The presence of both electron-electron and electron-phonon interactions suppresses the anomaly of the Seebeck coefficient

Fermi Surface Reconstruction

The sign change in the Seebeck coefficient is a hallmark of a change in carrier type, potentially associated with a Fermi surface reconstruction driven by electronic interactions This is evidenced by analyzing spectral functions A(k, ω = 0) at the Fermi level for U/t = 10 and different electronic densities For densities where the Seebeck coefficient changes sign, the peaks of A(k, ω = 0) shift in a manner consistent with a hole-like FS This behavior persists for other values of U and temperature, directly associating the sign change in the Seebeck coefficient with an FSR

Conclusion

The results establish the Seebeck coefficient as a sensitive probe of interaction-driven changes in electronic structure, including Mott gap formation and Fermi-surface reconstruction The onset of the Seebeck anomaly is closely connected to the emergence of insulating behavior and to the corresponding metal-insulator boundaries Additional interaction scales can significantly enhance the thermopower response, with nearest-neighbor interactions strongly amplifying the Seebeck anomaly near half-filling in the extended Hubbard model The study suggests that future research should investigate spin Seebeck effect in strongly correlated systems

Acknowledgements

The authors thank R.G. Pereira for enlightening discussions about this work The authors acknowledge financial support from the Brazilian funding agencies Conselho Nacional de Desenvolvimento Científicos e Tecnológico (CNPq), Coordenação de Aperfeiçoamento de Pessoal de Ensino Superior (CAPES), and Fundação de Amparo à Pesquisa do Estado do Rio Grande do Norte (FAPERJ) T.P. acknowledges support from FAPERJ Grants No. E-26/210.100/2023, E-26/210.781/2025, and E-26/200.230/26 N.C.C acknowledges support from FAPERJ Grants No. E-26/ 998/ 35 / 19, Serrapilheira Institute Grant No. R-2502-52037, and CNPq grant Nos. 308130/25-1, Serrapilheira Institute Grant No. R-2502-52037 CAPES for funding, the International Institute of Physics of UFRN (IIP-UFRN), where part of this work was completed (Grant No. 1699/24 IIF-FINEP), and CENAPAD-SP (National Center for High Performance Processing in São Paulo) for computational resources, project UNICAMP/FINEP-MCTI The paper is organized as follows. Section II introduces the model Hamiltonians, the determinant quantum Monte Carlo (DQMC) method, and the physical observables considered Section III presents results for (A) the attractive Hubbard model, (B) the extended Hubbard model, (C) the Holstein and Hubbard-Holstein models, and (D) the ionic Hubbard model In Section IV we discuss the relation between the Fermi surface reconstruction and the Seebeck anomaly Our conclusions and perspectives are given in Section V Appendix A: Cluster Perturbation Theory The results are presented below.

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The density profiles, local density of states, and entropy were shown in Figure 2 (a), including a dip at n = 1 for V /t = 1 Consequently, the Seebeck coefficient exhibits anomalies near half-filling for these parameters Notice that the thermopower anomaly is larger in the case with NN interactions than in the on-site case Interestingly, in the CDW phase, the charge gap directly measures the order parameter

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The results for temperature T /t = 0.5, with U/t = -6 and V /t = 0 the density profile varies mostly around µ = 0, thus we used δµ = 0.1 for µ < 2 and δµ = 0.2 otherwise For U/t = -6 and V /t = 1, we used δµ = 0.6 in the plateaus, where there is a small density variation with µ, and δµ = 0.

1. textbfFundamental Thermoelectric Property Prediction from Interactions: Develop a predictive model for Seebeck coefficient anomalies based on interaction type and doping level, as demonstrated by investigating attractive interactions versus repulsive interactions in the AHM (Section III A). This system can predict when an anomalous sign change in SKelvin is expected based on whether the interaction strength U/t is positive or negative at half-filling, noting that for attractive and noninteracting cases, the only sign change occurs at halffilling due to particle-hole symmetry.

2. textbfFermi Surface Reconstruction (FSR) Identification: Implement a method to diagnose Fermi surface topology changes using spectral weight analysis near the Fermi level, as shown in Section IV. This improved system can distinguish between an electron-like closed FS and a hole-like FS by analyzing shifts in spectral peaks of A(k, ω = 0) when the Seebeck coefficient changes sign.

3. textbf Charge Density Wave (CDW) Order Detection: Create an automated tool to detect incipient CDW order using the charge structure factor S CDW(q), as analyzed in Section III B. This system can map a metallic region (dark blue color) and a CDW region (dark green color) in the phase diagram of the Extended Hubbard Model, signaling potential thermoelectric enhancement linked to CDW order.

4. textbf Phonon-Mediated Transport Modeling: Develop a simulation capability to quantify how electron-phonon coupling suppresses or enhances thermopower, as analyzed in Section III C. This tool can predict that the presence of both electron-electron and electron-phonon interactions suppresses the anomaly of the Seebeck coefficient in the Holstein-Hubbard model, clarifying complex transport regimes.

5. textbf Phase Boundary Mapping for Insulating Regimes: Build a system capable of mapping phase transitions between metallic, bandinsulating, and Mott-insulating states using the ionic Hubbard model (Section III D). This allows the system to determine that the Seebeck coefficient clearly distinguishes the metallic, band-insulating, and Mott-insulating regimes, capturing transitions where the onset of the Seebeck anomaly is closely connected to the emergence of insulating behavior.

The Seebeck coefficient quantifies the voltage generated across a material in response to a temperature gradient. Recent studies have shown that strong electronic correlations can enhance this coefficient, producing anomalous behavior near half-filling associated with the Mott plateau. This raises the possibility that other interaction scales, not necessarily originating in Mott physics, could give rise to similar enhancements. Here, we investigate the Seebeck coefficient in the presence of attractive interactions, nearest-neighbor interactions, sublattice potentials, and electron-phonon coupling. The Seebeck coefficient is obtained via the Kelvin formula, using entropy data derived from density calculations within determinant quantum Monte Carlo (DQMC). We find that these additional interaction scales can indeed enhance the Seebeck coefficient and further induce multiple sign changes as a function of doping. We show that this anomalous behavior is associated with the opening of a gap in the ground state, as computed via cluster perturbation theory (CPT). Moreover, electron-phonon coupling alone-even in the absence of on-site repulsion-can produce a Seebeck anomaly. We relate these sign changes to a restructuring of the Fermi surface and an accompanying change in its topology, an effect commonly observed in cuprates.