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arXiv:2608.18529

Universal thermal breakdown of polaron coherence in one, two, and three dimensions

Jeet Shannigrahi, J. Bonča, Mona Berciu

Published Aug 19, 2026• 0 citations
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Abstract

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 $m^*$ and inverse lifetime $1/τ$ increase monotonically with temperature until, at $T \sim 0.5\,Ω$, the quasiparticle peak dissolves into a broad incoherent thermal continuum. The phonon frequency $Ω$ 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-$T$ Density Matrix Renormalization Group (DMRG).

DOI
10.48550/arxiv.2608.18529

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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).
We compute the Holstein polaron spectral function on a one-dimensional ring using the finite-temperature TTT Lanczos method. With increasing TTT additional features in the spectral function emerge even at temperatures below the phonon frequency. We observe a substantial spread of the spectral weight towards lower frequencies and the broadening of the quasiparticle (QP) peak. In the weak-coupling regime the QP peak merges with the continuum in the high-TTT limit. In the strong-coupling regime the main features of the low-TTT spectral function remain detectable up to the highest TTT used in our calculations. The effective polaron mass shows a nonmonotonic behavior as a function of TTT at small phonon frequency but increases with TTT at larger frequencies. The self-energy remains kkk independent even at elevated TTT in the frequency range corresponding to the polaron band, while at higher frequencies it develops a distinguishable kkk dependence. Analytical expressions for the first few frequency moments are derived, and they agree well with those extracted from numerical calculations in a wide-TTT regime.
Many condensed matter problems involve a particle coupled to its environment. The polaron, originally introduced to describe electrons in a polarizable medium, describes a particle coupled to a bosonic field. The Holstein polaron model, although simple, including only optical Einstein phonons and an interaction that couples them to the electron density, captures almost all of the standard polaronic properties. We herein investigate polarons that differ significantly from this behaviour. We study a model with phonon-modulated hopping, and find a radically different behaviour at strong couplings. We report a sharp transition, not a crossover, with a diverging effective mass at the critical coupling. We also look at a model with acoustic phonons, away from the perturbative limit, and again discover unusual polaron properties. Our work relies on the Bold Diagrammatic Monte Carlo (BDMC) method, which samples Feynman diagrammatic expansions efficiently, even those with weak sign problems. Proposed by Prokof'ev and Svistunov, it is extended to lattice polarons for the first time here. We also use the Momentum Average (MA) approximation, an analytical method proposed by Berciu, and find an excellent agreement with the BDMC results. A novel MA approximation able to treat dispersive phonons is also presented, along with a new exact solution for finite systems, inspired by the same formalism.
We provide a comprehensive solution to the lattice dynamics problem in the two dimensional Holstein model at finite electron density and finite temperature. We work in the physically relevant adiabatic regime and vary the electron-phonon interaction from the weak coupling perturbative window to the strong coupling polaronic regime. We explore three typical electron densities, dilute - where spatial correlations between polarons is weak, intermediate - where correlations are significant, and half-filling - where there is long range checkerboard order at low temperature. We use two methods both of which exploit the "slowness" of the phonons to handle the problem. These are (i)~a standard random phase approximation (RPA), adapted to capture small quantum fluctuations on Monte Carlo generated classical thermal backgrounds, and (ii)~a Langevin dynamics scheme, with a simplified ``thermal noise'', that can address large amplitude dynamical fluctuations. The Langevin scheme, as we argue in the paper, is the superior method in the strong coupling part of the phase diagram, where lattice distortions are large. It reveals a non trivial multi-peak momentum resolved spectrum with a high energy part, on the scale of the bare phonon frequency ΩΩΩ, and a low energy peak at ω≪Ωω\ll Ωω≪Ω. Below the polaronic threshold, the high energy dispersion changes only modestly with temperature TTT, while the broadening, arising from mode coupling, increases linearly with TTT at low temperature. The low energy peak shows up at strong coupling and finite temperature and arises from the slow tunneling of polarons. The tunneling events become spatially correlated as electron density increases towards half-filling, and the weight becomes strongly momentum and temperature dependent. We suggest the analytic basis of these results.
