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Physical review. B./Physical review. B

Thermal broadening of the phonon spectral function in classical lattice models: Projective truncation approximation

Hu-Wei Jia, Wen-Jun Liu, Yue-Hong Wu, Kou-Han Ma, Lei Wang, Ning-Hua Tong

Published Jan 24, 2025• 3 citations
Publisher

Abstract

Thermal broadening of the quasiparticle peak in the spectral function is an important physical feature in many statistical systems, but it is difficult to calculate. To tackle this problem, we propose the $H$-expanded basis within the projective truncation approximation (PTA) of the Green's function equation of motion. A zeros-removing technique is introduced to stabilize the iterative solution of the PTA equations. Benchmarking calculations on the classical one-variable anharmonic oscillator model and the one-dimensional ${\ensuremath{\phi}}^{4}$ lattice model show that the thermal broadened quasiparticle peak in the spectral function can be produced on a semiquantitative level. Using this method, we discuss the low- and high-temperature power-law behaviors of the spectral width ${\mathrm{\ensuremath{\Gamma}}}_{k}(T)$ of the one-dimensional ${\ensuremath{\phi}}^{4}$ model, finding it in contradiction with the assumption of effective phonon theory. A short-chain limit of this model is also discovered. Issues of extending the $H$-expanded basis to quantum systems and of the applicability of the Debye formula for thermal conductivity are discussed.

DOI
10.1103/physrevb.111.045153

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Thermal broadening of the quasiparticle peak in the spectral function is an important physical feature in many statistical systems, but it is difficult to calculate. To tackle this problem, we propose the HHH-expanded basis within the projective truncation approximation (PTA) of the Green's function equation of motion. A zeros-removing technique is introduced to stabilize the iterative solution of the PTA equations. Benchmarking calculations on the classical one-variable anharmonic oscillator model and the one-dimensional \ensuremathϕ4{\ensuremath{\phi}}^{4}\ensuremathϕ4 lattice model show that the thermal broadened quasiparticle peak in the spectral function can be produced on a semiquantitative level. Using this method, we discuss the low- and high-temperature power-law behaviors of the spectral width \ensuremathΓk(T){\mathrm{\ensuremath{\Gamma}}}_{k}(T)\ensuremathΓk​(T) of the one-dimensional \ensuremathϕ4{\ensuremath{\phi}}^{4}\ensuremathϕ4 model, finding it in contradiction with the assumption of effective phonon theory. A short-chain limit of this model is also discovered. Issues of extending the HHH-expanded basis to quantum systems and of the applicability of the Debye formula for thermal conductivity are discussed.
Thermal broadening of the quasi-particle peak in the spectral function is an important physical feature in many statistical systems, but it is difficult to calculate. To tackle this problem, we propose the HHH-expanded basis within the projective truncation approximation (PTA) of the Green's function equation of motion. A zeros-removing technique is introduced to stabilize the iterative solution of the PTA equations. Benchmarking calculations on the classical one-variable anharmonic oscillator model and the one-dimensional ϕ4ϕ^4ϕ4 lattice model show that the thermal broadened quasi-particle peak in the spectral function can be produced on a semi-quantitative level. Using this method, we discuss the low- and high- temperature power-law behaviors of the spectral width Γk(T)Γ_k(T)Γk​(T) of the one-dimensional ϕ4ϕ^4ϕ4 model, finding it in contradiction with the assumption of effective phonon theory. A short-chain limit of this model is also discovered. Issues of extending the HHH-expanded basis to quantum systems and of the applicability of the Debye formula for thermal conductivity are discussed.
In this paper, we first develop the projective truncation approximation (PTA) in the Green's function equation of motion (EOM) formalism for classical statistical models. To implement PTA for a given Hamiltonian, we choose a set of basis variables and projectively truncate the hierarchical EOM. We apply PTA to the one-dimensional ϕ^{4} lattice model. Phonon dispersion and static correlation functions are studied in detail. Using one- and two-dimensional bases, we obtain results identical to and beyond the quadratic variational approximation, respectively. In particular, we analyze the power-law temperature dependence of the static averages in the low- and high-temperature limits, and we give exact exponents.
In previous works, the projected truncation approximation (PTA) was developed as a systematic and controlled method to truncate the equation of motion of Green's functions (GFs) for a given quantum or classical many-body Hamiltonian. The static averages are obtained self-consistently with the GF through the spectral theorem. In this work, PTA is reformulated as a self-consistent theory for the reduced density matrices (RDMs) without reference to GF. We separately discuss the issues of determining the dynamical matrix M{\bf M}M and solving the physical quantities from it. The properties of M{\bf M}M is clarified and the solution of PTA equations is cast into an over-constrained optimization problem. This makes connection of the present theory to the variational RDM theory. We discuss various issues of PTA under this formalism, including the scheme of alternative inner product, the generalized virial theorem, the generalized Wick's theorem, and the static component problem of PTA.
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.
We introduce an alternative approach for obtaining smooth finite-temperature spectral functions of quantum impurity models using the numerical renormalization group (NRG) technique. It is based on calculating first the Green's function on the imaginary-frequency axis, followed by an analytic continuation to the real-frequency axis using Pad\'e approximants. The arbitrariness in choosing a suitable kernel in the conventional broadening approach is thereby removed and, furthermore, we find that the Pad\'e method is able to resolve fine details in spectral functions with less artifacts on the scale of $\ensuremath{\omega}\ensuremath{\sim}T$. We discuss the convergence properties with respect to the NRG calculation parameters (discretization $\ensuremath{\Lambda}$, zzz-averaging, truncation cutoff) and the number of Matsubara points taken into account in the analytic continuation. We test the technique on the single-impurity Anderson model and the Hubbard model (within the dynamical mean-field theory). For the Anderson impurity model, we discuss the shape of the Kondo resonance and its temperature dependence. For the Hubbard model, we discuss the inner structure of the Hubbard bands in metallic and insulating solutions at half-filling, as well as in the doped Mott insulator. Based on these test cases, we conclude that the Pad\'e approximant approach provides improved results for spectral functions at low-frequency scales of $\ensuremath{\omega}\ensuremath{\lesssim}T$ and that it is capable of resolving sharp spectral features also at high frequencies.
We study thermal broadening of the hole spectral function of the two-dimensional $t\text{\ensuremath{-}}J$ model (and its extensions) within the noncrossing approximation (NCA) with and without the contribution of optical phonons. We find that phonons at finite temperature broaden the lowest energy quasiparticle peak, however, the string excitations survive even for relatively strong electron-phonon coupling. Experimental angle-resolved photoemission spectroscopy (ARPES) results compare well with our calculations at finite temperature when we use strong electron-phonon coupling without any ad hoc broadening. In addition, we have studied the role of vertex corrections and we find that their contribution allows us to achieve the same overall agreement with the ARPES experimental results but using smaller values for the electron-phonon coupling $(\ensuremath{\gamma}\ensuremath{\sim}0.5)$ than that required $(\ensuremath{\gamma}\ensuremath{\sim}1)$ when restricting ourselves within NCA.