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Physical review. B, Condensed matter

Dynamic correlation functions in quantum many-body systems at zero temperature

Eduardo R. Gagliano, C. A. Balseiro

Published Dec 1, 1988• 52 citations
Publisher

Abstract

We present a simple numerical approach to compute dynamic correlation functions in quantum many-body systems at zero temperature. The method is based on a projective technique for the memory-function formalism, and requires as a starting point an independent evaluation of the ground-state energy and wave function. We illustrate the method with a one-dimensional model of spinless fermions and calculate the density-density, current-current, and single-particle correlation functions. The obtained spectral functions show different types of elementary excitations: particle-hole, soliton-antisoliton, exciton, etc. We compare our results with analytical results, and with those obtained through different approaches available in the literature.

DOI
10.1103/physrevb.38.11766

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We present a simple numerical approach to compute dynamic correlation functions in quantum many-body systems at zero temperature. The method is based on a projective technique for the memory-function formalism, and requires as a starting point an independent evaluation of the ground-state energy and wave function. We illustrate the method with a one-dimensional model of spinless fermions and calculate the density-density, current-current, and single-particle correlation functions. The obtained spectral functions show different types of elementary excitations: particle-hole, soliton-antisoliton, exciton, etc. We compare our results with analytical results, and with those obtained through different approaches available in the literature.
We present a method to compute dynamical correlation functions in quantum many-body systems at zero temperature. The ground state of a finite system is evaluated exactly by a modified Lanczos method and from it the real-frequency correlation functions are obtained by a projective technique for the memory-function formalism. We apply the method to the density-density correlation function of a one-dimensional spinless fermion system. Exact results are recovered in the noninteracting limit; in the strong-coupling limit our results are compared with Monte Carlo simulation and analytical approximations.
We present a method to obtain dynamical correlation functions of quantum many-body systems in the real-frequency domain from Monte Carlo data of the corresponding imaginary-time Green's functions. The method is based on a least-squares-fit procedure to solve the integral equation relating imaginary-time and real-frequency correlation functions. To demonstrate its feasibility, we have applied our method to imaginary-time Monte Carlo data for the density correlation function of a simple model of interacting spinless fermions in one dimension. We compare our results to the analytical results, available in the limits of zero and strong interaction. We find that despite the presence of noise in the input data, the real-frequency spectra obtained with our method are qualitatively correct. They reflect accurately the particle-hole and the soliton-antisoliton excitations, present in the noninteracting and in the strongly coupled fermion system, respectively. We also compare our method to other approaches that have been proposed for obtaining dynamical correlation functions via Monte Carlo simulation.
We propose a least-squares-fit method to extract real-frequency correlation functions from imaginary-time Monte Carlo data of a many-particle system. We apply it to the density correlation function in a one-dimensional model of spinless fermions. The spectral functions thus obtained reflect accurately the particle-hole and the soliton-antisoliton excitations of the system in the respective limits of noninteracting and strongly coupled fermions. They are in qualitative agreement with the available analytical results.
We study the correlations in a family of ground-state wave functions for one-dimensional quantum many-body systems. For a particular ordering of the M particles, these wave functions are given as the absolute value of the free-fermion ground-state wave function, raised to an arbitrary power \ensuremath{\lambda}. For other permutations of the particles, we use the particle statistics---boson or fermion---to construct the wave function. This prescription works equally well on the continuum line of length L or on the discrete lattice of size N. The theory of random matrices singles out the ``special'' values of \ensuremath{\lambda}=1/2, 1, and 2, and provides techniques for calculating all correlations. Results obtained in this work include analytic expressions for the lattice momentum distributions, accurate numerical results for all momentum distributions and their critical exponents, and explicit results for the lattice pair-correlation functions. We emphasize that this is only for the stated special values of \ensuremath{\lambda}. Finally, these wave functions are evaluated as candidates for variational wave functions.
