A formula for the electrical conductivity in lattice systems (original) (raw)

On the number of eigenvalues of a model operator related to a system of three particles on lattices

Journal of Physics A: Mathematical and Theoretical, 2011

We consider a quantum mechanical system on a lattice Z 3 in which three particles, two of them being identical, interact through a zero-range potential. We admit a very general form for the 'kinetic' part H 0 γ of the Hamiltonian, which contains a parameter γ to distinguish the two identical particles from the third one (in the continuum case this parameter would be the inverse of the mass). We prove that there is a value γ * of the parameter such that only for γ < γ * the Efimov effect (infinite number of bound states if the two-body interactions have a resonance) is absent for the sector of the Hilbert space which contains functions which are antisymmetric with respect to the two identical particles, while it is present for all values of γ on the symmetric sector. We comment briefly on the relation of this result with previous investigations on the Thomas effect. We also establish the following asymptotics for the number N(z) of eigenvalues z below E min , the lower limit of the essential spectrum of H 0. In the symmetric subspace lim z→E − min N s (z) | log |E min − z|| = U s 0 (γ), ∀ γ, whereas in the antisymmetric subspace lim z→E − min N as (z) | log |E min − z|| = U as 0 (γ), ∀ γ > γ * , where U as 0 (γ), U s 0 (γ) are written explicitly as a function of the integral kernel of operators acting on L 2 ((0, r) × (L 2 (S 2) ⊗ L 2 (S 2)) (S 2 is the unit sphere in R 3).

Elementary Introduction to the Hubbard Model

Exercise 4c: Compute the wavefunctions Ψ 1 (x) = x|1 and Ψ 2 (x) = x|2 of the first two excited states of the simple harmonic oscillator by using Ψ 0 (x) = x|0 = e −mωx 2 /2 and the expression forâ † in terms ofx andp.

Theory of random multiplicative transfer matrices and its implications for quantum transport

Journal de Physique, 1990

numériques indépendants. La distribution p(g) n'étant fonction que de 03C1(03BB) dans notre théorie, nous étudions dans la seconde partie de ce travail les propriétés d'échelle de 03C1(03BB) afin d'examiner la validité d'une théorie d'échelle à un paramètre. D'abord, nous généralisons pour chacun des niveaux 03B1i les lois d'échelle déjà obtenues pour la chaine désordonnée. Puis nous montrons que la convergence de chacun des 03B1i vers sa limite quasi-unidimensionnelle (N donné, Lz ~ oo) ne dépend que de la conductance moyenne ~g~. Nous en concluons que (g) satisfait à une loi d'échelle à un paramètre dans le domaine de paramètres examiné. Cependant, nous ne pouvons exclure que la dépendance en fonction de i des lois d'échelle obtenues n'introduise des paramètres d'échelle supplémentaires pour 03C1(g). 1 Phys. France 51 (1990) 587-609 ler AVRIL 1990, Classification Physics Abstracts 71.30 -71.55J -72.10 -72.15R

Note on the one-dimensional Hubbard model

Physical Review B, 1974

By use of a Jordan-Signer transformation, the one-dimensional Hubbard model for an itinerant magnet is expressed as an anisotropic Heisenberg model on a double chain with twoand three-spin interactions. The limiting cases={i) free electron, (ii) strong correlation, and (iii) zero-hopping rateare briefly examined in the spin representation. The Hubbard model' is perhaps the simplest model to account for the importance of Coulombic correlations in the theory of itinerant-electron magnetism. The general d-dimensional version of this model is intractableexcept in certain limiting cases. However, progress has been made toward complete characterization of the one-dimensional (1-D) modelnotably by Lieb and Wu.T he 1-D case is of more-than-academic interest, since it is believed to be a good model for various organic charge-transfer salts. s'4 The 1-D Hubba, rd Hamiltonian' may be written as 8= TQQ[C, (f+1)~C, (f) ty /=1 4+1 +C,(l)'C, (l+1)]+U Q n, (l) n, (f), where C, (l)~and C, (f) are the annihilation and creation operators, respectively, for a Fermi

Quantized transport induced by topology transfer between coupled one-dimensional lattice systems

Physical Review A, 2021

We show that a topological pump in a one-dimensional (1D) insulator can induce a strictly quantized transport in an auxiliary chain of non-interacting fermions weakly coupled to the first. The transported charge is determined by an integer topological invariant of the ficticious Hamiltonian of the insulator, given by the covariance matrix of single-particle correlations. If the original system consists of non-interacting fermions, this number is identical to the TKNN (Thouless, Kohmoto, Nightinghale, den Nijs) invariant of the original system and thus the coupling induces a transfer of topology to the auxiliary chain. When extended to particles with interactions, for which the TKNN number does not exist, the transported charge in the auxiliary chain defines a topological invariant for the interacting system. In certain cases this invariant agrees with the many-body generalization of the TKNN number introduced by Niu, Thouless, and Wu (NTW). We illustrate the topology transfer to the auxiliary system for the Rice-Mele model of non-interacting fermions at half filling and the extended superlattice Bose-Hubbard model at quarter filling. In the latter case the induced charge pump is fractional.

Schr�dinger Operators on Lattices. The Efimov Effect and Discrete Spectrum Asymptotics

Annales Henri Poincar�, 2004

The Hamiltonian of a system of three quantum mechanical particles moving on the three-dimensional lattice Z 3 and interacting via zero-range attractive potentials is considered. For the two-particle energy operator h(k), with k ∈ T 3 = (−π, π] 3 the two-particle quasi-momentum, the existence of a unique positive eigenvalue below the bottom of the continuous spectrum of h(k) for k = 0 is proven, provided that h(0) has a zero energy resonance. The location of the essential and discrete spectra of the threeparticle discrete Schrödinger operator H(K), K ∈ T 3 being the three-particle quasimomentum, is studied. The existence of infinitely many eigenvalues of H(0) is proven. It is found that for the number N (0, z) of eigenvalues of H(0) lying below z < 0 the following limit exists lim z→0− N (0, z) | log | z || = U 0 with U 0 > 0. Moreover, for all sufficiently small nonzero values of the three-particle quasi-momentum K the finiteness of the number N (K, τess(K)) of eigenvalues of H(K) below the essential spectrum is established and the asymptotics for the number N (K, 0) of eigenvalues lying below zero is given.

On the spectrum of Schrödinger-type operators on two dimensional lattices

Journal of Mathematical Analysis and Applications

We consider a family H a,b (µ) = H 0 + µ V a,b µ > 0, of Schrödinger-type operators on the two dimensional lattice Z 2 , where H 0 is a Laurent-Toeplitz-type convolution operator with a given Hopping matrix e and V a,b is a potential taking into account only the zero-range and one-range interactions, i.e., a multiplication operator by a function v such that v(0) = a, v(x) = b for |x| = 1 and v(x) = 0 for |x| ≥ 2, where a, b ∈ R \ {0}. Under certain conditions on the regularity of e we completely describe the discrete spectrum of H a,b (µ) lying above the essential spectrum and study the dependence of eigenvalues on parameters µ, a and b. Moreover, we characterize the threshold eigenfunctions and resonances.