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arXiv:2608.18466

Efficient calculation of real-space lattice propagators in the presence of a Fermi sea

Oliver Tong, Mona Berciu

Published Aug 19, 2026• 0 citations
arXivHTMLPDF

Abstract

We present an efficient method for computing the real-space propagators (lattice Green's functions) of any tight-binding Hamiltonian in the presence of a Fermi sea with carrier concentration $x$. The method is valid for any lattice, dispersion, and dimension, provided the corresponding $x=0$ propagators are known. We show that suitable combinations of the finite-$x$ particle-addition propagators have a real or imaginary part trivially related to their $x=0$ counterpart, while the remaining part follows from a Kramers-Kronig relation that can be evaluated for all energies at once using the fast Fourier transforms. The computational cost is therefore independent of the dimensionality, unlike that of direct Brillouin-zone integration. The particle-removal propagators follow from general identities. We validate the method against direct integration for hypercubic lattices in one, two, and three dimensions, and use the 2D square lattice to illustrate how the shape of the Fermi surface is imprinted on the spatial structure of the propagators. In particular, at energies far outside the band, the propagator maps converge to the Fraunhofer diffraction pattern whose aperture is the unoccupied part of the Brillouin zone.

DOI
10.48550/arxiv.2608.18466

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We present an efficient method for computing the real-space propagators (lattice Green's functions) of any tight-binding Hamiltonian in the presence of a Fermi sea with carrier concentration xxx. The method is valid for any lattice, dispersion, and dimension, provided the corresponding x=0x=0x=0 propagators are known. We show that suitable combinations of the finite-xxx particle-addition propagators have a real or imaginary part trivially related to their x=0x=0x=0 counterpart, while the remaining part follows from a Kramers-Kronig relation that can be evaluated for all energies at once using the fast Fourier transforms. The computational cost is therefore independent of the dimensionality, unlike that of direct Brillouin-zone integration. The particle-removal propagators follow from general identities. We validate the method against direct integration for hypercubic lattices in one, two, and three dimensions, and use the 2D square lattice to illustrate how the shape of the Fermi surface is imprinted on the spatial structure of the propagators. In particular, at energies far outside the band, the propagator maps converge to the Fraunhofer diffraction pattern whose aperture is the unoccupied part of the Brillouin zone.
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.
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We consider the expansion of wave packets governed by the free Schrödinger equation. This seemingly simple task plays an important role in simulations of various quantum experiments, especially in the field of matter-wave interferometry. The initial tight confinement of quantum particles results in a very fast expansion of the wave function at later times which significantly complicates an efficient and precise numerical evaluation. In many practical cases the expansion time is too short for the validity of the stationary phase approximation and too long for an efficient application of Fourier collocation-based methods. We develop an alternative method based on a discretization of the free-particle propagator. This simple approach yields highly accurate results which readily follows from the exceptionally fast convergence of the trapezoidal rule approximation of integrals involving smooth, rapidly decaying functions. We discuss and analyze our approach in detail and demonstrate how to estimate the numerical error in the one-dimensional setting. Furthermore, we show that by exploiting the separability of the Green's function, the numerical effort of the multi-dimensional approximation is considerably reduced. Our method is very fast, highly accurate, and easy to implement on modern hardware.
We derive exact general relations between various observables for NNN spin-1/2 fermions with zero-range or short-range interactions, in continuous space or on a lattice, in two or three dimensions, in an arbitrary external potential. Some of our results generalize known relations between the large-momentum behavior of the momentum distribution, the short-distance behaviors of the pair distribution function and of the one-body density matrix, the norm of the regular part of the wave function, the derivative of the energy with respect to the scattering length or to time, and the interaction energy (in the case of finite-range interactions). The expression relating the energy to a functional of the momentum distribution is also generalized. Moreover, we find expressions (in terms of the regular part of the wave function) for the derivative of the energy with respect to the effective range re{r}_{e}re​ in three dimensions (3D), and to the effective range squared in two dimensions (2D). They express the fact that the leading corrections to the eigenenergies due to a finite-interaction range are linear in the effective range in 3D (and in its square in 2D) with model-independent coefficients. There are subtleties in the validity condition of this conclusion, for the 2D continuous space (where it is saved by factors that are only logarithmically large in the zero-range limit) and for the 3D lattice models (where it applies only for some magic dispersion relations on the lattice that sufficiently weakly break Galilean invariance and that do not have cusps at the border of the first Brillouin zone; an example of such relations is constructed). Furthermore, the subleading short-distance behavior of the pair distribution function and the subleading 1/k61/{k}^{6}1/k6 tail of the momentum distribution are related to $\ensuremath{\partial}E/\ensuremath{\partial}{r}_{e}$ [or to $\ensuremath{\partial}E/\ensuremath{\partial}({r}_{e}^{2})$ in 2D]. The second-order derivative of energy with respect to the inverse (or the logarithm in the two-dimensional case) of the scattering length is found to be expressible for any eigenstate in terms of the eigen-wave-function's regular parts; this implies that, at thermal equilibrium, this second-order derivative, taken at fixed entropy, is negative. Applications of the general relations are presented: We compute corrections to exactly solvable two-body and three-body problems and find agreement with available numerics; for the unitary gas in an isotropic harmonic trap, we determine how the finite-1/a1/a1/a and finite-range energy corrections vary within each energy ladder (associated with the SO(2,1) dynamical symmetry) and we deduce the frequency shift and the collapse time of the breathing mode; for the bulk unitary gas, we compare to fixed-node Monte Carlo data, and we estimate the deviation from the Bertsch parameter due to the finite interaction-range in typical experiments.
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