This thesis is concerned with the numerical study of one-dimensional (1D) spin-1/2 quantum magnets and related method development. Its focus is on the calculation of dynamical spin correlation functions both at zero and finite temperature. This is motivated by the accessibility of dynamical quantities in experiments such as inelastic neutron scattering (INS) and electron spin resonance (ESR). The numerical methods used in this thesis are based on extensions of the density-matrix renormalization group (DMRG) and are formulated in the framework of matrix product states (MPS). While zero-temperature dynamical correlation functions are computed with existing MPS frequency-domain methods, an MPS frequency-domain approach for their calculation at finite temperature is developed in this thesis. The new method combines the Liouville-space formulation of the dynamics with a moment expansion of the dynamical correlation function. The majority of the results are obtained via MPS-based Chebyshev expansions. These numerical techniques are applied to two different model systems describing real materials. The first one is the material copper pyrimidine dinitrate (Cu-PM) which is modeled by a 1D spin-1/2 Heisenberg antiferromagnet with Dzyaloshinskii-Moriya interactions. The spin dynamics of this model is studied in an applied magnetic field and compared to ESR experiments. Zero-temperature calculations for momentum- and frequency-resolved dynamical quantities give direct access to the intensity of the elementary excitations and go beyond the low-energy description by the quantum sine-Gordon model. Thus, a deviation from the Lorentz invariant dispersion for the single-soliton resonance is found. The presence of the strongest boundary bound state previously predicted from a boundary sine-Gordon field theory is confirmed, while composite boundary-bulk excitations have too low intensities to be found in the numerical results. At finite temperature, there is a temperature-induced crossover of the soliton. Moreover, additional temperature effects such as interbreather transitions emerge, which is confirmed by accompanying ESR experiments on Cu-PM over a wide range of the applied field strength. The second system studied in this thesis is the compound BaCu2_22​V2_22​O8_88​. It is shown that the magnetic properties of this quasi-1D material can be described by a strongly alternating antiferromagnetic-ferromagnetic spin-1/2 Heisenberg chain. As found for other dimerized systems, the strong correlations in BaCu2_22​V2_22​O8_88​ persist even at elevated temperatures. Moreover, these correlations lead to an asymmetric lineshape broadening of the magnetic excitations at finite temperature. Upon raising the temperature, an increasingly asymmetric lineshape is observed in high-resolution INS experiments for BaCu2_22​V2_22​O8_88​. In this thesis, the lineshape is calculated by the MPS finite-temperature method developed as a part of this work. Comparing these results to the INS data and the lineshape obtained by a diagrammatic approach, excellent agreement is found over a broad temperature range. This demonstrates that coherent quantum behavior persists at elevated temperatures in BaCu2_22​V2_22​O8_88​ and that it can be predicted quantitatively.
The many-body problem is usually approached from one of two perspectives: the first originates from an action and is based on Feynman diagrams, the second is centered around a Hamiltonian and deals with quantum states and operators.The connection between results obtained in either way is made through spectral (or Lehmann) representations, well known for two-point correlation functions.Here, we complete this picture by deriving generalized spectral representations for multipoint correlation functions that apply in all of the commonly used many-body frameworks: the imaginary-frequency Matsubara and the realfrequency zero-temperature and Keldysh formalisms.Our approach separates spectral from time-ordering properties and thereby elucidates the relation between the three formalisms.The spectral representations of multipoint correlation functions consist of partial spectral functions and convolution kernels.The former are formalism independent but system specific; the latter are system independent but formalism specific.Using a numerical renormalization group method described in the accompanying paper, we present numerical results for selected quantum impurity models.We focus on the four-point vertex (effective interaction) obtained for the single-impurity Anderson model and for the dynamical mean-field theory solution of the one-band Hubbard model.In the Matsubara formalism, we analyze the evolution of the vertex down to very low temperatures and describe the crossover from strongly interacting particles to weakly interacting quasiparticles.In the Keldysh formalism, we first benchmark our results at weak and infinitely strong interaction and then reveal the rich real-frequency structure of the dynamical mean-field theory vertex in the coexistence regime of a metallic and insulating solution.