We examine the range of validity of the second-order cumulant expansion (CE) for the calculation of spectral functions, quasiparticle properties, and mobility of the Holstein polaron. We devise an efficient numerical implementation that allows us to make comparisons in a broad interval of temperature, electron-phonon coupling, and phonon frequency. For a benchmark, we use the dynamical mean-field theory which gives, as we have recently shown, rather accurate spectral functions in the whole parameter space, even in low dimensions. We find that in one dimension, the CE resolves well both the quasiparticle and the first satellite peak in a regime of intermediate coupling. At high temperatures, the charge mobility assumes a power law $\ensuremath{\mu}\ensuremath{\propto}{T}^{\ensuremath{-}2}$ in the limit of weak coupling and $\ensuremath{\mu}\ensuremath{\propto}{T}^{\ensuremath{-}3/2}$ for stronger coupling. We find that, for stronger coupling, the CE gives slightly better results than the self-consistent Migdal approximation (SCMA), while the one-shot Migdal approximation is appropriate only for a very weak electron-phonon interaction. We also analyze the atomic limit and the spectral sum rules. We derive an analytical expression for the moments in CE and find that they are exact up to the fourth order, as opposed to the SCMA where they are exact to the third order. Finally, we analyze the results in higher dimensions.
We present a systematic study of a mobile impurity immersed in a three-dimensional Fermi sea of fermions at finite temperature, by using the standard non-self-consistent many-body TTT-matrix theory that is equivalent to a finite-temperature variational approach with the inclusion of one-particle-hole excitation. The impurity spectral function is determined in the real-frequency domain, avoiding any potential errors due to the numerical analytic continuation in previous TTT-matrix calculations and the small spectral broadening parameter used in variational calculations. In the weak-coupling limit, we find that the quasiparticle decay rate of both attractive and repulsive polarons does not increase significantly with increasing temperature, and therefore Fermi polarons may remain well defined far above Fermi degeneracy. In contrast, near the unitary limit with strong coupling, the decay rate of Fermi polarons rapidly increases and the quasiparticle picture breaks down close to the Fermi temperature. We analyze in detail the recent ejection and injection radio-frequency (rf) spectroscopy measurements, performed at Massachusetts Institute of Technology (MIT) and at the European Laboratory for Non-Linear Spectroscopy (LENS), respectively. We show that the momentum average of the spectral function, which is necessary to account for the observed rf spectroscopy, has a sizable contribution to the width of the quasiparticle peak in spectroscopy. As a result, the measured decay rate of Fermi polarons could be significantly larger than the calculated quasiparticle decay rate at zero momentum. By taking this crucial contribution into account, we find that there is a reasonable agreement between theory and experiment for the lifetime of Fermi polarons in the strong-coupling regime, as long as they remain well defined.
We study the impact of the many-body effects on the phonon spectral function of Holstein polaron in one-dimension in the antiadiabatic regime by incorporating the contributions from the electron self-energy and vertex corrections within a weak-coupling approach that respects the charge-conserving Ward identity. We find that while the polaronic spectral weight is suppressed due to contribution from the electron self-energy, on the other hand, the same is enhanced due to contribution from the vertex corrections. While strength of both the contributions increases with increasing the wave vector ($\q$) of phonons, they nearly cancel each other for the small-$\q$ modes so that the polaronic spectral weight is weakly affected due to the many-body effects. For the large-$\q$ modes near the zone boundary, the net many-body correction is dominated by the contribution of the electron self-energy which increases faster in comparison to that of the vertex corrections with increasing the wave vector thereby resulting in a significant suppression of the polaronic spectral weight. We find that while the weak-coupling perturbative approach provides a reliable estimation of the impact of the many-body effects deep inside the antiadibatic regime, the renormalization of quasiparticle spectrum must be taken into account for an accurate estimation when the phonon energy approaches the electronic bandwidth.