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arXiv:2202.00963 · Physical review. E

Projective-truncation-approximation study of the one-dimensional ϕ4 lattice model

Kou-Han Ma, Yanjiang Guo, Lei Wang, Ning-Hua Tong

Published Jul 11, 2022• 4 citations
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Abstract

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.

DOI
10.1103/physreve.106.014110

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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.
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.
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 ${\ensuremath{\phi}}^{4}$ 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 HHH-expanded basis to quantum systems and of the applicability of the Debye formula for thermal conductivity are discussed.
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 derive the self-consistent random phase approximations (sc-RPA) from the projective truncation approximation (PTA) for the equation of motion of two-time Green's function. The obtained sc-RPA applies to arbitrary temperature and recovers the Rowe's formalism at zero temperature. The PTA formalism not only rationalize Rowe's formula, but also provides a general framework to extend sc-RPA. We implement the sc-RPA calculation for the one-dimensional spinless fermion model in the parameter regime of disordered ground state, with the N-representability constraints enforced. The obtained ground state energy, correlation function, and density spectral function agree well with existing results. The features of the Luttinger liquid ground state and the continuum/bound state in the spectral function are well captured. We discuss several issues concerning the approximations made in RPAs, difficulties of RPA for symmetric state, and the static component problem of PTA.
Recently, we developed the projective truncation approximation for the equation of motion of two-time Green's functions (Fan et al. , Phys. Rev. B 97 , 165140 (2018)). In that approximation, the precision of results depends on the selection of operator basis. Here, for three successively larger operator bases, we calculate the local static averages and the impurity density of states of the single-band Anderson impurity model. The results converge systematically towards those of numerical renormalization group as the basis size is enlarged. We also propose a quantitative gauge of the truncation error within this method and demonstrate its usefulness using the Hubbard-I basis. We thus confirm that the projective truncation approximation is a method of controllable precision for quantum many-body systems.
In this paper, we study the thermodynamical properties of the classical one-dimensional Klein-Gordan lattice model (n≥2n \ge 2n≥2) by using the cluster variation method with linear response theory. The results of this method are exact in the thermodynamical limit. We present the single site reduced density matrix ρ(1)(z)ρ^{(1)}(z)ρ(1)(z), averages such as ⟨z2⟩\langle z^2 \rangle⟨z2⟩, ⟨∣zn∣⟩\langle |z^n|\rangle⟨∣zn∣⟩, and ⟨(z1−z2)2⟩\langle (z_1-z_2)^2\rangle⟨(z1​−z2​)2⟩, the specific heat CvC_vCv​, and the static correlation functions. We analyzed the scaling behavior and obtained the exact scaling powers of these quantities in the low and high temperaures. Using these results, we gauge the accuracy of the projective truncation approximation for ϕ4ϕ^{4}ϕ4 lattice model.