We consider tight-binding electrons on a square lattice, with an anisotropic hopping intensity, and in a periodic external potential. Breaking of the translational symmetry of the system results in two bands, separated by a gap of width proportional to the unique energy parameter of the model. The off-diagonal matrix elements of the one-body reduced density matrix are exactly expressed by product of Euler's Γ-functions and Appell function. Their large-distance (σ) decay rate is derived to be of the form σ −1 exp (−σ/ξ). A control of the crossover from 2D to 1D system is achieved. In comparison with the isotropic case, the correlation length ξ varies with the gap in a strikingly different manner. In particular, it remains nonzero as the gap vanishes, in all the directions except the direction of stronger hopping intensity.
We provide a tight-binding model of insulator, for which we derive an exact analytic form of the one-body density matrix and its large-distance asymptotics in dimensions
D=1, 2. The system is built out of a band of single-particle orbitals in a periodic potential. Breaking of the translational symmetry of the system results in two bands, separated by a direct gap whose width is proportional to the unique energy parameter of the model. The form of the decay is a power law times an exponential. We determine the power in the power law and the correlation length in the exponential, versus the lattice direction, the direct-gap width, and the lattice dimension. In particular, the obtained exact formulas imply that in the diagonal direction of the square lattice the inverse correlation length vanishes linearly with the vanishing gap, while in nondiagonal directions, the linear scaling is replaced by the square root one. Independently of direction, for sufficiently large gaps the inverse correlation length grows logarithmically with the gap width.
In the framework of the band theory, we consider two tight-binding models of insulators. The first one, proposed recently by Taraskin et al, is a translationally invariant system, built out of two independent non-overlapping bands of single-particle orbitals that are coupled by a weak inter-band hybridization. This kind of insulator exhibits unphysical properties: we show, in particular, that the one-body density matrix does not depend on the width of the gap between the bands. Consequently, there is no delocalization effect with increasing metallicity. In the second model there are also two bands. However, they are not imposed by construction but are created from a band of single-particle orbitals due to the breaking of the translational symmetry by a periodic potential. These bands are separated by a gap for all nonzero values of the unique energy parameter of the model. We demonstrate that the one-body density matrix has the same structure as in the first model. As a result, the large distance asymptotic formulae derived by Taraskin et al in dimensions
D=1,2,3, apply as well, but only for very large gap widths. In D=1 and in the diagonal direction of D=2 cases, we derive a stronger asymptotic formula, valid for all gap widths. The both kinds of asymptotic formulae are composed of a dimension-dependent power-law factor and a gap-dependent exponentially decaying factor. The latter asymptotics implies that the exponential decay rate vanishes linearly with the vanishing gap. In non-diagonal directions, we have found numerically that the linear scaling is replaced by the square root one. Independently of the direction, the exponential decay rate grows logarithmically with the gap width, for sufficiently large gaps.
We study decay rates of one-body reduced density matrices in insulators, described by a tight-binding model, where not only an external potential but also hoppings are spatially modulated. We determine analytically the power in the power law and the correlation length in
D=1 case and in several lattice directions in
D=2 case. Unlike the uniform hopping case, in the
D=1 system and in some directions of the
D=2 system the correlation length is not determined uniquely by the gap. Moreover, a crossover from
D=2 decay rates to
D=1 ones is investigated.
Besides the chemical constituents, it is the lattice geometry that controls the most important material properties. In many interesting compounds, the arrangement of elements leads to pronounced anisotropies, which reflect into a varying degree of quasi-two-dimensionality of their low-energy excitations. Here we start by classifying important families of correlated materials according to a simple measure for the tetragonal anisotropy of their ab initio electronic (band) structure. Second, we investigate the impact of a progressively larger anisotropy in driving the nonlocality of many-body effects. To this end, we tune the Hubbard model from isotropic cubic in three dimensions to the two-dimensional limit and analyze it using the dynamical vertex approximation. For sufficiently isotropic hoppings, we find the low-energy self-energy to be well separable into a static nonlocal and a dynamical local contribution. While the latter could potentially be obtained from dynamical mean-field approaches, we find the former to be nonnegligible in all cases. Further, by increasing the model's anisotropy, we quantify the degree of quasi-two-dimensionality which causes this ``space-time separation'' to break down. Our systematic analysis improves the general understanding of electronic correlations in anisotropic or layered materials and heterostructures and provides useful guidance for future realistic studies.
We study the one-dimensional nearest-neighbor tight-binding model of electrons with independently distributed random hopping and no on-site potential (i.e., off-diagonal disorder with particle-hole symmetry, leading to sublattice symmetry, for each realization). For nonsingular distributions of the hopping, it is known that the model exhibits a universal, singular behavior of the density of states $\ensuremath{\rho}(E)\ensuremath{\sim}1/|E{ln}^{3}|E||$ and of the localization length $\ensuremath{\xi}(E)\ensuremath{\sim}|ln|E||$, near the band center
E=0. (This singular behavior is also applicable to random XY and Heisenberg spin chains; it was first obtained by Dyson for a specific random harmonic oscillator chain.) Simultaneously, the state at
E=0 shows a universal, subexponential decay at large distances $\ensuremath{\sim}exp[\ensuremath{-}\sqrt{r/{r}_{0}}]$. In this study, we consider singular, but normalizable, distributions of hopping, whose behavior at small
t is of the form $\ensuremath{\sim}1/[t{ln}^{\ensuremath{\lambda}+1}(1/t)]$, characterized by a single, continuously tunable parameter $\ensuremath{\lambda}>0$. We find, using a combination of analytic and numerical methods, that while the universal result applies for $\ensuremath{\lambda}>2$, it no longer holds in the interval $0<\ensuremath{\lambda}<2$. In particular, we find that the form of the density of states singularity is enhanced (relative to the Dyson result) in a continuous manner depending on the nonuniversal parameter $\ensuremath{\lambda}$; simultaneously, the localization length shows a less divergent form at low energies and ceases to diverge below $\ensuremath{\lambda}=1$. For $\ensuremath{\lambda}<2$, the fall-off of the
E=0 state at large distances also deviates from the universal result and is of the form $\ensuremath{\sim}exp[\ensuremath{-}{(r/{r}_{0})}^{1/\ensuremath{\lambda}}]$, which decays faster than an exponential for $\ensuremath{\lambda}<1$.