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Register a new userThe Greens Function for the Huckel (Tight Binding) Model
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Applications of the Huckel (tight binding) model are ubiquitous in quantum chemistry and solid state physics. The matrix representation of this model is isomorphic to an unoriented vertex adjacency matrix of a bipartite graph, which is also the Laplacian matrix plus twice the identity. In this paper, we analytically calculate the determinant and, when it exists, the inverse of this matrix in connection with the Greens function, $mathbf{G}$, of the $Ntimes N$ Huckel matrix. A corollary is a closed form expression for a Harmonic sum (Eq. 12). We then extend the results to $d-$dimensional lattices, whose linear size is $N$. The existence of the inverse becomes a question of number theory. We prove a new theorem in number theory pertaining to vanishing sums of cosines and use it to prove that the inverse exists if and only if $N+1$ and $d$ are odd and $d$ is smaller than the smallest divisor of $N+1$. We corroborate our results by demonstrating the entry patterns of the Greens function and discuss applications related to transport and conductivity.
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We consider a one-dimensional gas of spin-1/2 fermions interacting through $delta$-function repulsive potential of an arbitrary strength. For the case of all fermions but one having spin up, we calculate time-dependent two-point correlation function of the spin-down fermion. This impurity Greens function is represented in the thermodynamic limit as an integral of Fredholm determinants of integrable linear integral operators.
The Greens function has been an indispensable tool to study many-body systems that remain one of the biggest challenges in modern quantum physics for decades. The complicated calculation of Greens function impedes the research of many-body systems. The appearance of the noisy intermediate-scale quantum devices and quantum-classical hybrid algorithm inspire a new method to calculate Greens function. Here we design a programmable quantum circuit for photons with utilizing the polarization and the path degrees of freedom to construct a highly-precise variational quantum state of a photon, and first report the experimental realization for calculating the Greens function of the two-site Fermionic Hubbard model, a prototypical model for strongly-correlated materials, in photonic systems. We run the variational quantum eigensolver to obtain the ground state and excited states of the model, and then evaluate the transition amplitudes among the eigenstates. The experimental results present the spectral function of Greens function, which agrees well with the exact results. Our demonstration provides the further possibility of the photonic system in quantum simulation and applications in solving complicated problems in many-body systems, biological science, and so on.
In [BEI] we introduced a Levy process on a hierarchical lattice which is four dimensional, in the sense that the Greens function for the process equals 1/x^2. If the process is modified so as to be weakly self-repelling, it was shown that at the critical killing rate (mass-squared) beta^c, the Greens function behaves like the free one. - Now we analyze the end-to-end distance of the model and show that its expected value grows as a constant times sqrt{T} log^{1/8}T (1+O((log log T)/log T)), which is the same law as has been conjectured for self-avoiding walks on the simple cubic lattice Z^4. The proof uses inverse Laplace transforms to obtain the end-to-end distance from the Greens function, and requires detailed properties of the Greens function throughout a sector of the complex beta plane. These estimates are derived in a companion paper [math-ph/0205028].
We consider atomistic geometry relaxation in the context of linear tight binding models for point defects. A limiting model as Fermi-temperature is sent to zero is formulated, and an exponential rate of convergence for the nuclei configuration is established. We also formulate the thermodynamic limit model at zero Fermi-temperature, extending the results of [H. Chen, J. Lu, C. Ortner. Arch. Ration. Mech. Anal., 2018]. We discuss the non-trivial relationship between taking zero temperature and thermodynamic limits in the finite Fermi-temperature models.
In this work, results are presented of Hybrid-Monte-Carlo simulations of the tight-binding Hamiltonian of graphene, coupled to an instantaneous long-range two-body potential which is modeled by a Hubbard-Stratonovich auxiliary field. We present an investigation of the spontaneous breaking of the sublattice symmetry, which corresponds to a phase transition from a conducting to an insulating phase and which occurs when the effective fine-structure constant $alpha$ of the system crosses above a certain threshold $alpha_C$. Qualitative comparisons to earlier works on the subject (which used larger system sizes and higher statistics) are made and it is established that $alpha_C$ is of a plausible magnitude in our simulations. Also, we discuss differences between simulations using compact and non-compact variants of the Hubbard field and present a quantitative comparison of distinct discretization schemes of the Euclidean time-like dimension in the Fermion operator.
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