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arXiv:2103.15496 · Condensed Matter Physics

Spectral density of interacting pairs in low-dimensional finite lattices

Tirthaprasad Chattaraj

Published Mar 1, 2021• 0 citations
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Abstract

The spectral density of bound pairs in ideal 1D, 2D and Bethe lattices is computed for weak and strong interactions. The computations are performed with Green's functions by an efficient recursion method in real space. For the range of interaction strengths within which bound states are predominantly single pairs, the spectral profiles guide to the energy bandwidths where the bound pairs can be maximized.

DOI
10.5488/cmp.24.13302

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The spectral density of bound pairs in ideal 1D, 2D and Bethe lattices is computed for weak and strong interactions. The computations are performed with Green's functions by an efficient recursion method in real space. For the range of interaction strengths within which bound states are predominantly single pairs, the spectral profiles guide to the energy bandwidths where the bound pairs can be maximized.
In this article, the method of computing Greens functions recursively for interacting particles has been extended to square lattices and binary trees. The method allows for performing calculations for large lattices. Approximations, which make the method more efficient while maintaining accuracy are described for computations of dynamics and correlations of interacting particles in disordered systems. Direct informations related to spectral weights for different initial preparations of interacting particles in real space can be obtained using this method for larger system sizes with less finite size effects.
Correlation of interacting particles is studied in their dynamics and localization in ideal and disordered lattice systems with the help of numerical tools. Both 1D and 2D systems are considered. In 1D lattices with long-range hopping, differences between dynamics of attractively and repulsively interacting particles are noted. For calculations of 2D systems, a recursion based numerical algorithm for two-particle Greens functions is implemented which provides accurate results. Using this algorithm, spectral properties of 2D Hubbard and Hofstadter models is computed, along with localization parameters of 2D disordered systems.
An exact real-space rescaling transformation is used to calculate the spectral properties of tight-binding Hamiltonians with an arbitrary range of interaction of two- and three-dimensional lattices. The intrinsic problem of the proliferation of initial short-range interactions under rescaling is shown to be tractable numerically. This method provides a direct way of obtaining the local Green functions and local density of states. Singularities of the spectral functions are associated with fixed points or limit cycles of the rescaling transformation. Band edge fixed points and exponents have been found for the square, triangular and cubic lattices. In two and three dimensions there is rapid convergence to accurate numerical results for the Green function outside the band, while good results have been obtained for the in-band (e.g. density of states) spectra in two dimensions by suitable choices of cluster size and attenuation factor. Insight into the relationship between these factors follows from an investigation of finite-cluster approximations to the exact transformation. It is found that the zeros of the Green functions in the complex plane are Julia sets which have a fractal structure. For a given cluster size the attenuation factors must lie outside these zeros in order for the procedure to be convergent.
We study the spectral function of interacting one-dimensional fermions for an integrable lattice model away from half-filling. The divergent power-law singularity of the spectral function near the single-particle or single-hole energy is described by an effective x-ray edge type model. At low densities and for momentum near the zone boundary, we find a second divergent singularity at higher energies which is associated with a two-particle bound state. We use the Bethe ansatz solution of the model to calculate the exact singularity exponents for any momentum and for arbitrary values of chemical potential and interaction strength in the critical regime. We relate the singularities of the spectral function to the long-time decay of the fermion Green's function and compare our predictions with numerical results from the time-dependent density-matrix renormalization group (tDMRG). Our results show that the tDMRG method is able to provide accurate time decay exponents in the cases of power-law decay of the Green's function. Some implications for the line shape of the dynamical structure factor away from half-filling are also discussed. In addition, we show that 85% of the total spectral weight in the zero field Heisenberg model near wave-vector $q=\ensuremath{\pi}$ is given by the two-spinon contribution.
We study the exact solution of the two-body problem on a tight-binding one-dimensional lattice, with pairwise interaction potentials which have an arbitrary but finite range. We show how to obtain the full spectrum, the bound and scattering states, and the ``low-energy'' solutions by very efficient and easy-to-implement numerical means. All bound states are proven to be characterized by roots of a polynomial whose degree depends linearly on the range of the potential, and we discuss the connections between the number of bound states and the scattering lengths. ``Low-energy'' resonances can be located with great precision with the methods we introduce. Further generalizations to include more exotic interactions are also discussed.