Fermionic bound states on a one-dimensional lattice

📝 Abstract

We study bound states of two fermions with opposite spins in an extended Hubbard chain. The particles interact when located both on a site or on adjacent sites. We find three different types of bound states. Type U is predominantly formed of basis states with both fermions on the same site, while two states of type V originate from both fermions occupying neighbouring sites. Type U, and one of the states from type V, are symmetric with respect to spin flips. The remaining one from type V is antisymmetric. V-states are characterized by a diverging localization length below some critical wave number. All bound states become compact for wave numbers at the edge of the Brilloin zone.

💡 Analysis

We study bound states of two fermions with opposite spins in an extended Hubbard chain. The particles interact when located both on a site or on adjacent sites. We find three different types of bound states. Type U is predominantly formed of basis states with both fermions on the same site, while two states of type V originate from both fermions occupying neighbouring sites. Type U, and one of the states from type V, are symmetric with respect to spin flips. The remaining one from type V is antisymmetric. V-states are characterized by a diverging localization length below some critical wave number. All bound states become compact for wave numbers at the edge of the Brilloin zone.

📄 Content

arXiv:0905.1255v1 [cond-mat.quant-gas] 8 May 2009 Fermionic bound states on a one-dimensional lattice Jean-Pierre Nguenang1,2 and Sergej Flach1 1 Max-Planck-Institut f¨ur Physik komplexer Systeme, N¨othnitzer Str. 38, 01187 Dresden, Germany and 2 Fundamental Physics Laboratory: Group of Nonlinear physics and Complex systems, Department of Physics, University of Douala, P.O. Box 24157, Douala, Cameroon (Dated: October 28, 2018) We study bound states of two fermions with opposite spins in an extended Hubbard chain. The particles interact when located both on a site or on adjacent sites. We find three different types of bound states. Type U is predominantly formed of basis states with both fermions on the same site, while two states of type V originate from both fermions occupying neighbouring sites. Type U, and one of the states from type V, are symmetric with respect to spin flips. The remaining one from type V is antisymmetric. V-states are characterized by a diverging localization length below some critical wave number. All bound states become compact for wave numbers at the edge of the Brilloin zone. PACS numbers: 03.65.Ge, 34.50.Cx, 37.10.Jk Introduction. Advances in experimental techniques of manipulation of ultracold atoms in optical lattices make it feasible to explore the physics of few-body interactions. Systems with few quantum particles on lattices have new unexpected features as compared to the condensed mat- ter case of many-body interactions, where excitation en- ergies are typically small compared to the Fermi energy. In particular, a recent experiment explored the repulsive binding of bosonic atom pairs in an optical lattice [1], as predicted theoretically decades earlier [2, 3] (see also [4] for a review). In this work, we study binding properties of fermionic pairs with total spin zero. We use the extended Hubbard model, which contains two interaction scales

• the on site interaction U and the nearest neighbour intersite interaction V . The nonlocal interaction V is added in condensed matter physics to emulate remnants of the Coulomb interaction due to non-perfect screening of electronic charges. For fermionic ultracold atoms or molecules with magnetic or electric dipole-dipole inter- actions, it can be tuned with respect to the local interac- tion U by modifying the trap geometry of a condensate, additional external dc electric fields, combinations with fast rotating external fields, etc (for a review and relevant references see [5]). The paper is organized as follows. We first describe the model and introduce the basis we use to write down the Hamiltonian matrix to be diagonalized. Then we de- rive the quantum states of the lattice containing one and two fermions, with opposite spins in the latter case. We study the obtained bound states for two fermions. We obtain analytical expressions for the energy spectrum of the bound states. Some of the bound states are charac- terized by a critical momentum below which they dissolve with the two-particle continuum. Model and basis choice. We consider a one-dimensional lattice with f sites and periodic boundary conditions de- scribed by the extended Hubbard model with the follow- ing Hamiltonian: ˆH = ˆH0 + ˆHU + ˆHV , (1) where ˆH0 = − X j,σ ˆa+ j,σ(ˆaj−1,σ + ˆaj+1,σ) , (2) ˆHU = −U X j ˆnj,↑ˆnj,↓, ˆnj,σ = ˆa+ j,σˆaj,σ , (3) ˆHV = −V X j ˆnjˆnj+1 , ˆnj = ˆnj,↑+ ˆnj,↓. (4) ˆH0 describes the nearest-neighbor hopping of fermions along the lattice. Here the symbols σ =↑, ↓stand for a fermion with spin up or down. ˆHU describes the onsite interaction between the particles, and ˆHV the intersite interaction of fermions located at adjacent sites. ˆa+ j,σ and ˆaj,σ are the fermionic creation and annihilation operators satisfying the corresponding anticommutation relations: [ˆa+ j,σ, ˆal,σ′] = δj,lδσ,σ′, [ˆa+ j,σ, ˆa+ l,σ′] = [ˆaj,σ, ˆal,σ′] = 0. Note that throughout this work we consider U and V posi- tive, which leads to bound states located below the two- particle continuum. A change of the sign of U, V will simply swap the energies. The Hamiltonian (1) commutes with the number op- erator ˆ N = P j ˆnj whose eigenvalues are n = n↑+ n↓, i.e. the total number of fermions in the lattice. We con- sider n = 2, with n↑= 1 and n↓= 1. Therefore we construct a basis starting with the eigenstates of ˆN. We use a number state basis |Φn⟩= |n1; n2 · · · nf⟩[3], where ni = ni,↑+ ni,↓represents the number of fermions at the i-th site of the lattice. |Φn⟩is an eigenstate of the number operator ˆN with eigenvalue n = Pf j=1 nj. To observe the fermionic character of the considered states, any two-particle number state is generated from the vacuum |O⟩by first creating a particle with spin down, and then a particle with spin up: e.g. ˆa+ 2,↑ˆa+ 1,↓|O⟩ creates a particle with spin down on site 1 and one with 2 spin up on site 2, while ˆa+ 2,↑ˆa+ 2,↓|O⟩creates both particles with spin down and up on site 2. Due to periodic boundary conditions the Hamilto- nian (1) commutes also w

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